Finite Group Databasebeta

← #16854045

Subgroups — #16854045

Order 3968.

564 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 177 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
1C22×C4×Dic31C211yesmaximal, characteristic
2#14520956C212yesmaximal, complemented
3#14520945C212yesmaximal, complemented
4#14520930C214yesmaximal, complemented
5#14520921C212yesmaximal
6#14520910C212yesmaximal
7#14520909C211yesmaximal, characteristic
8C2×C31×C22⋊Q8C211yesmaximal, characteristic
9C22×Q8×C31C2211yescharacteristic
10C23×C4×C31C2211yescharacteristic
11C23×Dic31C2211yescharacteristic
12C23×Dic31C2211yescharacteristic
13C31×C22⋊Q8—24no—
14C2×C4⋊C4×C31C2212yes—
15C2×C4⋊C4×C31C2211yescharacteristic
16C2×C22⋊C4×C31C2212yes—
17C2×C22⋊Dic31C2212yes—
18C2×C4⋊Dic31C2212yes—
19C2×Dic31⋊C4C2214yes—
20C2×C4×Dic31—24no—
21C2×C4×Dic31C2212yes—
22C2×C4×Dic31C2212yes—
23C31⋊#95C2212yes—
24C31⋊#95C2214yes—
25C31⋊#95C2214yes—
26C31⋊#95C2214yes—
27C24×C31C2311yescharacteristic
28C2×Q8×C31—22no—
29C2×Q8×C31—24no—
30C22×C4×C31—24no—
31C22×C4×C31D414yes—
32C22×C4×C31C2312yes—
33C22×C4×C31C2312yes—
34C22×C4×C31C2311yescharacteristic
35C22×C4×C31C2311yescharacteristic
36C22×Dic31—22no—
37C22×Dic31—22no—
38C22×Dic31—22no—
39C22×Dic31—24no—
40C22×Dic31Q814yescomplemented
41C22×Dic31C2314yes—
42C22×Dic31C2312yes—
43C22×Dic31C2312yes—
44C4⋊C4×C31—22no—
45C4⋊C4×C31—24no—
46C22⋊C4×C31—24no—
47C22⋊Dic31—24no—
48C4⋊Dic31—24no—
49Dic31⋊C4—28no—
50C4×Dic31—24no—
51C4×Dic31—24no—
52C23×C31—22no—
53C23×C31—22no—
54C23×C31C2×D412yes—
55C23×C31C2×Q812yes—
56C23×C31#4112yes—
57C23×C31C2411yescharacteristic
58Q8×C31—44no—
59C2×C4×C31—22no—
60C2×C4×C31—28no—
61C2×C4×C31—22no—
62C2×C4×C31—22no—
63C2×C4×C31—24no—
64C2×C4×C31—24no—
65C2×C4×C31C2×D412yes—
66C2×C4×C31C2×D412yes—
67C2×C4×C31#4114yes—
68C2×Dic31—28no—
69C2×Dic31—28no—
70C2×Dic31—22no—
71C2×Dic31—22no—
72C2×Dic31—22no—
73C2×Dic31—22no—
74C2×Dic31—24no—
75C2×Dic31—24no—
76C2×Dic31—24no—
77C2×Dic31C2×Q814yes—
78C2×Dic31#4114yes—
79#2855—311nomaximal, Sylow
80C22×C31—22no—
81C22×C31—22no—
82C22×C31—24no—
83C22×C31—24no—
84C22×C31C22⋊Q814yes—
85C22×C31C22×D411yescharacteristic
86C22×C31C22×Q811yescharacteristic
87C22×C31C2×#4112yes—
88C22×C31C2×#4111yescharacteristic
89C22×C31C2×#4112yes—
90C4×C31—22no—
91C4×C31—42no—
92C4×C31—42no—
93C4×C31#12314yes—
94Dic31—24no—
95Dic31—24no—
96Dic31—44no—
97C2×C22⋊Q8D3111yescharacteristic
98C22×C42—311no—
99#391—312no—
100#389—312no—
101#387—314no—
102#385—312no—
103#381—312no—
104#380—311no—
105C2×C31—24no—
106C2×C31C2×C22⋊Q811yescharacteristic
107C2×C31C2×#12311yescharacteristic
108C2×C31#52511yescharacteristic
109C2×C31#52912yes—
110C2×C31#53312yes—
111C22×Q8C2×D3111yescharacteristic
112C23×C4—311no—
113C23×C4—311no—
114C23×C4C2×D3111yescharacteristic
115C22⋊Q8—24no—
116C2×C4⋊C4—312no—
117C2×C4⋊C4—314no—
118C2×C4⋊C4C2×D3112yes—
119C2×C4⋊C4C2×D3111yescharacteristic
120C2×C22⋊C4—312no—
121C2×C22⋊C4C2×D3112yes—
122C2×C42—312no—
123C2×C42—312no—
124C2×C42—624no—
125#95—312no—
126#95—314no—
127#95—314no—
128#95—314no—
129C31#285511yescharacteristic, complemented, Sylow
130C24C22×D3111yescharacteristic
131C2×Q8—24no—
132C2×Q8—22no—
133C22×C4—24no—
134C22×C4—314no—
135C22×C4—312no—
136C22×C4—312no—
137C22×C4—314no—
138C22×C4—624no—
139C22×C4—622no—
140C22×C4—622no—
141C22×C4—622no—
142C22×C4C22⋊D3114yes—
143C22×C4C22×D3112yes—
144C22×C4C22×D3112yes—
145C22×C4C22×D3111yescharacteristic
146C22×C4C22×D3111yescharacteristic
147C4⋊C4—22no—
148C4⋊C4—24no—
149C4⋊C4—628no—
150C4⋊C4—624no—
151C22⋊C4—24no—
152C22⋊C4—624no—
153C42—624no—
154C42—624no—
155C23—22no—
156C23—22no—
157C23Q8×D3112yes—
158C23#9249912yes—
159C23C2×C22⋊D3112yes—
160C23C23×D3111yescharacteristic
161Q8—44no—
162C2×C4—22no—
163C2×C4—28no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—314no—
169C2×C4—314no—
170C2×C4—628no—
171C2×C4—628no—
172C2×C4—622no—
173C2×C4—622no—
174C2×C4—622no—
175C2×C4—622no—
176C2×C4—624no—
177C2×C4—624no—
178C2×C4—624no—
179C2×C4#9249714yes—
180C2×C4C2×C22⋊D3112yes—
181C2×C4C2×C22⋊D3112yes—
182C22—24no—
183C22—22no—
184C22—22no—
185C22—24no—
186C22#1175852314yes—
187C22C2×#9249712yes—
188C22C2×Q8×D3111yescharacteristic
189C22C2×#9249911yescharacteristic
190C22#41×D3112yes—
191C22C22×C22⋊D3111yescharacteristic
192C4—22no—
193C4—42no—
194C4—42no—
195C4—624no—
196C4—624no—
197C4—1244no—
198C4#1175851214yes—
199C2—24no—
200C2C2×#1175852311yescharacteristic
201C2C2×#1175851211yescharacteristic
202C2#1452155611yescharacteristic
203C2#1452158512yes—
204C2#1452160112yes—
subgroupG/H
center, FrattiniC23C23×D31
commutator, socle, cosocleC23×C31C24
FittingC2×C31×C22⋊Q8C2
prime radicalC24×C31C23
hypercenterC2×C22⋊Q8D31
hypocenterC31#2855
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #14520956C2semidirect
3. #14520945C2semidirect
4. #14520930C2semidirect
40. C22×Dic31Q8semidirect
129. C31#2855semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic312
#145209562
#145209452
#145209302
#145209212
#145209102
#145209092
C2×C31×C22⋊Q82
#285531