Finite Group Databasebeta

← #10829242

Subgroups — #10829242

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×#5087C211yesmaximal, characteristic, complemented
3C2×#4711C211yesmaximal, characteristic, complemented
4#74239C211yesmaximal, characteristic, complemented
5C3×#1993C211yesmaximal, characteristic, complemented
6#69837C211yesmaximal, characteristic, complemented
7#69556C211yesmaximal, characteristic, complemented
8#68886C212yesmaximal, complemented
9#68881C212yesmaximal, complemented
10#68876C212yesmaximal, complemented
11#68822C212yesmaximal, complemented
12#24129—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×#947—22no—
15C22×D24C2211yescharacteristic
16#5734—24no—
17#5734—24no—
18C2×#895—21no—
19C2×#895C2212yes—
20C2×#895C2212yes—
21C2×#892—21no—
22C2×#892C2212yes—
23C2×C4×D12C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C2×C4×D12C2211yescharacteristic
26C2×C3×#97C2211yescharacteristic
27C3×#382C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×(C2×D3)⋊C8C2212yes—
31C2×(C2×D3)⋊C8C2212yes—
32C2×#817C2211yescharacteristic
33#5087—24no—
34#4987C2212yes—
35C2×#809C2212yes—
36C2×#809C2212yes—
37C2×#807C2212yes—
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39#4711—24no—
40#4674—21no—
41#4671—21no—
42#4665—22no—
43C3×#340C2212yes—
44C3×#340C2212yes—
45#4529C2212yes—
46C2×#555—31no—
47C2×#474—31no—
48C2×#442—31no—
49#2535—31no—
50#2033—32no—
51#1999—32no—
52#1993D311yescharacteristic, complemented
53#1955—31no—
54#1875—32no—
55#1854—32no—
56#1804—31no—
57C22×C3×Q8—21no—
58C22×D12—22no—
59C22×D12—22no—
60C22×D12D411yescharacteristic
61C22×D12C2311yescharacteristic
62C22×D12C2311yescharacteristic
63C22×C4×D3—21no—
64C22×C4×D3—21no—
65C22×C4×D3C2312yes—
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8D414yescomplemented
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C42C2311yescharacteristic
74#947—216no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—21no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—21no—
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4D412yes—
82C2×C4⋊Dic3D411yescharacteristic
83C2×C4⋊Dic3C2311yescharacteristic
84C2×C4⋊Dic3C2311yescharacteristic
85C2×C4×Dic3—22no—
86C22×C3⋊C8C2312yes—
87C2×D24—22no—
88C2×D24—22no—
89C2×D24—22no—
90#895—28no—
91#895—24no—
92#895—24no—
93#895—24no—
94#895—42no—
95#892—28no—
96#892—24no—
97#892—24no—
98C4×D12—22no—
99C4×D12—22no—
100C4×D12—22no—
101C4×D12—22no—
102C4×D12—22no—
103C4×D12D414yescomplemented
104C3×#97—22no—
105C3×#95—21no—
106C3×#95—21no—
107C3⋊#95—22no—
108#821—22no—
109#821—22no—
110(C2×D3)⋊C8—24no—
111(C2×D3)⋊C8—42no—
112#817—22no—
113#809—24no—
114#809—42no—
115#807—24no—
116C4⋊(C3⋊C8)—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×D4—31no—
128C2×#117—32no—
129C2×#117—61no—
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×#97C2×D311yescharacteristic
142#397—61no—
143#392—61no—
144#386—62no—
145#382C2×D311yescharacteristic
146#380—32no—
147#340—32no—
148#340C2×D312yes—
149#340C2×D312yes—
150C23×D3—21no—
151C23×D3—21no—
152C23×D3C2×D412yes—
153C2×C3×Q8—24no—
154C2×C3×Q8—82no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2411yescharacteristic
160C22×Dic3—21no—
161C22×Dic3—21no—
162C22×Dic3C2×D412yes—
163C2×D12—21no—
164C2×D12—21no—
165C2×D12—21no—
166C2×D12—21no—
167C2×D12—21no—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12—22no—
172C2×D12—21no—
173C2×D12—21no—
174C2×D12—28no—
175C2×D12—42no—
176C2×D12—44no—
177C2×D12—44no—
178C2×D12—82no—
179C2×D12—82no—
180C2×D12—82no—
181C2×D12C2×D412yes—
182C2×D12#4112yes—
183C2×C4×D3—24no—
184C2×C4×D3—22no—
185C2×C4×D3—22no—
186C2×C4×D3—22no—
187C2×C4×D3—22no—
188C2×C4×D3—22no—
189C2×C4×D3—22no—
190C2×C4×D3—44no—
191C2×C4×D3—42no—
192C2×C4×D3—82no—
193C2×C3×C8—21no—
194C2×C3×C8—21no—
195C2×C3×C8—21no—
196C2×C3×C8—21no—
197C2×C3×C8—21no—
198C2×C3×C8#4112yes—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4C2×D414yes—
204C3×C42C2×D412yes—
205C3×C42C2×D412yes—
206(C2×D3)⋊C4—28no—
207(C2×D3)⋊C4—28no—
208(C2×D3)⋊C4—24no—
209(C2×D3)⋊C4—24no—
210(C2×D3)⋊C4—44no—
211(C2×D3)⋊C4—42no—
212(C2×D3)⋊C4—82no—
213C4⋊Dic3—22no—
214C4⋊Dic3—21no—
215C4⋊Dic3—21no—
216C4⋊Dic3—21no—
217C4⋊Dic3C2×D412yes—
218C4×Dic3—28no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222D24—28no—
223C22×Q8—21no—
224C22×D4—31no—
225C22×D4—31no—
226C22×D4—31no—
227C22×D4—62no—
228C22×D4—62no—
229C23×C4—32no—
230C23×C4—61no—
231C23×C4—61no—
232C2×D8—62no—
233C2×D8—62no—
234C2×D8—62no—
235C22×C8—32no—
236C22×C8C22×D311yescharacteristic
237C22×C8C22×D311yescharacteristic
238C4⋊Q8—22no—
239C4⋊Q8D1214yescomplemented
240#124—616no—
241#121—64no—
242#121—68no—
243#121—64no—
244#121—64no—
245#121—122no—
246C4×D4—34no—
247C4×D4—62no—
248C4×D4—62no—
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—31no—
259C2×C4⋊C4C22×D312yes—
260C2×C4⋊C4C22×D312yes—
261C2×C22⋊C4—32no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—62no—
264C2×C22⋊C4—61no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4—61no—
267C2×C22⋊C4—62no—
268C2×C42—62no—
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—22no—
280#95—21no—
281#95—21no—
282#95—62no—
283C23×C3C22×D411yescharacteristic
284C22×D3—22no—
285C22×D3—22no—
286C22×D3—22no—
287C22×D3—22no—
288C22×D3—22no—
289C22×D3—22no—
290C22×D3—22no—
291C22×D3—22no—
292C22×D3—22no—
293C22×D3—41no—
294C22×D3—41no—
295C22×D3—41no—
296C22×D3—41no—
297C22×D3—42no—
298C22×D3—42no—
299C22×D3—42no—
300C22×D3—42no—
301C3×Q8—84no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—24no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4#12012yes—
311C2×C3×C4#12012yes—
312C2×C3×C4C22×D411yescharacteristic
313C2×C3×C4C22×D411yescharacteristic
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C4C2×#4111yescharacteristic
316C2×C3×C4H(5,𝔽2)11yescharacteristic
317C2×C3×C42−1+411yescharacteristic
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—42no—
323C2×Dic3—41no—
324C2×Dic3—42no—
325D12—22no—
326D12—22no—
327D12—22no—
328D12—22no—
329D12—44no—
330D12—41no—
331D12—41no—
332D12—41no—
333D12—41no—
334D12—88no—
335D12—88no—
336C4×D3—48no—
337C4×D3—42no—
338C4×D3—42no—
339C4×D3—42no—
340C4×D3—42no—
341C4×D3—84no—
342C3×C8—22no—
343C3×C8—41no—
344C3⋊C8—42no—
345C24—32no—
346C24—61no—
347C24—61no—
348C2×Q8—24no—
349C2×Q8—82no—
350C2×D4—32no—
351C2×D4—32no—
352C2×D4—68no—
353C2×D4—62no—
354C2×D4—61no—
355C2×D4—61no—
356C2×D4—61no—
357C2×D4—61no—
358C2×D4—61no—
359C2×D4—61no—
360C2×D4—61no—
361C2×D4—61no—
362C2×D4—61no—
363C2×D4—61no—
364C2×D4—122no—
365C2×D4—124no—
366C2×D4—124no—
367C2×D4—242no—
368C2×D4—242no—
369C2×D4—242no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—32no—
373C22×C4—61no—
374C22×C4—64no—
375C22×C4—62no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—62no—
380C22×C4—62no—
381C22×C4—61no—
382C22×C4—124no—
383C22×C4—122no—
384C22×C4—242no—
385C22×C4D3×D411yescharacteristic
386C22×C4D3×D411yescharacteristic
387C22×C4C23×D311yescharacteristic
388D8—68no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—62no—
395C2×C8—62no—
396C2×C8—62no—
397C2×C8#23912yes—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—32no—
403C4⋊C4—62no—
404C4⋊C4—61no—
405C4⋊C4—61no—
406C4⋊C4—61no—
407C4⋊C4C2×D1214yes—
408C22⋊C4—68no—
409C22⋊C4—68no—
410C22⋊C4—64no—
411C22⋊C4—64no—
412C22⋊C4—124no—
413C22⋊C4—122no—
414C22⋊C4—242no—
415C42—68no—
416C42C2×D1212yes—
417C42D3×D412yes—
418C22×C3C2×#12011yescharacteristic
419C22×C3#54611yescharacteristic
420C22×C3#54711yescharacteristic
421C22×C3#57512yes—
422C22×C3#57612yes—
423C2×D3—42no—
424C2×D3—42no—
425C2×D3—42no—
426C2×D3—42no—
427C2×D3—42no—
428C2×D3—42no—
429C2×D3—42no—
430C2×D3—42no—
431C2×D3—42no—
432C2×D3—42no—
433C2×D3—44no—
434C2×D3—42no—
435C2×D3—82no—
436C2×D3—82no—
437C2×D3—81no—
438C2×D3—81no—
439C2×D3—82no—
440C3×C4—22no—
441C3×C4—22no—
442C3×C4—44no—
443C3×C4—82no—
444C3×C4#52511yescharacteristic
445C3×C4#53711yescharacteristic
446C3×C4#54811yescharacteristic
447C3×C4#55011yescharacteristic
448Dic3—42no—
449Dic3—42no—
450Dic3—81no—
451C23—62no—
452C23—62no—
453C23—62no—
454C23—62no—
455C23—62no—
456C23—62no—
457C23—62no—
458C23—62no—
459C23—62no—
460C23—121no—
461C23—121no—
462C23—121no—
463C23—121no—
464C23—122no—
465C23—122no—
466C23—122no—
467C23—122no—
468C23C2×D3×D411yescharacteristic
469Q8—84no—
470D4—62no—
471D4—62no—
472D4—62no—
473D4—62no—
474D4—124no—
475D4—121no—
476D4—121no—
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—42no—
488C2×C4—42no—
489C2×C4—62no—
490C2×C4—62no—
491C2×C4—62no—
492C2×C4—62no—
493C2×C4—128no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—121no—
500C2×C4—122no—
501C2×C4—244no—
502C2×C4#88212yes—
503C2×C4#88212yes—
504C2×C4C22×D1211yescharacteristic
505C2×C4C2×D3×D411yescharacteristic
506C2×C4C2×#23911yescharacteristic
507C2×C4#100711yescharacteristic
508C2×C4D3×#4111yescharacteristic
509C2×C4#100911yescharacteristic
510C8—22no—
511C8—41no—
512C8—122no—
513C2×C3#240711yescharacteristic
514C2×C3#300511yescharacteristic
515C2×C3#300811yescharacteristic
516C2×C3#331912yes—
517C2×C3#334712yes—
518D3—82no—
519D3—82no—
520D3—82no—
521C22—122no—
522C22—124no—
523C22—122no—
524C22—122no—
525C22—122no—
526C22—122no—
527C22—122no—
528C22—122no—
529C22—122no—
530C22—122no—
531C22—122no—
532C22—122no—
533C22—242no—
534C22—242no—
535C22—242no—
536C22—241no—
537C22—241no—
538C22C2×#88211yescharacteristic
539C22#567611yescharacteristic
540C22#572611yescharacteristic
541C22#577412yes—
542C22#577712yes—
543C4—22no—
544C4—22no—
545C4—44no—
546C4—82no—
547C4—122no—
548C4—122no—
549C4—241no—
550C4#557611yescharacteristic
551C4#557711yescharacteristic
552C4#562711yescharacteristic
553C4#562911yescharacteristic
554C3#2412911yescharacteristic, complemented, Sylow
555C2—242no—
556C2—242no—
557C2—242no—
558C2#7661811yescharacteristic
559C2#7892811yescharacteristic
560C2#7895911yescharacteristic
561C2#8170212yes—
562C2#8199412yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1993C2
socleC23×C3C22×D4
hypercenter#1993D3
hypocenterC3#24129
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
1. C2×#5734C2semidirect
2. C2×#5087C2semidirect
3. C2×#4711C2semidirect
4. #74239C2semidirect
5. C3×#1993C2semidirect
6. #69837C2semidirect
7. #69556C2semidirect
8. #68886C2semidirect
9. #68881C2semidirect
10. #68876C2semidirect
11. #68822C2semidirect
52. #1993D3semidirect
69. C3×C4⋊Q8D4semidirect
103. C4×D12D4semidirect
239. C4⋊Q8D12semidirect
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×#50872
C2×#47112
#742392
C3×#19932
#698372
#695562
#688862
#688812
#688762
#688222
#241293