Finite Group Databasebeta

← #10824922

Subgroups — #10824922

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
1C4×#909C211yesmaximal, characteristic, complemented
2C4×#908C211yesmaximal, characteristic
3C2×#4710C211yesmaximal, characteristic, complemented
4#78489C211yesmaximal, characteristic, complemented
5C3×#1778C211yesmaximal, characteristic, complemented
6#70717C211yesmaximal, characteristic, complemented
7#70716C211yesmaximal, characteristic, complemented
8#70715C211yesmaximal, characteristic, complemented
9#70714C211yesmaximal, characteristic
10C4×#824C211yesmaximal, characteristic, complemented
11C4×#823C211yesmaximal, characteristic, complemented
12#68943C211yesmaximal, characteristic
13#68933C211yesmaximal, characteristic, complemented
14#68923C211yesmaximal, characteristic, complemented
15#68921C211yesmaximal, characteristic, complemented
16#23721—31nomaximal, Sylow
17C2×#909C2211yescharacteristic
18C2×#908C2211yescharacteristic
19C22×#204C2211yescharacteristic
20C4×#235C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×#108C2211yescharacteristic
24C2×C3×#108C2211yescharacteristic
25C3×C4×M16C2211yescharacteristic
26C3×C4×M16C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28C2×#824C2211yescharacteristic
29C2×#824C2211yescharacteristic
30#5133C2211yescharacteristic
31#5133C2211yescharacteristic
32#5131C2211yescharacteristic
33C2×#823C2211yescharacteristic
34C2×#823C2211yescharacteristic
35#5121C2211yescharacteristic
36#5121C2211yescharacteristic
37#5120C2211yescharacteristic
38C2×#821C2211yescharacteristic
39#5110C2211yescharacteristic
40#5110C2211yescharacteristic
41C2×#818C2211yescharacteristic
42C2×#816C2211yescharacteristic
43#5102C2211yescharacteristic
44#5102C2211yescharacteristic
45C4×#208C2211yescharacteristic
46#4716—21no—
47#4711—21no—
48#4710—21no—
49#4710—21no—
50#4710—22no—
51#4710—22no—
52#4709—22no—
53#4705—22no—
54C4×Dic12—21no—
55C4×D24—21no—
56C4×#204—21no—
57C4×#204—21no—
58C3×#341C2211yescharacteristic
59C3×#341C2211yescharacteristic
60#4554C2211yescharacteristic
61#4554C2211yescharacteristic
62#4554C2211yescharacteristic
63#4554C2211yescharacteristic
64C4×#137—31no—
65C4×AGL(1,ℤ/8)—31no—
66C2×#440—31no—
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#1778D311yescharacteristic, complemented
77C4×#101—31no—
78C4×#100—31no—
79C2×#235C2311yescharacteristic
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C22×Dic6C2311yescharacteristic
83C2×C3×M16C2311yescharacteristic
84C2×C3×M16C2311yescharacteristic
85C2×C3×M16C2311yescharacteristic
86C2×C3×M16C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C4×#67—21no—
91C4×#67—21no—
92C2×(C2×D3)⋊C4—21no—
93#926C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×Dic3⋊C4—21no—
97C2×C4×Dic3—21no—
98C2×#208C2311yescharacteristic
99C2×#208C2311yescharacteristic
100#909—22no—
101#909—22no—
102#908—22no—
103#908—22no—
104C2×Dic12—21no—
105C2×D24—21no—
106C2×#204—21no—
107C2×#204—22no—
108C2×#204—21no—
109C2×#204—22no—
110C2×#204—22no—
111C4×D12—21no—
112C4×D12—21no—
113C4×D12—21no—
114C4×D12—21no—
115C4×D12D411yescharacteristic
116C4×D12D411yescharacteristic
117#872—21no—
118C42×D3—21no—
119C4×Dic6—21no—
120C4×Dic6—21no—
121C4×Dic6—21no—
122C4×Dic6—21no—
123C4×Dic6D411yescharacteristic
124C4×Dic6D411yescharacteristic
125C3×#108—22no—
126C3×#108—22no—
127C3×#97—21no—
128C3×#97—21no—
129C3×#97D412yes—
130C3×#97D412yes—
131C3×C4×C8—21no—
132C3×C4×C8—21no—
133#824—22no—
134#824—22no—
135#823—22no—
136#823—22no—
137#821—21no—
138#821—21no—
139#821—22no—
140#821—21no—
141#821—21no—
142(C2×D3)⋊C8—22no—
143(C2×D3)⋊C8—22no—
144#818—22no—
145#818—21no—
146#818—21no—
147#817—21no—
148#817—21no—
149#816—21no—
150#816—21no—
151#816—22no—
152#816—21no—
153#816—21no—
154Dic3⋊C8—22no—
155Dic3⋊C8—22no—
156#803—21no—
157C4×C3⋊C8—21no—
158C2×#137—31no—
159C2×AGL(1,ℤ/8)—31no—
160C22×SD16—31no—
161C4×#41—31no—
162C2×C4×Q8—31no—
163C2×C4×D4—31no—
164#442—61no—
165#441—61no—
166#440—61no—
167#440—61no—
168#440—62no—
169#440—62no—
170C4×Q16—61no—
171C4×SD16—61no—
172C4×SD16—61no—
173C4×D8—61no—
174#433—62no—
175#432—62no—
176C2×#108C2×D311yescharacteristic
177C2×#108C2×D311yescharacteristic
178#428—31no—
179#428—31no—
180C2×#107—31no—
181#424—31no—
182#424—31no—
183#419—31no—
184#418—31no—
185#417—31no—
186#417—31no—
187C2×#103—31no—
188C2×#102—31no—
189C2×#101—31no—
190C2×#101—31no—
191C2×#100—31no—
192C2×#100—31no—
193#408—31no—
194#408—31no—
195C4×M16—31no—
196C4×M16C2×D311yescharacteristic
197C4×M16C2×D311yescharacteristic
198C2×#97C2×D311yescharacteristic
199#343—31no—
200#343—31no—
201#343—31no—
202#343—31no—
203#341C2×D311yescharacteristic
204#341C2×D311yescharacteristic
205C23×D3—21no—
206C22×C3×C4#4111yescharacteristic
207C22×C3×C4#4111yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C2×#67—21no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213#235—21no—
214#235—21no—
215#235—21no—
216#235—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12—21no—
221C2×D12—21no—
222C2×D12C2×D411yescharacteristic
223C2×D12#4112yes—
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3—21no—
227C2×C4×D3—41no—
228C2×C4×D3—41no—
229C2×C4×D3—82no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6C2×D411yescharacteristic
236C2×Dic6#4112yes—
237C3×M16—21no—
238C3×M16—21no—
239C3×M16—21no—
240C3×M16—21no—
241C3×M16—21no—
242C3×M16—21no—
243C3×M16#4112yes—
244C3×M16#4112yes—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8C2×D412yes—
254C2×C3×C8C2×D412yes—
255C3×C42C2×D411yescharacteristic
256C3×C42C2×D411yescharacteristic
257C3×C42C2×D411yescharacteristic
258C3×C42C2×D411yescharacteristic
259#217—21no—
260(C2×D3)⋊C4—21no—
261(C2×D3)⋊C4—21no—
262(C2×D3)⋊C4—41no—
263(C2×D3)⋊C4—41no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3C2×D411yescharacteristic
269C4⋊Dic3C2×D411yescharacteristic
270Dic3⋊C4—21no—
271Dic3⋊C4—21no—
272Dic3⋊C4—41no—
273Dic3⋊C4—41no—
274C4×Dic3—21no—
275C4×Dic3—21no—
276C4×Dic3—21no—
277C4×Dic3—41no—
278C4×Dic3—41no—
279#208—21no—
280#208—21no—
281#208—21no—
282#208—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285Dic12—24no—
286D24—24no—
287#204—28no—
288#204—24no—
289#204—24no—
290C2×#41—31no—
291C22×Q8—31no—
292C22×D4—31no—
293C23×C4—61no—
294#137—62no—
295#137—62no—
296AGL(1,ℤ/8)—62no—
297AGL(1,ℤ/8)—62no—
298C2×Q16—61no—
299C2×SD16—62no—
300C2×SD16—61no—
301C2×SD16—62no—
302C2×SD16—61no—
303C2×SD16—62no—
304C2×D8—61no—
305C2×M16—31no—
306C2×M16—31no—
307C2×M16C22×D311yescharacteristic
308C2×M16C22×D311yescharacteristic
309C2×M16C22×D311yescharacteristic
310C2×M16C22×D311yescharacteristic
311C22×C8C22×D311yescharacteristic
312C22×C8C22×D311yescharacteristic
313C4×Q8—31no—
314C4×Q8—31no—
315C4×Q8—61no—
316C4×Q8—61no—
317C4×Q8—61no—
318C4×Q8—61no—
319C4×D4—31no—
320C4×D4—31no—
321C4×D4—61no—
322C4×D4—61no—
323C4×D4—61no—
324C4×D4—61no—
325C4×D4—61no—
326C4×D4—61no—
327#117—31no—
328#117—61no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C42—61no—
334C2×C42—61no—
335C2×C42C22×D311yescharacteristic
336#108—22no—
337#108—22no—
338#107—61no—
339#107—61no—
340#107—62no—
341#106—61no—
342#106—61no—
343C4⋊C8—62no—
344C4⋊C8—62no—
345#103—61no—
346#103—61no—
347#103—61no—
348#103—61no—
349#103—62no—
350#102—61no—
351#102—61no—
352#102—61no—
353#102—61no—
354#102—62no—
355#101—62no—
356#101—62no—
357#100—62no—
358#100—62no—
359C22⋊C8—62no—
360C22⋊C8—62no—
361#97—21no—
362#97—21no—
363#97—61no—
364#97D1212yes—
365#97D1212yes—
366C4×C8—21no—
367C4×C8—21no—
368C4×C8—61no—
369C23×C3C2×#4111yescharacteristic
370C22×D3—21no—
371C22×D3—22no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—81no—
375C22×D3—81no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4#12111yescharacteristic
379C2×C3×C4#12111yescharacteristic
380C2×C3×C4#12111yescharacteristic
381C2×C3×C4#12111yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C22×D411yescharacteristic
384C2×C3×C4C2×#4111yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388#67—41no—
389#67—41no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—22no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—81no—
399C2×Dic3—81no—
400D12—22no—
401D12—21no—
402D12—22no—
403D12—21no—
404D12—41no—
405D12—41no—
406C4×D3—41no—
407C4×D3—41no—
408C4×D3—81no—
409C4×D3—81no—
410C4×D3—82no—
411C4×D3—82no—
412Dic6—22no—
413Dic6—21no—
414Dic6—22no—
415Dic6—21no—
416Dic6—41no—
417Dic6—41no—
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—61no—
427#41—61no—
428#41—61no—
429#41—61no—
430#41—61no—
431C2×Q8—31no—
432C2×Q8—32no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8—61no—
437C2×Q8—61no—
438C2×D4—31no—
439C2×D4—32no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—61no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—61no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—121no—
453C22×C4—121no—
454C22×C4—242no—
455C22×C4#23911yescharacteristic
456C22×C4#23911yescharacteristic
457C22×C4C23×D311yescharacteristic
458Q16—64no—
459SD16—68no—
460SD16—64no—
461SD16—64no—
462D8—64no—
463M16—21no—
464M16—21no—
465M16—21no—
466M16—21no—
467M16—21no—
468M16—21no—
469M16—61no—
470M16—61no—
471M16—61no—
472M16—61no—
473M16#23512yes—
474M16#23512yes—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8C2×D1212yes—
486C2×C8C2×D1212yes—
487C4⋊C4—31no—
488C4⋊C4—31no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—121no—
496C4⋊C4—121no—
497C22⋊C4—61no—
498C22⋊C4—61no—
499C22⋊C4—61no—
500C22⋊C4—121no—
501C22⋊C4—121no—
502C42—61no—
503C42—61no—
504C42—61no—
505C42—121no—
506C42—121no—
507C42C2×D1211yescharacteristic
508C42C2×D1211yescharacteristic
509C42D3×D411yescharacteristic
510C42D3×D411yescharacteristic
511C22×C3—21no—
512C22×C3—21no—
513C22×C3C2×#12111yescharacteristic
514C22×C3#52911yescharacteristic
515C22×C3#52911yescharacteristic
516C2×D3—41no—
517C2×D3—42no—
518C2×D3—41no—
519C2×D3—41no—
520C2×D3—82no—
521C2×D3—81no—
522C2×D3—81no—
523C2×D3—82no—
524C3×C4—21no—
525C3×C4—21no—
526C3×C4—21no—
527C3×C4—21no—
528C3×C4#52511yescharacteristic
529C3×C4#52511yescharacteristic
530C3×C4#52511yescharacteristic
531C3×C4#53211yescharacteristic
532Dic3—42no—
533Dic3—41no—
534Dic3—41no—
535Dic3—41no—
536Dic3—81no—
537Dic3—81no—
538C23—61no—
539C23—62no—
540C23—121no—
541C23—121no—
542C23—241no—
543C23—241no—
544C23C2×#23911yescharacteristic
545Q8—62no—
546Q8—61no—
547Q8—62no—
548Q8—61no—
549Q8—121no—
550Q8—121no—
551D4—62no—
552D4—61no—
553D4—62no—
554D4—61no—
555D4—121no—
556D4—121no—
557D4—121no—
558D4—121no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—62no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—241no—
572C2×C4—241no—
573C2×C4—242no—
574C2×C4—242no—
575C2×C4—241no—
576C2×C4—241no—
577C2×C4#89511yescharacteristic
578C2×C4#89511yescharacteristic
579C2×C4#89511yescharacteristic
580C2×C4#89511yescharacteristic
581C2×C4C22×D1211yescharacteristic
582C2×C4C2×#23511yescharacteristic
583C2×C4C2×#23511yescharacteristic
584C2×C4C2×D3×D411yescharacteristic
585C2×C4D3×#4111yescharacteristic
586C2×C4D3×#4111yescharacteristic
587C8—22no—
588C8—22no—
589C8—22no—
590C8—22no—
591C8—41no—
592C8—41no—
593C8—121no—
594C8—121no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3#239211yescharacteristic
598C2×C3#306811yescharacteristic
599C2×C3#306911yescharacteristic
600D3—81no—
601D3—82no—
602C22—21no—
603C22—21no—
604C22—121no—
605C22—122no—
606C22—121no—
607C22—121no—
608C22—242no—
609C22—242no—
610C22—241no—
611C22—241no—
612C22C2×#89511yescharacteristic
613C22#558011yescharacteristic
614C22#558011yescharacteristic
615C4—21no—
616C4—21no—
617C4—21no—
618C4—21no—
619C4—121no—
620C4—122no—
621C4—121no—
622C4—121no—
623C4—241no—
624C4—241no—
625C4#547711yescharacteristic
626C4#552411yescharacteristic
627C4#554711yescharacteristic
628C4#554711yescharacteristic
629C3#2372111yescharacteristic, complemented, Sylow
630C2—21no—
631C2—21no—
632C2—241no—
633C2—242no—
634C2#7377611yescharacteristic
635C2#7926111yescharacteristic
636C2#7926211yescharacteristic
subgroupG/H
centerC22C2×#895
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1778C2
socleC22×C3C2×#121
prime radicalC2×#908C22
hypercenter#1778D3
hypocenterC3#23721
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1778

Complements

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

NH ≅ G/Ntype
1. C4×#909C2semidirect
3. C2×#4710C2semidirect
4. #78489C2semidirect
5. C3×#1778C2semidirect
6. #70717C2semidirect
7. #70716C2semidirect
8. #70715C2semidirect
10. C4×#824C2semidirect
11. C4×#823C2semidirect
13. #68933C2semidirect
14. #68923C2semidirect
15. #68921C2semidirect
76. #1778D3semidirect
629. C3#23721semidirect

Maximal subgroups

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

subgroupindex
C4×#9092
C4×#9082
C2×#47102
#784892
C3×#17782
#707172
#707162
#707152
#707142
C4×#8242
C4×#8232
#689432
#689332
#689232
#689212
#237213