Finite Group Databasebeta

← #32780

Subgroups — #32780

Order 256.

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

normal subgroups only

This group has 326 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#2628C211yesmaximal, characteristic
2#2621C211yesmaximal, characteristic
3#2097C212yesmaximal, complemented
4#2085C212yesmaximal, complemented
5#2043C212yesmaximal, complemented
6#2036C212yesmaximal
7#2035C212yesmaximal
8#2024C212yesmaximal, complemented
9#1818C211yesmaximal, characteristic
10C2×C4⋊Q8C2211yescharacteristic
11C2×C4⋊Q8C2211yescharacteristic
12C2×C22⋊Q8C2212yes—
13C2×#117C2211yescharacteristic
14C2×#107C2212yes—
15C2×#106C2212yes—
16C2×#103C2212yes—
17C2×#103C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×C22⋊C8C2211yescharacteristic
21C2×C22⋊C8C2211yescharacteristic
22#400—21no—
23#399—21no—
24#398—21no—
25#398—21no—
26#397—21no—
27#397—21no—
28#395—21no—
29#393—21no—
30#391C2212yes—
31#389C2212yes—
32#382C2211yescharacteristic
33#382C2211yescharacteristic
34#380C2212yes—
35#340C2212yes—
36#340C2212yes—
37#336C2212yes—
38#336C2212yes—
39C22×Q8—21no—
40C22×Q8C2312yes—
41C23×C4C2311yescharacteristic
42C22×C8C2312yes—
43C22×C8C2312yes—
44C4⋊Q8—41no—
45C4⋊Q8—41no—
46C4⋊Q8—41no—
47C4⋊Q8—41no—
48C22⋊Q8—42no—
49C22⋊Q8—42no—
50#117—41no—
51#117—41no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4D411yescharacteristic
58C2×C4⋊C4D411yescharacteristic
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—42no—
69#106—42no—
70#103—42no—
71#103—42no—
72#103—42no—
73#103—42no—
74C22⋊C8—41no—
75C22⋊C8—41no—
76#95—22no—
77#95—21no—
78#95—21no—
79#95—21no—
80#95—21no—
81#95—22no—
82#95—22no—
83#95—22no—
84C24C2×D411yescharacteristic
85C2×Q8—22no—
86C2×Q8—22no—
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—42no—
92C2×Q8—82no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2411yescharacteristic
107C2×C8—22no—
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—22no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—41no—
120C4⋊C4—41no—
121C4⋊C4—44no—
122C4⋊C4—42no—
123C4⋊C4—42no—
124C4⋊C4—42no—
125C4⋊C4—82no—
126C22⋊C4—41no—
127C22⋊C4—41no—
128C22⋊C4—44no—
129C22⋊C4—82no—
130C42—21no—
131C42—21no—
132C42—21no—
133C42—21no—
134C23—21no—
135C23—21no—
136C23—21no—
137C23—22no—
138C23—22no—
139C23C22×D411yescharacteristic
140Q8—42no—
141Q8—42no—
142Q8—42no—
143Q8—42no—
144Q8—84no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4H(5,𝔽2)11yescharacteristic
171C2×C4H(5,𝔽2)11yescharacteristic
172C2×C4H(5,𝔽2)12yes—
173C2×C42−1+412yes—
174C8—42no—
175C8—42no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22—42no—
180C22—42no—
181C22#53411yescharacteristic
182C22#53511yescharacteristic
183C22#53611yescharacteristic
184C22#57712yes—
185C22#57812yes—
186C4—41no—
187C4—41no—
188C4—41no—
189C4—41no—
190C4—41no—
191C4—82no—
192C4—82no—
193C4—82no—
194C4—82no—
195C4#56212yes—
196C4#56212yes—
197C2—81no—
198C2#281611yescharacteristic
199C2#320011yescharacteristic
200C2#320211yescharacteristic
201C2#322512yes—
202C2#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
3. #2097C2semidirect
4. #2085C2semidirect
5. #2043C2semidirect
8. #2024C2semidirect

Maximal subgroups

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

subgroupindex
#26282
#26212
#20972
#20852
#20432
#20362
#20352
#20242
#18182