Finite Group Databasebeta

← #41003350

Subgroups — #41003350

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×#20269770C211yesmaximal, characteristic, complemented
2C22×C4×D421C211yesmaximal, characteristic, complemented
3#27212348C212yesmaximal, complemented
4#27212346C211yesmaximal, characteristic, complemented
5#27212345C211yesmaximal, characteristic, complemented
6#27212337C211yesmaximal, characteristic, complemented
7#27212334C212yesmaximal, complemented
8D4×Dic421C212yesmaximal, complemented
9#27212326C212yesmaximal, complemented
10#27212324C211yesmaximal, characteristic, complemented
11#121×C421C211yesmaximal, characteristic, complemented
12C23×D421C2211yescharacteristic
13#20269770—24no—
14C2×Dic842C2211yescharacteristic, complemented
15C2×C22⋊D421C2212yes—
16C2×C22⋊D421C2212yes—
17C22×Dic421C2211yescharacteristic
18C22×Dic421C2212yescomplemented
19C2×C4×D421—24no—
20C2×C4×D421C2211yescharacteristic
21C2×C4×D421C2211yescharacteristic
22C2×C4×D421C2211yescharacteristic
23C2×C4×D421C2211yescharacteristic
24C4⋊Dic421C2211yescharacteristic, complemented
25C4⋊Dic421C2212yescomplemented
26#20269750C2212yescomplemented
27C22⋊Dic421C2212yes—
28C22⋊Dic421C2212yescomplemented
29#20269748C2212yes—
30#20269748C2212yes—
31C4×Dic421C2212yescomplemented
32C2×D4×C421C2212yes—
33C2×D4×C421C2211yescharacteristic, complemented
34C22×C4×C421C2211yescharacteristic
35C4⋊C4×C421C2211yescharacteristic, complemented
36C22⋊C4×C421C2212yescomplemented
37C22×D421—22no—
38C22×D421—22no—
39C22×D421—22no—
40C22×D421C2311yescharacteristic
41C22×D421C2311yescharacteristic
42Dic842—22no—
43C22⋊D421—24no—
44C22⋊D421—24no—
45C2×Dic421—22no—
46C2×Dic421—24no—
47C2×Dic421C2312yes—
48C2×Dic421C2312yes—
49C2×Dic421C2311yescharacteristic
50C2×Dic421C2311yescharacteristic
51C4×D421—22no—
52C4×D421—22no—
53C4×D421—22no—
54C4×D421D414yescomplemented
55C23×C421C2312yes—
56C23×C421C2311yescharacteristic
57D4×C421—24no—
58D4×C421—22no—
59C2×C4×C421—22no—
60C2×C4×C421C2311yescharacteristic
61C2×C4×C421C2312yes—
62C2×C4×C421C2311yescharacteristic
63C22×C421—21no—
64C22×C421—21no—
65C22×C421—22no—
66C22×C421—22no—
67C22×C421—22no—
68C22×C421#4112yescomplemented
69C22×C421C2411yescharacteristic
70C2×D421—28no—
71C2×D421—21no—
72C2×D421—21no—
73C2×D421—21no—
74C2×D421—21no—
75C2×D421C2×D412yes—
76C2×D421#4112yescomplemented
77C4×C421—21no—
78C4×C421—22no—
79C4×C421C2×D412yes—
80Dic421—21no—
81Dic421—22no—
82Dic421—22no—
83Dic421C2×D412yes—
84C2×C421—22no—
85C2×C421—42no—
86C2×C421C22×D411yescharacteristic
87C2×C421C2×#4111yescharacteristic
88C2×C421C2×#4111yescharacteristic
89D421—22no—
90D421—22no—
91C421#52511yescharacteristic, complemented, Sylow
92#525—4211nomaximal, Sylow
93C2×#41—4211no—
94C23×C4—4211no—
95#123—4212no—
96C22⋊Q8—4211no—
97C22⋊Q8—4211no—
98#121—4211no—
99#121D42111yescharacteristic, complemented
100#120—4212no—
101C4×D4—4212no—
102C4×D4—4212no—
103#117—4211no—
104C24—4211no—
105#41—8424no—
106C2×Q8—4211no—
107C2×D4—4212no—
108C2×D4—4212no—
109C2×D4C2×D42112yes—
110C2×D4C2×D42111yescharacteristic, complemented
111C22×C4—8424no—
112C22×C4—4211no—
113C22×C4—4212no—
114C22×C4—4211no—
115C22×C4—4211no—
116C22×C4—4211no—
117C22×C4—4211no—
118C22×C4C2×D42111yescharacteristic
119C4⋊C4—4211no—
120C4⋊C4—4212no—
121C4⋊C4—4212no—
122C4⋊C4C2×D42111yescharacteristic, complemented
123C22⋊C4—4212no—
124C22⋊C4—4212no—
125C22⋊C4—4212no—
126C22⋊C4—4212no—
127C22⋊C4C2×D42112yescomplemented
128C42—4212no—
129C23—8422no—
130C23—8422no—
131C23—8422no—
132C23—4211no—
133C23—4211no—
134C23C22×D42112yes—
135C23C22×D42111yescharacteristic
136Q8—8422no—
137D4—24no—
138D4—22no—
139D4—8424no—
140D4—8424no—
141C2×C4—22no—
142C2×C4—8422no—
143C2×C4—8422no—
144C2×C4—8422no—
145C2×C4—8424no—
146C2×C4—8422no—
147C2×C4—4212no—
148C2×C4—4211no—
149C2×C4—4212no—
150C2×C4—4211no—
151C2×C4—4214no—
152C2×C4C22×D42112yes—
153C2×C4C22×D42111yescharacteristic
154C2×C4C22×D42111yescharacteristic
155C22—21no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—8428no—
161C22—8421no—
162C22—8421no—
163C22—8421no—
164C22—8421no—
165C22—4212no—
166C22—4212no—
167C22#2026977012yescomplemented
168C22C23×D42111yescharacteristic
169C4—21no—
170C4—22no—
171C4—8421no—
172C4—8422no—
173C4—8422no—
174C4—4212no—
175C4D4×D42112yes—
176C2—22no—
177C2—42no—
178C2—8422no—
179C2—8422no—
180C2C2×D4×D42111yescharacteristic
181C2#41×D42111yescharacteristic
182C2C2×#2026977011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D421
commutator, socle, cosocleC22×C421C24
Fitting#121×C421C2
hypercenter#121D421
hypocenterC421#525
LCSG ▷ C22×C421 ▷ C421
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#20269770C2semidirect
2. C22×C4×D421C2semidirect
3. #27212348C2semidirect
4. #27212346C2semidirect
5. #27212345C2semidirect
6. #27212337C2semidirect
7. #27212334C2semidirect
8. D4×Dic421C2semidirect
9. #27212326C2semidirect
10. #27212324C2semidirect
11. #121×C421C2semidirect
14. C2×Dic842C22semidirect
18. C22×Dic421C22semidirect
24. C4⋊Dic421C22semidirect
25. C4⋊Dic421C22semidirect
26. #20269750C22semidirect
28. C22⋊Dic421C22semidirect
31. C4×Dic421C22semidirect
33. C2×D4×C421C22semidirect
35. C4⋊C4×C421C22semidirect
36. C22⋊C4×C421C22semidirect
54. C4×D421D4semidirect
68. C22×C421#41semidirect
76. C2×D421#41semidirect
91. C421#525semidirect
99. #121D421semidirect
110. C2×D4C2×D421semidirect
122. C4⋊C4C2×D421semidirect
127. C22⋊C4C2×D421semidirect
167. C22#20269770semidirect

Maximal subgroups

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

subgroupindex
C2×#202697702
C22×C4×D4212
#272123482
#272123462
#272123452
#272123372
#272123342
D4×Dic4212
#272123262
#272123242
#121×C4212
#525421