Finite Group Databasebeta

← #32785

Subgroups — #32785

Order 256.

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

normal subgroups only

This group has 343 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, complemented
2#2620C211yesmaximal, characteristic
3#2096C211yesmaximal, characteristic, complemented
4#2094C211yesmaximal, characteristic, complemented
5#2084C212yesmaximal, complemented
6#2042C212yesmaximal, complemented
7#2035C212yesmaximal, complemented
8#2034C212yesmaximal
9#2026C211yesmaximal, characteristic, complemented
10#2025C211yesmaximal, characteristic, complemented
11#1818C211yesmaximal, characteristic, complemented
12C2×#125C2211yescharacteristic
13C2×#124C2211yescharacteristic
14C2×C22⋊Q8C2211yescharacteristic
15C2×#121C2211yescharacteristic
16C2×#117C2211yescharacteristic
17C2×#106C2212yes—
18C2×#106C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#102C2212yes—
22C2×#102C2212yes—
23C2×C22⋊C8C2212yes—
24#398—22no—
25#396—22no—
26#396—22no—
27#394—22no—
28#391C2211yescharacteristic
29#390C2211yescharacteristic
30#389C2212yes—
31#384C2211yescharacteristic
32#384C2211yescharacteristic
33#380C2212yes—
34#340C2212yes—
35#340C2212yes—
36#336C2212yes—
37#336C2211yescharacteristic
38#336C2211yescharacteristic
39C22×Q8C2311yescharacteristic
40C22×D4—21no—
41C22×D4C2311yescharacteristic
42C23×C4C2311yescharacteristic
43C22×C8C2312yes—
44C22×C8C2312yes—
45#125—42no—
46#124—42no—
47C22⋊Q8—42no—
48#121—42no—
49#117—42no—
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⋊C4—22no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4D412yes—
66C2×C42C2311yescharacteristic
67C2×C42C2311yescharacteristic
68#106—42no—
69#106—42no—
70#103—42no—
71#103—42no—
72#102—42no—
73#102—42no—
74C22⋊C8—42no—
75#95—22no—
76#95—22no—
77#95—22no—
78C24—21no—
79C24C2×D411yescharacteristic
80C2×Q8—22no—
81C2×Q8—22no—
82C2×Q8—21no—
83C2×Q8—21no—
84C2×D4—22no—
85C2×D4—22no—
86C2×D4—21no—
87C2×D4—21no—
88C2×D4—42no—
89C2×D4—82no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D412yes—
101C22×C4C2×D412yes—
102C22×C4C2411yescharacteristic
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—22no—
107C2×C8—22no—
108C2×C8—22no—
109C4⋊C4—22no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—42no—
116C4⋊C4—42no—
117C4⋊C4—44no—
118C4⋊C4—82no—
119C4⋊C4—82no—
120C4⋊C4—82no—
121C22⋊C4—42no—
122C22⋊C4—42no—
123C22⋊C4—44no—
124C22⋊C4—42no—
125C22⋊C4—82no—
126C22⋊C4—82no—
127C42—22no—
128C42—22no—
129C23—21no—
130C23—21no—
131C23—21no—
132C23—22no—
133C23—22no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23—42no—
138C23—42no—
139C23C22×D411yescharacteristic
140Q8—42no—
141Q8—42no—
142D4—42no—
143D4—42no—
144D4—84no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4H(5,𝔽2)12yes—
170C2×C4H(5,𝔽2)12yes—
171C2×C42−1+411yescharacteristic
172C2×C42−1+411yescharacteristic
173C8—42no—
174C8—42no—
175C22—41no—
176C22—42no—
177C22—41no—
178C22—41no—
179C22—42no—
180C22—81no—
181C22—82no—
182C22—81no—
183C22—81no—
184C22—82no—
185C22#53411yescharacteristic
186C22#53511yescharacteristic
187C22#53611yescharacteristic
188C22#57712yes—
189C22#57712yes—
190C4—41no—
191C4—42no—
192C4—42no—
193C4—81no—
194C4—82no—
195C4—82no—
196C4—81no—
197C4—81no—
198C4#56212yes—
199C4#56212yes—
200C2—81no—
201C2—161no—
202C2#281611yescharacteristic
203C2#320412yes—
204C2#322111yescharacteristic
205C2#322211yescharacteristic
206C2#324412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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. #2629C2semidirect
3. #2096C2semidirect
4. #2094C2semidirect
5. #2084C2semidirect
6. #2042C2semidirect
7. #2035C2semidirect
9. #2026C2semidirect
10. #2025C2semidirect
11. #1818C2semidirect

Maximal subgroups

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

subgroupindex
#26292
#26202
#20962
#20942
#20842
#20422
#20352
#20342
#20262
#20252
#18182