Finite Group Databasebeta

← C2×#79948 #11660292

Subgroups — C2×#79948 #11660292

Order 768.

1982 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5580C211yesmaximal, characteristic
2C2×C3×#446C211yesmaximal, characteristic, complemented
3C2×Q8×C3⋊C8C211yesmaximal, characteristic, complemented
4#79948C2116yesmaximal, direct product factor
5C2×#4839C212yesmaximal, complemented
6C2×D3×C4⋊C8C211yesmaximal, characteristic, complemented
7C2×#4838C211yesmaximal, characteristic, complemented
8C2×#4834C212yesmaximal, complemented
9C2×#4706C212yesmaximal, complemented
10C2×#4709C212yesmaximal, complemented
11C2×C8×D12C211yesmaximal, characteristic, complemented
12C2×#4685C211yesmaximal, characteristic, complemented
13C2×#2994—31nomaximal, Sylow
14C2×C3×C4×Q8C2211yescharacteristic
15C2×#947C411yescharacteristic
16C2×#946C411yescharacteristic
17C22×ΓL(1,𝔽25)C2212yes—
18C22×D3×C8C2212yes—
19C22×D3×C8C2212yes—
20#5580C418yes—
21#5580C2218yes—
22C2×#898C412yes—
23C2×#895C411yescharacteristic
24C2×#894C412yes—
25C2×#890C411yescharacteristic
26C2×C4×D12C2211yescharacteristic
27C2×C4×D12C2212yes—
28C2×#872C2212yes—
29C2×C42×D3C2211yescharacteristic
30C3×#446C2218yescomplemented
31C2×C3×C4⋊C8C2212yes—
32C2×C3×C4⋊C8C2211yescharacteristic
33C2×C3×#97C2212yes—
34C2×C3×C4×C8C2211yescharacteristic
35C2×(C2×D3)⋊C8C2211yescharacteristic
36C2×(C2×D3)⋊C8C2211yescharacteristic
37C2×(C2×D3)⋊C8C2212yes—
38C2×(C2×D3)⋊C8C2212yes—
39C2×Dic3⋊C8C2211yescharacteristic
40C2×Dic3⋊C8C2211yescharacteristic
41C2×Dic3⋊C8C2212yes—
42Q8×C3⋊C8C2218yescomplemented
43C2×C4⋊(C3⋊C8)C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2212yes—
45C2×C4×C3⋊C8C2211yescharacteristic
46C2×C4×C3⋊C8C2212yes—
47#4839C22116yescomplemented
48#4838C2218yescomplemented
49#4834C22116yescomplemented
50D3×C4⋊C8C2218yescomplemented
51#4709C22116yescomplemented
52#4706C22116yescomplemented
53C8×D12C2218yescomplemented
54#4685C2218yescomplemented
55C2×#529—31no—
56#2994—316no—
57C2×#446D311yescharacteristic, complemented
58C2×C8×Q8—31no—
59C2×#436—31no—
60C2×#435—32no—
61C2×C8×D4—32no—
62C2×C8×D4—31no—
63C2×#433—32no—
64C2×#431—31no—
65C2×#424—32no—
66C22×C4⋊C8—31no—
67C22×C3×Q8C2×C411yescharacteristic
68C22×D12C2×C412yes—
69C22×D12C2×C411yescharacteristic
70C22×C4×D3C2×C411yescharacteristic
71C22×C4×D3C2×C411yescharacteristic
72C22×C4×D3C2311yescharacteristic
73C22×C4×D3C2312yes—
74C22×C3×C8C2311yescharacteristic
75C22×C3×C8C2312yes—
76C22×C3×C8C2311yescharacteristic
77C3×C4×Q8C2×C414yes—
78C3×C4×Q8C2314yes—
79C2×C3×C4⋊C4C2×C411yescharacteristic
80C2×C3×C4⋊C4C2×C412yes—
81C2×C3×C42C2311yescharacteristic
82C2×C3×C42C2312yes—
83#947C2×C418yes—
84#946C2×C418yes—
85C2×(C2×D3)⋊C4C2×C412yes—
86C2×(C2×D3)⋊C4C2×C412yes—
87C2×(C2×D3)⋊C4C2×C412yes—
88C2×(C2×D3)⋊C4C2×C412yes—
89C2×(C2×D3)⋊C4C2×C412yes—
90C2×C4⋊Dic3C2×C412yes—
91C2×C4⋊Dic3C2×C411yescharacteristic
92C2×Dic3⋊C4C2×C412yes—
93C2×Dic3⋊C4C2×C412yes—
94C2×C4×Dic3C2×C411yescharacteristic
95C2×C4×Dic3C2×C411yescharacteristic
96C2×C4×Dic3C2311yescharacteristic
97C22×C3⋊C8C2311yescharacteristic
98C22×C3⋊C8C2312yes—
99C22×C3⋊C8C2311yescharacteristic
100C2×ΓL(1,𝔽25)—44no—
101C2×ΓL(1,𝔽25)C2318yes—
102C2×D3×C8—44no—
103C2×D3×C8—44no—
104C2×D3×C8C2318yes—
105C2×D3×C8C2318yes—
106#898C2×C4116yes—
107#895C2×C418yes—
108#894C2×C4116yes—
109#890C2×C418yes—
110C4×D12C2×C418yes—
111C4×D12C2×C414yes—
112C4×D12C2314yes—
113C4×D12C2318yes—
114#872C2×C418yes—
115#872C2318yes—
116C42×D3C2×C414yes—
117C42×D3C2314yes—
118C3×C4⋊C8C2314yes—
119C3×C4⋊C8C2318yes—
120C3×#97C2318yes—
121C3×C4×C8C2314yes—
122(C2×D3)⋊C8C2314yes—
123(C2×D3)⋊C8C2314yes—
124(C2×D3)⋊C8C2318yes—
125(C2×D3)⋊C8C2318yes—
126Dic3⋊C8C2314yes—
127Dic3⋊C8C2314yes—
128Dic3⋊C8C2318yes—
129C4⋊(C3⋊C8)C2314yes—
130C4⋊(C3⋊C8)C2318yes—
131C4×C3⋊C8C2314yes—
132C4×C3⋊C8C2318yes—
133C22×M16—32no—
134C23×C8—32no—
135C23×C8—32no—
136#529—38no—
137#529—38no—
138C2×#126—32no—
139C2×#125—31no—
140C2×#124—31no—
141C2×#123—32no—
142C2×C22⋊Q8—31no—
143C2×#121—31no—
144C2×C4×Q8C2×D311yescharacteristic
145C2×C4×D4—32no—
146C2×C4×D4—31no—
147C2×#117—32no—
148C22×C42—31no—
149#446C2×D318yescomplemented
150C8×Q8—38no—
151#436—38no—
152#435—316no—
153C8×D4—316no—
154C8×D4—38no—
155#433—316no—
156#431—38no—
157#424—316no—
158C2×C4⋊C8—32no—
159C2×C4⋊C8—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8—31no—
162C2×C4⋊C8—32no—
163C2×C4⋊C8—38no—
164C2×C4⋊C8C2×D311yescharacteristic
165C2×C4⋊C8C2×D312yes—
166C2×C22⋊C8—31no—
167C2×C22⋊C8—31no—
168C2×C22⋊C8—32no—
169C2×C22⋊C8—32no—
170C2×#97C2×D312yes—
171C2×C4×C8—31no—
172C2×C4×C8—32no—
173C2×C4×C8C2×D311yescharacteristic
174C23×D3C22×C412yes—
175C23×D3C22×C411yescharacteristic
176C2×C3×Q8—24no—
177C2×C3×Q8C22×C414yes—
178C22×C3×C4C22×C411yescharacteristic
179C22×C3×C4C22×C412yes—
180C22×C3×C4C22×C412yes—
181C22×C3×C4C22×C411yescharacteristic
182C22×C3×C4C2411yescharacteristic
183C22×Dic3C22×C411yescharacteristic
184C22×Dic3C22×C411yescharacteristic
185C22×Dic3C22×C411yescharacteristic
186C22×Dic3C22×C412yes—
187C2×D12—28no—
188C2×D12—24no—
189C2×D12C22×C418yes—
190C2×D12C22×C414yes—
191C2×C4×D3—24no—
192C2×C4×D3—24no—
193C2×C4×D3—22no—
194C2×C4×D3—24no—
195C2×C4×D3—22no—
196C2×C4×D3—24no—
197C2×C4×D3C22×C412yes—
198C2×C4×D3C22×C414yes—
199C2×C4×D3C22×C414yes—
200C2×C4×D3C22×C414yes—
201C2×C4×D3C2412yes—
202C2×C4×D3C2414yes—
203C2×C3×C8—22no—
204C2×C3×C8—24no—
205C2×C3×C8—22no—
206C2×C3×C8C2412yes—
207C2×C3×C8C2412yes—
208C2×C3×C8C2414yes—
209C3×C4⋊C4C22×C414yes—
210C3×C4⋊C4C22×C418yes—
211C3×C42C22×C412yes—
212C3×C42C22×C414yes—
213C3×C42C2412yes—
214C3×C42C2414yes—
215(C2×D3)⋊C4C22×C418yes—
216(C2×D3)⋊C4C22×C418yes—
217(C2×D3)⋊C4C22×C418yes—
218(C2×D3)⋊C4C22×C418yes—
219(C2×D3)⋊C4C22×C418yes—
220C4⋊Dic3C22×C418yes—
221C4⋊Dic3C22×C414yes—
222Dic3⋊C4C22×C418yes—
223Dic3⋊C4C22×C418yes—
224C4×Dic3C22×C412yes—
225C4×Dic3C22×C414yes—
226C4×Dic3C22×C414yes—
227C4×Dic3C2412yes—
228C2×C3⋊C8—24no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8C2412yes—
232C2×C3⋊C8C2414yes—
233C2×C3⋊C8C2412yes—
234ΓL(1,𝔽25)—48no—
235D3×C8—48no—
236D3×C8—48no—
237C22×Q8C4×D311yescharacteristic
238C22×D4—32no—
239C22×D4—31no—
240C23×C4—31no—
241C23×C4—31no—
242C23×C4—32no—
243C23×C4—31no—
244C2×M16—38no—
245C2×M16—124no—
246C22×C8—32no—
247C22×C8—31no—
248C22×C8—31no—
249C22×C8—38no—
250C22×C8—38no—
251C22×C8—124no—
252C22×C8—124no—
253C22×C8C22×D311yescharacteristic
254C22×C8C22×D312yes—
255C22×C8C22×D311yescharacteristic
256#126—316no—
257#125—38no—
258#124—38no—
259#123—316no—
260C22⋊Q8—38no—
261#121—38no—
262C4×Q8C4×D314yes—
263C4×Q8C22×D314yes—
264C4×D4—34no—
265C4×D4—38no—
266C4×D4—38no—
267C4×D4—34no—
268#117—38no—
269#117—38no—
270C2×C4⋊C4—32no—
271C2×C4⋊C4—32no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—32no—
274C2×C4⋊C4C4×D311yescharacteristic
275C2×C4⋊C4C4×D312yes—
276C2×C22⋊C4—32no—
277C2×C22⋊C4—32no—
278C2×C22⋊C4—32no—
279C2×C22⋊C4—32no—
280C2×C22⋊C4—32no—
281C2×C42—31no—
282C2×C42—31no—
283C2×C42—31no—
284C2×C42—34no—
285C2×C42—34no—
286C2×C42C22×D312yes—
287C2×C42C22×D311yescharacteristic
288C4⋊C8—38no—
289C4⋊C8—34no—
290C4⋊C8—34no—
291C4⋊C8—34no—
292C4⋊C8—38no—
293C4⋊C8C22×D314yes—
294C4⋊C8C22×D318yes—
295C22⋊C8—34no—
296C22⋊C8—34no—
297C22⋊C8—38no—
298C22⋊C8—38no—
299#97C22×D318yes—
300C4×C8—34no—
301C4×C8—38no—
302C4×C8C22×D314yes—
303C23×C3C23×C411yescharacteristic
304C22×D3—22no—
305C22×D3—22no—
306C22×D3—24no—
307C22×D3—24no—
308C22×D3—24no—
309C22×D3#13114yes—
310C22×D3C23×C414yes—
311C22×D3C23×C412yes—
312C3×Q8—28no—
313C2×C3×C4—22no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4#13114yes—
317C2×C3×C4C23×C412yes—
318C2×C3×C4C23×C414yes—
319C2×C3×C4C23×C414yes—
320C2×C3×C4C23×C411yescharacteristic
321C2×C3×C4C23×C412yes—
322C2×C3×C4H(5,𝔽2)12yes—
323C2×C3×C42−1+412yes—
324C2×C3×C4C2511yescharacteristic
325C2×Dic3—22no—
326C2×Dic3—24no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3C23×C412yes—
330C2×Dic3C23×C412yes—
331C2×Dic3C23×C412yes—
332C2×Dic3C23×C414yes—
333D12—216no—
334D12—28no—
335C4×D3—28no—
336C4×D3—24no—
337C4×D3—28no—
338C4×D3—28no—
339C4×D3—22no—
340C4×D3—24no—
341C4×D3—22no—
342C4×D3—24no—
343C3×C8—22no—
344C3×C8—24no—
345C3×C8—22no—
346C3⋊C8—24no—
347C3⋊C8—22no—
348C3⋊C8—22no—
349C24—31no—
350C24—32no—
351C2×Q8—24no—
352C2×Q8C2×C4×D314yes—
353C2×D4—38no—
354C2×D4—34no—
355C2×D4—68no—
356C2×D4—64no—
357C22×C4—31no—
358C22×C4—31no—
359C22×C4—31no—
360C22×C4—32no—
361C22×C4—32no—
362C22×C4—34no—
363C22×C4—34no—
364C22×C4—34no—
365C22×C4—32no—
366C22×C4—34no—
367C22×C4—64no—
368C22×C4—64no—
369C22×C4—62no—
370C22×C4—64no—
371C22×C4—62no—
372C22×C4—64no—
373C22×C4C2×C4×D311yescharacteristic
374C22×C4C2×C4×D312yes—
375C22×C4C2×C4×D312yes—
376C22×C4C2×C4×D311yescharacteristic
377C22×C4C23×D311yescharacteristic
378M16—128no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—24no—
382C2×C8—32no—
383C2×C8—34no—
384C2×C8—32no—
385C2×C8—62no—
386C2×C8—64no—
387C2×C8—62no—
388C2×C8—128no—
389C2×C8—128no—
390C2×C8C23×D312yes—
391C2×C8C23×D312yes—
392C2×C8C23×D314yes—
393C4⋊C4—38no—
394C4⋊C4—38no—
395C4⋊C4—34no—
396C4⋊C4—38no—
397C4⋊C4C2×C4×D314yes—
398C4⋊C4C2×C4×D318yes—
399C22⋊C4—38no—
400C22⋊C4—38no—
401C22⋊C4—38no—
402C22⋊C4—38no—
403C22⋊C4—38no—
404C42—32no—
405C42—34no—
406C42—34no—
407C42—32no—
408C42C2×C4×D312yes—
409C42C2×C4×D314yes—
410C42C23×D312yes—
411C42C23×D314yes—
412C22×C3#52012yes—
413C22×C3C2×#13112yes—
414C22×C3C2×#13112yes—
415C22×C3C24×C411yescharacteristic
416C2×D3—22no—
417C2×D3—28no—
418C2×D3—22no—
419C2×D3—24no—
420C2×D3—24no—
421C2×D3—24no—
422C2×D3—48no—
423C2×D3C2×#13114yes—
424C3×C4—22no—
425C3×C4—24no—
426C3×C4—24no—
427C3×C4#52012yes—
428C3×C4C2×#13114yes—
429C3×C4C2×H(5,𝔽2)11yescharacteristic
430C3×C4C2×2−1+411yescharacteristic
431Dic3—22no—
432Dic3—24no—
433Dic3—22no—
434Dic3—22no—
435C23—32no—
436C23—34no—
437C23—34no—
438C23—62no—
439C23—62no—
440C23—64no—
441C23—64no—
442C23—64no—
443C23C22×C4×D311yescharacteristic
444Q8—28no—
445D4—616no—
446D4—68no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—32no—
451C2×C4—32no—
452C2×C4—32no—
453C2×C4—34no—
454C2×C4—68no—
455C2×C4—62no—
456C2×C4—62no—
457C2×C4—62no—
458C2×C4—64no—
459C2×C4—62no—
460C2×C4—64no—
461C2×C4—62no—
462C2×C4—64no—
463C2×C4—68no—
464C2×C4—64no—
465C2×C4—68no—
466C2×C4#90714yes—
467C2×C4C22×C4×D312yes—
468C2×C4C22×C4×D314yes—
469C2×C4C22×C4×D314yes—
470C2×C4C22×C4×D312yes—
471C2×C4C22×C4×D311yescharacteristic
472C2×C4#100712yes—
473C2×C4#100912yes—
474C2×C4C24×D311yescharacteristic
475C8—22no—
476C8—22no—
477C8—24no—
478C8—62no—
479C8—64no—
480C8—62no—
481C2×C3#299414yescomplemented
482C2×C3C2×#52011yescharacteristic
483C2×C3C22×#13111yescharacteristic
484C2×C3C22×#13111yescharacteristic
485D3—24no—
486D3—44no—
487C22—34no—
488C22—62no—
489C22—68no—
490C22—62no—
491C22—64no—
492C22—64no—
493C22—64no—
494C22—128no—
495C22#557412yes—
496C22C2×#90712yes—
497C22D3×#13112yes—
498C22C23×C4×D311yescharacteristic
499C4—24no—
500C4—22no—
501C4—24no—
502C4—62no—
503C4—64no—
504C4—62no—
505C4—62no—
506C4#557412yes—
507C4C2×#90714yes—
508C4C2×#100711yescharacteristic
509C4C2×#100911yescharacteristic
510C3C2×#299411yescharacteristic, complemented, Sylow
511C2—64no—
512C2—124no—
513C2#7994814yesdirect product factor
514C2C2×#557411yescharacteristic
515C2C22×#90711yescharacteristic
516C2C2×D3×#13111yescharacteristic
subgroupG/H
centerC22×C4C23×D3
commutatorC22×C3C24×C4
FrattiniC2×C4C24×D3
FittingC2×C3×#446C2
socleC23×C3C23×C4
prime radicalC2×#895C4
cosocleC2×C3×C4C25
hypercenterC2×#446D3
hypocenterC3C2×#2994
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×#446

Complements

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

NH ≅ G/Ntype
2. C2×C3×#446C2semidirect
3. C2×Q8×C3⋊C8C2semidirect
4. #79948C2direct (N × H)
5. C2×#4839C2semidirect
6. C2×D3×C4⋊C8C2semidirect
7. C2×#4838C2semidirect
8. C2×#4834C2semidirect
9. C2×#4706C2semidirect
10. C2×#4709C2semidirect
11. C2×C8×D12C2semidirect
12. C2×#4685C2semidirect
30. C3×#446C22semidirect
42. Q8×C3⋊C8C22semidirect
47. #4839C22semidirect
48. #4838C22semidirect
49. #4834C22semidirect
50. D3×C4⋊C8C22semidirect
51. #4709C22semidirect
52. #4706C22semidirect
53. C8×D12C22semidirect
54. #4685C22semidirect
57. C2×#446D3semidirect
149. #446C2×D3semidirect
481. C2×C3#2994semidirect
510. C3C2×#2994semidirect
513. C2#79948direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#55802
C2×C3×#4462
C2×Q8×C3⋊C82
#799482
C2×#48392
C2×D3×C4⋊C82
C2×#48382
C2×#48342
C2×#47062
C2×#47092
C2×C8×D122
C2×#46852
C2×#29943