Finite Group Databasebeta

← #41003398

Subgroups — #41003398

Order 26944.

346 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 103 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×D4×D421C212yesmaximal, complemented
2#27212351C212yesmaximal, complemented
3#27212348C212yesmaximal, complemented
4#27212340C211yesmaximal, characteristic, complemented
5#27212338C212yesmaximal, complemented
6#27212336C212yesmaximal, complemented
7#27212334C212yesmaximal, complemented
8C2×#20269748C211yesmaximal, characteristic, complemented
9#121×C421C211yesmaximal, characteristic, complemented
10C23×D421C2211yescharacteristic
11C23×D421C2212yescomplemented
12C2×D1684C2212yes—
13C2×D1684C2211yescharacteristic, complemented
14C2×C22⋊D421C2212yes—
15C2×C22⋊D421C2212yescomplemented
16C2×C22⋊D421C2212yescomplemented
17D4×D421—28no—
18C22×Dic421C2211yescharacteristic, complemented
19C2×C4×D421C2212yescomplemented
20C4⋊Dic421C2211yescharacteristic, complemented
21#20269750C2212yescomplemented
22C22⋊Dic421C2212yescomplemented
23#20269748C2212yescomplemented
24#20269748C2212yescomplemented
25#20269748C2212yescomplemented
26#20269748C2212yescomplemented
27C2×D4×C421C2212yescomplemented
28C2×D4×C421C2211yescharacteristic, complemented
29C22×C4×C421C2211yescharacteristic, complemented
30C4⋊C4×C421C2211yescharacteristic, complemented
31C22⋊C4×C421C2212yescomplemented
32C22×D421—24no—
33C22×D421—41no—
34C22×D421—41no—
35C22×D421—42no—
36C22×D421—42no—
37C22×D421D414yescomplemented
38C22×D421C2312yes—
39C22×D421C2312yes—
40D1684—24no—
41D1684—41no—
42C22⋊D421—24no—
43C22⋊D421—24no—
44C22⋊D421—42no—
45C2×Dic421—41no—
46C2×Dic421C2312yes—
47C2×Dic421C2312yes—
48C4×D421—24no—
49C23×C421C2312yes—
50C23×C421C2311yescharacteristic
51D4×C421—24no—
52D4×C421—41no—
53C2×C4×C421—41no—
54C2×C4×C421C2311yescharacteristic
55C2×C4×C421C2312yes—
56C2×C4×C421C2311yescharacteristic
57C22×C421—21no—
58C22×C421—21no—
59C22×C421—22no—
60C22×C421—22no—
61C22×C421—22no—
62C22×C421C2×D412yes—
63C22×C421C2411yescharacteristic
64C2×D421—22no—
65C2×D421—22no—
66C2×D421—22no—
67C2×D421—22no—
68C2×D421—22no—
69C2×D421—44no—
70C2×D421—48no—
71C2×D421C2×D414yes—
72C4×C421—22no—
73C4×C421—21no—
74C4×C421—21no—
75Dic421—22no—
76Dic421—22no—
77C2×C421—22no—
78C2×C421—42no—
79C2×C421C22×D411yescharacteristic
80C2×C421H(5,𝔽2)11yescharacteristic
81C2×C421H(5,𝔽2)11yescharacteristic
82D421—24no—
83D421—42no—
84C421#53411yescharacteristic, complemented, Sylow
85#534—4211nomaximal, Sylow
86C22×D4—4212no—
87#123—4212no—
88#123—4212no—
89#121—4212no—
90#121—4211no—
91#121D42111yescharacteristic, complemented
92#120—4212no—
93#120—4212no—
94C2×C22⋊C4—4211no—
95C24—4212no—
96C24—4211no—
97C2×D4—8428no—
98C2×D4—4212no—
99C2×D4—4212no—
100C2×D4—4212no—
101C2×D4—4212no—
102C2×D4—4211no—
103C2×D4C2×D42112yescomplemented
104C2×D4C2×D42111yescharacteristic, complemented
105C22×C4—4212no—
106C22×C4—4211no—
107C22×C4C2×D42111yescharacteristic, complemented
108C4⋊C4—4212no—
109C4⋊C4—4211no—
110C4⋊C4C2×D42111yescharacteristic, complemented
111C22⋊C4—4212no—
112C22⋊C4—4212no—
113C22⋊C4—4212no—
114C22⋊C4—4212no—
115C22⋊C4—4212no—
116C22⋊C4C2×D42112yescomplemented
117C23—8424no—
118C23—16841no—
119C23—16841no—
120C23—16842no—
121C23—16842no—
122C23—4212no—
123C23—4212no—
124C23—4214no—
125C23C22×D42112yes—
126C23C22×D42111yescharacteristic
127D4—24no—
128D4—41no—
129D4—8424no—
130D4—8424no—
131D4—8424no—
132D4—16841no—
133D4—16842no—
134C2×C4—41no—
135C2×C4—8424no—
136C2×C4—16841no—
137C2×C4—4212no—
138C2×C4—4212no—
139C2×C4C22×D42111yescharacteristic
140C2×C4C22×D42111yescharacteristic
141C2×C4C22×D42112yes—
142C22—21no—
143C22—21no—
144C22—22no—
145C22—22no—
146C22—22no—
147C22—8422no—
148C22—8422no—
149C22—8422no—
150C22—8422no—
151C22—8422no—
152C22—16844no—
153C22—16848no—
154C22—4214no—
155C22D4×D42112yes—
156C22C23×D42111yescharacteristic
157C4—22no—
158C4—21no—
159C4—21no—
160C4—8422no—
161C4—8422no—
162C2—22no—
163C2—42no—
164C2—8424no—
165C2—16842no—
166C2C2×D4×D42111yescharacteristic
167C2#2721241111yescharacteristic
168C2#2721241211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D421
commutator, socle, cosocleC22×C421C24
Fitting#121×C421C2
hypercenter#121D421
hypocenterC421#534
LCSG ▷ C22×C421 ▷ C421
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×D4×D421C2semidirect
2. #27212351C2semidirect
3. #27212348C2semidirect
4. #27212340C2semidirect
5. #27212338C2semidirect
6. #27212336C2semidirect
7. #27212334C2semidirect
8. C2×#20269748C2semidirect
9. #121×C421C2semidirect
11. C23×D421C22semidirect
13. C2×D1684C22semidirect
15. C2×C22⋊D421C22semidirect
16. C2×C22⋊D421C22semidirect
18. C22×Dic421C22semidirect
19. C2×C4×D421C22semidirect
20. C4⋊Dic421C22semidirect
21. #20269750C22semidirect
22. C22⋊Dic421C22semidirect
23. #20269748C22semidirect
24. #20269748C22semidirect
25. #20269748C22semidirect
26. #20269748C22semidirect
27. C2×D4×C421C22semidirect
28. C2×D4×C421C22semidirect
29. C22×C4×C421C22semidirect
30. C4⋊C4×C421C22semidirect
31. C22⋊C4×C421C22semidirect
37. C22×D421D4semidirect
84. C421#534semidirect
91. #121D421semidirect
103. C2×D4C2×D421semidirect
104. C2×D4C2×D421semidirect
107. C22×C4C2×D421semidirect
110. C4⋊C4C2×D421semidirect
116. C22⋊C4C2×D421semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D4212
#272123512
#272123482
#272123402
#272123382
#272123362
#272123342
C2×#202697482
#121×C4212
#534421