Finite Group Databasebeta

← #10778206

Subgroups — #10778206

Order 768.

614 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2971C211yesmaximal, characteristic, complemented
2#82529C211yesmaximal, characteristic
3#82507C211yesmaximal, characteristic, complemented
4#78533C211yesmaximal, characteristic, complemented
5#70947C211yesmaximal, characteristic, complemented
6#70945C211yesmaximal, characteristic, complemented
7#70940C211yesmaximal, characteristic, complemented
8#70938C211yesmaximal, characteristic, complemented
9#70876C211yesmaximal, characteristic, complemented
10#70874C211yesmaximal, characteristic, complemented
11#70869C211yesmaximal, characteristic, complemented
12#70867C211yesmaximal, characteristic, complemented
13#70821C211yesmaximal, characteristic, complemented
14#70816C211yesmaximal, characteristic, complemented
15#69850C211yesmaximal, characteristic, complemented
16#19183—31nomaximal, Sylow
17C3×C4×#41C2211yescharacteristic
18#5477C2211yescharacteristic
19#5470C2211yescharacteristic
20C3×C8×Q8C2211yescharacteristic
21C3×C8×Q8C2211yescharacteristic
22C3×#436—21no—
23C3×#436—21no—
24C3×#436C2211yescharacteristic
25C3×#436C2211yescharacteristic
26C3×#432—21no—
27C3×#432C2211yescharacteristic
28C3×C4×M16—21no—
29C3×C4×M16C2211yescharacteristic
30#5238—21no—
31#5233—21no—
32#5134—21no—
33#5134—21no—
34#5027C2211yescharacteristic
35C4×#215—21no—
36#5022—21no—
37#5021C2211yescharacteristic
38#5021C2211yescharacteristic
39#5020C2211yescharacteristic
40#5015C2211yescharacteristic
41C4×#213—21no—
42#5009—21no—
43#5008C2211yescharacteristic
44#5008C2211yescharacteristic
45#5007C2211yescharacteristic
46#5003—21no—
47#4998—21no—
48#4925C2211yescharacteristic
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50#4710—21no—
51#4710—21no—
52#4694C2211yescharacteristic
53#4693C2211yescharacteristic
54#4682C2211yescharacteristic
55#4681C2211yescharacteristic
56#4485C2211yescharacteristic
57#4485C2211yescharacteristic
58#4482C2211yescharacteristic
59#4482C2211yescharacteristic
60#4481C2211yescharacteristic
61#4481C2211yescharacteristic
62#4481C2211yescharacteristic
63#4481C2211yescharacteristic
64#4458C2211yescharacteristic
65#4458C2211yescharacteristic
66#4457C2211yescharacteristic
67#4457C2211yescharacteristic
68#3182—31no—
69#3106—31no—
70#3076—31no—
71#2971D311yescharacteristic, complemented
72#1703—31no—
73#1697—31no—
74#1695—31no—
75#1685—31no—
76#1684—31no—
77#1678—31no—
78#1676—31no—
79#1666—31no—
80#1565—31no—
81#1554—31no—
82#1551—31no—
83C2×C3×#41D411yescharacteristic
84C2×#235—21no—
85C2×C3×M16—21no—
86C2×C3×M16—21no—
87C2×C3×M16D411yescharacteristic
88C2×C3×M16D411yescharacteristic
89C3×C4×Q8D411yescharacteristic
90C3×C4×Q8C2311yescharacteristic
91C3×C4×D4—21no—
92C3×C4×D4—21no—
93C3×C4×D4D411yescharacteristic
94C3×C4×D4C2311yescharacteristic
95C3×#117—21no—
96C3×#117D411yescharacteristic
97C2×C3×C42—21no—
98C2×C3×C42C2311yescharacteristic
99C2×#215—21no—
100C2×#213—21no—
101#930—21no—
102#930—21no—
103#926—21no—
104#924—21no—
105#924—21no—
106C22×C3⋊C8—21no—
107C2×#204—21no—
108C2×#204—21no—
109#875—21no—
110#874C2311yescharacteristic
111C4×D12—21no—
112C4×D12—21no—
113#870—21no—
114#869C2311yescharacteristic
115#869C2311yescharacteristic
116#869C2311yescharacteristic
117C4×Dic6—21no—
118C4×Dic6—21no—
119C3×C4⋊C8—21no—
120C3×C4⋊C8—21no—
121C3×C4⋊C8C2311yescharacteristic
122C3×C4⋊C8C2311yescharacteristic
123C3×C22⋊C8—21no—
124C3×C22⋊C8—21no—
125C3×C22⋊C8—21no—
126C3×#97—21no—
127C3×#97D411yescharacteristic
128C3×#97D411yescharacteristic
129C3×C4×C8—21no—
130C3×C4×C8—21no—
131C3×C4×C8C2311yescharacteristic
132C3×C4×C8C2311yescharacteristic
133#835—21no—
134#835—21no—
135#835—21no—
136#832—21no—
137#832—21no—
138#832—21no—
139#830—21no—
140#821—21no—
141#821—21no—
142#818—21no—
143#818—21no—
144#816—21no—
145#816—21no—
146#809—21no—
147#809—21no—
148#808—21no—
149#808—21no—
150#808—21no—
151#807—21no—
152#807—21no—
153#807—21no—
154C4⋊(C3⋊C8)—21no—
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4⋊(C3⋊C8)C2311yescharacteristic
158C4⋊(C3⋊C8)C2311yescharacteristic
159C4×C3⋊C8—21no—
160#533—31no—
161#532—31no—
162C4×#41C2×D311yescharacteristic
163#500—31no—
164#492—31no—
165#487—31no—
166#486—31no—
167#482—61no—
168#479—61no—
169#478—61no—
170#477—31no—
171#477—31no—
172#476—31no—
173#476—31no—
174#475—31no—
175#474—31no—
176#468—61no—
177#468—61no—
178#460—31no—
179#459—31no—
180#449—61no—
181C8×Q8C2×D311yescharacteristic
182C8×Q8C2×D311yescharacteristic
183#440—61no—
184#440—61no—
185C4×SD16—61no—
186C4×D8—61no—
187#436—21no—
188#436—21no—
189#436C2×D311yescharacteristic
190#436C2×D311yescharacteristic
191#432—21no—
192#432C2×D311yescharacteristic
193#431—31no—
194#427—61no—
195C2×C4⋊C8—31no—
196#416—61no—
197C4×M16—21no—
198C4×M16C2×D311yescharacteristic
199#335—31no—
200#335—31no—
201#334—31no—
202#334—31no—
203#333—31no—
204#333—31no—
205#333—31no—
206#333—31no—
207#332—31no—
208#332—31no—
209#331—31no—
210#331—31no—
211C3×#41—22no—
212C3×#41—41no—
213C2×C3×Q8C2×D411yescharacteristic
214C2×C3×D4—21no—
215C2×C3×D4C2×D411yescharacteristic
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4#4111yescharacteristic
222C2×#67—41no—
223#235—82no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×C4×D3—41no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C3×M16—22no—
232C3×M16—41no—
233C3×M16—42no—
234C3×M16—42no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8C2×D411yescharacteristic
238C2×C3×C8C2×D411yescharacteristic
239C2×C3×C8C2×D411yescharacteristic
240C2×C3×C8C2×D411yescharacteristic
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4C2×D411yescharacteristic
244C3×C4⋊C4C2×D411yescharacteristic
245C3×C22⋊C4—21no—
246C3×C22⋊C4—21no—
247C3×C22⋊C4—21no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—21no—
251C3×C42C2×D411yescharacteristic
252C3×C42C2×D411yescharacteristic
253C3×C42#4111yescharacteristic
254C3×C42C2411yescharacteristic
255#217—41no—
256#215—42no—
257#213—42no—
258(C2×D3)⋊C4—41no—
259(C2×D3)⋊C4—41no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266Dic3⋊C4—41no—
267Dic3⋊C4—41no—
268C4×Dic3—41no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—42no—
274#204—42no—
275#204—42no—
276C2×#41—61no—
277C2×#41D1211yescharacteristic
278C2×SD16—61no—
279C2×SD16—61no—
280C2×SD16—61no—
281C2×D8—61no—
282C2×M16—21no—
283C2×M16—21no—
284C2×M16#6711yescharacteristic
285C2×M16#6711yescharacteristic
286C22×C8—61no—
287C4⋊Q8—31no—
288C4⋊Q8—31no—
289C4⋊Q8—31no—
290#127—31no—
291#125—61no—
292#124—61no—
293C22⋊Q8—61no—
294C22⋊Q8—61no—
295#121—61no—
296#121—61no—
297C4×Q8—61no—
298C4×Q8—61no—
299C4×Q8D1211yescharacteristic
300C4×Q8C22×D311yescharacteristic
301C4×D4—21no—
302C4×D4—21no—
303C4×D4—61no—
304C4×D4—61no—
305C4×D4D1211yescharacteristic
306C4×D4C22×D311yescharacteristic
307#117—21no—
308#117—61no—
309#117D1211yescharacteristic
310C2×C42—21no—
311C2×C42C22×D311yescharacteristic
312#107—61no—
313#107—61no—
314#107—61no—
315#107—61no—
316#107—61no—
317#106—61no—
318#106—61no—
319#106—61no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8—31no—
323C4⋊C8—31no—
324C4⋊C8—31no—
325C4⋊C8—31no—
326C4⋊C8—61no—
327C4⋊C8C22×D311yescharacteristic
328C4⋊C8C22×D311yescharacteristic
329#103—61no—
330#103—61no—
331#103—61no—
332#103—61no—
333#103—61no—
334#102—61no—
335#102—61no—
336#102—61no—
337#102—61no—
338#102—61no—
339#102—61no—
340#102—61no—
341C22⋊C8—21no—
342C22⋊C8—21no—
343C22⋊C8—21no—
344C22⋊C8—61no—
345#97—21no—
346#97#6711yescharacteristic
347#97#6711yescharacteristic
348C4×C8—21no—
349C4×C8—21no—
350C4×C8—61no—
351C4×C8C22×D311yescharacteristic
352C4×C8C22×D311yescharacteristic
353C23×C3—21no—
354C23×C3#12111yescharacteristic
355C22×D3—41no—
356C3×Q8—21no—
357C3×Q8#13712yes—
358C3×D4—21no—
359C3×D4—21no—
360C3×D4—42no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—41no—
369C2×C3×C4—42no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—81no—
373C2×C3×C4#12111yescharacteristic
374C2×C3×C4#12111yescharacteristic
375C2×C3×C4#12111yescharacteristic
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379#67—82no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383D12—41no—
384D12—41no—
385D12—41no—
386D12—41no—
387D12—81no—
388C4×D3—82no—
389Dic6—41no—
390Dic6—41no—
391Dic6—81no—
392Dic6—81no—
393C3×C8—21no—
394C3×C8—21no—
395C3×C8—21no—
396C3×C8—41no—
397C3×C8—41no—
398C3×C8#13512yes—
399C3⋊C8—41no—
400C3⋊C8—41no—
401C3⋊C8—41no—
402C3⋊C8—41no—
403#41—22no—
404#41—41no—
405#41—242no—
406C2×Q8—61no—
407C2×Q8—61no—
408C2×Q8—61no—
409C2×Q8C2×D1211yescharacteristic
410C2×D4—21no—
411C2×D4—61no—
412C2×D4—61no—
413C2×D4—61no—
414C2×D4—121no—
415C2×D4C2×D1211yescharacteristic
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—121no—
420C22×C4C2×D1211yescharacteristic
421C22×C4#23511yescharacteristic
422C22×C4C2×#6711yescharacteristic
423SD16—122no—
424SD16—122no—
425SD16—122no—
426D8—122no—
427M16—22no—
428M16—41no—
429M16—42no—
430M16—42no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—122no—
438C2×C8C2×#6711yescharacteristic
439C2×C8C2×#6711yescharacteristic
440C2×C8C2×#6711yescharacteristic
441C2×C8C2×#6711yescharacteristic
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—121no—
451C4⋊C4—121no—
452C4⋊C4C2×D1211yescharacteristic
453C4⋊C4C2×D1211yescharacteristic
454C22⋊C4—21no—
455C22⋊C4—21no—
456C22⋊C4—21no—
457C22⋊C4—121no—
458C22⋊C4—121no—
459C22⋊C4—121no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—121no—
464C42C2×D1211yescharacteristic
465C42#23511yescharacteristic
466C42C2×#6711yescharacteristic
467C42C23×D311yescharacteristic
468C22×C3—21no—
469C22×C3—21no—
470C22×C3—21no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3C2×#12111yescharacteristic
475C2×D3—81no—
476C2×D3—81no—
477C2×D3—81no—
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—22no—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—41no—
484C3×C4—41no—
485C3×C4C2×#13511yescharacteristic
486C3×C4C2×#13711yescharacteristic
487C3×C4#57511yescharacteristic
488C3×C4#57511yescharacteristic
489Dic3—81no—
490Dic3—81no—
491Dic3—81no—
492C23—21no—
493C23—121no—
494C23#93011yescharacteristic
495Q8—21no—
496Q8—121no—
497Q8—121no—
498Q8—241no—
499Q8—241no—
500Q8#90912yes—
501D4—21no—
502D4—21no—
503D4—42no—
504D4—121no—
505D4—121no—
506D4—121no—
507D4—121no—
508D4—241no—
509D4—242no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—81no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—242no—
526C2×C4#93011yescharacteristic
527C2×C4#93011yescharacteristic
528C2×C4#93011yescharacteristic
529C2×C4C22×D1211yescharacteristic
530C2×C4C2×#23511yescharacteristic
531C2×C4C22×#6711yescharacteristic
532C8—21no—
533C8—21no—
534C8—21no—
535C8—41no—
536C8—41no—
537C8—121no—
538C8—121no—
539C8—121no—
540C8—121no—
541C8#95012yes—
542C2×C3—41no—
543C2×C3—81no—
544C2×C3#303511yescharacteristic
545C2×C3#305611yescharacteristic
546C2×C3#306811yescharacteristic
547D3—161no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—241no—
555C22—241no—
556C22—241no—
557C22C2×#93011yescharacteristic
558C4—21no—
559C4—22no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—41no—
564C4—41no—
565C4—241no—
566C4—241no—
567C4—241no—
568C4C2×#90911yescharacteristic
569C4#574811yescharacteristic
570C4C2×#95011yescharacteristic
571C4#582211yescharacteristic
572C3#1918311yescharacteristic, complemented, Sylow
573C2—41no—
574C2—81no—
575C2—481no—
576C2#8252511yescharacteristic
577C2#8377611yescharacteristic
578C2#8381411yescharacteristic
subgroupG/H
centerC22C2×#930
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#2971C2
socleC22×C3C2×#121
prime radical#82529C2
hypercenter#2971D3
hypocenterC3#19183
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2971

Complements

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

NH ≅ G/Ntype
1. C3×#2971C2semidirect
3. #82507C2semidirect
4. #78533C2semidirect
5. #70947C2semidirect
6. #70945C2semidirect
7. #70940C2semidirect
8. #70938C2semidirect
9. #70876C2semidirect
10. #70874C2semidirect
11. #70869C2semidirect
12. #70867C2semidirect
13. #70821C2semidirect
14. #70816C2semidirect
15. #69850C2semidirect
71. #2971D3semidirect
572. C3#19183semidirect

Maximal subgroups

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

subgroupindex
C3×#29712
#825292
#825072
#785332
#709472
#709452
#709402
#709382
#708762
#708742
#708692
#708672
#708212
#708162
#698502
#191833