Finite Group Databasebeta

← #27133308

Subgroups — #27133308

Order 12224.

334 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 113 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×#15690286C211yesmaximal, characteristic, complemented
2C2×D4×D191C211yesmaximal, characteristic, complemented
3#20259310C211yesmaximal, characteristic, complemented
4#20259306C212yesmaximal, complemented
5#20259302C211yesmaximal, characteristic, complemented
6#20259297C211yesmaximal, characteristic, complemented
7#20259296C212yesmaximal, complemented
8#20259292C212yesmaximal, complemented
9C4×D764C211yesmaximal, characteristic, complemented
10C2×(C2×D191)⋊C4C212yesmaximal, complemented
11C4×D4×C191C211yesmaximal, characteristic, complemented
12C23×D191C2212yes—
13#15690286—24no—
14C2×Dic382C2211yescharacteristic, complemented
15C2×D764C2211yescharacteristic
16C2×D764C2211yescharacteristic
17C2×D764C2212yescomplemented
18C2×C22⋊D191C2212yes—
19C2×C22⋊D191C2212yescomplemented
20D4×D191—24no—
21C22×Dic191C2212yescomplemented
22C2×C4×D191C2211yescharacteristic
23C2×C4×D191C2211yescharacteristic, complemented
24C4⋊Dic191C2211yescharacteristic, complemented
25C4⋊Dic191C2212yescomplemented
26(C2×D191)⋊C4C2211yescharacteristic
27(C2×D191)⋊C4C2212yes—
28(C2×D191)⋊C4C2212yescomplemented
29(C2×D191)⋊C4C2212yescomplemented
30(C2×D191)⋊C4C2211yescharacteristic, complemented
31(C2×D191)⋊C4C2212yescomplemented
32C2×D4×C191C2211yescharacteristic
33C22×C4×C191C2212yescomplemented
34C4⋊C4×C191C2211yescharacteristic, complemented
35C22⋊C4×C191C2212yescomplemented
36C42×C191C2211yescharacteristic, complemented
37C22×D191—24no—
38C22×D191—24no—
39C22×D191—42no—
40C22×D191C2312yes—
41C22×D191C2311yescharacteristic
42C22×D191C2311yescharacteristic
43Dic382—22no—
44D764—22no—
45D764—22no—
46D764—42no—
47C22⋊D191—24no—
48C22⋊D191—24no—
49C2×Dic191—24no—
50C2×Dic191C2311yescharacteristic
51C2×Dic191C2312yes—
52C2×Dic191C2311yescharacteristic
53C4×D191—22no—
54C4×D191—22no—
55C23×C191C2312yes—
56D4×C191D414yescomplemented
57C2×C4×C191—42no—
58C2×C4×C191C2311yescharacteristic
59C2×C4×C191C2311yescharacteristic
60C2×C4×C191C2312yes—
61C2×C4×C191C2311yescharacteristic
62C22×C191—22no—
63C22×C191—22no—
64C22×C191C2×D414yes—
65C22×C191C2411yescharacteristic
66C2×D191—21no—
67C2×D191—21no—
68C2×D191—21no—
69C2×D191—22no—
70C2×D191—21no—
71C2×D191—22no—
72C2×D191—21no—
73C2×D191—22no—
74C2×D191—48no—
75C2×D191#4112yescomplemented
76C4×C191—21no—
77C4×C191—21no—
78C4×C191—22no—
79C4×C191C2×D412yes—
80Dic191—21no—
81Dic191—21no—
82Dic191—22no—
83C2×C191—24no—
84C2×C191C22×D411yescharacteristic
85C2×C191C2×#4111yescharacteristic
86C2×C191H(5,𝔽2)11yescharacteristic
87D191—22no—
88D191—41no—
89D191—42no—
90C191#54611yescharacteristic, complemented, Sylow
91#546—1911nomaximal, Sylow
92C2×#41—1911no—
93C22×D4—1911no—
94#124—1911no—
95#123—1912no—
96C22⋊Q8—1911no—
97#121—1912no—
98#121—1911no—
99#120—1912no—
100C4×D4—1911no—
101C4×D4D19111yescharacteristic, complemented
102C2×C22⋊C4—1912no—
103C24—1912no—
104#41—3824no—
105C2×Q8—1911no—
106C2×D4—3824no—
107C2×D4—1912no—
108C2×D4—1912no—
109C2×D4—1911no—
110C2×D4—1911no—
111C2×D4—1912no—
112C2×D4C2×D19111yescharacteristic
113C22×C4—1912no—
114C22×C4—1911no—
115C22×C4—1911no—
116C22×C4C2×D19112yescomplemented
117C4⋊C4—1911no—
118C4⋊C4—1912no—
119C4⋊C4C2×D19111yescharacteristic, complemented
120C22⋊C4—1911no—
121C22⋊C4—1912no—
122C22⋊C4—1912no—
123C22⋊C4—1912no—
124C22⋊C4—1911no—
125C22⋊C4—1912no—
126C22⋊C4C2×D19112yescomplemented
127C42C2×D19111yescharacteristic, complemented
128C23—3824no—
129C23—3824no—
130C23—1912no—
131C23—1911no—
132C23—1911no—
133C23—7642no—
134C23C22×D19112yes—
135Q8—3822no—
136D4—3822no—
137D4—3824no—
138D4—3824no—
139D4—3822no—
140D4—7642no—
141D4D76414yescomplemented
142C2×C4—42no—
143C2×C4—3824no—
144C2×C4—3822no—
145C2×C4—3822no—
146C2×C4—1911no—
147C2×C4—1912no—
148C2×C4—1911no—
149C2×C4C22×D19111yescharacteristic
150C2×C4C22×D19111yescharacteristic
151C2×C4C22×D19112yes—
152C2×C4C22×D19111yescharacteristic
153C22—22no—
154C22—22no—
155C22—3821no—
156C22—3821no—
157C22—3821no—
158C22—3822no—
159C22—3821no—
160C22—3822no—
161C22—3821no—
162C22—3822no—
163C22—1912no—
164C22—7648no—
165C22C2×D76414yes—
166C22C23×D19111yescharacteristic
167C4—21no—
168C4—21no—
169C4—22no—
170C4—3821no—
171C4—3821no—
172C4—3822no—
173C4C2×D76412yes—
174C2—24no—
175C2—3822no—
176C2—7642no—
177C2—7641no—
178C2C22×D76411yescharacteristic
179C2#41×D19111yescharacteristic
180C2#2025936711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D191
commutator, socle, cosocleC22×C191C24
FittingC4×D4×C191C2
hypercenterC4×D4D191
hypocenterC191#546
LCSG ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#15690286C2semidirect
2. C2×D4×D191C2semidirect
3. #20259310C2semidirect
4. #20259306C2semidirect
5. #20259302C2semidirect
6. #20259297C2semidirect
7. #20259296C2semidirect
8. #20259292C2semidirect
9. C4×D764C2semidirect
10. C2×(C2×D191)⋊C4C2semidirect
11. C4×D4×C191C2semidirect
14. C2×Dic382C22semidirect
17. C2×D764C22semidirect
19. C2×C22⋊D191C22semidirect
21. C22×Dic191C22semidirect
23. C2×C4×D191C22semidirect
24. C4⋊Dic191C22semidirect
25. C4⋊Dic191C22semidirect
28. (C2×D191)⋊C4C22semidirect
29. (C2×D191)⋊C4C22semidirect
30. (C2×D191)⋊C4C22semidirect
31. (C2×D191)⋊C4C22semidirect
33. C22×C4×C191C22semidirect
34. C4⋊C4×C191C22semidirect
35. C22⋊C4×C191C22semidirect
36. C42×C191C22semidirect
56. D4×C191D4semidirect
75. C2×D191#41semidirect
90. C191#546semidirect
101. C4×D4D191semidirect
116. C22×C4C2×D191semidirect
119. C4⋊C4C2×D191semidirect
126. C22⋊C4C2×D191semidirect
127. C42C2×D191semidirect
141. D4D764semidirect

Maximal subgroups

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

subgroupindex
C2×#156902862
C2×D4×D1912
#202593102
#202593062
#202593022
#202592972
#202592962
#202592922
C4×D7642
C2×(C2×D191)⋊C42
C4×D4×C1912
#546191