Finite Group Databasebeta

← C22×#14581812 #24922189

Subgroups — C22×#14581812 #24922189

Order 10624.

1504 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1504 conjugacy classes of subgroups — too many to draw. Only its 565 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13050590C211yesmaximal, characteristic, complemented
2C2×#14581812C2124yesmaximal, direct product factor
3C22×#13050580C211yesmaximal, characteristic, complemented
4C22×#13050578C211yesmaximal, characteristic, complemented
5C22×#13050577C211yesmaximal, characteristic, complemented
6C22×#13050575C211yesmaximal, characteristic, complemented
7C23×C83⋊C8C211yesmaximal, characteristic, complemented
8C22×#41×C83C211yesmaximal, characteristic, complemented
9C2×#13050590—26no—
10C2×#13050590C22112yescomplemented
11C22×Dic166C2211yescharacteristic
12C22×D332C2211yescharacteristic
13C22×C22⋊D83—21no—
14C22×C4×D83—21no—
15#14581812C22164yesdirect product factor
16C2×#13050580C22112yescomplemented
17C2×#13050578C22112yescomplemented
18C2×#13050577C22112yescomplemented
19C2×#13050575C22112yescomplemented
20C22×C83⋊C8C2211yescharacteristic
21C22×C83⋊C8C2211yescharacteristic
22C22×C83⋊C8C22112yescomplemented
23C2×#41×C83—26no—
24C2×#41×C83C22112yescomplemented
25C22×Q8×C83C2211yescharacteristic
26C22×D4×C83—21no—
27C22×D4×C83C2211yescharacteristic
28C23×C4×C83—21no—
29C23×C4×C83C2211yescharacteristic
30C23×D83—21no—
31#13050590—224no—
32#13050590C23116yescomplemented
33C2×Dic166—23no—
34C2×Dic166C2316yes—
35C2×D332—23no—
36C2×D332C2316yes—
37C2×C22⋊D83—212no—
38C22×Dic83—21no—
39C2×C4×D83—212no—
40#13050580C23116yescomplemented
41#13050578C23116yescomplemented
42#13050577C23116yescomplemented
43#13050575C23116yescomplemented
44C2×C83⋊C8C2316yes—
45C2×C83⋊C8C2316yes—
46C2×C83⋊C8C23116yescomplemented
47C24×C83—21no—
48C24×C83D411yescharacteristic
49#41×C83—224no—
50#41×C83C23116yescomplemented
51C2×Q8×C83—23no—
52C2×Q8×C83C2316yes—
53C2×D4×C83—212no—
54C2×D4×C83—23no—
55C2×D4×C83C2316yes—
56C22×C4×C83—21no—
57C22×C4×C83—212no—
58C22×C4×C83D416yes—
59C22×C4×C83D411yescharacteristic
60C22×C4×C83C2311yescharacteristic
61C22×C4×C83C2316yes—
62C23×C83—24no—
63C23×C83—26no—
64C23×C83—44no—
65C23×C83C2×D416yes—
66C23×C83C2×D411yescharacteristic
67C22×D83—26no—
68C22×D83—44no—
69Q8×C83—26no—
70Q8×C83C2414yes—
71D4×C83—216no—
72D4×C83—26no—
73D4×C83C2414yes—
74C2×C4×C83—26no—
75C2×C4×C83—216no—
76C2×C4×C83C2×D416yes—
77C2×C4×C83C2×D413yes—
78C2×C4×C83C2×D4112yes—
79C2×C4×C83C2413yes—
80C2×C4×C83C2414yes—
81C22⋊D83—216no—
82C2×Dic83—26no—
83D332—26no—
84D332C2414yes—
85C4×D83—216no—
86Dic166—26no—
87Dic166C2414yes—
88C83⋊C8C2414yes—
89C83⋊C8C2414yes—
90C22×C83—212no—
91C22×C83—24no—
92C22×C83—412no—
93C22×C83#13514yescomplemented
94C22×C83C22×D414yes—
95C22×C83C22×D413yes—
96C2×D83—24no—
97C2×D83—412no—
98C4×C83—24no—
99C4×C83C22×D413yes—
100C4×C83C22×D414yes—
101C4×C83C2511yescharacteristic
102Dic83—24no—
103C2×C83—24no—
104C2×C83—44no—
105C2×C83C2×#13516yescomplemented
106C2×C83C23×D411yescharacteristic
107D83—44no—
108C22×#135—831nomaximal, Sylow
109C83C22×#13511yescharacteristic, complemented, Sylow
110C22×#41—831no—
111C22×#41D8311yescharacteristic, complemented
112C2×#135—8324no—
113C22×Q16—831no—
114C22×SD16—831no—
115C22×SD16—831no—
116C22×D8—831no—
117C23×C8—831no—
118C2×#41—26no—
119C2×#41—8312no—
120C2×#41—1666no—
121C2×#41C2×D83112yescomplemented
122C22×Q8—831no—
123C22×Q8C2×D8311yescharacteristic
124C22×D4—21no—
125C22×D4—831no—
126C22×D4—1661no—
127C22×D4C2×D8311yescharacteristic
128C23×C4—21no—
129C23×C4—1661no—
130C23×C4C2×D8311yescharacteristic
131#135—8364no—
132C2×Q16—8312no—
133C2×SD16—8312no—
134C2×SD16—8312no—
135C2×D8—8312no—
136C22×C8—831no—
137C22×C8—831no—
138C22×C8—8312no—
139C24—21no—
140C24—1661no—
141C24C22⋊D8311yescharacteristic
142#41—224no—
143#41—8316no—
144#41—16624no—
145#41C22×D83116yescomplemented
146C2×Q8—23no—
147C2×Q8—836no—
148C2×Q8—1663no—
149C2×Q8C22×D8316yes—
150C2×D4—212no—
151C2×D4—23no—
152C2×D4—836no—
153C2×D4—16612no—
154C2×D4—1663no—
155C2×D4C22×D8316yes—
156C22×C4—21no—
157C22×C4—212no—
158C22×C4—1661no—
159C22×C4—16612no—
160C22×C4C22⋊D8316yes—
161C22×C4C22⋊D8311yescharacteristic
162C22×C4C22×D8311yescharacteristic
163C22×C4C22×D8316yes—
164Q16—8316no—
165SD16—8316no—
166SD16—8316no—
167D8—8316no—
168C2×C8—836no—
169C2×C8—836no—
170C2×C8—8316no—
171C23—24no—
172C23—26no—
173C23—44no—
174C23—3324no—
175C23—1666no—
176C23C2×C22⋊D8316yes—
177C23C2×C22⋊D8311yescharacteristic
178Q8—26no—
179Q8—834no—
180Q8—1666no—
181Q8C23×D8314yes—
182D4—216no—
183D4—26no—
184D4—834no—
185D4—16616no—
186D4—1666no—
187D4C23×D8314yes—
188C2×C4—26no—
189C2×C4—216no—
190C2×C4—1666no—
191C2×C4—16616no—
192C2×C4C2×C22⋊D8316yes—
193C2×C4C2×C22⋊D8313yes—
194C2×C4C2×C22⋊D83112yes—
195C2×C4C23×D8313yes—
196C2×C4C23×D8314yes—
197C8—834no—
198C8—834no—
199C22—212no—
200C22—24no—
201C22—412no—
202C22—33212no—
203C22—1664no—
204C22#1458181214yesdirect product factor
205C22C22×C22⋊D8314yes—
206C22C22×C22⋊D8313yes—
207C4—24no—
208C4—1664no—
209C4C22×C22⋊D8314yes—
210C4C22×C22⋊D8313yes—
211C4C24×D8311yescharacteristic
212C2—24no—
213C2—44no—
214C2—3324no—
215C2C2×#1458181216yesdirect product factor
216C2C23×C22⋊D8311yescharacteristic
subgroupG/H
centerC22×C4C22⋊D83
commutator, cosocleC4×C83C25
FrattiniC4C24×D83
FittingC22×#41×C83C2
socleC23×C83C2×D4
hypercenterC22×#41D83
hypocenterC83C22×#135
LCSG ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C22×#13050590C2semidirect
2. C2×#14581812C2direct (N × H)
3. C22×#13050580C2semidirect
4. C22×#13050578C2semidirect
5. C22×#13050577C2semidirect
6. C22×#13050575C2semidirect
7. C23×C83⋊C8C2semidirect
8. C22×#41×C83C2semidirect
10. C2×#13050590C22semidirect
15. #14581812C22direct (N × H)
16. C2×#13050580C22semidirect
17. C2×#13050578C22semidirect
18. C2×#13050577C22semidirect
19. C2×#13050575C22semidirect
22. C22×C83⋊C8C22semidirect
24. C2×#41×C83C22semidirect
32. #13050590C23semidirect
40. #13050580C23semidirect
41. #13050578C23semidirect
42. #13050577C23semidirect
43. #13050575C23semidirect
46. C2×C83⋊C8C23semidirect
50. #41×C83C23semidirect
93. C22×C83#135semidirect
105. C2×C83C2×#135semidirect
109. C83C22×#135semidirect
111. C22×#41D83semidirect
121. C2×#41C2×D83semidirect
145. #41C22×D83semidirect
204. C22#14581812direct (N × H)
215. C2C2×#14581812direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130505902
C2×#145818122
C22×#130505802
C22×#130505782
C22×#130505772
C22×#130505752
C23×C83⋊C82
C22×#41×C832
C22×#13583