Finite Group Databasebeta

← #10824918

Subgroups — #10824918

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
1C3×#2923C211yesmaximal, characteristic, complemented
2C4×#951C211yesmaximal, characteristic, complemented
3C4×#949C211yesmaximal, characteristic
4C2×#4710C211yesmaximal, characteristic, complemented
5#70544C211yesmaximal, characteristic, complemented
6#70542C211yesmaximal, characteristic, complemented
7#70541C211yesmaximal, characteristic, complemented
8#70532C211yesmaximal, characteristic, complemented
9#70007C211yesmaximal, characteristic
10#70004C211yesmaximal, characteristic, complemented
11#70003C211yesmaximal, characteristic
12#69994C211yesmaximal, characteristic, complemented
13C4×#824C211yesmaximal, characteristic, complemented
14C4×#823C211yesmaximal, characteristic, complemented
15#69840C211yesmaximal, characteristic, complemented
16#23721—31nomaximal, Sylow
17C3×C4×#41C2211yescharacteristic
18C2×#951C2211yescharacteristic
19C2×#949C2211yescharacteristic
20C22×#204C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C3×#433—22no—
24C3×#432—22no—
25C3×#424C2211yescharacteristic
26C3×#424C2211yescharacteristic
27C3×#408C2211yescharacteristic
28C3×#408C2211yescharacteristic
29C3×C4×M16C2211yescharacteristic
30C2×C3×#97C2211yescharacteristic
31#5232C2211yescharacteristic
32#5232C2211yescharacteristic
33C2×#824C2211yescharacteristic
34C2×#824C2211yescharacteristic
35C2×#823C2211yescharacteristic
36C2×#823C2211yescharacteristic
37C2×#821C2211yescharacteristic
38C2×#819C2211yescharacteristic
39C2×#819C2211yescharacteristic
40C2×#818C2211yescharacteristic
41C2×#816C2211yescharacteristic
42#5030—21no—
43C4×#216—21no—
44#5026—21no—
45C4×#215—21no—
46#5018—21no—
47C4×#214—21no—
48#5014—21no—
49C4×#213—21no—
50#5004C2211yescharacteristic
51#5001C2211yescharacteristic
52#4995C2211yescharacteristic
53#4995C2211yescharacteristic
54C4×#208C2211yescharacteristic
55C4×#208C2211yescharacteristic
56#4710—22no—
57#4710—22no—
58#4554C2211yescharacteristic
59#4554C2211yescharacteristic
60#4554C2211yescharacteristic
61#4554C2211yescharacteristic
62#4552C2211yescharacteristic
63#4552C2211yescharacteristic
64C4×#137—31no—
65C4×AGL(1,ℤ/8)—31no—
66C2×#440—31no—
67#2923D311yescharacteristic, complemented
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—
79C2×C3×#41C2311yescharacteristic
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C22×Dic6C2311yescharacteristic
83C2×C3×M16C2311yescharacteristic
84C2×C3×M16C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×C4×Q8D411yescharacteristic
88C3×C4×Q8D411yescharacteristic
89C3×C4×D4—21no—
90C3×C4×D4—21no—
91C3×C4×D4D411yescharacteristic
92C3×C4×D4D411yescharacteristic
93C3×#117—21no—
94C3×#117C2311yescharacteristic
95C2×C3×C42—21no—
96C2×C3×C42C2311yescharacteristic
97#951—22no—
98#951—22no—
99#949—22no—
100#949—22no—
101C2×#216—21no—
102C2×#215—21no—
103C2×#214—21no—
104C2×#213—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×C4×Dic3—21no—
110C2×#208C2311yescharacteristic
111C2×#208C2311yescharacteristic
112C2×#208C2311yescharacteristic
113C2×#208C2311yescharacteristic
114C2×#204—22no—
115C2×#204—22no—
116C2×#204—22no—
117C4×D12—21no—
118C4×D12—21no—
119C4×D12—21no—
120C4×D12—21no—
121C4×Dic6—21no—
122C4×Dic6—21no—
123C4×Dic6—21no—
124C4×Dic6—21no—
125C3×C4⋊C8—22no—
126C3×C4⋊C8—22no—
127C3×C22⋊C8—22no—
128C3×C22⋊C8—22no—
129C3×#97—21no—
130C3×#97D412yes—
131C3×#97D412yes—
132C3×C4×C8—21no—
133#835—21no—
134#835—21no—
135#832—21no—
136#832—21no—
137#824—22no—
138#824—22no—
139#823—22no—
140#823—22no—
141#821—22no—
142#819—22no—
143#819—22no—
144#818—22no—
145#816—22no—
146#810—21no—
147#810—21no—
148#809—21no—
149#809—21no—
150#808—21no—
151#808—21no—
152#807—21no—
153#807—21no—
154#803—21no—
155#803—21no—
156C4×C3⋊C8—21no—
157C4×C3⋊C8—21no—
158C2×#137—31no—
159C2×AGL(1,ℤ/8)—31no—
160C22×SD16—31no—
161C4×#41C2×D311yescharacteristic
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—22no—
175#432—22no—
176C2×#108—31no—
177C2×#108—31no—
178#428—31no—
179#428—31no—
180C2×#107—31no—
181#424C2×D311yescharacteristic
182#424C2×D311yescharacteristic
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#408C2×D311yescharacteristic
194#408C2×D311yescharacteristic
195C4×M16—31no—
196C4×M16—31no—
197C4×M16C2×D311yescharacteristic
198C2×#97C2×D311yescharacteristic
199#343—31no—
200#343—31no—
201#343—31no—
202#343—31no—
203#341—31no—
204#341—31no—
205C23×D3—21no—
206C3×#41—21no—
207C3×#41—21no—
208C3×#41—21no—
209C3×#41—21no—
210C2×C3×Q8C2×D411yescharacteristic
211C2×C3×D4—21no—
212C2×C3×D4C2×D411yescharacteristic
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4#4111yescharacteristic
217C22×C3×C4#4111yescharacteristic
218C22×C3×C4C2411yescharacteristic
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12#4112yes—
228C2×C4×D3—41no—
229C2×C4×D3—41no—
230C2×C4×D3—82no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6#4112yes—
237C3×M16—21no—
238C3×M16—21no—
239C3×M16—21no—
240C3×M16—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8C2×D412yes—
248C2×C3×C8C2×D412yes—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4C2×D411yescharacteristic
252C3×C4⋊C4C2×D411yescharacteristic
253C3×C22⋊C4—21no—
254C3×C22⋊C4—21no—
255C3×C22⋊C4—21no—
256C3×C42—21no—
257C3×C42—21no—
258C3×C42—21no—
259C3×C42C2×D411yescharacteristic
260C3×C42C2×D411yescharacteristic
261C3×C42C2×D411yescharacteristic
262C3×C42C2×D411yescharacteristic
263#216—24no—
264#215—24no—
265#214—24no—
266#213—24no—
267(C2×D3)⋊C4—41no—
268(C2×D3)⋊C4—41no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273Dic3⋊C4—41no—
274Dic3⋊C4—41no—
275C4×Dic3—41no—
276C4×Dic3—41no—
277#208—21no—
278#208—21no—
279#208—21no—
280#208—21no—
281#208—21no—
282#208—21no—
283#208#4112yes—
284#208#4112yes—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289#204—28no—
290C2×#41C22×D311yescharacteristic
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—62no—
301C2×SD16—61no—
302C2×SD16—61no—
303C2×SD16—62no—
304C2×D8—61no—
305C2×M16—31no—
306C2×M16—31no—
307C2×M16—31no—
308C2×M16—31no—
309C2×M16C22×D311yescharacteristic
310C2×M16C22×D311yescharacteristic
311C22×C8C22×D311yescharacteristic
312C22×C8C22×D311yescharacteristic
313C4×Q8—61no—
314C4×Q8—61no—
315C4×Q8—61no—
316C4×Q8—61no—
317C4×Q8D1211yescharacteristic
318C4×Q8D1211yescharacteristic
319C4×D4—21no—
320C4×D4—21no—
321C4×D4—61no—
322C4×D4—61no—
323C4×D4—61no—
324C4×D4—61no—
325C4×D4D1211yescharacteristic
326C4×D4D1211yescharacteristic
327#117—21no—
328#117C22×D311yescharacteristic
329C2×C4⋊C4—31no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C42—21no—
334C2×C42—61no—
335C2×C42C22×D311yescharacteristic
336#108—62no—
337#108—62no—
338#107—62no—
339#107—61no—
340#107—61no—
341#106—61no—
342#106—61no—
343C4⋊C8—22no—
344C4⋊C8—22no—
345#103—61no—
346#103—61no—
347#103—62no—
348#103—61no—
349#103—61no—
350#102—61no—
351#102—61no—
352#102—62no—
353#102—61no—
354#102—61no—
355#101—62no—
356#101—62no—
357#100—62no—
358#100—62no—
359C22⋊C8—22no—
360C22⋊C8—22no—
361#97—21no—
362#97—61no—
363#97—61no—
364#97#6712yes—
365#97#6712yes—
366C4×C8—21no—
367C4×C8—61no—
368C4×C8—61no—
369C23×C3—21no—
370C23×C3C2×#4111yescharacteristic
371C22×D3—22no—
372C22×D3—41no—
373C22×D3—41no—
374C22×D3—81no—
375C22×D3—81no—
376C3×Q8—21no—
377C3×Q8—21no—
378C3×D4—21no—
379C3×D4—21no—
380C3×D4—41no—
381C3×D4—41no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—81no—
390C2×C3×C4—81no—
391C2×C3×C4#12111yescharacteristic
392C2×C3×C4#12111yescharacteristic
393C2×C3×C4#12111yescharacteristic
394C2×C3×C4#12111yescharacteristic
395C2×C3×C4C22×D411yescharacteristic
396C2×C3×C4C22×D411yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×Dic3—22no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—81no—
407C2×Dic3—81no—
408D12—22no—
409D12—22no—
410D12—41no—
411D12—41no—
412C4×D3—82no—
413C4×D3—82no—
414Dic6—22no—
415Dic6—22no—
416Dic6—41no—
417Dic6—41no—
418C3×C8—22no—
419C3×C8—22no—
420C3×C8—41no—
421C3×C8—41no—
422C3⋊C8—22no—
423C3⋊C8—22no—
424C3⋊C8—41no—
425C3⋊C8—41no—
426C24—61no—
427#41—21no—
428#41—21no—
429#41—21no—
430#41—21no—
431C2×Q8—32no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8—61no—
437C2×Q8C2×D1211yescharacteristic
438C2×D4—21no—
439C2×D4—32no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—61no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4C2×D1211yescharacteristic
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—61no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—121no—
453C22×C4—121no—
454C22×C4—242no—
455C22×C4#23511yescharacteristic
456C22×C4#23511yescharacteristic
457C22×C4C23×D311yescharacteristic
458Q16—64no—
459SD16—68no—
460SD16—64no—
461SD16—64no—
462D8—64no—
463M16—21no—
464M16—21no—
465M16—21no—
466M16—21no—
467M16—32no—
468M16—32no—
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—21no—
480C2×C8—21no—
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—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—121no—
494C4⋊C4—121no—
495C4⋊C4C2×D1211yescharacteristic
496C4⋊C4C2×D1211yescharacteristic
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—21no—
500C22⋊C4—121no—
501C22⋊C4—121no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—121no—
506C42—121no—
507C42C2×D1211yescharacteristic
508C42C2×D1211yescharacteristic
509C42C2×#6711yescharacteristic
510C42C2×#6711yescharacteristic
511C22×C3—21no—
512C22×C3—21no—
513C22×C3—41no—
514C22×C3—41no—
515C22×C3—41no—
516C22×C3C2×#12111yescharacteristic
517C22×C3#52911yescharacteristic
518C22×C3#52911yescharacteristic
519C2×D3—42no—
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—41no—
529C3×C4—41no—
530C3×C4—41no—
531C3×C4#52511yescharacteristic
532C3×C4#52511yescharacteristic
533C3×C4#52511yescharacteristic
534C3×C4#53211yescharacteristic
535Dic3—42no—
536Dic3—81no—
537Dic3—81no—
538C23—21no—
539C23—62no—
540C23—121no—
541C23—121no—
542C23—241no—
543C23—241no—
544C23C2×#23511yescharacteristic
545Q8—21no—
546Q8—21no—
547Q8—62no—
548Q8—62no—
549Q8—121no—
550Q8—121no—
551D4—21no—
552D4—21no—
553D4—41no—
554D4—41no—
555D4—62no—
556D4—62no—
557D4—121no—
558D4—121no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—62no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—242no—
574C2×C4—242no—
575C2×C4—241no—
576C2×C4—241no—
577C2×C4#93011yescharacteristic
578C2×C4#93011yescharacteristic
579C2×C4#93011yescharacteristic
580C2×C4#93011yescharacteristic
581C2×C4C22×D1211yescharacteristic
582C2×C4D3×#4111yescharacteristic
583C2×C4D3×#4111yescharacteristic
584C2×C4D3×#4111yescharacteristic
585C2×C4D3×#4111yescharacteristic
586C2×C4C22×#6711yescharacteristic
587C8—22no—
588C8—22no—
589C8—41no—
590C8—41no—
591C8—62no—
592C8—62no—
593C8—121no—
594C8—121no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3—81no—
598C2×C3#239211yescharacteristic
599C2×C3#306811yescharacteristic
600C2×C3#306911yescharacteristic
601D3—82no—
602C22—21no—
603C22—21no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—122no—
608C22—242no—
609C22—242no—
610C22—241no—
611C22—241no—
612C22#552811yescharacteristic
613C22#552811yescharacteristic
614C22C2×#93011yescharacteristic
615C4—21no—
616C4—21no—
617C4—21no—
618C4—21no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—122no—
623C4—241no—
624C4—241no—
625C4#552411yescharacteristic
626C4#552411yescharacteristic
627C4#582811yescharacteristic
628C4#583211yescharacteristic
629C3#2372111yescharacteristic, complemented, Sylow
630C2—21no—
631C2—21no—
632C2—81no—
633C2—242no—
634C2#8227311yescharacteristic
635C2#8381411yescharacteristic
636C2#8381611yescharacteristic
subgroupG/H
centerC22C2×#930
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2923C2
socleC22×C3C2×#121
prime radicalC2×#949C22
hypercenter#2923D3
hypocenterC3#23721
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2923

Complements

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

NH ≅ G/Ntype
1. C3×#2923C2semidirect
2. C4×#951C2semidirect
4. C2×#4710C2semidirect
5. #70544C2semidirect
6. #70542C2semidirect
7. #70541C2semidirect
8. #70532C2semidirect
10. #70004C2semidirect
12. #69994C2semidirect
13. C4×#824C2semidirect
14. C4×#823C2semidirect
15. #69840C2semidirect
67. #2923D3semidirect
629. C3#23721semidirect

Maximal subgroups

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

subgroupindex
C3×#29232
C4×#9512
C4×#9492
C2×#47102
#705442
#705422
#705412
#705322
#700072
#700042
#700032
#699942
C4×#8242
C4×#8232
#698402
#237213