Finite Group Databasebeta

← #24304

Subgroups — #24304

Order 256.

575 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 575 conjugacy classes of subgroups — too many to draw. Only its 147 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×#533C211yesmaximal, characteristic, complemented
2C2×#478C212yesmaximal, complemented
3C4×#121C211yesmaximal, characteristic
4#2058C211yesmaximal, characteristic, complemented
5#2057C211yesmaximal, characteristic, complemented
6#2023C211yesmaximal, characteristic, complemented
7#2023C211yesmaximal, characteristic, complemented
8#1989C212yesmaximal, complemented
9#1875C212yesmaximal
10#1854C212yesmaximal, complemented
11#1797C211yesmaximal, characteristic, complemented
12#533—28no—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#125C2212yes—
16C2×C22⋊Q8—21no—
17C2×C22⋊Q8—21no—
18C2×#121C2211yescharacteristic
19C2×C4×Q8—21no—
20C2×C4×Q8—21no—
21C2×C4×D4—21no—
22C2×C4×D4C2211yescharacteristic
23C2×#117C2212yes—
24C22×C42C2211yescharacteristic
25#478—28no—
26C2×#107C2212yes—
27C2×#106C2212yes—
28C2×C4⋊C8C2212yes—
29C2×#102C2212yes—
30C2×#102C2212yes—
31C2×C22⋊C8C2212yes—
32#390C2211yescharacteristic
33#389C2211yescharacteristic
34#388—21no—
35#386—21no—
36#380C2211yescharacteristic
37C4×C4⋊C4C2211yescharacteristic
38C4×C22⋊C4—21no—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43#336C2211yescharacteristic
44#336C2211yescharacteristic
45C22×Q8—21no—
46C22×Q8—21no—
47C22×D4—21no—
48C22×D4C2311yescharacteristic
49C23×C4—21no—
50C23×C4D411yescharacteristic
51C23×C4D411yescharacteristic
52C23×C4C2311yescharacteristic
53C22×C8C2312yes—
54C22×C8C2312yes—
55C4⋊Q8—24no—
56C4⋊Q8—24no—
57#125—24no—
58#125D418yescomplemented
59C22⋊Q8—28no—
60C22⋊Q8—28no—
61#121—22no—
62#121Q814yescomplemented
63C4×Q8—28no—
64C4×Q8—28no—
65C4×D4—24no—
66C4×D4—44no—
67#117—24no—
68#117—24no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2311yescharacteristic
76C2×C4⋊C4C2312yes—
77C2×C4⋊C4C2312yes—
78C2×C4⋊C4C2311yescharacteristic
79C2×C22⋊C4—22no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C42—22no—
86C2×C42—24no—
87C2×C42—21no—
88C2×C42D411yescharacteristic
89C2×C42D411yescharacteristic
90C2×C42C2311yescharacteristic
91C2×C42C2311yescharacteristic
92#107—24no—
93#106—24no—
94C4⋊C8—24no—
95#102—24no—
96#102—24no—
97C22⋊C8—24no—
98#95—21no—
99#95—21no—
100#95—21no—
101C24—21no—
102C24C2×D411yescharacteristic
103C2×Q8—24no—
104C2×Q8—24no—
105C2×Q8—42no—
106C2×Q8—42no—
107C2×Q8—42no—
108C2×Q8—42no—
109C2×D4—24no—
110C2×D4—22no—
111C2×D4—21no—
112C2×D4—22no—
113C2×D4—42no—
114C2×D4—42no—
115C2×D4C2×Q812yes—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137C22×C4C2×Q812yes—
138C22×C4#4112yes—
139C22×C4C2411yescharacteristic
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C4⋊C4—24no—
147C4⋊C4—24no—
148C4⋊C4—28no—
149C4⋊C4—28no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—21no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—42no—
156C4⋊C4C2×D414yes—
157C4⋊C4C2×D414yes—
158C4⋊C4C2×Q812yes—
159C22⋊C4—28no—
160C22⋊C4—24no—
161C22⋊C4—42no—
162C22⋊C4—42no—
163C22⋊C4—42no—
164C22⋊C4—42no—
165C42—28no—
166C42—22no—
167C42—22no—
168C42—22no—
169C42—42no—
170C42C2×D414yes—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—21no—
175C23—21no—
176C23—21no—
177C23—22no—
178C23—41no—
179C23—41no—
180C23—41no—
181C23—41no—
182C23—41no—
183C23—41no—
184C23C22⋊Q812yes—
185C23C22×D411yescharacteristic
186Q8—48no—
187Q8—48no—
188D4—24no—
189D4—48no—
190D4—42no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—24no—
205C2×C4—22no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—84no—
221C2×C4#12012yes—
222C2×C4#12012yes—
223C2×C4C22⋊Q812yes—
224C2×C4#13512yes—
225C2×C4AGL(1,ℤ/8)12yes—
226C2×C4C22×D412yes—
227C2×C4C22×Q811yescharacteristic
228C2×C4C2×#4111yescharacteristic
229C2×C42−1+412yes—
230C8—42no—
231C8—42no—
232C22—22no—
233C22—22no—
234C22—22no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—81no—
243C22—81no—
244C22—81no—
245C22—81no—
246C22C2×#12011yescharacteristic
247C22C2×C22⋊Q811yescharacteristic
248C22#53611yescharacteristic
249C22C2×#13511yescharacteristic
250C22C2×AGL(1,ℤ/8)11yescharacteristic
251C22#57511yescharacteristic
252C22#57511yescharacteristic
253C4—22no—
254C4—22no—
255C4—22no—
256C4—22no—
257C4—22no—
258C4—42no—
259C4—44no—
260C4—44no—
261C4—81no—
262C4—81no—
263C4#54812yes—
264C4D4×Q812yes—
265C2—42no—
266C2—82no—
267C2#240911yescharacteristic
268C2#300211yescharacteristic
269C2#300611yescharacteristic
270C2#307611yescharacteristic
271C2#307711yescharacteristic
272C2#322811yescharacteristic
273C2#324011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#533C2semidirect
2. C2×#478C2semidirect
4. #2058C2semidirect
5. #2057C2semidirect
6. #2023C2semidirect
7. #2023C2semidirect
8. #1989C2semidirect
10. #1854C2semidirect
11. #1797C2semidirect
58. #125D4semidirect
62. #121Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5332
C2×#4782
C4×#1212
#20582
#20572
#20232
#19892
#18752
#18542
#17972