Finite Group Databasebeta

← #11785359

Subgroups — #11785359

Order 1152.

570 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 103 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×#622C211yesmaximal, characteristic
2C2×#63742C212yesmaximal, complemented
3#10591720C212yesmaximal
4#10590075C212yesmaximal, complemented
5C23×C3⋊Dic3C2211yescharacteristic
6C22×#622—24no—
7C22×#622C414yescomplemented
8C22×#622C2211yescharacteristic
9C22×#622C2211yescharacteristic
10C2×C3×#217—22no—
11C2×#3724C2214yes—
12#63742—24no—
13#63579—22no—
14#4942—62no—
15C24×C32D411yescharacteristic
16C22×C3⋊Dic3—22no—
17C22×C3⋊Dic3—22no—
18C22×C3⋊Dic3C2×C412yes—
19C22×C3⋊Dic3D411yescharacteristic
20C22×C3⋊Dic3Q812yes—
21C22×C3⋊Dic3C2311yescharacteristic
22C22×C3×Dic3—24no—
23C2×#622—22no—
24C2×#622—28no—
25C2×#622C2×C412yes—
26C2×#622C2×C412yes—
27C2×#622D414yes—
28C3×#217—44no—
29#3724—24no—
30#3724—24no—
31#1807—91nomaximal, Sylow
32C23×Dic3—62no—
33C2×C3×C22⋊C4—62no—
34C2×#217—22no—
35C2×Dic3⋊C4—64no—
36C3⋊#95—64no—
37C23×C32—22no—
38C23×C32—22no—
39C23×C32C4⋊C412yes—
40C23×C32SD1614yes—
41C23×C32C2×D411yescharacteristic
42C2×C3⋊Dic3—22no—
43C2×C3⋊Dic3—22no—
44C2×C3⋊Dic3—22no—
45C2×C3⋊Dic3—24no—
46C2×C3⋊Dic3C4⋊C412yes—
47C2×C3⋊Dic3C22×C411yescharacteristic
48C2×C3⋊Dic3C2×D412yes—
49C2×C3⋊Dic3C2×Q811yescharacteristic
50C2×C3⋊Dic3#4112yes—
51C2×C3×Dic3—44no—
52C2×C3×Dic3—44no—
53C2×C3×Dic3—44no—
54#622—22no—
55#622C22⋊C414yescomplemented
56C23×C8—91no—
57C2×#106—92no—
58#380—92no—
59#340—92no—
60C24×C3—22no—
61C22×C3×C4—64no—
62C22×Dic3—24no—
63C22×Dic3—62no—
64C22×Dic3—62no—
65C22×Dic3—64no—
66C22×Dic3—64no—
67C22×Dic3—124no—
68C22×Dic3—124no—
69C3×C22⋊C4—124no—
70#217—44no—
71Dic3⋊C4—124no—
72Dic3⋊C4—124no—
73C22×C32—22no—
74C22×C32—22no—
75C22×C32—28no—
76C22×C32#10614yescomplemented
77C22×C32C2×C4⋊C411yescharacteristic
78C22×C32#12112yes—
79C22×C32C2×SD1612yes—
80C22×C32#13512yes—
81C3⋊Dic3—22no—
82C3⋊Dic3C2×C22⋊C411yescharacteristic
83C3⋊Dic3#11711yescharacteristic
84C3⋊Dic3C22⋊Q812yes—
85C3×Dic3—84no—
86C23×C4—91no—
87C22×C8—91no—
88C22×C8—94no—
89C22×C8—91no—
90C22×C8—184no—
91C2×C4⋊C4—94no—
92C2×C22⋊C4—62no—
93#106—184no—
94#95—182no—
95C23×C3—22no—
96C23×C3—24no—
97C23×C3—24no—
98C23×C3—24no—
99C23×C3—44no—
100C23×C3—44no—
101C2×C3×C4—124no—
102C2×C3×C4—124no—
103C2×C3×C4—124no—
104C2×Dic3—44no—
105C2×Dic3—44no—
106C2×Dic3—44no—
107C2×Dic3—62no—
108C2×Dic3—64no—
109C2×Dic3—62no—
110C2×Dic3—64no—
111C2×Dic3—64no—
112C2×Dic3—124no—
113C2×Dic3—124no—
114C2×Dic3—128no—
115C2×Dic3—124no—
116C2×C32—24no—
117C2×C32#38011yescharacteristic
118C2×C32C2×#10611yescharacteristic
119C2×C32#42711yescharacteristic
120C2×C32#46514yes—
121C24GO+(4,𝔽2)11yescharacteristic
122C22×C4—64no—
123C22×C4—91no—
124C22×C4—91no—
125C22×C4—92no—
126C22×C4—92no—
127C22×C4—182no—
128C22×C4—182no—
129C2×C8—94no—
130C2×C8—92no—
131C2×C8—92no—
132C2×C8—188no—
133C2×C8—182no—
134C4⋊C4—184no—
135C4⋊C4—184no—
136C22⋊C4—124no—
137C22×C3—22no—
138C22×C3—24no—
139C22×C3—22no—
140C22×C3—22no—
141C22×C3—24no—
142C22×C3—28no—
143C22×C3—44no—
144C22×C3—44no—
145C22×C3—48no—
146C22×C3—48no—
147C3×C4—244no—
148Dic3—62no—
149Dic3—62no—
150Dic3—64no—
151Dic3—84no—
152Dic3—124no—
153C32#180711yescharacteristic, complemented, Sylow
154C23—22no—
155C23—22no—
156C23#377312yes—
157C23#377514yes—
158C23C2×GO+(4,𝔽2)11yescharacteristic
159C2×C4—91no—
160C2×C4—91no—
161C2×C4—92no—
162C2×C4—92no—
163C2×C4—92no—
164C2×C4—124no—
165C2×C4—124no—
166C2×C4—124no—
167C2×C4—182no—
168C2×C4—182no—
169C2×C4—184no—
170C2×C4—182no—
171C8—94no—
172C8—182no—
173C2×C3—22no—
174C2×C3—22no—
175C2×C3—22no—
176C2×C3—24no—
177C2×C3—24no—
178C2×C3—48no—
179C22—22no—
180C22—22no—
181C22—28no—
182C22#6374214yescomplemented
183C22C2×#377311yescharacteristic
184C22#6423712yes—
185C22C2×#377512yes—
186C22#6424212yes—
187C4—91no—
188C4—91no—
189C4—92no—
190C4—182no—
191C4—244no—
192C3—22no—
193C2—24no—
194C2C2×#6374211yescharacteristic
195C2#1059351111yescharacteristic
196C2#1059351611yescharacteristic
197C2#1059352614yes—
subgroupG/H
center, FrattiniC23C2×GO+(4,𝔽2)
commutatorC2×C3⋊Dic3C22×C4
Fitting, prime radicalC24×C32D4
socleC23×C32C2×D4
cosocleC22×C3⋊Dic3C23
hypercenterC24GO+(4,𝔽2)
hypocenterC32#1807
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C2×#63742C2semidirect
4. #10590075C2semidirect
7. C22×#622C4semidirect
55. #622C22⋊C4semidirect
76. C22×C32#106semidirect
153. C32#1807semidirect
182. C22#63742semidirect

Maximal subgroups

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

subgroupindex
C23×#6222
C2×#637422
#105917202
#105900752
#18079