Finite Group Databasebeta

← #24338

Subgroups — #24338

Order 256.

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

normal subgroups only

This group has 648 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×D4×Q8C211yesmaximal, characteristic, complemented
2C2×#476C211yesmaximal, characteristic, complemented
3C2×#442C211yesmaximal, characteristic
4#2580C211yesmaximal, characteristic, complemented
5#2058C212yesmaximal, complemented
6#2023C212yesmaximal, complemented
7#1993C211yesmaximal, characteristic, complemented
8#1985C211yesmaximal, characteristic, complemented
9#1879C212yesmaximal
10#1864C212yesmaximal, complemented
11#1804C211yesmaximal, characteristic, complemented
12C23×Q8—21no—
13C22×D8C2211yescharacteristic
14D4×Q8—24no—
15D4×Q8—24no—
16C2×C4⋊Q8C2212yes—
17C2×C4⋊Q8C2211yescharacteristic
18C2×C22⋊Q8—22no—
19C2×C22⋊Q8—21no—
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2212yes—
23C2×C4×D4C2211yescharacteristic
24#476—24no—
25#442—24no—
26C2×#106C2211yescharacteristic
27C2×#106C2212yes—
28C2×C4⋊C8C2211yescharacteristic
29C2×#102C2212yes—
30C2×#102C2211yescharacteristic
31C2×#102C2212yes—
32C2×#102C2211yescharacteristic
33C2×#97C2211yescharacteristic
34#399—21no—
35#393—21no—
36#390C2212yes—
37#389C2212yes—
38#385—22no—
39#382C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43#336C2212yes—
44#336C2212yes—
45C22×Q8—28no—
46C22×Q8—21no—
47C22×Q8—22no—
48C22×Q8—48no—
49C22×Q8D411yescharacteristic
50C22×D4C2311yescharacteristic
51C22×D4C2311yescharacteristic
52C23×C4—22no—
53C23×C4—21no—
54C23×C4—21no—
55C2×D8—22no—
56C2×D8—22no—
57C2×D8Q814yescomplemented
58C22×C8C2311yescharacteristic
59C22×C8C2312yes—
60C22×C8C2311yescharacteristic
61C4⋊Q8—24no—
62C4⋊Q8—24no—
63C4⋊Q8—22no—
64C4⋊Q8D414yes—
65C22⋊Q8—28no—
66C22⋊Q8—216no—
67C4×Q8—22no—
68C4×Q8—22no—
69C4×D4—24no—
70C4×D4—24no—
71C4×D4—22no—
72C4×D4—22no—
73C4×D4—22no—
74C4×D4D414yes—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4D412yes—
79C2×C4⋊C4D411yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2312yes—
82C2×C4⋊C4C2312yes—
83C2×C4⋊C4C2311yescharacteristic
84C2×C22⋊C4—22no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C42C2312yes—
90C2×C42C2311yescharacteristic
91#106—22no—
92#106—24no—
93C4⋊C8—22no—
94#102—22no—
95#102—24no—
96#102—22no—
97#102—24no—
98#97—22no—
99#95—21no—
100#95—21no—
101#95—22no—
102C24—21no—
103C24—21no—
104C2×Q8—28no—
105C2×Q8—22no—
106C2×Q8—24no—
107C2×Q8—24no—
108C2×Q8—44no—
109C2×Q8—44no—
110C2×Q8—432no—
111C2×Q8—44no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—21no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—21no—
120C2×D4C2×D412yes—
121C2×D4C2×Q812yes—
122C2×D4C2×Q812yescomplemented
123C2×D4#4112yes—
124C22×C4—28no—
125C22×C4—22no—
126C22×C4—21no—
127C22×C4—24no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—44no—
131C22×C4—44no—
132C22×C4—44no—
133C22×C4—42no—
134C22×C4—82no—
135C22×C4C2×D412yes—
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D412yes—
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×Q812yescomplemented
150C4⋊C4—28no—
151C4⋊C4—24no—
152C4⋊C4—24no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—82no—
161C4⋊C4C2×D414yes—
162C4⋊C4C2×D412yes—
163C22⋊C4—28no—
164C22⋊C4—24no—
165C22⋊C4—42no—
166C22⋊C4—44no—
167C22⋊C4—82no—
168C42—22no—
169C42—22no—
170C42C2×D412yes—
171C42C2×D412yes—
172C23—22no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23—42no—
177C23—42no—
178C23—41no—
179C23—41no—
180C23—42no—
181C23—42no—
182C23C22×D411yescharacteristic
183Q8—28no—
184Q8—48no—
185Q8—416no—
186D4—22no—
187D4—22no—
188D4—22no—
189D4—22no—
190D4—22no—
191D4—41no—
192D4—41no—
193D4C22⋊Q814yes—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—24no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—48no—
204C2×C4—416no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—84no—
214C2×C4#12012yes—
215C2×C4#12012yes—
216C2×C4AGL(1,ℤ/8)14yes—
217C2×C4C22×D411yescharacteristic
218C2×C4C22×D411yescharacteristic
219C2×C4C22×Q811yescharacteristic
220C2×C4C2×#4111yescharacteristic
221C2×C42−1+411yescharacteristic
222C2×C42−1+411yescharacteristic
223C8—22no—
224C8—41no—
225C8—42no—
226C22—24no—
227C22—44no—
228C22—44no—
229C22—42no—
230C22—42no—
231C22—44no—
232C22—42no—
233C22—82no—
234C22—81no—
235C22—81no—
236C22C2×#12011yescharacteristic
237C22#54811yescharacteristic
238C22D4×Q811yescharacteristic
239C22C2×AGL(1,ℤ/8)12yes—
240C22#57712yes—
241C4—22no—
242C4—24no—
243C4—22no—
244C4—44no—
245C4—42no—
246C4—42no—
247C4—82no—
248C4—81no—
249C4C2×C22⋊Q811yescharacteristic
250C4#53611yescharacteristic
251C4#54811yescharacteristic
252C4D4×Q811yescharacteristic
253C2—44no—
254C2—82no—
255C2#240911yescharacteristic
256C2#300311yescharacteristic
257C2#301011yescharacteristic
258C2#337312yes—
259C2#338412yes—
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×D4×Q8C2semidirect
2. C2×#476C2semidirect
4. #2580C2semidirect
5. #2058C2semidirect
6. #2023C2semidirect
7. #1993C2semidirect
8. #1985C2semidirect
10. #1864C2semidirect
11. #1804C2semidirect
57. C2×D8Q8semidirect
122. C2×D4C2×Q8semidirect
149. C2×C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Q82
C2×#4762
C2×#4422
#25802
#20582
#20232
#19932
#19852
#18792
#18642
#18042