Finite Group Databasebeta

← C2×#79408 #11660278

Subgroups — C2×#79408 #11660278

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×#5557C211yesmaximal, characteristic
2C2×C3×C8×D4C211yesmaximal, characteristic, complemented
3C2×#5111C212yesmaximal, complemented
4C2×#5102C212yesmaximal, complemented
5C2×#5012C211yesmaximal, characteristic, complemented
6#79408C2116yesmaximal, direct product factor
7C2×#4837C211yesmaximal, characteristic, complemented
8C2×#4831C211yesmaximal, characteristic, complemented
9C2×#4729C212yesmaximal, complemented
10C2×#4720C212yesmaximal, complemented
11C2×#4685C211yesmaximal, characteristic, complemented
12C2×C8×Dic6C211yesmaximal, characteristic, complemented
13C2×#2993—31nomaximal, Sylow
14C23×C3×C8C2212yes—
15C2×C3×C4×D4C2211yescharacteristic
16C2×#938C411yescharacteristic
17C2×#936C411yescharacteristic
18C2×C4×#67C2212yes—
19C2×#926C2212yes—
20C22×#208C2212yes—
21C22×ΓL(1,𝔽25)C2212yes—
22#5557C418yes—
23#5557C2218yes—
24C2×#897C411yescharacteristic
25C2×#890C411yescharacteristic
26C2×#883C412yes—
27C2×#879C412yes—
28C2×C42×D3C2211yescharacteristic
29C2×C4×Dic6C2211yescharacteristic
30C3×C8×D4C2218yescomplemented
31C2×C3×C4⋊C8C2211yescharacteristic
32C2×C3×C22⋊C8C2212yes—
33C2×C3×C4×C8C2211yescharacteristic
34C2×#830C2212yes—
35#5111C22116yescomplemented
36C2×(C2×D3)⋊C8C2211yescharacteristic
37C2×(C2×D3)⋊C8C2211yescharacteristic
38#5102C22116yescomplemented
39C2×#815C2212yes—
40C2×Dic3⋊C8C2212yes—
41C2×Dic3⋊C8C2211yescharacteristic
42C2×Dic3⋊C8C2212yes—
43C2×Dic3⋊C8C2212yes—
44C2×Dic3⋊C8C2211yescharacteristic
45C2×C8×Dic3C2212yes—
46#5012C2218yescomplemented
47C2×C4⋊(C3⋊C8)C2211yescharacteristic
48C2×C4×C3⋊C8C2211yescharacteristic
49#4837C2218yescomplemented
50#4831C2218yescomplemented
51#4729C22116yescomplemented
52#4720C22116yescomplemented
53#4685C2218yescomplemented
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×D4D311yescharacteristic, complemented
63C2×#433—32no—
64C2×#431—31no—
65C2×#424—32no—
66C2×#423—31no—
67C22×C3×D4C2×C411yescharacteristic
68C23×C3×C4C2312yes—
69C22×#67C2×C412yes—
70C22×C4×D3C2×C411yescharacteristic
71C22×C4×D3C2×C411yescharacteristic
72C22×C4×D3C2311yescharacteristic
73C22×Dic6C2×C411yescharacteristic
74C22×C3×C8—44no—
75C22×C3×C8C2311yescharacteristic
76C22×C3×C8C2312yes—
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2318yes—
79C3×C4×D4C2×C414yes—
80C3×C4×D4C2314yes—
81C2×C3×C4⋊C4C2×C411yescharacteristic
82C2×C3×C22⋊C4C2×C412yes—
83C2×C3×C42C2311yescharacteristic
84C2×#217C2×C412yes—
85C2×#217C2×C412yes—
86C2×#217C2×C412yes—
87#938C2×C418yes—
88#936C2×C418yes—
89C4×#67C2×C418yes—
90C4×#67C2318yes—
91C2×(C2×D3)⋊C4C2×C412yes—
92#926C2×C418yes—
93#926C2318yes—
94C2×C4⋊Dic3C2×C411yescharacteristic
95C2×C4⋊Dic3C2×C412yes—
96C2×C4⋊Dic3C2×C412yes—
97C2×Dic3⋊C4C2×C412yes—
98C2×Dic3⋊C4C2×C412yes—
99C2×C4×Dic3C2×C411yescharacteristic
100C2×C4×Dic3C2×C411yescharacteristic
101C2×C4×Dic3C2312yes—
102C2×C4×Dic3C2311yescharacteristic
103C2×#208—44no—
104C2×#208C2318yes—
105C22×C3⋊C8C2311yescharacteristic
106C22×C3⋊C8C2311yescharacteristic
107C22×C3⋊C8C2312yes—
108C2×ΓL(1,𝔽25)—44no—
109C2×ΓL(1,𝔽25)C2318yes—
110#897C2×C418yes—
111#890C2×C418yes—
112#883C2×C4116yes—
113#879C2×C4116yes—
114C42×D3C2×C414yes—
115C42×D3C2314yes—
116C4×Dic6C2×C414yes—
117C4×Dic6C2314yes—
118C3×C4⋊C8C2314yes—
119C3×C22⋊C8C2318yes—
120C3×C4×C8C2314yes—
121#830C2318yes—
122(C2×D3)⋊C8C2314yes—
123(C2×D3)⋊C8C2314yes—
124#815C2318yes—
125Dic3⋊C8C2318yes—
126Dic3⋊C8C2314yes—
127Dic3⋊C8C2314yes—
128Dic3⋊C8C2318yes—
129Dic3⋊C8C2318yes—
130C8×Dic3C2318yes—
131C4⋊(C3⋊C8)C2314yes—
132C4×C3⋊C8C2314yes—
133C22×M16—32no—
134C22×M16—32no—
135C23×C8C2×D312yes—
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×D4C2×D311yescharacteristic
147C2×#117—32no—
148C22×C42—31no—
149#446—38no—
150C8×Q8—38no—
151#436—38no—
152#436—316no—
153#435—316no—
154C8×D4C2×D318yescomplemented
155#433—316no—
156#431—38no—
157#424—316no—
158#423—38no—
159C2×C4⋊C8—32no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8—32no—
162C2×C4⋊C8—32no—
163C2×C4⋊C8—31no—
164C2×C4⋊C8—31no—
165C2×C4⋊C8C2×D311yescharacteristic
166C2×C22⋊C8—32no—
167C2×C22⋊C8—31no—
168C2×C22⋊C8—31no—
169C2×C22⋊C8C2×D312yes—
170C2×#97—32no—
171C2×C4×C8—32no—
172C2×C4×C8—31no—
173C2×C4×C8C2×D311yescharacteristic
174C24×C3C22×C412yes—
175C23×D3C22×C411yescharacteristic
176C2×C3×D4—24no—
177C2×C3×D4C22×C414yes—
178C22×C3×C4—24no—
179C22×C3×C4—24no—
180C22×C3×C4C22×C411yescharacteristic
181C22×C3×C4C22×C412yes—
182C22×C3×C4C22×C414yes—
183C22×C3×C4C22×C411yescharacteristic
184C22×C3×C4C2414yes—
185C22×C3×C4C2411yescharacteristic
186C2×#67—44no—
187C2×#67C22×C418yes—
188C22×Dic3C22×C411yescharacteristic
189C22×Dic3C22×C411yescharacteristic
190C22×Dic3C22×C411yescharacteristic
191C22×Dic3C22×C412yes—
192C22×Dic3C22×C412yes—
193C2×C4×D3—24no—
194C2×C4×D3—22no—
195C2×C4×D3—22no—
196C2×C4×D3—42no—
197C2×C4×D3C22×C412yes—
198C2×C4×D3C22×C414yes—
199C2×C4×D3C22×C414yes—
200C2×C4×D3C2412yes—
201C2×Dic6—24no—
202C2×Dic6C22×C414yes—
203C2×C3×C8—24no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—48no—
207C2×C3×C8C2412yes—
208C2×C3×C8C2412yes—
209C2×C3×C8C2414yes—
210C3×C4⋊C4C22×C414yes—
211C3×C22⋊C4C22×C418yes—
212C3×C42C22×C412yes—
213C3×C42C2412yes—
214#217C22×C418yes—
215#217C22×C418yes—
216#217C22×C418yes—
217(C2×D3)⋊C4C22×C418yes—
218C4⋊Dic3C22×C414yes—
219C4⋊Dic3C22×C418yes—
220C4⋊Dic3C22×C418yes—
221Dic3⋊C4C22×C418yes—
222Dic3⋊C4C22×C418yes—
223C4×Dic3C22×C414yes—
224C4×Dic3C22×C412yes—
225C4×Dic3C22×C414yes—
226C4×Dic3C22×C414yes—
227C4×Dic3C2414yes—
228C4×Dic3C2412yes—
229#208—48no—
230C2×C3⋊C8—24no—
231C2×C3⋊C8—22no—
232C2×C3⋊C8—22no—
233C2×C3⋊C8C2412yes—
234C2×C3⋊C8C2412yes—
235C2×C3⋊C8C2414yes—
236ΓL(1,𝔽25)—48no—
237C22×Q8—31no—
238C22×D4—32no—
239C22×D4C4×D311yescharacteristic
240C23×C4—31no—
241C23×C4—31no—
242C23×C4—31no—
243C23×C4C22×D312yes—
244C2×M16—38no—
245C2×M16—38no—
246C2×M16—124no—
247C2×M16—124no—
248C22×C8—31no—
249C22×C8—32no—
250C22×C8—31no—
251C22×C8—44no—
252C22×C8C22×D311yescharacteristic
253C22×C8C22×D312yes—
254C22×C8C22×D311yescharacteristic
255C22×C8C22×D318yes—
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×D4C4×D314yes—
267C4×D4C22×D314yes—
268#117—38no—
269#117—38no—
270C2×C4⋊C4—32no—
271C2×C4⋊C4—32no—
272C2×C4⋊C4—32no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—32no—
275C2×C4⋊C4C4×D311yescharacteristic
276C2×C22⋊C4—32no—
277C2×C22⋊C4—32no—
278C2×C22⋊C4—32no—
279C2×C22⋊C4—32no—
280C2×C22⋊C4C4×D312yes—
281C2×C42—31no—
282C2×C42—31no—
283C2×C42—31no—
284C2×C42—32no—
285C2×C42—34no—
286C2×C42—34no—
287C2×C42C22×D311yescharacteristic
288C4⋊C8—38no—
289C4⋊C8—34no—
290C4⋊C8—38no—
291C4⋊C8—38no—
292C4⋊C8—34no—
293C4⋊C8—34no—
294C4⋊C8C22×D314yes—
295C22⋊C8—38no—
296C22⋊C8—34no—
297C22⋊C8—34no—
298C22⋊C8C22×D318yes—
299#97—38no—
300C4×C8—38no—
301C4×C8—34no—
302C4×C8C22×D314yes—
303C23×C3—24no—
304C23×C3—24no—
305C23×C3—24no—
306C23×C3C23×C414yes—
307C23×C3C23×C411yescharacteristic
308C22×D3—22no—
309C22×D3—22no—
310C22×D3—22no—
311C22×D3C23×C412yes—
312C3×D4—28no—
313C2×C3×C4—24no—
314C2×C3×C4—22no—
315C2×C3×C4—28no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4#13114yes—
319C2×C3×C4C23×C412yes—
320C2×C3×C4C23×C414yes—
321C2×C3×C4C23×C411yescharacteristic
322C2×C3×C4C23×C412yes—
323C2×C3×C4H(5,𝔽2)12yes—
324C2×C3×C42−1+412yes—
325C2×C3×C4C2511yescharacteristic
326#67—48no—
327C2×Dic3—22no—
328C2×Dic3—24no—
329C2×Dic3—22no—
330C2×Dic3—22no—
331C2×Dic3—24no—
332C2×Dic3C23×C412yes—
333C2×Dic3C23×C412yes—
334C2×Dic3C23×C412yes—
335C2×Dic3C23×C414yes—
336C2×Dic3C23×C414yes—
337C4×D3—24no—
338C4×D3—28no—
339C4×D3—22no—
340C4×D3—22no—
341C4×D3—44no—
342Dic6—28no—
343C3×C8—24no—
344C3×C8—22no—
345C3×C8—22no—
346C3⋊C8—24no—
347C3⋊C8—22no—
348C3⋊C8—22no—
349C24—31no—
350C24C2×C4×D312yes—
351C2×Q8—34no—
352C2×Q8—64no—
353C2×D4—24no—
354C2×D4—38no—
355C2×D4—124no—
356C2×D4C2×C4×D314yes—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—31no—
360C22×C4—31no—
361C22×C4—31no—
362C22×C4—32no—
363C22×C4—32no—
364C22×C4—32no—
365C22×C4—34no—
366C22×C4—32no—
367C22×C4—34no—
368C22×C4—64no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—122no—
372C22×C4C2×C4×D311yescharacteristic
373C22×C4C2×C4×D312yes—
374C22×C4C2×C4×D314yes—
375C22×C4C2×C4×D311yescharacteristic
376C22×C4C23×D314yes—
377C22×C4C23×D311yescharacteristic
378M16—128no—
379M16—128no—
380C2×C8—22no—
381C2×C8—24no—
382C2×C8—22no—
383C2×C8—32no—
384C2×C8—34no—
385C2×C8—32no—
386C2×C8—48no—
387C2×C8—64no—
388C2×C8—62no—
389C2×C8—62no—
390C2×C8C23×D312yes—
391C2×C8C23×D312yes—
392C2×C8C23×D314yes—
393C4⋊C4—38no—
394C4⋊C4—38no—
395C4⋊C4—38no—
396C4⋊C4—34no—
397C4⋊C4—38no—
398C4⋊C4C2×C4×D314yes—
399C22⋊C4—38no—
400C22⋊C4—38no—
401C22⋊C4—38no—
402C22⋊C4—38no—
403C22⋊C4C2×C4×D318yes—
404C42—32no—
405C42—34no—
406C42—34no—
407C42—34no—
408C42—34no—
409C42—32no—
410C42C2×C4×D312yes—
411C42C23×D312yes—
412C22×C3—24no—
413C22×C3—24no—
414C22×C3—24no—
415C22×C3—48no—
416C22×C3#52012yes—
417C22×C3C2×#13112yes—
418C22×C3#56812yes—
419C22×C3C24×C411yescharacteristic
420C2×D3—22no—
421C2×D3—22no—
422C2×D3—22no—
423C2×D3—44no—
424C3×C4—24no—
425C3×C4—22no—
426C3×C4#52012yes—
427C3×C4C2×#13114yes—
428C3×C4C2×H(5,𝔽2)11yescharacteristic
429C3×C4C2×2−1+411yescharacteristic
430Dic3—22no—
431Dic3—24no—
432Dic3—22no—
433Dic3—22no—
434Dic3—24no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—32no—
439C23—62no—
440C23—62no—
441C23—62no—
442C23C22×C4×D314yes—
443C23C22×C4×D311yescharacteristic
444Q8—68no—
445D4—28no—
446D4—128no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—28no—
452C2×C4—32no—
453C2×C4—32no—
454C2×C4—32no—
455C2×C4—34no—
456C2×C4—34no—
457C2×C4—62no—
458C2×C4—64no—
459C2×C4—62no—
460C2×C4—64no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4—62no—
464C2×C4—68no—
465C2×C4—64no—
466C2×C4—124no—
467C2×C4#90114yes—
468C2×C4C22×C4×D312yes—
469C2×C4C22×C4×D314yes—
470C2×C4C22×C4×D312yes—
471C2×C4C22×C4×D311yescharacteristic
472C2×C4#100412yes—
473C2×C4#101012yes—
474C2×C4C24×D311yescharacteristic
475C8—22no—
476C8—24no—
477C8—22no—
478C8—64no—
479C8—62no—
480C8—62no—
481C2×C3—44no—
482C2×C3#299314yescomplemented
483C2×C3C2×#52011yescharacteristic
484C2×C3C22×#13111yescharacteristic
485C2×C3C2×#56811yescharacteristic
486D3—42no—
487C22—24no—
488C22—24no—
489C22—24no—
490C22—48no—
491C22—62no—
492C22—62no—
493C22—62no—
494C22—124no—
495C22#554612yes—
496C22C2×#90112yes—
497C22#575012yes—
498C22C23×C4×D311yescharacteristic
499C4—22no—
500C4—24no—
501C4—62no—
502C4—64no—
503C4—62no—
504C4—64no—
505C4—62no—
506C4#554612yes—
507C4C2×#90114yes—
508C4C2×#100411yescharacteristic
509C4C2×#101011yescharacteristic
510C3C2×#299311yescharacteristic, complemented, Sylow
511C2—44no—
512C2—122no—
513C2#7940814yesdirect product factor
514C2C2×#554611yescharacteristic
515C2C22×#90111yescharacteristic
516C2C2×#575011yescharacteristic
subgroupG/H
centerC22×C4C23×D3
commutatorC22×C3C24×C4
FrattiniC2×C4C24×D3
FittingC2×C3×C8×D4C2
socleC23×C3C23×C4
prime radicalC2×#938C4
cosocleC2×C3×C4C25
hypercenterC2×C8×D4D3
hypocenterC3C2×#2993
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C8×D4C2semidirect
3. C2×#5111C2semidirect
4. C2×#5102C2semidirect
5. C2×#5012C2semidirect
6. #79408C2direct (N × H)
7. C2×#4837C2semidirect
8. C2×#4831C2semidirect
9. C2×#4729C2semidirect
10. C2×#4720C2semidirect
11. C2×#4685C2semidirect
12. C2×C8×Dic6C2semidirect
30. C3×C8×D4C22semidirect
35. #5111C22semidirect
38. #5102C22semidirect
46. #5012C22semidirect
49. #4837C22semidirect
50. #4831C22semidirect
51. #4729C22semidirect
52. #4720C22semidirect
53. #4685C22semidirect
54. C8×Dic6C22semidirect
62. C2×C8×D4D3semidirect
154. C8×D4C2×D3semidirect
482. C2×C3#2993semidirect
510. C3C2×#2993semidirect
513. C2#79408direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#55572
C2×C3×C8×D42
C2×#51112
C2×#51022
C2×#50122
#794082
C2×#48372
C2×#48312
C2×#47292
C2×#47202
C2×#46852
C2×C8×Dic62
C2×#29933