Finite Group Databasebeta

← #10829233

Subgroups — #10829233

Order 768.

1204 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#555C211yesmaximal, characteristic, complemented
2C2×#5087C211yesmaximal, characteristic, complemented
3#82415C211yesmaximal, characteristic, complemented
4C2×#5014C211yesmaximal, characteristic, complemented
5#71788C211yesmaximal, characteristic, complemented
6#71684C212yesmaximal, complemented
7#71679C212yesmaximal, complemented
8#71615C211yesmaximal, characteristic, complemented
9#71605C211yesmaximal, characteristic, complemented
10#71586C212yesmaximal, complemented
11#71576C212yesmaximal, complemented
12#24129—31nomaximal, Sylow
13C3×#555—24no—
14C3×#555—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#124—22no—
17C2×C3×#121—21no—
18C2×C3×#121C2212yes—
19C2×C3×C4×D4C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×C3×#117—21no—
22C2×C3×#117C2212yes—
23C22×#213C2211yescharacteristic
24C2×#930C2212yes—
25C2×C4×D12C2211yescharacteristic
26#5223—21no—
27C2×#832C2211yescharacteristic
28C2×#832C2212yes—
29#5212C2212yes—
30C2×#830C2212yes—
31C2×#830C2212yes—
32#5087—24no—
33#5014—24no—
34#4992—21no—
35C2×#809C2212yes—
36C2×#809C2211yescharacteristic
37C2×#808C2211yescharacteristic
38C2×#807C2212yes—
39#4956—22no—
40#4936C2211yescharacteristic
41C2×C4⋊(C3⋊C8)C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4529C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C2×#555D311yescharacteristic, complemented
47C2×#474—31no—
48C2×#442—31no—
49#2535—31no—
50#2033—32no—
51#1999—32no—
52#1993—31no—
53#1955—31no—
54#1875—32no—
55#1854—32no—
56#1804—31no—
57C22×C3×Q8—21no—
58C22×C3×D4—22no—
59C22×C3×D4D411yescharacteristic
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C23×C3×C4C2312yes—
63C22×#67—22no—
64C22×D12C2311yescharacteristic
65C22×C4×D3—21no—
66C3×C4⋊Q8—22no—
67C3×C4⋊Q8D414yescomplemented
68C3×#124—216no—
69C3×#121—28no—
70C3×#121—24no—
71C3×#121—24no—
72C3×C4×D4—22no—
73C3×C4×D4—22no—
74C3×C4×D4—22no—
75C3×C4×D4D414yescomplemented
76C3×#117—28no—
77C3×#117—24no—
78C3×#117—24no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4D411yescharacteristic
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4D412yes—
87C2×C3×C42—22no—
88C2×C3×C42C2311yescharacteristic
89C2×#217—22no—
90C2×#213—22no—
91C2×#213—22no—
92C2×#213—22no—
93#930—24no—
94#930—42no—
95C2×(C2×D3)⋊C4—22no—
96C2×(C2×D3)⋊C4—21no—
97C2×C4⋊Dic3C2312yes—
98C2×C4⋊Dic3C2311yescharacteristic
99C22×C3⋊C8C2311yescharacteristic
100C22×C3⋊C8C2312yes—
101C22×C3⋊C8C2311yescharacteristic
102C4×D12—22no—
103C4×D12—22no—
104#832—22no—
105#832—42no—
106C3⋊#95—21no—
107C3⋊#95—21no—
108C3⋊#95—22no—
109#830—24no—
110#830—42no—
111#809—22no—
112#809—24no—
113#808—22no—
114#807—24no—
115C4⋊(C3⋊C8)—22no—
116#803—22no—
117C22×D8—31no—
118#555—24no—
119#555—24no—
120C2×C4⋊Q8C2×D311yescharacteristic
121C2×#124—22no—
122C2×#121—21no—
123C2×#121—32no—
124C2×#121C2×D312yes—
125C2×C4×D4—31no—
126C2×C4×D4C2×D311yescharacteristic
127C2×C4×D4C2×D311yescharacteristic
128C2×#117—21no—
129C2×#117C2×D312yes—
130#474—64no—
131#442—64no—
132C2×#107—32no—
133C2×#106—31no—
134C2×C4⋊C8—31no—
135C2×#102—31no—
136C2×#102—31no—
137C2×#102—32no—
138C2×#102—32no—
139C2×C22⋊C8—32no—
140C2×C22⋊C8—32no—
141C2×#97—31no—
142#397—61no—
143#392—61no—
144#386—62no—
145#382—31no—
146#380—32no—
147#340—32no—
148#340—32no—
149#340—32no—
150C24×C3—21no—
151C24×C3C2×D412yes—
152C23×D3—21no—
153C2×C3×Q8—24no—
154C2×C3×Q8—82no—
155C2×C3×D4—28no—
156C2×C3×D4—22no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—42no—
163C2×C3×D4—82no—
164C2×C3×D4—82no—
165C2×C3×D4C2×D412yes—
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4—24no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—44no—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2411yescharacteristic
180C2×#67—44no—
181C2×#67—44no—
182C2×#67—82no—
183C22×Dic3—22no—
184C22×Dic3—21no—
185C2×D12—21no—
186C2×D12—21no—
187C2×D12—21no—
188C2×D12—21no—
189C2×D12—21no—
190C2×D12#4112yes—
191C2×C4×D3—42no—
192C2×C4×D3—82no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4C2×D414yes—
198C3×C4⋊C4C2×D412yes—
199C3×C22⋊C4—28no—
200C3×C22⋊C4—28no—
201C3×C22⋊C4—24no—
202C3×C22⋊C4—24no—
203C3×C42—28no—
204C3×C42C2×D412yes—
205C3×C42C2×D412yes—
206#217—82no—
207#213—28no—
208(C2×D3)⋊C4—44no—
209(C2×D3)⋊C4—42no—
210C4⋊Dic3—21no—
211C4⋊Dic3—21no—
212C4⋊Dic3—22no—
213C4⋊Dic3—22no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—22no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—21no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—21no—
222C2×C3⋊C8#4112yes—
223C22×Q8—21no—
224C22×D4—22no—
225C22×D4—31no—
226C22×D4—62no—
227C22×D4#6711yescharacteristic
228C22×D4C22×D311yescharacteristic
229C23×C4—21no—
230C23×C4—61no—
231C23×C4C22×D312yes—
232C2×D8—62no—
233C2×D8—62no—
234C2×D8—62no—
235C22×C8—31no—
236C22×C8—31no—
237C22×C8—32no—
238C4⋊Q8—22no—
239C4⋊Q8#6714yescomplemented
240#124—216no—
241#121—28no—
242#121—24no—
243#121—24no—
244#121—64no—
245#121—122no—
246C4×D4—22no—
247C4×D4—22no—
248C4×D4—22no—
249C4×D4—62no—
250C4×D4—62no—
251C4×D4#6714yescomplemented
252#117—28no—
253#117—24no—
254#117—24no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—31no—
257C2×C4⋊C4—32no—
258C2×C4⋊C4#6711yescharacteristic
259C2×C4⋊C4C22×D311yescharacteristic
260C2×C4⋊C4C22×D312yes—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—61no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4—62no—
267C2×C22⋊C4#6712yes—
268C2×C42—22no—
269C2×C42C22×D311yescharacteristic
270#107—64no—
271#106—62no—
272C4⋊C8—62no—
273#102—62no—
274#102—62no—
275#102—64no—
276#102—122no—
277C22⋊C8—64no—
278C22⋊C8—122no—
279#97—62no—
280#95—62no—
281#95—61no—
282#95—61no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—41no—
292C23×C3—41no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3C22×D411yescharacteristic
296C22×D3—22no—
297C22×D3—41no—
298C22×D3—41no—
299C22×D3—42no—
300C22×D3—42no—
301C3×Q8—84no—
302C3×D4—22no—
303C3×D4—22no—
304C3×D4—44no—
305C3×D4—41no—
306C3×D4—41no—
307C3×D4—88no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—24no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—48no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4#12012yes—
324C2×C3×C4#12012yes—
325C2×C3×C4C22×D411yescharacteristic
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C2×#4111yescharacteristic
328C2×C3×C4C2×#4111yescharacteristic
329C2×C3×C4H(5,𝔽2)11yescharacteristic
330C2×C3×C42−1+411yescharacteristic
331#67—88no—
332C2×Dic3—41no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337D12—22no—
338D12—22no—
339D12—41no—
340D12—41no—
341C4×D3—84no—
342C3⋊C8—22no—
343C3⋊C8—41no—
344C3⋊C8—42no—
345C24—21no—
346C24—61no—
347C24C2×#6712yes—
348C2×Q8—24no—
349C2×Q8—82no—
350C2×D4—28no—
351C2×D4—22no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—32no—
358C2×D4—42no—
359C2×D4—61no—
360C2×D4—61no—
361C2×D4—61no—
362C2×D4—61no—
363C2×D4—61no—
364C2×D4—82no—
365C2×D4—82no—
366C2×D4—124no—
367C2×D4—124no—
368C2×D4—242no—
369C2×D4C2×#6712yes—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—44no—
380C22×C4—62no—
381C22×C4—61no—
382C22×C4—122no—
383C22×C4—242no—
384C22×C4C2×#6712yes—
385C22×C4C2×#6711yescharacteristic
386C22×C4C2×#6711yescharacteristic
387C22×C4C23×D311yescharacteristic
388D8—68no—
389C2×C8—32no—
390C2×C8—61no—
391C2×C8—61no—
392C2×C8—62no—
393C2×C8—61no—
394C2×C8—62no—
395C2×C8—61no—
396C2×C8—61no—
397C2×C8—62no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—62no—
403C4⋊C4—62no—
404C4⋊C4—61no—
405C4⋊C4—61no—
406C4⋊C4C2×#6712yes—
407C4⋊C4C2×#6714yes—
408C22⋊C4—28no—
409C22⋊C4—28no—
410C22⋊C4—24no—
411C22⋊C4—24no—
412C22⋊C4—124no—
413C22⋊C4—122no—
414C22⋊C4—242no—
415C42—28no—
416C42C2×#6712yes—
417C42C2×#6712yes—
418C22×C3—42no—
419C22×C3—42no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3—82no—
429C22×C3—82no—
430C22×C3C2×#12011yescharacteristic
431C22×C3#54611yescharacteristic
432C22×C3#54711yescharacteristic
433C22×C3#57512yes—
434C22×C3#57612yes—
435C2×D3—42no—
436C2×D3—44no—
437C2×D3—82no—
438C2×D3—81no—
439C2×D3—81no—
440C3×C4—22no—
441C3×C4—22no—
442C3×C4—42no—
443C3×C4—42no—
444C3×C4—44no—
445C3×C4#52511yescharacteristic
446C3×C4#53711yescharacteristic
447C3×C4#54811yescharacteristic
448C3×C4#55011yescharacteristic
449Dic3—81no—
450Dic3—82no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—42no—
460C23—41no—
461C23—41no—
462C23—42no—
463C23—62no—
464C23—122no—
465C23—121no—
466C23—122no—
467C23—121no—
468C23C22×#6711yescharacteristic
469Q8—84no—
470D4—22no—
471D4—22no—
472D4—44no—
473D4—41no—
474D4—41no—
475D4—62no—
476D4—62no—
477D4—88no—
478D4—121no—
479D4—121no—
480D4—248no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—48no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—121no—
501C2×C4—244no—
502C2×C4#95312yes—
503C2×C4#95312yes—
504C2×C4D3×#4111yescharacteristic
505C2×C4D3×#4111yescharacteristic
506C2×C4#100911yescharacteristic
507C2×C4#101011yescharacteristic
508C2×C4C22×#6711yescharacteristic
509C2×C4C22×#6711yescharacteristic
510C8—62no—
511C8—121no—
512C8—122no—
513C2×C3—82no—
514C2×C3—82no—
515C2×C3#240711yescharacteristic
516C2×C3#300511yescharacteristic
517C2×C3#300811yescharacteristic
518C2×C3#331912yes—
519C2×C3#334712yes—
520D3—82no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—82no—
532C22—82no—
533C22—122no—
534C22—124no—
535C22—241no—
536C22—241no—
537C22—242no—
538C22#582212yes—
539C22#582411yescharacteristic
540C22#582911yescharacteristic
541C22#583512yes—
542C22C2×#95311yescharacteristic
543C4—22no—
544C4—22no—
545C4—42no—
546C4—42no—
547C4—44no—
548C4—241no—
549C4—242no—
550C4#582811yescharacteristic
551C4#583111yescharacteristic
552C4#583311yescharacteristic
553C4#583411yescharacteristic
554C3#2412911yescharacteristic, complemented, Sylow
555C2—82no—
556C2—82no—
557C2—242no—
558C2#8364812yes—
559C2#8365312yes—
560C2#8408111yescharacteristic
561C2#8409711yescharacteristic
562C2#8410111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#555C2
socleC23×C3C22×D4
hypercenterC2×#555D3
hypocenterC3#24129
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. C2×C3×#555C2semidirect
2. C2×#5087C2semidirect
3. #82415C2semidirect
4. C2×#5014C2semidirect
5. #71788C2semidirect
6. #71684C2semidirect
7. #71679C2semidirect
8. #71615C2semidirect
9. #71605C2semidirect
10. #71586C2semidirect
11. #71576C2semidirect
46. C2×#555D3semidirect
67. C3×C4⋊Q8D4semidirect
75. C3×C4×D4D4semidirect
239. C4⋊Q8#67semidirect
251. C4×D4#67semidirect
554. C3#24129semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5552
C2×#50872
#824152
C2×#50142
#717882
#716842
#716792
#716152
#716052
#715862
#715762
#241293