Finite Group Databasebeta

← #10848694

Subgroups — #10848694

Order 768.

1298 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5470C211yesmaximal, characteristic, complemented
2#82081C211yesmaximal, characteristic
3C2×#4693C211yesmaximal, characteristic, complemented
4C2×#4694C211yesmaximal, characteristic, complemented
5C3×C4×C22⋊C8C211yesmaximal, characteristic, complemented
6#69962C214yesmaximal, complemented
7#69777C211yesmaximal, characteristic, complemented
8#69771C211yesmaximal, characteristic, complemented
9#68686C214yesmaximal, complemented
10#25598—31nomaximal, Sylow
11C22×C3×C42C2211yescharacteristic
12C2×#930—21no—
13C2×#930—21no—
14C2×#930C2212yes—
15C2×#926—21no—
16C2×#924—21no—
17C2×#924C2212yes—
18C22×#204C2214yes—
19C2×#874C2211yescharacteristic
20#5470—24no—
21#5470—24no—
22C2×#870—21no—
23C2×#869C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C2×C4×Dic6—21no—
26C2×C4×Dic6C2212yes—
27C2×C3×C22⋊C8C2212yes—
28C2×C3×C4×C8C2211yescharacteristic
29C2×C3×C4×C8C2211yescharacteristic
30C2×#821C2214yes—
31C2×#816C2214yes—
32#4939—21no—
33#4939—21no—
34#4939C2212yes—
35#4935C2211yescharacteristic
36#4694—24no—
37#4693—24no—
38C3×#336C2212yes—
39#4551C2214yes—
40C2×#533—31no—
41C2×#492—31no—
42C2×#486—31no—
43#2856—31no—
44#2000—34no—
45#1990—31no—
46#1956—31no—
47#1880—34no—
48C4×C22⋊C8D311yescharacteristic, complemented
49C23×C3×C4D411yescharacteristic
50C23×C3×C4D411yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C22×#67—22no—
53C22×D12—21no—
54C22×D12—21no—
55C22×D12C2312yes—
56C22×Dic6—21no—
57C22×Dic6C2312yes—
58C22×C3×C8C2312yes—
59C22×C3×C8C2312yes—
60C2×C3×C42—22no—
61C2×C3×C42—22no—
62C2×C3×C42D411yescharacteristic
63C2×C3×C42D411yescharacteristic
64C2×C3×C42C2311yescharacteristic
65C2×C3×C42C2311yescharacteristic
66C2×#217—22no—
67#930—44no—
68#930—44no—
69#930—44no—
70C2×(C2×D3)⋊C4—22no—
71C2×(C2×D3)⋊C4—22no—
72C2×(C2×D3)⋊C4—22no—
73#926—28no—
74C2×C4⋊Dic3—21no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3—21no—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3C2312yes—
80C2×C4⋊Dic3C2312yes—
81#924—28no—
82#924—24no—
83#924—24no—
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4—22no—
86C2×C4×Dic3—22no—
87C2×#204—28no—
88C2×#204—28no—
89C2×#204—44no—
90#874—22no—
91#874—22no—
92#870—28no—
93#869—22no—
94#869—22no—
95#869D414yescomplemented
96#869D414yescomplemented
97C4×Dic6—28no—
98C4×Dic6—24no—
99C4×Dic6—24no—
100C3×C22⋊C8—24no—
101C3×C4×C8—22no—
102C3×C4×C8—22no—
103#821—28no—
104#816—44no—
105C22×SD16—34no—
106#533—64no—
107#533—64no—
108C2×C4⋊Q8—31no—
109C2×C4⋊Q8—31no—
110C2×#127—31no—
111C2×#125—61no—
112C2×C22⋊Q8—32no—
113C2×C22⋊Q8—61no—
114C2×#121—32no—
115C2×#121—61no—
116C2×#121—61no—
117C2×C4×Q8—32no—
118C2×C4×Q8—61no—
119C2×#117—61no—
120C22×C42C2×D311yescharacteristic
121#492—64no—
122#486—64no—
123C2×#103—34no—
124C2×#102—34no—
125C2×C22⋊C8C2×D312yes—
126C2×C4×C8C2×D311yescharacteristic
127C2×C4×C8C2×D311yescharacteristic
128#390—32no—
129#390—61no—
130#390—61no—
131#384—31no—
132#340—34no—
133#336C2×D312yes—
134C24×C3C2×D411yescharacteristic
135C23×D3—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4SD1614yescomplemented
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2411yescharacteristic
152C2×#67—44no—
153C2×#67—44no—
154C2×#67—44no—
155C22×Dic3—22no—
156C22×Dic3—22no—
157C22×Dic3—22no—
158C2×D12—22no—
159C2×D12—22no—
160C2×D12—24no—
161C2×D12—24no—
162C2×D12—42no—
163C2×D12—42no—
164C2×D12—42no—
165C2×D12—42no—
166C2×D12—42no—
167C2×D12—42no—
168C2×Dic6—24no—
169C2×Dic6—24no—
170C2×Dic6—24no—
171C2×Dic6—22no—
172C2×Dic6—42no—
173C2×Dic6—42no—
174C2×Dic6C2×D414yes—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C2×C3×C8#4114yes—
181C3×C42—22no—
182C3×C42—22no—
183C3×C42—21no—
184C3×C42—21no—
185C3×C42C2×D412yes—
186C3×C42C2×D412yes—
187#217—28no—
188(C2×D3)⋊C4—44no—
189(C2×D3)⋊C4—44no—
190(C2×D3)⋊C4—44no—
191C4⋊Dic3—24no—
192C4⋊Dic3—24no—
193C4⋊Dic3—24no—
194C4⋊Dic3—22no—
195C4⋊Dic3—24no—
196C4⋊Dic3—42no—
197C4⋊Dic3—42no—
198C4⋊Dic3C2×D414yes—
199Dic3⋊C4—28no—
200Dic3⋊C4—28no—
201C4×Dic3—28no—
202#204—416no—
203C22×Q8—32no—
204C22×Q8—61no—
205C22×D4—32no—
206C22×D4—62no—
207C22×D4—61no—
208C22×D4—61no—
209C23×C4D1211yescharacteristic
210C23×C4D1211yescharacteristic
211C23×C4C22×D311yescharacteristic
212C2×SD16—68no—
213C2×SD16—68no—
214C2×SD16—124no—
215C22×C8C22×D312yes—
216C22×C8C22×D312yes—
217C4⋊Q8—34no—
218C4⋊Q8—34no—
219C4⋊Q8—62no—
220C4⋊Q8—62no—
221#127—62no—
222#127—62no—
223#125—68no—
224C22⋊Q8—64no—
225C22⋊Q8—64no—
226C22⋊Q8—68no—
227#121—124no—
228#121—124no—
229#121—124no—
230C4×Q8—64no—
231C4×Q8—64no—
232C4×Q8—68no—
233#117—68no—
234C2×C4⋊C4—32no—
235C2×C4⋊C4—32no—
236C2×C4⋊C4—61no—
237C2×C4⋊C4—61no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4—61no—
241C2×C4⋊C4—61no—
242C2×C4⋊C4—61no—
243C2×C22⋊C4—62no—
244C2×C22⋊C4—62no—
245C2×C22⋊C4—62no—
246C2×C22⋊C4—62no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—62no—
250C2×C42D1211yescharacteristic
251C2×C42D1211yescharacteristic
252C2×C42C22×D311yescharacteristic
253C2×C42C22×D311yescharacteristic
254#103—124no—
255#102—68no—
256C22⋊C8—24no—
257C4×C8—22no—
258C4×C8—22no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—21no—
263C23×C3—21no—
264C23×C3—21no—
265C23×C3C2×SD1612yes—
266C23×C3C22×D411yescharacteristic
267C22×D3—42no—
268C22×D3—42no—
269C22×D3—42no—
270C22×D3—42no—
271C22×D3—42no—
272C22×D3—42no—
273C22×D3—42no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—28no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—44no—
286C2×C3×C4#12012yes—
287C2×C3×C4#12012yes—
288C2×C3×C4C2×SD1612yes—
289C2×C3×C4C2×SD1612yes—
290C2×C3×C4#13512yes—
291C2×C3×C4AGL(1,ℤ/8)12yes—
292C2×C3×C4#13712yes—
293C2×C3×C4C22×D411yescharacteristic
294C2×C3×C4C22×D411yescharacteristic
295C2×C3×C4C2×#4112yes—
296C2×C3×C4H(5,𝔽2)12yes—
297#67—88no—
298C2×Dic3—24no—
299C2×Dic3—24no—
300C2×Dic3—42no—
301C2×Dic3—42no—
302C2×Dic3—42no—
303C2×Dic3—42no—
304C2×Dic3—42no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307D12—44no—
308D12—44no—
309D12—84no—
310D12—84no—
311Dic6—28no—
312Dic6—44no—
313Dic6—48no—
314C3×C8—24no—
315C3×C8—42no—
316C24—62no—
317C24C2×D1211yescharacteristic
318C2×Q8—34no—
319C2×Q8—64no—
320C2×Q8—62no—
321C2×Q8—64no—
322C2×Q8—64no—
323C2×Q8—122no—
324C2×Q8—122no—
325C2×D4—64no—
326C2×D4—62no—
327C2×D4—62no—
328C2×D4—64no—
329C2×D4—124no—
330C2×D4—124no—
331C2×D4—124no—
332C2×D4—122no—
333C2×D4—122no—
334C2×D4—122no—
335C2×D4—122no—
336C2×D4—122no—
337C2×D4—122no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—62no—
348C22×C4—62no—
349C22×C4—62no—
350C22×C4#20414yescomplemented
351C22×C4C2×D1211yescharacteristic
352C22×C4C2×D1211yescharacteristic
353C22×C4C2×D1211yescharacteristic
354C22×C4C2×D1211yescharacteristic
355C22×C4C2×D1211yescharacteristic
356C22×C4C23×D311yescharacteristic
357SD16—1216no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8#23514yes—
364C4⋊C4—34no—
365C4⋊C4—68no—
366C4⋊C4—68no—
367C4⋊C4—64no—
368C4⋊C4—64no—
369C4⋊C4—64no—
370C4⋊C4—62no—
371C4⋊C4—64no—
372C4⋊C4—122no—
373C4⋊C4—122no—
374C22⋊C4—68no—
375C22⋊C4—124no—
376C22⋊C4—124no—
377C22⋊C4—124no—
378C42—22no—
379C42—22no—
380C42—21no—
381C42—21no—
382C42—68no—
383C42D3×D412yes—
384C42D3×D412yes—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—22no—
388C22×C3—41no—
389C22×C3—41no—
390C22×C3—41no—
391C22×C3—41no—
392C22×C3C2×#12011yescharacteristic
393C22×C3#53211yescharacteristic
394C22×C3#53511yescharacteristic
395C22×C3C22×SD1611yescharacteristic
396C22×C3C2×#13511yescharacteristic
397C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
398C22×C3C2×#13711yescharacteristic
399C2×D3—82no—
400C2×D3—82no—
401C2×D3—82no—
402C2×D3—82no—
403C2×D3—82no—
404C2×D3—82no—
405C2×D3—82no—
406C3×C4—22no—
407C3×C4—41no—
408C3×C4—41no—
409C3×C4#44814yes—
410C3×C4#44914yes—
411C3×C4#54612yes—
412C3×C4#55012yes—
413Dic3—44no—
414Dic3—44no—
415Dic3—82no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—122no—
423C23—122no—
424C23—122no—
425C23—122no—
426C23—122no—
427C23—122no—
428C23—122no—
429C23C2×#20412yes—
430C23C22×D1211yescharacteristic
431Q8—68no—
432Q8—128no—
433Q8—124no—
434D4—124no—
435D4—124no—
436D4—248no—
437D4—244no—
438D4—244no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—28no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—64no—
452C2×C4—64no—
453C2×C4—122no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4—122no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4#88212yes—
461C2×C4#88212yes—
462C2×C4C2×#20412yes—
463C2×C4C2×#20412yes—
464C2×C4#90412yes—
465C2×C4#90812yes—
466C2×C4#90912yes—
467C2×C4C2×#23512yes—
468C2×C4C2×D3×D411yescharacteristic
469C2×C4C2×D3×D411yescharacteristic
470C2×C4#100912yes—
471C8—24no—
472C8—42no—
473C2×C3—42no—
474C2×C3#243111yescharacteristic
475C2×C3C2×#44811yescharacteristic
476C2×C3C2×#44911yescharacteristic
477C2×C3#316711yescharacteristic
478C2×C3#318211yescharacteristic
479C2×C3#320811yescharacteristic
480C2×C3#321011yescharacteristic
481D3—162no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—242no—
490C22—242no—
491C22—242no—
492C22—242no—
493C22—242no—
494C22—242no—
495C22—242no—
496C22#547711yescharacteristic
497C22C2×#88211yescharacteristic
498C22#552511yescharacteristic
499C22C22×#20411yescharacteristic
500C22C2×#90411yescharacteristic
501C22C2×#90811yescharacteristic
502C22C2×#90911yescharacteristic
503C4—22no—
504C4—41no—
505C4—41no—
506C4—124no—
507C4—124no—
508C4—242no—
509C4#473414yes—
510C4#473814yes—
511C4#555012yes—
512C4#555312yes—
513C3#2559811yescharacteristic, complemented, Sylow
514C2—42no—
515C2—482no—
516C2#7358611yescharacteristic
517C2#7846811yescharacteristic
518C2#7853311yescharacteristic
519C2C2×#473411yescharacteristic
520C2C2×#473811yescharacteristic
521C2#7930511yescharacteristic
522C2#7930611yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×C22⋊C8C2
socleC23×C3C22×D4
prime radical#82081C2
hypercenterC4×C22⋊C8D3
hypocenterC3#25598
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊C8

Complements

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

NH ≅ G/Ntype
1. C2×#5470C2semidirect
3. C2×#4693C2semidirect
4. C2×#4694C2semidirect
5. C3×C4×C22⋊C8C2semidirect
6. #69962C2semidirect
7. #69777C2semidirect
8. #69771C2semidirect
9. #68686C2semidirect
48. C4×C22⋊C8D3semidirect
95. #869D4semidirect
96. #869D4semidirect
145. C22×C3×C4SD16semidirect
350. C22×C4#204semidirect
513. C3#25598semidirect

Maximal subgroups

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

subgroupindex
C2×#54702
#820812
C2×#46932
C2×#46942
C3×C4×C22⋊C82
#699622
#697772
#697712
#686862
#255983