Finite Group Databasebeta

← #24327

Subgroups — #24327

Order 256.

573 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 573 conjugacy classes of subgroups — too many to draw. Only its 151 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×#554C211yesmaximal, characteristic, complemented
2C2×#476C211yesmaximal, characteristic, complemented
3C2×C4×D8C211yesmaximal, characteristic
4#2538C211yesmaximal, characteristic, complemented
5#2059C212yesmaximal, complemented
6#2025C212yesmaximal, complemented
7#1990C211yesmaximal, characteristic, complemented
8#1984C211yesmaximal, characteristic, complemented
9#1879C212yesmaximal
10#1864C212yesmaximal, complemented
11#1802C211yesmaximal, characteristic, complemented
12C22×D8C2211yescharacteristic
13#554—24no—
14#554—24no—
15C2×C4⋊Q8C2211yescharacteristic
16C2×#125C2212yes—
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8—21no—
19C2×C22⋊Q8—22no—
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4C2212yes—
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C22×C4⋊C4—21no—
25#476—24no—
26C4×D8—24no—
27C2×#107C2211yescharacteristic
28C2×#106C2212yes—
29C2×C4⋊C8C2211yescharacteristic
30C2×#102C2212yes—
31C2×#102C2211yescharacteristic
32C2×#102C2212yes—
33C2×#102C2211yescharacteristic
34C2×C4×C8C2211yescharacteristic
35#396—22no—
36#390C2212yes—
37#389C2212yes—
38#384C2211yescharacteristic
39#380—21no—
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43#336C2212yes—
44#336C2212yes—
45C22×Q8—21no—
46C22×Q8D411yescharacteristic
47C22×D4C2311yescharacteristic
48C22×D4C2311yescharacteristic
49C23×C4—21no—
50C23×C4—22no—
51C23×C4—21no—
52C2×D8—22no—
53C2×D8—22no—
54C2×D8Q814yescomplemented
55C22×C8C2311yescharacteristic
56C22×C8C2312yes—
57C22×C8C2311yescharacteristic
58C4⋊Q8—22no—
59C4⋊Q8D414yes—
60#125—24no—
61#125—24no—
62C22⋊Q8—28no—
63C22⋊Q8—216no—
64C22⋊Q8—44no—
65C4×Q8—22no—
66C4×Q8—22no—
67C4×D4—24no—
68C4×D4—24no—
69C4×D4—22no—
70C4×D4—22no—
71C4×D4—22no—
72C4×D4D414yes—
73C2×C4⋊C4—28no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4D411yescharacteristic
81C2×C4⋊C4D412yes—
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2312yes—
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2312yes—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C42C2312yes—
92C2×C42C2311yescharacteristic
93#107—22no—
94#106—24no—
95C4⋊C8—22no—
96#102—22no—
97#102—24no—
98#102—22no—
99#102—24no—
100C4×C8—22no—
101#95—22no—
102C24—21no—
103C24—21no—
104C2×Q8—22no—
105C2×Q8—24no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—81no—
109C2×Q8—81no—
110C2×D4—21no—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—21no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4C2×D412yes—
119C2×D4C2×Q812yes—
120C2×D4C2×Q812yes—
121C2×D4#4112yes—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—28no—
125C22×C4—24no—
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—82no—
134C22×C4—82no—
135C22×C4C2×D412yes—
136C22×C4C2×D412yes—
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139C22×C4C2411yescharacteristic
140D8—28no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—22no—
149C2×C8C2×Q812yes—
150C4⋊C4—28no—
151C4⋊C4—28no—
152C4⋊C4—22no—
153C4⋊C4—24no—
154C4⋊C4—24no—
155C4⋊C4—21no—
156C4⋊C4—22no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—44no—
162C4⋊C4—42no—
163C4⋊C4—81no—
164C4⋊C4C2×D414yes—
165C4⋊C4C2×D412yes—
166C22⋊C4—28no—
167C22⋊C4—24no—
168C22⋊C4—44no—
169C22⋊C4—42no—
170C22⋊C4—82no—
171C42—22no—
172C42—22no—
173C42C2×D412yes—
174C42C2×D412yes—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—41no—
180C23—41no—
181C23—41no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23C22×D411yescharacteristic
190Q8—44no—
191Q8—84no—
192D4—22no—
193D4—22no—
194D4—22no—
195D4—22no—
196D4—22no—
197D4—41no—
198D4—41no—
199D4C22⋊Q814yescomplemented
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—24no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—48no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—88no—
222C2×C4—84no—
223C2×C4#12012yes—
224C2×C4#12012yes—
225C2×C4#13512yes—
226C2×C4AGL(1,ℤ/8)12yes—
227C2×C4C22×D411yescharacteristic
228C2×C4C22×D411yescharacteristic
229C2×C4C22×Q811yescharacteristic
230C2×C4C2×#4111yescharacteristic
231C2×C42−1+411yescharacteristic
232C2×C42−1+411yescharacteristic
233C8—22no—
234C8—41no—
235C8—42no—
236C22—24no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—81no—
247C22—81no—
248C22—81no—
249C22—81no—
250C22C2×#12011yescharacteristic
251C22#54811yescharacteristic
252C22D4×Q811yescharacteristic
253C22C2×#13511yescharacteristic
254C22C2×AGL(1,ℤ/8)11yescharacteristic
255C22#57611yescharacteristic
256C22#57711yescharacteristic
257C4—22no—
258C4—42no—
259C4—42no—
260C4—42no—
261C4—44no—
262C4—82no—
263C4—81no—
264C4#44914yes—
265C4C2×C22⋊Q811yescharacteristic
266C4#53611yescharacteristic
267C4#54811yescharacteristic
268C4D4×Q811yescharacteristic
269C2—44no—
270C2—82no—
271C2#240911yescharacteristic
272C2C2×#44911yescharacteristic
273C2#301211yescharacteristic
274C2#336011yescharacteristic
275C2#336611yescharacteristic
276C2#338011yescharacteristic
277C2#338411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#554C2semidirect
2. C2×#476C2semidirect
4. #2538C2semidirect
5. #2059C2semidirect
6. #2025C2semidirect
7. #1990C2semidirect
8. #1984C2semidirect
10. #1864C2semidirect
11. #1802C2semidirect
54. C2×D8Q8semidirect
199. D4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#4762
C2×C4×D82
#25382
#20592
#20252
#19902
#19842
#18792
#18642
#18022