Finite Group Databasebeta

← #10829241

Subgroups — #10829241

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×#5734C211yesmaximal, characteristic, complemented
2C2×C3×#474C211yesmaximal, characteristic, complemented
3#82416C211yesmaximal, characteristic, complemented
4C2×#5014C211yesmaximal, characteristic, complemented
5#71788C211yesmaximal, characteristic, complemented
6#70634C212yesmaximal, complemented
7#70633C212yesmaximal, complemented
8#70480C211yesmaximal, characteristic, complemented
9#70471C211yesmaximal, characteristic, complemented
10#70204C212yesmaximal, complemented
11#70201C212yesmaximal, complemented
12#24129—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C2×#947—22no—
16C2×#938C2212yes—
17C22×#213C2211yescharacteristic
18#5734—24no—
19#5734—24no—
20C2×#895—21no—
21C2×#895C2212yes—
22C2×#892—21no—
23C2×#892C2212yes—
24C2×C4×D12C2211yescharacteristic
25C2×C4×D12C2211yescharacteristic
26C3×#474—24no—
27C2×C3×#107C2212yes—
28C2×C3×C4⋊C8C2211yescharacteristic
29C2×C3×#102C2212yes—
30C2×#832C2211yescharacteristic
31#5213—22no—
32C2×#821C2212yes—
33C2×(C2×D3)⋊C8C2212yes—
34C2×(C2×D3)⋊C8C2212yes—
35#5014—24no—
36#4987C2212yes—
37C2×#809C2211yescharacteristic
38C2×#808C2211yescharacteristic
39#4960—21no—
40#4953—21no—
41#4936C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4551C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C2×#555—31no—
47C2×#474D311yescharacteristic, complemented
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×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C22×#67—22no—
61C22×D12—22no—
62C22×D12D411yescharacteristic
63C22×D12C2311yescharacteristic
64C22×C4×D3—21no—
65C22×C4×D3C2312yes—
66C22×C3×C8C2312yes—
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8D414yescomplemented
69C3×C4×D4—22no—
70C3×C4×D4—22no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C22⋊C4—21no—
75C2×C3×C42C2311yescharacteristic
76C2×#217—22no—
77#947—216no—
78#938—24no—
79#938—42no—
80C2×#213—22no—
81C2×#213—22no—
82C2×#213—22no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—21no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4D412yes—
88C2×C4⋊Dic3D411yescharacteristic
89C2×C4⋊Dic3C2312yes—
90C2×C4⋊Dic3C2311yescharacteristic
91C2×C4×Dic3—22no—
92C22×C3⋊C8C2311yescharacteristic
93C22×C3⋊C8C2311yescharacteristic
94#895—28no—
95#895—24no—
96#895—24no—
97#892—28no—
98#892—24no—
99#892—24no—
100C4×D12—22no—
101C4×D12—22no—
102C4×D12—22no—
103C4×D12D414yescomplemented
104C3×#107—24no—
105C3×C4⋊C8—22no—
106C3×#102—24no—
107#832—22no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110C3⋊#95—22no—
111#821—42no—
112(C2×D3)⋊C8—24no—
113(C2×D3)⋊C8—42no—
114#809—22no—
115#808—22no—
116#803—22no—
117C22×D8—31no—
118#555—64no—
119#555—64no—
120C2×C4⋊Q8C2×D311yescharacteristic
121C2×#124—62no—
122C2×#121—32no—
123C2×#121—32no—
124C2×#121—61no—
125C2×C4×D4—31no—
126C2×C4×D4—31no—
127C2×C4×D4C2×D311yescharacteristic
128C2×#117—32no—
129C2×#117—61no—
130#474—24no—
131#442—64no—
132C2×#107C2×D312yes—
133C2×#106—31no—
134C2×C4⋊C8C2×D311yescharacteristic
135C2×#102—31no—
136C2×#102—31no—
137C2×#102—32no—
138C2×#102C2×D312yes—
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—
151C23×D3—21no—
152C23×D3C2×D412yes—
153C2×C3×Q8—24no—
154C2×C3×Q8—82no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4#4112yes—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—42no—
164C22×C3×C4—82no—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2411yescharacteristic
168C2×#67—44no—
169C2×#67—44no—
170C2×#67—82no—
171C22×Dic3—21no—
172C22×Dic3—22no—
173C22×Dic3C2×D412yes—
174C2×D12—21no—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12—22no—
179C2×D12—21no—
180C2×D12—28no—
181C2×D12—42no—
182C2×D12—82no—
183C2×D12—82no—
184C2×D12C2×D412yes—
185C2×C4×D3—24no—
186C2×C4×D3—22no—
187C2×C4×D3—22no—
188C2×C4×D3—22no—
189C2×C4×D3—22no—
190C2×C4×D3—22no—
191C2×C4×D3—22no—
192C2×C4×D3—44no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C3×C4⋊C4—24no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4C2×D414yes—
201C3×C22⋊C4—42no—
202C3×C42C2×D412yes—
203C3×C42C2×D412yes—
204#217—44no—
205#213—28no—
206(C2×D3)⋊C4—28no—
207(C2×D3)⋊C4—28no—
208(C2×D3)⋊C4—24no—
209(C2×D3)⋊C4—24no—
210(C2×D3)⋊C4—82no—
211C4⋊Dic3—22no—
212C4⋊Dic3—21no—
213C4⋊Dic3—22no—
214C4⋊Dic3—22no—
215C4⋊Dic3C2×D412yes—
216C4×Dic3—28no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8—21no—
220C2×C3⋊C8—21no—
221C2×C3⋊C8—21no—
222C2×C3⋊C8#4112yes—
223C22×Q8—21no—
224C22×D4—31no—
225C22×D4—31no—
226C22×D4—62no—
227C22×D4—62no—
228C22×D4C22×D311yescharacteristic
229C23×C4—21no—
230C23×C4—32no—
231C23×C4—61no—
232C2×D8—62no—
233C2×D8—62no—
234C2×D8—62no—
235C22×C8—31no—
236C22×C8—31no—
237C22×C8C22×D312yes—
238C4⋊Q8—22no—
239C4⋊Q8#6714yescomplemented
240#124—616no—
241#121—64no—
242#121—68no—
243#121—64no—
244#121—64no—
245#121—122no—
246C4×D4—22no—
247C4×D4—22no—
248C4×D4—34no—
249C4×D4—62no—
250C4×D4—62no—
251C4×D4—62no—
252#117—64no—
253#117—68no—
254#117—64no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—31no—
257C2×C4⋊C4—31no—
258C2×C4⋊C4—32no—
259C2×C4⋊C4C22×D311yescharacteristic
260C2×C4⋊C4C22×D312yes—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—32no—
263C2×C22⋊C4—62no—
264C2×C22⋊C4—62no—
265C2×C22⋊C4—61no—
266C2×C22⋊C4—62no—
267C2×C22⋊C4—62no—
268C2×C42—62no—
269C2×C42C22×D311yescharacteristic
270#107—24no—
271#106—62no—
272C4⋊C8—22no—
273#102—24no—
274#102—62no—
275#102—62no—
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—41no—
285C23×C3—41no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3C22×D411yescharacteristic
289C22×D3—22no—
290C22×D3—22no—
291C22×D3—22no—
292C22×D3—22no—
293C22×D3—22no—
294C22×D3—22no—
295C22×D3—22no—
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—41no—
305C3×D4—41no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—24no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4—84no—
314C2×C3×C4#12012yes—
315C2×C3×C4#12012yes—
316C2×C3×C4C22×D411yescharacteristic
317C2×C3×C4C22×D411yescharacteristic
318C2×C3×C4C2×#4111yescharacteristic
319C2×C3×C4C2×#4111yescharacteristic
320C2×C3×C4H(5,𝔽2)11yescharacteristic
321C2×C3×C42−1+411yescharacteristic
322#67—88no—
323C2×Dic3—22no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331D12—22no—
332D12—22no—
333D12—44no—
334D12—41no—
335D12—41no—
336D12—88no—
337C4×D3—48no—
338C4×D3—42no—
339C4×D3—42no—
340C4×D3—42no—
341C4×D3—42no—
342C3×C8—42no—
343C3⋊C8—22no—
344C3⋊C8—41no—
345C24—21no—
346C24—32no—
347C24—61no—
348C2×Q8—24no—
349C2×Q8—82no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—32no—
356C2×D4—68no—
357C2×D4—62no—
358C2×D4—61no—
359C2×D4—61no—
360C2×D4—61no—
361C2×D4—61no—
362C2×D4—61no—
363C2×D4—122no—
364C2×D4—124no—
365C2×D4—124no—
366C2×D4—242no—
367C2×D4—242no—
368C2×D4—242no—
369C2×D4#23912yes—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—32no—
373C22×C4—42no—
374C22×C4—61no—
375C22×C4—64no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—82no—
384C22×C4—124no—
385C22×C4D3×D411yescharacteristic
386C22×C4D3×D411yescharacteristic
387C22×C4C23×D311yescharacteristic
388D8—68no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—32no—
393C2×C8—61no—
394C2×C8—61no—
395C2×C8—61no—
396C2×C8—61no—
397C2×C8—61no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—32no—
403C4⋊C4—62no—
404C4⋊C4—61no—
405C4⋊C4—62no—
406C4⋊C4—62no—
407C4⋊C4C2×#6714yes—
408C22⋊C4—42no—
409C22⋊C4—68no—
410C22⋊C4—68no—
411C22⋊C4—64no—
412C22⋊C4—64no—
413C22⋊C4—124no—
414C22⋊C4—242no—
415C42—68no—
416C42D3×D412yes—
417C42C2×#6712yes—
418C22×C3—42no—
419C22×C3—44no—
420C22×C3—82no—
421C22×C3—81no—
422C22×C3—81no—
423C22×C3C2×#12011yescharacteristic
424C22×C3#54611yescharacteristic
425C22×C3#54711yescharacteristic
426C22×C3#57512yes—
427C22×C3#57612yes—
428C2×D3—42no—
429C2×D3—42no—
430C2×D3—42no—
431C2×D3—42no—
432C2×D3—42no—
433C2×D3—42no—
434C2×D3—42no—
435C2×D3—42no—
436C2×D3—42no—
437C2×D3—42no—
438C2×D3—82no—
439C2×D3—82no—
440C3×C4—22no—
441C3×C4—22no—
442C3×C4—44no—
443C3×C4—81no—
444C3×C4#52511yescharacteristic
445C3×C4#53711yescharacteristic
446C3×C4#54811yescharacteristic
447C3×C4#55011yescharacteristic
448Dic3—42no—
449Dic3—42no—
450Dic3—82no—
451C23—22no—
452C23—42no—
453C23—42no—
454C23—41no—
455C23—41no—
456C23—62no—
457C23—62no—
458C23—62no—
459C23—62no—
460C23—62no—
461C23—62no—
462C23—62no—
463C23—62no—
464C23—122no—
465C23—122no—
466C23—121no—
467C23—121no—
468C23C2×D3×D411yescharacteristic
469Q8—84no—
470D4—22no—
471D4—22no—
472D4—41no—
473D4—41no—
474D4—62no—
475D4—62no—
476D4—124no—
477D4—121no—
478D4—121no—
479D4—248no—
480D4—248no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—62no—
489C2×C4—62no—
490C2×C4—62no—
491C2×C4—62no—
492C2×C4—84no—
493C2×C4—128no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—122no—
501C2×C4—122no—
502C2×C4#93712yes—
503C2×C4#93712yes—
504C2×C4C2×D3×D411yescharacteristic
505C2×C4C2×#23911yescharacteristic
506C2×C4#100711yescharacteristic
507C2×C4D3×#4111yescharacteristic
508C2×C4#100911yescharacteristic
509C2×C4C22×#6711yescharacteristic
510C8—42no—
511C8—62no—
512C8—121no—
513C2×C3—82no—
514C2×C3#240711yescharacteristic
515C2×C3#300511yescharacteristic
516C2×C3#300811yescharacteristic
517C2×C3#331912yes—
518C2×C3#334712yes—
519D3—82no—
520D3—82no—
521C22—44no—
522C22—42no—
523C22—81no—
524C22—82no—
525C22—81no—
526C22—122no—
527C22—122no—
528C22—122no—
529C22—122no—
530C22—122no—
531C22—122no—
532C22—122no—
533C22—122no—
534C22—122no—
535C22—122no—
536C22—242no—
537C22—242no—
538C22#563011yescharacteristic
539C22#563111yescharacteristic
540C22#575712yes—
541C22#577712yes—
542C22C2×#93711yescharacteristic
543C4—22no—
544C4—22no—
545C4—44no—
546C4—81no—
547C4—122no—
548C4—122no—
549C4—242no—
550C4#562711yescharacteristic
551C4#562911yescharacteristic
552C4#581711yescharacteristic
553C4#583411yescharacteristic
554C3#2412911yescharacteristic, complemented, Sylow
555C2—82no—
556C2—242no—
557C2—242no—
558C2#8098312yes—
559C2#8098912yes—
560C2#8281211yescharacteristic
561C2#8383311yescharacteristic
562C2#8398011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#474C2
socleC23×C3C22×D4
hypercenterC2×#474D3
hypocenterC3#24129
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. C2×#5734C2semidirect
2. C2×C3×#474C2semidirect
3. #82416C2semidirect
4. C2×#5014C2semidirect
5. #71788C2semidirect
6. #70634C2semidirect
7. #70633C2semidirect
8. #70480C2semidirect
9. #70471C2semidirect
10. #70204C2semidirect
11. #70201C2semidirect
47. C2×#474D3semidirect
68. C3×C4⋊Q8D4semidirect
103. C4×D12D4semidirect
239. C4⋊Q8#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×#57342
C2×C3×#4742
#824162
C2×#50142
#717882
#706342
#706332
#704802
#704712
#702042
#702012
#241293