Finite Group Databasebeta

← #32784

Subgroups — #32784

Order 256.

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

normal subgroups only

This group has 322 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#2629C211yesmaximal, characteristic
2#2621C211yesmaximal, characteristic
3#2096C211yesmaximal, characteristic, complemented
4#2095C211yesmaximal, characteristic, complemented
5#2085C212yesmaximal, complemented
6#2043C212yesmaximal, complemented
7#2036C212yesmaximal
8#2035C212yesmaximal
9#2026C211yesmaximal, characteristic, complemented
10#2026C211yesmaximal, characteristic, complemented
11#1818C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×#125C2211yescharacteristic
14C2×C22⋊Q8C2211yescharacteristic
15C2×C22⋊Q8C2211yescharacteristic
16C2×#117C2211yescharacteristic
17C2×#107C2212yes—
18C2×#106C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×C22⋊C8C2212yes—
24#398—22no—
25#398—22no—
26#397—22no—
27#396—22no—
28#391C2211yescharacteristic
29#391C2211yescharacteristic
30#389C2212yes—
31#384C2211yescharacteristic
32#382C2211yescharacteristic
33#380C2212yes—
34#340C2212yes—
35#340C2212yes—
36#336C2212yes—
37#336C2211yescharacteristic
38#336C2211yescharacteristic
39C22×Q8C2311yescharacteristic
40C22×Q8C2311yescharacteristic
41C23×C4C2311yescharacteristic
42C22×C8C2312yes—
43C22×C8C2312yes—
44C4⋊Q8—42no—
45#125—42no—
46C22⋊Q8—42no—
47C22⋊Q8—42no—
48#117—42no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4D412yes—
56C2×C4⋊C4C2311yescharacteristic
57C2×C4⋊C4C2311yescharacteristic
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2312yes—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—21no—
62C2×C22⋊C4—22no—
63C2×C22⋊C4D412yes—
64C2×C42C2311yescharacteristic
65C2×C42C2311yescharacteristic
66#107—42no—
67#106—42no—
68#103—42no—
69#103—42no—
70#103—42no—
71#103—42no—
72C22⋊C8—42no—
73#95—22no—
74#95—22no—
75#95—22no—
76#95—22no—
77#95—22no—
78C24C2×D411yescharacteristic
79C2×Q8—22no—
80C2×Q8—22no—
81C2×Q8—22no—
82C2×Q8—22no—
83C2×Q8—21no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8—21no—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4C2×D412yes—
98C22×C4C2×D412yes—
99C22×C4C2×D411yescharacteristic
100C22×C4C2411yescharacteristic
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—22no—
107C4⋊C4—22no—
108C4⋊C4—22no—
109C4⋊C4—22no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4—42no—
114C4⋊C4—42no—
115C4⋊C4—42no—
116C4⋊C4—44no—
117C4⋊C4—44no—
118C4⋊C4—82no—
119C4⋊C4—82no—
120C22⋊C4—42no—
121C22⋊C4—42no—
122C22⋊C4—42no—
123C22⋊C4—82no—
124C42—22no—
125C42—22no—
126C23—21no—
127C23—21no—
128C23—21no—
129C23—22no—
130C23—22no—
131C23C22×D411yescharacteristic
132Q8—42no—
133Q8—42no—
134Q8—42no—
135Q8—42no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—42no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4H(5,𝔽2)12yes—
164C2×C4H(5,𝔽2)12yes—
165C2×C42−1+411yescharacteristic
166C2×C42−1+411yescharacteristic
167C8—42no—
168C8—42no—
169C22—41no—
170C22—42no—
171C22—41no—
172C22—41no—
173C22—42no—
174C22#53411yescharacteristic
175C22#53511yescharacteristic
176C22#53611yescharacteristic
177C22#57712yes—
178C22#57812yes—
179C4—41no—
180C4—42no—
181C4—42no—
182C4—81no—
183C4—81no—
184C4—82no—
185C4—81no—
186C4—82no—
187C4—81no—
188C4#56212yes—
189C4#56212yes—
190C2—81no—
191C2#281611yescharacteristic
192C2#320112yes—
193C2#322111yescharacteristic
194C2#322311yescharacteristic
195C2#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. #2096C2semidirect
4. #2095C2semidirect
5. #2085C2semidirect
6. #2043C2semidirect
9. #2026C2semidirect
10. #2026C2semidirect

Maximal subgroups

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

subgroupindex
#26292
#26212
#20962
#20952
#20852
#20432
#20362
#20352
#20262
#18182