Finite Group Databasebeta

← #10829222

Subgroups — #10829222

Order 768.

1400 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#547C211yesmaximal, characteristic, complemented
2C2×#5084C211yesmaximal, characteristic, complemented
3#82415C211yesmaximal, characteristic, complemented
4C2×#5014C211yesmaximal, characteristic, complemented
5#71788C211yesmaximal, characteristic, complemented
6#71684C212yesmaximal, complemented
7#71680C212yesmaximal, complemented
8#71615C211yesmaximal, characteristic, complemented
9#71605C211yesmaximal, characteristic, complemented
10#71560C212yesmaximal, complemented
11#71556C212yesmaximal, complemented
12#24128—31nomaximal, Sylow
13C22×C3×#41—21no—
14C3×#547—24no—
15C3×#547—24no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#123—22no—
18C2×C3×C22⋊Q8—21no—
19C2×C3×#121C2212yes—
20C2×C3×C4×D4C2211yescharacteristic
21C2×C3×C4×D4C2211yescharacteristic
22C22×C3×C4⋊C4C2212yes—
23C22×#213C2211yescharacteristic
24C2×#930C2212yes—
25C2×C4×D12C2211yescharacteristic
26#5223—21no—
27C2×#832C2211yescharacteristic
28C2×#832C2212yes—
29#5212C2212yes—
30C2×#830C2212yes—
31C2×#830C2212yes—
32#5084—24no—
33#5014—24no—
34#4992—21no—
35C2×#809C2212yes—
36C2×#809C2211yescharacteristic
37C2×#808C2212yes—
38C2×#808C2211yescharacteristic
39#4956—22no—
40#4936C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C2×#547D311yescharacteristic, complemented
47C2×#476—31no—
48C2×#442—31no—
49#2535—31no—
50#2034—32no—
51#1999—32no—
52#1993—31no—
53#1955—31no—
54#1874—32no—
55#1853—32no—
56#1804—31no—
57C2×C3×#41—28no—
58C2×C3×#41—48no—
59C22×C3×Q8—21no—
60C22×C3×D4—22no—
61C22×C3×D4D411yescharacteristic
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—22no—
64C23×C3×C4—21no—
65C23×C3×C4C2312yes—
66C22×#67—22no—
67C22×D12C2311yescharacteristic
68C22×C4×D3—21no—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8D414yescomplemented
71C3×#123—216no—
72C3×C22⋊Q8—28no—
73C3×#121—24no—
74C3×#121—24no—
75C3×C4×D4—22no—
76C3×C4×D4—22no—
77C3×C4×D4—22no—
78C3×C4×D4D414yescomplemented
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—24no—
82C2×C3×C4⋊C4—24no—
83C2×C3×C4⋊C4D411yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2312yes—
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4D412yes—
89C2×C3×C42C2311yescharacteristic
90C2×#217—22no—
91C2×#213—22no—
92C2×#213—22no—
93C2×#213—22no—
94#930—24no—
95#930—42no—
96C2×(C2×D3)⋊C4—22no—
97C2×(C2×D3)⋊C4—21no—
98C2×C4⋊Dic3C2312yes—
99C2×C4⋊Dic3C2311yescharacteristic
100C22×C3⋊C8C2311yescharacteristic
101C22×C3⋊C8C2312yes—
102C22×C3⋊C8C2311yescharacteristic
103C4×D12—22no—
104C4×D12—22no—
105#832—22no—
106#832—42no—
107C3⋊#95—21no—
108C3⋊#95—21no—
109C3⋊#95—22no—
110#830—24no—
111#830—42no—
112#809—22no—
113#809—24no—
114#808—22no—
115#808—24no—
116C4⋊(C3⋊C8)—22no—
117#803—22no—
118C22×#41—21no—
119C22×D8—31no—
120#547—24no—
121#547—24no—
122C2×C4⋊Q8C2×D311yescharacteristic
123C2×#123—22no—
124C2×C22⋊Q8—21no—
125C2×#121—32no—
126C2×#121C2×D312yes—
127C2×C4×D4—31no—
128C2×C4×D4C2×D311yescharacteristic
129C2×C4×D4C2×D311yescharacteristic
130C22×C4⋊C4C2×D312yes—
131#476—64no—
132#442—64no—
133C2×#106—31no—
134C2×#106—32no—
135C2×C4⋊C8—31no—
136C2×#102—31no—
137C2×#102—31no—
138C2×#102—32no—
139C2×#102—32no—
140C2×C22⋊C8—32no—
141C2×C22⋊C8—32no—
142C2×#97—31no—
143#397—61no—
144#392—61no—
145#386—62no—
146#382—31no—
147#380—32no—
148#340—32no—
149#340—32no—
150#340—32no—
151C24×C3—21no—
152C24×C3C2×D412yes—
153C23×D3—21no—
154C3×#41—432no—
155C2×C3×Q8—24no—
156C2×C3×Q8—44no—
157C2×C3×D4—24no—
158C2×C3×D4—28no—
159C2×C3×D4—22no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4—21no—
165C2×C3×D4—44no—
166C2×C3×D4—44no—
167C2×C3×D4C2×D412yes—
168C22×C3×C4—28no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4—24no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—44no—
179C22×C3×C4—44no—
180C22×C3×C4—44no—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2411yescharacteristic
185C2×#67—44no—
186C2×#67—44no—
187C2×#67—82no—
188C22×Dic3—22no—
189C22×Dic3—21no—
190C2×D12—21no—
191C2×D12—21no—
192C2×D12—21no—
193C2×D12—21no—
194C2×D12—21no—
195C2×D12#4112yescomplemented
196C2×C4×D3—42no—
197C2×C4×D3—82no—
198C3×C4⋊C4—28no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4C2×D412yes—
204C3×C4⋊C4C2×D414yes—
205C3×C22⋊C4—28no—
206C3×C22⋊C4—24no—
207C3×C22⋊C4—24no—
208C3×C42C2×D412yes—
209C3×C42C2×D412yes—
210#217—82no—
211#213—28no—
212(C2×D3)⋊C4—44no—
213(C2×D3)⋊C4—42no—
214C4⋊Dic3—21no—
215C4⋊Dic3—21no—
216C4⋊Dic3—22no—
217C4⋊Dic3—22no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8—21no—
220C2×C3⋊C8—21no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—21no—
224C2×C3⋊C8—22no—
225C2×C3⋊C8—21no—
226C2×C3⋊C8#4112yescomplemented
227C2×#41—28no—
228C2×#41—48no—
229C22×Q8—21no—
230C22×D4—22no—
231C22×D4—31no—
232C22×D4—62no—
233C22×D4#6711yescharacteristic
234C22×D4C22×D311yescharacteristic
235C23×C4—22no—
236C23×C4—21no—
237C23×C4—61no—
238C23×C4C22×D312yes—
239C2×D8—62no—
240C2×D8—62no—
241C2×D8—62no—
242C22×C8—31no—
243C22×C8—31no—
244C22×C8—32no—
245C4⋊Q8—22no—
246C4⋊Q8#6714yescomplemented
247#123—216no—
248C22⋊Q8—28no—
249#121—24no—
250#121—24no—
251#121—64no—
252#121—122no—
253C4×D4—22no—
254C4×D4—22no—
255C4×D4—22no—
256C4×D4—62no—
257C4×D4—62no—
258C4×D4#6714yescomplemented
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—32no—
265C2×C4⋊C4#6711yescharacteristic
266C2×C4⋊C4C22×D311yescharacteristic
267C2×C4⋊C4C22×D312yes—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—61no—
271C2×C22⋊C4—62no—
272C2×C22⋊C4—62no—
273C2×C22⋊C4#6712yes—
274C2×C42C22×D311yescharacteristic
275#106—62no—
276#106—64no—
277C4⋊C8—62no—
278#102—62no—
279#102—62no—
280#102—64no—
281#102—122no—
282C22⋊C8—64no—
283C22⋊C8—122no—
284#97—62no—
285#95—62no—
286#95—61no—
287#95—61no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—41no—
296C23×C3—41no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3AGL(1,ℤ/8)14yes—
300C23×C3C22×D411yescharacteristic
301C22×D3—22no—
302C22×D3—41no—
303C22×D3—41no—
304C22×D3—42no—
305C22×D3—42no—
306C3×Q8—48no—
307C3×D4—28no—
308C3×D4—22no—
309C3×D4—22no—
310C3×D4—416no—
311C3×D4—41no—
312C3×D4—41no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—24no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—48no—
321C2×C3×C4—416no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4#12012yes—
330C2×C3×C4#12012yes—
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C2×#4111yescharacteristic
334C2×C3×C4C2×#4111yescharacteristic
335C2×C3×C4H(5,𝔽2)11yescharacteristic
336C2×C3×C42−1+411yescharacteristic
337#67—88no—
338C2×Dic3—41no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343D12—22no—
344D12—22no—
345D12—41no—
346D12—41no—
347C4×D3—84no—
348C3⋊C8—22no—
349C3⋊C8—41no—
350C3⋊C8—42no—
351C24—21no—
352C24—61no—
353C24C2×#6712yes—
354#41—432no—
355C2×Q8—24no—
356C2×Q8—44no—
357C2×D4—24no—
358C2×D4—28no—
359C2×D4—22no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—32no—
366C2×D4—44no—
367C2×D4—44no—
368C2×D4—61no—
369C2×D4—61no—
370C2×D4—61no—
371C2×D4—61no—
372C2×D4—61no—
373C2×D4—124no—
374C2×D4—124no—
375C2×D4—242no—
376C2×D4C2×#6712yes—
377C22×C4—28no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—62no—
391C22×C4—61no—
392C22×C4—122no—
393C22×C4—242no—
394C22×C4C2×#6712yes—
395C22×C4C2×#6711yescharacteristic
396C22×C4C2×#6711yescharacteristic
397C22×C4C23×D311yescharacteristic
398D8—68no—
399C2×C8—32no—
400C2×C8—61no—
401C2×C8—61no—
402C2×C8—62no—
403C2×C8—61no—
404C2×C8—62no—
405C2×C8—61no—
406C2×C8—61no—
407C2×C8—62no—
408C4⋊C4—28no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—21no—
412C4⋊C4—22no—
413C4⋊C4—62no—
414C4⋊C4—62no—
415C4⋊C4—61no—
416C4⋊C4—61no—
417C4⋊C4C2×#6712yes—
418C4⋊C4C2×#6714yes—
419C22⋊C4—28no—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—124no—
423C22⋊C4—122no—
424C22⋊C4—242no—
425C42C2×#6712yes—
426C42C2×#6712yes—
427C22×C3—24no—
428C22×C3—24no—
429C22×C3—24no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3—42no—
433C22×C3—42no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3—82no—
438C22×C3—82no—
439C22×C3C2×#12011yescharacteristic
440C22×C3#54611yescharacteristic
441C22×C3#54711yescharacteristic
442C22×C3C2×AGL(1,ℤ/8)12yes—
443C22×C3#57712yes—
444C2×D3—42no—
445C2×D3—44no—
446C2×D3—82no—
447C2×D3—81no—
448C2×D3—81no—
449C3×C4—22no—
450C3×C4—22no—
451C3×C4—42no—
452C3×C4—42no—
453C3×C4—44no—
454C3×C4#52511yescharacteristic
455C3×C4#53711yescharacteristic
456C3×C4#54811yescharacteristic
457C3×C4#55011yescharacteristic
458Dic3—81no—
459Dic3—82no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—42no—
468C23—41no—
469C23—41no—
470C23—42no—
471C23—62no—
472C23—122no—
473C23—121no—
474C23—122no—
475C23—121no—
476C23#93214yes—
477C23C22×#6711yescharacteristic
478Q8—48no—
479D4—28no—
480D4—22no—
481D4—22no—
482D4—416no—
483D4—41no—
484D4—41no—
485D4—62no—
486D4—62no—
487D4—121no—
488D4—121no—
489D4—248no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—48no—
498C2×C4—416no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4—122no—
509C2×C4—122no—
510C2×C4—121no—
511C2×C4—244no—
512C2×C4#95312yes—
513C2×C4#95312yes—
514C2×C4D3×#4111yescharacteristic
515C2×C4D3×#4111yescharacteristic
516C2×C4#100911yescharacteristic
517C2×C4#101011yescharacteristic
518C2×C4C22×#6711yescharacteristic
519C2×C4C22×#6711yescharacteristic
520C8—62no—
521C8—121no—
522C8—122no—
523C2×C3—44no—
524C2×C3—82no—
525C2×C3#240711yescharacteristic
526C2×C3#300311yescharacteristic
527C2×C3#301011yescharacteristic
528C2×C3#330612yes—
529C2×C3#334512yes—
530D3—82no—
531C22—24no—
532C22—24no—
533C22—24no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—82no—
542C22—82no—
543C22—124no—
544C22—122no—
545C22—242no—
546C22—241no—
547C22—241no—
548C22C2×#93212yes—
549C22#582411yescharacteristic
550C22#582911yescharacteristic
551C22#583712yes—
552C22C2×#95311yescharacteristic
553C4—22no—
554C4—22no—
555C4—42no—
556C4—44no—
557C4—42no—
558C4—242no—
559C4—241no—
560C4#582811yescharacteristic
561C4#583111yescharacteristic
562C4#583311yescharacteristic
563C4#583411yescharacteristic
564C3#2412811yescharacteristic, complemented, Sylow
565C2—44no—
566C2—82no—
567C2—242no—
568C2#8351912yes—
569C2#8352612yes—
570C2#8403811yescharacteristic
571C2#8409711yescharacteristic
572C2#8411311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#547C2
socleC23×C3C22×D4
hypercenterC2×#547D3
hypocenterC3#24128
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×C3×#547C2semidirect
2. C2×#5084C2semidirect
3. #82415C2semidirect
4. C2×#5014C2semidirect
5. #71788C2semidirect
6. #71684C2semidirect
7. #71680C2semidirect
8. #71615C2semidirect
9. #71605C2semidirect
10. #71560C2semidirect
11. #71556C2semidirect
46. C2×#547D3semidirect
70. C3×C4⋊Q8D4semidirect
78. C3×C4×D4D4semidirect
195. C2×D12#41semidirect
226. C2×C3⋊C8#41semidirect
246. C4⋊Q8#67semidirect
258. C4×D4#67semidirect
564. C3#24128semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5472
C2×#50842
#824152
C2×#50142
#717882
#716842
#716802
#716152
#716052
#715602
#715562
#241283