Finite Group Databasebeta

← #123×D41 #14580836

Subgroups — #123×D41 #14580836

Order 2624.

342 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 111 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×D4×D41C211yesmaximal, characteristic
2C22×C4×D41C211yesmaximal, characteristic, complemented
3#13050149C212yesmaximal, complemented
4#13050148C211yesmaximal, characteristic, complemented
5#13050147C211yesmaximal, characteristic, complemented
6#13050146C212yesmaximal, complemented
7#13050145C211yesmaximal, characteristic, complemented
8C4⋊C4×D41C212yesmaximal, complemented
9C22⋊C4×D41C211yesmaximal, characteristic, complemented
10C22⋊C4×D41C212yesmaximal, complemented
11#123×C41C211yesmaximal, characteristic, complemented
12C23×D41C2211yescharacteristic
13C23×D41C2211yescharacteristic
14C2×D4×C41C2211yescharacteristic
15C22×C4×C41C2211yescharacteristic, complemented
16C2×C22⋊D41C2211yescharacteristic
17C2×C22⋊D41C2211yescharacteristic
18C22×Dic41C2211yescharacteristic, complemented
19D4×D41—24no—
20C2×D164C2211yescharacteristic
21C2×C4×D41—24no—
22C2×C4×D41C2212yes—
23C2×C4×D41C2212yes—
24C2×C4×D41C2211yescharacteristic
25C2×C4×D41C2212yescomplemented
26C4⋊C4×C41C2212yescomplemented
27C22⋊C4×C41C2212yescomplemented
28C22⋊C4×C41C2211yescharacteristic, complemented
29C22⋊Dic41C2212yescomplemented
30C22⋊Dic41C2211yescharacteristic, complemented
31#10620281C2212yescomplemented
32#10620281C2212yescomplemented
33#10620281C2212yescomplemented
34C4⋊Dic41C2212yescomplemented
35#10620279C2212yescomplemented
36#10620279C2212yescomplemented
37C23×C41C2311yescharacteristic
38C23×C41C2311yescharacteristic
39C22×D41—24no—
40C22×D41—22no—
41C22×D41—24no—
42C22×D41D414yes—
43C22×D41C2311yescharacteristic
44C22×D41C2311yescharacteristic
45C22×D41C2311yescharacteristic
46D4×C41—22no—
47C2×C4×C41—22no—
48C2×C4×C41C2311yescharacteristic
49C2×C4×C41C2312yes—
50C2×C4×C41C2312yes—
51C22⋊D41—22no—
52C22⋊D41—22no—
53C2×Dic41—22no—
54C2×Dic41C2311yescharacteristic
55C2×Dic41C2312yes—
56C2×Dic41C2312yes—
57D164—22no—
58C4×D41—24no—
59C4×D41—24no—
60C4×D41—22no—
61C4×D41—24no—
62C22×C41—22no—
63C22×C41—21no—
64C22×C41—22no—
65C22×C41C2×D412yes—
66C22×C41C2411yescharacteristic
67C2×D41—22no—
68C2×D41—28no—
69C2×D41—21no—
70C2×D41—22no—
71C2×D41—44no—
72C2×D41C2×D412yes—
73C2×D41C2×D412yes—
74C2×D41#4114yes—
75C4×C41—22no—
76C4×C41—21no—
77C4×C41—22no—
78Dic41—22no—
79Dic41—21no—
80Dic41—22no—
81C2×C41—22no—
82C2×C41—41no—
83C2×C41C22×D411yescharacteristic
84C2×C41C2×#4112yes—
85D41—22no—
86D41—41no—
87D41#12314yesdirect product factor
88C2×#123—411nomaximal, Sylow
89C41C2×#12311yescharacteristic, complemented, Sylow
90C22×D4—411no—
91C23×C4—411no—
92#123—411no—
93#123—411no—
94#123—412no—
95#123—412no—
96#123—411no—
97#123D4111yescharacteristic, direct product factor
98C2×C4⋊C4—412no—
99C2×C22⋊C4—411no—
100C2×C22⋊C4—412no—
101C24—411no—
102C24—411no—
103C2×D4—411no—
104C2×D4—411no—
105C2×D4—411no—
106C2×D4—824no—
107C2×D4C2×D4111yescharacteristic
108C22×C4—411no—
109C22×C4—412no—
110C22×C4—412no—
111C22×C4—411no—
112C22×C4—412no—
113C22×C4—824no—
114C22×C4C2×D4111yescharacteristic, complemented
115C4⋊C4—412no—
116C4⋊C4—412no—
117C4⋊C4—412no—
118C4⋊C4C2×D4112yescomplemented
119C22⋊C4—411no—
120C22⋊C4—412no—
121C22⋊C4—412no—
122C22⋊C4—412no—
123C22⋊C4—412no—
124C22⋊C4C2×D4111yescharacteristic, complemented
125C22⋊C4C2×D4112yescomplemented
126C23—411no—
127C23—414no—
128C23—411no—
129C23—411no—
130C23—824no—
131C23—822no—
132C23—824no—
133C23C22×D4111yescharacteristic
134C23C22×D4111yescharacteristic
135D4—22no—
136D4—822no—
137D4—822no—
138D4—822no—
139C2×C4—22no—
140C2×C4—412no—
141C2×C4—412no—
142C2×C4—411no—
143C2×C4—824no—
144C2×C4—824no—
145C2×C4—822no—
146C2×C4—824no—
147C2×C4—822no—
148C2×C4C22×D4112yes—
149C2×C4C22×D4112yes—
150C2×C4C22×D4111yescharacteristic
151C22—22no—
152C22—21no—
153C22—22no—
154C22—412no—
155C22—414no—
156C22—412no—
157C22—828no—
158C22—822no—
159C22—821no—
160C22—822no—
161C22—1644no—
162C22D4×D4112yes—
163C22C23×D4111yescharacteristic
164C4—22no—
165C4—21no—
166C4—22no—
167C4—822no—
168C4—821no—
169C4—822no—
170C2—22no—
171C2—41no—
172C2—414no—
173C2—822no—
174C2—1641no—
175C2C2×D4×D4111yescharacteristic
176C2#41×D4112yes—
subgroupG/H
center, FrattiniC22C23×D41
commutator, socle, cosocleC22×C41C24
Fitting#123×C41C2
prime radicalC2×D4×D41C2
hypercenter#123D41
hypocenterC41C2×#123
LCSG ▷ C22×C41 ▷ C41
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
2. C22×C4×D41C2semidirect
3. #13050149C2semidirect
4. #13050148C2semidirect
5. #13050147C2semidirect
6. #13050146C2semidirect
7. #13050145C2semidirect
8. C4⋊C4×D41C2semidirect
9. C22⋊C4×D41C2semidirect
10. C22⋊C4×D41C2semidirect
11. #123×C41C2semidirect
15. C22×C4×C41C22semidirect
18. C22×Dic41C22semidirect
25. C2×C4×D41C22semidirect
26. C4⋊C4×C41C22semidirect
27. C22⋊C4×C41C22semidirect
28. C22⋊C4×C41C22semidirect
29. C22⋊Dic41C22semidirect
30. C22⋊Dic41C22semidirect
31. #10620281C22semidirect
32. #10620281C22semidirect
33. #10620281C22semidirect
34. C4⋊Dic41C22semidirect
35. #10620279C22semidirect
36. #10620279C22semidirect
87. D41#123direct (N × H)
89. C41C2×#123semidirect
97. #123D41direct (N × H)
114. C22×C4C2×D41semidirect
118. C4⋊C4C2×D41semidirect
124. C22⋊C4C2×D41semidirect
125. C22⋊C4C2×D41semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D412
C22×C4×D412
#130501492
#130501482
#130501472
#130501462
#130501452
C4⋊C4×D412
C22⋊C4×D412
#123×C412
C2×#12341