Finite Group Databasebeta

← #41003401

Subgroups — #41003401

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
1C22×D1684C211yesmaximal, characteristic, complemented
2C2×D4×D421C211yesmaximal, characteristic, complemented
3#27212351C212yesmaximal, complemented
4#27212349C211yesmaximal, characteristic, complemented
5#27212341C212yesmaximal, complemented
6#27212339C212yesmaximal, complemented
7#27212336C211yesmaximal, characteristic, complemented
8#27212336C212yesmaximal, complemented
9#27212334C211yesmaximal, characteristic, complemented
10C22⋊C4×D421C211yesmaximal, characteristic, complemented
11#123×C421C211yesmaximal, characteristic, complemented
12C23×D421C2211yescharacteristic
13C23×D421C2211yescharacteristic, complemented
14C23×D421C2211yescharacteristic, complemented
15C2×D1684—24no—
16C2×D1684C2212yes—
17C2×D1684C2212yescomplemented
18C2×D1684C2211yescharacteristic, complemented
19C2×D1684C2212yescomplemented
20C2×C22⋊D421C2211yescharacteristic
21C2×C22⋊D421C2211yescharacteristic, complemented
22C2×C22⋊D421C2212yescomplemented
23D4×D421—24no—
24C2×C4×D421C2211yescharacteristic, complemented
25C2×C4×D421C2212yescomplemented
26#20269750C2212yescomplemented
27C22⋊Dic421C2211yescharacteristic, complemented
28#20269748C2212yescomplemented
29#20269748C2212yescomplemented
30#20269748C2212yescomplemented
31#20269748C2212yescomplemented
32C2×D4×C421C2211yescharacteristic
33C22×C4×C421C2211yescharacteristic, complemented
34C4⋊C4×C421C2212yescomplemented
35C22⋊C4×C421C2211yescharacteristic, complemented
36C22⋊C4×C421C2212yescomplemented
37C22×D421—22no—
38C22×D421—22no—
39C22×D421—42no—
40C22×D421—42no—
41C22×D421—42no—
42C22×D421D414yescomplemented
43C22×D421C2311yescharacteristic
44C22×D421C2312yes—
45C22×D421C2311yescharacteristic
46C22×D421C2311yescharacteristic
47D1684—22no—
48D1684—24no—
49D1684—24no—
50D1684—42no—
51C22⋊D421—22no—
52C22⋊D421—22no—
53C22⋊D421—42no—
54C2×Dic421C2312yes—
55C2×Dic421C2311yescharacteristic
56C4×D421—22no—
57C4×D421—42no—
58C23×C421C2311yescharacteristic
59C23×C421C2311yescharacteristic
60D4×C421—22no—
61C2×C4×C421—22no—
62C2×C4×C421C2312yes—
63C2×C4×C421C2312yes—
64C2×C4×C421C2311yescharacteristic
65C22×C421—22no—
66C22×C421—21no—
67C22×C421—22no—
68C22×C421C2×D412yes—
69C22×C421C2411yescharacteristic
70C2×D421—22no—
71C2×D421—22no—
72C2×D421—21no—
73C2×D421—22no—
74C2×D421—22no—
75C2×D421—22no—
76C2×D421—22no—
77C2×D421—44no—
78C2×D421—44no—
79C2×D421—44no—
80C2×D421C2×D412yes—
81C2×D421C2×D412yes—
82C4×C421—22no—
83C4×C421—21no—
84C4×C421—22no—
85Dic421—21no—
86Dic421—22no—
87C2×C421—22no—
88C2×C421—41no—
89C2×C421C22×D411yescharacteristic
90C2×C421H(5,𝔽2)12yes—
91D421—22no—
92D421—22no—
93D421—41no—
94D421—42no—
95C421#53411yescharacteristic, complemented, Sylow
96#534—4211nomaximal, Sylow
97C22×D4—4211no—
98C22×D4—4211no—
99#123—4211no—
100#123—4212no—
101#123D42111yescharacteristic, complemented
102#121—4212no—
103#121—4212no—
104#120—4211no—
105#120—4212no—
106#120—4211no—
107C2×C22⋊C4—4211no—
108C24—4211no—
109C24—4211no—
110C24—4211no—
111C2×D4—8424no—
112C2×D4—8424no—
113C2×D4—4211no—
114C2×D4—4211no—
115C2×D4—4212no—
116C2×D4—4212no—
117C2×D4—4211no—
118C2×D4—4212no—
119C2×D4—4212no—
120C2×D4C2×D42111yescharacteristic
121C22×C4—4211no—
122C22×C4—4212no—
123C22×C4C2×D42111yescharacteristic, complemented
124C4⋊C4—4212no—
125C4⋊C4C2×D42112yescomplemented
126C22⋊C4—4211no—
127C22⋊C4—4212no—
128C22⋊C4—4212no—
129C22⋊C4—4212no—
130C22⋊C4—4212no—
131C22⋊C4C2×D42111yescharacteristic, complemented
132C22⋊C4C2×D42112yescomplemented
133C23—8422no—
134C23—8422no—
135C23—16842no—
136C23—16842no—
137C23—16842no—
138C23—4211no—
139C23—4212no—
140C23—4211no—
141C23—4211no—
142C23—4214no—
143C23C22×D42111yescharacteristic
144C23C22×D42111yescharacteristic
145D4—22no—
146D4—8424no—
147D4—8424no—
148D4—8422no—
149D4—8422no—
150D4—8422no—
151D4—16842no—
152D4—16842no—
153C2×C4—22no—
154C2×C4—8422no—
155C2×C4—16842no—
156C2×C4—4212no—
157C2×C4—4211no—
158C2×C4C22×D42112yes—
159C2×C4C22×D42111yescharacteristic
160C2×C4C22×D42112yes—
161C22—22no—
162C22—21no—
163C22—22no—
164C22—8422no—
165C22—8422no—
166C22—8422no—
167C22—8421no—
168C22—8422no—
169C22—8422no—
170C22—8422no—
171C22—16844no—
172C22—16844no—
173C22—16844no—
174C22—4212no—
175C22—4212no—
176C22D4×D42112yes—
177C22C23×D42111yescharacteristic
178C4—22no—
179C4—21no—
180C4—22no—
181C4—8421no—
182C4—8422no—
183C2—22no—
184C2—41no—
185C2—8422no—
186C2—8422no—
187C2—16842no—
188C2—16841no—
189C2C2×D4×D42111yescharacteristic
190C2#2721241212yes—
subgroupG/H
center, FrattiniC22C23×D421
commutator, socle, cosocleC22×C421C24
Fitting#123×C421C2
hypercenter#123D421
hypocenterC421#534
LCSG ▷ C22×C421 ▷ C421
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C22×D1684C2semidirect
2. C2×D4×D421C2semidirect
3. #27212351C2semidirect
4. #27212349C2semidirect
5. #27212341C2semidirect
6. #27212339C2semidirect
7. #27212336C2semidirect
8. #27212336C2semidirect
9. #27212334C2semidirect
10. C22⋊C4×D421C2semidirect
11. #123×C421C2semidirect
13. C23×D421C22semidirect
14. C23×D421C22semidirect
17. C2×D1684C22semidirect
18. C2×D1684C22semidirect
19. C2×D1684C22semidirect
21. C2×C22⋊D421C22semidirect
22. C2×C22⋊D421C22semidirect
24. C2×C4×D421C22semidirect
25. C2×C4×D421C22semidirect
26. #20269750C22semidirect
27. C22⋊Dic421C22semidirect
28. #20269748C22semidirect
29. #20269748C22semidirect
30. #20269748C22semidirect
31. #20269748C22semidirect
33. C22×C4×C421C22semidirect
34. C4⋊C4×C421C22semidirect
35. C22⋊C4×C421C22semidirect
36. C22⋊C4×C421C22semidirect
42. C22×D421D4semidirect
95. C421#534semidirect
101. #123D421semidirect
123. C22×C4C2×D421semidirect
125. C4⋊C4C2×D421semidirect
131. C22⋊C4C2×D421semidirect
132. C22⋊C4C2×D421semidirect

Maximal subgroups

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

subgroupindex
C22×D16842
C2×D4×D4212
#272123512
#272123492
#272123412
#272123392
#272123362
#272123342
C22⋊C4×D4212
#123×C4212
#534421