Finite Group Databasebeta

← #17616

Subgroups — #17616

Order 256.

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

normal subgroups only

This group has 459 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#3507C211yesmaximal, characteristic
2#2982C211yesmaximal, characteristic, complemented
3#2356C211yesmaximal, characteristic, complemented
4#1521C214yesmaximal, complemented
5#1518C211yesmaximal, characteristic, complemented
6#1517C211yesmaximal, characteristic
7#1503C211yesmaximal, characteristic, complemented
8#1502C211yesmaximal, characteristic
9#1500C214yesmaximal, complemented
10#559C412yescomplemented
11#557C412yescomplemented
12#556—22no—
13#556C412yescomplemented
14#554C412yescomplemented
15#552—22no—
16#539—22no—
17#536—21no—
18#533C2211yescharacteristic
19#532C2211yescharacteristic
20#529C2214yes—
21#520—21no—
22C2×C4×Q8C2211yescharacteristic
23C8×Q8—22no—
24C8×Q8—22no—
25#431C2214yes—
26#431C2211yescharacteristic
27#431C2211yescharacteristic
28#391—21no—
29#387—22no—
30#383C2212yes—
31#382C2211yescharacteristic
32C4×C4⋊C4—21no—
33C4×C4⋊C4C2211yescharacteristic
34#376C2212yes—
35#328C2212yes—
36#328C2214yes—
37#328C2212yes—
38#327C2212yes—
39#327C2214yes—
40#327C2212yes—
41C2×#41—21no—
42C22×Q8D411yescharacteristic
43C4⋊Q8—22no—
44C4⋊Q8C2×C411yescharacteristic
45C4⋊Q8C2×C411yescharacteristic
46C4⋊Q8C2×C411yescharacteristic
47#127C2×C411yescharacteristic
48#126—24no—
49#126—24no—
50#125—24no—
51#125—21no—
52#125C2×C414yes—
53#124—22no—
54#124—21no—
55#124C2×C414yes—
56#123—24no—
57#123—24no—
58C22⋊Q8—22no—
59C22⋊Q8—22no—
60C22⋊Q8—21no—
61C22⋊Q8—24no—
62C22⋊Q8C2312yes—
63#121—21no—
64#121C2312yes—
65C4×Q8—22no—
66C4×Q8—21no—
67C4×Q8C2×C414yes—
68C4×Q8C2×C412yes—
69C4×Q8C2×C412yes—
70C4×Q8C2×C412yes—
71C4×Q8D412yes—
72C4×Q8D412yes—
73C4×D4—24no—
74C4×D4—22no—
75C4×D4—21no—
76C4×D4C2×C414yes—
77C4×D4C2×C412yes—
78#117—21no—
79#117—22no—
80#117—24no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4D412yes—
86C2×C42—22no—
87C2×C42—21no—
88C2×C42C2312yes—
89C2×C42C2311yescharacteristic
90C4⋊C8—24no—
91C4⋊C8—21no—
92C4⋊C8—21no—
93C22⋊C8C2312yes—
94C22⋊C8C2312yes—
95C4×C8—24no—
96C4×C8—21no—
97C4×C8—21no—
98#95—22no—
99#95—22no—
100#95—21no—
101#95—22no—
102#95C2311yescharacteristic
103#95C2312yes—
104#95C2311yescharacteristic
105#41—42no—
106#41—42no—
107C2×Q8—21no—
108C2×Q8—21no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8C22⋊C412yes—
112C2×Q8C22⋊C412yes—
113C2×Q8C22×C412yes—
114C2×D4—22no—
115C2×D4—21no—
116C2×D4—21no—
117C2×D4C22×C412yes—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—24no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4C2×D412yes—
124C22×C4C2×D412yes—
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2411yescharacteristic
128C2×C8—22no—
129C2×C8—22no—
130C4⋊C4—21no—
131C4⋊C4—24no—
132C4⋊C4—21no—
133C4⋊C4—24no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—22no—
137C4⋊C4—24no—
138C4⋊C4—22no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C4⋊C4C22⋊C412yes—
142C4⋊C4C22⋊C412yes—
143C4⋊C4C22⋊C414yes—
144C4⋊C4C22⋊C414yes—
145C4⋊C4C22×C412yes—
146C4⋊C4C22×C412yes—
147C22⋊C4—22no—
148C22⋊C4—24no—
149C22⋊C4—24no—
150C22⋊C4—22no—
151C22⋊C4—24no—
152C22⋊C4—22no—
153C42—24no—
154C42—22no—
155C42—44no—
156C42—42no—
157C42C22×C411yescharacteristic
158C42C22×C414yes—
159C42C22×C411yescharacteristic
160C42C2×D414yes—
161C42C2×D411yescharacteristic
162C42C2×D411yescharacteristic
163C23—22no—
164C23C22×D411yescharacteristic
165Q8—24no—
166Q8—24no—
167Q8—42no—
168Q8—42no—
169Q8—81no—
170D4—42no—
171D4—42no—
172D4—42no—
173D4—42no—
174D4—81no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—42no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—44no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—84no—
194C2×C4C2×C22⋊C414yes—
195C2×C4C2×C22⋊C414yes—
196C2×C4C2×C22⋊C411yescharacteristic
197C2×C4C2×C22⋊C411yescharacteristic
198C2×C4C2×C22⋊C411yescharacteristic
199C2×C4C2×C22⋊C411yescharacteristic
200C2×C4#13512yes—
201C2×C4#13512yes—
202C2×C4C23×C411yescharacteristic
203C2×C4C22×D411yescharacteristic
204C8—42no—
205C8—42no—
206C22—21no—
207C22—21no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22C22×C22⋊C411yescharacteristic
212C22C2×#13511yescharacteristic
213C22C2×#13511yescharacteristic
214C4—21no—
215C4—21no—
216C4—24no—
217C4—24no—
218C4—21no—
219C4—21no—
220C4—42no—
221C4—42no—
222C4—42no—
223C4—42no—
224C4—81no—
225C4—82no—
226C4#41612yes—
227C4#41612yes—
228C2—21no—
229C2—21no—
230C2—82no—
231C2#288411yescharacteristic
232C2C2×#41611yescharacteristic
233C2#290111yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radical#532C22
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2982C2semidirect
3. #2356C2semidirect
4. #1521C2semidirect
5. #1518C2semidirect
7. #1503C2semidirect
9. #1500C2semidirect
10. #559C4semidirect
11. #557C4semidirect
13. #556C4semidirect
14. #554C4semidirect

Maximal subgroups

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

subgroupindex
#35072
#29822
#23562
#15212
#15182
#15172
#15032
#15022
#15002