Finite Group Databasebeta

← #36341818

Subgroups — #36341818

Order 21568.

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×#19129186C211yesmaximal, characteristic, complemented
2C22×C4×D337C211yesmaximal, characteristic, complemented
3#24924944C212yesmaximal, complemented
4#24924935C211yesmaximal, characteristic, complemented
5#24924934C211yesmaximal, characteristic, complemented
6#24924933C211yesmaximal, characteristic, complemented
7#24924929C212yesmaximal, complemented
8#24924921C212yesmaximal, complemented
9#24924919C212yesmaximal, complemented
10#24924917C211yesmaximal, characteristic, complemented
11C22⋊Q8×C337C211yesmaximal, characteristic, complemented
12C23×D337C2211yescharacteristic
13#19129186—24no—
14C2×D1348C2212yes—
15C2×D1348C2212yes—
16C2×D1348C2211yescharacteristic, complemented
17C2×C22⋊D337C2212yes—
18C22×Dic337C2211yescharacteristic
19C2×C4×D337—24no—
20C2×C4×D337C2211yescharacteristic
21C2×C4×D337C2211yescharacteristic
22C2×C4×D337C2211yescharacteristic
23C2×C4×D337C2211yescharacteristic
24C2×C4×D337C2212yescomplemented
25C4⋊Dic337C2211yescharacteristic, complemented
26#19129164C2212yescomplemented
27#19129162C2212yes—
28#19129162C2212yes—
29#19129162C2212yescomplemented
30#19129162C2212yescomplemented
31C4×Dic337C2212yescomplemented
32C2×Q8×C337C2211yescharacteristic, complemented
33C22×C4×C337C2211yescharacteristic
34C4⋊C4×C337C2211yescharacteristic, complemented
35C4⋊C4×C337C2212yescomplemented
36C22⋊C4×C337C2212yes—
37C22×D337—22no—
38C22×D337—22no—
39C22×D337—22no—
40C22×D337C2312yes—
41C22×D337C2311yescharacteristic
42C22×D337C2311yescharacteristic
43D1348—24no—
44D1348—24no—
45D1348—22no—
46C22⋊D337—24no—
47C2×Dic337—22no—
48C2×Dic337C2312yes—
49C2×Dic337C2311yescharacteristic
50C2×Dic337C2311yescharacteristic
51C4×D337—22no—
52C4×D337—22no—
53C4×D337—22no—
54C4×D337—24no—
55C4×D337D414yescomplemented
56C23×C337C2311yescharacteristic
57Q8×C337—22no—
58C2×C4×C337—22no—
59C2×C4×C337C2312yes—
60C2×C4×C337C2312yes—
61C2×C4×C337C2311yescharacteristic
62C2×C4×C337C2311yescharacteristic
63C22×C337—21no—
64C22×C337—21no—
65C22×C337#4112yescomplemented
66C22×C337C2411yescharacteristic
67C2×D337—28no—
68C2×D337—21no—
69C2×D337—21no—
70C2×D337—21no—
71C2×D337—21no—
72C2×D337—22no—
73C2×D337—22no—
74C2×D337—22no—
75C2×D337C2×D412yes—
76C2×D337#4112yescomplemented
77C4×C337—21no—
78C4×C337—22no—
79C4×C337—22no—
80C4×C337C2×D412yes—
81Dic337—21no—
82Dic337—22no—
83Dic337C2×D412yes—
84C2×C337—22no—
85C2×C337C22×D411yescharacteristic
86C2×C337C2×#4111yescharacteristic
87C2×C337C2×#4111yescharacteristic
88D337—22no—
89D337—22no—
90D337—42no—
91C337#52511yescharacteristic, complemented, Sylow
92#525—3371nomaximal, Sylow
93C2×#41—3371no—
94C23×C4—3371no—
95#123—3372no—
96C22⋊Q8—3371no—
97C22⋊Q8D33711yescharacteristic, complemented
98#121—3371no—
99#121—3371no—
100#120—3372no—
101C4×D4—3372no—
102C4×D4—3372no—
103#117—3371no—
104C24—3371no—
105#41—6744no—
106C2×Q8C2×D33711yescharacteristic, complemented
107C2×D4—3372no—
108C2×D4—3372no—
109C2×D4—3372no—
110C2×D4—3371no—
111C22×C4—3371no—
112C22×C4—3371no—
113C22×C4—3371no—
114C22×C4—3371no—
115C22×C4—3371no—
116C22×C4—3372no—
117C22×C4—6744no—
118C22×C4C2×D33711yescharacteristic
119C4⋊C4—3372no—
120C4⋊C4—3371no—
121C4⋊C4C2×D33711yescharacteristic, complemented
122C4⋊C4C2×D33712yescomplemented
123C22⋊C4—3372no—
124C22⋊C4—3372no—
125C22⋊C4—3372no—
126C22⋊C4—3372no—
127C22⋊C4C2×D33712yes—
128C42—3372no—
129C23—3371no—
130C23—3371no—
131C23—3372no—
132C23—6742no—
133C23—6742no—
134C23—6742no—
135C23C22×D33711yescharacteristic
136Q8—22no—
137D4—6744no—
138D4—6744no—
139D4—6744no—
140D4—6742no—
141C2×C4—22no—
142C2×C4—3371no—
143C2×C4—3374no—
144C2×C4—3372no—
145C2×C4—3371no—
146C2×C4—6742no—
147C2×C4—6744no—
148C2×C4—6742no—
149C2×C4—6742no—
150C2×C4—6742no—
151C2×C4C22×D33712yes—
152C2×C4C22×D33711yescharacteristic
153C2×C4C22×D33712yes—
154C2×C4C22×D33711yescharacteristic
155C22—21no—
156C22—21no—
157C22—3372no—
158C22—3372no—
159C22—6748no—
160C22—6742no—
161C22—6742no—
162C22—6742no—
163C22—6741no—
164C22—6741no—
165C22—6741no—
166C22—6741no—
167C22#1912918612yescomplemented
168C22C23×D33711yescharacteristic
169C4—21no—
170C4—22no—
171C4—22no—
172C4—3372no—
173C4—6741no—
174C4—6742no—
175C4D4×D33712yes—
176C2—22no—
177C2—13482no—
178C2—6742no—
179C2—6742no—
180C2C2×D4×D33711yescharacteristic
181C2#41×D33711yescharacteristic
182C2C2×#1912918611yescharacteristic
subgroupG/H
center, FrattiniC22C23×D337
commutator, socle, cosocleC22×C337C24
FittingC22⋊Q8×C337C2
hypercenterC22⋊Q8D337
hypocenterC337#525
LCSG ▷ C22×C337 ▷ C337
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#19129186C2semidirect
2. C22×C4×D337C2semidirect
3. #24924944C2semidirect
4. #24924935C2semidirect
5. #24924934C2semidirect
6. #24924933C2semidirect
7. #24924929C2semidirect
8. #24924921C2semidirect
9. #24924919C2semidirect
10. #24924917C2semidirect
11. C22⋊Q8×C337C2semidirect
16. C2×D1348C22semidirect
24. C2×C4×D337C22semidirect
25. C4⋊Dic337C22semidirect
26. #19129164C22semidirect
29. #19129162C22semidirect
30. #19129162C22semidirect
31. C4×Dic337C22semidirect
32. C2×Q8×C337C22semidirect
34. C4⋊C4×C337C22semidirect
35. C4⋊C4×C337C22semidirect
55. C4×D337D4semidirect
65. C22×C337#41semidirect
76. C2×D337#41semidirect
91. C337#525semidirect
97. C22⋊Q8D337semidirect
106. C2×Q8C2×D337semidirect
121. C4⋊C4C2×D337semidirect
122. C4⋊C4C2×D337semidirect
167. C22#19129186semidirect

Maximal subgroups

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

subgroupindex
C2×#191291862
C22×C4×D3372
#249249442
#249249352
#249249342
#249249332
#249249292
#249249212
#249249192
#249249172
C22⋊Q8×C3372
#525337