Finite Group Databasebeta

← #43272517

Subgroups — #43272517

Order 28736.

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×#20297735C211yesmaximal, characteristic, complemented
2C22×C4×D449C211yesmaximal, characteristic, complemented
3#28385322C212yesmaximal, complemented
4#28385320C211yesmaximal, characteristic, complemented
5#28385319C211yesmaximal, characteristic, complemented
6#28385311C211yesmaximal, characteristic, complemented
7#28385308C212yesmaximal, complemented
8D4×Dic449C212yesmaximal, complemented
9#28385300C212yesmaximal, complemented
10#28385298C211yesmaximal, characteristic, complemented
11#121×C449C211yesmaximal, characteristic, complemented
12C23×D449C2211yescharacteristic
13#20297735—24no—
14C2×Dic898C2211yescharacteristic, complemented
15C2×C22⋊D449C2212yes—
16C2×C22⋊D449C2212yes—
17C22×Dic449C2211yescharacteristic
18C22×Dic449C2212yescomplemented
19C2×C4×D449—24no—
20C2×C4×D449C2211yescharacteristic
21C2×C4×D449C2211yescharacteristic
22C2×C4×D449C2211yescharacteristic
23C2×C4×D449C2211yescharacteristic
24C4⋊Dic449C2211yescharacteristic, complemented
25C4⋊Dic449C2212yescomplemented
26#20297715C2212yescomplemented
27C22⋊Dic449C2212yes—
28C22⋊Dic449C2212yescomplemented
29#20297713C2212yes—
30#20297713C2212yes—
31C4×Dic449C2212yescomplemented
32C2×D4×C449C2212yes—
33C2×D4×C449C2211yescharacteristic, complemented
34C22×C4×C449C2211yescharacteristic
35C4⋊C4×C449C2211yescharacteristic, complemented
36C22⋊C4×C449C2212yescomplemented
37C22×D449—22no—
38C22×D449—22no—
39C22×D449—22no—
40C22×D449C2311yescharacteristic
41C22×D449C2311yescharacteristic
42Dic898—22no—
43C22⋊D449—24no—
44C22⋊D449—24no—
45C2×Dic449—22no—
46C2×Dic449—24no—
47C2×Dic449C2311yescharacteristic
48C2×Dic449C2312yes—
49C2×Dic449C2312yes—
50C2×Dic449C2311yescharacteristic
51C4×D449—22no—
52C4×D449—22no—
53C4×D449—22no—
54C4×D449D414yescomplemented
55C23×C449C2312yes—
56C23×C449C2311yescharacteristic
57D4×C449—24no—
58D4×C449—22no—
59C2×C4×C449—22no—
60C2×C4×C449C2311yescharacteristic
61C2×C4×C449C2312yes—
62C2×C4×C449C2311yescharacteristic
63C22×C449—21no—
64C22×C449—21no—
65C22×C449—22no—
66C22×C449—22no—
67C22×C449—22no—
68C22×C449#4112yescomplemented
69C22×C449C2411yescharacteristic
70C2×D449—28no—
71C2×D449—21no—
72C2×D449—21no—
73C2×D449—21no—
74C2×D449—21no—
75C2×D449C2×D412yes—
76C2×D449#4112yescomplemented
77C4×C449—21no—
78C4×C449—22no—
79C4×C449C2×D412yes—
80Dic449—21no—
81Dic449—22no—
82Dic449—22no—
83Dic449C2×D412yes—
84C2×C449—22no—
85C2×C449—42no—
86C2×C449C22×D411yescharacteristic
87C2×C449C2×#4111yescharacteristic
88C2×C449C2×#4111yescharacteristic
89D449—22no—
90D449—22no—
91C449#52511yescharacteristic, complemented, Sylow
92#525—4491nomaximal, Sylow
93C2×#41—4491no—
94C23×C4—4491no—
95#123—4492no—
96C22⋊Q8—4491no—
97C22⋊Q8—4491no—
98#121—4491no—
99#121D44911yescharacteristic, complemented
100#120—4492no—
101C4×D4—4492no—
102C4×D4—4492no—
103#117—4491no—
104C24—4491no—
105#41—8984no—
106C2×Q8—4491no—
107C2×D4—4492no—
108C2×D4—4492no—
109C2×D4C2×D44912yes—
110C2×D4C2×D44911yescharacteristic, complemented
111C22×C4—8984no—
112C22×C4—4491no—
113C22×C4—4492no—
114C22×C4—4491no—
115C22×C4—4491no—
116C22×C4—4491no—
117C22×C4—4491no—
118C22×C4C2×D44911yescharacteristic
119C4⋊C4—4491no—
120C4⋊C4—4492no—
121C4⋊C4—4492no—
122C4⋊C4C2×D44911yescharacteristic, complemented
123C22⋊C4—4492no—
124C22⋊C4—4492no—
125C22⋊C4—4492no—
126C22⋊C4—4492no—
127C22⋊C4C2×D44912yescomplemented
128C42—4492no—
129C23—8982no—
130C23—8982no—
131C23—8982no—
132C23—4491no—
133C23—4491no—
134C23C22×D44912yes—
135C23C22×D44911yescharacteristic
136Q8—8982no—
137D4—24no—
138D4—22no—
139D4—8984no—
140D4—8984no—
141C2×C4—22no—
142C2×C4—8982no—
143C2×C4—8982no—
144C2×C4—8982no—
145C2×C4—8984no—
146C2×C4—8982no—
147C2×C4—4492no—
148C2×C4—4491no—
149C2×C4—4492no—
150C2×C4—4491no—
151C2×C4—4494no—
152C2×C4C22×D44912yes—
153C2×C4C22×D44911yescharacteristic
154C2×C4C22×D44911yescharacteristic
155C22—21no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—8988no—
161C22—8981no—
162C22—8981no—
163C22—8981no—
164C22—8981no—
165C22—4492no—
166C22—4492no—
167C22#2029773512yescomplemented
168C22C23×D44911yescharacteristic
169C4—21no—
170C4—22no—
171C4—8981no—
172C4—8982no—
173C4—8982no—
174C4—4492no—
175C4D4×D44912yes—
176C2—22no—
177C2—42no—
178C2—8982no—
179C2—8982no—
180C2C2×D4×D44911yescharacteristic
181C2#41×D44911yescharacteristic
182C2C2×#2029773511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D449
commutator, socle, cosocleC22×C449C24
Fitting#121×C449C2
hypercenter#121D449
hypocenterC449#525
LCSG ▷ C22×C449 ▷ C449
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#20297735C2semidirect
2. C22×C4×D449C2semidirect
3. #28385322C2semidirect
4. #28385320C2semidirect
5. #28385319C2semidirect
6. #28385311C2semidirect
7. #28385308C2semidirect
8. D4×Dic449C2semidirect
9. #28385300C2semidirect
10. #28385298C2semidirect
11. #121×C449C2semidirect
14. C2×Dic898C22semidirect
18. C22×Dic449C22semidirect
24. C4⋊Dic449C22semidirect
25. C4⋊Dic449C22semidirect
26. #20297715C22semidirect
28. C22⋊Dic449C22semidirect
31. C4×Dic449C22semidirect
33. C2×D4×C449C22semidirect
35. C4⋊C4×C449C22semidirect
36. C22⋊C4×C449C22semidirect
54. C4×D449D4semidirect
68. C22×C449#41semidirect
76. C2×D449#41semidirect
91. C449#525semidirect
99. #121D449semidirect
110. C2×D4C2×D449semidirect
122. C4⋊C4C2×D449semidirect
127. C22⋊C4C2×D449semidirect
167. C22#20297735semidirect

Maximal subgroups

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

subgroupindex
C2×#202977352
C22×C4×D4492
#283853222
#283853202
#283853192
#283853112
#283853082
D4×Dic4492
#283853002
#283852982
#121×C4492
#525449