Finite Group Databasebeta

← #27135772

Subgroups — #27135772

Order 12352.

330 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 109 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×#15690556C211yesmaximal, characteristic, complemented
2C22×C4×D193C211yesmaximal, characteristic, complemented
3#20259884C212yesmaximal, complemented
4#20259875C211yesmaximal, characteristic, complemented
5#20259874C211yesmaximal, characteristic, complemented
6#20259873C211yesmaximal, characteristic, complemented
7#20259869C212yesmaximal, complemented
8#20259861C212yesmaximal, complemented
9#20259859C212yesmaximal, complemented
10#20259857C211yesmaximal, characteristic, complemented
11C22⋊Q8×C193C211yesmaximal, characteristic, complemented
12C23×D193C2211yescharacteristic
13#15690556—24no—
14C2×D772C2212yes—
15C2×D772C2212yes—
16C2×D772C2211yescharacteristic, complemented
17C2×C22⋊D193C2212yes—
18C22×Dic193C2211yescharacteristic
19C2×C4×D193—24no—
20C2×C4×D193C2211yescharacteristic
21C2×C4×D193C2211yescharacteristic
22C2×C4×D193C2211yescharacteristic
23C2×C4×D193C2211yescharacteristic
24C2×C4×D193C2212yescomplemented
25C4⋊Dic193C2211yescharacteristic, complemented
26#15690534C2212yescomplemented
27#15690532C2212yes—
28#15690532C2212yes—
29#15690532C2212yescomplemented
30#15690532C2212yescomplemented
31C4×Dic193C2212yescomplemented
32C2×Q8×C193C2211yescharacteristic, complemented
33C22×C4×C193C2211yescharacteristic
34C4⋊C4×C193C2211yescharacteristic, complemented
35C4⋊C4×C193C2212yescomplemented
36C22⋊C4×C193C2212yes—
37C22×D193—22no—
38C22×D193—22no—
39C22×D193—22no—
40C22×D193C2312yes—
41C22×D193C2311yescharacteristic
42C22×D193C2311yescharacteristic
43D772—24no—
44D772—24no—
45D772—22no—
46C22⋊D193—24no—
47C2×Dic193—22no—
48C2×Dic193C2312yes—
49C2×Dic193C2311yescharacteristic
50C2×Dic193C2311yescharacteristic
51C4×D193—22no—
52C4×D193—22no—
53C4×D193—22no—
54C4×D193—24no—
55C4×D193D414yescomplemented
56C23×C193C2311yescharacteristic
57Q8×C193—22no—
58C2×C4×C193—22no—
59C2×C4×C193C2312yes—
60C2×C4×C193C2312yes—
61C2×C4×C193C2311yescharacteristic
62C2×C4×C193C2311yescharacteristic
63C22×C193—21no—
64C22×C193—21no—
65C22×C193#4112yescomplemented
66C22×C193C2411yescharacteristic
67C2×D193—28no—
68C2×D193—21no—
69C2×D193—21no—
70C2×D193—21no—
71C2×D193—21no—
72C2×D193—22no—
73C2×D193—22no—
74C2×D193—22no—
75C2×D193C2×D412yes—
76C2×D193#4112yescomplemented
77C4×C193—21no—
78C4×C193—22no—
79C4×C193—22no—
80C4×C193C2×D412yes—
81Dic193—21no—
82Dic193—22no—
83Dic193C2×D412yes—
84C2×C193—22no—
85C2×C193C22×D411yescharacteristic
86C2×C193C2×#4111yescharacteristic
87C2×C193C2×#4111yescharacteristic
88D193—22no—
89D193—22no—
90D193—42no—
91C193#52511yescharacteristic, complemented, Sylow
92#525—1931nomaximal, Sylow
93C2×#41—1931no—
94C23×C4—1931no—
95#123—1932no—
96C22⋊Q8—1931no—
97C22⋊Q8D19311yescharacteristic, complemented
98#121—1931no—
99#121—1931no—
100#120—1932no—
101C4×D4—1932no—
102C4×D4—1932no—
103#117—1931no—
104C24—1931no—
105#41—3864no—
106C2×Q8C2×D19311yescharacteristic, complemented
107C2×D4—1932no—
108C2×D4—1932no—
109C2×D4—1932no—
110C2×D4—1931no—
111C22×C4—3864no—
112C22×C4—1931no—
113C22×C4—1931no—
114C22×C4—1931no—
115C22×C4—1931no—
116C22×C4—1932no—
117C22×C4—1931no—
118C22×C4C2×D19311yescharacteristic
119C4⋊C4—1932no—
120C4⋊C4—1931no—
121C4⋊C4C2×D19311yescharacteristic, complemented
122C4⋊C4C2×D19312yescomplemented
123C22⋊C4—1932no—
124C22⋊C4—1932no—
125C22⋊C4—1932no—
126C22⋊C4—1932no—
127C22⋊C4C2×D19312yes—
128C42—1932no—
129C23—3862no—
130C23—3862no—
131C23—3862no—
132C23—1932no—
133C23—1931no—
134C23—1931no—
135C23C22×D19311yescharacteristic
136Q8—22no—
137D4—3864no—
138D4—3864no—
139D4—3864no—
140D4—3862no—
141C2×C4—22no—
142C2×C4—3862no—
143C2×C4—3862no—
144C2×C4—3864no—
145C2×C4—3862no—
146C2×C4—3862no—
147C2×C4—1932no—
148C2×C4—1931no—
149C2×C4—1931no—
150C2×C4—1934no—
151C2×C4C22×D19312yes—
152C2×C4C22×D19311yescharacteristic
153C2×C4C22×D19312yes—
154C2×C4C22×D19311yescharacteristic
155C22—21no—
156C22—21no—
157C22—3868no—
158C22—3862no—
159C22—3862no—
160C22—3862no—
161C22—3861no—
162C22—3861no—
163C22—3861no—
164C22—3861no—
165C22—1932no—
166C22—1932no—
167C22#1569055612yescomplemented
168C22C23×D19311yescharacteristic
169C4—21no—
170C4—22no—
171C4—22no—
172C4—3861no—
173C4—3862no—
174C4—1932no—
175C4D4×D19312yes—
176C2—22no—
177C2—7722no—
178C2—3862no—
179C2—3862no—
180C2C2×D4×D19311yescharacteristic
181C2#41×D19311yescharacteristic
182C2C2×#1569055611yescharacteristic
subgroupG/H
center, FrattiniC22C23×D193
commutator, socle, cosocleC22×C193C24
FittingC22⋊Q8×C193C2
hypercenterC22⋊Q8D193
hypocenterC193#525
LCSG ▷ C22×C193 ▷ C193
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#15690556C2semidirect
2. C22×C4×D193C2semidirect
3. #20259884C2semidirect
4. #20259875C2semidirect
5. #20259874C2semidirect
6. #20259873C2semidirect
7. #20259869C2semidirect
8. #20259861C2semidirect
9. #20259859C2semidirect
10. #20259857C2semidirect
11. C22⋊Q8×C193C2semidirect
16. C2×D772C22semidirect
24. C2×C4×D193C22semidirect
25. C4⋊Dic193C22semidirect
26. #15690534C22semidirect
29. #15690532C22semidirect
30. #15690532C22semidirect
31. C4×Dic193C22semidirect
32. C2×Q8×C193C22semidirect
34. C4⋊C4×C193C22semidirect
35. C4⋊C4×C193C22semidirect
55. C4×D193D4semidirect
65. C22×C193#41semidirect
76. C2×D193#41semidirect
91. C193#525semidirect
97. C22⋊Q8D193semidirect
106. C2×Q8C2×D193semidirect
121. C4⋊C4C2×D193semidirect
122. C4⋊C4C2×D193semidirect
167. C22#15690556semidirect

Maximal subgroups

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

subgroupindex
C2×#156905562
C22×C4×D1932
#202598842
#202598752
#202598742
#202598732
#202598692
#202598612
#202598592
#202598572
C22⋊Q8×C1932
#525193