Finite Group Databasebeta

← #10883558

Subgroups — #10883558

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×#2949C211yesmaximal, characteristic, complemented
2C2×#5177C211yesmaximal, characteristic, complemented
3#83863C211yesmaximal, characteristic, complemented
4#78579C211yesmaximal, characteristic, complemented
5#71663C211yesmaximal, characteristic, complemented
6#71662C211yesmaximal, characteristic, complemented
7#71652C211yesmaximal, characteristic, complemented
8#71651C211yesmaximal, characteristic, complemented
9#70665C211yesmaximal, characteristic, complemented
10#70664C211yesmaximal, characteristic, complemented
11#70351C211yesmaximal, characteristic, complemented
12#70341C211yesmaximal, characteristic, complemented
13#70331C211yesmaximal, characteristic
14#70321C211yesmaximal, characteristic, complemented
15#69896C211yesmaximal, characteristic, complemented
16#28182—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C3×#519C2211yescharacteristic
19C3×#518C2211yescharacteristic
20#5816C2211yescharacteristic
21C2×#942C2211yescharacteristic
22#5805C2211yescharacteristic
23C2×#932C2211yescharacteristic
24#5748C2211yescharacteristic
25C2×#901C2211yescharacteristic
26C2×#901C2211yescharacteristic
27C3×#440—22no—
28C3×C4×SD16—22no—
29C3×#427C2211yescharacteristic
30C3×#419C2211yescharacteristic
31C3×#418C2211yescharacteristic
32C3×#405C2211yescharacteristic
33C2×#836C2211yescharacteristic
34#5225C2211yescharacteristic
35C2×#833C2211yescharacteristic
36#5215C2211yescharacteristic
37#5177—22no—
38#5177—22no—
39#5174—22no—
40#5171—22no—
41#5125C2211yescharacteristic
42C2×#819C2211yescharacteristic
43#5106C2211yescharacteristic
44#5006C2211yescharacteristic
45#5006C2211yescharacteristic
46#5005C2211yescharacteristic
47#5005C2211yescharacteristic
48#5005C2211yescharacteristic
49#5005C2211yescharacteristic
50#4717—22no—
51#4700—22no—
52#4556C2211yescharacteristic
53#4556C2211yescharacteristic
54#4533C2211yescharacteristic
55#4533C2211yescharacteristic
56#4531C2211yescharacteristic
57#4531C2211yescharacteristic
58#4531C2211yescharacteristic
59#4531C2211yescharacteristic
60C2×#471—31no—
61#3057—31no—
62#2955—31no—
63#2949D311yescharacteristic, complemented
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—
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C2×#235C2311yescharacteristic
78C2×#235C2311yescharacteristic
79C2×C3×SD16—21no—
80C2×C3×SD16—21no—
81C2×C3×SD16—21no—
82C2×C3×SD16—21no—
83C2×C3×SD16D412yes—
84C2×C3×SD16D412yes—
85C2×C3×M16C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×C4×Q8—22no—
88C3×C4×Q8—41no—
89C3×C4×D4—22no—
90C3×C4×D4—41no—
91C3×#117—41no—
92C3×#117C2311yescharacteristic
93C3×#117C2311yescharacteristic
94C3×#117C2311yescharacteristic
95C2×C3×C22⋊C4—41no—
96#953—41no—
97#947—21no—
98#946—22no—
99#944—21no—
100C2×#216—21no—
101#942—22no—
102#942—41no—
103C2×#215—21no—
104#940—21no—
105#938—22no—
106#936—21no—
107C2×#214—21no—
108#932—22no—
109#932—41no—
110C2×#213—21no—
111#929—41no—
112#926C2311yescharacteristic
113#926C2311yescharacteristic
114C2×#208C2311yescharacteristic
115C2×#208C2311yescharacteristic
116C2×#208C2311yescharacteristic
117C2×#208C2311yescharacteristic
118#909—21no—
119#909—21no—
120#908—21no—
121#908—21no—
122#904—22no—
123#901—21no—
124#901—21no—
125#901—21no—
126#901—21no—
127#901—21no—
128#901—21no—
129#901—21no—
130#901—21no—
131C2×ΓL(1,𝔽25)—21no—
132C2×ΓL(1,𝔽25)—21no—
133C2×D3×C8—21no—
134C2×D3×C8—21no—
135C3×#107—21no—
136C3×#106—21no—
137C3×#103—22no—
138C3×#102—22no—
139C3×#97D412yescomplemented
140C3×C4×C8D412yescomplemented
141#836—22no—
142#835—22no—
143#833—22no—
144#832—22no—
145#819—22no—
146#819—21no—
147#819—21no—
148#818—21no—
149#817—21no—
150#806—41no—
151#806—41no—
152#805—22no—
153#805—22no—
154#805—41no—
155#805—41no—
156#575—31no—
157C2×#137—31no—
158C2×AGL(1,ℤ/8)—31no—
159C22×SD16C2×D311yescharacteristic
160C2×#131—31no—
161C2×#131—31no—
162#536—31no—
163#535—31no—
164#519C2×D311yescharacteristic
165#518C2×D311yescharacteristic
166#471—62no—
167#471—62no—
168#468—62no—
169#465—62no—
170#444—62no—
171#443—62no—
172#440—22no—
173C4×SD16—22no—
174#430—31no—
175C2×#108—31no—
176#427—31no—
177#427C2×D311yescharacteristic
178#421—31no—
179#421—31no—
180#421—31no—
181#421—31no—
182#419—31no—
183#419C2×D311yescharacteristic
184#418—31no—
185#418C2×D311yescharacteristic
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—
199C24×C3—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8C2×D412yes—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4C2×D412yes—
212C22×C3×C4—21no—
213C22×C3×C4—42no—
214C22×C3×C4C2411yescharacteristic
215C2×#67—21no—
216C2×#67—21no—
217C2×#67—42no—
218#235—21no—
219#235—21no—
220#235—21no—
221#235—21no—
222#235—21no—
223#235—21no—
224#235—21no—
225#235—21no—
226C2×D12#4111yescharacteristic
227C2×D12#4111yescharacteristic
228C2×C4×D3—21no—
229C2×C4×D3—21no—
230C2×Dic6#4111yescharacteristic
231C2×Dic6#4111yescharacteristic
232C3×SD16—22no—
233C3×SD16—22no—
234C3×SD16—22no—
235C3×SD16—22no—
236C3×M16—21no—
237C3×M16—21no—
238C2×C3×C8C2×D411yescharacteristic
239C2×C3×C8C2×D411yescharacteristic
240C2×C3×C8C2×D412yes—
241C2×C3×C8C2×D411yescharacteristic
242C2×C3×C8C2×D411yescharacteristic
243C2×C3×C8#4111yescharacteristic
244C2×C3×C8#4111yescharacteristic
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—42no—
249C3×C22⋊C4—21no—
250C3×C22⋊C4—21no—
251C3×C22⋊C4—21no—
252C3×C22⋊C4—42no—
253C3×C42—21no—
254C3×C42—21no—
255C3×C42—42no—
256C3×C42C2×D412yes—
257#217—21no—
258#217—21no—
259#217—42no—
260#216—42no—
261#215—42no—
262#214—42no—
263#213—42no—
264(C2×D3)⋊C4—42no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267Dic3⋊C4—42no—
268C4×Dic3—21no—
269C4×Dic3—21no—
270#208—21no—
271#208—21no—
272#208—21no—
273#208—21no—
274#208—21no—
275#208—21no—
276#208—21no—
277#208—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282Dic12—22no—
283D24—22no—
284#204—22no—
285#204—22no—
286ΓL(1,𝔽25)—41no—
287ΓL(1,𝔽25)—41no—
288ΓL(1,𝔽25)—41no—
289ΓL(1,𝔽25)—41no—
290D3×C8—41no—
291D3×C8—41no—
292D3×C8—41no—
293D3×C8—41no—
294C2×#41—31no—
295C2×#41—31no—
296C22×Q8C22×D311yescharacteristic
297C22×D4C22×D311yescharacteristic
298#137—61no—
299#137—61no—
300#137—62no—
301#137—121no—
302AGL(1,ℤ/8)—61no—
303AGL(1,ℤ/8)—61no—
304AGL(1,ℤ/8)—62no—
305AGL(1,ℤ/8)—121no—
306#135—62no—
307C2×Q16—61no—
308C2×SD16—21no—
309C2×SD16—21no—
310C2×SD16—21no—
311C2×SD16—21no—
312C2×SD16—61no—
313C2×SD16—61no—
314C2×SD16D1212yes—
315C2×SD16D1212yes—
316C2×D8—61no—
317#131—61no—
318#131—61no—
319#131—61no—
320#131—61no—
321#131—61no—
322#131—61no—
323#131—61no—
324#131—61no—
325C2×M16—31no—
326C2×M16—31no—
327C2×M16—31no—
328C2×M16—31no—
329C2×M16—61no—
330C2×M16—61no—
331C2×M16C22×D311yescharacteristic
332C22×C8—61no—
333C22×C8—61no—
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—22no—
344C4×Q8—41no—
345C4×D4—22no—
346C4×D4—41no—
347#117—31no—
348#117—31no—
349#117—41no—
350#117C22×D311yescharacteristic
351#117C22×D311yescharacteristic
352#117C22×D311yescharacteristic
353C2×C22⋊C4—41no—
354#108—61no—
355#108—61no—
356#108—62no—
357#107—21no—
358#107—61no—
359#106—21no—
360#106—61no—
361#104—62no—
362#104—62no—
363#104—121no—
364#104—121no—
365#103—22no—
366#103—62no—
367#102—22no—
368#102—62no—
369#101—62no—
370#100—62no—
371#99—121no—
372#99—121no—
373#97#6712yescomplemented
374C4×C8#6712yescomplemented
375C23×C3—22no—
376C23×C3—41no—
377C23×C3—41no—
378C23×C3—161no—
379C23×C3H(5,𝔽2)11yescharacteristic
380C22×D3—21no—
381C22×D3—21no—
382C3×Q8—22no—
383C3×Q8—22no—
384C3×Q8—41no—
385C3×Q8—41no—
386C3×D4—22no—
387C3×D4—22no—
388C3×D4—41no—
389C3×D4—41no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—162no—
397C2×C3×C4C22×D411yescharacteristic
398C2×C3×C4C22×D411yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4H(5,𝔽2)11yescharacteristic
403#67—41no—
404#67—41no—
405#67—41no—
406#67—41no—
407#67—162no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412D12—21no—
413D12—21no—
414D12—21no—
415D12—21no—
416C4×D3—41no—
417C4×D3—41no—
418C4×D3—41no—
419C4×D3—41no—
420Dic6—21no—
421Dic6—21no—
422Dic6—21no—
423Dic6—21no—
424C3×C8—22no—
425C3×C8#12111yescharacteristic
426C3×C8#12111yescharacteristic
427C3×C8#12111yescharacteristic
428C3×C8#12111yescharacteristic
429C3⋊C8—41no—
430C3⋊C8—41no—
431C3⋊C8—41no—
432C3⋊C8—41no—
433C24—21no—
434#41—61no—
435#41—61no—
436#41—61no—
437#41—61no—
438#41—61no—
439#41—61no—
440#41—61no—
441#41—61no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—31no—
448C2×Q8—31no—
449C2×Q8C2×D1212yes—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—31no—
456C2×D4—31no—
457C2×D4—61no—
458C2×D4—61no—
459C2×D4—122no—
460C2×D4C2×D1212yes—
461C22×C4—21no—
462C22×C4—42no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4C23×D311yescharacteristic
466Q16—62no—
467Q16—122no—
468SD16—22no—
469SD16—22no—
470SD16—22no—
471SD16—22no—
472SD16—62no—
473SD16—62no—
474SD16—122no—
475SD16—122no—
476D8—62no—
477D8—122no—
478M16—21no—
479M16—21no—
480M16—61no—
481M16—61no—
482M16—61no—
483M16—61no—
484M16—61no—
485M16—61no—
486M16—61no—
487M16—61no—
488M16—121no—
489M16—121no—
490M16—121no—
491M16—121no—
492C2×C8—61no—
493C2×C8—61no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—121no—
497C2×C8—121no—
498C2×C8—121no—
499C2×C8—121no—
500C2×C8C2×D1211yescharacteristic
501C2×C8C2×D1211yescharacteristic
502C2×C8#23511yescharacteristic
503C2×C8#23511yescharacteristic
504C2×C8C2×#6712yes—
505C2×C8C2×#6711yescharacteristic
506C2×C8C2×#6711yescharacteristic
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—42no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—122no—
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—42no—
518C22⋊C4—61no—
519C22⋊C4—61no—
520C22⋊C4—122no—
521C22⋊C4—122no—
522C42—21no—
523C42—21no—
524C42—42no—
525C42—61no—
526C42—61no—
527C42C2×#6712yes—
528C22×C3—41no—
529C22×C3—42no—
530C22×C3—82no—
531C22×C3—82no—
532C22×C3—161no—
533C22×C3#53511yescharacteristic
534C22×C3#54611yescharacteristic
535C22×C3#55311yescharacteristic
536C2×D3—42no—
537C2×D3—41no—
538C2×D3—41no—
539C2×D3—81no—
540C3×C4—42no—
541C3×C4—42no—
542C3×C4—81no—
543C3×C4—81no—
544C3×C4C2×#12111yescharacteristic
545C3×C4#52511yescharacteristic
546C3×C4#54611yescharacteristic
547C3×C4#55311yescharacteristic
548Dic3—41no—
549Dic3—41no—
550Dic3—81no—
551Dic3—81no—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—61no—
556C23—61no—
557C23—161no—
558C23#100411yescharacteristic
559Q8—22no—
560Q8—22no—
561Q8—41no—
562Q8—41no—
563Q8—61no—
564Q8—61no—
565Q8—61no—
566Q8—61no—
567D4—22no—
568D4—22no—
569D4—41no—
570D4—41no—
571D4—61no—
572D4—61no—
573D4—61no—
574D4—61no—
575D4—121no—
576D4—121no—
577D4—121no—
578D4—121no—
579D4—482no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—162no—
595C2×C4C22×D1211yescharacteristic
596C2×C4C2×#23511yescharacteristic
597C2×C4#100411yescharacteristic
598C2×C4D3×#4111yescharacteristic
599C2×C4D3×#4111yescharacteristic
600C2×C4C22×#6711yescharacteristic
601C8—22no—
602C8—121no—
603C8—121no—
604C8—121no—
605C8—121no—
606C8#93011yescharacteristic
607C8#93011yescharacteristic
608C8#93011yescharacteristic
609C8#93011yescharacteristic
610C2×C3—21no—
611C2×C3—21no—
612C2×C3—21no—
613C2×C3—82no—
614C2×C3#253511yescharacteristic
615D3—81no—
616D3—81no—
617C22—41no—
618C22—42no—
619C22—82no—
620C22—82no—
621C22—122no—
622C22—121no—
623C22—121no—
624C22—161no—
625C22—241no—
626C22#555411yescharacteristic
627C22#555811yescharacteristic
628C22#580511yescharacteristic
629C4—42no—
630C4—42no—
631C4—81no—
632C4—81no—
633C4—121no—
634C4—121no—
635C4—241no—
636C4—241no—
637C4#555411yescharacteristic
638C4#555811yescharacteristic
639C4C2×#93011yescharacteristic
640C4#582811yescharacteristic
641C3#2818211yescharacteristic, complemented, Sylow
642C2—21no—
643C2—21no—
644C2—21no—
645C2—82no—
646C2—241no—
647C2—241no—
648C2#8241611yescharacteristic
subgroupG/H
centerC2#82416
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2949C2
socleC2×C3#2535
prime radical#70331C2
hypercenter#2949D3
hypocenterC3#28182
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #2949

Complements

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

NH ≅ G/Ntype
1. C3×#2949C2semidirect
2. C2×#5177C2semidirect
3. #83863C2semidirect
4. #78579C2semidirect
5. #71663C2semidirect
6. #71662C2semidirect
7. #71652C2semidirect
8. #71651C2semidirect
9. #70665C2semidirect
10. #70664C2semidirect
11. #70351C2semidirect
12. #70341C2semidirect
14. #70321C2semidirect
15. #69896C2semidirect
63. #2949D3semidirect
139. C3×#97D4semidirect
140. 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×#29492
C2×#51772
#838632
#785792
#716632
#716622
#716522
#716512
#706652
#706642
#703512
#703412
#703312
#703212
#698962
#281823