Finite Group Databasebeta

← #32830

Subgroups — #32830

Order 256.

306 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 103 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#2640C211yesmaximal, characteristic
3#2096C211yesmaximal, characteristic, complemented
4#2094C211yesmaximal, characteristic
5#2074C211yesmaximal, characteristic, complemented
6#2073C211yesmaximal, characteristic
7#2066C212yesmaximal, complemented
8#2065C212yesmaximal
9#2058C212yesmaximal, complemented
10#2042C212yesmaximal
11#1832C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×#124C2211yescharacteristic
14C2×#106C2212yes—
15C2×#106C2212yes—
16C2×C4⋊C8C2212yes—
17C2×#103C2212yes—
18C2×#103C2212yes—
19C2×#102C2212yes—
20C2×#102C2212yes—
21#400—22no—
22#398—22no—
23#397—22no—
24#393—22no—
25#391C2211yescharacteristic
26#391C2211yescharacteristic
27#390C2211yescharacteristic
28#390C2211yescharacteristic
29#389C2212yes—
30#389C2212yes—
31#384C2211yescharacteristic
32#382C2211yescharacteristic
33#376C2211yescharacteristic
34#340C2212yes—
35#340C2212yes—
36#336C2212yes—
37#336C2211yescharacteristic
38#336C2211yescharacteristic
39C22×Q8—21no—
40C22×Q8C2311yescharacteristic
41C22×D4C2311yescharacteristic
42C22×C8C2312yes—
43C22×C8C2312yes—
44C4⋊Q8—42no—
45#124—42no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—22no—
48C2×C4⋊C4—22no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4C2311yescharacteristic
51C2×C4⋊C4C2311yescharacteristic
52C2×C4⋊C4C2312yes—
53C2×C4⋊C4C2312yes—
54C2×C22⋊C4—21no—
55C2×C22⋊C4—21no—
56C2×C22⋊C4—22no—
57C2×C22⋊C4—22no—
58C2×C42C2311yescharacteristic
59C2×C42C2311yescharacteristic
60C2×C42C2311yescharacteristic
61#106—42no—
62#106—42no—
63C4⋊C8—42no—
64#103—42no—
65#103—42no—
66#102—42no—
67#102—42no—
68#95—22no—
69#95—22no—
70#95—22no—
71#95—22no—
72#95—22no—
73#95D412yes—
74#95D412yes—
75C24—21no—
76C2×Q8—21no—
77C2×Q8—22no—
78C2×Q8—22no—
79C2×Q8—21no—
80C2×Q8—42no—
81C2×Q8—82no—
82C2×D4—21no—
83C2×D4—22no—
84C2×D4—22no—
85C2×D4—21no—
86C22×C4—21no—
87C22×C4—22no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—22no—
91C22×C4C2×D412yes—
92C22×C4C2×D412yes—
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4C2411yescharacteristic
96C2×C8—22no—
97C2×C8—22no—
98C2×C8—22no—
99C2×C8—22no—
100C2×C8—22no—
101C2×C8—22no—
102C4⋊C4—22no—
103C4⋊C4—22no—
104C4⋊C4—22no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C4⋊C4—42no—
109C4⋊C4—82no—
110C4⋊C4—82no—
111C4⋊C4—82no—
112C22⋊C4—42no—
113C22⋊C4—42no—
114C22⋊C4—82no—
115C22⋊C4—82no—
116C42—21no—
117C42—22no—
118C42—22no—
119C42—21no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23—42no—
124C23—42no—
125C23C22×D411yescharacteristic
126Q8—42no—
127Q8—42no—
128Q8—84no—
129D4—42no—
130D4—42no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—42no—
145C2×C4—41no—
146C2×C4—42no—
147C2×C4—41no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4H(5,𝔽2)11yescharacteristic
155C2×C4H(5,𝔽2)12yes—
156C2×C42−1+412yes—
157C2×C42−1+411yescharacteristic
158C8—42no—
159C8—42no—
160C22—81no—
161C22—81no—
162C22—81no—
163C22—82no—
164C22—82no—
165C22#53511yescharacteristic
166C22#53611yescharacteristic
167C22#53711yescharacteristic
168C22#57712yes—
169C22#57712yes—
170C4—41no—
171C4—41no—
172C4—42no—
173C4—42no—
174C4—81no—
175C4—82no—
176C4—81no—
177C4—81no—
178C4—82no—
179C4#56212yes—
180C4#56312yes—
181C2—161no—
182C2#282011yescharacteristic
183C2#322111yescharacteristic
184C2#322211yescharacteristic
185C2#324412yes—
186C2#326312yes—
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. #2074C2semidirect
7. #2066C2semidirect
9. #2058C2semidirect
11. #1832C2semidirect

Maximal subgroups

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

subgroupindex
#26472
#26402
#20962
#20942
#20742
#20732
#20662
#20652
#20582
#20422
#18322