Finite Group Databasebeta

← #24830

Subgroups — #24830

Order 256.

450 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 135 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×#474C211yesmaximal, characteristic, complemented
2C2×#442C211yesmaximal, characteristic
3#2563C211yesmaximal, characteristic, complemented
4#2349C211yesmaximal, characteristic, complemented
5#2085C212yesmaximal, complemented
6#2067C212yesmaximal, complemented
7#1993C211yesmaximal, characteristic, complemented
8#1940C211yesmaximal, characteristic, complemented
9#1879C212yesmaximal
10#1804C211yesmaximal, characteristic, complemented
11#1804C212yesmaximal, complemented
12C22×D8C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4×D4C2211yescharacteristic
15C2×C4×D4C2212yes—
16C2×C4×D4C2211yescharacteristic
17#474—24no—
18#442—24no—
19C2×#107C2212yes—
20C2×#106C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2212yes—
23C2×#102C2211yescharacteristic
24C2×#102C2212yes—
25C2×#102C2211yescharacteristic
26C2×#102C2212yes—
27C2×#97C2211yescharacteristic
28#393—21no—
29#393—21no—
30#390C2212yes—
31#389C2212yes—
32#385—22no—
33#382C2212yes—
34#382C2211yescharacteristic
35#380—22no—
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—
57C4×D4—24no—
58C4×D4—22no—
59C4×D4—22no—
60C4×D4—22no—
61C4×D4—42no—
62C4×D4D414yes—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4D412yes—
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2311yescharacteristic
70C2×C22⋊C4—21no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C42C2312yes—
76C2×C42C2311yescharacteristic
77#107—24no—
78#106—22no—
79C4⋊C8—22no—
80C4⋊C8—42no—
81#102—22no—
82#102—24no—
83#102—22no—
84#102—42no—
85#97—22no—
86#95—21no—
87#95—22no—
88#95—22no—
89#95—22no—
90#95—21no—
91#95D411yescharacteristic
92#95D411yescharacteristic
93C24—21no—
94C24—21no—
95C2×Q8—42no—
96C2×Q8—82no—
97C2×D4—21no—
98C2×D4—21no—
99C2×D4—21no—
100C2×D4—21no—
101C2×D4—21no—
102C2×D4—21no—
103C2×D4—21no—
104C2×D4—21no—
105C2×D4C2×D412yes—
106C2×D4C2×Q812yes—
107C2×D4C2×Q812yes—
108C2×D4#4112yes—
109C22×C4—21no—
110C22×C4—24no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—44no—
115C22×C4—44no—
116C22×C4—44no—
117C22×C4—42no—
118C22×C4—82no—
119C22×C4—82no—
120C22×C4C2×D412yes—
121C22×C4C2×D412yes—
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2411yescharacteristic
125D8—28no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—21no—
132C2×C8—22no—
133C2×C8—21no—
134C2×C8C2×Q812yes—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—42no—
143C4⋊C4—42no—
144C4⋊C4—82no—
145C4⋊C4C2×D412yes—
146C22⋊C4—24no—
147C22⋊C4—44no—
148C22⋊C4—42no—
149C22⋊C4—82no—
150C22⋊C4—82no—
151C42—22no—
152C42—22no—
153C42C2×D412yes—
154C42#4112yes—
155C23—22no—
156C23—22no—
157C23—22no—
158C23—22no—
159C23—42no—
160C23—42no—
161C23—41no—
162C23—41no—
163C23—42no—
164C23—42no—
165C23C22×D411yescharacteristic
166Q8—84no—
167D4—22no—
168D4—22no—
169D4—22no—
170D4—22no—
171D4—22no—
172D4—41no—
173D4—41no—
174D4C22⋊Q814yes—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—84no—
194C2×C4—84no—
195C2×C4—88no—
196C2×C4#12112yes—
197C2×C4#12112yes—
198C2×C4C22×D411yescharacteristic
199C2×C4C22×Q811yescharacteristic
200C2×C4C2×#4111yescharacteristic
201C2×C4C2×#4111yescharacteristic
202C2×C4H(5,𝔽2)11yescharacteristic
203C2×C42−1+411yescharacteristic
204C8—22no—
205C8—41no—
206C8—42no—
207C22—24no—
208C22—44no—
209C22—42no—
210C22—42no—
211C22—44no—
212C22—44no—
213C22—42no—
214C22—81no—
215C22—81no—
216C22—82no—
217C22C2×#12111yescharacteristic
218C22#54611yescharacteristic
219C22D4×Q811yescharacteristic
220C22#57512yes—
221C22#57612yes—
222C4—22no—
223C4—22no—
224C4—42no—
225C4—42no—
226C4—42no—
227C4—81no—
228C4—82no—
229C4—82no—
230C4C2×C22⋊Q811yescharacteristic
231C4#53711yescharacteristic
232C4#55211yescharacteristic
233C4#55411yescharacteristic
234C2—44no—
235C2—82no—
236C2#241711yescharacteristic
237C2#304011yescharacteristic
238C2#304111yescharacteristic
239C2#331912yes—
240C2#338912yes—
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×#474C2semidirect
3. #2563C2semidirect
4. #2349C2semidirect
5. #2085C2semidirect
6. #2067C2semidirect
7. #1993C2semidirect
8. #1940C2semidirect
10. #1804C2semidirect
11. #1804C2semidirect
51. C2×D8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#4742
C2×#4422
#25632
#23492
#20852
#20672
#19932
#19402
#18792
#18042