Finite Group Databasebeta

← #32765

Subgroups — #32765

Order 256.

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

normal subgroups only

This group has 439 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#2620C211yesmaximal, characteristic, complemented
2#2619C211yesmaximal, characteristic, complemented
3#2074C211yesmaximal, characteristic, complemented
4#2073C211yesmaximal, characteristic, complemented
5#2071C212yesmaximal, complemented
6#2014C211yesmaximal, characteristic, complemented
7#2012C212yesmaximal, complemented
8#2011C211yesmaximal, characteristic, complemented
9#2000C212yesmaximal, complemented
10#1999C212yesmaximal, complemented
11#1818C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C22×D8C2212yes—
14C2×#127C2211yescharacteristic
15C2×#124C2211yescharacteristic
16C2×C22⋊Q8C2211yescharacteristic
17C2×#121C2212yes—
18C2×#121C2211yescharacteristic
19C2×#117C2211yescharacteristic
20C2×#103C2212yes—
21C2×#102C2212yes—
22C2×#102C2212yes—
23C2×#102C2212yes—
24C2×C22⋊C8C2212yes—
25#396—22no—
26#394—22no—
27#394—22no—
28#392—22no—
29#391C2211yescharacteristic
30#390C2212yes—
31#390C2211yescharacteristic
32#384C2211yescharacteristic
33#382C2211yescharacteristic
34#340C2212yes—
35#340C2212yes—
36#336C2212yes—
37#336C2211yescharacteristic
38#336C2211yescharacteristic
39C22×Q8C2311yescharacteristic
40C22×D4—22no—
41C22×D4—21no—
42C22×D4—22no—
43C22×D4C2311yescharacteristic
44C22×D4C2312yes—
45C23×C4C2311yescharacteristic
46C2×SD16—42no—
47C2×SD16—42no—
48C2×SD16—42no—
49C2×D8—42no—
50C2×D8—42no—
51C2×D8—42no—
52C22×C8C2312yes—
53C22×C8C2312yes—
54#127—42no—
55#124—42no—
56C22⋊Q8—42no—
57#121—42no—
58#121—42no—
59#121—42no—
60#117—42no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4D412yes—
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C22⋊C4—21no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4D412yes—
75C2×C42C2311yescharacteristic
76C2×C42C2311yescharacteristic
77#103—42no—
78#102—42no—
79#102—42no—
80#102—42no—
81C22⋊C8—42no—
82#95—22no—
83#95—22no—
84C24—21no—
85C24—22no—
86C24C2×D411yescharacteristic
87C2×Q8—21no—
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—21no—
91C2×D4—21no—
92C2×D4—22no—
93C2×D4—22no—
94C2×D4—22no—
95C2×D4—22no—
96C2×D4—22no—
97C2×D4—22no—
98C2×D4—22no—
99C2×D4—21no—
100C2×D4—22no—
101C2×D4—44no—
102C2×D4—42no—
103C2×D4—44no—
104C2×D4—82no—
105C2×D4—82no—
106C2×D4—82no—
107C2×D4—82no—
108C2×D4—82no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4C2×D412yes—
118C22×C4C2×D412yes—
119C22×C4C2×D411yescharacteristic
120C22×C4C2411yescharacteristic
121SD16—48no—
122D8—48no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—82no—
136C22⋊C4—42no—
137C22⋊C4—44no—
138C22⋊C4—42no—
139C22⋊C4—44no—
140C22⋊C4—42no—
141C22⋊C4—82no—
142C22⋊C4—82no—
143C22⋊C4—82no—
144C42—22no—
145C42—22no—
146C23—21no—
147C23—21no—
148C23—21no—
149C23—22no—
150C23—22no—
151C23—41no—
152C23—42no—
153C23—41no—
154C23—42no—
155C23—42no—
156C23—41no—
157C23—42no—
158C23—42no—
159C23—42no—
160C23—42no—
161C23—42no—
162C23—42no—
163C23C22×D411yescharacteristic
164Q8—42no—
165Q8—42no—
166D4—42no—
167D4—42no—
168D4—42no—
169D4—42no—
170D4—42no—
171D4—42no—
172D4—88no—
173D4—84no—
174D4—88no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4H(5,𝔽2)12yes—
197C2×C4H(5,𝔽2)12yes—
198C2×C4H(5,𝔽2)11yescharacteristic
199C2×C4H(5,𝔽2)11yescharacteristic
200C8—42no—
201C8—42no—
202C22—42no—
203C22—41no—
204C22—41no—
205C22—42no—
206C22—41no—
207C22—82no—
208C22—82no—
209C22—82no—
210C22—81no—
211C22—82no—
212C22—81no—
213C22—82no—
214C22—82no—
215C22—82no—
216C22—82no—
217C22—81no—
218C22—82no—
219C22#53411yescharacteristic
220C22#53511yescharacteristic
221C22#53511yescharacteristic
222C22#57612yes—
223C22#57712yes—
224C4—42no—
225C4—42no—
226C4—41no—
227C4—81no—
228C4—81no—
229C4—82no—
230C4—81no—
231C4#56012yes—
232C4#56012yes—
233C2—81no—
234C2—162no—
235C2—161no—
236C2#281511yescharacteristic
237C2#319812yes—
238C2#322011yescharacteristic
239C2#322111yescharacteristic
240C2#322412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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. #2620C2semidirect
2. #2619C2semidirect
3. #2074C2semidirect
4. #2073C2semidirect
5. #2071C2semidirect
6. #2014C2semidirect
7. #2012C2semidirect
8. #2011C2semidirect
9. #2000C2semidirect
10. #1999C2semidirect
11. #1818C2semidirect

Maximal subgroups

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

subgroupindex
#26202
#26192
#20742
#20732
#20712
#20142
#20122
#20112
#20002
#19992
#18182