Finite Group Databasebeta

← #18174

Subgroups — #18174

Order 256.

481 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 481 conjugacy classes of subgroups — too many to draw. Only its 117 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#3501C211yesmaximal, characteristic
2#3095C212yesmaximal, complemented
3#2395C211yesmaximal, characteristic
4#1617C212yesmaximal, complemented
5#1614C212yesmaximal, complemented
6#1611C212yesmaximal, complemented
7#1608C212yesmaximal, complemented
8#1508C212yesmaximal, complemented
9#1465C211yesmaximal, characteristic
10#554—24no—
11#554—22no—
12D4×Q8—22no—
13#544C2212yes—
14#543—21no—
15#543C2212yes—
16#533—21no—
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8C2212yes—
19#518—21no—
20C2×C4×D4C2211yescharacteristic
21C2×#117C2211yescharacteristic
22#485—24no—
23#484C2214yes—
24#483C2214yes—
25#428—22no—
26#419—22no—
27#407C2212yes—
28#400C2211yescharacteristic
29#399C2211yescharacteristic
30#398C2211yescharacteristic
31#397—21no—
32#397C2211yescharacteristic
33#394—21no—
34C4×C4⋊C4—21no—
35C4×C22⋊C4—21no—
36C4×C22⋊C4C2211yescharacteristic
37#328C2214yes—
38#328C2214yes—
39#324C2212yes—
40#323C2212yes—
41C22×Q8—21no—
42C22×Q8D412yes—
43C22×D4D411yescharacteristic
44C23×C4D411yescharacteristic
45C23×C4C2311yescharacteristic
46C2×M16—22no—
47C4⋊Q8—21no—
48C4⋊Q8—21no—
49C4⋊Q8—24no—
50#125—22no—
51#125—24no—
52C22⋊Q8—22no—
53C22⋊Q8—22no—
54C22⋊Q8—24no—
55C22⋊Q8—24no—
56C22⋊Q8—24no—
57C22⋊Q8—22no—
58C22⋊Q8—22no—
59C22⋊Q8—22no—
60C22⋊Q8—22no—
61C22⋊Q8—24no—
62C22⋊Q8D414yes—
63C22⋊Q8C2314yes—
64C4×Q8—22no—
65C4×Q8—22no—
66C4×D4—24no—
67C4×D4—24no—
68C4×D4—22no—
69C4×D4—22no—
70#117—22no—
71#117—22no—
72#117—22no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4D412yes—
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C22⋊C4—22no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4D411yescharacteristic
89C2×C22⋊C4C2311yescharacteristic
90C2×C42—21no—
91C2×C42—21no—
92C2×C42D411yescharacteristic
93#107—24no—
94#106—24no—
95#103—24no—
96#103—24no—
97C22⋊C8—22no—
98C22⋊C8C2314yes—
99#95—21no—
100#95—21no—
101#95—21no—
102#95—21no—
103#95C2311yescharacteristic
104#95C2311yescharacteristic
105C24C2×D411yescharacteristic
106C24C2×D411yescharacteristic
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—44no—
111C2×Q8—82no—
112C2×Q8C2×D414yes—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—22no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D412yes—
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D412yes—
134C22×C4C2411yescharacteristic
135M16—44no—
136C2×C8—24no—
137C4⋊C4—22no—
138C4⋊C4—24no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—24no—
142C4⋊C4—22no—
143C4⋊C4—24no—
144C4⋊C4—22no—
145C4⋊C4—24no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—42no—
150C4⋊C4—42no—
151C4⋊C4C2×D414yes—
152C22⋊C4—24no—
153C22⋊C4—24no—
154C22⋊C4—22no—
155C22⋊C4—24no—
156C22⋊C4—22no—
157C22⋊C4—42no—
158C22⋊C4—42no—
159C22⋊C4—42no—
160C22⋊C4—42no—
161C42—24no—
162C42—21no—
163C42—21no—
164C42—42no—
165C42—41no—
166C42—41no—
167C23—21no—
168C23—21no—
169C23—21no—
170C23—21no—
171C23—41no—
172C23—41no—
173C23—41no—
174C23—41no—
175C23#12012yes—
176C23#13712yes—
177C23C22×D411yescharacteristic
178Q8—44no—
179Q8—44no—
180Q8—82no—
181Q8—82no—
182D4—28no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—24no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—24no—
190C2×C4—24no—
191C2×C4—21no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—81no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4#12011yescharacteristic
214C2×C4#12011yescharacteristic
215C2×C4#13712yes—
216C2×C4C22×D412yes—
217C8—44no—
218C22—21no—
219C22—21no—
220C22—22no—
221C22—22no—
222C22—22no—
223C22—22no—
224C22—42no—
225C22—41no—
226C22—41no—
227C22—81no—
228C22—81no—
229C22—81no—
230C22C2×#12011yescharacteristic
231C22C2×#13711yescharacteristic
232C22C2×#13711yescharacteristic
233C4—22no—
234C4—22no—
235C4—21no—
236C4—42no—
237C4—41no—
238C4—44no—
239C4—41no—
240C4—44no—
241C4—81no—
242C4—81no—
243C4—81no—
244C4#45812yes—
245C2—21no—
246C2—21no—
247C2—42no—
248C2—42no—
249C2#300411yescharacteristic
250C2#301811yescharacteristic
251C2C2×#45811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #3095C2semidirect
4. #1617C2semidirect
5. #1614C2semidirect
6. #1611C2semidirect
7. #1608C2semidirect
8. #1508C2semidirect

Maximal subgroups

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

subgroupindex
#35012
#30952
#23952
#16172
#16142
#16112
#16082
#15082
#14652