Finite Group Databasebeta

← #17617

Subgroups — #17617

Order 256.

509 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 509 conjugacy classes of subgroups — too many to draw. Only its 183 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#3506C211yesmaximal, characteristic
2#2972C211yesmaximal, characteristic, complemented
3#2357C211yesmaximal, characteristic, complemented
4#1518C211yesmaximal, characteristic, complemented
5#1517C211yesmaximal, characteristic, complemented
6#1516C214yesmaximal, complemented
7#1503C211yesmaximal, characteristic, complemented
8#1502C211yesmaximal, characteristic, complemented
9#1501C214yesmaximal, complemented
10#559—22no—
11#556C412yescomplemented
12#555C412yescomplemented
13#553—22no—
14#550C412yescomplemented
15#547C412yescomplemented
16#538—22no—
17#535—21no—
18#533C2211yescharacteristic
19#532C2211yescharacteristic
20#525C2214yes—
21#520—21no—
22C2×C4×D4C2211yescharacteristic
23C8×D4—22no—
24C8×D4—22no—
25#431C2211yescharacteristic
26#431C2211yescharacteristic
27C2×C22⋊C8C2214yes—
28#390—21no—
29#388—22no—
30#382C2211yescharacteristic
31#381C2212yes—
32C4×C4⋊C4—21no—
33C4×C4⋊C4C2211yescharacteristic
34C4×C22⋊C4C2212yes—
35#328C2212yes—
36#328C2214yes—
37#328C2212yes—
38#327C2212yes—
39#327C2214yes—
40#327C2212yes—
41C2×#41—21no—
42C2×#41C2×C414yes—
43C22×D4D411yescharacteristic
44C23×C4C2312yes—
45C22×C8—24no—
46C4⋊Q8C2×C411yescharacteristic
47C4⋊Q8C2×C411yescharacteristic
48C4⋊Q8C2×C411yescharacteristic
49#127—22no—
50#127C2×C411yescharacteristic
51#126—24no—
52#125—21no—
53#124—22no—
54#124—24no—
55#124—21no—
56#123—24no—
57#123—24no—
58C22⋊Q8—21no—
59C22⋊Q8C2×C414yes—
60C22⋊Q8C2312yes—
61#121—22no—
62#121—22no—
63#121—21no—
64#121—24no—
65#121C2×C414yes—
66#121C2312yes—
67#120—24no—
68C4×Q8—22no—
69C4×Q8—21no—
70C4×Q8C2×C412yes—
71C4×D4—24no—
72C4×D4—24no—
73C4×D4—22no—
74C4×D4—21no—
75C4×D4C2×C412yes—
76C4×D4C2×C412yes—
77C4×D4C2×C412yes—
78C4×D4D412yes—
79C4×D4D412yes—
80#117—21no—
81#117—22no—
82#117C2×C414yes—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4D411yescharacteristic
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C22⋊C4D412yes—
90C2×C42—21no—
91C2×C42—22no—
92C2×C42C2311yescharacteristic
93C4⋊C8—21no—
94C4⋊C8—21no—
95C22⋊C8—24no—
96C22⋊C8C2312yes—
97C22⋊C8C2312yes—
98C4×C8—21no—
99C4×C8—21no—
100#95—22no—
101#95—21no—
102#95C2311yescharacteristic
103#95C2312yes—
104#95C2311yescharacteristic
105C24C2×D412yes—
106#41—42no—
107#41—42no—
108#41—84no—
109C2×Q8—21no—
110C2×Q8—21no—
111C2×Q8C22×C412yes—
112C2×D4—22no—
113C2×D4—24no—
114C2×D4—24no—
115C2×D4—21no—
116C2×D4—21no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4C22⋊C412yes—
120C2×D4C22⋊C412yes—
121C2×D4C22×C412yes—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—24no—
125C22×C4—22no—
126C22×C4—21no—
127C22×C4—82no—
128C22×C4C22×C414yes—
129C22×C4C2×D412yes—
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D414yes—
132C22×C4C2×D411yescharacteristic
133C22×C4C2411yescharacteristic
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—84no—
137C4⋊C4—21no—
138C4⋊C4—24no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4C22⋊C412yes—
147C4⋊C4C22⋊C412yes—
148C4⋊C4C22×C412yes—
149C4⋊C4C22×C412yes—
150C22⋊C4—24no—
151C22⋊C4—24no—
152C22⋊C4—22no—
153C22⋊C4—24no—
154C22⋊C4—22no—
155C22⋊C4—22no—
156C22⋊C4—44no—
157C22⋊C4—82no—
158C22⋊C4C22⋊C414yes—
159C22⋊C4C22⋊C414yes—
160C42—24no—
161C42—22no—
162C42—42no—
163C42—82no—
164C42C22×C411yescharacteristic
165C42C22×C411yescharacteristic
166C42C2×D411yescharacteristic
167C42C2×D411yescharacteristic
168C23—22no—
169C23—22no—
170C23—22no—
171C23—42no—
172C23—42no—
173C23C2×C22⋊C414yes—
174C23C22×D411yescharacteristic
175Q8—42no—
176Q8—42no—
177Q8—81no—
178D4—24no—
179D4—24no—
180D4—42no—
181D4—42no—
182D4—42no—
183D4—42no—
184D4—81no—
185D4—84no—
186D4—84no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—44no—
194C2×C4—41no—
195C2×C4—44no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—44no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—84no—
206C2×C4C2×C22⋊C414yes—
207C2×C4C2×C22⋊C411yescharacteristic
208C2×C4C2×C22⋊C411yescharacteristic
209C2×C4C2×C22⋊C411yescharacteristic
210C2×C4C2×C22⋊C411yescharacteristic
211C2×C4#13512yes—
212C2×C4#13512yes—
213C2×C4C23×C411yescharacteristic
214C2×C4C22×D411yescharacteristic
215C8—42no—
216C8—42no—
217C22—21no—
218C22—21no—
219C22—24no—
220C22—24no—
221C22—24no—
222C22—42no—
223C22—42no—
224C22—42no—
225C22—82no—
226C22—82no—
227C22C22×C22⋊C411yescharacteristic
228C22C2×#13511yescharacteristic
229C22C2×#13511yescharacteristic
230C4—21no—
231C4—21no—
232C4—24no—
233C4—21no—
234C4—21no—
235C4—42no—
236C4—42no—
237C4—42no—
238C4—42no—
239C4—81no—
240C4—82no—
241C4#41612yes—
242C4#41612yes—
243C2—21no—
244C2—21no—
245C2—44no—
246C2—82no—
247C2#288311yescharacteristic
248C2C2×#41611yescharacteristic
249C2#290111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radical#3506C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2972C2semidirect
3. #2357C2semidirect
4. #1518C2semidirect
5. #1517C2semidirect
6. #1516C2semidirect
7. #1503C2semidirect
8. #1502C2semidirect
9. #1501C2semidirect
11. #556C4semidirect
12. #555C4semidirect
14. #550C4semidirect
15. #547C4semidirect

Maximal subgroups

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

subgroupindex
#35062
#29722
#23572
#15182
#15172
#15162
#15032
#15022
#15012