Finite Group Databasebeta

← #43272520

Subgroups — #43272520

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×#20297737C211yesmaximal, characteristic, complemented
2C22×C4×D449C211yesmaximal, characteristic, complemented
3#28385325C212yesmaximal, complemented
4#28385316C211yesmaximal, characteristic, complemented
5#28385315C211yesmaximal, characteristic, complemented
6#28385314C211yesmaximal, characteristic, complemented
7#28385310C212yesmaximal, complemented
8#28385302C212yesmaximal, complemented
9#28385300C212yesmaximal, complemented
10#28385298C211yesmaximal, characteristic, complemented
11C22⋊Q8×C449C211yesmaximal, characteristic, complemented
12C23×D449C2211yescharacteristic
13#20297737—24no—
14C2×D1796C2212yes—
15C2×D1796C2212yes—
16C2×D1796C2211yescharacteristic, complemented
17C2×C22⋊D449C2212yes—
18C22×Dic449C2211yescharacteristic
19C2×C4×D449—24no—
20C2×C4×D449C2211yescharacteristic
21C2×C4×D449C2211yescharacteristic
22C2×C4×D449C2211yescharacteristic
23C2×C4×D449C2211yescharacteristic
24C2×C4×D449C2212yescomplemented
25C4⋊Dic449C2211yescharacteristic, complemented
26#20297715C2212yescomplemented
27#20297713C2212yes—
28#20297713C2212yes—
29#20297713C2212yescomplemented
30#20297713C2212yescomplemented
31C4×Dic449C2212yescomplemented
32C2×Q8×C449C2211yescharacteristic, complemented
33C22×C4×C449C2211yescharacteristic
34C4⋊C4×C449C2211yescharacteristic, complemented
35C4⋊C4×C449C2212yescomplemented
36C22⋊C4×C449C2212yes—
37C22×D449—22no—
38C22×D449—22no—
39C22×D449—22no—
40C22×D449C2312yes—
41C22×D449C2311yescharacteristic
42C22×D449C2311yescharacteristic
43D1796—24no—
44D1796—24no—
45D1796—22no—
46C22⋊D449—24no—
47C2×Dic449—22no—
48C2×Dic449C2311yescharacteristic
49C2×Dic449C2312yes—
50C2×Dic449C2311yescharacteristic
51C4×D449—22no—
52C4×D449—22no—
53C4×D449—22no—
54C4×D449—24no—
55C4×D449D414yescomplemented
56C23×C449C2311yescharacteristic
57Q8×C449—22no—
58C2×C4×C449—22no—
59C2×C4×C449C2311yescharacteristic
60C2×C4×C449C2312yes—
61C2×C4×C449C2312yes—
62C2×C4×C449C2311yescharacteristic
63C22×C449—21no—
64C22×C449—21no—
65C22×C449#4112yescomplemented
66C22×C449C2411yescharacteristic
67C2×D449—28no—
68C2×D449—21no—
69C2×D449—21no—
70C2×D449—21no—
71C2×D449—21no—
72C2×D449—22no—
73C2×D449—22no—
74C2×D449—22no—
75C2×D449C2×D412yes—
76C2×D449#4112yescomplemented
77C4×C449—21no—
78C4×C449—22no—
79C4×C449—22no—
80C4×C449C2×D412yes—
81Dic449—21no—
82Dic449—22no—
83Dic449C2×D412yes—
84C2×C449—22no—
85C2×C449C22×D411yescharacteristic
86C2×C449C2×#4111yescharacteristic
87C2×C449C2×#4111yescharacteristic
88D449—22no—
89D449—22no—
90D449—42no—
91C449#52511yescharacteristic, complemented, Sylow
92#525—4491nomaximal, Sylow
93C2×#41—4491no—
94C23×C4—4491no—
95#123—4492no—
96C22⋊Q8—4491no—
97C22⋊Q8D44911yescharacteristic, complemented
98#121—4491no—
99#121—4491no—
100#120—4492no—
101C4×D4—4492no—
102C4×D4—4492no—
103#117—4491no—
104C24—4491no—
105#41—8984no—
106C2×Q8C2×D44911yescharacteristic, complemented
107C2×D4—4492no—
108C2×D4—4492no—
109C2×D4—4492no—
110C2×D4—4491no—
111C22×C4—8984no—
112C22×C4—4491no—
113C22×C4—4491no—
114C22×C4—4491no—
115C22×C4—4491no—
116C22×C4—4492no—
117C22×C4—4491no—
118C22×C4C2×D44911yescharacteristic
119C4⋊C4—4491no—
120C4⋊C4—4492no—
121C4⋊C4C2×D44911yescharacteristic, complemented
122C4⋊C4C2×D44912yescomplemented
123C22⋊C4—4492no—
124C22⋊C4—4492no—
125C22⋊C4—4492no—
126C22⋊C4—4492no—
127C22⋊C4C2×D44912yes—
128C42—4492no—
129C23—8982no—
130C23—8982no—
131C23—8982no—
132C23—4492no—
133C23—4491no—
134C23—4491no—
135C23C22×D44911yescharacteristic
136Q8—22no—
137D4—8984no—
138D4—8984no—
139D4—8984no—
140D4—8982no—
141C2×C4—22no—
142C2×C4—8982no—
143C2×C4—8982no—
144C2×C4—8984no—
145C2×C4—8982no—
146C2×C4—8982no—
147C2×C4—4492no—
148C2×C4—4491no—
149C2×C4—4491no—
150C2×C4—4494no—
151C2×C4C22×D44912yes—
152C2×C4C22×D44911yescharacteristic
153C2×C4C22×D44912yes—
154C2×C4C22×D44911yescharacteristic
155C22—21no—
156C22—21no—
157C22—8988no—
158C22—8982no—
159C22—8982no—
160C22—8982no—
161C22—8981no—
162C22—8981no—
163C22—8981no—
164C22—8981no—
165C22—4492no—
166C22—4492no—
167C22#2029773712yescomplemented
168C22C23×D44911yescharacteristic
169C4—21no—
170C4—22no—
171C4—22no—
172C4—8981no—
173C4—8982no—
174C4—4492no—
175C4D4×D44912yes—
176C2—22no—
177C2—17962no—
178C2—8982no—
179C2—8982no—
180C2C2×D4×D44911yescharacteristic
181C2#41×D44911yescharacteristic
182C2C2×#2029773711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D449
commutator, socle, cosocleC22×C449C24
FittingC22⋊Q8×C449C2
hypercenterC22⋊Q8D449
hypocenterC449#525
LCSG ▷ C22×C449 ▷ C449
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#20297737C2semidirect
2. C22×C4×D449C2semidirect
3. #28385325C2semidirect
4. #28385316C2semidirect
5. #28385315C2semidirect
6. #28385314C2semidirect
7. #28385310C2semidirect
8. #28385302C2semidirect
9. #28385300C2semidirect
10. #28385298C2semidirect
11. C22⋊Q8×C449C2semidirect
16. C2×D1796C22semidirect
24. C2×C4×D449C22semidirect
25. C4⋊Dic449C22semidirect
26. #20297715C22semidirect
29. #20297713C22semidirect
30. #20297713C22semidirect
31. C4×Dic449C22semidirect
32. C2×Q8×C449C22semidirect
34. C4⋊C4×C449C22semidirect
35. C4⋊C4×C449C22semidirect
55. C4×D449D4semidirect
65. C22×C449#41semidirect
76. C2×D449#41semidirect
91. C449#525semidirect
97. C22⋊Q8D449semidirect
106. C2×Q8C2×D449semidirect
121. C4⋊C4C2×D449semidirect
122. C4⋊C4C2×D449semidirect
167. C22#20297737semidirect

Maximal subgroups

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

subgroupindex
C2×#202977372
C22×C4×D4492
#283853252
#283853162
#283853152
#283853142
#283853102
#283853022
#283853002
#283852982
C22⋊Q8×C4492
#525449