Finite Group Databasebeta

← #41003353

Subgroups — #41003353

Order 26944.

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×#20269772C211yesmaximal, characteristic, complemented
2C22×C4×D421C211yesmaximal, characteristic, complemented
3#27212351C212yesmaximal, complemented
4#27212342C211yesmaximal, characteristic, complemented
5#27212341C211yesmaximal, characteristic, complemented
6#27212340C211yesmaximal, characteristic, complemented
7#27212336C212yesmaximal, complemented
8#27212328C212yesmaximal, complemented
9#27212326C212yesmaximal, complemented
10#27212324C211yesmaximal, characteristic, complemented
11C22⋊Q8×C421C211yesmaximal, characteristic, complemented
12C23×D421C2211yescharacteristic
13#20269772—24no—
14C2×D1684C2212yes—
15C2×D1684C2212yes—
16C2×D1684C2211yescharacteristic, complemented
17C2×C22⋊D421C2212yes—
18C22×Dic421C2211yescharacteristic
19C2×C4×D421—24no—
20C2×C4×D421C2211yescharacteristic
21C2×C4×D421C2211yescharacteristic
22C2×C4×D421C2211yescharacteristic
23C2×C4×D421C2211yescharacteristic
24C2×C4×D421C2212yescomplemented
25C4⋊Dic421C2211yescharacteristic, complemented
26#20269750C2212yescomplemented
27#20269748C2212yes—
28#20269748C2212yes—
29#20269748C2212yescomplemented
30#20269748C2212yescomplemented
31C4×Dic421C2212yescomplemented
32C2×Q8×C421C2211yescharacteristic, complemented
33C22×C4×C421C2211yescharacteristic
34C4⋊C4×C421C2211yescharacteristic, complemented
35C4⋊C4×C421C2212yescomplemented
36C22⋊C4×C421C2212yes—
37C22×D421—22no—
38C22×D421—22no—
39C22×D421—22no—
40C22×D421C2312yes—
41C22×D421C2311yescharacteristic
42C22×D421C2311yescharacteristic
43D1684—24no—
44D1684—24no—
45D1684—22no—
46C22⋊D421—24no—
47C2×Dic421—22no—
48C2×Dic421C2312yes—
49C2×Dic421C2311yescharacteristic
50C2×Dic421C2311yescharacteristic
51C4×D421—22no—
52C4×D421—22no—
53C4×D421—22no—
54C4×D421—24no—
55C4×D421D414yescomplemented
56C23×C421C2311yescharacteristic
57Q8×C421—22no—
58C2×C4×C421—22no—
59C2×C4×C421C2312yes—
60C2×C4×C421C2312yes—
61C2×C4×C421C2311yescharacteristic
62C2×C4×C421C2311yescharacteristic
63C22×C421—21no—
64C22×C421—21no—
65C22×C421#4112yescomplemented
66C22×C421C2411yescharacteristic
67C2×D421—28no—
68C2×D421—21no—
69C2×D421—21no—
70C2×D421—21no—
71C2×D421—21no—
72C2×D421—22no—
73C2×D421—22no—
74C2×D421—22no—
75C2×D421C2×D412yes—
76C2×D421#4112yescomplemented
77C4×C421—21no—
78C4×C421—22no—
79C4×C421—22no—
80C4×C421C2×D412yes—
81Dic421—21no—
82Dic421—22no—
83Dic421C2×D412yes—
84C2×C421—22no—
85C2×C421C22×D411yescharacteristic
86C2×C421C2×#4111yescharacteristic
87C2×C421C2×#4111yescharacteristic
88D421—22no—
89D421—22no—
90D421—42no—
91C421#52511yescharacteristic, complemented, Sylow
92#525—4211nomaximal, Sylow
93C2×#41—4211no—
94C23×C4—4211no—
95#123—4212no—
96C22⋊Q8—4211no—
97C22⋊Q8D42111yescharacteristic, complemented
98#121—4211no—
99#121—4211no—
100#120—4212no—
101C4×D4—4212no—
102C4×D4—4212no—
103#117—4211no—
104C24—4211no—
105#41—8424no—
106C2×Q8C2×D42111yescharacteristic, complemented
107C2×D4—4212no—
108C2×D4—4212no—
109C2×D4—4212no—
110C2×D4—4211no—
111C22×C4—8424no—
112C22×C4—4211no—
113C22×C4—4211no—
114C22×C4—4212no—
115C22×C4—4211no—
116C22×C4—4211no—
117C22×C4—4211no—
118C22×C4C2×D42111yescharacteristic
119C4⋊C4—4211no—
120C4⋊C4—4212no—
121C4⋊C4C2×D42111yescharacteristic, complemented
122C4⋊C4C2×D42112yescomplemented
123C22⋊C4—4212no—
124C22⋊C4—4212no—
125C22⋊C4—4212no—
126C22⋊C4—4212no—
127C22⋊C4C2×D42112yes—
128C42—4212no—
129C23—8422no—
130C23—8422no—
131C23—8422no—
132C23—4212no—
133C23—4211no—
134C23—4211no—
135C23C22×D42111yescharacteristic
136Q8—22no—
137D4—8424no—
138D4—8424no—
139D4—8424no—
140D4—8422no—
141C2×C4—22no—
142C2×C4—8422no—
143C2×C4—8422no—
144C2×C4—8424no—
145C2×C4—8422no—
146C2×C4—8422no—
147C2×C4—4212no—
148C2×C4—4211no—
149C2×C4—4211no—
150C2×C4—4214no—
151C2×C4C22×D42112yes—
152C2×C4C22×D42111yescharacteristic
153C2×C4C22×D42112yes—
154C2×C4C22×D42111yescharacteristic
155C22—21no—
156C22—21no—
157C22—8428no—
158C22—8421no—
159C22—8421no—
160C22—8421no—
161C22—8421no—
162C22—8422no—
163C22—8422no—
164C22—8422no—
165C22—4212no—
166C22—4212no—
167C22#2026977212yescomplemented
168C22C23×D42111yescharacteristic
169C4—21no—
170C4—22no—
171C4—22no—
172C4—8421no—
173C4—8422no—
174C4—4212no—
175C4D4×D42112yes—
176C2—22no—
177C2—8422no—
178C2—8422no—
179C2—16842no—
180C2C2×D4×D42111yescharacteristic
181C2#41×D42111yescharacteristic
182C2C2×#2026977211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D421
commutator, socle, cosocleC22×C421C24
FittingC22⋊Q8×C421C2
hypercenterC22⋊Q8D421
hypocenterC421#525
LCSG ▷ C22×C421 ▷ C421
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#20269772C2semidirect
2. C22×C4×D421C2semidirect
3. #27212351C2semidirect
4. #27212342C2semidirect
5. #27212341C2semidirect
6. #27212340C2semidirect
7. #27212336C2semidirect
8. #27212328C2semidirect
9. #27212326C2semidirect
10. #27212324C2semidirect
11. C22⋊Q8×C421C2semidirect
16. C2×D1684C22semidirect
24. C2×C4×D421C22semidirect
25. C4⋊Dic421C22semidirect
26. #20269750C22semidirect
29. #20269748C22semidirect
30. #20269748C22semidirect
31. C4×Dic421C22semidirect
32. C2×Q8×C421C22semidirect
34. C4⋊C4×C421C22semidirect
35. C4⋊C4×C421C22semidirect
55. C4×D421D4semidirect
65. C22×C421#41semidirect
76. C2×D421#41semidirect
91. C421#525semidirect
97. C22⋊Q8D421semidirect
106. C2×Q8C2×D421semidirect
121. C4⋊C4C2×D421semidirect
122. C4⋊C4C2×D421semidirect
167. C22#20269772semidirect

Maximal subgroups

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

subgroupindex
C2×#202697722
C22×C4×D4212
#272123512
#272123422
#272123412
#272123402
#272123362
#272123282
#272123262
#272123242
C22⋊Q8×C4212
#525421