Finite Group Databasebeta

← #41003406

Subgroups — #41003406

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×#20269771C211yesmaximal, characteristic, complemented
2C2×D4×D421C211yesmaximal, characteristic, complemented
3#27212362C212yesmaximal, complemented
4#27212352C212yesmaximal, complemented
5#27212341C211yesmaximal, characteristic, complemented
6#27212340C211yesmaximal, characteristic, complemented
7#27212337C211yesmaximal, characteristic, complemented
8#27212336C212yesmaximal, complemented
9#27212334C212yesmaximal, complemented
10#27212324C211yesmaximal, characteristic, complemented
11#121×C421C211yesmaximal, characteristic, complemented
12C23×D421C2212yes—
13#20269771—24no—
14C2×Dic842C2211yescharacteristic, complemented
15C2×D1684C2211yescharacteristic
16C2×D1684C2211yescharacteristic
17C2×D1684C2212yescomplemented
18C2×C22⋊D421C2211yescharacteristic
19C2×C22⋊D421C2212yes—
20C2×C22⋊D421C2211yescharacteristic, complemented
21C2×C22⋊D421C2212yescomplemented
22C2×C22⋊D421C2212yescomplemented
23D4×D421—24no—
24C2×C4×D421C2211yescharacteristic
25C2×C4×D421C2211yescharacteristic, complemented
26C4⋊Dic421C2211yescharacteristic, complemented
27C22⋊Dic421C2212yescomplemented
28#20269748C2212yes—
29#20269748C2212yescomplemented
30#20269748C2212yescomplemented
31C4×Dic421C2212yescomplemented
32C2×D4×C421C2211yescharacteristic
33C2×D4×C421C2212yescomplemented
34C22×C4×C421C2211yescharacteristic, complemented
35C4⋊C4×C421C2211yescharacteristic, complemented
36C22⋊C4×C421C2212yescomplemented
37C22×D421—24no—
38C22×D421—42no—
39C22×D421—42no—
40C22×D421C2312yes—
41C22×D421C2311yescharacteristic
42C22×D421C2311yescharacteristic
43Dic842—22no—
44D1684—22no—
45D1684—22no—
46D1684—42no—
47C22⋊D421—22no—
48C22⋊D421—24no—
49C22⋊D421—22no—
50C22⋊D421—42no—
51C22⋊D421—42no—
52C2×Dic421C2312yes—
53C2×Dic421C2311yescharacteristic
54C2×Dic421C2311yescharacteristic
55C4×D421—22no—
56C4×D421D414yescomplemented
57C23×C421C2312yes—
58C23×C421C2311yescharacteristic
59D4×C421—22no—
60D4×C421—42no—
61C2×C4×C421—22no—
62C2×C4×C421C2311yescharacteristic
63C2×C4×C421C2312yes—
64C2×C4×C421C2311yescharacteristic
65C22×C421—21no—
66C22×C421—22no—
67C22×C421—21no—
68C22×C421—22no—
69C22×C421—21no—
70C22×C421—22no—
71C22×C421C2411yescharacteristic
72C2×D421—21no—
73C2×D421—21no—
74C2×D421—21no—
75C2×D421—22no—
76C2×D421—21no—
77C2×D421—22no—
78C2×D421—21no—
79C2×D421—22no—
80C2×D421—48no—
81C2×D421C2×D412yes—
82C4×C421—21no—
83C4×C421—22no—
84C4×C421C2×D412yes—
85Dic421—21no—
86Dic421—22no—
87Dic421C2×D412yes—
88C2×C421—42no—
89C2×C421—41no—
90C2×C421C22×D411yescharacteristic
91C2×C421H(5,𝔽2)11yescharacteristic
92C2×C421H(5,𝔽2)11yescharacteristic
93D421—22no—
94D421—41no—
95D421—42no—
96C421#53511yescharacteristic, complemented, Sylow
97#535—4211nomaximal, Sylow
98C2×#41—4211no—
99C22×D4—4211no—
100#127—4212no—
101#124—4212no—
102#121—4211no—
103#121—4211no—
104#121—4211no—
105#121D42111yescharacteristic, complemented
106#120—4212no—
107#120—4212no—
108#117—4211no—
109C24—4212no—
110#41—8424no—
111C2×Q8—4211no—
112C2×D4—8424no—
113C2×D4—4211no—
114C2×D4—4212no—
115C2×D4—4211no—
116C2×D4—4212no—
117C2×D4—4212no—
118C2×D4—4211no—
119C2×D4—4211no—
120C2×D4—4212no—
121C2×D4C2×D42111yescharacteristic
122C2×D4C2×D42112yescomplemented
123C22×C4—4211no—
124C22×C4—4211no—
125C22×C4C2×D42111yescharacteristic, complemented
126C4⋊C4—4211no—
127C4⋊C4C2×D42111yescharacteristic, complemented
128C22⋊C4—4212no—
129C22⋊C4—4212no—
130C22⋊C4—4212no—
131C22⋊C4—4212no—
132C22⋊C4C2×D42112yescomplemented
133C42—4212no—
134C23—8424no—
135C23—16842no—
136C23—16842no—
137C23—4212no—
138C23—4211no—
139C23—4211no—
140C23C22×D42112yes—
141C23C22×D42111yescharacteristic
142Q8—8422no—
143D4—22no—
144D4—42no—
145D4—8422no—
146D4—8422no—
147D4—8424no—
148D4—8422no—
149D4—8422no—
150D4—16842no—
151D4—16842no—
152D4—16842no—
153C2×C4—22no—
154C2×C4—8422no—
155C2×C4—4212no—
156C2×C4—4211no—
157C2×C4—4211no—
158C2×C4—4214no—
159C2×C4C22×D42112yes—
160C2×C4C22×D42111yescharacteristic
161C2×C4C22×D42111yescharacteristic
162C22—21no—
163C22—22no—
164C22—21no—
165C22—22no—
166C22—21no—
167C22—22no—
168C22—8421no—
169C22—8421no—
170C22—8421no—
171C22—8422no—
172C22—8421no—
173C22—8422no—
174C22—8421no—
175C22—8422no—
176C22—16848no—
177C22—4212no—
178C22C23×D42111yescharacteristic
179C4—21no—
180C4—22no—
181C4—8421no—
182C4—8422no—
183C4—4212no—
184C4D4×D42112yes—
185C2—42no—
186C2—41no—
187C2—8422no—
188C2—16842no—
189C2—16841no—
190C2C2×D4×D42111yescharacteristic
191C2#2721241111yescharacteristic
192C2#2721241211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D421
commutator, socle, cosocleC22×C421C24
Fitting#121×C421C2
hypercenter#121D421
hypocenterC421#535
LCSG ▷ C22×C421 ▷ C421
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#20269771C2semidirect
2. C2×D4×D421C2semidirect
3. #27212362C2semidirect
4. #27212352C2semidirect
5. #27212341C2semidirect
6. #27212340C2semidirect
7. #27212337C2semidirect
8. #27212336C2semidirect
9. #27212334C2semidirect
10. #27212324C2semidirect
11. #121×C421C2semidirect
14. C2×Dic842C22semidirect
17. C2×D1684C22semidirect
20. C2×C22⋊D421C22semidirect
21. C2×C22⋊D421C22semidirect
22. C2×C22⋊D421C22semidirect
25. C2×C4×D421C22semidirect
26. C4⋊Dic421C22semidirect
27. C22⋊Dic421C22semidirect
29. #20269748C22semidirect
30. #20269748C22semidirect
31. C4×Dic421C22semidirect
33. C2×D4×C421C22semidirect
34. C22×C4×C421C22semidirect
35. C4⋊C4×C421C22semidirect
36. C22⋊C4×C421C22semidirect
56. C4×D421D4semidirect
96. C421#535semidirect
105. #121D421semidirect
122. C2×D4C2×D421semidirect
125. C22×C4C2×D421semidirect
127. C4⋊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
C2×#202697712
C2×D4×D4212
#272123622
#272123522
#272123412
#272123402
#272123372
#272123362
#272123342
#272123242
#121×C4212
#535421