Finite Group Databasebeta

← #10829795

Subgroups — #10829795

Order 768.

1074 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85850C211yesmaximal, characteristic
2C3×C4×#137C211yesmaximal, characteristic, complemented
3#82478C211yesmaximal, characteristic, complemented
4#82441C211yesmaximal, characteristic, complemented
5#72110C212yesmaximal, complemented
6#71646C211yesmaximal, characteristic, complemented
7#70680C212yesmaximal, complemented
8#70348C212yesmaximal, complemented
9#70311C212yesmaximal, complemented
10#69879C211yesmaximal, characteristic, complemented
11#69867C211yesmaximal, characteristic, complemented
12#24173—31nomaximal, Sylow
13C2×C3×#137C2211yescharacteristic
14C3×C4×#41C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16#5833C2212yes—
17C2×#942C2212yes—
18#5584—21no—
19#5579—21no—
20#5565—22no—
21#5563—21no—
22#5552—22no—
23#5551—21no—
24#5534—22no—
25#5523C2212yes—
26#5477—21no—
27#5477C2211yescharacteristic
28#5470—21no—
29C2×#869C2211yescharacteristic
30C3×#441—21no—
31C3×#440—21no—
32C3×C4×Q16—21no—
33C3×C4×SD16—21no—
34C3×#428C2211yescharacteristic
35C3×#419C2211yescharacteristic
36C3×#417C2211yescharacteristic
37C3×C4×M16C2211yescharacteristic
38C2×#837C2212yes—
39#5229C2212yes—
40C2×#825C2212yes—
41C2×#825C2212yes—
42#5031—21no—
43#5028—22no—
44#5027—21no—
45#4979—22no—
46#4975C2212yes—
47#4966C2212yes—
48C4×C4⋊Dic3C2211yescharacteristic
49#4928—22no—
50#4924C2211yescharacteristic
51#4554C2212yes—
52#4554C2212yes—
53#4532C2212yes—
54#3511—31no—
55#3037—31no—
56C4×#137D311yescharacteristic, complemented
57#2567—31no—
58#2039—32no—
59#2007—32no—
60#1997—31no—
61#1980—31no—
62#1882—32no—
63#1859—32no—
64#1806—31no—
65C2×C3×#41C2311yescharacteristic
66C22×C3×Q8C2311yescharacteristic
67C2×#235—22no—
68C2×#235C2312yes—
69C22×Dic6D412yes—
70C3×#137—22no—
71C3×#137—22no—
72C2×C3×Q16—21no—
73C2×C3×SD16—21no—
74C2×C3×M16C2311yescharacteristic
75C2×C3×M16C2311yescharacteristic
76C3×C4×Q8—21no—
77C3×C4×Q8—21no—
78C3×C4×Q8—21no—
79C3×C4×Q8—21no—
80C3×C4×Q8D411yescharacteristic
81C3×C4×Q8D411yescharacteristic
82C3×C4×D4—21no—
83C3×C4×D4—21no—
84C3×C4×D4D411yescharacteristic
85C3×C4×D4D411yescharacteristic
86C3×#117—21no—
87C3×#117C2311yescharacteristic
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C42—21no—
91C2×C3×C42—21no—
92C2×C3×C42C2311yescharacteristic
93#946—22no—
94C2×#216—22no—
95#942—24no—
96#942—42no—
97C2×#215—22no—
98#939—24no—
99#938—22no—
100#930—22no—
101#930—21no—
102#930—21no—
103#930—22no—
104#926—22no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3D412yes—
107C2×C4⋊Dic3C2312yes—
108#924—22no—
109#924—22no—
110C2×Dic3⋊C4—24no—
111C2×C4×Dic3C2312yes—
112C2×#208C2312yes—
113#898—24no—
114#896—24no—
115#892—22no—
116D3×C4⋊C4—22no—
117#885—24no—
118#883—24no—
119#67⋊C4—24no—
120#875—22no—
121#875D412yes—
122#874D411yescharacteristic
123C4×D12—22no—
124C4×D12—22no—
125C4×D12—21no—
126C4×D12—21no—
127#870—22no—
128#869—21no—
129#869—21no—
130#869—21no—
131#869—21no—
132#869D411yescharacteristic
133C4×Dic6—22no—
134C4×Dic6—22no—
135C3×#107—21no—
136C3×#106—21no—
137C3×#103—21no—
138C3×#103—21no—
139C3×#103—21no—
140C3×#102—21no—
141C3×#97—21no—
142C3×C4×C8—21no—
143#837—42no—
144C3⋊#95—21no—
145C3⋊#95—24no—
146#825—24no—
147#825—42no—
148#810—22no—
149#809—22no—
150C4⋊(C3⋊C8)—21no—
151C4⋊(C3⋊C8)—21no—
152C2×#137—32no—
153C2×#137C2×D311yescharacteristic
154#556—61no—
155#555—61no—
156#554—61no—
157#553—62no—
158#548—62no—
159#547—61no—
160#542—62no—
161#537—32no—
162#533—61no—
163#532—31no—
164#532—61no—
165C2×C4⋊Q8—31no—
166#520—32no—
167C4×#41C2×D311yescharacteristic
168C2×C4×Q8C2×D311yescharacteristic
169#464—62no—
170#462—61no—
171#460—61no—
172#441—21no—
173#440—21no—
174C4×Q16—21no—
175C4×SD16—21no—
176#428C2×D311yescharacteristic
177#423—31no—
178C2×#104—32no—
179C2×#104—32no—
180C2×#104—32no—
181#419C2×D311yescharacteristic
182#417—32no—
183#417C2×D311yescharacteristic
184C4×M16C2×D311yescharacteristic
185#395—62no—
186#391—32no—
187#389—62no—
188#384—32no—
189C4×C4⋊C4—31no—
190#343—32no—
191#343—32no—
192#339—32no—
193C3×#41—21no—
194C3×#41—21no—
195C3×#41—21no—
196C3×#41—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8C2×D411yescharacteristic
203C2×C3×Q8#4112yescomplemented
204C2×C3×D4—21no—
205C2×C3×D4C2×D411yescharacteristic
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2411yescharacteristic
214C2×#67—22no—
215C2×#67—24no—
216C22×Dic3—22no—
217C22×Dic3C2×D414yes—
218#235—22no—
219#235—22no—
220#235—22no—
221#235—22no—
222#235—84no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12C2×D412yes—
226C2×C4×D3—22no—
227C2×C4×D3—24no—
228C2×Dic6—22no—
229C2×Dic6—22no—
230C2×Dic6—82no—
231C2×Dic6C2×D412yes—
232C3×Q16—24no—
233C3×SD16—24no—
234C3×M16—21no—
235C3×M16—21no—
236C3×M16—21no—
237C3×M16#4112yescomplemented
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—41no—
245C3×C4⋊C4—41no—
246C3×C4⋊C4C2×D411yescharacteristic
247C3×C4⋊C4C2×D411yescharacteristic
248C3×C22⋊C4—21no—
249C3×C22⋊C4—21no—
250C3×C22⋊C4—21no—
251C3×C42—21no—
252C3×C42—21no—
253C3×C42—21no—
254C3×C42—41no—
255C3×C42—41no—
256C3×C42C2×D411yescharacteristic
257C3×C42C2×D411yescharacteristic
258C3×C42C2×D411yescharacteristic
259C3×C42C2×D411yescharacteristic
260#217—24no—
261#216—44no—
262#215—44no—
263(C2×D3)⋊C4—24no—
264(C2×D3)⋊C4—24no—
265(C2×D3)⋊C4—22no—
266(C2×D3)⋊C4—22no—
267C4⋊Dic3—24no—
268C4⋊Dic3—21no—
269C4⋊Dic3—22no—
270C4⋊Dic3—22no—
271C4⋊Dic3—21no—
272C4⋊Dic3—22no—
273C4⋊Dic3—22no—
274Dic3⋊C4—24no—
275Dic3⋊C4—24no—
276Dic3⋊C4—84no—
277C4×Dic3—24no—
278C4×Dic3—22no—
279C4×Dic3—22no—
280#208—22no—
281#208—22no—
282C2×C3⋊C8—22no—
283C2×#41—32no—
284C2×#41—62no—
285C2×#41C22×D311yescharacteristic
286C22×Q8—32no—
287C22×Q8C22×D311yescharacteristic
288#137—22no—
289#137—22no—
290#137—64no—
291#137—122no—
292C2×Q16—21no—
293C2×Q16—62no—
294C2×SD16—21no—
295C2×SD16—62no—
296C2×M16—32no—
297C2×M16C22×D311yescharacteristic
298C2×M16C22×D311yescharacteristic
299C4⋊Q8—31no—
300C4⋊Q8—61no—
301C4⋊Q8—61no—
302C4⋊Q8—61no—
303C4⋊Q8—61no—
304#127—31no—
305#126—64no—
306#125—62no—
307#124—32no—
308#124—62no—
309#124—64no—
310#123—64no—
311C22⋊Q8—64no—
312C22⋊Q8—62no—
313C22⋊Q8—62no—
314C22⋊Q8—62no—
315#121—62no—
316#121—62no—
317#121—64no—
318#121—62no—
319#121—61no—
320#121—61no—
321C4×Q8—21no—
322C4×Q8—21no—
323C4×Q8—21no—
324C4×Q8—21no—
325C4×Q8—62no—
326C4×Q8—62no—
327C4×Q8#6711yescharacteristic
328C4×Q8#6711yescharacteristic
329C4×D4—21no—
330C4×D4—21no—
331C4×D4—62no—
332C4×D4—62no—
333C4×D4—64no—
334C4×D4—61no—
335C4×D4—61no—
336C4×D4#6711yescharacteristic
337C4×D4#6711yescharacteristic
338#117—21no—
339#117—62no—
340#117—62no—
341#117C22×D311yescharacteristic
342C2×C4⋊C4—21no—
343C2×C4⋊C4—32no—
344C2×C4⋊C4—32no—
345C2×C4⋊C4—61no—
346C2×C4⋊C4—62no—
347C2×C4⋊C4—64no—
348C2×C4⋊C4C22×D311yescharacteristic
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—32no—
352C2×C42C22×D311yescharacteristic
353#107—21no—
354#106—21no—
355C4⋊C8—61no—
356C4⋊C8—61no—
357#104—64no—
358#104—122no—
359#104—122no—
360#103—21no—
361#103—21no—
362#103—21no—
363#103—62no—
364#102—21no—
365#102—62no—
366#97—21no—
367C4×C8—21no—
368#95—64no—
369#95—61no—
370C23×C3—21no—
371C23×C3C22×D411yescharacteristic
372C22×D3—22no—
373C3×Q8—22no—
374C3×Q8—21no—
375C3×Q8—21no—
376C3×Q8—22no—
377C3×Q8—41no—
378C3×Q8—41no—
379C3×D4—21no—
380C3×D4—21no—
381C3×D4—41no—
382C3×D4—41no—
383C2×C3×C4—22no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—22no—
388C2×C3×C4—41no—
389C2×C3×C4—42no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—42no—
395C2×C3×C4—42no—
396C2×C3×C4—41no—
397C2×C3×C4#12011yescharacteristic
398C2×C3×C4#12011yescharacteristic
399C2×C3×C4#12011yescharacteristic
400C2×C3×C4#12011yescharacteristic
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4C22×D411yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4H(5,𝔽2)11yescharacteristic
406C2×C3×C42−1+411yescharacteristic
407#67—42no—
408#67—42no—
409#67—84no—
410C2×Dic3—24no—
411C2×Dic3—22no—
412C2×Dic3—24no—
413C2×Dic3—42no—
414C2×Dic3—42no—
415C2×Dic3—44no—
416D12—22no—
417D12—22no—
418D12—42no—
419D12—42no—
420C4×D3—42no—
421C4×D3—42no—
422C4×D3—84no—
423Dic6—22no—
424Dic6—22no—
425Dic6—44no—
426Dic6—82no—
427C3×C8—22no—
428C3×C8—41no—
429C3⋊C8—42no—
430#41—21no—
431#41—21no—
432#41—21no—
433#41—21no—
434#41—62no—
435#41—62no—
436#41—62no—
437#41—62no—
438#41—244no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—32no—
445C2×Q8—62no—
446C2×Q8—62no—
447C2×Q8—242no—
448C2×Q8#23512yescomplemented
449C2×Q8C2×#6711yescharacteristic
450C2×D4—21no—
451C2×D4—32no—
452C2×D4—62no—
453C2×D4—64no—
454C2×D4—61no—
455C2×D4—61no—
456C2×D4C2×#6711yescharacteristic
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—34no—
463C22×C4—64no—
464C22×C4—62no—
465C22×C4—62no—
466C22×C4D3×D411yescharacteristic
467C22×C4D3×D411yescharacteristic
468C22×C4C23×D311yescharacteristic
469Q16—24no—
470Q16—124no—
471SD16—24no—
472SD16—124no—
473M16—21no—
474M16—21no—
475M16—21no—
476M16—62no—
477M16—62no—
478M16#23512yescomplemented
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—62no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—64no—
489C4⋊C4—64no—
490C4⋊C4—64no—
491C4⋊C4—62no—
492C4⋊C4—62no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—62no—
496C4⋊C4—62no—
497C4⋊C4—244no—
498C4⋊C4C2×#6711yescharacteristic
499C4⋊C4C2×#6711yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—64no—
504C22⋊C4—64no—
505C22⋊C4—64no—
506C22⋊C4—62no—
507C22⋊C4—62no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—41no—
512C42—41no—
513C42—64no—
514C42—62no—
515C42—62no—
516C42D3×D411yescharacteristic
517C42D3×D411yescharacteristic
518C42C2×#6711yescharacteristic
519C42C2×#6711yescharacteristic
520C22×C3—21no—
521C22×C3—21no—
522C22×C3—41no—
523C22×C3—41no—
524C22×C3—41no—
525C22×C3C2×#12011yescharacteristic
526C22×C3#54811yescharacteristic
527C22×C3#55011yescharacteristic
528C2×D3—42no—
529C2×D3—42no—
530C2×D3—42no—
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—21no—
534C3×C4—41no—
535C3×C4—42no—
536C3×C4—41no—
537C3×C4—41no—
538C3×C4—81no—
539C3×C4#45612yes—
540C3×C4#52511yescharacteristic
541C3×C4#53711yescharacteristic
542C3×C4#54611yescharacteristic
543C3×C4#54711yescharacteristic
544Dic3—44no—
545Dic3—42no—
546Dic3—44no—
547C23—21no—
548C23—62no—
549C23C2×D3×D411yescharacteristic
550Q8—22no—
551Q8—21no—
552Q8—22no—
553Q8—21no—
554Q8—41no—
555Q8—41no—
556Q8—62no—
557Q8—62no—
558Q8—124no—
559Q8—242no—
560D4—21no—
561D4—21no—
562D4—41no—
563D4—41no—
564D4—62no—
565D4—62no—
566D4—122no—
567D4—122no—
568D4—122no—
569D4—122no—
570D4—244no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—64no—
586C2×C4—62no—
587C2×C4—64no—
588C2×C4—122no—
589C2×C4—122no—
590C2×C4—122no—
591C2×C4—122no—
592C2×C4—124no—
593C2×C4—244no—
594C2×C4#93711yescharacteristic
595C2×C4#93711yescharacteristic
596C2×C4#93711yescharacteristic
597C2×C4#93711yescharacteristic
598C2×C4C2×#23511yescharacteristic
599C2×C4C2×#23511yescharacteristic
600C2×C4C2×D3×D411yescharacteristic
601C2×C4#100911yescharacteristic
602C2×C4#101011yescharacteristic
603C2×C4C22×#6711yescharacteristic
604C8—22no—
605C8—41no—
606C8—122no—
607C2×C3—21no—
608C2×C3—21no—
609C2×C3—81no—
610C2×C3#240711yescharacteristic
611C2×C3C2×#45611yescharacteristic
612C2×C3#301911yescharacteristic
613D3—82no—
614C22—21no—
615C22—21no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—122no—
620C22—122no—
621C22—122no—
622C22#555211yescharacteristic
623C22#555311yescharacteristic
624C22C2×#93711yescharacteristic
625C4—21no—
626C4—21no—
627C4—21no—
628C4—41no—
629C4—41no—
630C4—41no—
631C4—42no—
632C4—81no—
633C4—124no—
634C4—124no—
635C4—122no—
636C4#520512yes—
637C4#555011yescharacteristic
638C4#555111yescharacteristic
639C4#579111yescharacteristic
640C4#583311yescharacteristic
641C3#2417311yescharacteristic, complemented, Sylow
642C2—21no—
643C2—21no—
644C2—81no—
645C2—242no—
646C2#8240211yescharacteristic
647C2C2×#520511yescharacteristic
648C2#8400511yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#137C2
socleC22×C3C2×#120
prime radical#85850C2
hypercenterC4×#137D3
hypocenterC3#24173
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
2. C3×C4×#137C2semidirect
3. #82478C2semidirect
4. #82441C2semidirect
5. #72110C2semidirect
6. #71646C2semidirect
7. #70680C2semidirect
8. #70348C2semidirect
9. #70311C2semidirect
10. #69879C2semidirect
11. #69867C2semidirect
56. C4×#137D3semidirect
203. C2×C3×Q8#41semidirect
237. C3×M16#41semidirect
448. C2×Q8#235semidirect
478. M16#235semidirect
641. C3#24173semidirect

Maximal subgroups

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

subgroupindex
#858502
C3×C4×#1372
#824782
#824412
#721102
#716462
#706802
#703482
#703112
#698792
#698672
#241733