Finite Group Databasebeta

← C2×#79944 #11660240

Subgroups — C2×#79944 #11660240

Order 768.

2154 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#239C211yesmaximal, characteristic
2C2×C3×#446C211yesmaximal, characteristic, complemented
3C2×#5024C211yesmaximal, characteristic, complemented
4#79944C2116yesmaximal, direct product factor
5C2×#4837C212yesmaximal, complemented
6C2×D3×C4⋊C8C211yesmaximal, characteristic, complemented
7C2×#4838C211yesmaximal, characteristic, complemented
8C2×#4836C212yesmaximal, complemented
9C2×#4706C212yesmaximal, complemented
10C2×#4705C212yesmaximal, complemented
11C2×#4688C211yesmaximal, characteristic, complemented
12C2×#4685C211yesmaximal, characteristic, complemented
13C2×#2989—31nomaximal, Sylow
14C22×#239C411yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×Q8×Dic3C411yescharacteristic
17C22×ΓL(1,𝔽25)C2212yes—
18C22×ΓL(1,𝔽25)C2212yes—
19C22×D3×C8C2212yes—
20C4×#239C418yes—
21C4×#239C2218yes—
22C2×#893C412yes—
23C2×#893C411yescharacteristic
24C2×#892C412yes—
25C2×#892C411yescharacteristic
26C2×C4×D12C2211yescharacteristic
27C2×C4×D12C2212yes—
28C2×C42×D3C2211yescharacteristic
29C2×C42×D3C2212yes—
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—
42#5024C2218yescomplemented
43C2×C4⋊(C3⋊C8)C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2212yes—
45C2×#803C2212yes—
46C2×C4×C3⋊C8C2211yescharacteristic
47#4838C2218yescomplemented
48#4837C22116yescomplemented
49#4836C22116yescomplemented
50D3×C4⋊C8C2218yescomplemented
51#4706C22116yescomplemented
52#4705C22116yescomplemented
53#4688C2218yescomplemented
54#4685C2218yescomplemented
55C2×C4×#41—31no—
56#2989—316no—
57C2×#446—31no—
58C2×#446D311yescharacteristic, complemented
59C2×#436—31no—
60C2×#436—31no—
61C2×#435—32no—
62C2×#435—32no—
63C2×#432—32no—
64C2×#431—31no—
65C2×#423—32no—
66C22×C4⋊C8—31no—
67C22×C3×Q8C2×C411yescharacteristic
68C2×#239—28no—
69C2×#239C2×C418yes—
70C22×D12C2×C412yes—
71C22×D12C2×C411yescharacteristic
72C22×C4×D3C2×C412yes—
73C22×C4×D3C2×C412yes—
74C22×C4×D3C2×C411yescharacteristic
75C22×C4×D3C2×C411yescharacteristic
76C22×C4×D3C2311yescharacteristic
77C22×C4×D3C2312yes—
78C22×C3×C8C2312yes—
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C3×C4×Q8C2×C414yes—
82C3×C4×Q8C2314yes—
83C2×C3×C4⋊C4C2×C412yes—
84C2×C3×C4⋊C4C2×C411yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×C3×C42C2312yes—
87Q8×Dic3C2×C418yes—
88C2×(C2×D3)⋊C4C2×C412yes—
89C2×(C2×D3)⋊C4C2×C412yes—
90C2×(C2×D3)⋊C4C2×C412yes—
91C2×C4⋊Dic3C2×C412yes—
92C2×C4⋊Dic3C2×C411yescharacteristic
93C2×C4×Dic3C2×C411yescharacteristic
94C2×C4×Dic3C2×C412yes—
95C2×C4×Dic3C2×C412yes—
96C2×C4×Dic3C2×C411yescharacteristic
97C2×C4×Dic3C2311yescharacteristic
98C22×C3⋊C8C2311yescharacteristic
99C22×C3⋊C8C2311yescharacteristic
100C22×C3⋊C8C2312yes—
101C2×ΓL(1,𝔽25)—44no—
102C2×ΓL(1,𝔽25)—44no—
103C2×ΓL(1,𝔽25)C2318yes—
104C2×ΓL(1,𝔽25)C2318yes—
105C2×D3×C8—44no—
106C2×D3×C8C2318yes—
107#893C2×C4116yes—
108#893C2×C418yes—
109#892C2×C4116yes—
110#892C2×C418yes—
111C4×D12C2×C414yes—
112C4×D12C2×C418yes—
113C4×D12C2314yes—
114C4×D12C2318yes—
115C42×D3C2×C418yes—
116C42×D3C2×C414yes—
117C42×D3C2314yes—
118C42×D3C2318yes—
119C3×C4⋊C8C2314yes—
120C3×C4⋊C8C2318yes—
121C3×#97C2318yes—
122C3×C4×C8C2314yes—
123(C2×D3)⋊C8C2314yes—
124(C2×D3)⋊C8C2314yes—
125(C2×D3)⋊C8C2318yes—
126(C2×D3)⋊C8C2318yes—
127Dic3⋊C8C2314yes—
128Dic3⋊C8C2314yes—
129Dic3⋊C8C2318yes—
130C4⋊(C3⋊C8)C2314yes—
131C4⋊(C3⋊C8)C2318yes—
132#803C2318yes—
133C4×C3⋊C8C2314yes—
134C22×#41—31no—
135C22×M16—32no—
136C22×M16—32no—
137C23×C8—32no—
138C4×#41—38no—
139C4×#41—38no—
140C2×C4×Q8—31no—
141C2×C4×Q8C2×D311yescharacteristic
142C2×C4×D4—32no—
143C2×C4×D4—31no—
144C2×C4×D4—32no—
145C2×C4×D4—31no—
146C2×#117—31no—
147C2×#117—32no—
148C22×C42—31no—
149C22×C42—32no—
150#446—38no—
151#446C2×D318yescomplemented
152#436—38no—
153#436—38no—
154#435—316no—
155#435—316no—
156#432—316no—
157#431—38no—
158#423—316no—
159C2×C4⋊C8—32no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8—31no—
162C2×C4⋊C8—31no—
163C2×C4⋊C8—32no—
164C2×C4⋊C8—38no—
165C2×C4⋊C8C2×D311yescharacteristic
166C2×C4⋊C8C2×D312yes—
167C2×C22⋊C8—31no—
168C2×C22⋊C8—31no—
169C2×C22⋊C8—32no—
170C2×C22⋊C8—32no—
171C2×#97—32no—
172C2×#97C2×D312yes—
173C2×C4×C8—31no—
174C2×C4×C8C2×D311yescharacteristic
175C23×D3C22×C412yes—
176C23×D3C22×C411yescharacteristic
177C2×C3×Q8—24no—
178C2×C3×Q8C22×C414yes—
179C22×C3×C4C22×C412yes—
180C22×C3×C4C22×C411yescharacteristic
181C22×C3×C4C22×C412yes—
182C22×C3×C4C22×C411yescharacteristic
183C22×C3×C4C2411yescharacteristic
184C22×Dic3C22×C411yescharacteristic
185C22×Dic3C22×C411yescharacteristic
186C22×Dic3C22×C412yes—
187C22×Dic3C22×C411yescharacteristic
188#239—232no—
189C2×D12—28no—
190C2×D12—24no—
191C2×D12C22×C418yes—
192C2×D12C22×C414yes—
193C2×C4×D3—22no—
194C2×C4×D3—24no—
195C2×C4×D3—22no—
196C2×C4×D3—24no—
197C2×C4×D3—28no—
198C2×C4×D3—42no—
199C2×C4×D3—44no—
200C2×C4×D3M1618yescomplemented
201C2×C4×D3C22×C414yes—
202C2×C4×D3C22×C412yes—
203C2×C4×D3C22×C418yes—
204C2×C4×D3C22×C418yes—
205C2×C4×D3C22×C414yes—
206C2×C4×D3C22×C414yes—
207C2×C4×D3C2412yes—
208C2×C4×D3C2414yes—
209C2×C3×C8—24no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8C2412yes—
213C2×C3×C8C2412yes—
214C2×C3×C8C2414yes—
215C3×C4⋊C4C22×C418yes—
216C3×C4⋊C4C22×C414yes—
217C3×C42C22×C414yes—
218C3×C42C22×C412yes—
219C3×C42C2412yes—
220C3×C42C2414yes—
221(C2×D3)⋊C4C22×C418yes—
222(C2×D3)⋊C4C22×C418yes—
223(C2×D3)⋊C4C22×C418yes—
224C4⋊Dic3C22×C418yes—
225C4⋊Dic3C22×C414yes—
226C4×Dic3C22×C412yes—
227C4×Dic3C22×C414yes—
228C4×Dic3C22×C418yes—
229C4×Dic3C22×C418yes—
230C4×Dic3C22×C414yes—
231C4×Dic3C2412yes—
232C2×C3⋊C8—24no—
233C2×C3⋊C8—22no—
234C2×C3⋊C8—22no—
235C2×C3⋊C8C2412yes—
236C2×C3⋊C8C2412yes—
237C2×C3⋊C8C2414yes—
238ΓL(1,𝔽25)—48no—
239ΓL(1,𝔽25)—48no—
240D3×C8—48no—
241C2×#41—38no—
242C2×#41—68no—
243C22×Q8C4×D311yescharacteristic
244C22×D4—32no—
245C22×D4—31no—
246C23×C4—31no—
247C23×C4—32no—
248C23×C4—31no—
249C23×C4—32no—
250C23×C4—31no—
251C23×C4—32no—
252C2×M16—38no—
253C2×M16—38no—
254C2×M16—124no—
255C2×M16—124no—
256C22×C8—32no—
257C22×C8—31no—
258C22×C8—31no—
259C22×C8—38no—
260C22×C8—124no—
261C22×C8C22×D312yes—
262C22×C8C22×D311yescharacteristic
263C22×C8C22×D311yescharacteristic
264C4×Q8—38no—
265C4×Q8C4×D314yes—
266C4×Q8C22×D314yes—
267C4×D4—316no—
268C4×D4—38no—
269C4×D4—38no—
270C4×D4—34no—
271C4×D4—38no—
272C4×D4—34no—
273#117—38no—
274#117—316no—
275C2×C4⋊C4—32no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4C4×D312yes—
278C2×C4⋊C4C4×D311yescharacteristic
279C2×C22⋊C4—32no—
280C2×C22⋊C4—32no—
281C2×C22⋊C4—32no—
282C2×C42—31no—
283C2×C42—32no—
284C2×C42—32no—
285C2×C42—31no—
286C2×C42—31no—
287C2×C42—34no—
288C2×C42—38no—
289C2×C42—34no—
290C2×C42—38no—
291C2×C42C22×D311yescharacteristic
292C2×C42C22×D312yes—
293C4⋊C8—38no—
294C4⋊C8—34no—
295C4⋊C8—34no—
296C4⋊C8—34no—
297C4⋊C8—38no—
298C4⋊C8C22×D314yes—
299C4⋊C8C22×D318yes—
300C22⋊C8—34no—
301C22⋊C8—34no—
302C22⋊C8—38no—
303C22⋊C8—38no—
304#97—38no—
305#97C22×D318yes—
306C4×C8—34no—
307C4×C8C22×D314yes—
308C23×C3C23×C411yescharacteristic
309C22×D3—22no—
310C22×D3—22no—
311C22×D3—24no—
312C22×D3—24no—
313C22×D3—24no—
314C22×D3C2×M1614yes—
315C22×D3C23×C414yes—
316C22×D3C23×C412yes—
317C3×Q8—28no—
318C2×C3×C4—24no—
319C2×C3×C4—22no—
320C2×C3×C4—24no—
321C2×C3×C4C2×M1614yes—
322C2×C3×C4C23×C414yes—
323C2×C3×C4C23×C412yes—
324C2×C3×C4C23×C414yes—
325C2×C3×C4C23×C411yescharacteristic
326C2×C3×C4C23×C412yes—
327C2×C3×C4H(5,𝔽2)12yes—
328C2×C3×C42−1+412yes—
329C2×C3×C4C2511yescharacteristic
330C2×Dic3—22no—
331C2×Dic3—24no—
332C2×Dic3—22no—
333C2×Dic3C2×M1614yes—
334C2×Dic3C23×C412yes—
335C2×Dic3C23×C412yes—
336C2×Dic3C23×C414yes—
337C2×Dic3C23×C412yes—
338D12—216no—
339D12—28no—
340C4×D3—24no—
341C4×D3—28no—
342C4×D3—22no—
343C4×D3—24no—
344C4×D3—22no—
345C4×D3—24no—
346C4×D3—216no—
347C4×D3—44no—
348C4×D3—48no—
349C4×D3C2×M16116yescomplemented
350C3×C8—24no—
351C3×C8—22no—
352C3×C8—22no—
353C3⋊C8—24no—
354C3⋊C8—22no—
355C3⋊C8—22no—
356C24—31no—
357C24—32no—
358#41—632no—
359C2×Q8—24no—
360C2×Q8C2×C4×D314yes—
361C2×D4—38no—
362C2×D4—34no—
363C2×D4—68no—
364C2×D4—64no—
365C22×C4—31no—
366C22×C4—38no—
367C22×C4—34no—
368C22×C4—31no—
369C22×C4—31no—
370C22×C4—32no—
371C22×C4—38no—
372C22×C4—32no—
373C22×C4—34no—
374C22×C4—32no—
375C22×C4—34no—
376C22×C4—34no—
377C22×C4—38no—
378C22×C4—68no—
379C22×C4—62no—
380C22×C4—64no—
381C22×C4—62no—
382C22×C4—64no—
383C22×C4—122no—
384C22×C4—124no—
385C22×C4C2×C4×D311yescharacteristic
386C22×C4C2×C4×D312yes—
387C22×C4C2×C4×D312yes—
388C22×C4C2×C4×D311yescharacteristic
389C22×C4C23×D311yescharacteristic
390M16—128no—
391M16—128no—
392C2×C8—22no—
393C2×C8—24no—
394C2×C8—22no—
395C2×C8—32no—
396C2×C8—32no—
397C2×C8—34no—
398C2×C8—62no—
399C2×C8—64no—
400C2×C8—62no—
401C2×C8—128no—
402C2×C8C23×D312yes—
403C2×C8C23×D312yes—
404C2×C8C23×D314yes—
405C4⋊C4—38no—
406C4⋊C4—34no—
407C4⋊C4C2×C4×D318yes—
408C4⋊C4C2×C4×D314yes—
409C22⋊C4—38no—
410C22⋊C4—38no—
411C22⋊C4—38no—
412C42—32no—
413C42—34no—
414C42—38no—
415C42—38no—
416C42—34no—
417C42—32no—
418C42C2×C4×D314yes—
419C42C2×C4×D312yes—
420C42C23×D312yes—
421C42C23×D314yes—
422C22×C3#52012yes—
423C22×C3C22×M1612yes—
424C22×C3#56812yes—
425C22×C3C24×C411yescharacteristic
426C2×D3—22no—
427C2×D3—28no—
428C2×D3—22no—
429C2×D3—24no—
430C2×D3—24no—
431C2×D3—24no—
432C2×D3—48no—
433C2×D3C22×M1614yes—
434C3×C4—24no—
435C3×C4—22no—
436C3×C4—24no—
437C3×C4#52012yes—
438C3×C4C22×M1614yes—
439C3×C4C2×H(5,𝔽2)11yescharacteristic
440C3×C4C2×2−1+411yescharacteristic
441Dic3—22no—
442Dic3—24no—
443Dic3—22no—
444Dic3C22×M1614yes—
445C23—32no—
446C23—34no—
447C23—34no—
448C23—62no—
449C23—62no—
450C23—64no—
451C23—64no—
452C23—64no—
453C23C22×C4×D311yescharacteristic
454Q8—28no—
455D4—616no—
456D4—68no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—32no—
461C2×C4—34no—
462C2×C4—316no—
463C2×C4—32no—
464C2×C4—32no—
465C2×C4—34no—
466C2×C4—616no—
467C2×C4—62no—
468C2×C4—62no—
469C2×C4—64no—
470C2×C4—62no—
471C2×C4—64no—
472C2×C4—62no—
473C2×C4—64no—
474C2×C4—64no—
475C2×C4—68no—
476C2×C4—124no—
477C2×C4—128no—
478C2×C4D3×M1614yes—
479C2×C4C22×C4×D312yes—
480C2×C4C22×C4×D314yes—
481C2×C4C22×C4×D314yes—
482C2×C4C22×C4×D312yes—
483C2×C4C22×C4×D311yescharacteristic
484C2×C4#100712yes—
485C2×C4#100912yes—
486C2×C4C24×D311yescharacteristic
487C8—22no—
488C8—24no—
489C8—22no—
490C8—62no—
491C8—64no—
492C8—62no—
493C2×C3#298914yescomplemented
494C2×C3C2×#52011yescharacteristic
495C2×C3C23×M1611yescharacteristic
496C2×C3C2×#56811yescharacteristic
497D3—24no—
498D3—44no—
499C22—34no—
500C22—62no—
501C22—68no—
502C22—62no—
503C22—64no—
504C22—64no—
505C22—64no—
506C22—128no—
507C22#557412yes—
508C22C2×D3×M1612yes—
509C22#575012yes—
510C22C23×C4×D311yescharacteristic
511C4—24no—
512C4—22no—
513C4—24no—
514C4—34no—
515C4—62no—
516C4—64no—
517C4—62no—
518C4#557412yes—
519C4C2×D3×M1614yes—
520C4C2×#100711yescharacteristic
521C4C2×#100911yescharacteristic
522C3C2×#298911yescharacteristic, complemented, Sylow
523C2—64no—
524C2—124no—
525C2#7994414yesdirect product factor
526C2C2×#557411yescharacteristic
527C2C22×D3×M1611yescharacteristic
528C2C2×#575011yescharacteristic
subgroupG/H
centerC22×C4C23×D3
commutatorC22×C3C24×C4
FrattiniC2×C4C24×D3
FittingC2×C3×#446C2
socleC23×C3C23×C4
prime radicalC22×#239C4
cosocleC2×C3×C4C25
hypercenterC2×#446D3
hypocenterC3C2×#2989
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×#5024C2semidirect
4. #79944C2direct (N × H)
5. C2×#4837C2semidirect
6. C2×D3×C4⋊C8C2semidirect
7. C2×#4838C2semidirect
8. C2×#4836C2semidirect
9. C2×#4706C2semidirect
10. C2×#4705C2semidirect
11. C2×#4688C2semidirect
12. C2×#4685C2semidirect
30. C3×#446C22semidirect
42. #5024C22semidirect
47. #4838C22semidirect
48. #4837C22semidirect
49. #4836C22semidirect
50. D3×C4⋊C8C22semidirect
51. #4706C22semidirect
52. #4705C22semidirect
53. #4688C22semidirect
54. #4685C22semidirect
58. C2×#446D3semidirect
151. #446C2×D3semidirect
200. C2×C4×D3M16semidirect
349. C4×D3C2×M16semidirect
493. C2×C3#2989semidirect
522. C3C2×#2989semidirect
525. C2#79944direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#2392
C2×C3×#4462
C2×#50242
#799442
C2×#48372
C2×D3×C4⋊C82
C2×#48382
C2×#48362
C2×#47062
C2×#47052
C2×#46882
C2×#46852
C2×#29893