Finite Group Databasebeta

← #10778210

Subgroups — #10778210

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×#3106C211yesmaximal, characteristic
2#83294C211yesmaximal, characteristic, complemented
3#82519C211yesmaximal, characteristic, complemented
4#82507C211yesmaximal, characteristic, complemented
5#71807C211yesmaximal, characteristic, complemented
6#71151C211yesmaximal, characteristic, complemented
7#71149C211yesmaximal, characteristic, complemented
8#71139C211yesmaximal, characteristic, complemented
9#71137C211yesmaximal, characteristic, complemented
10#71022C211yesmaximal, characteristic, complemented
11#71020C211yesmaximal, characteristic, complemented
12#71010C211yesmaximal, characteristic, complemented
13#71008C211yesmaximal, characteristic, complemented
14#70781C211yesmaximal, characteristic, complemented
15#70746C211yesmaximal, characteristic, complemented
16#19183—31nomaximal, Sylow
17C3×#532C2211yescharacteristic
18C3×C4×#41C2211yescharacteristic
19#5470C2211yescharacteristic
20C3×#482—21no—
21C3×#477C2211yescharacteristic
22C3×#476C2211yescharacteristic
23C3×#460C2211yescharacteristic
24C3×#459C2211yescharacteristic
25C3×#449—21no—
26C3×C4×SD16—21no—
27C3×C4×D8—21no—
28C3×#431C2211yescharacteristic
29#5233—21no—
30#5093C2211yescharacteristic
31#5078C2211yescharacteristic
32#5076C2211yescharacteristic
33#5070C2211yescharacteristic
34#5042—21no—
35#5042—21no—
36Q8×C3⋊C8C2211yescharacteristic
37Q8×C3⋊C8C2211yescharacteristic
38#5022—21no—
39#5021C2211yescharacteristic
40#5020C2211yescharacteristic
41#5018—21no—
42#5018—21no—
43#5012—21no—
44#5012—21no—
45#5012C2211yescharacteristic
46#5012C2211yescharacteristic
47#5009—21no—
48#5008C2211yescharacteristic
49#5007C2211yescharacteristic
50#4998—21no—
51#4926—21no—
52#4926C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54C4×#208—21no—
55C4×#208C2211yescharacteristic
56#4546C2211yescharacteristic
57#4546C2211yescharacteristic
58#4544C2211yescharacteristic
59#4544C2211yescharacteristic
60#4486C2211yescharacteristic
61#4486C2211yescharacteristic
62#4481C2211yescharacteristic
63#4481C2211yescharacteristic
64#4480C2211yescharacteristic
65#4480C2211yescharacteristic
66#4479C2211yescharacteristic
67#4479C2211yescharacteristic
68#3182—31no—
69#3106D311yescharacteristic
70#3076—31no—
71#2971—31no—
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×#41—21no—
84C2×C3×#41D411yescharacteristic
85C2×C3×SD16—21no—
86C2×C3×D8—21no—
87C3×C4⋊Q8C2311yescharacteristic
88C3×#127C2311yescharacteristic
89C3×#124—21no—
90C3×#121—21no—
91C3×#121—21no—
92C3×C4×Q8D411yescharacteristic
93C3×C4×Q8C2311yescharacteristic
94C3×C4×D4—21no—
95C3×C4×D4—21no—
96C3×C4×D4—21no—
97C3×C4×D4—21no—
98C3×C4×D4D411yescharacteristic
99C3×C4×D4C2311yescharacteristic
100C3×#117—21no—
101C3×#117D411yescharacteristic
102C2×C3×C42—21no—
103C2×C3×C42C2311yescharacteristic
104C2×#214—21no—
105C2×#214—21no—
106#926—21no—
107#924—21no—
108#924—21no—
109C2×#208—21no—
110C2×#208—21no—
111C2×#208D411yescharacteristic
112C2×#208D411yescharacteristic
113C22×C3⋊C8—21no—
114#870—21no—
115#869C2311yescharacteristic
116#869C2311yescharacteristic
117C4×Dic6—21no—
118C4×Dic6—21no—
119C3×#107—21no—
120C3×#106—21no—
121C3×C4⋊C8C2311yescharacteristic
122C3×C4⋊C8C2311yescharacteristic
123C3×#103—21no—
124C3×#102—21no—
125C3×#102—21no—
126C3×#102—21no—
127C3×C22⋊C8—21no—
128C3×C4×C8—21no—
129#835—21no—
130#835—21no—
131#832—21no—
132#832—21no—
133#832—21no—
134#832—21no—
135#830—21no—
136#830—21no—
137#830—21no—
138#810—21no—
139#810—21no—
140#808—21no—
141#808—21no—
142#807—21no—
143#807—21no—
144#807—21no—
145#807—21no—
146C4⋊(C3⋊C8)—21no—
147C4⋊(C3⋊C8)—21no—
148C4⋊(C3⋊C8)—21no—
149C4⋊(C3⋊C8)C2311yescharacteristic
150C4⋊(C3⋊C8)C2311yescharacteristic
151C4⋊(C3⋊C8)C2311yescharacteristic
152C4⋊(C3⋊C8)C2311yescharacteristic
153#803—21no—
154#803D411yescharacteristic
155#803D411yescharacteristic
156C4×C3⋊C8—21no—
157C4×C3⋊C8—21no—
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#533—31no—
161#532C2×D311yescharacteristic
162C4×#41C2×D311yescharacteristic
163#500—31no—
164#492—31no—
165#487—31no—
166#486—31no—
167#482—21no—
168#479—61no—
169#478—61no—
170#477—31no—
171#477C2×D311yescharacteristic
172#476—31no—
173#476C2×D311yescharacteristic
174#475—31no—
175#474—31no—
176#468—61no—
177#468—61no—
178#460C2×D311yescharacteristic
179#459C2×D311yescharacteristic
180#449—21no—
181C8×Q8—31no—
182C8×Q8—31no—
183#440—61no—
184#440—61no—
185C4×SD16—21no—
186C4×D8—21no—
187#436—31no—
188#436—31no—
189#436—61no—
190#436—61no—
191#432—31no—
192#432—61no—
193#431C2×D311yescharacteristic
194#427—61no—
195C2×C4⋊C8—31no—
196#416—61no—
197C4×M16—31no—
198C4×M16—61no—
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—
213C3×#41—82no—
214C2×C3×Q8—21no—
215C2×C3×Q8C2×D411yescharacteristic
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4—21no—
219C2×C3×D4—21no—
220C2×C3×D4—41no—
221C2×C3×D4C2×D411yescharacteristic
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4—21no—
225C22×C3×C4—41no—
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4C2×D411yescharacteristic
228C22×C3×C4#4111yescharacteristic
229C2×Dic6—21no—
230C2×Dic6—21no—
231C3×SD16—42no—
232C3×D8—42no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4C2×D411yescharacteristic
240C3×C4⋊C4C2×D411yescharacteristic
241C3×C22⋊C4—21no—
242C3×C22⋊C4—21no—
243C3×C22⋊C4—21no—
244C3×C22⋊C4—41no—
245C3×C22⋊C4—41no—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42C2×D411yescharacteristic
250C3×C42C2×D411yescharacteristic
251C3×C42#4111yescharacteristic
252C3×C42C2411yescharacteristic
253#217—41no—
254#214—42no—
255#214—42no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260Dic3⋊C4—41no—
261Dic3⋊C4—41no—
262C4×Dic3—41no—
263#208—22no—
264#208—41no—
265#208—42no—
266#208—42no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—42no—
272C2×C3⋊C8C2×D411yescharacteristic
273C2×C3⋊C8C2×D411yescharacteristic
274C2×C3⋊C8C2×D411yescharacteristic
275C2×C3⋊C8C2×D411yescharacteristic
276C2×#41—21no—
277C2×#41#6711yescharacteristic
278C2×SD16—21no—
279C2×SD16—61no—
280C2×SD16—61no—
281C2×D8—21no—
282C2×M16—31no—
283C2×M16—31no—
284C2×M16—61no—
285C2×M16—61no—
286C22×C8—61no—
287C4⋊Q8—31no—
288C4⋊Q8—31no—
289C4⋊Q8C22×D311yescharacteristic
290#127C22×D311yescharacteristic
291#125—61no—
292#124—21no—
293C22⋊Q8—61no—
294C22⋊Q8—61no—
295#121—21no—
296#121—21no—
297C4×Q8—61no—
298C4×Q8—61no—
299C4×Q8#6711yescharacteristic
300C4×Q8C22×D311yescharacteristic
301C4×D4—21no—
302C4×D4—21no—
303C4×D4—21no—
304C4×D4—21no—
305C4×D4#6711yescharacteristic
306C4×D4C22×D311yescharacteristic
307#117—21no—
308#117—61no—
309#117#6711yescharacteristic
310C2×C42—21no—
311C2×C42C22×D311yescharacteristic
312#107—21no—
313#107—61no—
314#107—61no—
315#107—61no—
316#107—61no—
317#106—21no—
318#106—61no—
319#106—61no—
320C4⋊C8—31no—
321C4⋊C8—31no—
322C4⋊C8—31no—
323C4⋊C8—31no—
324C4⋊C8—61no—
325C4⋊C8—61no—
326C4⋊C8—61no—
327C4⋊C8C22×D311yescharacteristic
328C4⋊C8C22×D311yescharacteristic
329#103—21no—
330#103—61no—
331#103—61no—
332#103—61no—
333#103—61no—
334#102—21no—
335#102—21no—
336#102—21no—
337#102—61no—
338#102—61no—
339#102—61no—
340#102—61no—
341C22⋊C8—21no—
342C22⋊C8—61no—
343C22⋊C8—61no—
344C22⋊C8—61no—
345#97—31no—
346#97—31no—
347#97—61no—
348C4×C8—21no—
349C4×C8—31no—
350C4×C8—31no—
351C4×C8—61no—
352C4×C8—61no—
353C23×C3—21no—
354C23×C3—41no—
355C23×C3#12111yescharacteristic
356C3×Q8—21no—
357C3×Q8—81no—
358C3×Q8#13712yes—
359C3×D4—21no—
360C3×D4—21no—
361C3×D4—42no—
362C3×D4—41no—
363C3×D4—41no—
364C3×D4—41no—
365C3×D4—41no—
366C3×D4—81no—
367C3×D4—82no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—41no—
376C2×C3×C4—42no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—81no—
381C2×C3×C4—82no—
382C2×C3×C4#12111yescharacteristic
383C2×C3×C4#12111yescharacteristic
384C2×C3×C4#12111yescharacteristic
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C4C22×D411yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×Dic3—41no—
389C2×Dic3—41no—
390Dic6—41no—
391Dic6—41no—
392Dic6—81no—
393C3×C8—41no—
394C3×C8—41no—
395C3⋊C8—21no—
396C3⋊C8—21no—
397C3⋊C8—21no—
398C3⋊C8—41no—
399C3⋊C8—41no—
400C3⋊C8—41no—
401C3⋊C8—41no—
402C3⋊C8#13512yes—
403#41—22no—
404#41—41no—
405#41—82no—
406C2×Q8—21no—
407C2×Q8—61no—
408C2×Q8—61no—
409C2×Q8C2×#6711yescharacteristic
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—41no—
415C2×D4C2×#6711yescharacteristic
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—41no—
420C22×C4D3×D411yescharacteristic
421C22×C4#23711yescharacteristic
422C22×C4C2×#6711yescharacteristic
423SD16—42no—
424SD16—122no—
425SD16—122no—
426D8—42no—
427M16—62no—
428M16—122no—
429M16—121no—
430M16—122no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—31no—
434C2×C8—31no—
435C2×C8—31no—
436C2×C8—31no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—122no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—121no—
451C4⋊C4—121no—
452C4⋊C4C2×#6711yescharacteristic
453C4⋊C4C2×#6711yescharacteristic
454C22⋊C4—21no—
455C22⋊C4—21no—
456C22⋊C4—21no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C22⋊C4—121no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—121no—
464C42D3×D411yescharacteristic
465C42#23711yescharacteristic
466C42C2×#6711yescharacteristic
467C42C23×D311yescharacteristic
468C22×C3—21no—
469C22×C3—21no—
470C22×C3—21no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3—81no—
475C22×C3—81no—
476C22×C3—81no—
477C22×C3C2×#12111yescharacteristic
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—22no—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—41no—
484C3×C4—41no—
485C3×C4—81no—
486C3×C4C2×#13511yescharacteristic
487C3×C4C2×#13711yescharacteristic
488C3×C4#57511yescharacteristic
489C3×C4#57511yescharacteristic
490Dic3—81no—
491Dic3—81no—
492C23—21no—
493C23—41no—
494C23#93911yescharacteristic
495Q8—21no—
496Q8—81no—
497Q8—121no—
498Q8—121no—
499Q8—241no—
500Q8#95112yes—
501D4—21no—
502D4—21no—
503D4—42no—
504D4—41no—
505D4—41no—
506D4—41no—
507D4—41no—
508D4—81no—
509D4—82no—
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—41no—
522C2×C4—81no—
523C2×C4—82no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4#93911yescharacteristic
527C2×C4#93911yescharacteristic
528C2×C4#93911yescharacteristic
529C2×C4C2×D3×D411yescharacteristic
530C2×C4C2×#23711yescharacteristic
531C2×C4C22×#6711yescharacteristic
532C8—32no—
533C8—41no—
534C8—41no—
535C8—61no—
536C8—61no—
537C8—61no—
538C8—121no—
539C8—121no—
540C8—121no—
541C8—121no—
542C2×C3—41no—
543C2×C3—81no—
544C2×C3—161no—
545C2×C3#303511yescharacteristic
546C2×C3#305611yescharacteristic
547C2×C3#306811yescharacteristic
548C22—21no—
549C22—21no—
550C22—21no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C2×#93911yescharacteristic
558C4—21no—
559C4—21no—
560C4—21no—
561C4—21no—
562C4—22no—
563C4—41no—
564C4—41no—
565C4—81no—
566C4—241no—
567C4—241no—
568C4D3×#13511yescharacteristic
569C4#576811yescharacteristic
570C4#582211yescharacteristic
571C4C2×#95111yescharacteristic
572C3#1918311yescharacteristic, complemented, Sylow
573C2—41no—
574C2—81no—
575C2—161no—
576C2#8292511yescharacteristic
577C2#8390011yescharacteristic
578C2#8394311yescharacteristic
subgroupG/H
centerC22C2×#939
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
Fitting, prime radicalC3×#3106C2
socleC22×C3C2×#121
hypercenter#3106D3
hypocenterC3#19183
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3106

Complements

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

NH ≅ G/Ntype
2. #83294C2semidirect
3. #82519C2semidirect
4. #82507C2semidirect
5. #71807C2semidirect
6. #71151C2semidirect
7. #71149C2semidirect
8. #71139C2semidirect
9. #71137C2semidirect
10. #71022C2semidirect
11. #71020C2semidirect
12. #71010C2semidirect
13. #71008C2semidirect
14. #70781C2semidirect
15. #70746C2semidirect
572. C3#19183semidirect

Maximal subgroups

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

subgroupindex
C3×#31062
#832942
#825192
#825072
#718072
#711512
#711492
#711392
#711372
#710222
#710202
#710102
#710082
#707812
#707462
#191833