Finite Group Databasebeta

← #29699721

Subgroups — #29699721

Order 15616.

710 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454617C211yesmaximal, characteristic
2#21454487C211yesmaximal, characteristic, complemented
3#21454485C211yesmaximal, characteristic, complemented
4#21446709C211yesmaximal, characteristic
5#21446704C211yesmaximal, characteristic
6#21446470C211yesmaximal, characteristic, complemented
7#21446470C211yesmaximal, characteristic, complemented
8#21446465C211yesmaximal, characteristic, complemented
9#21446465C211yesmaximal, characteristic, complemented
10#21445993C211yesmaximal, characteristic, complemented
11#21445840C211yesmaximal, characteristic, complemented
12#21445702C211yesmaximal, characteristic, complemented
13#21445697C211yesmaximal, characteristic, complemented
14#21445694C211yesmaximal, characteristic, complemented
15C61×#1806C211yesmaximal, characteristic, complemented
16C2×#14519797C2211yescharacteristic
17C4×#11758222C2211yescharacteristic
18#16839872C411yescharacteristic, complemented
19#16839872C411yescharacteristic, complemented
20#16839867C411yescharacteristic, complemented
21#16839867C411yescharacteristic, complemented
22#16839862C411yescharacteristic, complemented
23#16839862C411yescharacteristic, complemented
24#16839860C411yescharacteristic, complemented
25#16839860C411yescharacteristic, complemented
26#16839823—21no—
27#16839823—21no—
28#16839822—21no—
29#16839822—21no—
30#16839819—21no—
31#16839818—21no—
32#16839815—21no—
33#16839814—21no—
34C2×#14519712C2211yescharacteristic
35C2×#14519712C2211yescharacteristic
36C2×#14519712C2211yescharacteristic
37C2×#14519712C2211yescharacteristic
38#16839798C2211yescharacteristic
39#16839786C2211yescharacteristic
40C2×C4⋊#92418C2211yescharacteristic
41C2×#14519705C2211yescharacteristic
42C2×#14519705C2211yescharacteristic
43#16839758C2211yescharacteristic
44#16839756C2211yescharacteristic
45#16839730C2211yescharacteristic
46#16839730C2211yescharacteristic
47#16839729C2211yescharacteristic
48#16839729C2211yescharacteristic
49#16839720C2211yescharacteristic
50#16839720C2211yescharacteristic
51#16839718C2211yescharacteristic
52#16839718C2211yescharacteristic
53C2×#14519688C2211yescharacteristic
54C2×#14519688C2211yescharacteristic
55C2×C4×#92418C2211yescharacteristic
56#16839522C2211yescharacteristic
57#16839522C2211yescharacteristic
58#16839517C2211yescharacteristic
59#16839517C2211yescharacteristic
60C4×#41×C61C2211yescharacteristic
61C61×#423C2211yescharacteristic
62C2×#104×C61C2211yescharacteristic
63C2×#104×C61C2211yescharacteristic
64C4×C4⋊C4×C61C2211yescharacteristic
65C61×#339C2211yescharacteristic
66C61×#339C2211yescharacteristic
67C2×#11758222C2311yescharacteristic
68#14519797—22no—
69#14519797—22no—
70C2×#11758202C2×C411yescharacteristic
71C2×#11758200C2×C411yescharacteristic
72C2×#11758201C2×C411yescharacteristic
73C2×#11758199C2×C411yescharacteristic
74C2×#11758194C2311yescharacteristic
75C2×#11758194C2311yescharacteristic
76C22×#92418C2311yescharacteristic
77C22×#92418C2311yescharacteristic
78C22×#92418C2311yescharacteristic
79C22×#92418C2311yescharacteristic
80C4×Dic122C2×C411yescharacteristic
81C4×Dic122C2×C411yescharacteristic
82C4×D244C2×C411yescharacteristic
83C4×D244C2×C411yescharacteristic
84C4×C22⋊D61—21no—
85C4×C22⋊D61—21no—
86#14519736C2311yescharacteristic
87#14519734—21no—
88C42×D61—21no—
89#14519712—22no—
90#14519712—22no—
91#14519712—41no—
92#14519712—41no—
93#14519710C2×C411yescharacteristic
94#14519710C2×C411yescharacteristic
95#14519708C2×C411yescharacteristic
96#14519708C2×C411yescharacteristic
97C4⋊#92418—21no—
98C4⋊#92418—21no—
99C4⋊#92418D411yescharacteristic
100C4⋊#92418D411yescharacteristic
101C4⋊#92418D411yescharacteristic
102C4⋊#92418D411yescharacteristic
103#14519705—41no—
104#14519705—41no—
105#14519701C2×C411yescharacteristic
106#14519701C2×C411yescharacteristic
107#14519700C2×C411yescharacteristic
108#14519700C2×C411yescharacteristic
109#14519698C2×C411yescharacteristic
110#14519698C2×C411yescharacteristic
111#14519697C2×C411yescharacteristic
112#14519697C2×C411yescharacteristic
113C22⋊#92418—21no—
114C22⋊#92418—21no—
115C22⋊#92418—21no—
116C22⋊#92418—21no—
117#14519689—21no—
118#14519689—21no—
119#14519689—21no—
120#14519689—21no—
121#14519688—21no—
122#14519688—21no—
123#14519688—22no—
124#14519688C2×C411yescharacteristic
125#14519688C2×C411yescharacteristic
126#14519688C2×C411yescharacteristic
127#14519688C2×C411yescharacteristic
128#14519687—21no—
129#14519687—21no—
130C4×#92418—21no—
131C4×#92418—21no—
132C4×#92418—22no—
133C8×Dic61—21no—
134C8×Dic61—21no—
135C2×#41×C61C2311yescharacteristic
136C2×M16×C61C2311yescharacteristic
137C2×M16×C61C2311yescharacteristic
138C4×Q8×C61C2×C411yescharacteristic
139C4×Q8×C61C2×C411yescharacteristic
140C4×D4×C61—21no—
141C4×D4×C61—21no—
142C4×D4×C61C2×C411yescharacteristic
143C4×D4×C61C2×C411yescharacteristic
144#117×C61—21no—
145#117×C61C2311yescharacteristic
146C2×C4⋊C4×C61—21no—
147C2×C42×C61—21no—
148C2×C42×C61C2311yescharacteristic
149C2×C42×C61C2311yescharacteristic
150C2×C42×C61C2311yescharacteristic
151C4⋊C8×C61—21no—
152C4⋊C8×C61—21no—
153#104×C61—41no—
154#104×C61—41no—
155#95×C61—21no—
156#41×C61—21no—
157#41×C61—21no—
158#41×C61—21no—
159#41×C61—21no—
160C2×Q8×C61C22×C411yescharacteristic
161C2×D4×C61—21no—
162C2×D4×C61C22×C411yescharacteristic
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61—21no—
166C22×C4×C61—21no—
167C22×C4×C61—21no—
168C22×C4×C61#4111yescharacteristic
169C22×C4×C61#4111yescharacteristic
170C22×C4×C61C2411yescharacteristic
171C2×C22⋊D61—21no—
172#11758222—21no—
173#11758222—21no—
174#11758222—21no—
175#11758222—21no—
176C2×D244C22×C411yescharacteristic
177C2×C4×D61—21no—
178C2×C4×D61—21no—
179C2×C4×D61—21no—
180C2×Dic122C22×C411yescharacteristic
181M16×C61—21no—
182M16×C61—21no—
183M16×C61—21no—
184M16×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—42no—
190C4⋊C4×C61C22×C411yescharacteristic
191C4⋊C4×C61C22×C411yescharacteristic
192C22⋊C4×C61—21no—
193C22⋊C4×C61—21no—
194C22⋊C4×C61—21no—
195C42×C61—21no—
196C42×C61—21no—
197C42×C61—21no—
198C42×C61—21no—
199C42×C61—21no—
200C42×C61—21no—
201C42×C61—21no—
202C42×C61C22×C411yescharacteristic
203C42×C61C22×C411yescharacteristic
204C42×C61C2×D411yescharacteristic
205C42×C61C2×D411yescharacteristic
206C22⋊Dic61—21no—
207#11758202—22no—
208#11758201—22no—
209#11758200—22no—
210#11758199—22no—
211#11758198—21no—
212#11758198—21no—
213C4⋊Dic61C22×C411yescharacteristic
214C4⋊Dic61C22×C411yescharacteristic
215#11758196—21no—
216#11758196—21no—
217C4×Dic61—21no—
218C4×Dic61—21no—
219C4×Dic61—21no—
220#11758194—21no—
221#11758194—21no—
222#11758194—21no—
223#11758194—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418—21no—
231C2×#92418—21no—
232C2×#92418C22×C411yescharacteristic
233C2×#92418C22×C411yescharacteristic
234C2×#92418C22×C411yescharacteristic
235C2×#92418C22×C411yescharacteristic
236C2×#92418C2×D411yescharacteristic
237C2×#92418C2×D411yescharacteristic
238C2×#92418C2×D411yescharacteristic
239C2×#92418C2×D411yescharacteristic
240C2×#92418#4112yes—
241C2×#92418#4112yes—
242C23×C61—21no—
243C23×C61C2×#4111yescharacteristic
244C22×D61—21no—
245Q8×C61—21no—
246Q8×C61—21no—
247D4×C61—21no—
248D4×C61—21no—
249D4×C61—41no—
250D4×C61—41no—
251C2×C4×C61—22no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—42no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61—81no—
266C2×C4×C61C4×D411yescharacteristic
267C2×C4×C61C4×D411yescharacteristic
268C2×C4×C61C4×D411yescharacteristic
269C2×C4×C61C4×D411yescharacteristic
270C2×C4×C61C23×C411yescharacteristic
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C2×#4111yescharacteristic
273C2×C4×C61C2×#4111yescharacteristic
274C2×C4×C61H(5,𝔽2)11yescharacteristic
275C2×C4×C612−1+411yescharacteristic
276C22⋊D61—41no—
277C22⋊D61—41no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—21no—
281D244—21no—
282D244—21no—
283C4×D61—42no—
284C4×D61—41no—
285C4×D61—41no—
286C4×D61—81no—
287Dic122—21no—
288Dic122—21no—
289C8×C61—41no—
290C8×C61—41no—
291#92418—21no—
292#92418—21no—
293#92418—21no—
294#92418—21no—
295#92418—21no—
296#92418—21no—
297#92418—41no—
298#92418—41no—
299#92418#11712yes—
300#92418#11712yes—
301#22804—611nomaximal, Sylow
302C22×C61—21no—
303C22×C61—21no—
304C22×C61—41no—
305C22×C61—41no—
306C22×C61—41no—
307C22×C61C2×C4×D411yescharacteristic
308C22×C61#55511yescharacteristic
309C22×C61#55611yescharacteristic
310C2×D61—41no—
311C2×D61—41no—
312C2×D61—41no—
313C4×C61—21no—
314C4×C61—21no—
315C4×C61—21no—
316C4×C61—21no—
317C4×C61—42no—
318C4×C61—41no—
319C4×C61—41no—
320C4×C61—41no—
321C4×C61—81no—
322C4×C61C2×#11711yescharacteristic
323C4×C61#52011yescharacteristic
324C4×C61#54711yescharacteristic
325C4×C61#55011yescharacteristic
326Dic61—41no—
327Dic61—41no—
328Dic61—41no—
329#2943—611no—
330#2925—611no—
331#2925—611no—
332#1975—611no—
333#1974—611no—
334#1929—611no—
335#1929—611no—
336#1929—611no—
337#1929—611no—
338#1849—611no—
339#1806—611no—
340#1806D6111yescharacteristic, complemented
341#1773—611no—
342#1772—611no—
343#1771—611no—
344C2×C61—21no—
345C2×C61—21no—
346C2×C61—81no—
347C2×C61#235411yescharacteristic
348C2×C61#295511yescharacteristic
349C2×C61#295611yescharacteristic
350D61—81no—
351C2×#135—611no—
352C4×#41—611no—
353C4×#41C2×D6111yescharacteristic
354#442—611no—
355#442—611no—
356#441—611no—
357#441—611no—
358#440—611no—
359#440—611no—
360#440—611no—
361#440—611no—
362#433—1221no—
363#433—1221no—
364#433—1221no—
365#433—1221no—
366#432—1221no—
367#432—1221no—
368#431—1221no—
369#431—1221no—
370C2×#108—611no—
371C2×#108—611no—
372C2×#108—611no—
373C2×#108—611no—
374#427—611no—
375#423—611no—
376#423C2×D6111yescharacteristic
377C2×C4⋊C8—611no—
378C2×#104—611no—
379C2×#104—611no—
380C2×#104C2×D6111yescharacteristic
381C2×#104C2×D6111yescharacteristic
382#416—611no—
383#416—611no—
384#408—611no—
385#408—611no—
386#408—611no—
387#408—611no—
388#405—611no—
389#405—611no—
390#405—611no—
391#405—611no—
392C2×#97—611no—
393C2×#97—611no—
394C2×C4×C8—611no—
395C4×C4⋊C4C2×D6111yescharacteristic
396#339—611no—
397#339—611no—
398#339C2×D6111yescharacteristic
399#339C2×D6111yescharacteristic
400#336—611no—
401#336—611no—
402C61#2280411yescharacteristic, complemented, Sylow
403C2×#41—611no—
404C2×#41C22×D6111yescharacteristic
405#135—1222no—
406#135—1222no—
407C2×Q16—611no—
408C2×SD16—611no—
409C2×SD16—611no—
410C2×D8—611no—
411C2×M16—611no—
412C2×M16—611no—
413C2×M16C22×D6111yescharacteristic
414C2×M16C22×D6111yescharacteristic
415C22×C8—611no—
416C22×C8—611no—
417C22×C8—611no—
418C22×C8—611no—
419C4×Q8—611no—
420C4×Q8—611no—
421C4×Q8C4×D6111yescharacteristic
422C4×Q8C4×D6111yescharacteristic
423C4×D4—21no—
424C4×D4—21no—
425C4×D4—611no—
426C4×D4—611no—
427C4×D4—1221no—
428C4×D4—1221no—
429C4×D4C4×D6111yescharacteristic
430C4×D4C4×D6111yescharacteristic
431#117—21no—
432#117—611no—
433#117—1221no—
434#117C22×D6111yescharacteristic
435C2×C4⋊C4—21no—
436C2×C42—21no—
437C2×C42—1221no—
438C2×C42C22×D6111yescharacteristic
439C2×C42C22×D6111yescharacteristic
440C2×C42C22×D6111yescharacteristic
441#108—1222no—
442#108—1222no—
443#108—2441no—
444#108—2441no—
445#107—611no—
446#107—611no—
447#106—611no—
448#106—611no—
449C4⋊C8—21no—
450C4⋊C8—21no—
451C4⋊C8—611no—
452C4⋊C8—611no—
453C4⋊C8—611no—
454C4⋊C8—611no—
455C4⋊C8—1221no—
456C4⋊C8—1221no—
457#104—41no—
458#104—41no—
459#104—2441no—
460#104—2441no—
461#103—611no—
462#103—611no—
463#103—611no—
464#103—611no—
465#102—611no—
466#102—611no—
467#102—611no—
468#102—611no—
469C22⋊C8—1221no—
470C22⋊C8—1221no—
471C22⋊C8—1221no—
472C22⋊C8—1221no—
473C22⋊C8—1221no—
474C22⋊C8—1221no—
475C22⋊C8—1221no—
476C22⋊C8—1221no—
477#97—611no—
478#97—611no—
479#97—611no—
480#97—611no—
481#97—1222no—
482#97—1221no—
483#97—1221no—
484#97—1221no—
485#97—1221no—
486C4×C8—1222no—
487C4×C8—1221no—
488C4×C8—1221no—
489C4×C8—1221no—
490C4×C8—1221no—
491#95—21no—
492#41—21no—
493#41—21no—
494#41—21no—
495#41—21no—
496#41—1221no—
497#41—1221no—
498#41—1221no—
499#41—1221no—
500C2×Q8—611no—
501C2×Q8C2×C4×D6111yescharacteristic
502C2×D4—21no—
503C2×D4—611no—
504C2×D4—1221no—
505C2×D4C2×C4×D6111yescharacteristic
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4#1175822411yescharacteristic
515C22×C4#1175822411yescharacteristic
516C22×C4C23×D6111yescharacteristic
517Q16—1222no—
518SD16—1222no—
519SD16—1222no—
520D8—1222no—
521M16—21no—
522M16—21no—
523M16—21no—
524M16—21no—
525M16—1221no—
526M16—1221no—
527M16—1221no—
528M16—1221no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—611no—
532C2×C8—611no—
533C2×C8—612no—
534C2×C8—612no—
535C2×C8—611no—
536C2×C8—611no—
537C2×C8—611no—
538C2×C8—611no—
539C2×C8—611no—
540C2×C8—611no—
541C2×C8—1221no—
542C2×C8—1221no—
543C2×C8—1221no—
544C2×C8—1221no—
545C2×C8—1221no—
546C2×C8—1221no—
547C2×C8—1221no—
548C2×C8—1221no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—42no—
552C4⋊C4—611no—
553C4⋊C4—611no—
554C4⋊C4—1221no—
555C4⋊C4—1221no—
556C4⋊C4C2×C4×D6111yescharacteristic
557C4⋊C4C2×C4×D6111yescharacteristic
558C22⋊C4—21no—
559C22⋊C4—21no—
560C22⋊C4—21no—
561C22⋊C4—1221no—
562C22⋊C4—1221no—
563C22⋊C4—1221no—
564C42—21no—
565C42—21no—
566C42—21no—
567C42—21no—
568C42—21no—
569C42—21no—
570C42—21no—
571C42—1221no—
572C42—1221no—
573C42—1221no—
574C42C2×C4×D6111yescharacteristic
575C42C2×C4×D6111yescharacteristic
576C42D4×D6111yescharacteristic
577C42D4×D6111yescharacteristic
578C23—21no—
579C23—1221no—
580C23C2×#1175822411yescharacteristic
581Q8—21no—
582Q8—21no—
583Q8—1221no—
584Q8—1221no—
585D4—21no—
586D4—21no—
587D4—41no—
588D4—41no—
589D4—1221no—
590D4—1221no—
591D4—2441no—
592D4—2441no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—81no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—4881no—
612C2×C4—2442no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4#1451973711yescharacteristic
616C2×C4#1451973711yescharacteristic
617C2×C4#1451973711yescharacteristic
618C2×C4#1451973711yescharacteristic
619C2×C4C22×C4×D6111yescharacteristic
620C2×C4C2×D4×D6111yescharacteristic
621C2×C4#41×D6111yescharacteristic
622C2×C4#41×D6111yescharacteristic
623C2×C4#1451982211yescharacteristic
624C2×C4#1451982511yescharacteristic
625C8—41no—
626C8—41no—
627C8—612no—
628C8—612no—
629C8—1221no—
630C8—1221no—
631C8—1221no—
632C8—1221no—
633C8—1221no—
634C8—1221no—
635C8—2441no—
636C8—2441no—
637C22—21no—
638C22—21no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—2441no—
643C22—2441no—
644C22—2441no—
645C22C2×#1451973711yescharacteristic
646C22#1684047611yescharacteristic
647C22#1684048311yescharacteristic
648C4—21no—
649C4—21no—
650C4—21no—
651C4—21no—
652C4—42no—
653C4—41no—
654C4—41no—
655C4—41no—
656C4—81no—
657C4—2441no—
658C4—2441no—
659C4—2441no—
660C4#117×D6111yescharacteristic
661C4#1684024311yescharacteristic
662C4#1684041411yescharacteristic
663C4#1684043011yescharacteristic
664C2—21no—
665C2—21no—
666C2—81no—
667C2—4881no—
668C2#2144876311yescharacteristic
669C2#2145471811yescharacteristic
670C2#2145472511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1806C2
socleC22×C61C2×C4×D4
prime radicalC2×#14519797C22
cosocleC22×C4×C61C24
hypercenter#1806D61
hypocenterC61#22804
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1806

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21454487C2semidirect
3. #21454485C2semidirect
6. #21446470C2semidirect
7. #21446470C2semidirect
8. #21446465C2semidirect
9. #21446465C2semidirect
10. #21445993C2semidirect
11. #21445840C2semidirect
12. #21445702C2semidirect
13. #21445697C2semidirect
14. #21445694C2semidirect
15. C61×#1806C2semidirect
18. #16839872C4semidirect
19. #16839872C4semidirect
20. #16839867C4semidirect
21. #16839867C4semidirect
22. #16839862C4semidirect
23. #16839862C4semidirect
24. #16839860C4semidirect
25. #16839860C4semidirect
340. #1806D61semidirect
402. C61#22804semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214546172
#214544872
#214544852
#214467092
#214467042
#214464702
#214464652
#214459932
#214458402
#214457022
#214456972
#214456942
C61×#18062
#2280461