Finite Group Databasebeta

← #24829

Subgroups — #24829

Order 256.

478 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 139 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×#476C211yesmaximal, characteristic, complemented
2C2×#442C211yesmaximal, characteristic
3#2542C211yesmaximal, characteristic, complemented
4#2349C211yesmaximal, characteristic, complemented
5#2084C212yesmaximal, complemented
6#2065C212yesmaximal, complemented
7#1993C211yesmaximal, characteristic, complemented
8#1940C211yesmaximal, characteristic, complemented
9#1879C212yesmaximal
10#1804C211yesmaximal, characteristic, complemented
11#1802C212yesmaximal, complemented
12C22×D8C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C22⋊Q8—22no—
15C2×C4×D4C2211yescharacteristic
16C2×C4×D4C2212yes—
17C2×C4×D4C2211yescharacteristic
18#476—24no—
19#442—24no—
20C2×#106C2212yes—
21C2×#106C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×C4⋊C8C2212yes—
24C2×#102C2211yescharacteristic
25C2×#102C2212yes—
26C2×#102C2211yescharacteristic
27C2×#102C2212yes—
28C2×#97C2211yescharacteristic
29#399—21no—
30#399—21no—
31#390C2212yes—
32#389C2212yes—
33#385—22no—
34#384C2212yes—
35#382C2211yescharacteristic
36#380—21no—
37C4×C4⋊C4C2211yescharacteristic
38C2×#95—21no—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8—21no—
44C22×D4C2311yescharacteristic
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C23×C4—21no—
48C23×C4—22no—
49C2×D8—22no—
50C2×D8—22no—
51C2×D8Q814yescomplemented
52C22×C8C2312yes—
53C22×C8C2311yescharacteristic
54C22×C8C2311yescharacteristic
55C4⋊Q8—22no—
56C4⋊Q8—22no—
57C22⋊Q8—48no—
58C4×D4—24no—
59C4×D4—22no—
60C4×D4—22no—
61C4×D4—22no—
62C4×D4—42no—
63C4×D4D414yes—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4D412yes—
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2312yes—
72C2×C4⋊C4C2311yescharacteristic
73C2×C22⋊C4—21no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C42C2312yes—
79C2×C42C2311yescharacteristic
80#106—22no—
81#106—24no—
82C4⋊C8—22no—
83C4⋊C8—42no—
84#102—22no—
85#102—24no—
86#102—22no—
87#102—42no—
88#97—22no—
89#95—21no—
90#95—22no—
91#95—21no—
92#95D411yescharacteristic
93#95D411yescharacteristic
94C24—21no—
95C24—21no—
96C2×Q8—44no—
97C2×Q8—42no—
98C2×D4—21no—
99C2×D4—21no—
100C2×D4—21no—
101C2×D4—21no—
102C2×D4—21no—
103C2×D4—21no—
104C2×D4—21no—
105C2×D4—21no—
106C2×D4C2×D412yes—
107C2×D4C2×Q812yes—
108C2×D4C2×Q812yes—
109C2×D4#4112yes—
110C22×C4—21no—
111C22×C4—24no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—44no—
116C22×C4—44no—
117C22×C4—42no—
118C22×C4—44no—
119C22×C4—82no—
120C22×C4—82no—
121C22×C4C2×D412yes—
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D412yes—
125C22×C4C2411yescharacteristic
126D8—28no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—21no—
133C2×C8—22no—
134C2×C8—21no—
135C2×C8C2×Q812yes—
136C4⋊C4—24no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—21no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—44no—
145C4⋊C4—42no—
146C4⋊C4—44no—
147C4⋊C4—82no—
148C4⋊C4C2×D412yes—
149C22⋊C4—24no—
150C22⋊C4—44no—
151C22⋊C4—44no—
152C22⋊C4—42no—
153C22⋊C4—44no—
154C42—22no—
155C42—22no—
156C42C2×D412yes—
157C42#4112yes—
158C23—22no—
159C23—22no—
160C23—22no—
161C23—22no—
162C23—42no—
163C23—42no—
164C23—41no—
165C23—41no—
166C23—42no—
167C23—42no—
168C23C22×D411yescharacteristic
169Q8—84no—
170D4—22no—
171D4—22no—
172D4—22no—
173D4—22no—
174D4—22no—
175D4—41no—
176D4—41no—
177D4C22⋊Q814yescomplemented
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—84no—
196C2×C4—88no—
197C2×C4—84no—
198C2×C4#12112yes—
199C2×C4#12112yes—
200C2×C4AGL(1,ℤ/8)14yes—
201C2×C4C22×D411yescharacteristic
202C2×C4C22×Q811yescharacteristic
203C2×C4C2×#4111yescharacteristic
204C2×C4C2×#4111yescharacteristic
205C2×C4H(5,𝔽2)11yescharacteristic
206C2×C42−1+411yescharacteristic
207C8—22no—
208C8—41no—
209C8—42no—
210C22—24no—
211C22—44no—
212C22—42no—
213C22—42no—
214C22—44no—
215C22—44no—
216C22—42no—
217C22—81no—
218C22—81no—
219C22—82no—
220C22C2×#12111yescharacteristic
221C22#54611yescharacteristic
222C22D4×Q811yescharacteristic
223C22C2×AGL(1,ℤ/8)12yes—
224C22#57712yes—
225C4—22no—
226C4—22no—
227C4—24no—
228C4—42no—
229C4—42no—
230C4—81no—
231C4—82no—
232C4—82no—
233C4C2×C22⋊Q811yescharacteristic
234C4#53711yescharacteristic
235C4#55211yescharacteristic
236C4#55411yescharacteristic
237C2—44no—
238C2—82no—
239C2#241711yescharacteristic
240C2#303811yescharacteristic
241C2#304211yescharacteristic
242C2#330612yes—
243C2#338412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#476C2semidirect
3. #2542C2semidirect
4. #2349C2semidirect
5. #2084C2semidirect
6. #2065C2semidirect
7. #1993C2semidirect
8. #1940C2semidirect
10. #1804C2semidirect
11. #1802C2semidirect
51. C2×D8Q8semidirect
177. D4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#4762
C2×#4422
#25422
#23492
#20842
#20652
#19932
#19402
#18792
#18042
#18022