Finite Group Databasebeta

← #11783148

Subgroups — #11783148

Order 1152.

522 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 125 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×#64286C212yesmaximal
2C2×#64283C211yesmaximal, characteristic
3#10590091C214yesmaximal, complemented
4C22×C4×C3⋊D3C2211yescharacteristic
5C2×C22⋊#157—22no—
6C2×#3792C412yescomplemented
7#64286—22no—
8#64286—22no—
9#64286C414yescomplemented
10#64286C2214yes—
11C2×C4⋊#157—22no—
12C2×C4×#157C414yescomplemented
13#64283—22no—
14#64283—22no—
15C2×#3788C2212yes—
16C2×#3787C412yescomplemented
17C23×C3⋊D3D411yescharacteristic
18C22×#157—24no—
19C22×C32×C4Q811yescharacteristic
20C22×C3⋊Dic3C2×C411yescharacteristic
21C2×C4×C3⋊D3C2×C411yescharacteristic
22C2×C4×C3⋊D3C2×C412yes—
23C2×C4×C3⋊D3C2×C412yes—
24C2×C4×C3⋊D3D411yescharacteristic
25C2×C4×C3⋊D3D412yes—
26C2×C4×C3⋊D3D412yes—
27C2×C4×C3⋊D3Q811yescharacteristic
28C2×C4×C3⋊D3C2311yescharacteristic
29C22⋊#157—24no—
30C22⋊#157—24no—
31#3792—24no—
32C2×#622C2×C414yescomplemented
33C4⋊#157—24no—
34C4⋊#157—24no—
35C4×#157—24no—
36C4×#157C2×C418yescomplemented
37#3788—24no—
38#3787—22no—
39#3787—22no—
40C2×H(3,ℤ/4)—91nomaximal, Sylow
41C22×C4×D3—121no—
42C23×C32C4⋊C411yescharacteristic
43C22×C3⋊D3—42no—
44C22×C3⋊D3C22⋊C412yes—
45C22×C3⋊D3C22⋊C412yes—
46C22×C3⋊D3C4⋊C411yescharacteristic
47C22×C3⋊D3C2×D411yescharacteristic
48C2×#157—28no—
49C2×#157—48no—
50C2×C32×C4C4⋊C411yescharacteristic
51C2×C32×C4C4⋊C412yes—
52C2×C32×C4C4⋊C412yes—
53C2×C32×C4C2×Q811yescharacteristic
54C2×C3⋊Dic3C4212yes—
55C2×C3⋊Dic3C22⋊C412yes—
56C2×C3⋊Dic3C4⋊C411yescharacteristic
57C2×C3⋊Dic3C22×C411yescharacteristic
58C4×C3⋊D3C4211yescharacteristic
59C4×C3⋊D3C4211yescharacteristic
60C4×C3⋊D3C22⋊C412yes—
61C4×C3⋊D3C22⋊C412yes—
62C4×C3⋊D3C22⋊C411yescharacteristic
63C4×C3⋊D3C22⋊C411yescharacteristic
64C4×C3⋊D3C4⋊C412yes—
65C4×C3⋊D3C4⋊C412yes—
66C4×C3⋊D3C22×C412yes—
67C4×C3⋊D3C2×D412yes—
68#622—24no—
69C22×M16—91no—
70C2×#117—92no—
71H(3,ℤ/4)—94no—
72C23×D3—121no—
73C22×C3×C4—41no—
74C22×Dic3—121no—
75C2×C4×D3—122no—
76C2×C4×D3—122no—
77C2×C4×D3—121no—
78C2×C4×D3—122no—
79C2×C4×D3—121no—
80C2×C4×D3—122no—
81C2×C4×D3—121no—
82C2×C4×D3—121no—
83C22×C32—41no—
84C22×C32#9512yes—
85C22×C32C2×C4⋊C411yescharacteristic
86C2×C3⋊D3—24no—
87C2×C3⋊D3—22no—
88C2×C3⋊D3—42no—
89C2×C3⋊D3—41no—
90C2×C3⋊D3#9511yescharacteristic
91C2×C3⋊D3#9511yescharacteristic
92C2×C3⋊D3C2×C22⋊C412yes—
93#157—48no—
94C32×C4#9511yescharacteristic
95C32×C4#9511yescharacteristic
96C32×C4C2×C4⋊C412yes—
97C3⋊Dic3#9512yes—
98C3⋊Dic3C2×C4211yescharacteristic
99C3⋊Dic3C2×C22⋊C411yescharacteristic
100C23×C4—91no—
101C2×M16—92no—
102C2×M16—92no—
103C2×M16—182no—
104C2×M16—182no—
105C22×C8—92no—
106#117—94no—
107#117—94no—
108#117—182no—
109#117—182no—
110C2×C4⋊C4—182no—
111C2×C22⋊C4—182no—
112C2×C42—94no—
113C23×C3—41no—
114C22×D3—122no—
115C22×D3—122no—
116C22×D3—121no—
117C22×D3—121no—
118C22×D3—128no—
119C2×C3×C4—41no—
120C2×C3×C4—41no—
121C2×C3×C4—42no—
122C2×C3×C4—42no—
123C2×Dic3—122no—
124C2×Dic3—122no—
125C2×Dic3—121no—
126C2×Dic3—121no—
127C4×D3—121no—
128C4×D3—121no—
129C4×D3—122no—
130C4×D3—121no—
131C4×D3—122no—
132C4×D3—121no—
133C4×D3—122no—
134C4×D3—122no—
135C4×D3—122no—
136C4×D3—122no—
137C2×C32—22no—
138C2×C32—21no—
139C2×C32C2×#9511yescharacteristic
140C3⋊D3—22no—
141C3⋊D3—21no—
142C3⋊D3H(3,ℤ/4)12yescomplemented
143C24—91no—
144C22×C4—91no—
145C22×C4—91no—
146C22×C4—91no—
147C22×C4—92no—
148C22×C4—92no—
149C22×C4—92no—
150C22×C4—91no—
151C22×C4—92no—
152C22×C4—91no—
153C22×C4—184no—
154C22×C4PSU(3,𝔽2)11yescharacteristic
155M16—184no—
156M16—184no—
157C2×C8—94no—
158C2×C8—182no—
159C2×C8—182no—
160C4⋊C4—184no—
161C4⋊C4—184no—
162C22⋊C4—184no—
163C22⋊C4—184no—
164C42—98no—
165C42—184no—
166C22×C3—44no—
167C22×C3—42no—
168C22×C3—41no—
169C2×D3—121no—
170C2×D3—121no—
171C2×D3—122no—
172C2×D3—128no—
173C2×D3—124no—
174C2×D3—128no—
175C2×D3—124no—
176C3×C4—41no—
177C3×C4—41no—
178C3×C4—42no—
179Dic3—122no—
180Dic3—121no—
181Dic3—121no—
182C32C2×H(3,ℤ/4)11yescharacteristic, complemented, Sylow
183C23—92no—
184C23—91no—
185C23—92no—
186C23—91no—
187C23—362no—
188C23#377711yescharacteristic
189C2×C4—92no—
190C2×C4—92no—
191C2×C4—91no—
192C2×C4—91no—
193C2×C4—92no—
194C2×C4—91no—
195C2×C4—91no—
196C2×C4—92no—
197C2×C4—91no—
198C2×C4—91no—
199C2×C4—92no—
200C2×C4—92no—
201C2×C4—92no—
202C2×C4—92no—
203C2×C4—188no—
204C2×C4—368no—
205C2×C4#377711yescharacteristic
206C2×C4#377712yes—
207C2×C4#377712yes—
208C2×C4C2×PSU(3,𝔽2)11yescharacteristic
209C8—184no—
210C2×C3—44no—
211C2×C3—42no—
212C2×C3—41no—
213D3—122no—
214D3—124no—
215D3—122no—
216C22—41no—
217C22—91no—
218C22—92no—
219C22—91no—
220C22—184no—
221C22—182no—
222C22—362no—
223C22—361no—
224C22#6374712yes—
225C22C2×#377711yescharacteristic
226C4—91no—
227C4—92no—
228C4—91no—
229C4—368no—
230C4#6374711yescharacteristic
231C4#6374711yescharacteristic
232C4C2×#377712yes—
233C3—41no—
234C2—22no—
235C2—21no—
236C2—92no—
237C2—182no—
238C2—181no—
239C2C2×#6374711yescharacteristic
subgroupG/H
centerC4#63747
commutatorC3⋊Dic3C2×C42
FrattiniC2×C4C2×PSU(3,𝔽2)
FittingC22×C32×C4Q8
socleC2×C32C2×#95
prime radicalC23×C3⋊D3D4
cosocleC2×C4×C3⋊D3C23
hypercenterC22×C4PSU(3,𝔽2)
hypocenterC32C2×H(3,ℤ/4)
LCSG ▷ C3⋊Dic3 ▷ C32
UCSC1 ◁ C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #10590091C2semidirect
6. C2×#3792C4semidirect
9. #64286C4semidirect
12. C2×C4×#157C4semidirect
16. C2×#3787C4semidirect
32. C2×#622C2×C4semidirect
36. C4×#157C2×C4semidirect
142. C3⋊D3H(3,ℤ/4)semidirect
182. C32C2×H(3,ℤ/4)semidirect

Maximal subgroups

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

subgroupindex
C2×#642862
C2×#642832
#105900912
C2×H(3,ℤ/4)9