Finite Group Databasebeta

← #11794296

Subgroups — #11794296

Order 1152.

264 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 53 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
1#10596543C211yesmaximal, characteristic
2#10593774C211yesmaximal, characteristic
3#10593545C211yesmaximal, characteristic
4#10593544C211yesmaximal, characteristic
5#10590189C211yesmaximal, characteristic
6#10590188C211yesmaximal, characteristic
7#10590092C211yesmaximal, characteristic
8#64286C2211yescharacteristic
9#64286C2211yescharacteristic
10C2×C4⋊#157C2211yescharacteristic
11C2×C4⋊#157C2211yescharacteristic
12C2×C4×#157C2211yescharacteristic
13#64283C2211yescharacteristic
14C22⋊PSU(3,𝔽2)—22no—
15C4⋊PSU(3,𝔽2)—21no—
16C4⋊PSU(3,𝔽2)—21no—
17#64243—21no—
18#64243—21no—
19M16×C3⋊D3C2211yescharacteristic
20#63769—21no—
21#63768—21no—
22#63747—22no—
23#63746—21no—
24#63745—21no—
25C22×#157—21no—
26C22×#157—21no—
27C22×#157D412yes—
28C2×PSU(3,𝔽2)—22no—
29C2×C4×C3⋊D3C2311yescharacteristic
30C22⋊#157—21no—
31C22⋊#157—21no—
32#3792Q812yescomplemented
33C4⋊#157—21no—
34C4⋊#157—21no—
35C4⋊#157—21no—
36C4⋊#157—21no—
37C4⋊#157D411yescharacteristic
38C4⋊#157D411yescharacteristic
39C4⋊#157D411yescharacteristic
40C4⋊#157D411yescharacteristic
41C4×#157—21no—
42C4×#157—21no—
43C4×#157—21no—
44C4×#157Q812yescomplemented
45#3788—21no—
46#3787—21no—
47#3777—22no—
48#3777—22no—
49#3777—22no—
50C32×M16Q811yescharacteristic, complemented
51#3748Q811yescharacteristic, complemented
52#3743—21no—
53C8×C3⋊D3—21no—
54#2061—91nomaximal, Sylow
55D3×M16—121no—
56C22×C3⋊D3C2×D411yescharacteristic
57C2×#157—21no—
58C2×#157—21no—
59C2×#157—22no—
60C2×#157—21no—
61C2×#157—21no—
62C2×#157—22no—
63C2×#157—41no—
64C2×#157—41no—
65C2×#157—42no—
66C2×#157—42no—
67C2×#157—41no—
68PSU(3,𝔽2)—82no—
69C2×C32×C4C2×Q811yescharacteristic
70C2×C3⋊Dic3C2×Q811yescharacteristic
71C4×C3⋊D3C2×D411yescharacteristic
72C4×C3⋊D3C2×D411yescharacteristic
73C4×C3⋊D3C2×Q811yescharacteristic
74C4×C3⋊D3#4111yescharacteristic
75#622—22no—
76C32×C8—21no—
77C3⋊(C3⋊C8)—21no—
78#544—91no—
79#428—91no—
80#428—91no—
81#382—91no—
82#343—91no—
83#339—91no—
84#339—91no—
85C2×C4×D3—121no—
86C3×M16—41no—
87#208—121no—
88ΓL(1,𝔽25)—122no—
89D3×C8—122no—
90C22×C32C4⋊Q811yescharacteristic
91C2×C3⋊D3—21no—
92C2×C3⋊D3—21no—
93C2×C3⋊D3#12011yescharacteristic
94C2×C3⋊D3C22⋊Q811yescharacteristic
95#157—42no—
96#157—41no—
97#157—41no—
98#157—41no—
99#157—41no—
100#157—42no—
101#157—81no—
102C32×C4C22⋊Q811yescharacteristic
103C32×C4C4⋊Q811yescharacteristic
104C3⋊Dic3C22⋊Q811yescharacteristic
105C3⋊Dic3C4⋊Q811yescharacteristic
106C2×M16—91no—
107C2×M16—91no—
108C4⋊Q8—181no—
109C4⋊Q8—181no—
110#125—181no—
111#125—181no—
112C22⋊Q8—182no—
113#117—91no—
114#117—91no—
115C2×C4⋊C4—91no—
116C2×C4⋊C4—91no—
117C2×C42—91no—
118#107—181no—
119#107—181no—
120#106—181no—
121#106—181no—
122#95—182no—
123C22×D3—121no—
124C2×C3×C4—41no—
125C2×Dic3—121no—
126C4×D3—121no—
127C4×D3—121no—
128C4×D3—121no—
129C4×D3—121no—
130C3×C8—42no—
131C3⋊C8—122no—
132C2×C32—21no—
133C2×C32#39511yescharacteristic
134C3⋊D3—21no—
135C3⋊D3#45411yescharacteristic
136C3⋊D3#45511yescharacteristic
137C2×Q8—182no—
138C22×C4—91no—
139C22×C4—92no—
140C22×C4—181no—
141C22×C4—181no—
142M16—92no—
143M16—91no—
144M16—181no—
145M16—181no—
146M16PSU(3,𝔽2)11yescharacteristic, complemented
147C2×C8—181no—
148C2×C8—181no—
149C4⋊C4—91no—
150C4⋊C4—91no—
151C4⋊C4—91no—
152C4⋊C4—91no—
153C4⋊C4—181no—
154C4⋊C4—181no—
155C4⋊C4—181no—
156C4⋊C4—181no—
157C4⋊C4—182no—
158C4⋊C4—182no—
159C4⋊C4—182no—
160C22⋊C4—181no—
161C22⋊C4—181no—
162C42—92no—
163C42—181no—
164C42—181no—
165C42—181no—
166C22×C3—41no—
167C2×D3—121no—
168C2×D3—121no—
169C2×D3—241no—
170C2×D3—241no—
171C3×C4—41no—
172C3×C4—41no—
173Dic3—121no—
174Dic3—121no—
175C32#206111yescharacteristic, complemented, Sylow
176C23—91no—
177Q8—722no—
178C2×C4—91no—
179C2×C4—91no—
180C2×C4—91no—
181C2×C4—91no—
182C2×C4—91no—
183C2×C4—182no—
184C2×C4—181no—
185C2×C4—181no—
186C2×C4—181no—
187C2×C4—181no—
188C2×C4—182no—
189C2×C4—361no—
190C2×C4—361no—
191C2×C4—362no—
192C2×C4—361no—
193C2×C4—362no—
194C2×C4C2×PSU(3,𝔽2)11yescharacteristic
195C8—21no—
196C8—182no—
197C8—181no—
198C2×C3—41no—
199C2×C3—81no—
200D3—121no—
201D3—121no—
202D3—241no—
203C22—91no—
204C22—91no—
205C22—181no—
206C22—181no—
207C22C4⋊PSU(3,𝔽2)11yescharacteristic
208C4—91no—
209C4—91no—
210C4—362no—
211C4—361no—
212C4—361no—
213C4—361no—
214C4—361no—
215C4—362no—
216C4—721no—
217C4C4⋊PSU(3,𝔽2)11yescharacteristic
218C4C22⋊PSU(3,𝔽2)11yescharacteristic
219C3—41no—
220C2—21no—
221C2—91no—
222C2—91no—
223C2—181no—
224C2#1059355611yescharacteristic
subgroupG/H
centerC2#10593556
commutator, cosocleC2×C4×C3⋊D3C23
FrattiniC2×C4C2×PSU(3,𝔽2)
FittingC32×M16Q8
socleC2×C32#395
prime radicalC22×C3⋊D3C2×D4
hypercenterM16PSU(3,𝔽2)
hypocenterC32#2061
LCSG ▷ C2×C4×C3⋊D3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
32. #3792Q8semidirect
44. C4×#157Q8semidirect
50. C32×M16Q8semidirect
51. #3748Q8semidirect
146. M16PSU(3,𝔽2)semidirect
175. C32#2061semidirect

Maximal subgroups

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

subgroupindex
#105965432
#105937742
#105935452
#105935442
#105901892
#105901882
#105900922
#20619