Finite Group Databasebeta

← #39909760

Subgroups — #39909760

Order 26176.

330 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C2×#20266989C211yesmaximal, characteristic, complemented
2C22×C4×D409C211yesmaximal, characteristic, complemented
3#27189702C212yesmaximal, complemented
4#27189693C211yesmaximal, characteristic, complemented
5#27189692C211yesmaximal, characteristic, complemented
6#27189691C211yesmaximal, characteristic, complemented
7#27189687C212yesmaximal, complemented
8#27189679C212yesmaximal, complemented
9#27189677C212yesmaximal, complemented
10#27189675C211yesmaximal, characteristic, complemented
11C22⋊Q8×C409C211yesmaximal, characteristic, complemented
12C23×D409C2211yescharacteristic
13#20266989—24no—
14C2×D1636C2212yes—
15C2×D1636C2212yes—
16C2×D1636C2211yescharacteristic, complemented
17C2×C22⋊D409C2212yes—
18C22×Dic409C2211yescharacteristic
19C2×C4×D409—24no—
20C2×C4×D409C2211yescharacteristic
21C2×C4×D409C2211yescharacteristic
22C2×C4×D409C2211yescharacteristic
23C2×C4×D409C2211yescharacteristic
24C2×C4×D409C2212yescomplemented
25C4⋊Dic409C2211yescharacteristic, complemented
26#20266967C2212yescomplemented
27#20266965C2212yes—
28#20266965C2212yes—
29#20266965C2212yescomplemented
30#20266965C2212yescomplemented
31C4×Dic409C2212yescomplemented
32C2×Q8×C409C2211yescharacteristic, complemented
33C22×C4×C409C2211yescharacteristic
34C4⋊C4×C409C2211yescharacteristic, complemented
35C4⋊C4×C409C2212yescomplemented
36C22⋊C4×C409C2212yes—
37C22×D409—22no—
38C22×D409—22no—
39C22×D409—22no—
40C22×D409C2312yes—
41C22×D409C2311yescharacteristic
42C22×D409C2311yescharacteristic
43D1636—24no—
44D1636—24no—
45D1636—22no—
46C22⋊D409—24no—
47C2×Dic409—22no—
48C2×Dic409C2312yes—
49C2×Dic409C2311yescharacteristic
50C2×Dic409C2311yescharacteristic
51C4×D409—22no—
52C4×D409—22no—
53C4×D409—22no—
54C4×D409—24no—
55C4×D409D414yescomplemented
56C23×C409C2311yescharacteristic
57Q8×C409—22no—
58C2×C4×C409—22no—
59C2×C4×C409C2312yes—
60C2×C4×C409C2312yes—
61C2×C4×C409C2311yescharacteristic
62C2×C4×C409C2311yescharacteristic
63C22×C409—21no—
64C22×C409—21no—
65C22×C409#4112yescomplemented
66C22×C409C2411yescharacteristic
67C2×D409—28no—
68C2×D409—21no—
69C2×D409—21no—
70C2×D409—21no—
71C2×D409—21no—
72C2×D409—22no—
73C2×D409—22no—
74C2×D409—22no—
75C2×D409C2×D412yes—
76C2×D409#4112yescomplemented
77C4×C409—21no—
78C4×C409—22no—
79C4×C409—22no—
80C4×C409C2×D412yes—
81Dic409—21no—
82Dic409—22no—
83Dic409C2×D412yes—
84C2×C409—22no—
85C2×C409C22×D411yescharacteristic
86C2×C409C2×#4111yescharacteristic
87C2×C409C2×#4111yescharacteristic
88D409—22no—
89D409—22no—
90D409—42no—
91C409#52511yescharacteristic, complemented, Sylow
92#525—4091nomaximal, Sylow
93C2×#41—4091no—
94C23×C4—4091no—
95#123—4092no—
96C22⋊Q8—4091no—
97C22⋊Q8D40911yescharacteristic, complemented
98#121—4091no—
99#121—4091no—
100#120—4092no—
101C4×D4—4092no—
102C4×D4—4092no—
103#117—4091no—
104C24—4091no—
105#41—8184no—
106C2×Q8C2×D40911yescharacteristic, complemented
107C2×D4—4092no—
108C2×D4—4092no—
109C2×D4—4092no—
110C2×D4—4091no—
111C22×C4—8184no—
112C22×C4—4091no—
113C22×C4—4091no—
114C22×C4—4092no—
115C22×C4—4091no—
116C22×C4—4091no—
117C22×C4—4091no—
118C22×C4C2×D40911yescharacteristic
119C4⋊C4—4091no—
120C4⋊C4—4092no—
121C4⋊C4C2×D40911yescharacteristic, complemented
122C4⋊C4C2×D40912yescomplemented
123C22⋊C4—4092no—
124C22⋊C4—4092no—
125C22⋊C4—4092no—
126C22⋊C4—4092no—
127C22⋊C4C2×D40912yes—
128C42—4092no—
129C23—8182no—
130C23—8182no—
131C23—8182no—
132C23—4092no—
133C23—4091no—
134C23—4091no—
135C23C22×D40911yescharacteristic
136Q8—22no—
137D4—8184no—
138D4—8184no—
139D4—8184no—
140D4—8182no—
141C2×C4—22no—
142C2×C4—8182no—
143C2×C4—8182no—
144C2×C4—8184no—
145C2×C4—8182no—
146C2×C4—8182no—
147C2×C4—4092no—
148C2×C4—4091no—
149C2×C4—4091no—
150C2×C4—4094no—
151C2×C4C22×D40912yes—
152C2×C4C22×D40911yescharacteristic
153C2×C4C22×D40912yes—
154C2×C4C22×D40911yescharacteristic
155C22—21no—
156C22—21no—
157C22—8188no—
158C22—8181no—
159C22—8181no—
160C22—8181no—
161C22—8181no—
162C22—8182no—
163C22—8182no—
164C22—8182no—
165C22—4092no—
166C22—4092no—
167C22#2026698912yescomplemented
168C22C23×D40911yescharacteristic
169C4—21no—
170C4—22no—
171C4—22no—
172C4—8182no—
173C4—8181no—
174C4—4092no—
175C4D4×D40912yes—
176C2—22no—
177C2—8182no—
178C2—8182no—
179C2—16362no—
180C2C2×D4×D40911yescharacteristic
181C2#41×D40911yescharacteristic
182C2C2×#2026698911yescharacteristic
subgroupG/H
center, FrattiniC22C23×D409
commutator, socle, cosocleC22×C409C24
FittingC22⋊Q8×C409C2
hypercenterC22⋊Q8D409
hypocenterC409#525
LCSG ▷ C22×C409 ▷ C409
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#20266989C2semidirect
2. C22×C4×D409C2semidirect
3. #27189702C2semidirect
4. #27189693C2semidirect
5. #27189692C2semidirect
6. #27189691C2semidirect
7. #27189687C2semidirect
8. #27189679C2semidirect
9. #27189677C2semidirect
10. #27189675C2semidirect
11. C22⋊Q8×C409C2semidirect
16. C2×D1636C22semidirect
24. C2×C4×D409C22semidirect
25. C4⋊Dic409C22semidirect
26. #20266967C22semidirect
29. #20266965C22semidirect
30. #20266965C22semidirect
31. C4×Dic409C22semidirect
32. C2×Q8×C409C22semidirect
34. C4⋊C4×C409C22semidirect
35. C4⋊C4×C409C22semidirect
55. C4×D409D4semidirect
65. C22×C409#41semidirect
76. C2×D409#41semidirect
91. C409#525semidirect
97. C22⋊Q8D409semidirect
106. C2×Q8C2×D409semidirect
121. C4⋊C4C2×D409semidirect
122. C4⋊C4C2×D409semidirect
167. C22#20266989semidirect

Maximal subgroups

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

subgroupindex
C2×#202669892
C22×C4×D4092
#271897022
#271896932
#271896922
#271896912
#271896872
#271896792
#271896772
#271896752
C22⋊Q8×C4092
#525409