Finite Group Databasebeta

← #10824920

Subgroups — #10824920

Order 768.

776 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#440C211yesmaximal, characteristic, complemented
2C4×#942C211yesmaximal, characteristic, complemented
3C4×#932C211yesmaximal, characteristic
4#78489C211yesmaximal, characteristic, complemented
5#72059C211yesmaximal, characteristic
6#72053C211yesmaximal, characteristic, complemented
7C4×#836C211yesmaximal, characteristic, complemented
8C4×#833C211yesmaximal, characteristic, complemented
9#70647C211yesmaximal, characteristic, complemented
10#70626C211yesmaximal, characteristic, complemented
11#70265C211yesmaximal, characteristic, complemented
12#70264C211yesmaximal, characteristic, complemented
13#70197C211yesmaximal, characteristic, complemented
14#70196C211yesmaximal, characteristic
15#69840C211yesmaximal, characteristic, complemented
16#23721—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×#942C2211yescharacteristic
21C2×#932C2211yescharacteristic
22C4×#235C2211yescharacteristic
23C3×#440—22no—
24C3×#440—22no—
25C2×C3×#107C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#97C2211yescharacteristic
29C2×#836C2211yescharacteristic
30C2×#836C2211yescharacteristic
31#5225C2211yescharacteristic
32C2×#833C2211yescharacteristic
33C2×#833C2211yescharacteristic
34#5215C2211yescharacteristic
35#5110C2211yescharacteristic
36#5110C2211yescharacteristic
37C2×#819C2211yescharacteristic
38C2×#819C2211yescharacteristic
39#5102C2211yescharacteristic
40#5102C2211yescharacteristic
41#5030—21no—
42C4×#216—21no—
43#5026—21no—
44C4×#215—21no—
45#5018—21no—
46C4×#214—21no—
47#5014—21no—
48C4×#213—21no—
49#4966C2211yescharacteristic
50#4966C2211yescharacteristic
51#4964C2211yescharacteristic
52#4964C2211yescharacteristic
53C4×#208C2211yescharacteristic
54C4×#208C2211yescharacteristic
55C4×#208C2211yescharacteristic
56#4709—22no—
57#4705—22no—
58#4552C2211yescharacteristic
59#4552C2211yescharacteristic
60#4530C2211yescharacteristic
61#4530C2211yescharacteristic
62#4530C2211yescharacteristic
63#4530C2211yescharacteristic
64C4×#137—31no—
65C4×AGL(1,ℤ/8)—31no—
66C2×#440D311yescharacteristic, complemented
67#2923—31no—
68#2089—31no—
69#2078—31no—
70#2064—31no—
71#2063—31no—
72#2052—31no—
73#2051—31no—
74#2046—31no—
75#2019—31no—
76#1778—31no—
77C4×#101—31no—
78C4×#100—31no—
79C22×C3×Q8C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C23×C3×C4—21no—
82C2×#235C2311yescharacteristic
83C2×C3×SD16—22no—
84C2×C3×SD16—22no—
85C2×C3×SD16—22no—
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×C4×Q8—21no—
89C3×C4×Q8—21no—
90C3×C4×Q8—21no—
91C3×C4×Q8—21no—
92C3×C4×D4—21no—
93C3×C4×D4—21no—
94C3×C4×D4—21no—
95C3×C4×D4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C22⋊C4—21no—
100C2×C3×C42—21no—
101C2×C3×C42C2311yescharacteristic
102C2×#216—21no—
103#942—22no—
104#942—22no—
105C2×#215—21no—
106C2×#214—21no—
107#932—22no—
108#932—22no—
109C2×#213—21no—
110C4×#67—21no—
111C4×#67—21no—
112#926C2311yescharacteristic
113C2×#208C2311yescharacteristic
114C2×#208C2311yescharacteristic
115C2×#208C2311yescharacteristic
116C2×#208C2311yescharacteristic
117C2×#208C2311yescharacteristic
118C2×#208C2311yescharacteristic
119C4×D12D411yescharacteristic
120C4×D12D411yescharacteristic
121#872—21no—
122C42×D3—21no—
123C4×Dic6D411yescharacteristic
124C4×Dic6D411yescharacteristic
125C3×#107—22no—
126C3×#103—22no—
127C3×#102—22no—
128C3×#97D412yes—
129C3×#97D412yes—
130#836—22no—
131#836—22no—
132#835—21no—
133#835—21no—
134#833—22no—
135#833—22no—
136#832—21no—
137#832—21no—
138(C2×D3)⋊C8—22no—
139(C2×D3)⋊C8—22no—
140#819—22no—
141#819—22no—
142Dic3⋊C8—22no—
143Dic3⋊C8—22no—
144#810—21no—
145#810—21no—
146#809—21no—
147#809—21no—
148#808—21no—
149#808—21no—
150#807—21no—
151#807—21no—
152#803—21no—
153#803—21no—
154#803—21no—
155C4×C3⋊C8—21no—
156C4×C3⋊C8—21no—
157C4×C3⋊C8—21no—
158C2×#137—31no—
159C2×AGL(1,ℤ/8)—31no—
160C22×SD16C2×D311yescharacteristic
161C4×#41—31no—
162C2×C4×Q8C2×D311yescharacteristic
163C2×C4×D4C2×D311yescharacteristic
164#442—61no—
165#441—61no—
166#440—22no—
167#440—22no—
168#440—61no—
169#440—61no—
170C4×Q16—61no—
171C4×SD16—61no—
172C4×SD16—61no—
173C4×D8—61no—
174#433—62no—
175#432—62no—
176C2×#108—31no—
177C2×#108—31no—
178#428—31no—
179#428—31no—
180C2×#107C2×D311yescharacteristic
181#424—31no—
182#424—31no—
183#419—31no—
184#418—31no—
185#417—31no—
186#417—31no—
187C2×#103C2×D311yescharacteristic
188C2×#102C2×D311yescharacteristic
189C2×#101—31no—
190C2×#101—31no—
191C2×#100—31no—
192C2×#100—31no—
193#408—31no—
194#408—31no—
195C4×M16—31no—
196C4×M16—31no—
197C4×M16—31no—
198C2×#97C2×D311yescharacteristic
199#343—31no—
200#343—31no—
201#343—31no—
202#343—31no—
203#341—31no—
204#341—31no—
205C24×C3—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8#4112yes—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4#4112yes—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—41no—
222C22×C3×C4—41no—
223C22×C3×C4—82no—
224C22×C3×C4#4111yescharacteristic
225C22×C3×C4#4111yescharacteristic
226C22×C3×C4C2411yescharacteristic
227C2×#67—21no—
228#235—21no—
229#235—21no—
230#235—21no—
231#235—21no—
232C2×D12C2×D411yescharacteristic
233C2×C4×D3—21no—
234C2×C4×D3—21no—
235C2×C4×D3—21no—
236C2×Dic6C2×D411yescharacteristic
237C3×SD16—28no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8C2×D412yes—
243C2×C3×C8C2×D412yes—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—41no—
249C3×C4⋊C4—41no—
250C3×C22⋊C4—41no—
251C3×C22⋊C4—41no—
252C3×C42—41no—
253C3×C42—41no—
254C3×C42C2×D411yescharacteristic
255C3×C42C2×D411yescharacteristic
256C3×C42C2×D411yescharacteristic
257C3×C42C2×D411yescharacteristic
258#217—21no—
259#216—24no—
260#215—24no—
261#214—24no—
262#213—24no—
263(C2×D3)⋊C4—21no—
264(C2×D3)⋊C4—21no—
265C4⋊Dic3C2×D411yescharacteristic
266C4⋊Dic3C2×D411yescharacteristic
267Dic3⋊C4—21no—
268Dic3⋊C4—21no—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C4×Dic3—21no—
272#208—21no—
273#208—21no—
274#208—21no—
275#208—21no—
276#208—21no—
277#208—21no—
278#208—21no—
279#208—21no—
280#208—21no—
281#208—21no—
282#208#4112yes—
283#208#4112yes—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×#41—31no—
291C22×Q8C22×D311yescharacteristic
292C22×D4C22×D311yescharacteristic
293C23×C4—21no—
294#137—62no—
295#137—62no—
296AGL(1,ℤ/8)—62no—
297AGL(1,ℤ/8)—62no—
298C2×Q16—61no—
299C2×SD16—22no—
300C2×SD16—22no—
301C2×SD16—22no—
302C2×SD16—61no—
303C2×SD16—61no—
304C2×D8—61no—
305C2×M16—31no—
306C2×M16—31no—
307C2×M16—31no—
308C2×M16—31no—
309C2×M16—31no—
310C2×M16—31no—
311C22×C8C22×D311yescharacteristic
312C22×C8C22×D311yescharacteristic
313C4×Q8—21no—
314C4×Q8—21no—
315C4×Q8—21no—
316C4×Q8—21no—
317C4×Q8—31no—
318C4×Q8—31no—
319C4×D4—21no—
320C4×D4—21no—
321C4×D4—21no—
322C4×D4—21no—
323C4×D4—31no—
324C4×D4—31no—
325C4×D4—61no—
326C4×D4—61no—
327#117—31no—
328#117—61no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4C22×D311yescharacteristic
331C2×C4⋊C4C22×D311yescharacteristic
332C2×C22⋊C4—21no—
333C2×C42—21no—
334C2×C42—61no—
335C2×C42C22×D311yescharacteristic
336#108—62no—
337#108—62no—
338#107—22no—
339#107—61no—
340#107—61no—
341#106—61no—
342#106—61no—
343C4⋊C8—62no—
344C4⋊C8—62no—
345#103—22no—
346#103—61no—
347#103—61no—
348#103—61no—
349#103—61no—
350#102—22no—
351#102—61no—
352#102—61no—
353#102—61no—
354#102—61no—
355#101—62no—
356#101—62no—
357#100—62no—
358#100—62no—
359C22⋊C8—62no—
360C22⋊C8—62no—
361#97—61no—
362#97—61no—
363#97—61no—
364#97#6712yes—
365#97#6712yes—
366C4×C8—61no—
367C4×C8—61no—
368C4×C8—61no—
369C23×C3—22no—
370C23×C3—41no—
371C23×C3—41no—
372C23×C3—81no—
373C23×C3—81no—
374C23×C3C2×#4111yescharacteristic
375C22×D3—21no—
376C3×Q8—22no—
377C3×Q8—22no—
378C3×Q8—41no—
379C3×Q8—41no—
380C3×D4—22no—
381C3×D4—22no—
382C3×D4—41no—
383C3×D4—41no—
384C2×C3×C4—22no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—82no—
392C2×C3×C4—82no—
393C2×C3×C4—81no—
394C2×C3×C4—81no—
395C2×C3×C4#12111yescharacteristic
396C2×C3×C4#12111yescharacteristic
397C2×C3×C4#12111yescharacteristic
398C2×C3×C4#12111yescharacteristic
399C2×C3×C4C22×D411yescharacteristic
400C2×C3×C4C22×D411yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405#67—41no—
406#67—41no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410D12—21no—
411D12—21no—
412C4×D3—41no—
413C4×D3—41no—
414C4×D3—81no—
415C4×D3—81no—
416Dic6—21no—
417Dic6—21no—
418C3×C8—22no—
419C3×C8—22no—
420C3⋊C8—22no—
421C3⋊C8—22no—
422C3⋊C8—41no—
423C3⋊C8—41no—
424C3⋊C8—41no—
425C3⋊C8—41no—
426C24—21no—
427#41—61no—
428#41—61no—
429#41—61no—
430#41—61no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—31no—
437C2×Q8#23512yes—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—31no—
444C2×D4—61no—
445C2×D4#23512yes—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—82no—
455C22×C4#23711yescharacteristic
456C22×C4#23711yescharacteristic
457C22×C4C23×D311yescharacteristic
458Q16—64no—
459SD16—28no—
460SD16—64no—
461SD16—64no—
462D8—64no—
463M16—32no—
464M16—32no—
465M16—61no—
466M16—61no—
467M16—61no—
468M16—61no—
469M16—61no—
470M16—61no—
471M16—61no—
472M16—61no—
473M16—61no—
474M16—61no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8C2×#6712yes—
486C2×C8C2×#6712yes—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—31no—
492C4⋊C4—31no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—61no—
496C4⋊C4—61no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—61no—
500C22⋊C4—61no—
501C22⋊C4—61no—
502C42—41no—
503C42—41no—
504C42—61no—
505C42—61no—
506C42—61no—
507C42D3×D411yescharacteristic
508C42D3×D411yescharacteristic
509C42C2×#6711yescharacteristic
510C42C2×#6711yescharacteristic
511C22×C3—21no—
512C22×C3—21no—
513C22×C3—42no—
514C22×C3—82no—
515C22×C3—82no—
516C22×C3—81no—
517C22×C3—81no—
518C22×C3C2×#12111yescharacteristic
519C22×C3#52911yescharacteristic
520C22×C3#52911yescharacteristic
521C2×D3—41no—
522C2×D3—41no—
523C2×D3—41no—
524C3×C4—21no—
525C3×C4—21no—
526C3×C4—21no—
527C3×C4—21no—
528C3×C4—42no—
529C3×C4—81no—
530C3×C4—81no—
531C3×C4#52511yescharacteristic
532C3×C4#52511yescharacteristic
533C3×C4#52511yescharacteristic
534C3×C4#53211yescharacteristic
535Dic3—41no—
536Dic3—41no—
537Dic3—41no—
538C23—22no—
539C23—41no—
540C23—41no—
541C23—61no—
542C23—81no—
543C23—81no—
544C23C2×#23711yescharacteristic
545Q8—22no—
546Q8—22no—
547Q8—41no—
548Q8—41no—
549Q8—61no—
550Q8—61no—
551D4—22no—
552D4—22no—
553D4—41no—
554D4—41no—
555D4—61no—
556D4—61no—
557D4—121no—
558D4—121no—
559C2×C4—22no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—241no—
576C2×C4—241no—
577C2×C4#93811yescharacteristic
578C2×C4#93811yescharacteristic
579C2×C4#93811yescharacteristic
580C2×C4#93811yescharacteristic
581C2×C4C2×#23511yescharacteristic
582C2×C4C2×#23511yescharacteristic
583C2×C4C2×D3×D411yescharacteristic
584C2×C4D3×#4111yescharacteristic
585C2×C4D3×#4111yescharacteristic
586C2×C4C22×#6711yescharacteristic
587C8—22no—
588C8—22no—
589C8—62no—
590C8—62no—
591C8—121no—
592C8—121no—
593C8—121no—
594C8—121no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3—82no—
598C2×C3#239211yescharacteristic
599C2×C3#306811yescharacteristic
600C2×C3#306911yescharacteristic
601D3—81no—
602C22—21no—
603C22—21no—
604C22—42no—
605C22—82no—
606C22—82no—
607C22—81no—
608C22—81no—
609C22—121no—
610C22—121no—
611C22—121no—
612C22#553811yescharacteristic
613C22#553811yescharacteristic
614C22C2×#93811yescharacteristic
615C4—21no—
616C4—21no—
617C4—21no—
618C4—21no—
619C4—42no—
620C4—81no—
621C4—81no—
622C4—121no—
623C4—121no—
624C4—121no—
625C4#554711yescharacteristic
626C4#554711yescharacteristic
627C4#579111yescharacteristic
628C4#583211yescharacteristic
629C3#2372111yescharacteristic, complemented, Sylow
630C2—21no—
631C2—21no—
632C2—82no—
633C2—241no—
634C2#8240111yescharacteristic
635C2#8396311yescharacteristic
636C2#8399811yescharacteristic
subgroupG/H
centerC22C2×#938
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#440C2
socleC22×C3C2×#121
prime radicalC2×#932C22
hypercenterC2×#440D3
hypocenterC3#23721
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
1. C2×C3×#440C2semidirect
2. C4×#942C2semidirect
4. #78489C2semidirect
6. #72053C2semidirect
7. C4×#836C2semidirect
8. C4×#833C2semidirect
9. #70647C2semidirect
10. #70626C2semidirect
11. #70265C2semidirect
12. #70264C2semidirect
13. #70197C2semidirect
15. #69840C2semidirect
66. C2×#440D3semidirect
629. C3#23721semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4402
C4×#9422
C4×#9322
#784892
#720592
#720532
C4×#8362
C4×#8332
#706472
#706262
#702652
#702642
#701972
#701962
#698402
#237213