Finite Group Databasebeta

← #10829902

Subgroups — #10829902

Order 768.

1082 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85788C211yesmaximal, characteristic
2C3×#2326C211yesmaximal, characteristic, complemented
3#82441C211yesmaximal, characteristic, complemented
4#82431C211yesmaximal, characteristic, complemented
5#71731C212yesmaximal, complemented
6#71617C211yesmaximal, characteristic, complemented
7#69693C212yesmaximal, complemented
8#69665C212yesmaximal, complemented
9#69592C212yesmaximal, complemented
10#69441C211yesmaximal, characteristic, complemented
11#69440C211yesmaximal, characteristic, complemented
12#24182—31nomaximal, Sylow
13C2×C3×#124C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#942C2212yes—
17#5805C2212yes—
18#5791—22no—
19#5563—21no—
20#5559—22no—
21#5558—22no—
22#5553—21no—
23#5551—21no—
24#5542—21no—
25#5504C2212yes—
26#5494—22no—
27#5477C2211yescharacteristic
28#5470C2211yescharacteristic
29C3×#391C2211yescharacteristic
30C3×#390C2211yescharacteristic
31C3×#388—22no—
32C3×#385—22no—
33C3×#384C2211yescharacteristic
34C3×#376C2211yescharacteristic
35#5225C2212yes—
36C2×#833C2212yes—
37#5031—21no—
38#5028—22no—
39#5027—21no—
40#5022—22no—
41#5021—21no—
42#5020—21no—
43#4966C2212yes—
44#4964C2212yes—
45C2×#805C2212yes—
46C2×#805C2212yes—
47#4924C2211yescharacteristic
48#4924C2211yescharacteristic
49#4530C2212yes—
50#4523C2212yes—
51#4523C2212yes—
52#3506—31no—
53#3078—31no—
54#3037—31no—
55#2326D311yescharacteristic, complemented
56#2037—32no—
57#2004—32no—
58#1995—31no—
59#1963—31no—
60#1882—32no—
61#1859—32no—
62#1800—31no—
63C22×C3×Q8C2311yescharacteristic
64C22×C3×D4C2311yescharacteristic
65C23×C3×C4D412yes—
66C2×#235—22no—
67C2×#235C2312yes—
68C3×#124—22no—
69C3×#124—22no—
70C3×C4×Q8—21no—
71C3×C4×Q8—21no—
72C3×C4×Q8—21no—
73C3×C4×Q8—21no—
74C3×C4×D4—21no—
75C3×C4×D4—21no—
76C3×C4×D4—21no—
77C3×C4×D4—21no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4D412yes—
85C2×C3×C42—21no—
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89#953—24no—
90C2×#216—22no—
91#942—24no—
92#942—42no—
93C2×#215—22no—
94#940—22no—
95#938—22no—
96#938—22no—
97#936—22no—
98#930—22no—
99#930—21no—
100#930—21no—
101#929—24no—
102C4×#67—24no—
103C4×#67—24no—
104#926—22no—
105#926C2312yes—
106#924—22no—
107#924—21no—
108#924—21no—
109C2×#208C2312yes—
110C2×#208C2312yes—
111#893—22no—
112#892—22no—
113D3×C4⋊C4—22no—
114#887—22no—
115#885—24no—
116#884—24no—
117#883—24no—
118#879—24no—
119#875D412yes—
120#874D411yescharacteristic
121C4×D12—21no—
122C4×D12—21no—
123#870D412yes—
124#869D411yescharacteristic
125#869D411yescharacteristic
126#869D411yescharacteristic
127C4×Dic6—21no—
128C4×Dic6—21no—
129C3×#95—21no—
130C3×#95—21no—
131C3×#95—22no—
132#835—22no—
133#835—22no—
134#833—42no—
135#810—22no—
136#809—22no—
137#808—22no—
138#807—22no—
139#805—24no—
140#805—42no—
141C4⋊(C3⋊C8)—21no—
142C4⋊(C3⋊C8)—21no—
143C4⋊(C3⋊C8)—21no—
144C4⋊(C3⋊C8)—21no—
145C2×#137—32no—
146#559—62no—
147#556—61no—
148#555—61no—
149#553—62no—
150#550—61no—
151#547—61no—
152#538—62no—
153#535—32no—
154#533—31no—
155#532—31no—
156C2×#124C2×D311yescharacteristic
157#525—62no—
158#520—32no—
159C2×C4×Q8C2×D311yescharacteristic
160C2×C4×D4C2×D311yescharacteristic
161#479—62no—
162#477—61no—
163#475—61no—
164#464—62no—
165#462—61no—
166#460—61no—
167#428—32no—
168#423—31no—
169#423—31no—
170C2×#104—32no—
171C2×#104—32no—
172#419—32no—
173#417—32no—
174C2×#100—32no—
175#391C2×D311yescharacteristic
176#390C2×D311yescharacteristic
177#388—22no—
178#385—22no—
179#384C2×D311yescharacteristic
180#376C2×D311yescharacteristic
181#343—32no—
182#339—32no—
183#339—32no—
184C24×C3C2×D412yes—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8#4112yescomplemented
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—22no—
200C22×C3×C4—22no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—82no—
204C22×C3×C4C2×D412yes—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2411yescharacteristic
208C2×#67—22no—
209C2×#67—24no—
210C2×#67—24no—
211#235—22no—
212#235—22no—
213#235—22no—
214#235—22no—
215#235—84no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12C2×D412yes—
219C2×C4×D3—22no—
220C2×C4×D3—24no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6C2×D412yes—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—41no—
229C3×C4⋊C4—41no—
230C3×C4⋊C4—82no—
231C3×C22⋊C4—28no—
232C3×C22⋊C4—22no—
233C3×C22⋊C4—22no—
234C3×C22⋊C4—82no—
235C3×C42—21no—
236C3×C42—21no—
237C3×C42—21no—
238C3×C42—41no—
239C3×C42—41no—
240C3×C42C2×D411yescharacteristic
241C3×C42C2×D411yescharacteristic
242C3×C42C2×D411yescharacteristic
243C3×C42C2×D411yescharacteristic
244C3×C42#4112yescomplemented
245#217—24no—
246#217—24no—
247#217—22no—
248#216—44no—
249#215—44no—
250(C2×D3)⋊C4—24no—
251(C2×D3)⋊C4—22no—
252(C2×D3)⋊C4—22no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3C2×D412yes—
258C4⋊Dic3C2×D412yes—
259Dic3⋊C4—24no—
260Dic3⋊C4—22no—
261Dic3⋊C4—22no—
262C4×Dic3—24no—
263C4×Dic3—22no—
264#208—22no—
265#208—22no—
266#208—22no—
267#208—22no—
268C2×C3⋊C8—22no—
269C2×C3⋊C8—22no—
270C2×#41—32no—
271C2×#41—62no—
272C22×Q8C22×D311yescharacteristic
273C22×D4C22×D311yescharacteristic
274C23×C4#6712yes—
275#137—64no—
276#137—122no—
277C2×Q16—62no—
278C2×SD16—62no—
279C2×M16—32no—
280C2×M16—32no—
281C4⋊Q8—31no—
282C4⋊Q8—31no—
283C4⋊Q8—31no—
284#127—31no—
285#127—62no—
286#126—64no—
287#125—32no—
288#124—22no—
289#124—22no—
290#124—32no—
291#124—64no—
292#124—62no—
293#123—64no—
294#123—64no—
295C22⋊Q8—62no—
296C22⋊Q8—61no—
297C22⋊Q8—61no—
298#121—64no—
299#121—62no—
300#121—62no—
301#121—62no—
302#121—61no—
303#121—61no—
304#120—64no—
305C4×Q8—21no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×Q8—21no—
309C4×Q8—62no—
310C4×Q8—61no—
311C4×Q8—61no—
312C4×D4—21no—
313C4×D4—21no—
314C4×D4—21no—
315C4×D4—21no—
316C4×D4—62no—
317C4×D4—64no—
318C4×D4—64no—
319C4×D4—61no—
320C4×D4—61no—
321#117—32no—
322#117—62no—
323#117—62no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—62no—
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C4⋊C4C22×D311yescharacteristic
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4#6712yes—
332C2×C42—21no—
333C2×C42C22×D311yescharacteristic
334C2×C42C22×D311yescharacteristic
335C2×C42C22×D311yescharacteristic
336#107—62no—
337#106—62no—
338C4⋊C8—61no—
339C4⋊C8—61no—
340C4⋊C8—61no—
341C4⋊C8—61no—
342#104—64no—
343#104—122no—
344#103—62no—
345#103—62no—
346#103—62no—
347#102—62no—
348#100—122no—
349#95—21no—
350#95—21no—
351#95—22no—
352C23×C3—22no—
353C23×C3—22no—
354C23×C3—22no—
355C23×C3—42no—
356C23×C3—42no—
357C23×C3C22×D411yescharacteristic
358C22×D3—22no—
359C3×Q8—22no—
360C3×Q8—22no—
361C3×Q8—41no—
362C3×Q8—41no—
363C3×D4—41no—
364C3×D4—41no—
365C3×D4—41no—
366C3×D4—41no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—44no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—42no—
378C2×C3×C4—41no—
379C2×C3×C4—42no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—82no—
384C2×C3×C4#12011yescharacteristic
385C2×C3×C4#12011yescharacteristic
386C2×C3×C4#12011yescharacteristic
387C2×C3×C4#12011yescharacteristic
388C2×C3×C4C22×D411yescharacteristic
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4H(5,𝔽2)11yescharacteristic
393C2×C3×C42−1+411yescharacteristic
394#67—42no—
395#67—42no—
396#67—84no—
397#67—84no—
398C2×Dic3—22no—
399C2×Dic3—22no—
400C2×Dic3—22no—
401D12—22no—
402D12—22no—
403D12—42no—
404D12—81no—
405C4×D3—42no—
406C4×D3—42no—
407C4×D3—84no—
408Dic6—22no—
409Dic6—22no—
410Dic6—42no—
411Dic6—81no—
412C3⋊C8—42no—
413C3⋊C8—42no—
414C24C2×#6712yes—
415#41—62no—
416#41—62no—
417#41—62no—
418#41—62no—
419#41—244no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—32no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8#23512yescomplemented
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—32no—
436C2×D4—62no—
437C2×D4—64no—
438C2×D4—64no—
439C2×D4—61no—
440C2×D4—61no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—64no—
448C22×C4—62no—
449C22×C4—82no—
450C22×C4C2×#6712yes—
451C22×C4C2×#6711yescharacteristic
452C22×C4C2×#6711yescharacteristic
453C22×C4C23×D311yescharacteristic
454Q16—124no—
455SD16—124no—
456M16—62no—
457M16—62no—
458M16—62no—
459M16—62no—
460C2×C8—62no—
461C2×C8—62no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—32no—
467C4⋊C4—32no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—64no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—62no—
474C4⋊C4—61no—
475C4⋊C4—62no—
476C4⋊C4—61no—
477C4⋊C4—82no—
478C22⋊C4—28no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—64no—
482C22⋊C4—64no—
483C22⋊C4—64no—
484C22⋊C4—62no—
485C22⋊C4—62no—
486C22⋊C4—62no—
487C22⋊C4—82no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—41no—
492C42—41no—
493C42—64no—
494C42—62no—
495C42#23512yescomplemented
496C42D3×D411yescharacteristic
497C42D3×D411yescharacteristic
498C42D3×D411yescharacteristic
499C42D3×D411yescharacteristic
500C22×C3—21no—
501C22×C3—21no—
502C22×C3—44no—
503C22×C3—42no—
504C22×C3—42no—
505C22×C3—42no—
506C22×C3—82no—
507C22×C3C2×#12011yescharacteristic
508C22×C3#52511yescharacteristic
509C22×C3#53711yescharacteristic
510C2×D3—42no—
511C2×D3—42no—
512C2×D3—42no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4—21no—
517C3×C4—42no—
518C3×C4—42no—
519C3×C4—42no—
520C3×C4—81no—
521C3×C4—81no—
522C3×C4#54611yescharacteristic
523C3×C4#54711yescharacteristic
524C3×C4#54811yescharacteristic
525C3×C4#55011yescharacteristic
526Dic3—42no—
527Dic3—42no—
528Dic3—42no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—42no—
533C23—42no—
534C23—62no—
535C23C22×#6711yescharacteristic
536Q8—22no—
537Q8—22no—
538Q8—41no—
539Q8—41no—
540Q8—62no—
541Q8—62no—
542Q8—122no—
543Q8—241no—
544D4—41no—
545D4—41no—
546D4—41no—
547D4—41no—
548D4—62no—
549D4—62no—
550D4—122no—
551D4—122no—
552D4—122no—
553D4—241no—
554D4—244no—
555D4—244no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—44no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—62no—
573C2×C4—62no—
574C2×C4—62no—
575C2×C4—82no—
576C2×C4—122no—
577C2×C4—122no—
578C2×C4—244no—
579C2×C4#93711yescharacteristic
580C2×C4#93711yescharacteristic
581C2×C4#93711yescharacteristic
582C2×C4#93711yescharacteristic
583C2×C4C2×#23511yescharacteristic
584C2×C4C2×#23511yescharacteristic
585C2×C4C2×D3×D411yescharacteristic
586C2×C4C2×D3×D411yescharacteristic
587C2×C4#100911yescharacteristic
588C2×C4#101011yescharacteristic
589C8—122no—
590C8—122no—
591C2×C3—21no—
592C2×C3—21no—
593C2×C3—82no—
594C2×C3#240711yescharacteristic
595C2×C3#301911yescharacteristic
596C2×C3#302211yescharacteristic
597D3—82no—
598C22—21no—
599C22—21no—
600C22—44no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—82no—
605C22—122no—
606C22—122no—
607C22—122no—
608C22#579111yescharacteristic
609C22C2×#93711yescharacteristic
610C22#583311yescharacteristic
611C4—21no—
612C4—21no—
613C4—21no—
614C4—21no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—81no—
619C4—81no—
620C4—122no—
621C4—122no—
622C4—122no—
623C4#555011yescharacteristic
624C4#555111yescharacteristic
625C4#555211yescharacteristic
626C4#555311yescharacteristic
627C3#2418211yescharacteristic, complemented, Sylow
628C2—21no—
629C2—21no—
630C2—82no—
631C2—242no—
632C2#8240211yescharacteristic
633C2#8339411yescharacteristic
634C2#8341611yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2326C2
socleC22×C3C2×#120
prime radical#85788C2
hypercenter#2326D3
hypocenterC3#24182
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2326

Complements

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

NH ≅ G/Ntype
2. C3×#2326C2semidirect
3. #82441C2semidirect
4. #82431C2semidirect
5. #71731C2semidirect
6. #71617C2semidirect
7. #69693C2semidirect
8. #69665C2semidirect
9. #69592C2semidirect
10. #69441C2semidirect
11. #69440C2semidirect
55. #2326D3semidirect
190. C2×C3×Q8#41semidirect
244. C3×C42#41semidirect
428. C2×Q8#235semidirect
495. C42#235semidirect
627. C3#24182semidirect

Maximal subgroups

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

subgroupindex
#857882
C3×#23262
#824412
#824312
#717312
#716172
#696932
#696652
#695922
#694412
#694402
#241823