Finite Group Databasebeta

← #13072261

Subgroups — #13072261

Order 1408.

800 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 197 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
1C23×Dic22C211yesmaximal, characteristic
2C2×C11×C22⋊Q8C211yesmaximal, characteristic
3C2×#66997C212yesmaximal, complemented
4C4×C22⋊Dic11C211yesmaximal, characteristic
5#10622592C214yesmaximal, complemented
6#10622587C212yesmaximal, complemented
7#10622559C214yesmaximal
8C22×Q8×C11C2211yescharacteristic
9C23×C4×C11C2211yescharacteristic
10C23×Dic11C2212yes—
11C22×Dic22—24no—
12C22×Dic22—28no—
13C22×Dic22C2212yes—
14C22×Dic22C2212yes—
15C11×C22⋊Q8—24no—
16C2×C11×C4⋊C4C2211yescharacteristic
17C2×C11×C4⋊C4C2212yes—
18C2×C11×C22⋊C4C2212yes—
19C2×C22⋊Dic11C2212yes—
20#66997—28no—
21C2×Dic11⋊C4C2218yes—
22C2×C4×Dic11C2212yes—
23C11⋊#95C2218yes—
24C11⋊#95C2212yes—
25C24×C11C2311yescharacteristic
26C2×Q8×C11—24no—
27C2×Q8×C11—22no—
28C22×C4×C11—24no—
29C22×C4×C11D414yes—
30C22×C4×C11C2312yes—
31C22×C4×C11C2312yes—
32C22×C4×C11C2311yescharacteristic
33C22×C4×C11C2311yescharacteristic
34C22×Dic11—28no—
35C22×Dic11—24no—
36C22×Dic11C2314yes—
37C22×Dic11C2314yes—
38C2×Dic22—232no—
39C2×Dic22—28no—
40C2×Dic22—24no—
41C2×Dic22—24no—
42C2×Dic22—24no—
43C2×Dic22D418yes—
44C11×C4⋊C4—22no—
45C11×C4⋊C4—24no—
46C11×C22⋊C4—24no—
47C22⋊Dic11Q818yescomplemented
48Dic11⋊C4—216no—
49C4×Dic11—24no—
50#2434—111nomaximal, Sylow
51C23×C11—22no—
52C23×C11—22no—
53C23×C11C2×D412yes—
54C23×C11C2×Q814yes—
55C23×C11C2411yescharacteristic
56Q8×C11—44no—
57C2×C4×C11—22no—
58C2×C4×C11—28no—
59C2×C4×C11—24no—
60C2×C4×C11—22no—
61C2×C4×C11—22no—
62C2×C4×C11—24no—
63C2×C4×C11C2×D414yes—
64C2×C4×C11C2×D412yes—
65C2×C4×C11C2×D412yes—
66C2×Dic11—216no—
67C2×Dic11—24no—
68C2×Dic11—24no—
69C2×Dic11—24no—
70C2×Dic11—24no—
71C2×Dic11C2×D418yes—
72C2×Dic11C2×Q818yescomplemented
73Dic22—216no—
74Dic22—216no—
75C23×Q8—111no—
76C2×C4⋊Q8—112no—
77C2×C22⋊Q8D1111yescharacteristic
78#395—114no—
79#391—112no—
80#385—114no—
81C4×C22⋊C4—111no—
82C22×C11—22no—
83C22×C11—22no—
84C22×C11—28no—
85C22×C11C4⋊Q814yescomplemented
86C22×C11C22×D412yes—
87C22×C11C22×D411yescharacteristic
88C22×C11C22×Q812yes—
89C22×C112−1+412yes—
90C4×C11—22no—
91C4×C11—42no—
92C4×C11—42no—
93C4×C11#12014yes—
94Dic11—28no—
95Dic11—28no—
96C22×Q8—112no—
97C22×Q8—112no—
98C22×Q8—228no—
99C22×Q8—224no—
100C22×Q8C2×D1111yescharacteristic
101C23×C4—112no—
102C23×C4C2×D1111yescharacteristic
103C4⋊Q8—228no—
104C22⋊Q8—24no—
105C2×C4⋊C4—118no—
106C2×C4⋊C4C2×D1111yescharacteristic
107C2×C4⋊C4C2×D1112yes—
108C2×C22⋊C4—112no—
109C2×C22⋊C4C2×D1112yes—
110C2×C42—112no—
111#95—112no—
112#95—118no—
113C2×C11—24no—
114C2×C11C2×#12011yescharacteristic
115C2×C11C2×C4⋊Q811yescharacteristic
116C2×C11#53611yescharacteristic
117C2×C11D4×Q814yes—
118C24C22×D1111yescharacteristic
119C2×Q8—24no—
120C2×Q8—22no—
121C2×Q8—118no—
122C2×Q8—228no—
123C2×Q8—2232no—
124C2×Q8—224no—
125C2×Q8—224no—
126C2×Q8—224no—
127C22×C4—24no—
128C22×C4—114no—
129C22×C4—114no—
130C22×C4—228no—
131C22×C4—224no—
132C22×C4C22⋊D1114yes—
133C22×C4C22×D1112yes—
134C22×C4C22×D1112yes—
135C22×C4C22×D1111yescharacteristic
136C22×C4C22×D1111yescharacteristic
137C4⋊C4—22no—
138C4⋊C4—24no—
139C4⋊C4—2216no—
140C22⋊C4—24no—
141C22⋊C4—118no—
142C42—224no—
143C11#243411yescharacteristic, complemented, Sylow
144C23—22no—
145C23—22no—
146C23Q8×D1114yes—
147C23C2×C22⋊D1112yes—
148C23C23×D1111yescharacteristic
149Q8—44no—
150Q8—2216no—
151Q8—2216no—
152C2×C4—22no—
153C2×C4—28no—
154C2×C4—24no—
155C2×C4—24no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—118no—
159C2×C4—118no—
160C2×C4—2216no—
161C2×C4—224no—
162C2×C4—224no—
163C2×C4—224no—
164C2×C4—224no—
165C2×C4D4×D1114yes—
166C2×C4C2×C22⋊D1112yes—
167C2×C4C2×C22⋊D1112yes—
168C22—22no—
169C22—22no—
170C22—28no—
171C22#6699714yescomplemented
172C22C2×D4×D1112yes—
173C22C2×Q8×D1112yes—
174C22#6704312yes—
175C22C22×C22⋊D1111yescharacteristic
176C4—22no—
177C4—42no—
178C4—42no—
179C4—228no—
180C4—228no—
181C4#6699014yes—
182C2—24no—
183C2#1062316314yes—
184C2C2×#6699011yescharacteristic
185C2C2×#6699711yescharacteristic
186C2#1062335611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D11
commutator, socle, cosocleC23×C11C24
FittingC2×C11×C22⋊Q8C2
prime radicalC24×C11C23
hypercenterC2×C22⋊Q8D11
hypocenterC11#2434
LCSG ▷ C23×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#66997C2semidirect
5. #10622592C2semidirect
6. #10622587C2semidirect
47. C22⋊Dic11Q8semidirect
72. C2×Dic11C2×Q8semidirect
85. C22×C11C4⋊Q8semidirect
143. C11#2434semidirect
171. C22#66997semidirect

Maximal subgroups

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

subgroupindex
C23×Dic222
C2×C11×C22⋊Q82
C2×#669972
C4×C22⋊Dic112
#106225922
#106225872
#106225592
#243411