Finite Group Databasebeta

← #16854058

Subgroups — #16854058

Order 3968.

644 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 209 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×#11758523C212yesmaximal
2C2×C42×D31C211yesmaximal, characteristic
3#14520956C212yesmaximal, complemented
4#14520945C214yesmaximal, complemented
5#14520919C211yesmaximal, characteristic, complemented
6#14520906C214yesmaximal
7C2×C31×C4⋊Q8C211yesmaximal, characteristic, complemented
8C22×Q8×C31C2212yes—
9C22×C4×D31C2211yescharacteristic
10C22×C4×D31C2212yes—
11C31×C4⋊Q8—24no—
12C2×C4⋊C4×C31C2214yes—
13C2×C42×C31C2211yescharacteristic
14#11758523—28no—
15C2×(C2×D31)⋊C4C2214yes—
16C2×C4⋊Dic31C2214yes—
17C2×C4⋊Dic31C2212yes—
18C2×Dic31⋊C4C2214yes—
19C2×C4×Dic31C2212yes—
20C2×C4×Dic31C2211yescharacteristic
21C42×D31—24no—
22C31⋊#95C2218yes—
23C23×D31C2311yescharacteristic
24C2×Q8×C31—24no—
25C2×Q8×C31—24no—
26C2×Q8×C31—24no—
27C22×C4×C31C2314yes—
28C22×C4×C31C2311yescharacteristic
29C22×C4×C31C2312yes—
30C22×Dic31C2314yes—
31C22×Dic31C2311yescharacteristic
32C22×Dic31C2312yes—
33C2×C4×D31—22no—
34C2×C4×D31—24no—
35C2×C4×D31—22no—
36C2×C4×D31—24no—
37C2×C4×D31D414yes—
38C2×C4×D31Q818yescomplemented
39C4⋊C4×C31—28no—
40C42×C31D414yes—
41(C2×D31)⋊C4—28no—
42C4⋊Dic31—24no—
43C4⋊Dic31—28no—
44Dic31⋊C4—28no—
45C4×Dic31—22no—
46C4×Dic31—24no—
47C23×C31C2411yescharacteristic
48C22×D31—21no—
49C22×D31—21no—
50C22×D31—22no—
51C22×D31C2×D412yes—
52C22×D31C2×Q814yes—
53Q8×C31—48no—
54C2×C4×C31—24no—
55C2×C4×C31—24no—
56C2×C4×C31—24no—
57C2×C4×C31C2×D412yes—
58C2×C4×C31C2×D412yes—
59C2×C4×C31C2×D414yes—
60C2×C4×C31C2×Q814yes—
61C2×C4×C31#4112yes—
62C2×C4×C31#4114yes—
63C2×Dic31—21no—
64C2×Dic31—22no—
65C2×Dic31—21no—
66C2×Dic31—22no—
67C2×Dic31—24no—
68C2×Dic31—28no—
69C2×Dic31C2×D412yes—
70C2×Dic31C2×Q814yes—
71C4×D31—28no—
72C4×D31—216no—
73#2857—311nomaximal, Sylow
74C22×C31C22×D411yescharacteristic
75C22×C31C22×D411yescharacteristic
76C22×C31C22×Q812yes—
77C22×C31C2×#4111yescharacteristic
78C22×C31C2×#4112yes—
79C2×D31—22no—
80C2×D31—24no—
81C2×D31—22no—
82C2×D31—24no—
83C2×D31C4⋊Q814yescomplemented
84C4×C31—44no—
85C4×C31#12114yes—
86C4×C31C22⋊Q818yescomplemented
87Dic31—22no—
88Dic31—24no—
89Dic31—44no—
90C2×C4⋊Q8D3111yescharacteristic, complemented
91C2×C22⋊Q8—312no—
92C22×C42—311no—
93#391—312no—
94#389—314no—
95#384—311no—
96#380—314no—
97C2×C31C2×#12111yescharacteristic
98C2×C31C2×C22⋊Q812yes—
99C2×C31C2×C4⋊Q811yescharacteristic
100C2×C31#53211yescharacteristic
101C2×C31#53312yes—
102D31—24no—
103C22×Q8C2×D3112yes—
104C23×C4—311no—
105C23×C4—312no—
106C4⋊Q8—24no—
107C22⋊Q8—628no—
108C2×C4⋊C4—312no—
109C2×C4⋊C4—314no—
110C2×C4⋊C4—314no—
111C2×C4⋊C4C2×D3114yes—
112C2×C22⋊C4—314no—
113C2×C42—312no—
114C2×C42—311no—
115C2×C42—624no—
116C2×C42C2×D3111yescharacteristic
117#95—318no—
118C31#285711yescharacteristic, complemented, Sylow
119C24—311no—
120C2×Q8—24no—
121C2×Q8—24no—
122C2×Q8—24no—
123C22×C4—312no—
124C22×C4—311no—
125C22×C4—314no—
126C22×C4—318no—
127C22×C4—314no—
128C22×C4—622no—
129C22×C4—624no—
130C22×C4—622no—
131C22×C4—624no—
132C22×C4C22×D3114yes—
133C22×C4C22×D3111yescharacteristic
134C22×C4C22×D3112yes—
135C4⋊C4—28no—
136C4⋊C4—628no—
137C4⋊C4—624no—
138C4⋊C4—628no—
139C22⋊C4—628no—
140C42—622no—
141C42—624no—
142C42C22⋊D3114yes—
143C23—312no—
144C23—314no—
145C23—621no—
146C23—622no—
147C23—621no—
148C23C23×D3111yescharacteristic
149Q8—48no—
150C2×C4—24no—
151C2×C4—24no—
152C2×C4—24no—
153C2×C4—312no—
154C2×C4—314no—
155C2×C4—621no—
156C2×C4—622no—
157C2×C4—621no—
158C2×C4—622no—
159C2×C4—628no—
160C2×C4—6216no—
161C2×C4—624no—
162C2×C4—628no—
163C2×C4D4×D3112yes—
164C2×C4#9249712yes—
165C2×C4Q8×D3114yes—
166C2×C4#9249914yes—
167C2×C4C2×C22⋊D3112yes—
168C2×C4C2×C22⋊D3114yes—
169C22—314no—
170C22—622no—
171C22—624no—
172C22—622no—
173C22—624no—
174C22C2×D4×D3111yescharacteristic
175C22C2×#9249711yescharacteristic
176C22C2×Q8×D3112yes—
177C22C2×#9249912yes—
178C22C22×C22⋊D3111yescharacteristic
179C4—44no—
180C4—622no—
181C4—624no—
182C4—1244no—
183C4#1175851514yes—
184C4#1175852318yescomplemented
185C2—624no—
186C2C2×#1175851511yescharacteristic
187C2C2×#1175852312yes—
188C2C4⋊Q8×D3111yescharacteristic
189C2#1452160011yescharacteristic
190C2#1452160412yes—
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC2×C31×C4⋊Q8C2
prime radicalC23×D31C23
hypercenterC2×C4⋊Q8D31
hypocenterC31#2857
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #14520956C2semidirect
4. #14520945C2semidirect
5. #14520919C2semidirect
7. C2×C31×C4⋊Q8C2semidirect
38. C2×C4×D31Q8semidirect
83. C2×D31C4⋊Q8semidirect
86. C4×C31C22⋊Q8semidirect
90. C2×C4⋊Q8D31semidirect
118. C31#2857semidirect
184. C4#11758523semidirect

Maximal subgroups

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

subgroupindex
C2×#117585232
C2×C42×D312
#145209562
#145209452
#145209192
#145209062
C2×C31×C4⋊Q82
#285731