Finite Group Databasebeta

← #8911

Subgroups — #8911

Order 256.

242 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 67 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#2379C211yesmaximal, characteristic
2#1959C211yesmaximal, characteristic
3#1847C211yesmaximal, characteristic, complemented
4#1836C211yesmaximal, characteristic, complemented
5C2×#329C211yesmaximal, characteristic
6#1295C211yesmaximal, characteristic, complemented
7#1295C211yesmaximal, characteristic, complemented
8C2×#124C2211yescharacteristic
9C2×#107—21no—
10C2×#106—21no—
11C2×#103—21no—
12C2×#102—21no—
13C2×#97C2211yescharacteristic
14C2×#97C2211yescharacteristic
15C2×#97C2211yescharacteristic
16#391—21no—
17#388—21no—
18#388—21no—
19#386—21no—
20#385—21no—
21#385—21no—
22#384C2211yescharacteristic
23#382C2211yescharacteristic
24#382C2211yescharacteristic
25#340—21no—
26#340—21no—
27#329C414yescomplemented
28C22×Q8D411yescharacteristic
29C22×D4D411yescharacteristic
30C23×C4—21no—
31C22×C8—21no—
32C22×C8—21no—
33C22×C8—21no—
34#124—21no—
35#124—21no—
36#124C2×C412yes—
37#124Q812yes—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4D411yescharacteristic
43C2×C4⋊C4D411yescharacteristic
44C2×C4⋊C4D411yescharacteristic
45C2×C4⋊C4D411yescharacteristic
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4—21no—
49C2×C22⋊C4—21no—
50C2×C22⋊C4—21no—
51C2×C42—21no—
52C2×C42C2311yescharacteristic
53#107—42no—
54#106—42no—
55#103—42no—
56#102—42no—
57#97—21no—
58#97—21no—
59#97—21no—
60#97—21no—
61#97C2×C412yescomplemented
62#97C2×C412yescomplemented
63#95—21no—
64#95—21no—
65#95—21no—
66#95—21no—
67#95—21no—
68#95—21no—
69C24—21no—
70C2×Q8—21no—
71C2×Q8—41no—
72C2×Q8—41no—
73C2×Q8C4⋊C412yes—
74C2×D4—21no—
75C2×D4—41no—
76C2×D4—41no—
77C2×D4C4⋊C412yes—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—41no—
84C22×C4—81no—
85C22×C4—81no—
86C22×C4—81no—
87C22×C4C2×D411yescharacteristic
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D411yescharacteristic
90C2×C8—22no—
91C2×C8—22no—
92C2×C8—22no—
93C2×C8—22no—
94C2×C8—22no—
95C2×C8—41no—
96C2×C8—41no—
97C2×C8—41no—
98C2×C8—41no—
99C4⋊C4—41no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C4⋊C4—41no—
105C4⋊C4—41no—
106C4⋊C4—41no—
107C4⋊C4—42no—
108C4⋊C4—81no—
109C22⋊C4—22no—
110C22⋊C4—22no—
111C22⋊C4—22no—
112C22⋊C4—22no—
113C22⋊C4—81no—
114C22⋊C4—81no—
115C22⋊C4—81no—
116C42—81no—
117C42C22×C411yescharacteristic
118C42C2×Q811yescharacteristic
119C42#4111yescharacteristic
120C42#4111yescharacteristic
121C23—22no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23—41no—
127C23—41no—
128C23#12011yescharacteristic
129Q8—42no—
130Q8—81no—
131D4—42no—
132D4—81no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—22no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—84no—
155C2×C4—81no—
156C2×C4—81no—
157C2×C4—81no—
158C2×C4C2×C4⋊C411yescharacteristic
159C2×C4C4×D411yescharacteristic
160C2×C4C4×D411yescharacteristic
161C2×C4C22⋊Q811yescharacteristic
162C2×C4C22⋊Q811yescharacteristic
163C2×C4#12311yescharacteristic
164C8—24no—
165C8—42no—
166C8—42no—
167C22—42no—
168C22—42no—
169C22—42no—
170C22—81no—
171C22—81no—
172C22—81no—
173C22—81no—
174C22#38511yescharacteristic
175C22#45311yescharacteristic
176C22#45311yescharacteristic
177C22#45411yescharacteristic
178C22#45511yescharacteristic
179C22#45511yescharacteristic
180C22#45611yescharacteristic
181C4—21no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—21no—
186C4—21no—
187C4—42no—
188C4—81no—
189C4—81no—
190C4—81no—
191C4—81no—
192C4—161no—
193C2—82no—
194C2#165611yescharacteristic
195C2#165911yescharacteristic
196C2#166111yescharacteristic
197C2#166411yescharacteristic
198C2#185611yescharacteristic
199C2#185911yescharacteristic
200C2#186011yescharacteristic
subgroupG/H
center, socleC23#120
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1847C2semidirect
4. #1836C2semidirect
6. #1295C2semidirect
7. #1295C2semidirect
27. #329C4semidirect
61. #97C2×C4semidirect
62. #97C2×C4semidirect

Maximal subgroups

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

subgroupindex
#23792
#19592
#18472
#18362
C2×#3292
#12952