Finite Group Databasebeta

← #10824926

Subgroups — #10824926

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×C4×#137C211yesmaximal, characteristic, complemented
2C4×#949C211yesmaximal, characteristic
3#82514C211yesmaximal, characteristic, complemented
4C2×#5026C211yesmaximal, characteristic, complemented
5#72107C211yesmaximal, characteristic
6#72105C211yesmaximal, characteristic, complemented
7C4×#836C211yesmaximal, characteristic, complemented
8#70676C211yesmaximal, characteristic, complemented
9#70670C211yesmaximal, characteristic, complemented
10#70339C211yesmaximal, characteristic, complemented
11#70337C211yesmaximal, characteristic, complemented
12#70307C211yesmaximal, characteristic
13#70301C211yesmaximal, characteristic, complemented
14#69873C211yesmaximal, characteristic, complemented
15C4×#823C211yesmaximal, characteristic, complemented
16#23721—31nomaximal, Sylow
17C2×C3×#137C2211yescharacteristic
18C3×C4×#41C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×#949C2211yescharacteristic
21C22×#215C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C3×#441—21no—
24C3×#440—21no—
25C3×C4×Q16—21no—
26C3×C4×SD16—21no—
27C3×#428C2211yescharacteristic
28C3×#419C2211yescharacteristic
29C3×#417C2211yescharacteristic
30C3×C4×M16C2211yescharacteristic
31#5234C2211yescharacteristic
32#5234C2211yescharacteristic
33#5232C2211yescharacteristic
34C2×#836C2211yescharacteristic
35C2×#836C2211yescharacteristic
36C2×#835C2211yescharacteristic
37C2×#823C2211yescharacteristic
38C2×#823C2211yescharacteristic
39C2×#822C2211yescharacteristic
40C2×#822C2211yescharacteristic
41#5026—21no—
42#5026—22no—
43#5026—22no—
44C4×#215—21no—
45#5014—21no—
46C4×#213—21no—
47#5001C2211yescharacteristic
48#5000—22no—
49#4999C2211yescharacteristic
50#4999C2211yescharacteristic
51#4995C2211yescharacteristic
52C2×#809C2211yescharacteristic
53C2×#807C2211yescharacteristic
54#4926—22no—
55C4×#208C2211yescharacteristic
56C4×#208C2211yescharacteristic
57C2×#803C2211yescharacteristic
58#4558C2211yescharacteristic
59#4558C2211yescharacteristic
60#4554C2211yescharacteristic
61#4554C2211yescharacteristic
62#4530C2211yescharacteristic
63#4530C2211yescharacteristic
64C4×#137D311yescharacteristic, complemented
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#1778—31no—
77C4×#101—31no—
78C4×#100—31no—
79C2×C3×#41C2311yescharacteristic
80C22×C3×Q8C2311yescharacteristic
81C22×D12C2311yescharacteristic
82C22×C4×D3—21no—
83C3×#137—22no—
84C3×#137—22no—
85C2×C3×Q16—21no—
86C2×C3×SD16—21no—
87C2×C3×M16C2311yescharacteristic
88C2×C3×M16C2311yescharacteristic
89C3×C4×Q8—21no—
90C3×C4×Q8—21no—
91C3×C4×Q8—21no—
92C3×C4×Q8—21no—
93C3×C4×Q8D411yescharacteristic
94C3×C4×Q8D411yescharacteristic
95C3×C4×D4—21no—
96C3×C4×D4—21no—
97C3×C4×D4D411yescharacteristic
98C3×C4×D4D411yescharacteristic
99C3×#117—21no—
100C3×#117C2311yescharacteristic
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C42—21no—
104C2×C3×C42—21no—
105C2×C3×C42C2311yescharacteristic
106#949—22no—
107#949—22no—
108C2×#215—21no—
109C2×#215—22no—
110C2×#215—22no—
111C2×#215—22no—
112C2×#213—21no—
113C2×(C2×D3)⋊C4—21no—
114C2×C4⋊Dic3C2311yescharacteristic
115C2×#208C2311yescharacteristic
116C2×#208C2311yescharacteristic
117C2×#208C2311yescharacteristic
118C2×#208C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C4×D12—21no—
122C4×D12—21no—
123C4×D12—21no—
124C4×D12—21no—
125C3×#107—21no—
126C3×#106—21no—
127C3×#103—21no—
128C3×#103—21no—
129C3×#103—21no—
130C3×#102—21no—
131C3×#97—21no—
132C3×C4×C8—21no—
133#836—22no—
134#836—22no—
135#835—21no—
136#835—22no—
137#832—21no—
138#830—22no—
139#830—22no—
140#823—22no—
141#823—22no—
142#822—22no—
143#822—22no—
144#809—21no—
145#809—21no—
146#809—22no—
147#808—21no—
148#807—21no—
149#807—22no—
150C4⋊(C3⋊C8)—22no—
151C4⋊(C3⋊C8)—22no—
152#803—21no—
153#803—21no—
154#803D412yes—
155#803D412yes—
156C4×C3⋊C8—21no—
157C4×C3⋊C8—21no—
158C2×#137C2×D311yescharacteristic
159C2×AGL(1,ℤ/8)—31no—
160C22×SD16—31no—
161C4×#41C2×D311yescharacteristic
162C2×C4×Q8C2×D311yescharacteristic
163C2×C4×D4—31no—
164#442—61no—
165#441—21no—
166#440—21no—
167#440—62no—
168#440—62no—
169#440—61no—
170C4×Q16—21no—
171C4×SD16—21no—
172C4×SD16—61no—
173C4×D8—61no—
174#433—62no—
175#432—62no—
176C2×#108—31no—
177C2×#108—31no—
178#428—31no—
179#428C2×D311yescharacteristic
180C2×#107—31no—
181#424—31no—
182#424—31no—
183#419C2×D311yescharacteristic
184#418—31no—
185#417—31no—
186#417C2×D311yescharacteristic
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×M16—31no—
197C4×M16C2×D311yescharacteristic
198C2×#97—31no—
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×Q8—21no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8C2×D411yescharacteristic
216C2×C3×Q8#4112yes—
217C2×C3×D4—21no—
218C2×C3×D4C2×D411yescharacteristic
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4#4111yescharacteristic
225C22×C3×C4#4111yescharacteristic
226C22×C3×C4C2411yescharacteristic
227C22×Dic3—21no—
228C2×D12—21no—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12—21no—
232C2×D12—21no—
233C2×D12#4112yes—
234C2×C4×D3—41no—
235C2×C4×D3—41no—
236C2×C4×D3—82no—
237C3×Q16—24no—
238C3×SD16—24no—
239C3×M16—21no—
240C3×M16—21no—
241C3×M16—21no—
242C3×M16#4112yes—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—41no—
250C3×C4⋊C4—41no—
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×C42—41no—
260C3×C42—41no—
261C3×C42C2×D411yescharacteristic
262C3×C42C2×D411yescharacteristic
263C3×C42C2×D411yescharacteristic
264C3×C42C2×D411yescharacteristic
265#215—28no—
266#215—24no—
267#213—24no—
268(C2×D3)⋊C4—41no—
269(C2×D3)⋊C4—41no—
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#4112yes—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8C2×D412yes—
289C2×C3⋊C8C2×D412yes—
290C2×#41C22×D311yescharacteristic
291C22×Q8C22×D311yescharacteristic
292C22×D4—31no—
293C23×C4—61no—
294#137—22no—
295#137—22no—
296AGL(1,ℤ/8)—62no—
297AGL(1,ℤ/8)—62no—
298C2×Q16—21no—
299C2×SD16—21no—
300C2×SD16—62no—
301C2×SD16—62no—
302C2×SD16—62no—
303C2×SD16—61no—
304C2×D8—61no—
305C2×M16—31no—
306C2×M16—31no—
307C2×M16—31no—
308C2×M16—31no—
309C2×M16C22×D311yescharacteristic
310C2×M16C22×D311yescharacteristic
311C22×C8—31no—
312C22×C8—31no—
313C4×Q8—21no—
314C4×Q8—21no—
315C4×Q8—21no—
316C4×Q8—21no—
317C4×Q8#6711yescharacteristic
318C4×Q8#6711yescharacteristic
319C4×D4—21no—
320C4×D4—21no—
321C4×D4—61no—
322C4×D4—61no—
323C4×D4—61no—
324C4×D4—61no—
325C4×D4#6711yescharacteristic
326C4×D4#6711yescharacteristic
327#117—21no—
328#117C22×D311yescharacteristic
329C2×C4⋊C4—21no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4C22×D311yescharacteristic
332C2×C22⋊C4—61no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42C22×D311yescharacteristic
336#108—62no—
337#108—62no—
338#107—21no—
339#107—62no—
340#107—61no—
341#106—21no—
342#106—61no—
343C4⋊C8—62no—
344C4⋊C8—62no—
345#103—21no—
346#103—21no—
347#103—21no—
348#103—62no—
349#103—61no—
350#102—21no—
351#102—62no—
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—21no—
362#97—32no—
363#97—32no—
364#97—61no—
365#97—61no—
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—22no—
377C3×Q8—21no—
378C3×Q8—22no—
379C3×Q8—21no—
380C3×Q8—41no—
381C3×Q8—41no—
382C3×D4—21no—
383C3×D4—21no—
384C3×D4—41no—
385C3×D4—41no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—22no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—81no—
398C2×C3×C4—81no—
399C2×C3×C4—81no—
400C2×C3×C4#12111yescharacteristic
401C2×C3×C4#12111yescharacteristic
402C2×C3×C4#12111yescharacteristic
403C2×C3×C4#12111yescharacteristic
404C2×C3×C4C22×D411yescharacteristic
405C2×C3×C4C22×D411yescharacteristic
406C2×C3×C4C2×#4111yescharacteristic
407C2×C3×C4C2×#4111yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C4C2×#4111yescharacteristic
410C2×Dic3—41no—
411C2×Dic3—81no—
412D12—22no—
413D12—22no—
414D12—41no—
415D12—41no—
416C4×D3—82no—
417C4×D3—82no—
418C3×C8—22no—
419C3×C8—41no—
420C3⋊C8—22no—
421C3⋊C8—22no—
422C3⋊C8—22no—
423C3⋊C8—41no—
424C3⋊C8—41no—
425C3⋊C8—41no—
426C24—61no—
427#41—21no—
428#41—21no—
429#41—21no—
430#41—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8#23512yes—
437C2×Q8C2×#6711yescharacteristic
438C2×D4—21no—
439C2×D4—32no—
440C2×D4—61no—
441C2×D4—61no—
442C2×D4—61no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4C2×#6711yescharacteristic
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—61no—
452C22×C4—121no—
453C22×C4—121no—
454C22×C4—242no—
455C22×C4#23711yescharacteristic
456C22×C4#23711yescharacteristic
457C22×C4C23×D311yescharacteristic
458Q16—24no—
459SD16—24no—
460SD16—68no—
461SD16—64no—
462D8—64no—
463M16—21no—
464M16—21no—
465M16—21no—
466M16—32no—
467M16—61no—
468M16—61no—
469M16—61no—
470M16—61no—
471M16—61no—
472M16—61no—
473M16—61no—
474M16#23512yes—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—32no—
478C2×C8—32no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4C2×#6711yescharacteristic
496C4⋊C4C2×#6711yescharacteristic
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—21no—
500C22⋊C4—121no—
501C22⋊C4—121no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—41no—
506C42—41no—
507C42D3×D411yescharacteristic
508C42D3×D411yescharacteristic
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—42no—
530C3×C4—41no—
531C3×C4—41no—
532C3×C4—81no—
533C3×C4#52511yescharacteristic
534C3×C4#52511yescharacteristic
535C3×C4#52511yescharacteristic
536C3×C4#53211yescharacteristic
537Dic3—81no—
538C23—21no—
539C23—62no—
540C23—121no—
541C23—121no—
542C23—241no—
543C23—241no—
544C23C2×#23711yescharacteristic
545Q8—22no—
546Q8—21no—
547Q8—22no—
548Q8—21no—
549Q8—41no—
550Q8—41no—
551D4—21no—
552D4—21no—
553D4—41no—
554D4—41no—
555D4—62no—
556D4—62no—
557D4—121no—
558D4—121no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—121no—
574C2×C4—242no—
575C2×C4—242no—
576C2×C4—241no—
577C2×C4#93911yescharacteristic
578C2×C4#93911yescharacteristic
579C2×C4#93911yescharacteristic
580C2×C4#93911yescharacteristic
581C2×C4C2×#23511yescharacteristic
582C2×C4C2×#23511yescharacteristic
583C2×C4C2×D3×D411yescharacteristic
584C2×C4D3×#4111yescharacteristic
585C2×C4D3×#4111yescharacteristic
586C2×C4C22×#6711yescharacteristic
587C8—22no—
588C8—41no—
589C8—62no—
590C8—62no—
591C8—62no—
592C8—121no—
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#553811yescharacteristic
613C22#555711yescharacteristic
614C22C2×#93911yescharacteristic
615C4—21no—
616C4—21no—
617C4—21no—
618C4—21no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—42no—
623C4—81no—
624C4—241no—
625C4#554711yescharacteristic
626C4#554811yescharacteristic
627C4#579111yescharacteristic
628C4#582811yescharacteristic
629C3#2372111yescharacteristic, complemented, Sylow
630C2—21no—
631C2—21no—
632C2—81no—
633C2—242no—
634C2#8234511yescharacteristic
635C2#8397811yescharacteristic
636C2#8400211yescharacteristic
subgroupG/H
centerC22C2×#939
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#137C2
socleC22×C3C2×#121
prime radicalC2×#949C22
hypercenterC4×#137D3
hypocenterC3#23721
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#137

Complements

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

NH ≅ G/Ntype
1. C3×C4×#137C2semidirect
3. #82514C2semidirect
4. C2×#5026C2semidirect
6. #72105C2semidirect
7. C4×#836C2semidirect
8. #70676C2semidirect
9. #70670C2semidirect
10. #70339C2semidirect
11. #70337C2semidirect
13. #70301C2semidirect
14. #69873C2semidirect
15. C4×#823C2semidirect
64. C4×#137D3semidirect
629. C3#23721semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#1372
C4×#9492
#825142
C2×#50262
#721072
#721052
C4×#8362
#706762
#706702
#703392
#703372
#703072
#703012
#698732
C4×#8232
#237213