Finite Group Databasebeta

← #41003400

Subgroups — #41003400

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×C22⋊D421C211yesmaximal, characteristic, complemented
2C2×D4×D421C211yesmaximal, characteristic, complemented
3#27212350C211yesmaximal, characteristic, complemented
4#27212348C212yesmaximal, complemented
5#27212347C211yesmaximal, characteristic, complemented
6#27212339C212yesmaximal, complemented
7#27212337C212yesmaximal, complemented
8#27212335C211yesmaximal, characteristic, complemented
9#27212334C212yesmaximal, complemented
10C22⋊C4×D421C211yesmaximal, characteristic, complemented
11#120×C421C211yesmaximal, characteristic, complemented
12C23×D421C2211yescharacteristic
13C23×D421C2211yescharacteristic, complemented
14C2×D1684C2211yescharacteristic, complemented
15C2×C22⋊D421—24no—
16C2×C22⋊D421C2212yes—
17C2×C22⋊D421C2211yescharacteristic
18C2×C22⋊D421C2211yescharacteristic, complemented
19C2×C22⋊D421C2212yescomplemented
20C2×C22⋊D421C2212yescomplemented
21D4×D421—24no—
22C22×Dic421C2211yescharacteristic, complemented
23C2×C4×D421C2211yescharacteristic, complemented
24C2×C4×D421C2212yescomplemented
25C4⋊Dic421C2212yescomplemented
26#20269750C2212yescomplemented
27C22⋊Dic421C2211yescharacteristic, complemented
28C22⋊Dic421C2212yescomplemented
29C22⋊Dic421C2212yescomplemented
30#20269748C2212yescomplemented
31#20269748C2212yescomplemented
32C24×C421C2211yescharacteristic, complemented
33C2×D4×C421C2211yescharacteristic
34C2×D4×C421C2212yescomplemented
35C22⋊C4×C421C2211yescharacteristic, complemented
36C22⋊C4×C421C2212yescomplemented
37C22×D421—22no—
38C22×D421—42no—
39C22×D421—42no—
40C22×D421D414yescomplemented
41C22×D421C2311yescharacteristic
42C22×D421C2311yescharacteristic
43C22×D421C2311yescharacteristic
44D1684—22no—
45C22⋊D421—22no—
46C22⋊D421—22no—
47C22⋊D421—24no—
48C22⋊D421—24no—
49C22⋊D421—42no—
50C2×Dic421—22no—
51C2×Dic421C2312yes—
52C2×Dic421C2312yes—
53C2×Dic421C2311yescharacteristic
54C4×D421—22no—
55C4×D421—42no—
56C23×C421—22no—
57C23×C421—42no—
58C23×C421C2312yes—
59C23×C421C2311yescharacteristic
60C23×C421C2311yescharacteristic
61D4×C421—22no—
62D4×C421—42no—
63C2×C4×C421C2311yescharacteristic
64C2×C4×C421C2312yes—
65C22×C421—22no—
66C22×C421—21no—
67C22×C421—22no—
68C22×C421—22no—
69C22×C421—22no—
70C22×C421—22no—
71C22×C421—44no—
72C22×C421C2×D412yes—
73C22×C421C2411yescharacteristic
74C2×D421—22no—
75C2×D421—22no—
76C2×D421—21no—
77C2×D421—22no—
78C2×D421—44no—
79C2×D421—44no—
80C2×D421C2×D412yes—
81C2×D421C2×D412yes—
82C4×C421—21no—
83C4×C421—22no—
84Dic421—21no—
85Dic421—22no—
86Dic421—22no—
87C2×C421—22no—
88C2×C421—41no—
89C2×C421—42no—
90C2×C421C22×D411yescharacteristic
91C2×C421H(5,𝔽2)12yes—
92D421—22no—
93D421—22no—
94D421—41no—
95C421#53411yescharacteristic, complemented, Sylow
96#534—4211nomaximal, Sylow
97C22×D4—4211no—
98C22×D4—4211no—
99#123—4211no—
100#123—4212no—
101#123—4211no—
102#121—4212no—
103#121—4212no—
104#120—4211no—
105#120—4212no—
106#120D42111yescharacteristic, complemented
107C2×C22⋊C4—4211no—
108C24—4211no—
109C24—4211no—
110C24C2×D42111yescharacteristic, complemented
111C2×D4—8424no—
112C2×D4—8424no—
113C2×D4—4212no—
114C2×D4—4211no—
115C2×D4—4211no—
116C2×D4—4212no—
117C2×D4—4212no—
118C2×D4—4211no—
119C2×D4C2×D42111yescharacteristic
120C2×D4C2×D42112yescomplemented
121C22×C4—4211no—
122C22×C4—4211no—
123C22×C4—4212no—
124C4⋊C4—4212no—
125C4⋊C4—4212no—
126C22⋊C4—4211no—
127C22⋊C4—4212no—
128C22⋊C4—4212no—
129C22⋊C4—4212no—
130C22⋊C4—4212no—
131C22⋊C4C2×D42111yescharacteristic, complemented
132C22⋊C4C2×D42112yescomplemented
133C23—22no—
134C23—42no—
135C23—8422no—
136C23—16842no—
137C23—16842no—
138C23—4211no—
139C23—4211no—
140C23—4211no—
141C23—4214no—
142C23C22×D42111yescharacteristic
143C23C22×D42112yes—
144C23C22×D42111yescharacteristic
145D4—22no—
146D4—42no—
147D4—8422no—
148D4—8422no—
149D4—8424no—
150D4—8424no—
151D4—8422no—
152D4—16842no—
153C2×C4—8422no—
154C2×C4—8422no—
155C2×C4—16842no—
156C2×C4—4212no—
157C2×C4—4211no—
158C2×C4—4212no—
159C2×C4C22×D42112yes—
160C2×C4C22×D42111yescharacteristic
161C22—22no—
162C22—21no—
163C22—22no—
164C22—22no—
165C22—22no—
166C22—22no—
167C22—44no—
168C22—8422no—
169C22—8422no—
170C22—8421no—
171C22—8422no—
172C22—16844no—
173C22—16844no—
174C22—4212no—
175C22—4212no—
176C22D4×D42112yes—
177C22C23×D42111yescharacteristic
178C4—21no—
179C4—22no—
180C4—8421no—
181C4—8422no—
182C4—8422no—
183C2—22no—
184C2—42no—
185C2—41no—
186C2—8422no—
187C2—8422no—
188C2—16841no—
189C2C2×D4×D42111yescharacteristic
190C2#2721241112yes—
subgroupG/H
center, FrattiniC22C23×D421
commutator, socle, cosocleC22×C421C24
Fitting#120×C421C2
hypercenter#120D421
hypocenterC421#534
LCSG ▷ C22×C421 ▷ C421
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D421C2semidirect
2. C2×D4×D421C2semidirect
3. #27212350C2semidirect
4. #27212348C2semidirect
5. #27212347C2semidirect
6. #27212339C2semidirect
7. #27212337C2semidirect
8. #27212335C2semidirect
9. #27212334C2semidirect
10. C22⋊C4×D421C2semidirect
11. #120×C421C2semidirect
13. C23×D421C22semidirect
14. C2×D1684C22semidirect
18. C2×C22⋊D421C22semidirect
19. C2×C22⋊D421C22semidirect
20. C2×C22⋊D421C22semidirect
22. C22×Dic421C22semidirect
23. C2×C4×D421C22semidirect
24. C2×C4×D421C22semidirect
25. C4⋊Dic421C22semidirect
26. #20269750C22semidirect
27. C22⋊Dic421C22semidirect
28. C22⋊Dic421C22semidirect
29. C22⋊Dic421C22semidirect
30. #20269748C22semidirect
31. #20269748C22semidirect
32. C24×C421C22semidirect
34. C2×D4×C421C22semidirect
35. C22⋊C4×C421C22semidirect
36. C22⋊C4×C421C22semidirect
40. C22×D421D4semidirect
95. C421#534semidirect
106. #120D421semidirect
110. C24C2×D421semidirect
120. C2×D4C2×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×C22⋊D4212
C2×D4×D4212
#272123502
#272123482
#272123472
#272123392
#272123372
#272123352
#272123342
C22⋊C4×D4212
#120×C4212
#534421