Finite Group Databasebeta

← #41003487

Subgroups — #41003487

Order 26944.

334 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 105 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×#20269771C211yesmaximal, characteristic, complemented
2C2×D4×D421C211yesmaximal, characteristic, complemented
3#27212353C211yesmaximal, characteristic, complemented
4#27212348C212yesmaximal, complemented
5#27212346C211yesmaximal, characteristic, complemented
6#27212337C212yesmaximal, complemented
7#27212334C212yesmaximal, complemented
8#27212328C211yesmaximal, characteristic, complemented
9C4×C22⋊D421C211yesmaximal, characteristic, complemented
10C22⋊C4×D421C212yesmaximal, complemented
11#121×C421C211yesmaximal, characteristic, complemented
12C23×D421C2212yes—
13#20269771—24no—
14C2×Dic842C2211yescharacteristic, complemented
15C2×D1684C2211yescharacteristic
16C2×C22⋊D421C2211yescharacteristic
17C2×C22⋊D421C2212yes—
18C2×C22⋊D421C2212yescomplemented
19D4×D421—24no—
20C2×C4×D421C2211yescharacteristic
21C2×C4×D421C2212yescomplemented
22C2×C4×D421C2212yescomplemented
23C4⋊Dic421C2212yescomplemented
24#20269750C2211yescharacteristic, complemented
25C22⋊Dic421C2212yescomplemented
26C22⋊Dic421C2211yescharacteristic, complemented
27C22⋊Dic421C2212yescomplemented
28#20269748C2211yescharacteristic
29#20269748C2212yes—
30#20269748C2212yescomplemented
31C4×Dic421C2211yescharacteristic, complemented
32C2×D4×C421C2211yescharacteristic
33C2×D4×C421C2212yescomplemented
34C22×C4×C421C2211yescharacteristic, complemented
35C4⋊C4×C421C2211yescharacteristic, complemented
36C22⋊C4×C421C2212yescomplemented
37C22×D421—24no—
38C22×D421—24no—
39C22×D421—42no—
40C22×D421C2312yes—
41C22×D421C2311yescharacteristic
42Dic842—22no—
43D1684D414yescomplemented
44C22⋊D421—24no—
45C22⋊D421—24no—
46C22⋊D421—22no—
47C2×Dic421C2311yescharacteristic
48C2×Dic421C2311yescharacteristic
49C2×Dic421C2312yes—
50C2×Dic421C2311yescharacteristic
51C4×D421—24no—
52C4×D421—22no—
53C4×D421—42no—
54C23×C421C2312yes—
55C23×C421C2311yescharacteristic
56D4×C421—22no—
57D4×C421—42no—
58C2×C4×C421—22no—
59C2×C4×C421C2311yescharacteristic
60C2×C4×C421C2312yes—
61C2×C4×C421C2311yescharacteristic
62C22×C421—21no—
63C22×C421—22no—
64C22×C421—21no—
65C22×C421—22no—
66C22×C421—21no—
67C22×C421—22no—
68C22×C421C2411yescharacteristic
69C2×D421—21no—
70C2×D421—22no—
71C2×D421—22no—
72C2×D421—21no—
73C2×D421—48no—
74C2×D421C2×D414yes—
75C2×D421#4112yescomplemented
76C4×C421—21no—
77C4×C421—22no—
78C4×C421C2×D412yes—
79Dic421—21no—
80Dic421—22no—
81Dic421—21no—
82Dic421—21no—
83C2×C421—42no—
84C2×C421—41no—
85C2×C421C22×D411yescharacteristic
86C2×C421C2×#4111yescharacteristic
87C2×C421H(5,𝔽2)11yescharacteristic
88D421—22no—
89D421—24no—
90C421#54611yescharacteristic, complemented, Sylow
91#546—4211nomaximal, Sylow
92C2×#41—4211no—
93C22×D4—4211no—
94#124—4211no—
95#123—4212no—
96C22⋊Q8—4211no—
97#121—4212no—
98#121D42111yescharacteristic, complemented
99#120—4212no—
100C4×D4—4211no—
101C4×D4—4211no—
102C2×C22⋊C4—4212no—
103C24—4212no—
104#41—8424no—
105C2×Q8—4211no—
106C2×D4—8424no—
107C2×D4—4211no—
108C2×D4—4212no—
109C2×D4—4212no—
110C2×D4—4211no—
111C2×D4C2×D42111yescharacteristic
112C2×D4C2×D42112yescomplemented
113C22×C4—4212no—
114C22×C4—4211no—
115C22×C4—4212no—
116C22×C4C2×D42111yescharacteristic, complemented
117C4⋊C4—4211no—
118C4⋊C4—4212no—
119C4⋊C4C2×D42111yescharacteristic, complemented
120C22⋊C4—4212no—
121C22⋊C4—4211no—
122C22⋊C4—4212no—
123C22⋊C4—4211no—
124C22⋊C4—4212no—
125C22⋊C4—4212no—
126C22⋊C4C2×D42112yescomplemented
127C42—4211no—
128C23—8424no—
129C23—8424no—
130C23—16842no—
131C23—4211no—
132C23—4212no—
133C23C22×D42112yes—
134C23C22×D42111yescharacteristic
135Q8—8422no—
136D4—22no—
137D4—42no—
138D4—8424no—
139D4—8424no—
140D4—8422no—
141D4—4214no—
142C2×C4—22no—
143C2×C4—8424no—
144C2×C4—8422no—
145C2×C4—16842no—
146C2×C4—4211no—
147C2×C4—4211no—
148C2×C4—4212no—
149C2×C4—4211no—
150C2×C4C22×D42112yes—
151C2×C4C22×D42111yescharacteristic
152C2×C4C22×D42111yescharacteristic
153C22—21no—
154C22—22no—
155C22—21no—
156C22—22no—
157C22—21no—
158C22—22no—
159C22—8421no—
160C22—8422no—
161C22—8422no—
162C22—8421no—
163C22—16848no—
164C22—4214no—
165C22—4212no—
166C22C23×D42111yescharacteristic
167C4—21no—
168C4—22no—
169C4—8421no—
170C4—8422no—
171C4—8421no—
172C4—8421no—
173C4D4×D42112yes—
174C2—42no—
175C2—41no—
176C2—8422no—
177C2—8424no—
178C2C2×D4×D42111yescharacteristic
179C2#41×D42111yescharacteristic
180C2#2721241111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D421
commutator, socle, cosocleC22×C421C24
Fitting#121×C421C2
hypercenter#121D421
hypocenterC421#546
LCSG ▷ C22×C421 ▷ C421
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#20269771C2semidirect
2. C2×D4×D421C2semidirect
3. #27212353C2semidirect
4. #27212348C2semidirect
5. #27212346C2semidirect
6. #27212337C2semidirect
7. #27212334C2semidirect
8. #27212328C2semidirect
9. C4×C22⋊D421C2semidirect
10. C22⋊C4×D421C2semidirect
11. #121×C421C2semidirect
14. C2×Dic842C22semidirect
18. C2×C22⋊D421C22semidirect
21. C2×C4×D421C22semidirect
22. C2×C4×D421C22semidirect
23. C4⋊Dic421C22semidirect
24. #20269750C22semidirect
25. C22⋊Dic421C22semidirect
26. C22⋊Dic421C22semidirect
27. C22⋊Dic421C22semidirect
30. #20269748C22semidirect
31. C4×Dic421C22semidirect
33. C2×D4×C421C22semidirect
34. C22×C4×C421C22semidirect
35. C4⋊C4×C421C22semidirect
36. C22⋊C4×C421C22semidirect
43. D1684D4semidirect
75. C2×D421#41semidirect
90. C421#546semidirect
98. #121D421semidirect
112. C2×D4C2×D421semidirect
116. C22×C4C2×D421semidirect
119. C4⋊C4C2×D421semidirect
126. C22⋊C4C2×D421semidirect

Maximal subgroups

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

subgroupindex
C2×#202697712
C2×D4×D4212
#272123532
#272123482
#272123462
#272123372
#272123342
#272123282
C4×C22⋊D4212
C22⋊C4×D4212
#121×C4212
#546421