Finite Group Databasebeta

← #10829971

Subgroups — #10829971

Order 768.

1394 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5577C211yesmaximal, characteristic
2#82482C211yesmaximal, characteristic, complemented
3C2×#4848C211yesmaximal, characteristic, complemented
4C2×C4×#204C211yesmaximal, characteristic, complemented
5C3×#1984C211yesmaximal, characteristic, complemented
6#70479C211yesmaximal, characteristic, complemented
7#69772C211yesmaximal, characteristic, complemented
8#68676C212yesmaximal, complemented
9#68672C212yesmaximal, complemented
10#68671C212yesmaximal, complemented
11#68629C212yesmaximal, complemented
12#24188—31nomaximal, Sylow
13C22×#239—21no—
14C2×C3×C4×Q8C2211yescharacteristic
15C22×#204C2211yescharacteristic
16C22×#204C2212yes—
17#5577—24no—
18#5577—24no—
19C2×#895—21no—
20C2×#895—22no—
21C2×#874C2212yes—
22C2×#874C2211yescharacteristic
23C2×C4×D12C2212yes—
24C2×C4×D12C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×C4×C8C2211yescharacteristic
28#5229—21no—
29C2×#821C2211yescharacteristic
30C2×#821C2212yes—
31C2×#821C2212yes—
32C2×#821C2212yes—
33C2×#821C2212yes—
34C2×#817C2211yescharacteristic
35C2×#816C2211yescharacteristic
36#4982—22no—
37#4972—21no—
38#4935C2211yescharacteristic
39#4929C2212yes—
40#4928C2212yes—
41#4848—24no—
42C4×#204—24no—
43C3×#336C2212yes—
44C3×#336C2212yes—
45#4551C2212yes—
46C2×#550—31no—
47C2×#460—31no—
48C2×C4×SD16—31no—
49#2582—31no—
50#2041—32no—
51#2014—32no—
52#1984D311yescharacteristic, complemented
53#1957—31no—
54#1877—32no—
55#1864—32no—
56#1802—31no—
57C22×C3×Q8D411yescharacteristic
58C2×#239—28no—
59C2×#239—44no—
60C2×#239—44no—
61C22×D12—22no—
62C22×D12—21no—
63C22×D12—22no—
64C22×D12—21no—
65C22×D12C2312yes—
66C22×D12C2311yescharacteristic
67C22×C4×D3—22no—
68C22×C4×D3—21no—
69C22×Dic6C2311yescharacteristic
70C22×C3×C8C2312yes—
71C22×C3×C8C2311yescharacteristic
72C22×C3×C8C2311yescharacteristic
73C3×C4×Q8—22no—
74C3×C4×Q8—22no—
75C2×C3×C4⋊C4D411yescharacteristic
76C2×C3×C4⋊C4D412yes—
77C2×C3×C42C2312yes—
78C2×C3×C42C2311yescharacteristic
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4—21no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3C2311yescharacteristic
84C2×C4⋊Dic3C2312yes—
85C2×C4⋊Dic3C2311yescharacteristic
86C2×Dic3⋊C4—21no—
87C2×Dic3⋊C4—22no—
88C2×C4×Dic3—21no—
89C2×#204—22no—
90C2×#204—22no—
91C2×#204—22no—
92C2×#204—24no—
93C2×#204—24no—
94C2×#204—42no—
95#895—28no—
96#895—216no—
97#874—24no—
98#874—24no—
99#874—22no—
100#874D414yes—
101C4×D12—24no—
102C4×D12—22no—
103C4×D12—24no—
104C4×D12D414yes—
105C4×Dic6—22no—
106C4×Dic6—22no—
107C3×C4⋊C8—22no—
108C3×C4×C8—22no—
109C3⋊#95—22no—
110C3⋊#95—22no—
111C3⋊#95—22no—
112#821—22no—
113#821—24no—
114#821—24no—
115#821—42no—
116#821—42no—
117#817—22no—
118#816—22no—
119C22×#41—61no—
120C22×SD16—31no—
121C22×SD16—32no—
122#550—64no—
123#550—64no—
124C2×#127—31no—
125C2×#127—32no—
126C2×#121—61no—
127C2×#121—62no—
128C2×C4×Q8—31no—
129C2×C4×Q8C2×D311yescharacteristic
130C2×C4×D4—31no—
131C2×C4×D4—32no—
132#460—64no—
133C4×SD16—64no—
134C2×#106—31no—
135C2×C4⋊C8C2×D311yescharacteristic
136C2×#103—31no—
137C2×#102—32no—
138C2×#102—32no—
139C2×#102—31no—
140C2×#102—32no—
141C2×#102—32no—
142C2×C4×C8C2×D311yescharacteristic
143#398—62no—
144#391—32no—
145#391—61no—
146#389—32no—
147#389—61no—
148#384—31no—
149#340—32no—
150#336C2×D312yes—
151#336C2×D312yes—
152C23×D3—22no—
153C23×D3—21no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8SD1614yescomplemented
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2411yescharacteristic
162C22×Dic3—21no—
163C22×Dic3—21no—
164C22×Dic3—22no—
165C22×Dic3—21no—
166#239—432no—
167C2×D12—22no—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12—22no—
171C2×D12—21no—
172C2×D12—22no—
173C2×D12—21no—
174C2×D12—22no—
175C2×D12—28no—
176C2×D12—24no—
177C2×D12—22no—
178C2×D12—21no—
179C2×D12—28no—
180C2×D12—24no—
181C2×D12—44no—
182C2×D12—44no—
183C2×D12—42no—
184C2×D12—42no—
185C2×D12—81no—
186C2×D12—81no—
187C2×D12—82no—
188C2×D12—82no—
189C2×D12C2×D414yes—
190C2×D12C2×D412yes—
191C2×C4×D3—28no—
192C2×C4×D3—24no—
193C2×C4×D3—44no—
194C2×C4×D3—44no—
195C2×C4×D3—42no—
196C2×C4×D3—42no—
197C2×Dic6—21no—
198C2×Dic6—21no—
199C2×Dic6—21no—
200C2×Dic6—21no—
201C2×Dic6—21no—
202C2×Dic6#4112yescomplemented
203C2×C3×C8—21no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C2×C3×C8—21no—
209C2×C3×C8—22no—
210C2×C3×C8—21no—
211C2×C3×C8#4112yescomplemented
212C3×C4⋊C4—24no—
213C3×C4⋊C4—22no—
214C3×C42—22no—
215C3×C42—22no—
216C3×C42C2×D412yes—
217C3×C42C2×D412yes—
218(C2×D3)⋊C4—28no—
219(C2×D3)⋊C4—24no—
220C4⋊Dic3—21no—
221C4⋊Dic3—21no—
222C4⋊Dic3—21no—
223C4⋊Dic3—22no—
224C4⋊Dic3—22no—
225C4⋊Dic3—42no—
226C4⋊Dic3—81no—
227C4⋊Dic3C2×D412yes—
228Dic3⋊C4—42no—
229Dic3⋊C4—82no—
230C4×Dic3—42no—
231#204—28no—
232#204—48no—
233C2×#41—68no—
234C2×#41—124no—
235C2×#41—124no—
236C22×Q8—31no—
237C22×Q8D1211yescharacteristic
238C22×D4—31no—
239C22×D4—32no—
240C22×D4—62no—
241C22×D4—61no—
242C22×D4—62no—
243C22×D4—61no—
244C23×C4—62no—
245C23×C4—61no—
246C2×SD16—62no—
247C2×SD16—62no—
248C2×SD16—64no—
249C2×SD16—64no—
250C2×SD16—62no—
251C2×SD16—122no—
252C22×C8C22×D312yes—
253C22×C8C22×D311yescharacteristic
254C22×C8C22×D311yescharacteristic
255#127—34no—
256#127—64no—
257#127—64no—
258#127—62no—
259#121—68no—
260#121—616no—
261C4×Q8—22no—
262C4×Q8—22no—
263C4×Q8—62no—
264C4×Q8—62no—
265C4×D4—34no—
266C4×D4—64no—
267C4×D4—64no—
268C4×D4—62no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—32no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4—62no—
276C2×C4⋊C4D1211yescharacteristic
277C2×C4⋊C4D1212yes—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—61no—
280C2×C42—61no—
281C2×C42C22×D312yes—
282C2×C42C22×D311yescharacteristic
283#106—62no—
284C4⋊C8—22no—
285#103—62no—
286#102—62no—
287#102—64no—
288#102—64no—
289#102—122no—
290#102—122no—
291C4×C8—22no—
292#95—62no—
293#95—62no—
294#95—62no—
295C23×C3C22×D411yescharacteristic
296C22×D3—24no—
297C22×D3—22no—
298C22×D3—41no—
299C22×D3—42no—
300C22×D3—42no—
301C22×D3—41no—
302C22×D3—42no—
303C22×D3—41no—
304C22×D3—42no—
305C22×D3—41no—
306C22×D3—42no—
307C22×D3—41no—
308C22×D3—42no—
309C22×D3—41no—
310C3×Q8—28no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4#12012yes—
319C2×C3×C4#12012yes—
320C2×C3×C4C2×SD1614yes—
321C2×C3×C4C2×SD1612yes—
322C2×C3×C4AGL(1,ℤ/8)12yes—
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C22×D411yescharacteristic
325C2×C3×C4C2×#4111yescharacteristic
326C2×C3×C4C2×#4111yescharacteristic
327C2×C3×C4H(5,𝔽2)11yescharacteristic
328C2×C3×C42−1+411yescharacteristic
329C2×Dic3—22no—
330C2×Dic3—22no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—41no—
334C2×Dic3—41no—
335C2×Dic3—41no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338C2×Dic3—41no—
339C2×Dic3—42no—
340C2×Dic3—41no—
341D12—22no—
342D12—22no—
343D12—24no—
344D12—24no—
345D12—416no—
346D12—42no—
347D12—42no—
348D12—41no—
349D12—41no—
350D12—48no—
351D12—84no—
352D12—88no—
353C4×D3—416no—
354C4×D3—48no—
355Dic6—22no—
356Dic6—22no—
357Dic6—41no—
358Dic6—41no—
359C3×C8—22no—
360C3×C8—41no—
361C3×C8—42no—
362C24—62no—
363C24—61no—
364#41—1232no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—32no—
368C2×Q8—61no—
369C2×Q8—61no—
370C2×Q8—61no—
371C2×Q8—61no—
372C2×Q8—61no—
373C2×Q8#20414yescomplemented
374C2×D4—32no—
375C2×D4—34no—
376C2×D4—68no—
377C2×D4—68no—
378C2×D4—64no—
379C2×D4—64no—
380C2×D4—61no—
381C2×D4—62no—
382C2×D4—62no—
383C2×D4—61no—
384C2×D4—62no—
385C2×D4—61no—
386C2×D4—62no—
387C2×D4—61no—
388C2×D4—61no—
389C2×D4—62no—
390C2×D4—124no—
391C2×D4—124no—
392C2×D4—122no—
393C2×D4—122no—
394C2×D4—241no—
395C2×D4—241no—
396C2×D4—242no—
397C2×D4—242no—
398C22×C4—68no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—64no—
402C22×C4—62no—
403C22×C4—61no—
404C22×C4—124no—
405C22×C4—124no—
406C22×C4—122no—
407C22×C4—122no—
408C22×C4C2×D1212yes—
409C22×C4C2×D1211yescharacteristic
410C22×C4C2×D1212yes—
411C22×C4C2×D1211yescharacteristic
412C22×C4C23×D311yescharacteristic
413SD16—68no—
414SD16—128no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—22no—
419C2×C8—21no—
420C2×C8—22no—
421C2×C8—21no—
422C2×C8—22no—
423C2×C8#23912yescomplemented
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—32no—
427C4⋊C4—61no—
428C4⋊C4—61no—
429C4⋊C4—61no—
430C4⋊C4—62no—
431C4⋊C4—62no—
432C4⋊C4—122no—
433C4⋊C4—122no—
434C4⋊C4—242no—
435C4⋊C4—241no—
436C22⋊C4—68no—
437C22⋊C4—64no—
438C42—22no—
439C42—22no—
440C42—122no—
441C42D3×D412yes—
442C42D3×D412yes—
443C22×C3C2×#12011yescharacteristic
444C22×C3#54811yescharacteristic
445C22×C3#55011yescharacteristic
446C22×C3C22×SD1611yescharacteristic
447C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
448C22×C3#57611yescharacteristic
449C22×C3#57711yescharacteristic
450C2×D3—42no—
451C2×D3—44no—
452C2×D3—44no—
453C2×D3—42no—
454C2×D3—44no—
455C2×D3—42no—
456C2×D3—81no—
457C2×D3—82no—
458C2×D3—82no—
459C2×D3—81no—
460C2×D3—82no—
461C2×D3—81no—
462C2×D3—82no—
463C2×D3—81no—
464C3×C4—24no—
465C3×C4—22no—
466C3×C4—42no—
467C3×C4#45014yes—
468C3×C4#52511yescharacteristic
469C3×C4#53711yescharacteristic
470C3×C4#54611yescharacteristic
471C3×C4#54711yescharacteristic
472Dic3—42no—
473Dic3—42no—
474Dic3—82no—
475Dic3—81no—
476C23—64no—
477C23—62no—
478C23—122no—
479C23—121no—
480C23—122no—
481C23—121no—
482C23—122no—
483C23—121no—
484C23—122no—
485C23—121no—
486C23—122no—
487C23—121no—
488C23—122no—
489C23—121no—
490C23C22×D1211yescharacteristic
491Q8—28no—
492Q8—62no—
493Q8—62no—
494Q8—121no—
495Q8—121no—
496D4—64no—
497D4—64no—
498D4—62no—
499D4—62no—
500D4—1216no—
501D4—121no—
502D4—121no—
503D4—128no—
504D4—122no—
505D4—122no—
506D4—244no—
507D4—248no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—62no—
516C2×C4—62no—
517C2×C4—1216no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—128no—
523C2×C4—121no—
524C2×C4—122no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—122no—
528C2×C4—122no—
529C2×C4#88212yes—
530C2×C4#88212yes—
531C2×C4C2×#20414yes—
532C2×C4C2×#20412yes—
533C2×C4#90812yes—
534C2×C4C2×D3×D411yescharacteristic
535C2×C4C2×D3×D411yescharacteristic
536C2×C4C2×#23911yescharacteristic
537C2×C4#100711yescharacteristic
538C2×C4D3×#4111yescharacteristic
539C2×C4#100911yescharacteristic
540C8—22no—
541C8—42no—
542C8—41no—
543C2×C3#240711yescharacteristic
544C2×C3C2×#45011yescharacteristic
545C2×C3#301211yescharacteristic
546C2×C3#336211yescharacteristic
547C2×C3#336811yescharacteristic
548C2×C3#339211yescharacteristic
549C2×C3#339911yescharacteristic
550D3—82no—
551D3—84no—
552C22—124no—
553C22—122no—
554C22—124no—
555C22—122no—
556C22—124no—
557C22—122no—
558C22—242no—
559C22—241no—
560C22—242no—
561C22—241no—
562C22—242no—
563C22—241no—
564C22—242no—
565C22—241no—
566C22C2×#88211yescharacteristic
567C22#557611yescharacteristic
568C22#557711yescharacteristic
569C22C22×#20411yescharacteristic
570C22C2×#90811yescharacteristic
571C22#575111yescharacteristic
572C22#575211yescharacteristic
573C4—24no—
574C4—22no—
575C4—42no—
576C4—122no—
577C4—122no—
578C4—241no—
579C4—242no—
580C4#473114yes—
581C4#562711yescharacteristic
582C4#562911yescharacteristic
583C4#567611yescharacteristic
584C4#572611yescharacteristic
585C3#2418811yescharacteristic, complemented, Sylow
586C2—244no—
587C2—242no—
588C2#7661811yescharacteristic
589C2C2×#473111yescharacteristic
590C2#7867211yescharacteristic
591C2#7991111yescharacteristic
592C2#7993211yescharacteristic
593C2#7995211yescharacteristic
594C2#7998011yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1984C2
socleC23×C3C22×D4
prime radicalC2×#5577C2
hypercenter#1984D3
hypocenterC3#24188
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1984

Complements

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

NH ≅ G/Ntype
2. #82482C2semidirect
3. C2×#4848C2semidirect
4. C2×C4×#204C2semidirect
5. C3×#1984C2semidirect
6. #70479C2semidirect
7. #69772C2semidirect
8. #68676C2semidirect
9. #68672C2semidirect
10. #68671C2semidirect
11. #68629C2semidirect
52. #1984D3semidirect
156. C2×C3×Q8SD16semidirect
202. C2×Dic6#41semidirect
211. C2×C3×C8#41semidirect
373. C2×Q8#204semidirect
423. C2×C8#239semidirect
585. C3#24188semidirect

Maximal subgroups

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

subgroupindex
C2×#55772
#824822
C2×#48482
C2×C4×#2042
C3×#19842
#704792
#697722
#686762
#686722
#686712
#686292
#241883