Finite Group Databasebeta

← C2×#72342683 #72342745

Subgroups — C2×#72342683 #72342745

Order 6144.

At least 14535 conjugacy classes of subgroups; only the 964 normal subgroups are shown.

The subgroup lattice has 292 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#72342681C211yesmaximal, characteristic, complemented
2C2×#72342679C211yesmaximal, characteristic, complemented
3C2×#72342677C211yesmaximal, characteristic, complemented
4C2×#72342675C211yesmaximal, characteristic, complemented
5C2×#72342672C212yesmaximal, complemented
6C2×#72342671C212yesmaximal, complemented
7C2×#11715670C2212yescomplemented
8C2×#21440⋊C3C2212yescomplemented
9C2×#72342670C212yesmaximal, complemented
10C2×#11711115C2212yescomplemented
11C2×C8×#1020⋊C2C212yesmaximal, complemented
12#72342683C2116yesmaximal, direct product factor
13#72342681C2218yescomplemented
14#72342679C2218yescomplemented
15#72342677C2218yescomplemented
16#72342675C2218yescomplemented
17#72342672C22116yescomplemented
18#72342671C22116yescomplemented
19#72342670C22116yescomplemented
20C2×#11711113C2212yescomplemented
21C2×C8×#120⋊C3C2212yescomplemented
22C2×#72252249C211yesmaximal, characteristic, complemented
23C2×#20579⋊C3C2211yescharacteristic, complemented
24C2×#1020⋊C8C2212yes—
25C2×#1020⋊C8C2212yescomplemented
26C2×#11711114C2212yescomplemented
27C2×#11711114C2212yescomplemented
28C22×C8×#1020C211yesmaximal, characteristic, complemented
29C22×#85561C2211yescharacteristic, complemented
30C22×C8×C22⋊A4C2211yescharacteristic, complemented
31C2×#6372601D311yescharacteristic, complemented
32#72252249C2218yescomplemented
33C2×C8×#1020C2211yescharacteristic
34C2×C8×#1020C2211yescharacteristic
35C2×C8×#1020C2212yescomplemented
36C2×C8×#1020C2211yescharacteristic, complemented
37C2×C8×#1020C2211yescharacteristic, complemented
38C2×C8×#1020C2218yescomplemented
39C8×#1020⋊C2C22116yescomplemented
40C2×#72220637C211yesmaximal, characteristic
41C2×#11713216C412yes—
42C2×#11715809C2211yescharacteristic
43C2×#36614⋊C3C2211yescharacteristic
44C2×#11713244C411yescharacteristic
45C2×#1020⋊Q8C411yescharacteristic
46C2×#11713243C411yescharacteristic
47C2×#11713151C2212yes—
48C2×#11713271C411yescharacteristic
49C22×C4×#1020C2211yescharacteristic
50C2×C4×#1020⋊C2C2212yes—
51C2×#11715649C2211yescharacteristic, complemented
52#72220637C418yes—
53#72220637C2218yes—
54C2×#11713200C412yes—
55C23×#1020C2×C411yescharacteristic
56#11715809C2×C414yes—
57#11715809C2314yes—
58C2×#87502C2×C411yescharacteristic
59C2×C4⋊#1020C2×C411yescharacteristic
60C2×#87524C2×C412yes—
61C2×C4×#1020C2×C411yescharacteristic
62C2×C4×#1020C2×C411yescharacteristic
63C2×C4×#1020C2×C414yes—
64C2×C4×#1020C2314yes—
65C2×C4×#1020C2311yescharacteristic
66C2×C4×#1020C2311yescharacteristic
67C2×C4×#1020C2312yes—
68C22×#5936C2×C411yescharacteristic
69#11715670C2318yescomplemented
70C8×#1020C2314yes—
71C8×#1020C2314yes—
72C8×#1020C2318yescomplemented
73C8×#1020C2314yescomplemented
74C8×#1020C2314yescomplemented
75#11715649C2314yescomplemented
76C2×#85561C2311yescharacteristic
77C2×#85561C2312yes—
78C2×#85561C2311yescharacteristic
79C2×#85561C2314yescomplemented
80C22×#1020⋊C2C2×C412yes—
81#1020⋊Q8C2×C418yes—
82#11713271C2×C418yes—
83#11713244C2×C418yes—
84#11713243C2×C418yes—
85#11713216C2×C4116yes—
86#11713200C2×C4116yes—
87#11713151C2×C418yes—
88#11713151C2318yes—
89C2×#72989C2×C412yes—
90C2×#72991C2×C412yes—
91C2×#73081C2×C412yes—
92C2×#73081C2×C412yes—
93C4×#1020⋊C2C2×C418yes—
94C4×#1020⋊C2C2318yes—
95C2×#73080C2×C412yes—
96C2×#1020⋊C4C2×C412yes—
97C2×#1020⋊C4C2×C412yes—
98#11711115C2318yescomplemented
99#11711114C2318yescomplemented
100#11711114C2318yescomplemented
101#11711113C2318yescomplemented
102#1020⋊C8C2318yes—
103#1020⋊C8C2318yescomplemented
104C22×C4×C22⋊A4C2311yescharacteristic
105C2×C8×C22⋊A4C2311yescharacteristic
106C2×C8×C22⋊A4C2311yescharacteristic
107C2×C8×C22⋊A4C2314yescomplemented
108#36614⋊C3C2×C414yes—
109#36614⋊C3C2314yes—
110C2×#2848⋊C3C2×C411yescharacteristic
111C2×C4⋊(#120⋊C3)C2×C411yescharacteristic
112C2×AΣL(1,𝔽64)C2×C412yes—
113C2×C4×#120⋊C3C2312yes—
114#21440⋊C3C2318yescomplemented
115C8×#120⋊C3C2318yescomplemented
116#20579⋊C3C2314yescomplemented
117C2×#1779⋊C3C2312yes—
118C26×C8C2×D311yescharacteristic, complemented
119C2×#36614C2×D311yescharacteristic
120#6372601C2×D318yescomplemented
121C2×#21440C2×D312yescomplemented
122C2×C8×#120C2×D312yescomplemented
123C2×#20579C2×D311yescharacteristic, complemented
124C23×C22⋊A4C22×C411yescharacteristic
125C22×#1020C22×C411yescharacteristic
126C22×#1020C22×C414yes—
127C22×#1020C22×C412yes—
128C22×#1020C22×C411yescharacteristic
129C2×C4×C22⋊A4C22×C411yescharacteristic
130C2×C4×C22⋊A4C22×C412yes—
131C2×C4×C22⋊A4C2411yescharacteristic
132C2×C4×C22⋊A4C2412yes—
133#87524C22×C418yes—
134C2×#5936C22×C411yescharacteristic
135C2×#5936C22×C412yes—
136C2×#5936C22×C414yes—
137C2×#5936C22×C411yescharacteristic
138C4⋊#1020C22×C414yes—
139C4×#1020C22×C414yes—
140C4×#1020C22×C412yes—
141C4×#1020C22×C414yes—
142C4×#1020C22×C412yes—
143C4×#1020C22×C414yes—
144C4×#1020C2412yes—
145C4×#1020C2414yes—
146C4×#1020C2412yes—
147#87502C22×C414yes—
148C8×C22⋊A4C2412yes—
149C8×C22⋊A4C2412yes—
150C22×#120⋊C3C22×C412yes—
151#85561C2412yes—
152#85561C2412yes—
153#85561C2414yes—
154C2×#1020⋊C2C22×C418yes—
155#2848⋊C3C22×C414yes—
156C4⋊(#120⋊C3)C22×C414yes—
157AΣL(1,𝔽64)C22×C418yes—
158C4×#120⋊C3C22×C414yes—
159C4×#120⋊C3C2414yes—
160C2×#379⋊C3C22×C412yes—
161#73081C22×C418yes—
162#73081C22×C418yes—
163#73080C22×C418yes—
164#72991C22×C418yes—
165#1020⋊C4C22×C418yes—
166#1020⋊C4C22×C418yes—
167#72989C22×C418yes—
168#1779⋊C3C2414yes—
169C26×C4C22×D311yescharacteristic
170C23×#131S411yescharacteristic, complemented
171C25×C8C22×D311yescharacteristic
172C25×C8C22×D311yescharacteristic
173C25×C8C22×D314yescomplemented
174#36614C4×D314yes—
175#36614C22×D314yes—
176C2×#2848C4×D311yescharacteristic
177C2×#2847C4×D311yescharacteristic
178C2×#2846C4×D312yes—
179C2×C4×#120C22×D312yes—
180#21440C22×D318yescomplemented
181C8×#120C22×D318yescomplemented
182#20579C22×D314yescomplemented
183C2×#1779C22×D312yes—
184C22×C22⋊A4C23×C412yes—
185C22×C22⋊A4C23×C411yescharacteristic
186C2×#1020#13114yescomplemented
187C2×#1020C23×C414yes—
188C2×#1020C23×C412yes—
189C2×#1020C23×C412yes—
190C4×C22⋊A4C23×C412yes—
191C4×C22⋊A4C23×C411yescharacteristic
192C4×C22⋊A4C2511yescharacteristic
193#5936C23×C412yes—
194#5936C23×C414yes—
195#5936C23×C412yes—
196C2×#120⋊C3C23×C414yes—
197#379⋊C3C23×C414yes—
198C27C2×C4×D311yescharacteristic
199C23×#41C2×S411yescharacteristic
200C25×C4C2×C4×D311yescharacteristic
201C25×C4C2×C4×D312yes—
202C25×C4C23×D312yes—
203C25×C4C23×D311yescharacteristic
204C22×#131C2×S418yescomplemented
205C23×M16C2×S411yescharacteristic
206C23×M16C2×S412yescomplemented
207C24×C8C2×S411yescharacteristic
208C24×C8C2×S412yescomplemented
209C24×C8C23×D312yes—
210C24×C8C23×D312yes—
211C22×#120C2×C4×D312yes—
212#2848C2×C4×D314yes—
213#2847C2×C4×D314yes—
214#2846C2×C4×D318yes—
215C4×#120C2×C4×D314yes—
216C4×#120C23×D314yes—
217C2×#379C2×C4×D312yes—
218#1779C23×D314yes—
219C2×C22⋊A4C2×#13112yescomplemented
220C2×C22⋊A4C24×C411yescharacteristic
221#1020C2×#13114yescomplemented
222C26C22×C4×D312yes—
223C26C22×C4×D311yescharacteristic
224C22×#41C4×S414yes—
225C22×#41C22×S414yes—
226C23×Q8C4×S411yescharacteristic
227C23×D4C4×S412yes—
228C23×D4C4×S411yescharacteristic
229C24×C4C22×C4×D312yes—
230C24×C4C22×C4×D311yescharacteristic
231C24×C4C22×S411yescharacteristic
232C24×C4C22×S412yes—
233C24×C4C24×D311yescharacteristic
234C22×M16C22×S414yes—
235C22×M16C22×S418yescomplemented
236C23×C8C22×S411yescharacteristic
237C23×C8C22×S411yescharacteristic
238C23×C8C22×S414yes—
239C23×C8C22×S412yes—
240C23×C8C22×S418yescomplemented
241C2×#120C22×C4×D314yes—
242#379C22×C4×D314yes—
243C22⋊A4C22×#13111yescharacteristic, complemented
244C25D3×#13112yescomplemented
245C25C2×C4×S411yescharacteristic
246C25C2×C4×S412yes—
247C25C23×C4×D311yescharacteristic
248C22×Q8C2×C4×S414yes—
249C22×D4C2×C4×S418yes—
250C22×D4C2×C4×S414yes—
251C23×C4C2×C4×S411yescharacteristic
252C23×C4C2×C4×S412yes—
253C23×C4C2×C4×S412yes—
254C23×C4C2×C4×S414yes—
255C23×C4C23×S412yes—
256C23×C4C23×S411yescharacteristic
257C23×C4C23×S414yes—
258C22×C8#1020⋊C211yescharacteristic, complemented
259C22×C8C23×S412yes—
260C22×C8C23×S412yes—
261C22×C8C23×S414yes—
262C24C2×D3×#13111yescharacteristic, complemented
263C24C22×C4×S412yes—
264C24C22×C4×S414yes—
265C24C22×C4×S411yescharacteristic
266C22×C4C2×#1020⋊C211yescharacteristic
267C22×C4C22×C4×S411yescharacteristic
268C22×C4C22×C4×S412yes—
269C22×C4C22×C4×S414yes—
270C22×C4C24×S411yescharacteristic
271C2×C8C2×#1020⋊C211yescharacteristic
272C2×C8C2×#1020⋊C211yescharacteristic
273C2×C8C2×#1020⋊C214yescomplemented
274C23C4×#1020⋊C211yescharacteristic
275C23S4×#13112yes—
276C23C23×C4×S411yescharacteristic
277C2×C4C4×#1020⋊C211yescharacteristic
278C2×C4C4×#1020⋊C212yes—
279C2×C4C22×#1020⋊C211yescharacteristic
280C2×C4C22×#1020⋊C212yes—
281C8C22×#1020⋊C212yes—
282C8C22×#1020⋊C212yes—
283C22C2×C4×#1020⋊C212yes—
284C22C2×C4×#1020⋊C211yescharacteristic
285C22C2×S4×#13111yescharacteristic
286C4C23×#1020⋊C211yescharacteristic
287C4C2×C4×#1020⋊C211yescharacteristic
288C4C2×C4×#1020⋊C212yes—
289C2C22×C4×#1020⋊C211yescharacteristic
290C2#7234268312yesdirect product factor
subgroupG/H
centerC2×C8C2×#1020⋊C2
commutatorC2×C22⋊A4C24×C4
FrattiniC22×C4C24×S4
FittingC2×#6372601D3
socleC24C22×C4×S4
prime radicalC2×#72220637C2
cosocleC4×C22⋊A4C25
hypercenterC22×C8#1020⋊C2
hypocenterC22⋊A4C22×#131
LCSG ▷ C2×C22⋊A4 ▷ C22⋊A4
UCSC1 ◁ C2×C8 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
1. C2×#72342681C2semidirect
2. C2×#72342679C2semidirect
3. C2×#72342677C2semidirect
4. C2×#72342675C2semidirect
5. C2×#72342672C2semidirect
6. C2×#72342671C2semidirect
7. C2×#11715670C22semidirect
8. C2×#21440⋊C3C22semidirect
9. C2×#72342670C2semidirect
10. C2×#11711115C22semidirect
11. C2×C8×#1020⋊C2C2semidirect
12. #72342683C2direct (N × H)
13. #72342681C22semidirect
14. #72342679C22semidirect
15. #72342677C22semidirect
16. #72342675C22semidirect
17. #72342672C22semidirect
18. #72342671C22semidirect
19. #72342670C22semidirect
20. C2×#11711113C22semidirect
21. C2×C8×#120⋊C3C22semidirect
22. C2×#72252249C2semidirect
23. C2×#20579⋊C3C22semidirect
25. C2×#1020⋊C8C22semidirect
26. C2×#11711114C22semidirect
27. C2×#11711114C22semidirect
28. C22×C8×#1020C2semidirect
29. C22×#85561C22semidirect
30. C22×C8×C22⋊A4C22semidirect
31. C2×#6372601D3semidirect
32. #72252249C22semidirect
35. C2×C8×#1020C22semidirect
36. C2×C8×#1020C22semidirect
37. C2×C8×#1020C22semidirect
38. C2×C8×#1020C22semidirect
39. C8×#1020⋊C2C22semidirect
51. C2×#11715649C22semidirect
69. #11715670C23semidirect
72. C8×#1020C23semidirect
73. C8×#1020C23semidirect
74. C8×#1020C23semidirect
75. #11715649C23semidirect
79. C2×#85561C23semidirect
98. #11711115C23semidirect
99. #11711114C23semidirect
100. #11711114C23semidirect
101. #11711113C23semidirect
103. #1020⋊C8C23semidirect
107. C2×C8×C22⋊A4C23semidirect
114. #21440⋊C3C23semidirect
115. C8×#120⋊C3C23semidirect
116. #20579⋊C3C23semidirect
118. C26×C8C2×D3semidirect
120. #6372601C2×D3semidirect
121. C2×#21440C2×D3semidirect
122. C2×C8×#120C2×D3semidirect
123. C2×#20579C2×D3semidirect
170. C23×#131S4semidirect
173. C25×C8C22×D3semidirect
180. #21440C22×D3semidirect
181. C8×#120C22×D3semidirect
182. #20579C22×D3semidirect
186. C2×#1020#131semidirect
204. C22×#131C2×S4semidirect
206. C23×M16C2×S4semidirect
208. C24×C8C2×S4semidirect
219. C2×C22⋊A4C2×#131semidirect
221. #1020C2×#131semidirect
235. C22×M16C22×S4semidirect
240. C23×C8C22×S4semidirect
243. C22⋊A4C22×#131semidirect
244. C25D3×#131semidirect
258. C22×C8#1020⋊C2semidirect
262. C24C2×D3×#131semidirect
273. C2×C8C2×#1020⋊C2semidirect
290. C2#72342683direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#723426812
C2×#723426792
C2×#723426772
C2×#723426752
C2×#723426722
C2×#723426712
C2×#723426702
C2×C8×#1020⋊C22
#723426832
C2×S4×#1314
C2×#722522492
C22×C8×#10202
C2×#722839433
C2×#722206372