Finite Group Databasebeta

← C2×#79295 #11660279

Subgroups — C2×#79295 #11660279

Order 768.

1926 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×#5479C211yesmaximal, characteristic
2C2×C3×#431C211yesmaximal, characteristic, complemented
3C2×#4924C211yesmaximal, characteristic, complemented
4#79295C2116yesmaximal, direct product factor
5C2×#4839C212yesmaximal, complemented
6C2×#4840C212yesmaximal, complemented
7C2×#4729C212yesmaximal, complemented
8C2×#4728C212yesmaximal, complemented
9C2×#4688C211yesmaximal, characteristic, complemented
10C2×C8×D12C211yesmaximal, characteristic, complemented
11C2×#4679C211yesmaximal, characteristic, complemented
12C2×C8×Dic6C211yesmaximal, characteristic, complemented
13C2×#2993—31nomaximal, Sylow
14C22×C3×C42C2211yescharacteristic
15C2×#930C411yescharacteristic
16C2×#929C412yes—
17C2×C4×#67C2212yes—
18C2×#924C411yescharacteristic
19C22×#208C2212yes—
20C22×ΓL(1,𝔽25)C2212yes—
21C22×D3×C8C2212yes—
22#5479C418yes—
23#5479C2218yes—
24C2×#876C412yes—
25C2×#875C411yescharacteristic
26C2×C4×D12C2211yescharacteristic
27C2×#872C2212yes—
28C2×#870C411yescharacteristic
29C2×C4×Dic6C2211yescharacteristic
30C3×#431C2218yescomplemented
31C2×C3×C4⋊C8C2211yescharacteristic
32C2×C3×C4⋊C8C2211yescharacteristic
33C2×C3×C22⋊C8C2211yescharacteristic
34C2×C3×C22⋊C8C2211yescharacteristic
35C2×C3×C4×C8C2211yescharacteristic
36C2×C3×C4×C8C2211yescharacteristic
37C2×(C2×D3)⋊C8C2212yes—
38C2×(C2×D3)⋊C8C2212yes—
39C2×#815C2212yes—
40C2×Dic3⋊C8C2212yes—
41C2×Dic3⋊C8C2212yes—
42C2×C8×Dic3C2212yes—
43#4924C2218yescomplemented
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46C2×C4⋊(C3⋊C8)C2212yes—
47#4840C22116yescomplemented
48#4839C22116yescomplemented
49#4729C22116yescomplemented
50#4728C22116yescomplemented
51#4688C2218yescomplemented
52C8×D12C2218yescomplemented
53#4679C2218yescomplemented
54C8×Dic6C2218yescomplemented
55C2×#529—31no—
56#2993—316no—
57C2×#446—31no—
58C2×C8×Q8—31no—
59C2×#436—32no—
60C2×#436—31no—
61C2×#435—32no—
62C2×C8×D4—31no—
63C2×#433—32no—
64C2×#431D311yescharacteristic, complemented
65C2×#424—32no—
66C2×#423—31no—
67C23×C3×C4C2×C411yescharacteristic
68C23×C3×C4C2×C411yescharacteristic
69C23×C3×C4C2311yescharacteristic
70C22×#67C2×C412yes—
71C22×D12C2×C411yescharacteristic
72C22×C4×D3C2312yes—
73C22×Dic6C2×C411yescharacteristic
74C22×C3×C8C2311yescharacteristic
75C22×C3×C8C2311yescharacteristic
76C22×C3×C8C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C2×C3×C42C2×C411yescharacteristic
79C2×C3×C42C2×C411yescharacteristic
80C2×C3×C42C2×C414yes—
81C2×C3×C42C2311yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C2×C3×C42C2314yes—
84C2×#217C2×C412yes—
85#930C2×C418yes—
86#929C2×C4116yes—
87C4×#67C2×C418yes—
88C4×#67C2318yes—
89C2×(C2×D3)⋊C4C2×C412yes—
90C2×(C2×D3)⋊C4C2×C412yes—
91C2×(C2×D3)⋊C4C2×C412yes—
92C2×(C2×D3)⋊C4C2×C412yes—
93C2×C4⋊Dic3C2×C411yescharacteristic
94C2×C4⋊Dic3C2×C411yescharacteristic
95#924C2×C418yes—
96C2×Dic3⋊C4C2×C412yes—
97C2×Dic3⋊C4C2×C412yes—
98C2×Dic3⋊C4C2×C412yes—
99C2×Dic3⋊C4C2×C412yes—
100C2×C4×Dic3C2312yes—
101C2×#208—44no—
102C2×#208C2318yes—
103C22×C3⋊C8C2312yes—
104C22×C3⋊C8C2312yes—
105C2×ΓL(1,𝔽25)—44no—
106C2×ΓL(1,𝔽25)C2318yes—
107C2×D3×C8—44no—
108C2×D3×C8C2318yes—
109#876C2×C4116yes—
110#875C2×C418yes—
111C4×D12C2×C414yes—
112C4×D12C2314yes—
113#872C2×C418yes—
114#872C2318yes—
115#870C2×C418yes—
116C4×Dic6C2×C414yes—
117C4×Dic6C2314yes—
118C3×C4⋊C8C2314yes—
119C3×C4⋊C8C2314yes—
120C3×C22⋊C8C2314yes—
121C3×C22⋊C8C2314yes—
122C3×C4×C8C2314yes—
123C3×C4×C8C2314yes—
124(C2×D3)⋊C8C2318yes—
125(C2×D3)⋊C8C2318yes—
126#815C2318yes—
127Dic3⋊C8C2318yes—
128Dic3⋊C8C2318yes—
129C8×Dic3C2318yes—
130C4⋊(C3⋊C8)C2314yes—
131C4⋊(C3⋊C8)C2314yes—
132C4⋊(C3⋊C8)C2318yes—
133C22×M16—32no—
134C22×M16—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×Q8—31no—
145C2×C4×D4—32no—
146C2×C4×D4—31no—
147C2×#117—32no—
148C22×C42C2×D311yescharacteristic
149#446—38no—
150C8×Q8—38no—
151#436—316no—
152#436—38no—
153#435—316no—
154C8×D4—38no—
155#433—316no—
156#431C2×D318yescomplemented
157#424—316no—
158#423—38no—
159C2×C4⋊C8—32no—
160C2×C4⋊C8—32no—
161C2×C4⋊C8—32no—
162C2×C4⋊C8—31no—
163C2×C4⋊C8—31no—
164C2×C4⋊C8C2×D311yescharacteristic
165C2×C4⋊C8C2×D311yescharacteristic
166C2×C22⋊C8—32no—
167C2×C22⋊C8—32no—
168C2×C22⋊C8C2×D311yescharacteristic
169C2×C22⋊C8C2×D311yescharacteristic
170C2×#97—32no—
171C2×C4×C8—32no—
172C2×C4×C8C2×D311yescharacteristic
173C2×C4×C8C2×D311yescharacteristic
174C24×C3C22×C411yescharacteristic
175C23×D3C22×C412yes—
176C22×C3×C4—24no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—42no—
180C22×C3×C4C22×C411yescharacteristic
181C22×C3×C4C22×C411yescharacteristic
182C22×C3×C4C22×C411yescharacteristic
183C22×C3×C4C22×C411yescharacteristic
184C22×C3×C4C22×C414yes—
185C22×C3×C4C22×C412yes—
186C22×C3×C4C22×C414yes—
187C22×C3×C4C22×C411yescharacteristic
188C22×C3×C4C2412yes—
189C22×C3×C4C2411yescharacteristic
190C2×#67—44no—
191C2×#67C22×C418yes—
192C22×Dic3C22×C412yes—
193C22×Dic3C22×C412yes—
194C22×Dic3C22×C412yes—
195C2×D12—24no—
196C2×D12C22×C414yes—
197C2×C4×D3—24no—
198C2×C4×D3—24no—
199C2×C4×D3C22×C414yes—
200C2×C4×D3C2414yes—
201C2×Dic6—24no—
202C2×Dic6C22×C414yes—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8C2412yes—
208C2×C3×C8C2412yes—
209C2×C3×C8C2412yes—
210C2×C3×C8C2412yes—
211C3×C42C22×C412yes—
212C3×C42C22×C414yes—
213C3×C42C22×C412yes—
214C3×C42C22×C414yes—
215C3×C42C2412yes—
216C3×C42C2412yes—
217#217C22×C418yes—
218(C2×D3)⋊C4C22×C418yes—
219(C2×D3)⋊C4C22×C418yes—
220(C2×D3)⋊C4C22×C418yes—
221(C2×D3)⋊C4C22×C418yes—
222C4⋊Dic3C22×C414yes—
223C4⋊Dic3C22×C414yes—
224Dic3⋊C4C22×C418yes—
225Dic3⋊C4C22×C418yes—
226Dic3⋊C4C22×C418yes—
227Dic3⋊C4C22×C418yes—
228C4×Dic3C22×C414yes—
229C4×Dic3C2414yes—
230#208—48no—
231C2×C3⋊C8—24no—
232C2×C3⋊C8—24no—
233C2×C3⋊C8C2414yes—
234C2×C3⋊C8C2414yes—
235ΓL(1,𝔽25)—48no—
236D3×C8—48no—
237C22×Q8—31no—
238C22×D4—32no—
239C22×D4—31no—
240C23×C4—32no—
241C23×C4C4×D311yescharacteristic
242C23×C4C4×D311yescharacteristic
243C23×C4C22×D311yescharacteristic
244C2×M16—38no—
245C2×M16—38no—
246C2×M16—124no—
247C2×M16—124no—
248C22×C8—32no—
249C22×C8—32no—
250C22×C8—38no—
251C22×C8—124no—
252C22×C8C22×D311yescharacteristic
253C22×C8C22×D311yescharacteristic
254C22×C8C22×D311yescharacteristic
255C22×C8C22×D311yescharacteristic
256#126—316no—
257#125—38no—
258#124—38no—
259#123—316no—
260C22⋊Q8—38no—
261#121—38no—
262C4×Q8—34no—
263C4×Q8—34no—
264C4×D4—38no—
265C4×D4—38no—
266C4×D4—34no—
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⋊C4—31no—
275C2×C4⋊C4—32no—
276C2×C22⋊C4—32no—
277C2×C22⋊C4—32no—
278C2×C22⋊C4—32no—
279C2×C22⋊C4—32no—
280C2×C22⋊C4—32no—
281C2×C42—32no—
282C2×C42C4×D311yescharacteristic
283C2×C42C4×D311yescharacteristic
284C2×C42C4×D314yes—
285C2×C42C22×D314yes—
286C2×C42C22×D311yescharacteristic
287C2×C42C22×D311yescharacteristic
288C4⋊C8—38no—
289C4⋊C8—38no—
290C4⋊C8—38no—
291C4⋊C8—34no—
292C4⋊C8—34no—
293C4⋊C8C22×D314yes—
294C4⋊C8C22×D314yes—
295C22⋊C8—38no—
296C22⋊C8—38no—
297C22⋊C8C22×D314yes—
298C22⋊C8C22×D314yes—
299#97—38no—
300C4×C8—38no—
301C4×C8C22×D314yes—
302C4×C8C22×D314yes—
303C23×C3—22no—
304C23×C3—22no—
305C23×C3—22no—
306C23×C3C23×C412yes—
307C23×C3C23×C411yescharacteristic
308C22×D3—24no—
309C22×D3—24no—
310C22×D3—24no—
311C22×D3C23×C414yes—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—24no—
317C2×C3×C4—28no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—44no—
321C2×C3×C4#13114yes—
322C2×C3×C4C23×C412yes—
323C2×C3×C4C23×C412yes—
324C2×C3×C4C23×C412yes—
325C2×C3×C4C23×C412yes—
326C2×C3×C4C23×C411yescharacteristic
327C2×C3×C4C23×C412yes—
328C2×C3×C4H(5,𝔽2)12yes—
329C2×C3×C42−1+412yes—
330C2×C3×C4C2511yescharacteristic
331#67—48no—
332C2×Dic3—24no—
333C2×Dic3—24no—
334C2×Dic3—24no—
335C2×Dic3C23×C414yes—
336C2×Dic3C23×C414yes—
337C2×Dic3C23×C414yes—
338D12—28no—
339C4×D3—28no—
340C4×D3—24no—
341C4×D3—24no—
342Dic6—28no—
343C3×C8—22no—
344C3×C8—22no—
345C3×C8—22no—
346C3×C8—22no—
347C3⋊C8—24no—
348C3⋊C8—24no—
349C24—32no—
350C24C2×C4×D311yescharacteristic
351C2×Q8—34no—
352C2×Q8—64no—
353C2×D4—38no—
354C2×D4—34no—
355C2×D4—64no—
356C2×D4—124no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—32no—
361C22×C4—32no—
362C22×C4—32no—
363C22×C4—34no—
364C22×C4—34no—
365C22×C4—42no—
366C22×C4—64no—
367C22×C4—64no—
368C22×C4C2×C4×D311yescharacteristic
369C22×C4C2×C4×D312yes—
370C22×C4C2×C4×D311yescharacteristic
371C22×C4C2×C4×D311yescharacteristic
372C22×C4C2×C4×D311yescharacteristic
373C22×C4C2×C4×D314yes—
374C22×C4C2×C4×D314yes—
375C22×C4C2×C4×D311yescharacteristic
376C22×C4C23×D312yes—
377C22×C4C23×D311yescharacteristic
378M16—128no—
379M16—128no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—34no—
385C2×C8—34no—
386C2×C8—64no—
387C2×C8—64no—
388C2×C8—128no—
389C2×C8C23×D312yes—
390C2×C8C23×D312yes—
391C2×C8C23×D312yes—
392C2×C8C23×D312yes—
393C4⋊C4—38no—
394C4⋊C4—38no—
395C4⋊C4—34no—
396C4⋊C4—38no—
397C4⋊C4—34no—
398C4⋊C4—38no—
399C22⋊C4—38no—
400C22⋊C4—38no—
401C22⋊C4—38no—
402C22⋊C4—38no—
403C22⋊C4—38no—
404C42—34no—
405C42—34no—
406C42C2×C4×D312yes—
407C42C2×C4×D312yes—
408C42C2×C4×D314yes—
409C42C2×C4×D314yes—
410C42C23×D312yes—
411C42C23×D312yes—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—22no—
415C22×C3—44no—
416C22×C3#52012yes—
417C22×C3C2×#13112yes—
418C22×C3#56812yes—
419C22×C3C24×C411yescharacteristic
420C2×D3—24no—
421C2×D3—24no—
422C2×D3—24no—
423C2×D3—48no—
424C3×C4—22no—
425C3×C4—22no—
426C3×C4—22no—
427C3×C4—22no—
428C3×C4#52012yes—
429C3×C4C2×#13114yes—
430C3×C4C2×H(5,𝔽2)11yescharacteristic
431C3×C4C2×2−1+411yescharacteristic
432Dic3—24no—
433Dic3—24no—
434Dic3—24no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—34no—
439C23—64no—
440C23—64no—
441C23—64no—
442C23C22×C4×D312yes—
443C23C22×C4×D311yescharacteristic
444Q8—68no—
445D4—68no—
446D4—128no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—28no—
454C2×C4—24no—
455C2×C4—34no—
456C2×C4—34no—
457C2×C4—34no—
458C2×C4—44no—
459C2×C4—64no—
460C2×C4—64no—
461C2×C4—64no—
462C2×C4—64no—
463C2×C4—64no—
464C2×C4—68no—
465C2×C4#90114yes—
466C2×C4C22×C4×D312yes—
467C2×C4C22×C4×D312yes—
468C2×C4C22×C4×D312yes—
469C2×C4C22×C4×D312yes—
470C2×C4C22×C4×D312yes—
471C2×C4C22×C4×D311yescharacteristic
472C2×C4#100912yes—
473C2×C4#101012yes—
474C2×C4C24×D311yescharacteristic
475C8—22no—
476C8—22no—
477C8—22no—
478C8—22no—
479C8—64no—
480C8—64no—
481C2×C3—42no—
482C2×C3#299314yescomplemented
483C2×C3C2×#52011yescharacteristic
484C2×C3C22×#13111yescharacteristic
485C2×C3C2×#56811yescharacteristic
486D3—44no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—44no—
491C22—64no—
492C22—64no—
493C22—64no—
494C22—128no—
495C22#552312yes—
496C22C2×#90112yes—
497C22#574512yes—
498C22C23×C4×D311yescharacteristic
499C4—22no—
500C4—22no—
501C4—22no—
502C4—22no—
503C4—64no—
504C4—64no—
505C4—64no—
506C4#552312yes—
507C4C2×#90114yes—
508C4C2×#100911yescharacteristic
509C4C2×#101011yescharacteristic
510C3C2×#299311yescharacteristic, complemented, Sylow
511C2—42no—
512C2—124no—
513C2#7929514yesdirect product factor
514C2C2×#552311yescharacteristic
515C2C22×#90111yescharacteristic
516C2C2×#574511yescharacteristic
subgroupG/H
centerC22×C4C23×D3
commutatorC22×C3C24×C4
FrattiniC2×C4C24×D3
FittingC2×C3×#431C2
socleC23×C3C23×C4
prime radicalC2×#930C4
cosocleC2×C3×C4C25
hypercenterC2×#431D3
hypocenterC3C2×#2993
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×#431

Complements

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

NH ≅ G/Ntype
2. C2×C3×#431C2semidirect
3. C2×#4924C2semidirect
4. #79295C2direct (N × H)
5. C2×#4839C2semidirect
6. C2×#4840C2semidirect
7. C2×#4729C2semidirect
8. C2×#4728C2semidirect
9. C2×#4688C2semidirect
10. C2×C8×D12C2semidirect
11. C2×#4679C2semidirect
12. C2×C8×Dic6C2semidirect
30. C3×#431C22semidirect
43. #4924C22semidirect
47. #4840C22semidirect
48. #4839C22semidirect
49. #4729C22semidirect
50. #4728C22semidirect
51. #4688C22semidirect
52. C8×D12C22semidirect
53. #4679C22semidirect
54. C8×Dic6C22semidirect
64. C2×#431D3semidirect
156. #431C2×D3semidirect
482. C2×C3#2993semidirect
510. C3C2×#2993semidirect
513. C2#79295direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#54792
C2×C3×#4312
C2×#49242
#792952
C2×#48392
C2×#48402
C2×#47292
C2×#47282
C2×#46882
C2×C8×D122
C2×#46792
C2×C8×Dic62
C2×#29933