Finite Group Databasebeta

← #41003405

Subgroups — #41003405

Order 26944.

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

normal subgroups only

This group has 334 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×#20269770C211yesmaximal, characteristic, complemented
2C22×C22⋊D421C211yesmaximal, characteristic, complemented
3#27212362C212yesmaximal, complemented
4#27212352C212yesmaximal, complemented
5#27212339C212yesmaximal, complemented
6#27212338C212yesmaximal, complemented
7#27212336C211yesmaximal, characteristic, complemented
8#27212335C211yesmaximal, characteristic, complemented
9#27212334C211yesmaximal, characteristic, complemented
10#27212322C211yesmaximal, characteristic, complemented
11#120×C421C211yesmaximal, characteristic, complemented
12C23×D421C2211yescharacteristic
13#20269770—24no—
14C2×Dic842C2211yescharacteristic, complemented
15C2×D1684C2212yescomplemented
16C2×C22⋊D421—24no—
17C2×C22⋊D421C2211yescharacteristic
18C2×C22⋊D421C2212yes—
19C2×C22⋊D421C2212yes—
20C2×C22⋊D421C2211yescharacteristic, complemented
21C2×C22⋊D421C2212yescomplemented
22C2×C22⋊D421C2212yescomplemented
23C22×Dic421C2211yescharacteristic
24C22×Dic421C2211yescharacteristic, complemented
25C2×C4×D421C2211yescharacteristic, complemented
26#20269750C2212yescomplemented
27C22⋊Dic421C2211yescharacteristic
28C22⋊Dic421C2212yescomplemented
29#20269748C2212yescomplemented
30#20269748C2212yescomplemented
31C4×Dic421C2212yescomplemented
32C24×C421C2211yescharacteristic
33C2×D4×C421C2211yescharacteristic
34C2×D4×C421C2212yescomplemented
35C22⋊C4×C421C2211yescharacteristic
36C22⋊C4×C421C2212yescomplemented
37C22×D421—22no—
38C22×D421—42no—
39C22×D421C2312yes—
40C22×D421C2311yescharacteristic
41Dic842—22no—
42D1684—42no—
43C22⋊D421—22no—
44C22⋊D421—24no—
45C22⋊D421—24no—
46C22⋊D421—22no—
47C22⋊D421—42no—
48C22⋊D421—42no—
49C2×Dic421—22no—
50C2×Dic421D414yescomplemented
51C2×Dic421C2312yes—
52C2×Dic421C2311yescharacteristic
53C2×Dic421C2311yescharacteristic
54C2×Dic421C2311yescharacteristic
55C4×D421—22no—
56C23×C421—22no—
57C23×C421—42no—
58C23×C421C2312yes—
59C23×C421C2311yescharacteristic
60C23×C421C2311yescharacteristic
61D4×C421—22no—
62D4×C421—42no—
63C2×C4×C421C2312yes—
64C2×C4×C421C2311yescharacteristic
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—21no—
75C2×D421—22no—
76C2×D421—22no—
77C2×D421—22no—
78C2×D421—22no—
79C2×D421—44no—
80C4×C421—21no—
81C4×C421—22no—
82Dic421—21no—
83Dic421—22no—
84Dic421C2×D412yes—
85Dic421C2×D412yes—
86C2×C421—22no—
87C2×C421—41no—
88C2×C421—42no—
89C2×C421C22×D411yescharacteristic
90C2×C421H(5,𝔽2)12yes—
91D421—41no—
92D421—42no—
93C421#53511yescharacteristic, complemented, Sylow
94#535—4211nomaximal, Sylow
95C2×#41—4211no—
96C22×D4—4211no—
97#127—4212no—
98#124—4212no—
99#121—4212no—
100#121—4212no—
101#120—4211no—
102#120—4211no—
103#120—4211no—
104#120D42111yescharacteristic, complemented
105#117—4211no—
106C24—4211no—
107C24C2×D42111yescharacteristic
108#41—8424no—
109C2×Q8—4211no—
110C2×D4—8424no—
111C2×D4—4211no—
112C2×D4—4212no—
113C2×D4—4212no—
114C2×D4—4211no—
115C2×D4—4212no—
116C2×D4—4212no—
117C2×D4—4212no—
118C2×D4C2×D42111yescharacteristic
119C2×D4C2×D42112yescomplemented
120C22×C4—4211no—
121C22×C4—4211no—
122C22×C4—4211no—
123C4⋊C4—4212no—
124C22⋊C4—4211no—
125C22⋊C4—4212no—
126C22⋊C4—4212no—
127C22⋊C4—4212no—
128C22⋊C4C2×D42111yescharacteristic
129C22⋊C4C2×D42112yescomplemented
130C42—4212no—
131C23—22no—
132C23—42no—
133C23—8422no—
134C23—16842no—
135C23—4212no—
136C23—4211no—
137C23C22×D42112yes—
138C23C22×D42111yescharacteristic
139C23C22×D42111yescharacteristic
140Q8—8422no—
141D4—22no—
142D4—42no—
143D4—8422no—
144D4—8424no—
145D4—8424no—
146D4—8422no—
147D4—16842no—
148D4—16842no—
149D4—16842no—
150C2×C4—8422no—
151C2×C4—8422no—
152C2×C4—4212no—
153C2×C4—4211no—
154C2×C4—4214no—
155C2×C4—4211no—
156C2×C4—4211no—
157C2×C4C22×D42112yes—
158C2×C4C22×D42111yescharacteristic
159C22—22no—
160C22—21no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—44no—
166C22—8421no—
167C22—8422no—
168C22—8422no—
169C22—8422no—
170C22—8422no—
171C22—16844no—
172C22D4×D42112yes—
173C22C23×D42111yescharacteristic
174C4—21no—
175C4—22no—
176C4—8421no—
177C4—8422no—
178C4—4212no—
179C4—4212no—
180C2—22no—
181C2—42no—
182C2—41no—
183C2—16842no—
184C2—16841no—
185C2C2×D4×D42111yescharacteristic
186C2#2721241112yes—
subgroupG/H
center, FrattiniC22C23×D421
commutator, socle, cosocleC22×C421C24
Fitting#120×C421C2
hypercenter#120D421
hypocenterC421#535
LCSG ▷ C22×C421 ▷ C421
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#20269770C2semidirect
2. C22×C22⋊D421C2semidirect
3. #27212362C2semidirect
4. #27212352C2semidirect
5. #27212339C2semidirect
6. #27212338C2semidirect
7. #27212336C2semidirect
8. #27212335C2semidirect
9. #27212334C2semidirect
10. #27212322C2semidirect
11. #120×C421C2semidirect
14. C2×Dic842C22semidirect
15. C2×D1684C22semidirect
20. C2×C22⋊D421C22semidirect
21. C2×C22⋊D421C22semidirect
22. C2×C22⋊D421C22semidirect
24. C22×Dic421C22semidirect
25. C2×C4×D421C22semidirect
26. #20269750C22semidirect
28. C22⋊Dic421C22semidirect
29. #20269748C22semidirect
30. #20269748C22semidirect
31. C4×Dic421C22semidirect
34. C2×D4×C421C22semidirect
36. C22⋊C4×C421C22semidirect
50. C2×Dic421D4semidirect
93. C421#535semidirect
104. #120D421semidirect
119. C2×D4C2×D421semidirect
129. C22⋊C4C2×D421semidirect

Maximal subgroups

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

subgroupindex
C2×#202697702
C22×C22⋊D4212
#272123622
#272123522
#272123392
#272123382
#272123362
#272123352
#272123342
#272123222
#120×C4212
#535421