Finite Group Databasebeta

← #32782

Subgroups — #32782

Order 256.

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

normal subgroups only

This group has 372 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#2619C211yesmaximal, characteristic
3#2094C211yesmaximal, characteristic, complemented
4#2093C211yesmaximal, characteristic, complemented
5#2085C212yesmaximal, complemented
6#2041C212yesmaximal, complemented
7#2034C212yesmaximal, complemented
8#2033C212yesmaximal, complemented
9#2025C211yesmaximal, characteristic, complemented
10#2025C211yesmaximal, characteristic, complemented
11#1818C211yesmaximal, characteristic, complemented
12C2×#127C2211yescharacteristic
13C2×#125C2211yescharacteristic
14C2×#121C2211yescharacteristic
15C2×#121C2211yescharacteristic
16C2×#117C2211yescharacteristic
17C2×#107C2212yes—
18C2×#106C2212yes—
19C2×#102C2212yes—
20C2×#102C2212yes—
21C2×#102C2212yes—
22C2×#102C2212yes—
23C2×C22⋊C8C2212yes—
24#398—22no—
25#396—22no—
26#394—22no—
27#392—22no—
28#390C2211yescharacteristic
29#390C2211yescharacteristic
30#389C2212yes—
31#384C2211yescharacteristic
32#382C2211yescharacteristic
33#380C2212yes—
34#340C2212yes—
35#340C2212yes—
36#336C2212yes—
37#336C2211yescharacteristic
38#336C2211yescharacteristic
39C22×D4—21no—
40C22×D4—21no—
41C22×D4—22no—
42C22×D4C2311yescharacteristic
43C22×D4C2311yescharacteristic
44C23×C4C2311yescharacteristic
45C22×C8C2312yes—
46C22×C8C2312yes—
47#127—42no—
48#125—42no—
49#121—42no—
50#121—42no—
51#117—42no—
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#107—42no—
69#106—42no—
70#102—42no—
71#102—42no—
72#102—42no—
73#102—42no—
74C22⋊C8—42no—
75#95—22no—
76#95—22no—
77#95—22no—
78C24—21no—
79C24—21no—
80C24C2×D411yescharacteristic
81C2×D4—22no—
82C2×D4—22no—
83C2×D4—22no—
84C2×D4—22no—
85C2×D4—21no—
86C2×D4—21no—
87C2×D4—21no—
88C2×D4—21no—
89C2×D4—44no—
90C2×D4—42no—
91C2×D4—42no—
92C2×D4—82no—
93C2×D4—82no—
94C2×D4—82no—
95C2×D4—82no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4C2×D412yes—
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D412yes—
107C22×C4C2411yescharacteristic
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—42no—
121C4⋊C4—44no—
122C4⋊C4—82no—
123C4⋊C4—82no—
124C22⋊C4—42no—
125C22⋊C4—42no—
126C22⋊C4—42no—
127C22⋊C4—82no—
128C22⋊C4—82no—
129C22⋊C4—82no—
130C42—22no—
131C42—22no—
132C23—21no—
133C23—21no—
134C23—21no—
135C23—22no—
136C23—22no—
137C23—41no—
138C23—41no—
139C23—41no—
140C23—41no—
141C23—41no—
142C23—41no—
143C23—42no—
144C23—42no—
145C23—42no—
146C23—42no—
147C23C22×D411yescharacteristic
148D4—42no—
149D4—42no—
150D4—42no—
151D4—42no—
152D4—88no—
153D4—84no—
154D4—84no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4H(5,𝔽2)12yes—
177C2×C4H(5,𝔽2)12yes—
178C2×C42−1+411yescharacteristic
179C2×C42−1+411yescharacteristic
180C8—42no—
181C8—42no—
182C22—41no—
183C22—42no—
184C22—41no—
185C22—41no—
186C22—42no—
187C22—81no—
188C22—81no—
189C22—82no—
190C22—82no—
191C22—81no—
192C22—81no—
193C22—81no—
194C22—81no—
195C22—82no—
196C22—82no—
197C22#53411yescharacteristic
198C22#53511yescharacteristic
199C22#53611yescharacteristic
200C22#57612yes—
201C22#57712yes—
202C4—41no—
203C4—42no—
204C4—42no—
205C4—82no—
206C4—82no—
207C4—81no—
208C4—81no—
209C4#56212yes—
210C4#56212yes—
211C2—81no—
212C2—161no—
213C2—161no—
214C2#281611yescharacteristic
215C2#319812yes—
216C2#322011yescharacteristic
217C2#322211yescharacteristic
218C2#324112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2619C2
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. #2094C2semidirect
4. #2093C2semidirect
5. #2085C2semidirect
6. #2041C2semidirect
7. #2034C2semidirect
8. #2033C2semidirect
9. #2025C2semidirect
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
#26192
#20942
#20932
#20852
#20412
#20342
#20332
#20252
#18182