Finite Group Databasebeta

← #8425

Subgroups — #8425

Order 256.

375 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 375 conjugacy classes of subgroups — too many to draw. Only its 89 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#3503C211yesmaximal, characteristic
2#1981C211yesmaximal, characteristic, complemented
3#1806C211yesmaximal, characteristic
4#1550C211yesmaximal, characteristic, complemented
5#1548C211yesmaximal, characteristic, complemented
6#1503C211yesmaximal, characteristic, complemented
7#1477C211yesmaximal, characteristic
8Q82C412yescomplemented
9#557—21no—
10#554—22no—
11#554C412yescomplemented
12D4×Q8—22no—
13D4×Q8—21no—
14#544—22no—
15#544—21no—
16#533—21no—
17#533C2211yescharacteristic
18C2×C4⋊Q8—21no—
19C2×C4⋊Q8C2211yescharacteristic
20#520—21no—
21C4×#41C2211yescharacteristic
22#431C2211yescharacteristic
23#423C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#104—21no—
26C2×#103—22no—
27C4×C4⋊C4C2211yescharacteristic
28#339—21no—
29#333—22no—
30#333—21no—
31#332—21no—
32#328—22no—
33Q8⋊C8—21no—
34D4⋊C8—21no—
35C2×#41D411yescharacteristic
36C22×Q8—22no—
37C22×Q8D412yes—
38C2×M16—21no—
39C22×C8—21no—
40C4⋊Q8—22no—
41C4⋊Q8—22no—
42C4⋊Q8—24no—
43C4⋊Q8—22no—
44C4⋊Q8—22no—
45C4⋊Q8—21no—
46C4⋊Q8—21no—
47C4⋊Q8C2×C412yes—
48C4⋊Q8D412yes—
49C4⋊Q8D411yescharacteristic
50C4⋊Q8D411yescharacteristic
51#125—22no—
52#125—24no—
53#125—22no—
54#125—21no—
55C22⋊Q8—22no—
56C22⋊Q8—24no—
57C22⋊Q8—24no—
58C22⋊Q8—22no—
59C22⋊Q8—24no—
60C22⋊Q8—21no—
61C22⋊Q8D412yes—
62C4×Q8—22no—
63C4×Q8—22no—
64C4×Q8—22no—
65C4×Q8—21no—
66C4×Q8C2×C411yescharacteristic
67C4×Q8C2×C412yes—
68C4×Q8D411yescharacteristic
69C4×D4—24no—
70C4×D4—22no—
71C4×D4—21no—
72C4×D4—21no—
73C4×D4C2×C411yescharacteristic
74C4×D4D411yescharacteristic
75#117—22no—
76#117—21no—
77#117—21no—
78#117D411yescharacteristic
79C2×C4⋊C4—22no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—21no—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42C2311yescharacteristic
87C4⋊C8—21no—
88C4⋊C8—21no—
89C4⋊C8—21no—
90C4⋊C8—21no—
91C4⋊C8—21no—
92#104—42no—
93#103—44no—
94C22⋊C8—21no—
95C4×C8—21no—
96#95—21no—
97#41—21no—
98#41—21no—
99#41SD1612yes—
100#41Q1612yes—
101C2×Q8—24no—
102C2×Q8—22no—
103C2×Q8—21no—
104C2×Q8—42no—
105C2×Q8—42no—
106C2×Q8—82no—
107C2×Q8—82no—
108C2×Q8C22⋊C411yescharacteristic
109C2×Q8C22⋊C412yes—
110C2×Q8C22⋊C414yes—
111C2×D4—21no—
112C2×D4C22⋊C411yescharacteristic
113C22×C4—21no—
114C22×C4—24no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—41no—
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122M16—41no—
123M16—41no—
124C2×C8—42no—
125C2×C8—41no—
126C2×C8—41no—
127C2×C8—41no—
128C4⋊C4—24no—
129C4⋊C4—24no—
130C4⋊C4—24no—
131C4⋊C4—24no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—21no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—21no—
138C4⋊C4—22no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—41no—
145C4⋊C4C22⋊C412yes—
146C4⋊C4C22⋊C411yescharacteristic
147C4⋊C4C22⋊C411yescharacteristic
148C22⋊C4—24no—
149C22⋊C4—22no—
150C22⋊C4—21no—
151C22⋊C4—21no—
152C22⋊C4—21no—
153C42—24no—
154C42—21no—
155C42—22no—
156C42—21no—
157C42—21no—
158C42—41no—
159C42—41no—
160C42C22×C411yescharacteristic
161C42C2×D411yescharacteristic
162C42C2×D411yescharacteristic
163C42C2×D411yescharacteristic
164C23—21no—
165C23#12011yescharacteristic
166Q8—21no—
167Q8—44no—
168Q8—42no—
169Q8—42no—
170Q8—41no—
171Q8—42no—
172Q8—41no—
173Q8—84no—
174Q8#10312yescomplemented
175D4—22no—
176D4—22no—
177D4—21no—
178D4#10312yescomplemented
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—82no—
195C2×C4—84no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4C2×C22⋊C411yescharacteristic
199C2×C4C2×C22⋊C411yescharacteristic
200C2×C4C2×C22⋊C411yescharacteristic
201C2×C4#12011yescharacteristic
202C2×C4#12011yescharacteristic
203C2×C4#12011yescharacteristic
204C2×C4C2×SD1611yescharacteristic
205C2×C4C2×Q1611yescharacteristic
206C2×C4#13711yescharacteristic
207C2×C4#13711yescharacteristic
208C8—81no—
209C8—81no—
210C8—81no—
211C22—21no—
212C22—21no—
213C22—22no—
214C22—41no—
215C22—41no—
216C22#37911yescharacteristic
217C22#44811yescharacteristic
218C22#45111yescharacteristic
219C4—21no—
220C4—21no—
221C4—21no—
222C4—22no—
223C4—41no—
224C4—44no—
225C4—42no—
226C4—41no—
227C4—42no—
228C4—82no—
229C4C2×#10311yescharacteristic
230C4#41911yescharacteristic
231C4#44811yescharacteristic
232C4#45111yescharacteristic
233C4#45311yescharacteristic
234C4#45511yescharacteristic
235C2—21no—
236C2—21no—
237C2—42no—
238C2#178711yescharacteristic
239C2#179211yescharacteristic
240C2#179611yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC2×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1981C2semidirect
4. #1550C2semidirect
5. #1548C2semidirect
6. #1503C2semidirect
8. Q82C4semidirect
11. #554C4semidirect
174. Q8#103semidirect
178. D4#103semidirect

Maximal subgroups

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

subgroupindex
#35032
#19812
#18062
#15502
#15482
#15032
#14772