Finite Group Databasebeta

← #10829224

Subgroups — #10829224

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×#5555C211yesmaximal, characteristic, complemented
2C3×#2535C211yesmaximal, characteristic, complemented
3C2×#5008C211yesmaximal, characteristic, complemented
4C2×#5014C211yesmaximal, characteristic, complemented
5#71684C212yesmaximal, complemented
6#71677C212yesmaximal, complemented
7#71615C211yesmaximal, characteristic, complemented
8#69556C211yesmaximal, characteristic, complemented
9#69555C211yesmaximal, characteristic, complemented
10#69524C212yesmaximal, complemented
11#69505C212yesmaximal, complemented
12#24128—31nomaximal, Sylow
13C22×#237—21no—
14C2×C3×#121C2212yes—
15C2×C3×C4×D4C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C22×#213C2211yescharacteristic
18C2×#930C2212yes—
19C22×C4⋊Dic3C2212yes—
20#5555—24no—
21#5555—24no—
22C2×#897—21no—
23C2×#886—22no—
24C2×C4×D12C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C3×#397—21no—
27C3×#392—21no—
28C3×#386—22no—
29C3×#382C2211yescharacteristic
30C3×#380C2212yes—
31C2×#832C2212yes—
32C2×#832C2211yescharacteristic
33C2×#830C2212yes—
34C2×#830C2212yes—
35#5014—24no—
36#5008—24no—
37C2×#809C2211yescharacteristic
38C2×#809C2212yes—
39C2×#808C2211yescharacteristic
40C2×#808C2212yes—
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C2×#547—31no—
47C2×#476—31no—
48C2×#442—31no—
49#2535D311yescharacteristic, complemented
50#2034—32no—
51#1999—32no—
52#1993—31no—
53#1955—31no—
54#1874—32no—
55#1853—32no—
56#1804—31no—
57C22×C3×D4—22no—
58C22×C3×D4D411yescharacteristic
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4—21no—
61C23×C3×C4C2312yes—
62C22×#67—22no—
63C23×Dic3—22no—
64C2×#237—28no—
65C2×#237—48no—
66C22×D12C2311yescharacteristic
67C22×C4×D3—21no—
68C22×Dic6—21no—
69C3×#121—24no—
70C3×#121—42no—
71C3×C4×D4—22no—
72C3×C4×D4—22no—
73C3×C4×D4—22no—
74C3×C4×D4—22no—
75C2×C3×C4⋊C4D411yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4D412yes—
82C2×C3×C42C2311yescharacteristic
83C2×#213—22no—
84C2×#213—22no—
85C2×#213—22no—
86#930—24no—
87#930—24no—
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—22no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3—22no—
92C2×C4⋊Dic3—24no—
93C2×C4⋊Dic3—24no—
94C2×C4⋊Dic3C2312yes—
95C2×C4⋊Dic3C2311yescharacteristic
96C22×C3⋊C8C2311yescharacteristic
97C22×C3⋊C8C2312yes—
98C22×C3⋊C8C2311yescharacteristic
99#897—28no—
100#886—216no—
101C4×D12—22no—
102C4×D12D414yescomplemented
103#869—22no—
104#869D414yescomplemented
105C3×#95—22no—
106C3×#95—21no—
107C3×#95—21no—
108#832—24no—
109#832—22no—
110#830—24no—
111#830—42no—
112#809—22no—
113#809—42no—
114#808—24no—
115#808—22no—
116C4⋊(C3⋊C8)—22no—
117#803—22no—
118C22×#41—61no—
119C22×D8—31no—
120#547—64no—
121#547—64no—
122C2×C4⋊Q8—31no—
123C2×#123—62no—
124C2×C22⋊Q8—61no—
125C2×#121—32no—
126C2×#121C2×D312yes—
127C2×C4×D4—31no—
128C2×C4×D4C2×D311yescharacteristic
129C2×C4×D4C2×D311yescharacteristic
130C22×C4⋊C4—32no—
131#476—64no—
132#442—64no—
133C2×#106—31no—
134C2×#106—32no—
135C2×C4⋊C8—31no—
136C2×#102—31no—
137C2×#102—32no—
138C2×#102—31no—
139C2×#102—32no—
140C2×C22⋊C8—32no—
141C2×C22⋊C8—32no—
142C2×#97—31no—
143#397—21no—
144#392—21no—
145#386—22no—
146#382C2×D311yescharacteristic
147#380C2×D312yes—
148#340—32no—
149#340—32no—
150#340—32no—
151C24×C3—21no—
152C24×C3C2×D412yes—
153C23×D3—21no—
154C2×C3×D4—24no—
155C2×C3×D4—22no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—44no—
162C2×C3×D4—44no—
163C2×C3×D4—82no—
164C2×C3×D4#4112yescomplemented
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—42no—
174C22×C3×C4—82no—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2411yescharacteristic
179C2×#67—28no—
180C2×#67—44no—
181C2×#67—44no—
182C22×Dic3—28no—
183C22×Dic3—22no—
184C22×Dic3—21no—
185C22×Dic3—44no—
186C22×Dic3—44no—
187#237—432no—
188C2×D12—21no—
189C2×D12—21no—
190C2×D12—21no—
191C2×D12—21no—
192C2×D12—21no—
193C2×D12C2×D412yes—
194C2×C4×D3—24no—
195C2×C4×D3—44no—
196C2×Dic6—24no—
197C2×Dic6—44no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C22⋊C4—24no—
204C3×C22⋊C4—44no—
205C3×C22⋊C4—42no—
206C3×C22⋊C4—82no—
207C3×C42C2×D412yes—
208C3×C42C2×D412yes—
209#213—28no—
210(C2×D3)⋊C4—28no—
211(C2×D3)⋊C4—24no—
212C4⋊Dic3—28no—
213C4⋊Dic3—24no—
214C4⋊Dic3—22no—
215C4⋊Dic3—21no—
216C4⋊Dic3C2×D414yes—
217C4⋊Dic3C2×D412yes—
218C2×C3⋊C8—21no—
219C2×C3⋊C8—21no—
220C2×C3⋊C8—21no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8—21no—
225C2×C3⋊C8—21no—
226C2×C3⋊C8#4112yescomplemented
227C2×#41—68no—
228C2×#41—128no—
229C22×Q8—61no—
230C22×D4—22no—
231C22×D4—31no—
232C22×D4—62no—
233C22×D4#6711yescharacteristic
234C22×D4C22×D311yescharacteristic
235C23×C4—21no—
236C23×C4—62no—
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—34no—
246C4⋊Q8—62no—
247#123—616no—
248C22⋊Q8—68no—
249#121—24no—
250#121—42no—
251#121—64no—
252#121—64no—
253C4×D4—22no—
254C4×D4—22no—
255C4×D4—22no—
256C4×D4—22no—
257C4×D4—34no—
258C4×D4—62no—
259C2×C4⋊C4—31no—
260C2×C4⋊C4—32no—
261C2×C4⋊C4—64no—
262C2×C4⋊C4—64no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4#6711yescharacteristic
266C2×C4⋊C4C22×D311yescharacteristic
267C2×C4⋊C4C22×D312yes—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—61no—
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—22no—
286#95—21no—
287#95—21no—
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×D4—28no—
307C3×D4—22no—
308C3×D4—22no—
309C3×D4—41no—
310C3×D4—41no—
311C3×D4—88no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—41no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—84no—
327C2×C3×C4#12012yes—
328C2×C3×C4#12012yes—
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4C22×D411yescharacteristic
331C2×C3×C4C2×#4111yescharacteristic
332C2×C3×C4C2×#4111yescharacteristic
333C2×C3×C4H(5,𝔽2)11yescharacteristic
334C2×C3×C42−1+411yescharacteristic
335#67—416no—
336C2×Dic3—24no—
337C2×Dic3—22no—
338C2×Dic3—416no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C2×Dic3—42no—
342D12—22no—
343D12—22no—
344D12—41no—
345D12—41no—
346C4×D3—48no—
347Dic6—48no—
348C3⋊C8—22no—
349C3⋊C8—41no—
350C3⋊C8—42no—
351C24—21no—
352C24—61no—
353C24C2×#6712yes—
354#41—1232no—
355C2×Q8—64no—
356C2×Q8—124no—
357C2×D4—24no—
358C2×D4—22no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—32no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—68no—
368C2×D4—61no—
369C2×D4—61no—
370C2×D4—61no—
371C2×D4—61no—
372C2×D4—61no—
373C2×D4—82no—
374C2×D4—124no—
375C2×D4—124no—
376C2×D4#23712yescomplemented
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—42no—
386C22×C4—68no—
387C22×C4—61no—
388C22×C4—62no—
389C22×C4—64no—
390C22×C4—82no—
391C22×C4—124no—
392C22×C4—124no—
393C22×C4—124no—
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—61no—
403C2×C8—62no—
404C2×C8—61no—
405C2×C8—62no—
406C2×C8—62no—
407C2×C8—61no—
408C4⋊C4—22no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—32no—
414C4⋊C4—34no—
415C4⋊C4—68no—
416C4⋊C4—64no—
417C4⋊C4—61no—
418C4⋊C4—62no—
419C22⋊C4—24no—
420C22⋊C4—44no—
421C22⋊C4—42no—
422C22⋊C4—68no—
423C22⋊C4—64no—
424C22⋊C4—82no—
425C42D3×D412yes—
426C42D3×D412yes—
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—44no—
436C22×C3—82no—
437C22×C3—81no—
438C22×C3—81no—
439C22×C3C2×#12011yescharacteristic
440C22×C3#54611yescharacteristic
441C22×C3#54711yescharacteristic
442C22×C3C2×AGL(1,ℤ/8)12yes—
443C22×C3#57712yes—
444C2×D3—42no—
445C2×D3—42no—
446C2×D3—42no—
447C2×D3—82no—
448C2×D3—82no—
449C3×C4—22no—
450C3×C4—22no—
451C3×C4—42no—
452C3×C4—81no—
453C3×C4—82no—
454C3×C4#52511yescharacteristic
455C3×C4#53711yescharacteristic
456C3×C4#54811yescharacteristic
457C3×C4#55011yescharacteristic
458Dic3—44no—
459Dic3—42no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—41no—
468C23—42no—
469C23—41no—
470C23—42no—
471C23—62no—
472C23—121no—
473C23—122no—
474C23—121no—
475C23—122no—
476C23#94914yes—
477C23C22×#6711yescharacteristic
478Q8—128no—
479D4—28no—
480D4—22no—
481D4—22no—
482D4—41no—
483D4—41no—
484D4—62no—
485D4—62no—
486D4—88no—
487D4—1216no—
488D4—121no—
489D4—121no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—62no—
505C2×C4—64no—
506C2×C4—84no—
507C2×C4—128no—
508C2×C4—1216no—
509C2×C4—122no—
510C2×C4—122no—
511C2×C4—122no—
512C2×C4#93712yes—
513C2×C4#93712yes—
514C2×C4C2×D3×D411yescharacteristic
515C2×C4C2×D3×D411yescharacteristic
516C2×C4C2×#23711yescharacteristic
517C2×C4#100411yescharacteristic
518C2×C4D3×#4111yescharacteristic
519C2×C4#101011yescharacteristic
520C8—62no—
521C8—122no—
522C8—121no—
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—44no—
540C22—82no—
541C22—81no—
542C22—81no—
543C22—122no—
544C22—122no—
545C22—122no—
546C22—242no—
547C22—242no—
548C22C2×#93711yescharacteristic
549C22#580611yescharacteristic
550C22C2×#94912yes—
551C22#582411yescharacteristic
552C22#583612yes—
553C4—22no—
554C4—22no—
555C4—42no—
556C4—82no—
557C4—81no—
558C4—122no—
559C4—124no—
560C4#559811yescharacteristic
561C4#559911yescharacteristic
562C4#560611yescharacteristic
563C4#560811yescharacteristic
564C3#2412811yescharacteristic, complemented, Sylow
565C2—44no—
566C2—82no—
567C2—242no—
568C2#8270111yescharacteristic
569C2#8292711yescharacteristic
570C2#8339011yescharacteristic
571C2#8349612yes—
572C2#8360812yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2535C2
socleC23×C3C22×D4
hypercenter#2535D3
hypocenterC3#24128
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2535

Complements

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

NH ≅ G/Ntype
1. C2×#5555C2semidirect
2. C3×#2535C2semidirect
3. C2×#5008C2semidirect
4. C2×#5014C2semidirect
5. #71684C2semidirect
6. #71677C2semidirect
7. #71615C2semidirect
8. #69556C2semidirect
9. #69555C2semidirect
10. #69524C2semidirect
11. #69505C2semidirect
49. #2535D3semidirect
102. C4×D12D4semidirect
104. #869D4semidirect
164. C2×C3×D4#41semidirect
226. C2×C3⋊C8#41semidirect
376. C2×D4#237semidirect
564. C3#24128semidirect

Maximal subgroups

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

subgroupindex
C2×#55552
C3×#25352
C2×#50082
C2×#50142
#716842
#716772
#716152
#695562
#695552
#695242
#695052
#241283