Finite Group Databasebeta

← #29882

Subgroups — #29882

Order 256.

336 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 119 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×#499C211yesmaximal, characteristic, complemented
2C2×#433C211yesmaximal, characteristic
3#2629C212yesmaximal
4#2088C212yesmaximal, complemented
5#2087C211yesmaximal, characteristic, complemented
6#2086C211yesmaximal, characteristic, complemented
7#2077C212yesmaximal
8#2076C211yesmaximal, characteristic
9#2075C211yesmaximal, characteristic
10#1992C212yesmaximal, complemented
11#1834C211yesmaximal, characteristic, complemented
12C2×#125C2212yes—
13C2×#117C2211yescharacteristic
14#499—24no—
15#433—24no—
16C2×#107C2211yescharacteristic
17C2×#107C2211yescharacteristic
18C2×#106C2212yes—
19C2×C4⋊C8C2211yescharacteristic
20C2×C4⋊C8C2211yescharacteristic
21C2×C22⋊C8C2211yescharacteristic
22C2×C22⋊C8C2211yescharacteristic
23C2×#97C2211yescharacteristic
24C2×C4×C8C2211yescharacteristic
25#398—22no—
26#398—22no—
27#396—22no—
28#396—22no—
29#389C2212yes—
30#389C2212yes—
31#384C2212yes—
32#380C2212yes—
33#380C2212yes—
34#340C2211yescharacteristic
35#340C2212yes—
36#340C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41C23×C4C2311yescharacteristic
42C22×C8C2311yescharacteristic
43C22×C8C2311yescharacteristic
44C22×C8C2311yescharacteristic
45C22×C8C2311yescharacteristic
46#125—24no—
47#125—24no—
48#117—22no—
49#117—22no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4D411yescharacteristic
57C2×C4⋊C4D411yescharacteristic
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2312yes—
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C22⋊C4—22no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4D411yescharacteristic
65C2×C22⋊C4D411yescharacteristic
66C2×C42C2311yescharacteristic
67C2×C42C2311yescharacteristic
68#107—22no—
69#107—22no—
70#106—24no—
71C4⋊C8—22no—
72C4⋊C8—22no—
73C22⋊C8—22no—
74C22⋊C8—22no—
75#97—22no—
76C4×C8D414yes—
77#95—22no—
78#95—22no—
79#95—22no—
80#95—22no—
81C24C2×D411yescharacteristic
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4C2×D411yescharacteristic
91C22×C4C2×D411yescharacteristic
92C22×C4C2×D411yescharacteristic
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4C2411yescharacteristic
96C2×C8—22no—
97C2×C8—22no—
98C2×C8—21no—
99C2×C8—21no—
100C2×C8—22no—
101C2×C8—22no—
102C2×C8C2×D412yes—
103C2×C8C2×D412yes—
104C4⋊C4—22no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C4⋊C4—22no—
109C4⋊C4—22no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—44no—
115C4⋊C4—44no—
116C4⋊C4—82no—
117C4⋊C4—82no—
118C4⋊C4—82no—
119C4⋊C4—82no—
120C22⋊C4—22no—
121C22⋊C4—22no—
122C22⋊C4—82no—
123C22⋊C4—82no—
124C42—21no—
125C42—21no—
126C42C2×D412yes—
127C42#4112yes—
128C23—21no—
129C23—22no—
130C23—21no—
131C23—22no—
132C23—21no—
133C23C22×D411yescharacteristic
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—22no—
141C2×C4—21no—
142C2×C4—22no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4#12112yes—
160C2×C4#12112yes—
161C2×C4C22×D411yescharacteristic
162C2×C4C2×#4111yescharacteristic
163C2×C42−1+412yes—
164C2×C42−1+412yes—
165C8—22no—
166C8—22no—
167C8—41no—
168C8—41no—
169C22—41no—
170C22—42no—
171C22—42no—
172C22—41no—
173C22—41no—
174C22C2×#12111yescharacteristic
175C22#53612yes—
176C22#57611yescharacteristic
177C22#57712yes—
178C22#57811yescharacteristic
179C4—22no—
180C4—22no—
181C4—41no—
182C4—41no—
183C4—41no—
184C4—82no—
185C4—82no—
186C4—82no—
187C4—82no—
188C4#53611yescharacteristic
189C4#53611yescharacteristic
190C4#54112yes—
191C2—81no—
192C2#262111yescharacteristic
193C2#306011yescharacteristic
194C2#306111yescharacteristic
195C2#324112yes—
196C2#324212yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#499C2semidirect
4. #2088C2semidirect
5. #2087C2semidirect
6. #2086C2semidirect
10. #1992C2semidirect
11. #1834C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4992
C2×#4332
#26292
#20882
#20872
#20862
#20772
#20762
#20752
#19922
#18342