Finite Group Databasebeta

← #10865152

Subgroups — #10865152

Order 768.

710 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3097C211yesmaximal, characteristic
2#83401C211yesmaximal, characteristic
3#82571C211yesmaximal, characteristic
4#79198C211yesmaximal, characteristic, complemented
5#71868C211yesmaximal, characteristic
6#71664C211yesmaximal, characteristic
7#70672C211yesmaximal, characteristic, complemented
8#70666C211yesmaximal, characteristic, complemented
9#70659C211yesmaximal, characteristic, complemented
10#70345C211yesmaximal, characteristic, complemented
11#70338C211yesmaximal, characteristic, complemented
12#70265C211yesmaximal, characteristic, complemented
13#70225C211yesmaximal, characteristic, complemented
14#70004C211yesmaximal, characteristic
15#69997C211yesmaximal, characteristic
16#26849—31nomaximal, Sylow
17C3×#536C2211yescharacteristic
18C3×#520C2211yescharacteristic
19C2×#951C2211yescharacteristic
20C2×#944C2211yescharacteristic
21#5519C2211yescharacteristic
22#5516C2211yescharacteristic
23C3×#485—21no—
24C3×#485—21no—
25C3×#484—21no—
26C3×#483—21no—
27C3×#428C2211yescharacteristic
28C3×#427C2211yescharacteristic
29C3×#419C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31C3×#408C2211yescharacteristic
32C2×#837C2211yescharacteristic
33#5233C2211yescharacteristic
34#5232C2211yescharacteristic
35C2×#836C2211yescharacteristic
36#5225C2211yescharacteristic
37C2×#835C2211yescharacteristic
38C2×#824C2211yescharacteristic
39C2×#819C2211yescharacteristic
40C2×#816C2211yescharacteristic
41#5102C2211yescharacteristic
42C2×C8×Dic3C2211yescharacteristic
43#5082—21no—
44#5079—21no—
45#5056—21no—
46#5055—21no—
47#5030—21no—
48C4×#216—21no—
49#5018—21no—
50C4×#214—21no—
51#5004C2211yescharacteristic
52#4997C2211yescharacteristic
53#4997C2211yescharacteristic
54#4995C2211yescharacteristic
55#4975C2211yescharacteristic
56C2×#810C2211yescharacteristic
57#4964C2211yescharacteristic
58#4877—21no—
59#4875—21no—
60#4857—21no—
61#4854—21no—
62#4558C2211yescharacteristic
63#4555C2211yescharacteristic
64#4554C2211yescharacteristic
65#4532C2211yescharacteristic
66#4532C2211yescharacteristic
67#4530C2211yescharacteristic
68#3137—31no—
69#3097D311yescharacteristic
70#3082—31no—
71#2947—31no—
72#2064—31no—
73#2060—31no—
74#2052—31no—
75#2005—31no—
76#1956—31no—
77#1937—31no—
78#1930—31no—
79#1930—31no—
80#1886—31no—
81#1859—31no—
82#1772—31no—
83C2×C3×#41C2311yescharacteristic
84C22×C3×Q8C2311yescharacteristic
85C22×Dic6C2311yescharacteristic
86C2×C3×M16C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×C4⋊Q8—21no—
89C3×#124—21no—
90C3×#123—41no—
91C3×C22⋊Q8—21no—
92C3×C22⋊Q8—21no—
93C3×C4×Q8—21no—
94C3×C4×D4—21no—
95C3×C4×D4—41no—
96C3×#117—21no—
97C3×#117C2311yescharacteristic
98C3×#117C2311yescharacteristic
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4C2311yescharacteristic
101#951—22no—
102#951—22no—
103#944—22no—
104#944—22no—
105C2×#216—21no—
106C2×#214—21no—
107#926C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×Dic3⋊C4—41no—
110C2×Dic3⋊C4—41no—
111C2×C4×Dic3C2311yescharacteristic
112C2×#208C2311yescharacteristic
113C2×#208C2311yescharacteristic
114C2×#208C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116#890—21no—
117#888—21no—
118#887—21no—
119#887—21no—
120#878—41no—
121#877—41no—
122#870—21no—
123#869—21no—
124C4×Dic6—21no—
125C4×Dic6—21no—
126C3×#107—21no—
127C3×#107D411yescharacteristic
128C3×#107D411yescharacteristic
129C3×#106—21no—
130C3×#106D411yescharacteristic
131C3×#106D411yescharacteristic
132C3×#103—21no—
133C3×#103—21no—
134C3×#103—21no—
135C3×#103—21no—
136C3×C22⋊C8—21no—
137C3×C22⋊C8—21no—
138#837—22no—
139#836—22no—
140#835—21no—
141#835—21no—
142#835—21no—
143#835—21no—
144#835—21no—
145#835D411yescharacteristic
146#835D411yescharacteristic
147#832—21no—
148#832D411yescharacteristic
149#832D411yescharacteristic
150#824—22no—
151#819—22no—
152#816—21no—
153#816—21no—
154Dic3⋊C8—21no—
155Dic3⋊C8—21no—
156C8×Dic3—22no—
157#810—21no—
158#810—21no—
159#810—21no—
160#810—21no—
161#808—21no—
162#808—21no—
163#807—21no—
164#807—21no—
165#803—21no—
166#803—21no—
167C4×C3⋊C8—21no—
168C4×C3⋊C8—21no—
169C2×#137—31no—
170#544—31no—
171#536C2×D311yescharacteristic
172C2×C4⋊Q8—31no—
173#520C2×D311yescharacteristic
174#519—31no—
175#491—61no—
176#489—61no—
177#488—61no—
178#487—61no—
179#485—21no—
180#485—21no—
181#484—21no—
182#483—21no—
183#479—61no—
184#479—61no—
185#477—61no—
186#475—61no—
187#441—61no—
188#440—61no—
189C4×Q16—61no—
190C4×SD16—61no—
191C2×#108—31no—
192#428—31no—
193#428—31no—
194#428C2×D311yescharacteristic
195#427C2×D311yescharacteristic
196#424—31no—
197C2×#104—31no—
198#419—31no—
199#419—31no—
200#419C2×D311yescharacteristic
201#417—31no—
202#416—31no—
203C2×#103—31no—
204C2×#103—31no—
205C2×#103—31no—
206C2×#103C2×D311yescharacteristic
207C2×#101—31no—
208C2×#101—31no—
209#408C2×D311yescharacteristic
210#405—31no—
211#405—31no—
212C2×C4×C8—31no—
213#384—31no—
214#343—31no—
215#343—31no—
216#341—31no—
217#339—31no—
218#339—31no—
219#339—31no—
220C3×#41—21no—
221C3×#41—21no—
222C3×#41—21no—
223C3×#41—21no—
224C2×C3×Q8—21no—
225C2×C3×Q8—21no—
226C2×C3×Q8—21no—
227C2×C3×Q8—21no—
228C2×C3×Q8—21no—
229C2×C3×Q8—21no—
230C2×C3×Q8C2×D411yescharacteristic
231C2×C3×D4—21no—
232C2×C3×D4C2×D411yescharacteristic
233C22×C3×C4—21no—
234C22×C3×C4—21no—
235C22×C3×C4—21no—
236C22×C3×C4—41no—
237C22×C3×C4C2411yescharacteristic
238C22×Dic3—21no—
239C22×Dic3—21no—
240C22×Dic3—21no—
241C2×Dic6—21no—
242C2×Dic6—21no—
243C2×Dic6—21no—
244C2×Dic6—21no—
245C2×Dic6—21no—
246C2×Dic6—41no—
247C2×Dic6#4112yes—
248C3×M16—21no—
249C3×M16—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8C2×D411yescharacteristic
254C2×C3×C8C2×D411yescharacteristic
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—41no—
258C3×C4⋊C4—41no—
259C3×C4⋊C4—41no—
260C3×C4⋊C4C2×D411yescharacteristic
261C3×C4⋊C4C2×D411yescharacteristic
262C3×C4⋊C4C2×D411yescharacteristic
263C3×C4⋊C4C2×D411yescharacteristic
264C3×C22⋊C4—21no—
265C3×C22⋊C4—21no—
266C3×C22⋊C4—41no—
267C3×C22⋊C4—41no—
268C3×C22⋊C4—41no—
269C3×C42—21no—
270C3×C42—21no—
271C3×C42—41no—
272#217—21no—
273#216—24no—
274#214—24no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277C4⋊Dic3—41no—
278C4⋊Dic3C2×D411yescharacteristic
279C4⋊Dic3C2×D411yescharacteristic
280Dic3⋊C4—44no—
281Dic3⋊C4—41no—
282Dic3⋊C4—41no—
283Dic3⋊C4—41no—
284Dic3⋊C4—41no—
285Dic3⋊C4—82no—
286C4×Dic3—21no—
287C4×Dic3—21no—
288C4×Dic3—41no—
289C4×Dic3—41no—
290C4×Dic3#4112yes—
291#208—21no—
292#208—21no—
293#208—21no—
294#208—21no—
295#208—21no—
296#208#4112yes—
297C2×C3⋊C8—21no—
298C2×C3⋊C8—21no—
299C2×C3⋊C8—21no—
300C2×C3⋊C8—21no—
301C2×C3⋊C8—21no—
302C2×C3⋊C8C2×D411yescharacteristic
303C2×C3⋊C8C2×D411yescharacteristic
304C2×#41C22×D311yescharacteristic
305C22×Q8—31no—
306C22×Q8C22×D311yescharacteristic
307#137—62no—
308#137—62no—
309C2×Q16—61no—
310C2×SD16—61no—
311C2×M16—31no—
312C2×M16—31no—
313C2×M16—31no—
314C2×M16C22×D311yescharacteristic
315C22×C8—31no—
316C22×C8C22×D311yescharacteristic
317C4⋊Q8—21no—
318C4⋊Q8—61no—
319C4⋊Q8—62no—
320C4⋊Q8—61no—
321C4⋊Q8—62no—
322#125—61no—
323#125—61no—
324#124—21no—
325#123—41no—
326C22⋊Q8—21no—
327C22⋊Q8—21no—
328C22⋊Q8—121no—
329C4×Q8—21no—
330C4×Q8—61no—
331C4×Q8—61no—
332C4×Q8—61no—
333C4×Q8—61no—
334C4×D4—21no—
335C4×D4—41no—
336#117—21no—
337#117—31no—
338#117—121no—
339#117C22×D311yescharacteristic
340#117C22×D311yescharacteristic
341C2×C4⋊C4—21no—
342C2×C4⋊C4—31no—
343C2×C4⋊C4—121no—
344C2×C4⋊C4—121no—
345C2×C4⋊C4C22×D311yescharacteristic
346C2×C42—31no—
347#108—62no—
348#107—21no—
349#107—61no—
350#107—61no—
351#107#6711yescharacteristic
352#107#6711yescharacteristic
353#106—21no—
354#106—61no—
355#106—61no—
356#106#6711yescharacteristic
357#106#6711yescharacteristic
358C4⋊C8—61no—
359C4⋊C8—61no—
360#104—62no—
361#103—21no—
362#103—21no—
363#103—21no—
364#103—21no—
365#103—31no—
366#103—31no—
367#103—61no—
368#103—61no—
369#103—61no—
370#103—61no—
371#103—61no—
372#103—61no—
373#103—61no—
374#103—61no—
375#103—61no—
376#103—61no—
377#103—61no—
378#102—31no—
379#102—31no—
380#102—61no—
381#101—62no—
382#101—62no—
383C22⋊C8—21no—
384C22⋊C8—21no—
385#97—61no—
386#97—61no—
387C4×C8—61no—
388C4×C8—61no—
389C4×C8—62no—
390C23×C3—21no—
391C23×C3H(5,𝔽2)11yescharacteristic
392C3×Q8—21no—
393C3×Q8—21no—
394C3×Q8—41no—
395C3×Q8—41no—
396C3×Q8—41no—
397C3×Q8—41no—
398C3×D4—21no—
399C3×D4—21no—
400C3×D4—41no—
401C3×D4—41no—
402C2×C3×C4—21no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—21no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4—41no—
413C2×C3×C4—81no—
414C2×C3×C4—161no—
415C2×C3×C4C22×D411yescharacteristic
416C2×C3×C4C22×D411yescharacteristic
417C2×C3×C4C2×#4111yescharacteristic
418C2×C3×C4C2×#4111yescharacteristic
419C2×C3×C4C2×#4111yescharacteristic
420C2×C3×C42−1+411yescharacteristic
421C2×Dic3—22no—
422C2×Dic3—22no—
423C2×Dic3—22no—
424C2×Dic3—21no—
425C2×Dic3—21no—
426C2×Dic3—22no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—81no—
431Dic6—22no—
432Dic6—22no—
433Dic6—41no—
434Dic6—41no—
435Dic6—161no—
436C3×C8—21no—
437C3×C8—21no—
438C3×C8—41no—
439C3⋊C8—22no—
440C3⋊C8—21no—
441C3⋊C8—21no—
442C3⋊C8—41no—
443C3⋊C8—41no—
444#41—21no—
445#41—21no—
446#41—21no—
447#41—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—32no—
455C2×Q8—61no—
456C2×Q8—61no—
457C2×Q8—61no—
458C2×Q8—61no—
459C2×Q8—61no—
460C2×Q8—121no—
461C2×Q8D3×D411yescharacteristic
462C2×D4—21no—
463C2×D4D3×D411yescharacteristic
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—41no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4C23×D311yescharacteristic
472Q16—64no—
473SD16—64no—
474M16—21no—
475M16—21no—
476M16—32no—
477M16—61no—
478M16—61no—
479M16—61no—
480M16—61no—
481M16—61no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—31no—
486C2×C8—31no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8C2×#6711yescharacteristic
493C2×C8C2×#6711yescharacteristic
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—31no—
497C4⋊C4—31no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C4⋊C4—121no—
504C4⋊C4—121no—
505C4⋊C4—121no—
506C4⋊C4—121no—
507C4⋊C4—121no—
508C4⋊C4—124no—
509C4⋊C4—242no—
510C4⋊C4C2×#6711yescharacteristic
511C4⋊C4C2×#6711yescharacteristic
512C4⋊C4C2×#6711yescharacteristic
513C4⋊C4C2×#6711yescharacteristic
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—41no—
517C22⋊C4—41no—
518C22⋊C4—41no—
519C22⋊C4—61no—
520C42—21no—
521C42—21no—
522C42—32no—
523C42—41no—
524C42—61no—
525C42—61no—
526C42—121no—
527C42—121no—
528C22×C3—21no—
529C22×C3—21no—
530C22×C3—41no—
531C22×C3—41no—
532C22×C3—41no—
533C22×C3#54211yescharacteristic
534C22×C3D4211yescharacteristic
535C22×C3#55111yescharacteristic
536C3×C4—41no—
537C3×C4—41no—
538C3×C4—41no—
539C3×C4—41no—
540C3×C4—41no—
541C3×C4—81no—
542C3×C4—81no—
543C3×C4#52511yescharacteristic
544C3×C4#53211yescharacteristic
545C3×C4#54711yescharacteristic
546C3×C4#54811yescharacteristic
547Dic3—24no—
548Dic3—42no—
549Dic3—41no—
550Dic3—41no—
551Dic3—81no—
552C23—21no—
553C23#100411yescharacteristic
554Q8—21no—
555Q8—21no—
556Q8—41no—
557Q8—41no—
558Q8—41no—
559Q8—41no—
560Q8—62no—
561Q8—62no—
562Q8—121no—
563Q8—121no—
564Q8—481no—
565D4—21no—
566D4—21no—
567D4—41no—
568D4—41no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—62no—
581C2×C4—62no—
582C2×C4—62no—
583C2×C4—62no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—81no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—161no—
591C2×C4—241no—
592C2×C4C2×D3×D411yescharacteristic
593C2×C4D3×#4111yescharacteristic
594C2×C4D3×#4111yescharacteristic
595C2×C4D3×#4111yescharacteristic
596C2×C4#101011yescharacteristic
597C2×C4C22×#6711yescharacteristic
598C8—21no—
599C8—21no—
600C8—41no—
601C8—61no—
602C8—61no—
603C8—62no—
604C8—121no—
605C8—121no—
606C2×C3—21no—
607C2×C3—21no—
608C2×C3—81no—
609C2×C3#246111yescharacteristic
610C2×C3#329111yescharacteristic
611C2×C3#329311yescharacteristic
612C22—21no—
613C22—21no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22#565911yescharacteristic
618C22#566211yescharacteristic
619C22D4×#6711yescharacteristic
620C4—41no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—64no—
626C4—81no—
627C4—81no—
628C4—121no—
629C4—121no—
630C4—122no—
631C4—241no—
632C4#565511yescharacteristic
633C4#565611yescharacteristic
634C4#583111yescharacteristic
635C4#583211yescharacteristic
636C3#2684911yescharacteristic, complemented, Sylow
637C2—21no—
638C2—21no—
639C2—81no—
640C2#8285211yescharacteristic
641C2#8401411yescharacteristic
642C2#8401811yescharacteristic
subgroupG/H
centerC22D4×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3097C2
socleC22×C3D42
prime radicalC2×C3×#41C23
hypercenter#3097D3
hypocenterC3#26849
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #3097

Complements

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

NH ≅ G/Ntype
4. #79198C2semidirect
7. #70672C2semidirect
8. #70666C2semidirect
9. #70659C2semidirect
10. #70345C2semidirect
11. #70338C2semidirect
12. #70265C2semidirect
13. #70225C2semidirect
636. C3#26849semidirect

Maximal subgroups

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

subgroupindex
C3×#30972
#834012
#825712
#791982
#718682
#716642
#706722
#706662
#706592
#703452
#703382
#702652
#702252
#700042
#699972
#268493