Finite Group Databasebeta

← C2×#82832 #11001342

Subgroups — C2×#82832 #11001342

Order 768.

2674 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C22⋊Q8C211yesmaximal, characteristic, complemented
2C22×#929C211yesmaximal, characteristic
3C2×#5231C211yesmaximal, characteristic, complemented
4C2×#5230C211yesmaximal, characteristic, complemented
5C2×#5212C211yesmaximal, characteristic, complemented
6#82832C2116yesmaximal, direct product factor
7C2×#4992C211yesmaximal, characteristic, complemented
8C2×#4994C212yesmaximal, complemented
9C2×#4983C211yesmaximal, characteristic, complemented
10C2×#4982C212yesmaximal, complemented
11C2×#4959C212yesmaximal
12C2×#4952C212yesmaximal, complemented
13C2×#2814—31nomaximal, Sylow
14C23×C3×Q8C2211yescharacteristic
15C24×C3×C4C2211yescharacteristic
16C23×#67C2211yescharacteristic
17C2×C3×C22⋊Q8—28no—
18C2×C3×C22⋊Q8C2218yescomplemented
19C22×C3×C4⋊C4C2212yes—
20C22×C3×C4⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4C2212yes—
22C22×#217C2211yescharacteristic
23C22×#217C2212yes—
24C2×#929—28no—
25C2×#929C2218yes—
26C22×(C2×D3)⋊C4C2211yescharacteristic
27C22×(C2×D3)⋊C4C2211yescharacteristic
28C22×(C2×D3)⋊C4C2212yes—
29C22×(C2×D3)⋊C4C2212yes—
30C22×C4⋊Dic3C2212yes—
31C22×Dic3⋊C4C2211yescharacteristic
32C22×Dic3⋊C4C2211yescharacteristic
33C22×Dic3⋊C4C2212yes—
34C22×Dic3⋊C4C2212yes—
35#5231C2218yescomplemented
36#5230C2218yescomplemented
37#5212C2218yescomplemented
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2211yescharacteristic
40C2×C3⋊#95C2212yes—
41C2×C3⋊#95C2211yescharacteristic
42C2×C3⋊#95C2212yes—
43C2×C3⋊#95C2212yes—
44#4994C22116yescomplemented
45#4992C2218yescomplemented
46#4983C2218yescomplemented
47#4982C22116yescomplemented
48#4959C22116yes—
49#4952C22116yescomplemented
50C22×#123—31no—
51C22×C22⋊Q8D311yescharacteristic, complemented
52#2814—316no—
53C2×#398—32no—
54C2×#398—31no—
55C2×#397—31no—
56C2×#397—32no—
57C2×#396—32no—
58C2×#395—31no—
59C2×#394—32no—
60C2×#393—31no—
61C2×#380—31no—
62C25×C3C2311yescharacteristic
63C24×D3C2311yescharacteristic
64C22×C3×Q8—24no—
65C22×C3×Q8—44no—
66C22×C3×Q8C2314yes—
67C23×C3×C4—44no—
68C23×C3×C4D418yes—
69C23×C3×C4C2312yes—
70C23×C3×C4C2312yes—
71C23×C3×C4C2311yescharacteristic
72C23×C3×C4C2311yescharacteristic
73C23×C3×C4C2314yes—
74C22×#67—24no—
75C22×#67—44no—
76C22×#67C2314yes—
77C23×Dic3C2311yescharacteristic
78C23×Dic3C2311yescharacteristic
79C23×Dic3C2312yes—
80C23×Dic3C2312yes—
81C23×Dic3C2311yescharacteristic
82C3×C22⋊Q8—232no—
83C2×C3×C4⋊C4—24no—
84C2×C3×C4⋊C4—28no—
85C2×C3×C4⋊C4C2318yes—
86C2×C3×C4⋊C4C2314yes—
87C2×C3×C22⋊C4—28no—
88C2×C3×C22⋊C4C2318yes—
89C2×#217—24no—
90C2×#217—44no—
91C2×#217C2314yes—
92C2×#217C2318yes—
93#929—232no—
94C2×(C2×D3)⋊C4—24no—
95C2×(C2×D3)⋊C4—24no—
96C2×(C2×D3)⋊C4—44no—
97C2×(C2×D3)⋊C4—44no—
98C2×(C2×D3)⋊C4C2314yes—
99C2×(C2×D3)⋊C4C2314yes—
100C2×(C2×D3)⋊C4C2318yes—
101C2×(C2×D3)⋊C4C2318yes—
102C2×C4⋊Dic3—44no—
103C2×C4⋊Dic3C2318yes—
104C2×Dic3⋊C4—24no—
105C2×Dic3⋊C4—24no—
106C2×Dic3⋊C4—44no—
107C2×Dic3⋊C4—44no—
108C2×Dic3⋊C4C2314yes—
109C2×Dic3⋊C4C2314yes—
110C2×Dic3⋊C4C2318yes—
111C2×Dic3⋊C4C2318yes—
112C3⋊#95C2318yes—
113C3⋊#95C2318yes—
114C3⋊#95C2318yes—
115C3⋊#95C2314yes—
116C3⋊#95C2314yes—
117C3⋊#95C2318yes—
118C23×Q8C2×D311yescharacteristic
119C23×D4—31no—
120C24×C4C2×D311yescharacteristic
121C2×#123—38no—
122C2×#123—68no—
123C2×C22⋊Q8—28no—
124C2×C22⋊Q8C2×D318yescomplemented
125C22×C4⋊C4—31no—
126C22×C4⋊C4—31no—
127C22×C4⋊C4—32no—
128C22×C4⋊C4—32no—
129C22×C4⋊C4—32no—
130C22×C4⋊C4C2×D312yes—
131C22×C4⋊C4C2×D311yescharacteristic
132C22×C22⋊C4—31no—
133C22×C22⋊C4—31no—
134C22×C22⋊C4—31no—
135C22×C22⋊C4—32no—
136C22×C22⋊C4—32no—
137C22×C22⋊C4—32no—
138C22×C22⋊C4C2×D312yes—
139#398—316no—
140#398—38no—
141#397—38no—
142#397—316no—
143#396—316no—
144#395—38no—
145#394—316no—
146#393—38no—
147#380—38no—
148C2×#95—32no—
149C2×#95—31no—
150C2×#95—32no—
151C2×#95—32no—
152C2×#95—31no—
153C2×#95—32no—
154C24×C3—22no—
155C24×C3—24no—
156C24×C3—24no—
157C24×C3—22no—
158C24×C3C2×D414yes—
159C24×C3C2412yes—
160C24×C3C2411yescharacteristic
161C23×D3—22no—
162C23×D3—22no—
163C23×D3—24no—
164C23×D3—24no—
165C23×D3—22no—
166C23×D3C2412yes—
167C2×C3×Q8—28no—
168C2×C3×Q8—48no—
169C2×C3×Q8—416no—
170C22×C3×C4—24no—
171C22×C3×C4—24no—
172C22×C3×C4—24no—
173C22×C3×C4—24no—
174C22×C3×C4—24no—
175C22×C3×C4—24no—
176C22×C3×C4—24no—
177C22×C3×C4—24no—
178C22×C3×C4—416no—
179C22×C3×C4—48no—
180C22×C3×C4C2×D414yes—
181C22×C3×C4C2×D414yes—
182C22×C3×C4C2×D4116yes—
183C22×C3×C4C2414yes—
184C22×C3×C4C2414yes—
185C22×C3×C4C2412yes—
186C22×C3×C4C2412yes—
187C2×#67—28no—
188C2×#67—48no—
189C2×#67—416no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—24no—
195C22×Dic3—24no—
196C22×Dic3—24no—
197C22×Dic3—24no—
198C22×Dic3—24no—
199C22×Dic3—24no—
200C22×Dic3—24no—
201C22×Dic3—24no—
202C22×Dic3—22no—
203C22×Dic3C2412yes—
204C22×Dic3C2414yes—
205C22×Dic3C2414yes—
206C22×Dic3C2412yes—
207C22×Dic3C2412yes—
208C3×C4⋊C4—28no—
209C3×C4⋊C4—216no—
210C3×C22⋊C4—216no—
211#217—28no—
212#217—48no—
213(C2×D3)⋊C4—28no—
214(C2×D3)⋊C4—28no—
215(C2×D3)⋊C4—48no—
216(C2×D3)⋊C4—48no—
217C4⋊Dic3—48no—
218Dic3⋊C4—28no—
219Dic3⋊C4—28no—
220Dic3⋊C4—48no—
221Dic3⋊C4—48no—
222C25—31no—
223C25C22×D311yescharacteristic
224C22×Q8—24no—
225C22×Q8—44no—
226C22×Q8C22×D314yes—
227C22×D4—34no—
228C22×D4—64no—
229C22×D4—124no—
230C23×C4—31no—
231C23×C4—32no—
232C23×C4—32no—
233C23×C4—31no—
234C23×C4—31no—
235C23×C4—44no—
236C23×C4#6718yes—
237C23×C4C22×D312yes—
238C23×C4C22×D312yes—
239C23×C4C22×D311yescharacteristic
240C23×C4C22×D311yescharacteristic
241C23×C4C22×D314yes—
242#123—632no—
243C22⋊Q8—232no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—34no—
247C2×C4⋊C4—34no—
248C2×C4⋊C4—38no—
249C2×C4⋊C4—38no—
250C2×C4⋊C4—38no—
251C2×C4⋊C4—64no—
252C2×C4⋊C4—64no—
253C2×C4⋊C4—124no—
254C2×C4⋊C4—124no—
255C2×C4⋊C4—124no—
256C2×C4⋊C4C22×D318yes—
257C2×C4⋊C4C22×D314yes—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—34no—
260C2×C22⋊C4—34no—
261C2×C22⋊C4—34no—
262C2×C22⋊C4—38no—
263C2×C22⋊C4—38no—
264C2×C22⋊C4—38no—
265C2×C22⋊C4—64no—
266C2×C22⋊C4—64no—
267C2×C22⋊C4—64no—
268C2×C22⋊C4—124no—
269C2×C22⋊C4—124no—
270C2×C22⋊C4—124no—
271C2×C22⋊C4C22×D318yes—
272#95—34no—
273#95—34no—
274#95—38no—
275#95—38no—
276#95—38no—
277#95—38no—
278C23×C3—28no—
279C23×C3—28no—
280C23×C3—28no—
281C23×C3—22no—
282C23×C3—24no—
283C23×C3—24no—
284C23×C3—22no—
285C23×C3—48no—
286C23×C3—48no—
287C23×C3C22×D414yes—
288C23×C3C22×D412yes—
289C23×C3H(5,𝔽2)14yes—
290C23×C3H(5,𝔽2)12yes—
291C23×C32−1+414yes—
292C23×C32−1+412yes—
293C23×C3C2511yescharacteristic
294C22×D3—22no—
295C22×D3—22no—
296C22×D3—24no—
297C22×D3—24no—
298C22×D3—22no—
299C22×D3—44no—
300C22×D3—44no—
301C22×D3—44no—
302C22×D3—48no—
303C22×D3—48no—
304C3×Q8—416no—
305C2×C3×C4—28no—
306C2×C3×C4—28no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—24no—
315C2×C3×C4—416no—
316C2×C3×C4—48no—
317C2×C3×C4—48no—
318C2×C3×C4C22×D414yes—
319C2×C3×C4C22×D414yes—
320#67—416no—
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—22no—
325C2×Dic3—24no—
326C2×Dic3—24no—
327C2×Dic3—24no—
328C2×Dic3—24no—
329C2×Dic3—24no—
330C2×Dic3—24no—
331C2×Dic3—24no—
332C2×Dic3—24no—
333C2×Dic3—22no—
334C2×Dic3—44no—
335C2×Dic3—44no—
336C2×Dic3—44no—
337C2×Dic3—48no—
338C2×Dic3—48no—
339C24—24no—
340C24—24no—
341C24—22no—
342C24—22no—
343C24—32no—
344C24—64no—
345C24—64no—
346C24—62no—
347C24—62no—
348C24—62no—
349C24C2×#6714yes—
350C24C23×D312yes—
351C24C23×D311yescharacteristic
352C2×Q8—28no—
353C2×Q8—48no—
354C2×Q8—416no—
355C2×D4—68no—
356C2×D4—128no—
357C2×D4—1216no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—32no—
367C22×C4—34no—
368C22×C4—34no—
369C22×C4—32no—
370C22×C4—32no—
371C22×C4—416no—
372C22×C4—48no—
373C22×C4—64no—
374C22×C4—64no—
375C22×C4—64no—
376C22×C4—64no—
377C22×C4—64no—
378C22×C4—64no—
379C22×C4—64no—
380C22×C4—64no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4C2×#6714yes—
387C22×C4C2×#6714yes—
388C22×C4C2×#67116yes—
389C22×C4C23×D314yes—
390C22×C4C23×D314yes—
391C22×C4C23×D312yes—
392C22×C4C23×D312yes—
393C4⋊C4—28no—
394C4⋊C4—216no—
395C4⋊C4—68no—
396C4⋊C4—68no—
397C4⋊C4—128no—
398C4⋊C4—128no—
399C4⋊C4—128no—
400C22⋊C4—216no—
401C22⋊C4—68no—
402C22⋊C4—68no—
403C22⋊C4—68no—
404C22⋊C4—128no—
405C22⋊C4—128no—
406C22⋊C4—128no—
407C22×C3—24no—
408C22×C3—24no—
409C22×C3—24no—
410C22×C3—44no—
411C22×C3—44no—
412C22×C3—416no—
413C22×C3#53414yes—
414C22×C3#53614yes—
415C22×C3#53714yes—
416C22×C3#56218yes—
417C22×C3#56318yes—
418C22×C3C23×D411yescharacteristic
419C22×C3C2×H(5,𝔽2)12yes—
420C22×C3C2×H(5,𝔽2)11yescharacteristic
421C22×C3C2×2−1+412yes—
422C22×C3C2×2−1+411yescharacteristic
423C2×D3—42no—
424C2×D3—42no—
425C2×D3—42no—
426C2×D3—44no—
427C2×D3—44no—
428C2×D3—88no—
429C3×C4—24no—
430C3×C4—24no—
431C3×C4—44no—
432C3×C4—44no—
433Dic3—42no—
434Dic3—42no—
435Dic3—42no—
436Dic3—44no—
437Dic3—44no—
438C23—24no—
439C23—24no—
440C23—22no—
441C23—22no—
442C23—28no—
443C23—28no—
444C23—28no—
445C23—48no—
446C23—48no—
447C23—64no—
448C23—64no—
449C23—62no—
450C23—62no—
451C23—62no—
452C23—128no—
453C23—128no—
454C23—124no—
455C23—124no—
456C23—124no—
457C23#100414yes—
458C23#100714yes—
459C23#100912yes—
460C23#101012yes—
461C23C22×#6714yes—
462C23C22×#6712yes—
463C23C24×D311yescharacteristic
464Q8—416no—
465D4—1216no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—28no—
469C2×C4—28no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—416no—
477C2×C4—48no—
478C2×C4—48no—
479C2×C4—64no—
480C2×C4—64no—
481C2×C4—64no—
482C2×C4—64no—
483C2×C4—64no—
484C2×C4—64no—
485C2×C4—64no—
486C2×C4—64no—
487C2×C4—62no—
488C2×C4—62no—
489C2×C4—62no—
490C2×C4—62no—
491C2×C4—62no—
492C2×C4—128no—
493C2×C4—128no—
494C2×C4—124no—
495C2×C4—124no—
496C2×C4—124no—
497C2×C4C22×#6714yes—
498C2×C4C22×#6714yes—
499C2×C3—44no—
500C2×C3#281418yescomplemented
501C2×C3C2×#53411yescharacteristic
502C2×C3C2×#53611yescharacteristic
503C2×C3C2×#53711yescharacteristic
504C2×C3C2×#56212yes—
505C2×C3C2×#56312yes—
506D3—82no—
507C22—24no—
508C22—24no—
509C22—24no—
510C22—416no—
511C22—44no—
512C22—44no—
513C22—122no—
514C22—122no—
515C22—124no—
516C22—122no—
517C22—124no—
518C22—248no—
519C22#564618yes—
520C22#564718yes—
521C22#580414yes—
522C22#581614yes—
523C22#583314yes—
524C22C2×#100412yes—
525C22C2×#100712yes—
526C22C2×#100911yescharacteristic
527C22C2×#101011yescharacteristic
528C22C23×#6711yescharacteristic
529C4—24no—
530C4—24no—
531C4—44no—
532C4—44no—
533C4—122no—
534C4—122no—
535C4—122no—
536C4—124no—
537C4—124no—
538C3C2×#281411yescharacteristic, complemented, Sylow
539C2—44no—
540C2—242no—
541C2#8283218yesdirect product factor
542C2C2×#564612yes—
543C2C2×#564712yes—
544C2C2×#580411yescharacteristic
545C2C2×#581611yescharacteristic
546C2C2×#583311yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC22×C3×C22⋊Q8C2
socleC24×C3C24
prime radicalC23×#67C22
hypercenterC22×C22⋊Q8D3
hypocenterC3C2×#2814
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×C3×C22⋊Q8C2semidirect
3. C2×#5231C2semidirect
4. C2×#5230C2semidirect
5. C2×#5212C2semidirect
6. #82832C2direct (N × H)
7. C2×#4992C2semidirect
8. C2×#4994C2semidirect
9. C2×#4983C2semidirect
10. C2×#4982C2semidirect
12. C2×#4952C2semidirect
18. C2×C3×C22⋊Q8C22semidirect
35. #5231C22semidirect
36. #5230C22semidirect
37. #5212C22semidirect
44. #4994C22semidirect
45. #4992C22semidirect
46. #4983C22semidirect
47. #4982C22semidirect
49. #4952C22semidirect
51. C22×C22⋊Q8D3semidirect
124. C2×C22⋊Q8C2×D3semidirect
500. C2×C3#2814semidirect
538. C3C2×#2814semidirect
541. C2#82832direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊Q82
C22×#9292
C2×#52312
C2×#52302
C2×#52122
#828322
C2×#49922
C2×#49942
C2×#49832
C2×#49822
C2×#49592
C2×#49522
C2×#28143