Finite Group Databasebeta

← #29698514

Subgroups — #29698514

Order 15616.

850 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839882C211yesmaximal, characteristic
2#21454698C211yesmaximal, characteristic
3#21454690C211yesmaximal, characteristic
4#21454419C211yesmaximal, characteristic, complemented
5#21454410C211yesmaximal, characteristic, complemented
6#21446603C211yesmaximal, characteristic, complemented
7#21446598C211yesmaximal, characteristic, complemented
8#21446055C211yesmaximal, characteristic, complemented
9#21446050C211yesmaximal, characteristic, complemented
10#21446048C211yesmaximal, characteristic, complemented
11#21446043C211yesmaximal, characteristic, complemented
12#21445984C211yesmaximal, characteristic, complemented
13#21445880C211yesmaximal, characteristic, complemented
14#21445706C211yesmaximal, characteristic, complemented
15C61×#1815C211yesmaximal, characteristic, complemented
16C2×#14519797C2211yescharacteristic
17C2×#14519783C2211yescharacteristic
18#16840245C2211yescharacteristic
19#16840245C2211yescharacteristic
20#16840243C2211yescharacteristic
21#16840243C2211yescharacteristic
22#16839882C418yescomplemented
23#16839871—21no—
24#16839866—21no—
25#16839865—21no—
26#16839859—21no—
27#16839854—21no—
28#16839853—21no—
29#16839847—21no—
30#16839842—21no—
31C2×#14519712C2211yescharacteristic
32#16839804—21no—
33#16839804—21no—
34#16839804C2211yescharacteristic
35#16839804C2211yescharacteristic
36#16839802C2211yescharacteristic
37#16839802C2211yescharacteristic
38#16839801—21no—
39#16839801—21no—
40#16839800—21no—
41#16839800C2211yescharacteristic
42#16839797—21no—
43C2×#14519711—21no—
44C2×#14519711C2211yescharacteristic
45C2×#14519710C2211yescharacteristic
46C2×#14519708C2211yescharacteristic
47#106×D61—21no—
48C2×#14519704C2211yescharacteristic
49C2×#14519703C2211yescharacteristic
50#16839764C2211yescharacteristic
51#16839764C2211yescharacteristic
52#16839720C2211yescharacteristic
53#16839720C2211yescharacteristic
54C2×#14519687C2211yescharacteristic
55#16839631C2211yescharacteristic
56#16839627C2211yescharacteristic
57#16839530C2211yescharacteristic
58#16839530C2211yescharacteristic
59#16839523C2211yescharacteristic
60#16839523C2211yescharacteristic
61C2×#131×C61C2211yescharacteristic
62C2×#108×C61C2211yescharacteristic
63C2×#106×C61C2211yescharacteristic
64C61×#417C2211yescharacteristic
65C61×#417C2211yescharacteristic
66C61×#343C2211yescharacteristic
67C61×#343C2211yescharacteristic
68C2×#11758222C2311yescharacteristic
69#14519797C2×C414yes—
70#14519797Q814yes—
71C2×#11758202D411yescharacteristic
72C2×#11758200D411yescharacteristic
73C2×#11758201D411yescharacteristic
74C2×#11758199D411yescharacteristic
75#14519783—22no—
76#14519783C2×C414yescomplemented
77C2×#11758194C2311yescharacteristic
78C2×#11758189—21no—
79C22×#92418C2311yescharacteristic
80C2×C8×D61—21no—
81Q8×Dic61—21no—
82Q8×Dic61—21no—
83#14519742—21no—
84#14519742—21no—
85D4×Dic61—21no—
86D4×Dic61—21no—
87D4×Dic61—41no—
88D4×Dic61—41no—
89#14519739—21no—
90#14519739—21no—
91C4×C22⋊D61—41no—
92C4×C22⋊D61—41no—
93#14519736—21no—
94#14519736—21no—
95#14519736C2311yescharacteristic
96#14519736C2311yescharacteristic
97#14519735—21no—
98#14519735—21no—
99C2×C4⋊Dic61—21no—
100C2×C4⋊Dic61—21no—
101C2×C4⋊Dic61C2311yescharacteristic
102C2×C4⋊Dic61C2311yescharacteristic
103C2×#11758196—41no—
104C4⋊C4×D61—21no—
105C4⋊C4×D61—21no—
106C2×C4×Dic61C2311yescharacteristic
107#14519712C2×C414yescomplemented
108#14519711—21no—
109#14519711—21no—
110#14519711—21no—
111#14519711—21no—
112#14519711D411yescharacteristic
113#14519711D411yescharacteristic
114#14519710—21no—
115#14519710—21no—
116#14519710—21no—
117#14519710—21no—
118#14519709—21no—
119#14519709—21no—
120#14519709D411yescharacteristic
121#14519709D411yescharacteristic
122#14519708—21no—
123#14519708—21no—
124#14519708—21no—
125#14519708—21no—
126#14519704C2×C414yescomplemented
127#14519703C2×C414yescomplemented
128#14519702—21no—
129#14519702—21no—
130#14519699—21no—
131#14519699—21no—
132#14519687—21no—
133#14519687—21no—
134#14519687C2×C414yescomplemented
135C8×Dic61—21no—
136C8×Dic61—21no—
137#14519684—41no—
138C2×#41×C61C2311yescharacteristic
139#131×C61—22no—
140#131×C61C2×C414yescomplemented
141C2×M16×C61—21no—
142C2×M16×C61C2311yescharacteristic
143C2×M16×C61C2311yescharacteristic
144C2×M16×C61C2311yescharacteristic
145C22×C8×C61—21no—
146C22×C8×C61C2311yescharacteristic
147C2×C4⋊C4×C61C2311yescharacteristic
148C2×C4⋊C4×C61C2311yescharacteristic
149#108×C61—22no—
150#106×C61—21no—
151#106×C61—21no—
152#103×C61—21no—
153#103×C61—21no—
154#102×C61—21no—
155#102×C61—21no—
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×C61C2411yescharacteristic
167C2×C22⋊D61—21no—
168C22×Dic61—21no—
169C22×Dic61—21no—
170C22×Dic61—21no—
171C22×Dic61—41no—
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—24no—
184M16×C61—21no—
185M16×C61—21no—
186M16×C61—21no—
187M16×C61—21no—
188M16×C61C22×C412yes—
189C2×C8×C61—21no—
190C2×C8×C61—24no—
191C2×C8×C61—21no—
192C2×C8×C61—21no—
193C2×C8×C61C22×C412yes—
194C2×C8×C61C2×D411yescharacteristic
195C2×C8×C61C2×D411yescharacteristic
196C2×C8×C61#4112yes—
197C4⋊C4×C61—21no—
198C4⋊C4×C61—21no—
199C4⋊C4×C61—21no—
200C4⋊C4×C61—21no—
201C22⋊Dic61—21no—
202C22⋊Dic61—21no—
203C22⋊Dic61—41no—
204C22⋊Dic61—41no—
205#11758202C4⋊C414yescomplemented
206#11758201C4⋊C414yescomplemented
207#11758200C4⋊C414yescomplemented
208#11758199C4⋊C414yescomplemented
209#11758198—41no—
210#11758198—41no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214C4⋊Dic61—21no—
215C4⋊Dic61—41no—
216C4⋊Dic61—41no—
217C4⋊Dic61C2×D411yescharacteristic
218C4⋊Dic61C2×D411yescharacteristic
219C4⋊Dic61C2×D411yescharacteristic
220C4⋊Dic61C2×D411yescharacteristic
221#11758196—41no—
222#11758196—41no—
223#11758196—82no—
224C4×Dic61—21no—
225C4×Dic61—21no—
226C4×Dic61—21no—
227C4×Dic61—41no—
228C4×Dic61—41no—
229C4×Dic61—41no—
230C4×Dic61—41no—
231C4×Dic61C22×C412yes—
232#11758194—21no—
233#11758194C22×C412yes—
234C2×#92418—21no—
235C2×#92418C22×C412yes—
236C2×#92418C2×D411yescharacteristic
237C2×#92418C2×D411yescharacteristic
238C2×#92418C2×Q812yes—
239#11758189—24no—
240C8×D61—24no—
241C23×C61—21no—
242C23×C61H(5,𝔽2)11yescharacteristic
243C22×D61—21no—
244Q8×C61—21no—
245Q8×C61C2×C4⋊C412yes—
246D4×C61—22no—
247D4×C61—22no—
248D4×C61—21no—
249D4×C61C2×C4⋊C412yes—
250C2×C4×C61—21no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—81no—
256C2×C4×C61—81no—
257C2×C4×C61C23×C411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C22×Q811yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C612−1+411yescharacteristic
263C22⋊D61—22no—
264C22⋊D61—22no—
265C2×Dic61—21no—
266C2×Dic61—22no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—81no—
276C2×Dic61—81no—
277C2×Dic61—161no—
278C2×Dic61—161no—
279D244—21no—
280D244C2×C4⋊C412yes—
281C4×D61—22no—
282C4×D61—22no—
283C4×D61—161no—
284C4×D61—161no—
285Dic122—21no—
286Dic122C2×C4⋊C412yes—
287C8×C61—22no—
288C8×C61—41no—
289C8×C61—41no—
290C8×C61C4×D412yes—
291C8×C61C4×D412yes—
292#92418—22no—
293#92418C2×C4⋊C412yes—
294#92418C2×C4⋊C412yes—
295#22693—611nomaximal, Sylow
296C22×C61—21no—
297C22×C61—21no—
298C22×C61—22no—
299C22×C61—41no—
300C22×C61—41no—
301C22×C61#52011yescharacteristic
302C22×C61D4211yescharacteristic
303C22×C61#55411yescharacteristic
304C2×D61—22no—
305C2×D61—41no—
306C2×D61—41no—
307C4×C61—22no—
308C4×C61—81no—
309C4×C61—81no—
310C4×C61C22×C4⋊C411yescharacteristic
311C4×C61C2×C4×D411yescharacteristic
312C4×C61#54711yescharacteristic
313C4×C61D4×Q811yescharacteristic
314Dic61—22no—
315Dic61—42no—
316Dic61—41no—
317Dic61—41no—
318Dic61—41no—
319Dic61—41no—
320Dic61—81no—
321Dic61—81no—
322C2×#444—611no—
323#2952—611no—
324#2951—611no—
325#2913—611no—
326#2912—611no—
327#1952—611no—
328#1951—611no—
329#1860—611no—
330#1860—611no—
331#1859—611no—
332#1859—611no—
333#1847—611no—
334#1816—611no—
335#1815D6111yescharacteristic, complemented
336#1773—611no—
337C2×C61—21no—
338C2×C61—21no—
339C2×C61—42no—
340C2×C61D4×C4⋊C411yescharacteristic
341C2×C61#328811yescharacteristic
342C2×C61#328911yescharacteristic
343D61—42no—
344C2×#135—611no—
345C2×#131—611no—
346C2×#131C2×D6111yescharacteristic
347#520—611no—
348#520—611no—
349#520—611no—
350#520—611no—
351#444—618no—
352#442—1221no—
353#441—1221no—
354#440—1221no—
355#440—1221no—
356C4×Q16—1221no—
357C4×SD16—1221no—
358C4×SD16—1221no—
359C4×D8—1221no—
360C2×#108—611no—
361C2×#108C2×D6111yescharacteristic
362#428—611no—
363#428—611no—
364#428—611no—
365#428—611no—
366#428—1221no—
367#428—1221no—
368#428—1221no—
369#428—1221no—
370#427—611no—
371#427—1221no—
372#427—1221no—
373C2×#107—611no—
374C2×#107—611no—
375C2×#107—1221no—
376C2×#106—611no—
377C2×#106—1221no—
378C2×#106C2×D6111yescharacteristic
379C2×#104—611no—
380C2×#104—611no—
381#417—611no—
382#417—611no—
383#417C2×D6111yescharacteristic
384#417C2×D6111yescharacteristic
385#405—611no—
386#405—611no—
387C2×#97—611no—
388#384—611no—
389#382—611no—
390#343—611no—
391#343—611no—
392#343C2×D6111yescharacteristic
393#343C2×D6111yescharacteristic
394#339—611no—
395#339—611no—
396C61#2269311yescharacteristic, complemented, Sylow
397C2×#41—611no—
398C2×#41C22×D6111yescharacteristic
399#135—614no—
400#135—614no—
401C2×Q16—611no—
402C2×SD16—611no—
403C2×SD16—611no—
404C2×D8—611no—
405#131—22no—
406#131—614no—
407#131—1222no—
408#131C4×D6114yescomplemented
409C2×M16—21no—
410C2×M16—611no—
411C2×M16—1221no—
412C2×M16C22×D6111yescharacteristic
413C2×M16C22×D6111yescharacteristic
414C2×M16C22×D6111yescharacteristic
415C22×C8—21no—
416C22×C8—611no—
417C22×C8—1221no—
418C22×C8C22×D6111yescharacteristic
419C4×Q8—1221no—
420C4×Q8—1221no—
421C4×Q8—1221no—
422C4×Q8—1221no—
423C4×D4—1221no—
424C4×D4—1221no—
425C4×D4—1221no—
426C4×D4—1221no—
427C4×D4—2441no—
428C4×D4—2441no—
429C4×D4—2441no—
430C4×D4—2441no—
431#117—611no—
432#117—611no—
433#117—1221no—
434#117—1221no—
435#117—1221no—
436#117—1221no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—2441no—
444C2×C4⋊C4C22×D6111yescharacteristic
445C2×C4⋊C4C22×D6111yescharacteristic
446C2×C42—611no—
447#108—22no—
448#108—614no—
449#107—611no—
450#107—611no—
451#107—1221no—
452#107—1221no—
453#107—1221no—
454#107—1221no—
455#107—1221no—
456#107—1221no—
457#107—1221no—
458#107—1221no—
459#106—21no—
460#106—21no—
461#106—611no—
462#106—611no—
463#106—1221no—
464#106—1221no—
465#106—1221no—
466#106—1221no—
467#106—1221no—
468#106—1221no—
469#104—614no—
470#104—614no—
471#103—21no—
472#103—21no—
473#103—1221no—
474#103—1221no—
475#102—21no—
476#102—21no—
477#102—1221no—
478#102—1221no—
479#97—614no—
480#97—1221no—
481#97—1221no—
482C4×C8—1221no—
483C4×C8—1221no—
484#95—2441no—
485#41—21no—
486#41—21no—
487#41—612no—
488#41—612no—
489#41—1221no—
490#41—1221no—
491#41C2×C4×D6112yes—
492#41Q8×D6112yes—
493C2×Q8—611no—
494C2×Q8D4×D6111yescharacteristic
495C2×D4—21no—
496C2×D4—611no—
497C2×D4—1221no—
498C2×D4D4×D6111yescharacteristic
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—2441no—
507C22×C4—2441no—
508C22×C4—2441no—
509C22×C4—2441no—
510C22×C4C23×D6111yescharacteristic
511Q16—614no—
512SD16—614no—
513SD16—614no—
514D8—614no—
515M16—21no—
516M16—24no—
517M16—21no—
518M16—21no—
519M16—21no—
520M16—21no—
521M16—612no—
522M16—1221no—
523M16—1224no—
524M16C2×C4×D6112yes—
525C2×C8—21no—
526C2×C8—24no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—612no—
530C2×C8—611no—
531C2×C8—611no—
532C2×C8—612no—
533C2×C8—1221no—
534C2×C8—1224no—
535C2×C8C2×C4×D6112yes—
536C2×C8D4×D6111yescharacteristic
537C2×C8D4×D6111yescharacteristic
538C2×C8#1175822412yes—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—611no—
544C4⋊C4—611no—
545C4⋊C4—611no—
546C4⋊C4—611no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—4882no—
552C4⋊C4—2441no—
553C4⋊C4—2441no—
554C4⋊C4—2441no—
555C4⋊C4—2441no—
556C22⋊C4—1221no—
557C22⋊C4—1221no—
558C22⋊C4—2441no—
559C22⋊C4—2441no—
560C22⋊C4—2441no—
561C22⋊C4—2441no—
562C42—612no—
563C42—1221no—
564C42—1221no—
565C42—1221no—
566C42—2441no—
567C42—2441no—
568C42—2441no—
569C42—2441no—
570C23—21no—
571C23—1221no—
572C23#1451982311yescharacteristic
573Q8—21no—
574Q8—612no—
575Q8—1221no—
576Q8C4⋊C4×D6112yes—
577D4—22no—
578D4—22no—
579D4—21no—
580D4—612no—
581D4—1222no—
582D4—1222no—
583D4—1221no—
584D4C4⋊C4×D6112yes—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—1221no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—9761no—
601C2×C4—9761no—
602C2×C4—9761no—
603C2×C4—9761no—
604C2×C4—4881no—
605C2×C4—4881no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4C22×C4×D6111yescharacteristic
611C2×C4C2×D4×D6111yescharacteristic
612C2×C4C2×D4×D6111yescharacteristic
613C2×C4C2×Q8×D6111yescharacteristic
614C2×C4C2×#1175822411yescharacteristic
615C2×C4#1451982411yescharacteristic
616C8—22no—
617C8—41no—
618C8—41no—
619C8—612no—
620C8—612no—
621C8—1222no—
622C8#1451973712yes—
623C8#1451973712yes—
624C22—21no—
625C22—21no—
626C22—22no—
627C22—41no—
628C22—41no—
629C22—1222no—
630C22—2441no—
631C22—2441no—
632C22#1684024211yescharacteristic
633C22#1684039611yescharacteristic
634C22#1684047011yescharacteristic
635C4—22no—
636C4—81no—
637C4—81no—
638C4—1222no—
639C4—4881no—
640C4—4881no—
641C4—2442no—
642C4—2441no—
643C4—2441no—
644C4—2441no—
645C4—2441no—
646C4C2×C4⋊C4×D6111yescharacteristic
647C4C2×#1451973711yescharacteristic
648C4#1684041211yescharacteristic
649C4#1684042511yescharacteristic
650C2—21no—
651C2—21no—
652C2—42no—
653C2—2442no—
654C2#2144871511yescharacteristic
655C2#2145758411yescharacteristic
656C2#2145759111yescharacteristic
subgroupG/H
centerC22#16840396
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1815C2
socleC22×C61D42
prime radicalC2×#14519797C22
cosocleC22×C4×C61C24
hypercenter#1815D61
hypocenterC61#22693
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1815

Complements

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

NH ≅ G/Ntype
4. #21454419C2semidirect
5. #21454410C2semidirect
6. #21446603C2semidirect
7. #21446598C2semidirect
8. #21446055C2semidirect
9. #21446050C2semidirect
10. #21446048C2semidirect
11. #21446043C2semidirect
12. #21445984C2semidirect
13. #21445880C2semidirect
14. #21445706C2semidirect
15. C61×#1815C2semidirect
22. #16839882C4semidirect
76. #14519783C2×C4semidirect
107. #14519712C2×C4semidirect
126. #14519704C2×C4semidirect
127. #14519703C2×C4semidirect
134. #14519687C2×C4semidirect
140. #131×C61C2×C4semidirect
205. #11758202C4⋊C4semidirect
206. #11758201C4⋊C4semidirect
207. #11758200C4⋊C4semidirect
208. #11758199C4⋊C4semidirect
335. #1815D61semidirect
396. C61#22693semidirect
408. #131C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398822
#214546982
#214546902
#214544192
#214544102
#214466032
#214465982
#214460552
#214460502
#214460482
#214460432
#214459842
#214458802
#214457062
C61×#18152
#2269361