Finite Group Databasebeta

← #29698492

Subgroups — #29698492

Order 15616.

846 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839876C211yesmaximal, characteristic
2#21454692C211yesmaximal, characteristic
3#21454690C211yesmaximal, characteristic
4#21454413C211yesmaximal, characteristic, complemented
5#21454412C211yesmaximal, characteristic, complemented
6#21446603C211yesmaximal, characteristic, complemented
7#21446601C211yesmaximal, characteristic, complemented
8#21446054C211yesmaximal, characteristic, complemented
9#21446053C211yesmaximal, characteristic, complemented
10#21446051C211yesmaximal, characteristic, complemented
11#21445981C211yesmaximal, characteristic, complemented
12#21445877C211yesmaximal, characteristic, complemented
13#21445876C211yesmaximal, characteristic, complemented
14#21445697C211yesmaximal, characteristic, complemented
15C61×#1860C211yesmaximal, characteristic, complemented
16C2×#14519797C2211yescharacteristic
17C2×#14519785C2211yescharacteristic
18C2×#14519784C2211yescharacteristic
19#16840245C2211yescharacteristic
20#16840243C2211yescharacteristic
21#16840242C2211yescharacteristic
22#16839876C418yescomplemented
23#16839865—21no—
24#16839862—21no—
25#16839860—21no—
26#16839859—21no—
27C4×#11758202—21no—
28#16839854—21no—
29C4×#11758199—21no—
30#16839842—21no—
31C2×#14519712C2211yescharacteristic
32C2×#14519712C2211yescharacteristic
33#16839805—21no—
34#16839805C2211yescharacteristic
35#16839804C2211yescharacteristic
36#16839803—21no—
37#16839802—21no—
38#16839802C2211yescharacteristic
39#16839801—21no—
40#16839799—21no—
41#16839798—21no—
42#16839798C2211yescharacteristic
43C2×#14519710C2211yescharacteristic
44C2×#14519710C2211yescharacteristic
45C2×#14519708—21no—
46C2×#14519708C2211yescharacteristic
47C2×#14519708C2211yescharacteristic
48#106×D61—21no—
49C2×#14519705C2211yescharacteristic
50#16839764C2211yescharacteristic
51#16839763C2211yescharacteristic
52#16839761C2211yescharacteristic
53#16839720C2211yescharacteristic
54#16839718C2211yescharacteristic
55C2×C4×#92418C2211yescharacteristic
56#16839628C2211yescharacteristic
57#16839531C2211yescharacteristic
58#16839530C2211yescharacteristic
59#16839530C2211yescharacteristic
60#16839522C2211yescharacteristic
61C61×#520C2211yescharacteristic
62C61×#428C2211yescharacteristic
63C2×#104×C61C2211yescharacteristic
64C61×#417C2211yescharacteristic
65C61×#382C2211yescharacteristic
66C61×#343C2211yescharacteristic
67C61×#339C2211yescharacteristic
68C2×#11758222C2311yescharacteristic
69#14519797C2×C414yes—
70#14519797Q814yes—
71C2×#11758202D411yescharacteristic
72C2×#11758200D411yescharacteristic
73C2×#11758201D411yescharacteristic
74C2×#11758199D411yescharacteristic
75#14519785—22no—
76#14519785C2×C414yescomplemented
77#14519784—22no—
78#14519784C2×C414yescomplemented
79C2×#11758194—21no—
80C2×#11758194C2311yescharacteristic
81C2×#11758194C2311yescharacteristic
82C2×#11758189—21no—
83C22×#92418—21no—
84C22×#92418C2311yescharacteristic
85C22×#92418C2311yescharacteristic
86C2×C8×D61—21no—
87Q8×Dic61—21no—
88C4×Dic122—21no—
89#14519742—21no—
90C4×D244—21no—
91D4×Dic61—21no—
92D4×Dic61—41no—
93#14519739—21no—
94C4×C22⋊D61—41no—
95#14519737—41no—
96#14519736—21no—
97#14519736C2311yescharacteristic
98#14519735—21no—
99#14519735—21no—
100C2×C4⋊Dic61—21no—
101C2×C4⋊Dic61C2311yescharacteristic
102C2×C4⋊Dic61C2311yescharacteristic
103C4⋊C4×D61—21no—
104C4⋊C4×D61—21no—
105#14519712—22no—
106#14519712C2×C414yescomplemented
107#14519711—21no—
108#14519710—21no—
109#14519710—21no—
110#14519710—21no—
111#14519710—21no—
112#14519710—21no—
113#14519710—21no—
114#14519710D411yescharacteristic
115#14519710D411yescharacteristic
116#14519709—21no—
117#14519708—21no—
118#14519708—21no—
119#14519708—21no—
120#14519708—21no—
121#14519708—21no—
122#14519708—21no—
123#14519708D411yescharacteristic
124#14519708D411yescharacteristic
125#14519705C2×C414yescomplemented
126#14519702—21no—
127#14519701—21no—
128#14519700—21no—
129#14519699—21no—
130#14519698—21no—
131#14519697—21no—
132#14519688—21no—
133#14519687—21no—
134C4×#92418—21no—
135C4×#92418C2×C414yescomplemented
136C8×Dic61—21no—
137#14519684—41no—
138C2×#41×C61C2311yescharacteristic
139C2×M16×C61C2311yescharacteristic
140C2×M16×C61C2311yescharacteristic
141C4×Q8×C61—21no—
142C4×D4×C61—21no—
143C4×D4×C61—41no—
144#117×C61—21no—
145#117×C61C2311yescharacteristic
146C2×C4⋊C4×C61—21no—
147C2×C4⋊C4×C61C2311yescharacteristic
148C2×C4⋊C4×C61C2311yescharacteristic
149C2×C42×C61C2311yescharacteristic
150#107×C61—21no—
151#106×C61—21no—
152#104×C61C2×C414yescomplemented
153#103×C61—21no—
154#102×C61—21no—
155#95×C61—41no—
156#41×C61—21no—
157#41×C61—21no—
158#41×C61C22×C412yes—
159#41×C61C2×Q812yes—
160C2×Q8×C61C2×D411yescharacteristic
161C2×D4×C61—21no—
162C2×D4×C61C2×D411yescharacteristic
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61—21no—
166C22×C4×C61—21no—
167C22×C4×C61—41no—
168C22×C4×C61C2411yescharacteristic
169C2×C22⋊D61—21no—
170C22×Dic61—21no—
171C22×Dic61—21no—
172C22×Dic61—41no—
173#11758222—21no—
174#11758222—21no—
175#11758222C22×C412yes—
176#11758222C2×Q812yes—
177C2×D244C2×D411yescharacteristic
178C2×C4×D61—21no—
179C2×C4×D61—41no—
180C2×C4×D61—41no—
181C2×Dic122C2×D411yescharacteristic
182M16×C61—21no—
183M16×C61—21no—
184M16×C61—21no—
185M16×C61C22×C412yes—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—41no—
193C4⋊C4×C61C2×D411yescharacteristic
194C4⋊C4×C61C2×D411yescharacteristic
195C22⋊C4×C61—21no—
196C22⋊C4×C61—41no—
197C42×C61—21no—
198C42×C61—21no—
199C42×C61—41no—
200C42×C61C22×C412yes—
201C22⋊Dic61—21no—
202C22⋊Dic61—41no—
203#11758202C4⋊C414yes—
204#11758201C4⋊C414yes—
205#11758200C4⋊C414yes—
206#11758199C4⋊C414yes—
207#11758198—41no—
208#11758198—41no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—41no—
214C4⋊Dic61C2×D411yescharacteristic
215C4⋊Dic61C2×D411yescharacteristic
216#11758196—41no—
217#11758196—41no—
218C4×Dic61—21no—
219C4×Dic61—41no—
220C4×Dic61—41no—
221C4×Dic61—41no—
222#11758194—21no—
223#11758194—24no—
224#11758194—21no—
225#11758194—21no—
226#11758194C22×C412yes—
227C2×#92418—21no—
228C2×#92418—24no—
229C2×#92418—21no—
230C2×#92418C22×C412yes—
231C2×#92418C22×C412yes—
232C2×#92418C2×D411yescharacteristic
233C2×#92418C2×D411yescharacteristic
234C2×#92418C2×D411yescharacteristic
235C2×#92418C2×D411yescharacteristic
236C2×#92418C2×Q812yes—
237C2×#92418#4112yes—
238#11758189—24no—
239C8×D61—24no—
240C23×C61—21no—
241C23×C61H(5,𝔽2)11yescharacteristic
242C22×D61—21no—
243Q8×C61—21no—
244Q8×C61C2×C4⋊C412yes—
245D4×C61—22no—
246D4×C61—22no—
247D4×C61—21no—
248D4×C61C2×C4⋊C412yes—
249C2×C4×C61—21no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—81no—
260C2×C4×C61—81no—
261C2×C4×C61—161no—
262C2×C4×C61C23×C411yescharacteristic
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C22×Q811yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C612−1+411yescharacteristic
268C22⋊D61—22no—
269C22⋊D61—22no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—81no—
276C2×Dic61—81no—
277C2×Dic61—161no—
278D244—21no—
279D244C2×C4⋊C412yes—
280C4×D61—22no—
281C4×D61—22no—
282C4×D61—161no—
283C4×D61—161no—
284Dic122—21no—
285Dic122C2×C4⋊C412yes—
286C8×C61—22no—
287C8×C61—41no—
288#92418—22no—
289#92418—41no—
290#92418C2×C4⋊C412yes—
291#92418C2×C4⋊C412yes—
292#92418C4×D412yes—
293#92418C4×D412yes—
294#22690—611nomaximal, Sylow
295C22×C61—21no—
296C22×C61—21no—
297C22×C61—22no—
298C22×C61—41no—
299C22×C61—41no—
300C22×C61#52011yescharacteristic
301C22×C61D4211yescharacteristic
302C22×C61#55411yescharacteristic
303C2×D61—22no—
304C2×D61—41no—
305C2×D61—41no—
306C4×C61—22no—
307C4×C61—42no—
308C4×C61—41no—
309C4×C61—41no—
310C4×C61—81no—
311C4×C61—81no—
312C4×C61C22×C4⋊C411yescharacteristic
313C4×C61C2×C4×D411yescharacteristic
314C4×C61#54711yescharacteristic
315C4×C61D4×Q811yescharacteristic
316Dic61—22no—
317Dic61—41no—
318Dic61—41no—
319Dic61—81no—
320Dic61—81no—
321C2×#443—611no—
322#2951—611no—
323#2951—611no—
324#2912—611no—
325#2912—611no—
326#1952—611no—
327#1952—611no—
328#1860—611no—
329#1860—611no—
330#1860—611no—
331#1860D6111yescharacteristic, complemented
332#1846—611no—
333#1815—611no—
334#1815—611no—
335#1772—611no—
336C2×C61—21no—
337C2×C61—21no—
338C2×C61—42no—
339C2×C61D4×C4⋊C411yescharacteristic
340C2×C61#329011yescharacteristic
341C2×C61#329011yescharacteristic
342D61—42no—
343C2×#135—611no—
344C2×#131—611no—
345C2×#131—611no—
346#520—611no—
347#520—611no—
348#520—611no—
349#520C2×D6111yescharacteristic
350#443—618no—
351#440—1221no—
352#440—1221no—
353#440—1221no—
354#440—1221no—
355C4×Q16—1221no—
356C4×Q16—1221no—
357C4×D8—1221no—
358C4×D8—1221no—
359C2×#108—611no—
360C2×#108—611no—
361#428—611no—
362#428—611no—
363#428—611no—
364#428—1221no—
365#428—1221no—
366#428—1221no—
367#428—1221no—
368#428C2×D6111yescharacteristic
369#427—611no—
370#427—1221no—
371#427—1221no—
372C2×#107—611no—
373C2×#107—611no—
374C2×#106—611no—
375C2×#106—611no—
376C2×#106—1221no—
377C2×#106—1221no—
378C2×#104—611no—
379C2×#104C2×D6111yescharacteristic
380#417—611no—
381#417—611no—
382#417—611no—
383#417C2×D6111yescharacteristic
384#405—611no—
385#405—611no—
386C2×C4×C8—611no—
387#382—611no—
388#382C2×D6111yescharacteristic
389#343—611no—
390#343—611no—
391#343—611no—
392#343C2×D6111yescharacteristic
393#339—611no—
394#339C2×D6111yescharacteristic
395C61#2269011yescharacteristic, complemented, Sylow
396C2×#41—611no—
397C2×#41C22×D6111yescharacteristic
398#135—614no—
399#135—614no—
400C2×Q16—611no—
401C2×SD16—611no—
402C2×SD16—611no—
403C2×D8—611no—
404#131—614no—
405#131—614no—
406#131—1222no—
407#131—1222no—
408C2×M16—611no—
409C2×M16—611no—
410C2×M16—1221no—
411C2×M16—1221no—
412C2×M16C22×D6111yescharacteristic
413C2×M16C22×D6111yescharacteristic
414C22×C8—611no—
415C22×C8—611no—
416C22×C8—1221no—
417C22×C8—1221no—
418C4×Q8—21no—
419C4×Q8—1221no—
420C4×Q8—1221no—
421C4×Q8—1221no—
422C4×D4—21no—
423C4×D4—41no—
424C4×D4—1221no—
425C4×D4—1221no—
426C4×D4—1221no—
427C4×D4—2441no—
428C4×D4—2441no—
429C4×D4—2441no—
430#117—21no—
431#117—611no—
432#117—1221no—
433#117—1221no—
434#117—1221no—
435#117C22×D6111yescharacteristic
436C2×C4⋊C4—21no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4C22×D6111yescharacteristic
443C2×C4⋊C4C22×D6111yescharacteristic
444C2×C42C22×D6111yescharacteristic
445#108—614no—
446#108—1222no—
447#107—21no—
448#107—611no—
449#107—611no—
450#107—1221no—
451#107—1221no—
452#107—1221no—
453#107—1221no—
454#107—1221no—
455#107—1221no—
456#107—1221no—
457#106—21no—
458#106—611no—
459#106—611no—
460#106—1221no—
461#106—1221no—
462#106—1221no—
463#106—1221no—
464#106—1221no—
465#106—1221no—
466#106—1221no—
467#104—614no—
468#104C4×D6114yescomplemented
469#103—21no—
470#103—1221no—
471#103—1221no—
472#103—1221no—
473#102—21no—
474#102—1221no—
475#102—1221no—
476#102—1221no—
477#97—1221no—
478#97—1221no—
479C4×C8—614no—
480C4×C8—1221no—
481C4×C8—1221no—
482#95—41no—
483#95—2441no—
484#41—21no—
485#41—21no—
486#41—612no—
487#41—612no—
488#41—1221no—
489#41—1221no—
490#41C2×C4×D6112yes—
491#41Q8×D6112yes—
492C2×Q8—611no—
493C2×Q8D4×D6111yescharacteristic
494C2×D4—21no—
495C2×D4—611no—
496C2×D4—1221no—
497C2×D4D4×D6111yescharacteristic
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—41no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—2441no—
507C22×C4—2441no—
508C22×C4—2441no—
509C22×C4C23×D6111yescharacteristic
510Q16—614no—
511SD16—614no—
512SD16—614no—
513D8—614no—
514M16—21no—
515M16—21no—
516M16—21no—
517M16—612no—
518M16—1224no—
519M16—1224no—
520M16—1221no—
521M16—1221no—
522M16—1221no—
523M16C2×C4×D6112yes—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—612no—
527C2×C8—612no—
528C2×C8—611no—
529C2×C8—611no—
530C2×C8—611no—
531C2×C8—611no—
532C2×C8—612no—
533C2×C8—612no—
534C2×C8—1224no—
535C2×C8—1224no—
536C2×C8—1221no—
537C2×C8—1221no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—41no—
543C4⋊C4—611no—
544C4⋊C4—611no—
545C4⋊C4—1221no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—2441no—
550C4⋊C4—2441no—
551C4⋊C4—2441no—
552C4⋊C4D4×D6111yescharacteristic
553C4⋊C4D4×D6111yescharacteristic
554C22⋊C4—21no—
555C22⋊C4—41no—
556C22⋊C4—1221no—
557C22⋊C4—2441no—
558C22⋊C4—2441no—
559C22⋊C4—2441no—
560C42—21no—
561C42—21no—
562C42—41no—
563C42—1221no—
564C42—2441no—
565C42—2441no—
566C42—2441no—
567C42C2×C4×D6112yes—
568C23—21no—
569C23—1221no—
570C23#1451982211yescharacteristic
571Q8—21no—
572Q8—612no—
573Q8—1221no—
574Q8C4⋊C4×D6112yes—
575D4—22no—
576D4—22no—
577D4—21no—
578D4—612no—
579D4—1222no—
580D4—1222no—
581D4—1221no—
582D4C4⋊C4×D6112yes—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—161no—
596C2×C4—1221no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—9761no—
602C2×C4—9761no—
603C2×C4—9761no—
604C2×C4—4881no—
605C2×C4—4881no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4C22×C4×D6111yescharacteristic
609C2×C4C2×D4×D6111yescharacteristic
610C2×C4C2×D4×D6111yescharacteristic
611C2×C4C2×Q8×D6111yescharacteristic
612C2×C4#41×D6111yescharacteristic
613C2×C4#1451982411yescharacteristic
614C8—22no—
615C8—41no—
616C8—612no—
617C8—612no—
618C8—612no—
619C8—612no—
620C8—1222no—
621C8—2441no—
622C22—21no—
623C22—21no—
624C22—22no—
625C22—41no—
626C22—41no—
627C22—1222no—
628C22—2441no—
629C22—2441no—
630C22#1684024111yescharacteristic
631C22#1684039411yescharacteristic
632C22#1684046611yescharacteristic
633C4—22no—
634C4—42no—
635C4—41no—
636C4—41no—
637C4—81no—
638C4—81no—
639C4—1222no—
640C4—4881no—
641C4—4881no—
642C4—2441no—
643C4—2441no—
644C4C2×C4⋊C4×D6111yescharacteristic
645C4C4×D4×D6111yescharacteristic
646C4#1684041011yescharacteristic
647C4#1684042511yescharacteristic
648C2—21no—
649C2—21no—
650C2—42no—
651C2—2442no—
652C2#2144871711yescharacteristic
653C2#2145759811yescharacteristic
654C2#2145760311yescharacteristic
subgroupG/H
centerC22#16840394
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1860C2
socleC22×C61D42
prime radicalC2×#14519797C22
cosocleC22×C4×C61C24
hypercenter#1860D61
hypocenterC61#22690
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1860

Complements

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

NH ≅ G/Ntype
4. #21454413C2semidirect
5. #21454412C2semidirect
6. #21446603C2semidirect
7. #21446601C2semidirect
8. #21446054C2semidirect
9. #21446053C2semidirect
10. #21446051C2semidirect
11. #21445981C2semidirect
12. #21445877C2semidirect
13. #21445876C2semidirect
14. #21445697C2semidirect
15. C61×#1860C2semidirect
22. #16839876C4semidirect
76. #14519785C2×C4semidirect
78. #14519784C2×C4semidirect
106. #14519712C2×C4semidirect
125. #14519705C2×C4semidirect
135. C4×#92418C2×C4semidirect
152. #104×C61C2×C4semidirect
331. #1860D61semidirect
395. C61#22690semidirect
468. #104C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398762
#214546922
#214546902
#214544132
#214544122
#214466032
#214466012
#214460542
#214460532
#214460512
#214459812
#214458772
#214458762
#214456972
C61×#18602
#2269061