Finite Group Databasebeta

← #10883564

Subgroups — #10883564

Order 768.

788 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2955C211yesmaximal, characteristic, complemented
2C2×#5141C211yesmaximal, characteristic, complemented
3#83798C211yesmaximal, characteristic, complemented
4#78572C211yesmaximal, characteristic, complemented
5#72120C211yesmaximal, characteristic, complemented
6#72119C211yesmaximal, characteristic
7#72013C211yesmaximal, characteristic, complemented
8#72012C211yesmaximal, characteristic, complemented
9#70733C211yesmaximal, characteristic, complemented
10#70731C211yesmaximal, characteristic, complemented
11#70411C211yesmaximal, characteristic, complemented
12#70410C211yesmaximal, characteristic, complemented
13#70408C211yesmaximal, characteristic, complemented
14#70407C211yesmaximal, characteristic, complemented
15#69896C211yesmaximal, characteristic, complemented
16#28182—31nomaximal, Sylow
17C3×#575C2211yescharacteristic
18C2×C3×#131C2211yescharacteristic
19C2×C3×#131C2211yescharacteristic
20#5830C2211yescharacteristic
21#5825C2211yescharacteristic
22C2×#951C2211yescharacteristic
23C2×#949C2211yescharacteristic
24C22×#204C2211yescharacteristic
25#5521C2211yescharacteristic
26#5516C2211yescharacteristic
27C3×#444—22no—
28C3×#443—22no—
29C3×#430C2211yescharacteristic
30C3×#421C2211yescharacteristic
31C3×#421C2211yescharacteristic
32C3×#405C2211yescharacteristic
33#5240C2211yescharacteristic
34#5240C2211yescharacteristic
35#5236C2211yescharacteristic
36#5236C2211yescharacteristic
37#5141—22no—
38#5141—22no—
39C2×#824C2211yescharacteristic
40#5134—22no—
41#5131C2211yescharacteristic
42#5131C2211yescharacteristic
43C2×#823C2211yescharacteristic
44#5120C2211yescharacteristic
45#5120C2211yescharacteristic
46#5114—22no—
47C2×#819C2211yescharacteristic
48#5106C2211yescharacteristic
49#5106C2211yescharacteristic
50#4710—22no—
51C4×#204—22no—
52#4559C2211yescharacteristic
53#4559C2211yescharacteristic
54#4557C2211yescharacteristic
55#4557C2211yescharacteristic
56#4556C2211yescharacteristic
57#4556C2211yescharacteristic
58#4531C2211yescharacteristic
59#4531C2211yescharacteristic
60C2×#471—31no—
61#3057—31no—
62#2955D311yescharacteristic, complemented
63#2949—31no—
64#2082—31no—
65#2081—31no—
66#2009—31no—
67#2008—31no—
68#1912—31no—
69#1911—31no—
70#1905—31no—
71#1904—31no—
72#1833—31no—
73#1814—31no—
74#1812—31no—
75C2×C3×#41C2311yescharacteristic
76C2×C3×#41C2311yescharacteristic
77C22×D12C2311yescharacteristic
78C22×Dic6C2311yescharacteristic
79C3×#137—21no—
80C3×#137—21no—
81C3×AGL(1,ℤ/8)—21no—
82C3×AGL(1,ℤ/8)—21no—
83C3×#135—22no—
84C3×#131—21no—
85C3×#131—21no—
86C3×#131—21no—
87C3×#131—21no—
88C3×#131—21no—
89C3×#131—21no—
90C3×#131—21no—
91C3×#131—21no—
92C2×C3×M16—21no—
93C2×C3×M16—21no—
94C2×C3×M16C2311yescharacteristic
95C2×C3×M16C2311yescharacteristic
96C2×C3×M16C2311yescharacteristic
97C22×C3×C8—21no—
98C22×C3×C8—21no—
99C22×C3×C8C2311yescharacteristic
100C3×#117C2311yescharacteristic
101#951—22no—
102#951—41no—
103#949—22no—
104#949—41no—
105#947—21no—
106#944—21no—
107C2×#216—21no—
108C2×#215—21no—
109#940—21no—
110#937—41no—
111#936—21no—
112#935—41no—
113C2×#214—21no—
114C2×#213—21no—
115#930—22no—
116#926C2311yescharacteristic
117#926C2311yescharacteristic
118#926C2311yescharacteristic
119#926C2311yescharacteristic
120#924—22no—
121C2×#208C2311yescharacteristic
122C2×#208C2311yescharacteristic
123C2×#204—21no—
124C2×#204—21no—
125C2×#204—21no—
126C2×#204—21no—
127C2×#204D412yes—
128C2×#204D412yes—
129#893—41no—
130#887—41no—
131D3×C22⋊C4—41no—
132#877—41no—
133C4×D12—22no—
134C4×Dic6—22no—
135C3×#108—22no—
136C3×#104—22no—
137C3×#104—22no—
138C3×#97D412yescomplemented
139C3×C4×C8D412yescomplemented
140#837—41no—
141#837—41no—
142#834—41no—
143#834—41no—
144#824—22no—
145#823—22no—
146#821—22no—
147#821—22no—
148#819—21no—
149#819—21no—
150#818—21no—
151#818—21no—
152#817—21no—
153#817—21no—
154#816—22no—
155#816—22no—
156#575C2×D311yescharacteristic
157C2×#137—31no—
158C2×AGL(1,ℤ/8)—31no—
159C22×SD16—31no—
160C2×#131C2×D311yescharacteristic
161C2×#131C2×D311yescharacteristic
162#536—31no—
163#535—31no—
164#519—31no—
165#518—31no—
166#471—62no—
167#471—62no—
168#468—62no—
169#465—62no—
170#444—22no—
171#443—22no—
172#440—62no—
173C4×SD16—62no—
174#430C2×D311yescharacteristic
175C2×#108—31no—
176#427—31no—
177#427—31no—
178#421—31no—
179#421—31no—
180#421C2×D311yescharacteristic
181#421C2×D311yescharacteristic
182#419—31no—
183#419—31no—
184#418—31no—
185#418—31no—
186C2×#101—31no—
187C2×#100—31no—
188#410—31no—
189#410—31no—
190#405C2×D311yescharacteristic
191#344—31no—
192#344—31no—
193#344—31no—
194#344—31no—
195H(3,ℤ/4)—31no—
196H(3,ℤ/4)—31no—
197H(3,ℤ/4)—31no—
198H(3,ℤ/4)—31no—
199C23×D3—21no—
200C3×#41—21no—
201C3×#41—21no—
202C3×#41—21no—
203C3×#41—21no—
204C3×#41—21no—
205C3×#41—21no—
206C3×#41—21no—
207C3×#41—21no—
208C2×C3×Q8#4111yescharacteristic
209C2×C3×Q8#4111yescharacteristic
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4#4111yescharacteristic
213C2×C3×D4#4111yescharacteristic
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4C2411yescharacteristic
217C2×#67—42no—
218C22×Dic3—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12C2×D412yes—
225C2×C4×D3—42no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6C2×D412yes—
232C3×Q16—22no—
233C3×SD16—22no—
234C3×SD16—22no—
235C3×D8—22no—
236C3×M16—21no—
237C3×M16—21no—
238C3×M16—21no—
239C3×M16—21no—
240C3×M16—21no—
241C3×M16—21no—
242C3×M16—41no—
243C3×M16—41no—
244C3×M16—41no—
245C3×M16—41no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8—41no—
249C2×C3×C8—41no—
250C2×C3×C8—41no—
251C2×C3×C8—41no—
252C2×C3×C8C2×D411yescharacteristic
253C2×C3×C8C2×D411yescharacteristic
254C2×C3×C8C2×D412yes—
255C2×C3×C8C2×D411yescharacteristic
256C2×C3×C8C2×D411yescharacteristic
257C2×C3×C8#4111yescharacteristic
258C2×C3×C8#4111yescharacteristic
259C3×C4⋊C4—21no—
260C3×C22⋊C4—21no—
261C3×C42C2×D412yes—
262#217—21no—
263#217—21no—
264#217—21no—
265#217—21no—
266#217—42no—
267#216—42no—
268#215—42no—
269#214—42no—
270#213—42no—
271(C2×D3)⋊C4—42no—
272(C2×D3)⋊C4—42no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277Dic3⋊C4—42no—
278Dic3⋊C4—42no—
279C4×Dic3—21no—
280C4×Dic3—21no—
281C4×Dic3—21no—
282C4×Dic3—21no—
283C4×Dic3—42no—
284#208—21no—
285#208—21no—
286#208—21no—
287#208—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290#204—22no—
291#204—22no—
292#204—22no—
293#204—22no—
294C2×#41C22×D311yescharacteristic
295C2×#41C22×D311yescharacteristic
296C22×Q8—31no—
297C22×D4—31no—
298#137—21no—
299#137—21no—
300#137—62no—
301#137—121no—
302AGL(1,ℤ/8)—21no—
303AGL(1,ℤ/8)—21no—
304AGL(1,ℤ/8)—62no—
305AGL(1,ℤ/8)—121no—
306#135—22no—
307C2×Q16—61no—
308C2×SD16—32no—
309C2×SD16—32no—
310C2×SD16—61no—
311C2×SD16—61no—
312C2×SD16—61no—
313C2×SD16—61no—
314C2×SD16—61no—
315C2×SD16—61no—
316C2×D8—61no—
317#131—21no—
318#131—21no—
319#131—21no—
320#131—21no—
321#131—21no—
322#131—21no—
323#131—21no—
324#131—21no—
325C2×M16—21no—
326C2×M16—21no—
327C2×M16—31no—
328C2×M16—31no—
329C2×M16C22×D311yescharacteristic
330C2×M16C22×D311yescharacteristic
331C2×M16C22×D311yescharacteristic
332C22×C8—21no—
333C22×C8—21no—
334C22×C8C22×D311yescharacteristic
335C4⋊Q8—61no—
336#127—61no—
337#124—61no—
338#124—61no—
339#123—121no—
340C22⋊Q8—62no—
341#121—62no—
342#120—121no—
343C4×Q8—62no—
344C4×Q8—121no—
345C4×D4—62no—
346C4×D4—121no—
347#117—31no—
348#117—31no—
349#117—31no—
350#117—31no—
351#117—121no—
352#117C22×D311yescharacteristic
353C2×C22⋊C4—121no—
354#108—22no—
355#108—61no—
356#108—61no—
357#107—61no—
358#107—61no—
359#106—61no—
360#106—61no—
361#104—22no—
362#104—22no—
363#104—121no—
364#104—121no—
365#103—62no—
366#103—62no—
367#102—62no—
368#102—62no—
369#101—62no—
370#100—62no—
371#99—121no—
372#99—121no—
373#97#6712yescomplemented
374C4×C8#6712yescomplemented
375C23×C3—21no—
376C23×C3—21no—
377C23×C3H(5,𝔽2)11yescharacteristic
378C22×D3—22no—
379C22×D3—41no—
380C22×D3—41no—
381C22×D3—161no—
382C3×Q8—21no—
383C3×Q8—21no—
384C3×Q8—21no—
385C3×Q8—21no—
386C3×D4—21no—
387C3×D4—21no—
388C3×D4—21no—
389C3×D4—21no—
390C3×D4—41no—
391C3×D4—41no—
392C3×D4—41no—
393C3×D4—41no—
394C2×C3×C4—22no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4C22×D411yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4C2×#4111yescharacteristic
406C2×C3×C4H(5,𝔽2)11yescharacteristic
407#67—162no—
408C2×Dic3—22no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415D12—22no—
416D12—22no—
417D12—41no—
418D12—41no—
419C4×D3—162no—
420Dic6—22no—
421Dic6—22no—
422Dic6—41no—
423Dic6—41no—
424C3×C8—22no—
425C3×C8—41no—
426C3×C8—41no—
427C3×C8#12111yescharacteristic
428C3×C8#12111yescharacteristic
429C3×C8#12111yescharacteristic
430C3×C8#12111yescharacteristic
431C3⋊C8—41no—
432C3⋊C8—41no—
433C24—61no—
434#41—21no—
435#41—21no—
436#41—21no—
437#41—21no—
438#41—21no—
439#41—21no—
440#41—21no—
441#41—21no—
442C2×Q8—32no—
443C2×Q8—61no—
444C2×Q8—61no—
445C2×Q8—61no—
446C2×Q8—61no—
447C2×Q8—61no—
448C2×Q8#23511yescharacteristic
449C2×Q8#23511yescharacteristic
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—32no—
453C2×D4—61no—
454C2×D4—61no—
455C2×D4—61no—
456C2×D4—61no—
457C2×D4—61no—
458C2×D4—122no—
459C2×D4#23511yescharacteristic
460C2×D4#23511yescharacteristic
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—61no—
464C22×C4—122no—
465C22×C4C23×D311yescharacteristic
466Q16—22no—
467Q16—122no—
468SD16—22no—
469SD16—22no—
470SD16—62no—
471SD16—62no—
472SD16—62no—
473SD16—62no—
474SD16—122no—
475SD16—122no—
476D8—22no—
477D8—122no—
478M16—21no—
479M16—21no—
480M16—21no—
481M16—21no—
482M16—21no—
483M16—21no—
484M16—41no—
485M16—41no—
486M16—41no—
487M16—41no—
488M16—61no—
489M16—61no—
490M16—61no—
491M16—61no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—41no—
497C2×C8—41no—
498C2×C8—61no—
499C2×C8—61no—
500C2×C8D3×D411yescharacteristic
501C2×C8D3×D411yescharacteristic
502C2×C8#23711yescharacteristic
503C2×C8#23711yescharacteristic
504C2×C8C2×#6712yes—
505C2×C8C2×#6711yescharacteristic
506C2×C8C2×#6711yescharacteristic
507C4⋊C4—21no—
508C4⋊C4—61no—
509C4⋊C4—61no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—122no—
513C4⋊C4—122no—
514C22⋊C4—21no—
515C22⋊C4—61no—
516C22⋊C4—61no—
517C22⋊C4—61no—
518C22⋊C4—61no—
519C22⋊C4—122no—
520C22⋊C4—122no—
521C22⋊C4—122no—
522C42—61no—
523C42—61no—
524C42—61no—
525C42—61no—
526C42—122no—
527C42C2×#6712yes—
528C22×C3—41no—
529C22×C3—42no—
530C22×C3—41no—
531C22×C3—41no—
532C22×C3—81no—
533C22×C3#53511yescharacteristic
534C22×C3#54611yescharacteristic
535C22×C3#55311yescharacteristic
536C2×D3—42no—
537C2×D3—82no—
538C2×D3—82no—
539C2×D3—161no—
540C3×C4—42no—
541C3×C4—41no—
542C3×C4—41no—
543C3×C4C2×#12111yescharacteristic
544C3×C4#52511yescharacteristic
545C3×C4#54611yescharacteristic
546C3×C4#55311yescharacteristic
547Dic3—42no—
548Dic3—81no—
549Dic3—81no—
550Dic3—81no—
551Dic3—81no—
552C23—21no—
553C23—21no—
554C23—62no—
555C23—121no—
556C23—121no—
557C23—481no—
558C23#100911yescharacteristic
559Q8—21no—
560Q8—21no—
561Q8—21no—
562Q8—21no—
563Q8—62no—
564Q8—62no—
565Q8—121no—
566Q8—121no—
567D4—21no—
568D4—21no—
569D4—21no—
570D4—21no—
571D4—41no—
572D4—41no—
573D4—41no—
574D4—41no—
575D4—62no—
576D4—62no—
577D4—121no—
578D4—121no—
579D4—482no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—62no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—482no—
595C2×C4C2×#23511yescharacteristic
596C2×C4C2×#23511yescharacteristic
597C2×C4C2×D3×D411yescharacteristic
598C2×C4C2×#23711yescharacteristic
599C2×C4#100911yescharacteristic
600C2×C4C22×#6711yescharacteristic
601C8—22no—
602C8—41no—
603C8—41no—
604C8—121no—
605C8—121no—
606C8#93811yescharacteristic
607C8#93811yescharacteristic
608C8#93811yescharacteristic
609C8#93811yescharacteristic
610C2×C3—21no—
611C2×C3—21no—
612C2×C3—21no—
613C2×C3—81no—
614C2×C3—81no—
615C2×C3#253511yescharacteristic
616D3—82no—
617C22—41no—
618C22—42no—
619C22—41no—
620C22—41no—
621C22—81no—
622C22—122no—
623C22—242no—
624C22—242no—
625C22—481no—
626C22#555011yescharacteristic
627C22#556511yescharacteristic
628C22#583011yescharacteristic
629C4—42no—
630C4—41no—
631C4—41no—
632C4—122no—
633C4—241no—
634C4—241no—
635C4—241no—
636C4—241no—
637C4#555011yescharacteristic
638C4#556511yescharacteristic
639C4#579111yescharacteristic
640C4C2×#93811yescharacteristic
641C3#2818211yescharacteristic, complemented, Sylow
642C2—21no—
643C2—21no—
644C2—21no—
645C2—81no—
646C2—81no—
647C2—242no—
648C2#8241511yescharacteristic
subgroupG/H
centerC2#82415
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2955C2
socleC2×C3#2535
prime radical#72119C2
hypercenter#2955D3
hypocenterC3#28182
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #2955

Complements

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

NH ≅ G/Ntype
1. C3×#2955C2semidirect
2. C2×#5141C2semidirect
3. #83798C2semidirect
4. #78572C2semidirect
5. #72120C2semidirect
7. #72013C2semidirect
8. #72012C2semidirect
9. #70733C2semidirect
10. #70731C2semidirect
11. #70411C2semidirect
12. #70410C2semidirect
13. #70408C2semidirect
14. #70407C2semidirect
15. #69896C2semidirect
62. #2955D3semidirect
138. C3×#97D4semidirect
139. C3×C4×C8D4semidirect
373. #97#67semidirect
374. C4×C8#67semidirect
641. C3#28182semidirect

Maximal subgroups

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

subgroupindex
C3×#29552
C2×#51412
#837982
#785722
#721202
#721192
#720132
#720122
#707332
#707312
#704112
#704102
#704082
#704072
#698962
#281823