Finite Group Databasebeta

← C2×#78550 #11658596

Subgroups — C2×#78550 #11658596

Order 768.

3278 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#907C211yesmaximal, characteristic
2C22×#901C211yesmaximal, characteristic
3C2×#5522C211yesmaximal, characteristic
4C2×C3×#405C211yesmaximal, characteristic, complemented
5C2×Dic3×M16C211yesmaximal, characteristic, complemented
6C22×#815C211yesmaximal, characteristic, complemented
7C2×#4997C211yesmaximal, characteristic, complemented
8#78550C2116yesmaximal, direct product factor
9C2×#4708C212yesmaximal, complemented
10C2×D3×#97C212yesmaximal, complemented
11C2×#4689C212yesmaximal, complemented
12C2×C4×ΓL(1,𝔽25)C212yesmaximal, complemented
13C2×#2875—31nomaximal, Sylow
14C22×#235C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C23×C3×C8C2211yescharacteristic
17C2×C3×#117C2211yescharacteristic
18C22×C4×Dic3C2211yescharacteristic
19C22×#208C2211yescharacteristic
20C23×C3⋊C8C2211yescharacteristic
21C2×#907C4116yescomplemented
22C2×#907C2218yes—
23C2×#901C4116yescomplemented
24C2×#901C2218yes—
25C22×ΓL(1,𝔽25)C2212yes—
26C22×ΓL(1,𝔽25)C2212yes—
27C22×D3×C8C2212yes—
28C22×D3×C8C2212yes—
29#5522C418yes—
30#5522C2218yes—
31C2×#893C412yes—
32C2×#887C412yes—
33C2×#67⋊C4C414yes—
34C2×#872C2212yes—
35C2×C42×D3C2212yes—
36C3×#405C2218yescomplemented
37C2×C3×#97C2212yes—
38C2×C3×C4×C8C2212yes—
39Dic3×M16C2218yescomplemented
40C2×#815C2211yescharacteristic
41C2×#815C2211yescharacteristic
42C2×#815C2211yescharacteristic
43C2×#815C2211yescharacteristic
44C2×#815C2212yes—
45C2×#815C2218yescomplemented
46C2×C8×Dic3C2212yes—
47#4997C2218yescomplemented
48C2×#803C2212yes—
49C2×C4×C3⋊C8C2212yes—
50#4708C22116yescomplemented
51D3×#97C22116yescomplemented
52#4689C22116yescomplemented
53C4×ΓL(1,𝔽25)C22116yescomplemented
54C22×#131—31no—
55C22×#131—31no—
56C2×C4×#41—31no—
57#2875—316no—
58C2×#405—32no—
59C2×#405—32no—
60C2×#405—31no—
61C2×#405D311yescharacteristic, complemented
62C2×C4×M16—32no—
63C2×C4×M16—31no—
64C22×#97—32no—
65C22×#97—31no—
66C23×C3×C4C2311yescharacteristic
67C22×#67C2×C412yes—
68C23×Dic3C2×C412yes—
69C2×#235C2×C418yes—
70C2×#235C2×C418yes—
71C2×#235C2×C414yes—
72C2×#235C2314yes—
73C22×D12C2×C411yescharacteristic
74C22×C4×D3C2×C414yes—
75C22×C4×D3C2312yes—
76C22×Dic6C2×C411yescharacteristic
77C2×C3×M16C2×C418yescomplemented
78C2×C3×M16C2314yes—
79C22×C3×C8C2×C418yescomplemented
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2312yes—
83C22×C3×C8C2314yes—
84C3×#117C2×C414yes—
85C3×#117C2314yes—
86C2×C3×C4⋊C4C2×C412yes—
87C2×C3×C22⋊C4C2×C412yes—
88C2×C3×C42C2312yes—
89C2×(C2×D3)⋊C4C2×C414yes—
90C2×Dic3⋊C4C2×C414yes—
91C2×C4×Dic3C2×C414yes—
92C2×C4×Dic3C2×C412yes—
93C2×C4×Dic3C2×C414yes—
94C2×C4×Dic3C2311yescharacteristic
95C2×C4×Dic3C2311yescharacteristic
96C2×C4×Dic3C2314yes—
97C2×#208C2×C418yescomplemented
98C2×#208C2314yes—
99C22×C3⋊C8C2×C418yescomplemented
100C22×C3⋊C8C2311yescharacteristic
101C22×C3⋊C8C2312yes—
102C22×C3⋊C8C2311yescharacteristic
103C22×C3⋊C8C2314yes—
104#907C2×C4164yescomplemented
105#901C2×C4164yescomplemented
106C2×ΓL(1,𝔽25)C2×C4116yescomplemented
107C2×ΓL(1,𝔽25)C2×C4116yescomplemented
108C2×ΓL(1,𝔽25)C2318yes—
109C2×ΓL(1,𝔽25)C2318yes—
110C2×D3×C8C2×C4116yescomplemented
111C2×D3×C8C2×C4116yescomplemented
112C2×D3×C8C2318yes—
113C2×D3×C8C2318yes—
114#893C2×C4116yes—
115#887C2×C4116yes—
116#67⋊C4C2×C4132yes—
117#872C2×C418yes—
118#872C2318yes—
119C42×D3C2×C418yes—
120C42×D3C2318yes—
121C3×#97C2318yes—
122C3×C4×C8C2318yes—
123#815C2314yes—
124#815C2318yes—
125#815C2314yes—
126#815C2314yes—
127#815C2314yes—
128C8×Dic3C2318yes—
129#803C2318yes—
130C4×C3⋊C8C2318yes—
131C22×#41—31no—
132C2×#131—316no—
133C2×#131—38no—
134C2×#131—316no—
135C2×#131—38no—
136C22×M16—31no—
137C22×M16—32no—
138C22×M16—32no—
139C22×M16C2×D311yescharacteristic
140C23×C8—31no—
141C23×C8—32no—
142C23×C8—32no—
143C23×C8C2×D311yescharacteristic
144C4×#41—38no—
145C4×#41—38no—
146C2×C4×Q8—32no—
147C2×C4×D4—32no—
148C2×C4×D4—34no—
149C2×#117—32no—
150C2×#117C2×D311yescharacteristic
151C22×C42—31no—
152C22×C42—32no—
153#405—38no—
154#405—316no—
155#405—316no—
156#405C2×D318yescomplemented
157C4×M16—38no—
158C4×M16—316no—
159C2×#97—31no—
160C2×#97—32no—
161C2×#97—31no—
162C2×#97—31no—
163C2×#97—31no—
164C2×#97—38no—
165C2×#97—32no—
166C2×#97—316no—
167C2×#97C2×D312yes—
168C2×C4×C8—32no—
169C2×C4×C8—32no—
170C2×C4×C8C2×D312yes—
171C24×C3C22×C411yescharacteristic
172C23×D3C22×C412yes—
173C22×C3×C4C22×C414yes—
174C22×C3×C4C22×C411yescharacteristic
175C22×C3×C4C22×C412yes—
176C22×C3×C4C22×C414yes—
177C22×C3×C4C22×C414yes—
178C22×C3×C4C2412yes—
179C22×C3×C4C2411yescharacteristic
180C2×#67C42116yes—
181C2×#67C22×C418yes—
182C22×Dic3—44no—
183C22×Dic3C22×C412yes—
184C22×Dic3C22×C412yes—
185C22×Dic3C22×C418yes—
186C22×Dic3C22×C412yes—
187#235C42132yes—
188#235C22×C4116yes—
189#235C22×C4116yes—
190C2×D12C4218yes—
191C2×D12C22×C414yes—
192C2×C4×D3—48no—
193C2×C4×D3C22×C4116yes—
194C2×C4×D3C22×C414yes—
195C2×C4×D3C22×C418yes—
196C2×C4×D3C22×C418yes—
197C2×C4×D3C2414yes—
198C2×Dic6C4218yes—
199C2×Dic6C22×C414yes—
200C3×M16C22×C4116yescomplemented
201C2×C3×C8C22×C414yes—
202C2×C3×C8C22×C418yes—
203C2×C3×C8C22×C414yes—
204C2×C3×C8C22×C4116yescomplemented
205C2×C3×C8C2414yes—
206C2×C3×C8C2412yes—
207C2×C3×C8C2412yes—
208C3×C4⋊C4C22×C418yes—
209C3×C22⋊C4C22×C418yes—
210C3×C42C22×C414yes—
211C3×C42C2414yes—
212(C2×D3)⋊C4C22×C4116yes—
213Dic3⋊C4C22×C4116yes—
214C4×Dic3C22×C412yes—
215C4×Dic3C22×C4116yes—
216C4×Dic3C22×C412yes—
217C4×Dic3C22×C418yes—
218C4×Dic3C2412yes—
219C4×Dic3C2412yes—
220#208C22×C4116yescomplemented
221C2×C3⋊C8C22×C414yes—
222C2×C3⋊C8C22×C418yes—
223C2×C3⋊C8C22×C414yes—
224C2×C3⋊C8C22×C4116yescomplemented
225C2×C3⋊C8C2412yes—
226C2×C3⋊C8C2414yes—
227C2×C3⋊C8C2412yes—
228ΓL(1,𝔽25)C22×C4132yescomplemented
229ΓL(1,𝔽25)C22×C4132yescomplemented
230D3×C8C22×C4132yescomplemented
231D3×C8C22×C4132yescomplemented
232C2×#41—34no—
233C2×#41—38no—
234C2×#41—38no—
235C2×#41—34no—
236C22×Q8—31no—
237C22×D4—32no—
238C22×D4—31no—
239C23×C4—32no—
240C23×C4—32no—
241C23×C4—34no—
242C23×C4C22×D311yescharacteristic
243#131—364no—
244#131—364no—
245C2×M16—38no—
246C2×M16—34no—
247C2×M16—316no—
248C2×M16—316no—
249C2×M16—38no—
250C2×M16—38no—
251C2×M16C4×D318yescomplemented
252C2×M16C22×D314yes—
253C22×C8—38no—
254C22×C8—34no—
255C22×C8—31no—
256C22×C8—31no—
257C22×C8—32no—
258C22×C8—316no—
259C22×C8—316no—
260C22×C8—38no—
261C22×C8—38no—
262C22×C8C4×D318yescomplemented
263C22×C8C22×D311yescharacteristic
264C22×C8C22×D312yes—
265C22×C8C22×D311yescharacteristic
266C22×C8C22×D314yes—
267C4×Q8—316no—
268C4×D4—316no—
269C4×D4—332no—
270#117—38no—
271#117—38no—
272#117C4×D314yes—
273#117C22×D314yes—
274C2×C4⋊C4—34no—
275C2×C4⋊C4C4×D312yes—
276C2×C22⋊C4—34no—
277C2×C22⋊C4C4×D312yes—
278C2×C42—34no—
279C2×C42—32no—
280C2×C42—34no—
281C2×C42—34no—
282C2×C42—31no—
283C2×C42—31no—
284C2×C42—38no—
285C2×C42—38no—
286C2×C42C22×D312yes—
287#97—34no—
288#97—38no—
289#97—34no—
290#97—34no—
291#97—34no—
292#97—38no—
293#97C22×D318yes—
294C4×C8—38no—
295C4×C8—38no—
296C4×C8C22×D318yes—
297C23×C3—24no—
298C23×C3C2×C4214yes—
299C23×C3C23×C412yes—
300C23×C3C23×C411yescharacteristic
301C22×D3—28no—
302C22×D3C2×C4218yes—
303C22×D3C23×C414yes—
304C2×C3×C4—28no—
305C2×C3×C4C2×C4214yes—
306C2×C3×C4C2×C4218yes—
307C2×C3×C4C23×C418yes—
308C2×C3×C4C23×C412yes—
309C2×C3×C4C23×C412yes—
310C2×C3×C4C23×C414yes—
311C2×C3×C4C23×C414yes—
312C2×C3×C4C23×C411yescharacteristic
313C2×C3×C4C23×C412yes—
314C2×C3×C4C2511yescharacteristic
315#67C2×C42132yes—
316C2×Dic3—24no—
317C2×Dic3—24no—
318C2×Dic3—48no—
319C2×Dic3C2×C4218yes—
320C2×Dic3C23×C414yes—
321C2×Dic3C23×C414yes—
322C2×Dic3C23×C414yes—
323D12C2×C42116yes—
324C4×D3—416no—
325C4×D3C2×C42116yes—
326C4×D3C23×C418yes—
327C4×D3C23×C418yes—
328Dic6C2×C42116yes—
329C3×C8C23×C414yes—
330C3×C8C23×C418yes—
331C3×C8C23×C414yes—
332C3⋊C8C23×C414yes—
333C3⋊C8C23×C418yes—
334C3⋊C8C23×C414yes—
335C24—32no—
336C24C2×C4×D311yescharacteristic
337#41—332no—
338#41—316no—
339#41—316no—
340C2×Q8—38no—
341C2×Q8—34no—
342C2×D4—316no—
343C2×D4—38no—
344C2×D4—38no—
345C2×D4—34no—
346C22×C4—32no—
347C22×C4—34no—
348C22×C4—38no—
349C22×C4—38no—
350C22×C4—34no—
351C22×C4—32no—
352C22×C4—32no—
353C22×C4—38no—
354C22×C4—316no—
355C22×C4—128no—
356C22×C4—124no—
357C22×C4C2×C4×D314yes—
358C22×C4C2×C4×D311yescharacteristic
359C22×C4C2×C4×D312yes—
360C22×C4C2×C4×D314yes—
361C22×C4C2×C4×D314yes—
362C22×C4C23×D312yes—
363C22×C4C23×D311yescharacteristic
364M16—316no—
365M16—332no—
366M16—332no—
367M16C2×C4×D3116yescomplemented
368C2×C8—34no—
369C2×C8—34no—
370C2×C8—38no—
371C2×C8—316no—
372C2×C8—34no—
373C2×C8—32no—
374C2×C8—32no—
375C2×C8—332no—
376C2×C8—332no—
377C2×C8C2×C4×D314yes—
378C2×C8C2×C4×D318yes—
379C2×C8C2×C4×D314yes—
380C2×C8C2×C4×D3116yescomplemented
381C2×C8C23×D314yes—
382C2×C8C23×D312yes—
383C2×C8C23×D312yes—
384C4⋊C4—316no—
385C4⋊C4C2×C4×D318yes—
386C22⋊C4—316no—
387C22⋊C4C2×C4×D318yes—
388C42—32no—
389C42—32no—
390C42—316no—
391C42—38no—
392C42—32no—
393C42—32no—
394C42C2×C4×D314yes—
395C42C23×D314yes—
396C22×C3—28no—
397C22×C3—24no—
398C22×C3C22×C4214yes—
399C22×C3C22×C4212yes—
400C22×C3#56814yes—
401C22×C3C24×C411yescharacteristic
402C2×D3—216no—
403C2×D3—28no—
404C2×D3C22×C4218yes—
405C3×C4—28no—
406C3×C4C22×C4214yes—
407C3×C4C22×C4212yes—
408C3×C4C24×C411yescharacteristic
409C3×C4C24×C411yescharacteristic
410Dic3—24no—
411Dic3—24no—
412Dic3C22×C4218yes—
413C23—24no—
414C23—34no—
415C23—38no—
416C23—68no—
417C23C42×D314yes—
418C23C22×C4×D312yes—
419C23C22×C4×D311yescharacteristic
420Q8—316no—
421D4—332no—
422D4—316no—
423C2×C4—28no—
424C2×C4—34no—
425C2×C4—38no—
426C2×C4—38no—
427C2×C4—38no—
428C2×C4—316no—
429C2×C4—34no—
430C2×C4—34no—
431C2×C4—64no—
432C2×C4—64no—
433C2×C4—1216no—
434C2×C4—128no—
435C2×C4C42×D314yes—
436C2×C4C42×D318yes—
437C2×C4C22×C4×D318yes—
438C2×C4C22×C4×D312yes—
439C2×C4C22×C4×D312yes—
440C2×C4C22×C4×D314yes—
441C2×C4C22×C4×D314yes—
442C2×C4C22×C4×D311yescharacteristic
443C2×C4C22×C4×D312yes—
444C2×C4C24×D311yescharacteristic
445C8—34no—
446C8—34no—
447C8—38no—
448C8C22×C4×D314yes—
449C8C22×C4×D318yes—
450C8C22×C4×D314yes—
451C2×C3—24no—
452C2×C3#287514yescomplemented
453C2×C3C23×C4211yescharacteristic
454C2×C3C2×#56812yes—
455D3—28no—
456C22—24no—
457C22—28no—
458C22—38no—
459C22—68no—
460C22—616no—
461C22C2×C42×D314yes—
462C22C2×C42×D312yes—
463C22#575014yes—
464C22C23×C4×D311yescharacteristic
465C4—28no—
466C4—38no—
467C4—64no—
468C4—64no—
469C4C2×C42×D312yes—
470C4C2×C42×D314yes—
471C4C23×C4×D311yescharacteristic
472C4C23×C4×D311yescharacteristic
473C3C2×#287511yescharacteristic, complemented, Sylow
474C2—24no—
475C2—68no—
476C2#7855014yesdirect product factor
477C2C22×C42×D311yescharacteristic
478C2C2×#575012yes—
subgroupG/H
centerC22×C4C23×D3
commutatorC2×C3C23×C42
FrattiniC2×C4C24×D3
FittingC2×C3×#405C2
socleC23×C3C23×C4
prime radicalC22×#235C22
cosocleC2×C3×C4C25
hypercenterC2×#405D3
hypocenterC3C2×#2875
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
4. C2×C3×#405C2semidirect
5. C2×Dic3×M16C2semidirect
6. C22×#815C2semidirect
7. C2×#4997C2semidirect
8. #78550C2direct (N × H)
9. C2×#4708C2semidirect
10. C2×D3×#97C2semidirect
11. C2×#4689C2semidirect
12. C2×C4×ΓL(1,𝔽25)C2semidirect
21. C2×#907C4semidirect
23. C2×#901C4semidirect
36. C3×#405C22semidirect
39. Dic3×M16C22semidirect
45. C2×#815C22semidirect
47. #4997C22semidirect
50. #4708C22semidirect
51. D3×#97C22semidirect
52. #4689C22semidirect
53. C4×ΓL(1,𝔽25)C22semidirect
61. C2×#405D3semidirect
77. C2×C3×M16C2×C4semidirect
79. C22×C3×C8C2×C4semidirect
97. C2×#208C2×C4semidirect
99. C22×C3⋊C8C2×C4semidirect
104. #907C2×C4semidirect
105. #901C2×C4semidirect
106. C2×ΓL(1,𝔽25)C2×C4semidirect
107. C2×ΓL(1,𝔽25)C2×C4semidirect
110. C2×D3×C8C2×C4semidirect
111. C2×D3×C8C2×C4semidirect
156. #405C2×D3semidirect
200. C3×M16C22×C4semidirect
204. C2×C3×C8C22×C4semidirect
220. #208C22×C4semidirect
224. C2×C3⋊C8C22×C4semidirect
228. ΓL(1,𝔽25)C22×C4semidirect
229. ΓL(1,𝔽25)C22×C4semidirect
230. D3×C8C22×C4semidirect
231. D3×C8C22×C4semidirect
251. C2×M16C4×D3semidirect
262. C22×C8C4×D3semidirect
367. M16C2×C4×D3semidirect
380. C2×C8C2×C4×D3semidirect
452. C2×C3#2875semidirect
473. C3C2×#2875semidirect
476. C2#78550direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9072
C22×#9012
C2×#55222
C2×C3×#4052
C2×Dic3×M162
C22×#8152
C2×#49972
#785502
C2×#47082
C2×D3×#972
C2×#46892
C2×C4×ΓL(1,𝔽25)2
C2×#28753