Finite Group Databasebeta

← #29508362

Subgroups — #29508362

Order 15424.

334 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 113 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×#16838769C211yesmaximal, characteristic, complemented
2C2×D4×D241C211yesmaximal, characteristic, complemented
3#21440630C211yesmaximal, characteristic, complemented
4#21440626C212yesmaximal, complemented
5#21440622C211yesmaximal, characteristic, complemented
6#21440617C211yesmaximal, characteristic, complemented
7#21440616C212yesmaximal, complemented
8#21440612C212yesmaximal, complemented
9C4×D964C211yesmaximal, characteristic, complemented
10C2×#16838747C212yesmaximal, complemented
11C4×D4×C241C211yesmaximal, characteristic, complemented
12C23×D241C2212yes—
13#16838769—24no—
14C2×Dic482C2211yescharacteristic, complemented
15C2×D964C2211yescharacteristic
16C2×D964C2211yescharacteristic
17C2×D964C2212yescomplemented
18C2×C22⋊D241C2212yes—
19C2×C22⋊D241C2212yescomplemented
20D4×D241—24no—
21C22×Dic241C2212yescomplemented
22C2×C4×D241C2211yescharacteristic
23C2×C4×D241C2211yescharacteristic, complemented
24C4⋊Dic241C2211yescharacteristic, complemented
25C4⋊Dic241C2212yescomplemented
26#16838747C2211yescharacteristic
27#16838747C2212yes—
28#16838747C2212yescomplemented
29#16838747C2212yescomplemented
30#16838747C2211yescharacteristic, complemented
31#16838747C2212yescomplemented
32C2×D4×C241C2211yescharacteristic
33C22×C4×C241C2212yescomplemented
34C4⋊C4×C241C2211yescharacteristic, complemented
35C22⋊C4×C241C2212yescomplemented
36C42×C241C2211yescharacteristic, complemented
37C22×D241—24no—
38C22×D241—24no—
39C22×D241—42no—
40C22×D241C2312yes—
41C22×D241C2311yescharacteristic
42C22×D241C2311yescharacteristic
43Dic482—22no—
44D964—22no—
45D964—22no—
46D964—42no—
47C22⋊D241—24no—
48C22⋊D241—24no—
49C2×Dic241—24no—
50C2×Dic241C2311yescharacteristic
51C2×Dic241C2312yes—
52C2×Dic241C2311yescharacteristic
53C4×D241—22no—
54C4×D241—22no—
55C23×C241C2312yes—
56D4×C241D414yescomplemented
57C2×C4×C241—42no—
58C2×C4×C241C2311yescharacteristic
59C2×C4×C241C2311yescharacteristic
60C2×C4×C241C2312yes—
61C2×C4×C241C2311yescharacteristic
62C22×C241—22no—
63C22×C241—22no—
64C22×C241C2×D414yes—
65C22×C241C2411yescharacteristic
66C2×D241—21no—
67C2×D241—21no—
68C2×D241—21no—
69C2×D241—22no—
70C2×D241—21no—
71C2×D241—22no—
72C2×D241—21no—
73C2×D241—22no—
74C2×D241—48no—
75C2×D241#4112yescomplemented
76C4×C241—21no—
77C4×C241—21no—
78C4×C241—22no—
79C4×C241C2×D412yes—
80Dic241—21no—
81Dic241—21no—
82Dic241—22no—
83C2×C241—24no—
84C2×C241C22×D411yescharacteristic
85C2×C241C2×#4111yescharacteristic
86C2×C241H(5,𝔽2)11yescharacteristic
87D241—22no—
88D241—41no—
89D241—42no—
90C241#54611yescharacteristic, complemented, Sylow
91#546—2411nomaximal, Sylow
92C2×#41—2411no—
93C22×D4—2411no—
94#124—2411no—
95#123—2412no—
96C22⋊Q8—2411no—
97#121—2412no—
98#121—2411no—
99#120—2412no—
100C4×D4—2411no—
101C4×D4D24111yescharacteristic, complemented
102C2×C22⋊C4—2412no—
103C24—2412no—
104#41—4824no—
105C2×Q8—2411no—
106C2×D4—4824no—
107C2×D4—2412no—
108C2×D4—2412no—
109C2×D4—2412no—
110C2×D4—2411no—
111C2×D4—2411no—
112C2×D4C2×D24111yescharacteristic
113C22×C4—2412no—
114C22×C4—2411no—
115C22×C4—2411no—
116C22×C4C2×D24112yescomplemented
117C4⋊C4—2411no—
118C4⋊C4—2412no—
119C4⋊C4C2×D24111yescharacteristic, complemented
120C22⋊C4—2411no—
121C22⋊C4—2412no—
122C22⋊C4—2412no—
123C22⋊C4—2412no—
124C22⋊C4—2411no—
125C22⋊C4—2412no—
126C22⋊C4C2×D24112yescomplemented
127C42C2×D24111yescharacteristic, complemented
128C23—9642no—
129C23—4824no—
130C23—4824no—
131C23—2412no—
132C23—2411no—
133C23—2411no—
134C23C22×D24112yes—
135Q8—4822no—
136D4—9642no—
137D4—4822no—
138D4—4824no—
139D4—4824no—
140D4—4822no—
141D4D96414yescomplemented
142C2×C4—42no—
143C2×C4—4824no—
144C2×C4—4822no—
145C2×C4—4822no—
146C2×C4—2411no—
147C2×C4—2412no—
148C2×C4—2411no—
149C2×C4C22×D24111yescharacteristic
150C2×C4C22×D24111yescharacteristic
151C2×C4C22×D24112yes—
152C2×C4C22×D24111yescharacteristic
153C22—22no—
154C22—22no—
155C22—9648no—
156C22—4821no—
157C22—4822no—
158C22—4821no—
159C22—4822no—
160C22—4821no—
161C22—4822no—
162C22—4821no—
163C22—4821no—
164C22—2412no—
165C22C2×D96414yes—
166C22C23×D24111yescharacteristic
167C4—21no—
168C4—21no—
169C4—22no—
170C4—4821no—
171C4—4821no—
172C4—4822no—
173C4C2×D96412yes—
174C2—24no—
175C2—9642no—
176C2—9641no—
177C2—4822no—
178C2C22×D96411yescharacteristic
179C2#41×D24111yescharacteristic
180C2#2144068711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D241
commutator, socle, cosocleC22×C241C24
FittingC4×D4×C241C2
hypercenterC4×D4D241
hypocenterC241#546
LCSG ▷ C22×C241 ▷ C241
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16838769C2semidirect
2. C2×D4×D241C2semidirect
3. #21440630C2semidirect
4. #21440626C2semidirect
5. #21440622C2semidirect
6. #21440617C2semidirect
7. #21440616C2semidirect
8. #21440612C2semidirect
9. C4×D964C2semidirect
10. C2×#16838747C2semidirect
11. C4×D4×C241C2semidirect
14. C2×Dic482C22semidirect
17. C2×D964C22semidirect
19. C2×C22⋊D241C22semidirect
21. C22×Dic241C22semidirect
23. C2×C4×D241C22semidirect
24. C4⋊Dic241C22semidirect
25. C4⋊Dic241C22semidirect
28. #16838747C22semidirect
29. #16838747C22semidirect
30. #16838747C22semidirect
31. #16838747C22semidirect
33. C22×C4×C241C22semidirect
34. C4⋊C4×C241C22semidirect
35. C22⋊C4×C241C22semidirect
36. C42×C241C22semidirect
56. D4×C241D4semidirect
75. C2×D241#41semidirect
90. C241#546semidirect
101. C4×D4D241semidirect
116. C22×C4C2×D241semidirect
119. C4⋊C4C2×D241semidirect
126. C22⋊C4C2×D241semidirect
127. C42C2×D241semidirect
141. D4D964semidirect

Maximal subgroups

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

subgroupindex
C2×#168387692
C2×D4×D2412
#214406302
#214406262
#214406222
#214406172
#214406162
#214406122
C4×D9642
C2×#168387472
C4×D4×C2412
#546241