Finite Group Databasebeta

← #42122321

Subgroups — #42122321

Order 27712.

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×#20294096C211yesmaximal, characteristic, complemented
2C22×C4×D433C211yesmaximal, characteristic, complemented
3#28357202C212yesmaximal, complemented
4#28357200C211yesmaximal, characteristic, complemented
5#28357199C211yesmaximal, characteristic, complemented
6#28357191C211yesmaximal, characteristic, complemented
7#28357188C212yesmaximal, complemented
8D4×Dic433C212yesmaximal, complemented
9#28357180C212yesmaximal, complemented
10#28357178C211yesmaximal, characteristic, complemented
11#121×C433C211yesmaximal, characteristic, complemented
12C23×D433C2211yescharacteristic
13#20294096—24no—
14C2×Dic866C2211yescharacteristic, complemented
15C2×C22⋊D433C2212yes—
16C2×C22⋊D433C2212yes—
17C22×Dic433C2211yescharacteristic
18C22×Dic433C2212yescomplemented
19C2×C4×D433—24no—
20C2×C4×D433C2211yescharacteristic
21C2×C4×D433C2211yescharacteristic
22C2×C4×D433C2211yescharacteristic
23C2×C4×D433C2211yescharacteristic
24C4⋊Dic433C2211yescharacteristic, complemented
25C4⋊Dic433C2212yescomplemented
26#20294076C2212yescomplemented
27C22⋊Dic433C2212yes—
28C22⋊Dic433C2212yescomplemented
29#20294074C2212yes—
30#20294074C2212yes—
31C4×Dic433C2212yescomplemented
32C2×D4×C433C2212yes—
33C2×D4×C433C2211yescharacteristic, complemented
34C22×C4×C433C2211yescharacteristic
35C4⋊C4×C433C2211yescharacteristic, complemented
36C22⋊C4×C433C2212yescomplemented
37C22×D433—22no—
38C22×D433—22no—
39C22×D433—22no—
40C22×D433C2311yescharacteristic
41C22×D433C2311yescharacteristic
42Dic866—22no—
43C22⋊D433—24no—
44C22⋊D433—24no—
45C2×Dic433—22no—
46C2×Dic433—24no—
47C2×Dic433C2312yes—
48C2×Dic433C2312yes—
49C2×Dic433C2311yescharacteristic
50C2×Dic433C2311yescharacteristic
51C4×D433—22no—
52C4×D433—22no—
53C4×D433—22no—
54C4×D433D414yescomplemented
55C23×C433C2312yes—
56C23×C433C2311yescharacteristic
57D4×C433—24no—
58D4×C433—22no—
59C2×C4×C433—22no—
60C2×C4×C433C2311yescharacteristic
61C2×C4×C433C2312yes—
62C2×C4×C433C2311yescharacteristic
63C22×C433—21no—
64C22×C433—21no—
65C22×C433—22no—
66C22×C433—22no—
67C22×C433—22no—
68C22×C433#4112yescomplemented
69C22×C433C2411yescharacteristic
70C2×D433—28no—
71C2×D433—21no—
72C2×D433—21no—
73C2×D433—21no—
74C2×D433—21no—
75C2×D433C2×D412yes—
76C2×D433#4112yescomplemented
77C4×C433—21no—
78C4×C433—22no—
79C4×C433C2×D412yes—
80Dic433—21no—
81Dic433—22no—
82Dic433—22no—
83Dic433C2×D412yes—
84C2×C433—22no—
85C2×C433—42no—
86C2×C433C22×D411yescharacteristic
87C2×C433C2×#4111yescharacteristic
88C2×C433C2×#4111yescharacteristic
89D433—22no—
90D433—22no—
91C433#52511yescharacteristic, complemented, Sylow
92#525—4331nomaximal, Sylow
93C2×#41—4331no—
94C23×C4—4331no—
95#123—4332no—
96C22⋊Q8—4331no—
97C22⋊Q8—4331no—
98#121—4331no—
99#121D43311yescharacteristic, complemented
100#120—4332no—
101C4×D4—4332no—
102C4×D4—4332no—
103#117—4331no—
104C24—4331no—
105#41—8664no—
106C2×Q8—4331no—
107C2×D4—4332no—
108C2×D4—4332no—
109C2×D4C2×D43312yes—
110C2×D4C2×D43311yescharacteristic, complemented
111C22×C4—8664no—
112C22×C4—4331no—
113C22×C4—4332no—
114C22×C4—4331no—
115C22×C4—4331no—
116C22×C4—4331no—
117C22×C4—4331no—
118C22×C4C2×D43311yescharacteristic
119C4⋊C4—4331no—
120C4⋊C4—4332no—
121C4⋊C4—4332no—
122C4⋊C4C2×D43311yescharacteristic, complemented
123C22⋊C4—4332no—
124C22⋊C4—4332no—
125C22⋊C4—4332no—
126C22⋊C4—4332no—
127C22⋊C4C2×D43312yescomplemented
128C42—4332no—
129C23—8662no—
130C23—8662no—
131C23—8662no—
132C23—4331no—
133C23—4331no—
134C23C22×D43312yes—
135C23C22×D43311yescharacteristic
136Q8—8662no—
137D4—24no—
138D4—22no—
139D4—8664no—
140D4—8664no—
141C2×C4—22no—
142C2×C4—8662no—
143C2×C4—8662no—
144C2×C4—8662no—
145C2×C4—8664no—
146C2×C4—8662no—
147C2×C4—4332no—
148C2×C4—4331no—
149C2×C4—4332no—
150C2×C4—4331no—
151C2×C4—4334no—
152C2×C4C22×D43312yes—
153C2×C4C22×D43311yescharacteristic
154C2×C4C22×D43311yescharacteristic
155C22—21no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—8668no—
161C22—8661no—
162C22—8661no—
163C22—8661no—
164C22—8661no—
165C22—4332no—
166C22—4332no—
167C22#2029409612yescomplemented
168C22C23×D43311yescharacteristic
169C4—21no—
170C4—22no—
171C4—8661no—
172C4—8662no—
173C4—8662no—
174C4—4332no—
175C4D4×D43312yes—
176C2—22no—
177C2—42no—
178C2—8662no—
179C2—8662no—
180C2C2×D4×D43311yescharacteristic
181C2#41×D43311yescharacteristic
182C2C2×#2029409611yescharacteristic
subgroupG/H
center, FrattiniC22C23×D433
commutator, socle, cosocleC22×C433C24
Fitting#121×C433C2
hypercenter#121D433
hypocenterC433#525
LCSG ▷ C22×C433 ▷ C433
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#20294096C2semidirect
2. C22×C4×D433C2semidirect
3. #28357202C2semidirect
4. #28357200C2semidirect
5. #28357199C2semidirect
6. #28357191C2semidirect
7. #28357188C2semidirect
8. D4×Dic433C2semidirect
9. #28357180C2semidirect
10. #28357178C2semidirect
11. #121×C433C2semidirect
14. C2×Dic866C22semidirect
18. C22×Dic433C22semidirect
24. C4⋊Dic433C22semidirect
25. C4⋊Dic433C22semidirect
26. #20294076C22semidirect
28. C22⋊Dic433C22semidirect
31. C4×Dic433C22semidirect
33. C2×D4×C433C22semidirect
35. C4⋊C4×C433C22semidirect
36. C22⋊C4×C433C22semidirect
54. C4×D433D4semidirect
68. C22×C433#41semidirect
76. C2×D433#41semidirect
91. C433#525semidirect
99. #121D433semidirect
110. C2×D4C2×D433semidirect
122. C4⋊C4C2×D433semidirect
127. C22⋊C4C2×D433semidirect
167. C22#20294096semidirect

Maximal subgroups

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

subgroupindex
C2×#202940962
C22×C4×D4332
#283572022
#283572002
#283571992
#283571912
#283571882
D4×Dic4332
#283571802
#283571782
#121×C4332
#525433