Finite Group Databasebeta

← #8312

Subgroups — #8312

Order 256.

560 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C2×C4×#41C211yesmaximal, characteristic
2#1797C211yesmaximal, characteristic, complemented
3C4×C22⋊C8C211yesmaximal, characteristic, complemented
4C2×Q8⋊C8C212yesmaximal, complemented
5C2×D4⋊C8C212yesmaximal, complemented
6C22×#41C411yescharacteristic
7C4×#41—24no—
8C4×#41—24no—
9C2×C4×Q8C411yescharacteristic
10C2×C4×Q8C2211yescharacteristic
11C2×C4×D4—21no—
12C2×C4×D4—21no—
13C2×C4×D4C411yescharacteristic
14C2×C4×D4C2211yescharacteristic
15C2×#117—21no—
16C2×#117C411yescharacteristic
17C22×C42—21no—
18C22×C42C2211yescharacteristic
19C2×C4⋊C8C2212yes—
20C2×C22⋊C8—21no—
21C2×C22⋊C8—21no—
22C2×C4×C8C2212yes—
23#336—21no—
24#336—21no—
25Q8⋊C8—24no—
26D4⋊C8—24no—
27C2×#41—22no—
28C2×#41—22no—
29C2×#41—22no—
30C2×#41—22no—
31C2×#41—22no—
32C2×#41C814yescomplemented
33C22×Q8C2×C411yescharacteristic
34C22×D4—21no—
35C22×D4C2×C411yescharacteristic
36C23×C4—21no—
37C23×C4—21no—
38C23×C4—21no—
39C23×C4C2×C411yescharacteristic
40C23×C4D411yescharacteristic
41C23×C4D411yescharacteristic
42C22×C8—22no—
43C22×C8—22no—
44C4×Q8—22no—
45C4×Q8—22no—
46C4×Q8—22no—
47C4×Q8D414yes—
48C4×D4—28no—
49C4×D4—28no—
50C4×D4—22no—
51C4×D4—22no—
52C4×D4—22no—
53C4×D4D414yes—
54#117—28no—
55#117—22no—
56#117—22no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4C2×C411yescharacteristic
60C2×C4⋊C4C2×C411yescharacteristic
61C2×C22⋊C4—21no—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—21no—
64C2×C42—28no—
65C2×C42—21no—
66C2×C42—21no—
67C2×C42—21no—
68C2×C42—22no—
69C2×C42—22no—
70C2×C42C2×C411yescharacteristic
71C2×C42D411yescharacteristic
72C2×C42D411yescharacteristic
73C2×C42C2311yescharacteristic
74C4⋊C8—22no—
75C4⋊C8—22no—
76C22⋊C8—24no—
77C22⋊C8—42no—
78C4×C8—22no—
79C4×C8—22no—
80C24—21no—
81C24C22⋊C411yescharacteristic
82#41—28no—
83#41—28no—
84#41—28no—
85#41—28no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×Q8—21no—
89C2×Q8C22⋊C412yes—
90C2×Q8C2×C812yes—
91C2×Q8M1612yes—
92C2×D4—24no—
93C2×D4—24no—
94C2×D4—24no—
95C2×D4—21no—
96C2×D4—21no—
97C2×D4—21no—
98C2×D4C22⋊C412yes—
99C2×D4C2×C812yes—
100C2×D4M1612yes—
101C22×C4—24no—
102C22×C4—24no—
103C22×C4—21no—
104C22×C4—24no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—24no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—24no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—41no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4C22⋊C411yescharacteristic
126C22×C4C22⋊C411yescharacteristic
127C22×C4C22⋊C411yescharacteristic
128C22×C4C2×C812yes—
129C22×C4M1612yes—
130C22×C4C22×C411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C2×C8—24no—
134C2×C8—42no—
135C2×C8—42no—
136C2×C8—42no—
137C2×C8—42no—
138C2×C8—42no—
139C4⋊C4—24no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—24no—
143C4⋊C4C22⋊C412yes—
144C4⋊C4C22⋊C412yes—
145C22⋊C4—24no—
146C22⋊C4—24no—
147C22⋊C4—24no—
148C42—24no—
149C42—24no—
150C42—21no—
151C42—21no—
152C42—22no—
153C42—24no—
154C42—22no—
155C42C2×D412yes—
156C42C2×D412yes—
157C23—21no—
158C23—22no—
159C23—22no—
160C23—21no—
161C23—22no—
162C23—21no—
163C23—21no—
164C23—21no—
165C23—21no—
166C23—41no—
167C23—41no—
168C23—41no—
169C23—41no—
170C23C22⋊C812yes—
171C23C2×C22⋊C411yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—22no—
175Q8C22⋊C814yescomplemented
176D4—216no—
177D4—22no—
178D4—22no—
179D4—22no—
180D4C22⋊C814yescomplemented
181C2×C4—216no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—44no—
201C2×C4—48no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—44no—
208C2×C4—42no—
209C2×C4—44no—
210C2×C4C22⋊C812yes—
211C2×C4#10412yes—
212C2×C4C2×C22⋊C411yescharacteristic
213C2×C4C2×C22⋊C411yescharacteristic
214C2×C4#12012yes—
215C2×C4#12012yes—
216C2×C4C22×C811yescharacteristic
217C2×C4C2×M1611yescharacteristic
218C2×C4C2×M1611yescharacteristic
219C2×C4C2×M1611yescharacteristic
220C2×C4#13512yes—
221C2×C4AGL(1,ℤ/8)11yescharacteristic
222C2×C4#13711yescharacteristic
223C8—44no—
224C8—82no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—24no—
229C22—42no—
230C22—42no—
231C22—41no—
232C22—42no—
233C22—41no—
234C22—42no—
235C22—41no—
236C22—42no—
237C22—41no—
238C22—42no—
239C22#37911yescharacteristic
240C22C2×C22⋊C811yescharacteristic
241C22#40711yescharacteristic
242C22#41611yescharacteristic
243C22#41711yescharacteristic
244C22C2×#10411yescharacteristic
245C22#42111yescharacteristic
246C4—22no—
247C4—24no—
248C4—22no—
249C4—22no—
250C4—41no—
251C4—42no—
252C4—42no—
253C4C2×C22⋊C811yescharacteristic
254C4C2×C22⋊C811yescharacteristic
255C4#40711yescharacteristic
256C4#40711yescharacteristic
257C4#44912yes—
258C4#45212yes—
259C2—42no—
260C2—44no—
261C2#147611yescharacteristic
262C2#147711yescharacteristic
263C2#148811yescharacteristic
264C2#148911yescharacteristic
265C2#177911yescharacteristic
266C2#178811yescharacteristic
267C2#178911yescharacteristic
subgroupG/H
centerC22×C4C2×D4
commutatorC2×C4C22×C8
Frattini, cosocleC2×C42C23
socleC23C2×C22⋊C4
prime radicalC22×#41C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1797C2semidirect
3. C4×C22⋊C8C2semidirect
4. C2×Q8⋊C8C2semidirect
5. C2×D4⋊C8C2semidirect
32. C2×#41C8semidirect
175. Q8C22⋊C8semidirect
180. D4C22⋊C8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#412
#17972
C4×C22⋊C82
C2×Q8⋊C82
C2×D4⋊C82