Finite Group Databasebeta

← #11791640

Subgroups — #11791640

Order 1152.

338 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 81 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×C4⋊PSU(3,𝔽2)C211yesmaximal, characteristic
2C4⋊C4×#157C211yesmaximal, characteristic
3C2×#63744C211yesmaximal, characteristic
4#10590163C212yesmaximal
5#10590093C212yesmaximal
6C22×PSU(3,𝔽2)—21no—
7C2×C4⋊#157C2212yes—
8C2×C4⋊#157C2211yescharacteristic
9C2×C4×#157—21no—
10C2×C4×#157C2211yescharacteristic
11C2×#3787C2212yes—
12C2×#3777—21no—
13C4⋊PSU(3,𝔽2)—22no—
14C4⋊PSU(3,𝔽2)C414yescomplemented
15C4⋊C4×C3⋊D3C2211yescharacteristic
16#63781—21no—
17#63744—22no—
18C22×#157—21no—
19C22×#157—22no—
20C22×#157D411yescharacteristic
21C22×#157D411yescharacteristic
22C2×PSU(3,𝔽2)—24no—
23C2×PSU(3,𝔽2)—42no—
24C2×PSU(3,𝔽2)—42no—
25C2×C4×C3⋊D3—21no—
26C2×C4×C3⋊D3C2311yescharacteristic
27C2×#622—22no—
28C4⋊#157—22no—
29C4⋊#157—21no—
30C4⋊#157C2×C414yes—
31C4⋊#157Q812yescomplemented
32C4×#157—42no—
33C4×#157C2×C412yes—
34C4×#157D412yes—
35#3787—22no—
36#3787—22no—
37#3777—24no—
38C32×C4⋊C4Q811yescharacteristic, complemented
39C4⋊(C3⋊Dic3)Q811yescharacteristic, complemented
40#3750—21no—
41#1988—91nomaximal, Sylow
42D3×C4⋊C4—121no—
43C22×C3⋊D3C2×D411yescharacteristic
44C2×#157—21no—
45C2×#157—21no—
46C2×#157—24no—
47C2×#157—22no—
48C2×#157—42no—
49C2×#157—41no—
50C2×#157—42no—
51C2×#157—41no—
52C2×#157C22⋊C412yes—
53C2×#157C22⋊C412yes—
54C2×#157D812yes—
55C2×#157SD1612yes—
56PSU(3,𝔽2)—48no—
57C2×C32×C4—21no—
58C2×C32×C4C2×Q811yescharacteristic
59C2×C3⋊Dic3—21no—
60C2×C3⋊Dic3#4111yescharacteristic
61C4×C3⋊D3—41no—
62C4×C3⋊D3—41no—
63C4×C3⋊D3C22×C411yescharacteristic
64C4×C3⋊D3C2×D411yescharacteristic
65C4×C3⋊D3C2×Q811yescharacteristic
66C4×C3⋊D3#4111yescharacteristic
67#622—42no—
68C2×C4⋊Q8—91no—
69C2×C4⋊C8—91no—
70C4×C4⋊C4—91no—
71#340—92no—
72#340—92no—
73C2×C4×D3—121no—
74C2×C4×D3—122no—
75C3×C4⋊C4—41no—
76C4⋊Dic3—121no—
77Dic3⋊C4—122no—
78C22×C32C22⋊Q811yescharacteristic
79C2×C3⋊D3C2×C22⋊C411yescharacteristic
80C2×C3⋊D3C22⋊Q811yescharacteristic
81C2×C3⋊D3C2×D811yescharacteristic
82C2×C3⋊D3C2×SD1611yescharacteristic
83C2×C3⋊D3#13711yescharacteristic
84C2×C3⋊D3#13711yescharacteristic
85#157—22no—
86#157—44no—
87#157—42no—
88#157#10214yes—
89C32×C4—41no—
90C32×C4C4×Q811yescharacteristic
91C32×C4C4⋊Q811yescharacteristic
92C3⋊Dic3—41no—
93C3⋊Dic3C4×Q811yescharacteristic
94C3⋊Dic3#12411yescharacteristic
95C22×Q8—181no—
96C22×C8—92no—
97C4⋊Q8—94no—
98C4⋊Q8—182no—
99C2×C4⋊C4—92no—
100C2×C4⋊C4—91no—
101C2×C4⋊C4—91no—
102C2×C4⋊C4—181no—
103C2×C42—91no—
104C2×C42—181no—
105C4⋊C8—182no—
106#95—181no—
107C22×D3—121no—
108C2×C3×C4—42no—
109C2×C3×C4—41no—
110C2×Dic3—121no—
111C2×Dic3—122no—
112C4×D3—121no—
113C4×D3—121no—
114C4×D3—121no—
115C4×D3—121no—
116C4×D3—242no—
117C4×D3—242no—
118C2×C32#39111yescharacteristic
119C2×C32#47411yescharacteristic
120C2×C32#47511yescharacteristic
121C3⋊D3C2×#10211yescharacteristic
122C3⋊D3#41911yescharacteristic
123C3⋊D3#47411yescharacteristic
124C3⋊D3#47511yescharacteristic
125C2×Q8—184no—
126C2×Q8—362no—
127C2×Q8—362no—
128C22×C4—91no—
129C22×C4—91no—
130C22×C4—91no—
131C22×C4—181no—
132C22×C4—181no—
133C22×C4—182no—
134C2×C8—182no—
135C2×C8—182no—
136C2×C8—182no—
137C4⋊C4—94no—
138C4⋊C4—92no—
139C4⋊C4—91no—
140C4⋊C4—181no—
141C4⋊C4—182no—
142C4⋊C4—181no—
143C4⋊C4—184no—
144C4⋊C4PSU(3,𝔽2)11yescharacteristic, complemented
145C42—92no—
146C42—92no—
147C42—362no—
148C22×C3—41no—
149C2×D3—121no—
150C2×D3—121no—
151C2×D3—121no—
152C2×D3—121no—
153C2×D3—121no—
154C2×D3—121no—
155C3×C4—41no—
156C3×C4—41no—
157C3×C4—82no—
158Dic3—121no—
159Dic3—121no—
160Dic3—242no—
161C32#198811yescharacteristic, complemented, Sylow
162C23—91no—
163Q8—368no—
164C2×C4—21no—
165C2×C4—92no—
166C2×C4—92no—
167C2×C4—92no—
168C2×C4—91no—
169C2×C4—91no—
170C2×C4—91no—
171C2×C4—91no—
172C2×C4—91no—
173C2×C4—92no—
174C2×C4—181no—
175C2×C4—181no—
176C2×C4—184no—
177C2×C4—182no—
178C2×C4—181no—
179C2×C4—361no—
180C2×C4—361no—
181C2×C4—361no—
182C2×C4—362no—
183C2×C4—362no—
184C2×C4—361no—
185C2×C4C2×PSU(3,𝔽2)11yescharacteristic
186C8—362no—
187C2×C3—41no—
188C2×C3—41no—
189C2×C3—41no—
190D3—121no—
191D3—121no—
192D3—121no—
193D3—121no—
194C22—91no—
195C22—91no—
196C22—91no—
197C22—91no—
198C22—91no—
199C22—91no—
200C22C22⋊PSU(3,𝔽2)11yescharacteristic
201C4—41no—
202C4—94no—
203C4—91no—
204C4—91no—
205C4—182no—
206C4—364no—
207C4—362no—
208C4—361no—
209C4C4×PSU(3,𝔽2)11yescharacteristic
210C4C4⋊PSU(3,𝔽2)11yescharacteristic
211C3—41no—
212C2—91no—
213C2—91no—
214C2—91no—
215C2—91no—
216C2#1059355511yescharacteristic
217C2#1059355811yescharacteristic
218C2#1059356111yescharacteristic
subgroupG/H
centerC22C22⋊PSU(3,𝔽2)
commutatorC4×C3⋊D3C22×C4
FrattiniC2×C4C2×PSU(3,𝔽2)
FittingC32×C4⋊C4Q8
socleC22×C32C22⋊Q8
prime radicalC22×C3⋊D3C2×D4
cosocleC2×C4×C3⋊D3C23
hypercenterC4⋊C4PSU(3,𝔽2)
hypocenterC32#1988
LCSG ▷ C4×C3⋊D3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
14. C4⋊PSU(3,𝔽2)C4semidirect
31. C4⋊#157Q8semidirect
38. C32×C4⋊C4Q8semidirect
39. C4⋊(C3⋊Dic3)Q8semidirect
144. C4⋊C4PSU(3,𝔽2)semidirect
161. C32#1988semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊PSU(3,𝔽2)2
C4⋊C4×#1572
C2×#637442
#105901632
#105900932
#19889