Finite Group Databasebeta

← #10762166

Subgroups — #10762166

Order 768.

1020 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86278C211yesmaximal, characteristic
2#79294C211yesmaximal, characteristic, complemented
3C4×#907C211yesmaximal, characteristic, complemented
4C3×#1485C211yesmaximal, characteristic, complemented
5#71826C211yesmaximal, characteristic, complemented
6#70754C211yesmaximal, characteristic, complemented
7#69854C211yesmaximal, characteristic, complemented
8#68240C212yesmaximal, complemented
9#68238C212yesmaximal, complemented
10#68235C212yesmaximal, complemented
11#68229C212yesmaximal, complemented
12#17518—31nomaximal, Sylow
13C3×#533C2211yescharacteristic
14C2×#907—21no—
15#5732C411yescharacteristic
16#5727C411yescharacteristic
17#5726C411yescharacteristic
18#5721C411yescharacteristic
19#5693—22no—
20#5641C412yes—
21#5639—21no—
22#5633—21no—
23#5631C412yes—
24#5580—21no—
25#5578C2212yes—
26C4×#239C2212yes—
27C4×D3×Q8—21no—
28#5528—22no—
29C4×#235C2211yescharacteristic
30C4×#235C2211yescharacteristic
31C3×#431C2211yescharacteristic
32C3×C4×M16C2211yescharacteristic
33#5122—21no—
34#4924C2211yescharacteristic
35C2×C4×C3⋊C8C2211yescharacteristic
36#4837C2212yes—
37#4833C2212yes—
38#4729—22no—
39#4708—22no—
40#4688—21no—
41C4×ΓL(1,𝔽25)C2212yes—
42C4×D3×C8C2212yes—
43C8×Dic6—21no—
44C3×Q8⋊C8C2212yes—
45C3×Q8⋊C8C2212yes—
46#4535C2212yes—
47#4535C2212yes—
48#4484C2211yescharacteristic
49#4484C2211yescharacteristic
50#4483C2211yescharacteristic
51#4483C2211yescharacteristic
52#4459C2212yes—
53#4456C2212yes—
54#3482—31no—
55#2991—31no—
56C4×#131—31no—
57#1493—32no—
58#1493—31no—
59#1489—31no—
60#1489—32no—
61#1485—32no—
62#1485D311yescharacteristic, complemented
63#1476—31no—
64#1476—32no—
65C2×#239C2×C412yes—
66C2×D3×Q8—21no—
67C2×#235C2×C411yescharacteristic
68C2×#235D411yescharacteristic
69C2×C3×M16—21no—
70C3×C4⋊Q8C2×C411yescharacteristic
71C3×C4⋊Q8C2×C411yescharacteristic
72C3×#125—21no—
73C3×C22⋊Q8—21no—
74C3×C22⋊Q8C2×C412yes—
75C3×C4×Q8—21no—
76C3×C4×Q8C2312yes—
77C3×#117—21no—
78C2×C3×C42C2311yescharacteristic
79#947—21no—
80#946—21no—
81#946C2×C412yes—
82Q8×Dic3—21no—
83Q8×Dic3C2×C412yes—
84#944C2×C412yes—
85C4×#67—21no—
86C4×#67—21no—
87C4×#67C2×C412yes—
88C4×#67D412yes—
89#926C2×C411yescharacteristic
90#926D411yescharacteristic
91C2×#208—21no—
92C22×C3⋊C8—21no—
93#907—28no—
94C2×ΓL(1,𝔽25)—22no—
95C2×ΓL(1,𝔽25)—22no—
96C2×D3×C8—22no—
97#898—22no—
98#898—22no—
99#897C2×C412yes—
100#896—22no—
101#896—22no—
102#895—21no—
103#894—22no—
104#894—22no—
105#893—22no—
106#893C2×C412yes—
107#892—22no—
108#892C2×C412yes—
109D3×C4⋊C4—21no—
110D3×C4⋊C4—22no—
111#890—21no—
112#889—22no—
113#889—22no—
114#888C2×C412yes—
115#887—21no—
116#887—22no—
117#885—22no—
118#884—22no—
119#883—22no—
120#879—22no—
121C4×D12—22no—
122C4×D12C2×C411yescharacteristic
123C4×D12C2×C411yescharacteristic
124C4×D12D411yescharacteristic
125C4×D12C2311yescharacteristic
126#872—22no—
127#872—21no—
128#872D412yes—
129C42×D3—21no—
130C42×D3—22no—
131C42×D3C2312yes—
132C4×Dic6—22no—
133C4×Dic6C2×C411yescharacteristic
134C4×Dic6C2×C411yescharacteristic
135C4×Dic6D411yescharacteristic
136C4×Dic6C2311yescharacteristic
137C3×C4⋊C8C2311yescharacteristic
138C3×C4⋊C8C2311yescharacteristic
139C3×C22⋊C8—21no—
140C3×#97—21no—
141C3×C4×C8—21no—
142C3×C4×C8C2312yes—
143(C2×D3)⋊C8—22no—
144#815—22no—
145Dic3⋊C8—22no—
146C8×Dic3—22no—
147C8×Dic3—22no—
148C4⋊(C3⋊C8)—21no—
149C4⋊(C3⋊C8)C2312yes—
150C4×C3⋊C8—21no—
151C4×C3⋊C8C2311yescharacteristic
152C4×C3⋊C8C2311yescharacteristic
153C2×#131—61no—
154#557—31no—
155#556—31no—
156#556—32no—
157#552—61no—
158#552—62no—
159D4×Q8—31no—
160#548—61no—
161#547—31no—
162#547—32no—
163#533—32no—
164#533C2×D311yescharacteristic
165#529—62no—
166#529—61no—
167C4×#41—31no—
168C4×#41—32no—
169C4×#41—31no—
170C2×C4×Q8—61no—
171C8×Q8—61no—
172#436—61no—
173#436—62no—
174#431—32no—
175#431C2×D311yescharacteristic
176#423—31no—
177#423—32no—
178#405—62no—
179#405—61no—
180C4×M16—32no—
181C4×M16C2×D311yescharacteristic
182C2×C4×C8—31no—
183C2×C4×C8—32no—
184Q8⋊C8—32no—
185Q8⋊C8—32no—
186Q8⋊C8—31no—
187Q8⋊C8—31no—
188Q8⋊C8—32no—
189Q8⋊C8C2×D312yes—
190Q8⋊C8C2×D312yes—
191D4⋊C8—31no—
192D4⋊C8—31no—
193D4⋊C8—32no—
194C2×C3×Q8—21no—
195C2×C3×Q8C22×C412yes—
196C22×C3×C4C22×C411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C2×#67—21no—
200C2×#67C22⋊C412yes—
201#239—24no—
202#239—42no—
203D3×Q8—44no—
204#235—22no—
205#235—24no—
206#235—41no—
207C2×D12—22no—
208C2×D12C22⋊C411yescharacteristic
209C2×D12C22×C411yescharacteristic
210C2×C4×D3—21no—
211C2×C4×D3—21no—
212C2×C4×D3—21no—
213C2×C4×D3—22no—
214C2×C4×D3—22no—
215C2×C4×D3C22×C412yes—
216C2×C4×D3C2×D412yes—
217C2×C4×D3C2×D412yes—
218C2×Dic6—22no—
219C2×Dic6C22⋊C411yescharacteristic
220C2×Dic6C22×C411yescharacteristic
221C3×M16—24no—
222C2×C3×C8—22no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4C22×C412yes—
231C3×C22⋊C4—22no—
232C3×C42—22no—
233C3×C42C22×C411yescharacteristic
234C3×C42C2×D411yescharacteristic
235C3×C42C2×D411yescharacteristic
236C3×C42C2411yescharacteristic
237#217—21no—
238#217C22⋊C412yes—
239(C2×D3)⋊C4—22no—
240(C2×D3)⋊C4—22no—
241(C2×D3)⋊C4—22no—
242(C2×D3)⋊C4—21no—
243(C2×D3)⋊C4—21no—
244(C2×D3)⋊C4—22no—
245(C2×D3)⋊C4C22⋊C412yes—
246(C2×D3)⋊C4C22⋊C412yes—
247C4⋊Dic3—22no—
248C4⋊Dic3—22no—
249C4⋊Dic3C22⋊C411yescharacteristic
250C4⋊Dic3C22⋊C411yescharacteristic
251C4⋊Dic3C22×C411yescharacteristic
252C4⋊Dic3C22×C411yescharacteristic
253Dic3⋊C4—21no—
254Dic3⋊C4—21no—
255Dic3⋊C4—22no—
256Dic3⋊C4—22no—
257Dic3⋊C4—22no—
258Dic3⋊C4—22no—
259Dic3⋊C4C22⋊C412yes—
260Dic3⋊C4C22⋊C412yes—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—22no—
265C4×Dic3—22no—
266C4×Dic3C22×C412yes—
267C4×Dic3C2×D412yes—
268C4×Dic3C2×D412yes—
269#208—81no—
270C2×C3⋊C8—24no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—22no—
274ΓL(1,𝔽25)—28no—
275ΓL(1,𝔽25)—82no—
276D3×C8—28no—
277C2×#41—31no—
278C2×#41—32no—
279C2×#41—31no—
280C22×Q8—61no—
281#131—68no—
282C2×M16—21no—
283C2×M16—62no—
284C2×M16—62no—
285C2×M16—61no—
286C22×C8—62no—
287C22×C8—61no—
288C4⋊Q8—32no—
289C4⋊Q8—32no—
290C4⋊Q8C4×D311yescharacteristic
291C4⋊Q8C4×D311yescharacteristic
292#126—62no—
293#126—62no—
294#126—62no—
295#125—21no—
296#125—62no—
297#125—62no—
298#125—61no—
299#124—62no—
300#124—61no—
301#123—62no—
302#123—62no—
303#123—62no—
304C22⋊Q8—21no—
305C22⋊Q8—32no—
306C22⋊Q8—32no—
307C22⋊Q8—62no—
308C22⋊Q8—62no—
309C22⋊Q8—61no—
310C22⋊Q8C4×D312yes—
311#121—62no—
312#121—61no—
313C4×Q8—21no—
314C4×Q8—31no—
315C4×Q8—31no—
316C4×Q8—32no—
317C4×Q8—31no—
318C4×Q8—31no—
319C4×Q8—62no—
320C4×Q8—62no—
321C4×Q8—61no—
322C4×Q8—61no—
323C4×Q8C22×D312yes—
324C4×D4—32no—
325C4×D4—32no—
326C4×D4—32no—
327C4×D4—31no—
328C4×D4—31no—
329C4×D4—31no—
330C4×D4—31no—
331C4×D4—62no—
332C4×D4—62no—
333C4×D4—61no—
334C4×D4—61no—
335#117—21no—
336#117—31no—
337#117—32no—
338#117—31no—
339#117—32no—
340#117—62no—
341#117—62no—
342#117—61no—
343C2×C4⋊C4—61no—
344C2×C4⋊C4—62no—
345C2×C42—32no—
346C2×C42—62no—
347C2×C42—61no—
348C2×C42C22×D311yescharacteristic
349C4⋊C8—32no—
350C4⋊C8—62no—
351C4⋊C8—61no—
352C4⋊C8C22×D311yescharacteristic
353C4⋊C8C22×D311yescharacteristic
354C22⋊C8—21no—
355C22⋊C8—62no—
356#97—21no—
357#97—62no—
358C4×C8—21no—
359C4×C8—31no—
360C4×C8—31no—
361C4×C8—62no—
362C4×C8—62no—
363C4×C8—61no—
364C4×C8C22×D312yes—
365C23×C3C2×C22⋊C411yescharacteristic
366C22×D3—21no—
367C22×D3C2×C22⋊C412yes—
368C3×Q8—22no—
369C3×Q8—22no—
370C3×Q8—42no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4C2×C22⋊C411yescharacteristic
379C2×C3×C4C2×C22⋊C411yescharacteristic
380C2×C3×C4C2×C22⋊C411yescharacteristic
381C2×C3×C4C23×C411yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C22×D411yescharacteristic
384#67—24no—
385#67—42no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3C2×C22⋊C412yes—
390C2×Dic3C2×C22⋊C412yes—
391C2×Dic3C2×C22⋊C412yes—
392D12—21no—
393D12—22no—
394D12—44no—
395D12#13112yescomplemented
396C4×D3—22no—
397C4×D3—22no—
398C4×D3—22no—
399C4×D3—22no—
400C4×D3—24no—
401C4×D3—44no—
402C4×D3—42no—
403C4×D3—41no—
404C4×D3—41no—
405C4×D3—42no—
406C4×D3—44no—
407Dic6—21no—
408Dic6—21no—
409Dic6—44no—
410Dic6#10414yes—
411C3×C8—24no—
412C3×C8—41no—
413C3×C8—41no—
414C3⋊C8—22no—
415C3⋊C8—22no—
416C3⋊C8—42no—
417#41—64no—
418#41—62no—
419#41—64no—
420#41—121no—
421#41—122no—
422C2×Q8—21no—
423C2×Q8—31no—
424C2×Q8—31no—
425C2×Q8—62no—
426C2×Q8—124no—
427C2×Q8C2×C4×D312yes—
428C2×D4—32no—
429C2×D4—31no—
430C2×D4—31no—
431C2×D4—62no—
432C2×D4—61no—
433C22×C4—32no—
434C22×C4—32no—
435C22×C4—32no—
436C22×C4—62no—
437C22×C4—62no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4C2×C4×D311yescharacteristic
442C22×C4D3×D411yescharacteristic
443C22×C4D3×D411yescharacteristic
444M16—24no—
445M16—68no—
446M16—242no—
447M16—241no—
448C2×C8—22no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—68no—
452C2×C8—64no—
453C2×C8—61no—
454C2×C8—61no—
455C2×C8—62no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—32no—
462C4⋊C4—31no—
463C4⋊C4—32no—
464C4⋊C4—31no—
465C4⋊C4—31no—
466C4⋊C4—31no—
467C4⋊C4—62no—
468C4⋊C4—62no—
469C4⋊C4—62no—
470C4⋊C4—62no—
471C4⋊C4—62no—
472C4⋊C4—62no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4C2×C4×D312yes—
476C22⋊C4—22no—
477C22⋊C4—32no—
478C22⋊C4—32no—
479C22⋊C4—32no—
480C22⋊C4—62no—
481C22⋊C4—62no—
482C22⋊C4—62no—
483C22⋊C4—62no—
484C22⋊C4—61no—
485C22⋊C4—61no—
486C22⋊C4—61no—
487C42—22no—
488C42—32no—
489C42—32no—
490C42—32no—
491C42—62no—
492C42—62no—
493C42—61no—
494C42—61no—
495C42—61no—
496C42C2×C4×D311yescharacteristic
497C42D3×D411yescharacteristic
498C42D3×D411yescharacteristic
499C42C23×D311yescharacteristic
500C22×C3—21no—
501C22×C3—21no—
502C22×C3—21no—
503C22×C3C22×C22⋊C411yescharacteristic
504C2×D3—22no—
505C2×D3—22no—
506C2×D3—22no—
507C2×D3—22no—
508C2×D3—41no—
509C2×D3—41no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—42no—
514C3×C4—42no—
515C3×C4C2×#10412yes—
516C3×C4C2×#13111yescharacteristic
517C3×C4#56811yescharacteristic
518C3×C4#57711yescharacteristic
519C3×C4#57811yescharacteristic
520Dic3—22no—
521Dic3—22no—
522Dic3—41no—
523Dic3—41no—
524Dic3—41no—
525Dic3C2×#10414yes—
526C23—32no—
527C23—61no—
528C23D3×C22⋊C411yescharacteristic
529Q8—22no—
530Q8—22no—
531Q8—34no—
532Q8—42no—
533Q8—61no—
534Q8—61no—
535Q8—124no—
536D4—32no—
537D4—64no—
538D4—61no—
539D4—62no—
540D4—124no—
541D4—122no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—32no—
550C2×C4—32no—
551C2×C4—32no—
552C2×C4—64no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—122no—
561C2×C4—124no—
562C2×C4—124no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—122no—
566C2×C4D3×C22⋊C411yescharacteristic
567C2×C4D3×C22⋊C411yescharacteristic
568C2×C4D3×C22⋊C411yescharacteristic
569C2×C4C22×C4×D311yescharacteristic
570C2×C4C2×D3×D411yescharacteristic
571C2×C4C2×D3×D411yescharacteristic
572C8—24no—
573C8—41no—
574C8—41no—
575C8—62no—
576C8—62no—
577C8—122no—
578C2×C3—41no—
579C2×C3#288011yescharacteristic
580C2×C3#289811yescharacteristic
581C2×C3C22×#10411yescharacteristic
582D3—42no—
583D3—42no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—62no—
588C22—62no—
589C22—62no—
590C22—62no—
591C22—121no—
592C22—121no—
593C22C2×D3×C22⋊C411yescharacteristic
594C4—21no—
595C4—21no—
596C4—21no—
597C4—34no—
598C4—42no—
599C4—42no—
600C4—62no—
601C4—62no—
602C4—121no—
603C4—121no—
604C4—121no—
605C4D3×#10412yes—
606C4D3×#13111yescharacteristic
607C4#575011yescharacteristic
608C4#577811yescharacteristic
609C4#577911yescharacteristic
610C3#1751811yescharacteristic, complemented, Sylow
611C2—41no—
612C2—122no—
613C2—122no—
614C2#7866311yescharacteristic
615C2#7895711yescharacteristic
616C2C2×D3×#10411yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1485C2
socleC22×C3C22×C22⋊C4
prime radical#86278C2
cosocleC3×C42C24
hypercenter#1485D3
hypocenterC3#17518
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1485

Complements

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

NH ≅ G/Ntype
2. #79294C2semidirect
3. C4×#907C2semidirect
4. C3×#1485C2semidirect
5. #71826C2semidirect
6. #70754C2semidirect
7. #69854C2semidirect
8. #68240C2semidirect
9. #68238C2semidirect
10. #68235C2semidirect
11. #68229C2semidirect
62. #1485D3semidirect
395. D12#131semidirect
610. C3#17518semidirect

Maximal subgroups

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

subgroupindex
#862782
#792942
C4×#9072
C3×#14852
#718262
#707542
#698542
#682402
#682382
#682352
#682292
#175183