Finite Group Databasebeta

← #30090

Subgroups — #30090

Order 256.

333 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 333 conjugacy classes of subgroups — too many to draw. Only its 107 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#2647C211yesmaximal, characteristic, complemented
2#2326C211yesmaximal, characteristic
3#2096C211yesmaximal, characteristic, complemented
4#2094C211yesmaximal, characteristic
5#2084C212yesmaximal, complemented
6#2074C211yesmaximal, characteristic, complemented
7#2073C211yesmaximal, characteristic
8#2042C212yesmaximal
9#1865C212yesmaximal, complemented
10#1864C212yesmaximal
11#1840C211yesmaximal, characteristic, complemented
12C2×#124C2211yescharacteristic
13C2×C4×Q8C2211yescharacteristic
14C2×C4×D4C2211yescharacteristic
15C2×#106C2212yes—
16C2×#106C2212yes—
17C2×#103C2212yes—
18C2×#103C2212yes—
19C2×#102C2212yes—
20C2×#102C2212yes—
21#400—22no—
22#398—22no—
23#391C2211yescharacteristic
24#390C2211yescharacteristic
25#389C2212yes—
26#389C2212yes—
27#388—22no—
28#385—22no—
29#384C2211yescharacteristic
30#384C2211yescharacteristic
31#382C2211yescharacteristic
32#376C2211yescharacteristic
33#340C2212yes—
34#340C2212yes—
35#336C2211yescharacteristic
36#336C2211yescharacteristic
37#336C2212yes—
38#336C2212yes—
39C22×Q8C2311yescharacteristic
40C22×D4C2311yescharacteristic
41C23×C4—21no—
42C22×C8C2312yes—
43C22×C8C2312yes—
44#124—22no—
45#124—22no—
46C4×Q8—42no—
47C4×D4—42no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4C2311yescharacteristic
54C2×C4⋊C4C2311yescharacteristic
55C2×C4⋊C4C2312yes—
56C2×C4⋊C4C2312yes—
57C2×C22⋊C4—22no—
58C2×C22⋊C4—21no—
59C2×C22⋊C4—22no—
60C2×C42—21no—
61C2×C42C2311yescharacteristic
62C2×C42C2311yescharacteristic
63C2×C42C2311yescharacteristic
64#106—42no—
65#106—42no—
66#103—24no—
67#103—42no—
68#102—24no—
69#102—42no—
70#95—22no—
71#95—22no—
72#95—22no—
73#95—22no—
74#95D412yes—
75#95D412yes—
76C24—21no—
77C2×Q8—21no—
78C2×Q8—22no—
79C2×Q8—22no—
80C2×Q8#4112yes—
81C2×D4—21no—
82C2×D4—22no—
83C2×D4—22no—
84C2×D4#4112yes—
85C22×C4—21no—
86C22×C4—22no—
87C22×C4—21no—
88C22×C4—22no—
89C22×C4—21no—
90C22×C4—42no—
91C22×C4—82no—
92C22×C4C2×D412yes—
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D412yes—
95C22×C4C2×D411yescharacteristic
96C22×C4C2411yescharacteristic
97C2×C8—22no—
98C2×C8—22no—
99C2×C8—22no—
100C2×C8—22no—
101C2×C8—22no—
102C2×C8—22no—
103C4⋊C4—22no—
104C4⋊C4—22no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C4⋊C4—22no—
109C4⋊C4—42no—
110C4⋊C4—82no—
111C4⋊C4—82no—
112C4⋊C4—82no—
113C4⋊C4—82no—
114C22⋊C4—28no—
115C22⋊C4—42no—
116C22⋊C4—82no—
117C42—21no—
118C42—21no—
119C42—22no—
120C42—22no—
121C42—42no—
122C23—22no—
123C23—41no—
124C23—41no—
125C23—42no—
126C23—42no—
127C23C22×D411yescharacteristic
128Q8—24no—
129Q8—42no—
130D4—24no—
131D4—42no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—84no—
156C2×C4C2×#4111yescharacteristic
157C2×C4C2×#4111yescharacteristic
158C2×C4H(5,𝔽2)12yes—
159C2×C42−1+412yes—
160C8—42no—
161C8—42no—
162C22—44no—
163C22—42no—
164C22—81no—
165C22—81no—
166C22—82no—
167C22#52511yescharacteristic
168C22#53511yescharacteristic
169C22#53611yescharacteristic
170C22#57712yes—
171C22#57712yes—
172C4—42no—
173C4—42no—
174C4—41no—
175C4—41no—
176C4—42no—
177C4—82no—
178C4—82no—
179C4—81no—
180C4—81no—
181C4#53912yes—
182C4#53912yes—
183C2—82no—
184C2#263311yescharacteristic
185C2#312512yes—
186C2#322111yescharacteristic
187C2#322211yescharacteristic
188C2#324412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2647C2semidirect
3. #2096C2semidirect
5. #2084C2semidirect
6. #2074C2semidirect
9. #1865C2semidirect
11. #1840C2semidirect

Maximal subgroups

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

subgroupindex
#26472
#23262
#20962
#20942
#20842
#20742
#20732
#20422
#18652
#18642
#18402