Finite Group Databasebeta

← #24828

Subgroups — #24828

Order 256.

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

normal subgroups only

This group has 412 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×#475C211yesmaximal, characteristic, complemented
2C2×#440C211yesmaximal, characteristic
3#2563C211yesmaximal, characteristic
4#2351C211yesmaximal, characteristic, complemented
5#2085C212yesmaximal, complemented
6#2065C212yesmaximal
7#1993C211yesmaximal, characteristic, complemented
8#1940C211yesmaximal, characteristic, complemented
9#1880C212yesmaximal
10#1805C212yesmaximal, complemented
11#1804C211yesmaximal, characteristic
12C22×SD16C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4×Q8C2212yes—
15C2×C4×Q8C2211yescharacteristic
16C2×C4×D4C2211yescharacteristic
17#475—24no—
18#440—24no—
19C2×#107C2211yescharacteristic
20C2×#106C2212yes—
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2212yes—
23C2×#103C2212yes—
24C2×#103C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2212yes—
27C2×#97C2211yescharacteristic
28#393—21no—
29#393—21no—
30#390C2212yes—
31#389—22no—
32#389C2212yes—
33#382—21no—
34#382C2212yes—
35#382C2211yescharacteristic
36#380—22no—
37C4×C4⋊C4—21no—
38C4×C4⋊C4C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8C2311yescharacteristic
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C2×SD16—22no—
48C2×SD16—22no—
49C2×SD16Q814yescomplemented
50C22×C8C2312yes—
51C22×C8C2311yescharacteristic
52C22×C8C2311yescharacteristic
53C4⋊Q8—22no—
54C4⋊Q8—22no—
55C4×Q8—22no—
56C4×Q8—24no—
57C4×Q8—42no—
58C4×Q8D414yes—
59C4×D4—22no—
60C4×D4—22no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4D412yes—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2311yescharacteristic
70C2×C22⋊C4—21no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C42—21no—
74C2×C42—22no—
75C2×C42C2312yes—
76C2×C42C2311yescharacteristic
77#107—22no—
78#106—24no—
79C4⋊C8—22no—
80C4⋊C8—42no—
81#103—22no—
82#103—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—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—42no—
98C2×Q8—82no—
99C2×Q8C2×D412yes—
100C2×Q8C2×Q812yes—
101C2×Q8#4112yes—
102C2×D4—21no—
103C2×D4—21no—
104C2×D4—21no—
105C2×D4—21no—
106C2×D4—21no—
107C2×D4C2×Q812yes—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—44no—
114C22×C4—42no—
115C22×C4C2×D412yes—
116C22×C4C2×D412yes—
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2411yescharacteristic
120SD16—28no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—21no—
127C2×C8—22no—
128C2×C8—21no—
129C2×C8C2×Q812yes—
130C4⋊C4—24no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—42no—
139C4⋊C4—44no—
140C4⋊C4—44no—
141C4⋊C4—42no—
142C4⋊C4C2×D412yes—
143C22⋊C4—42no—
144C22⋊C4—82no—
145C22⋊C4—82no—
146C42—24no—
147C42—22no—
148C42—22no—
149C42—44no—
150C42C2×D412yes—
151C42#4112yes—
152C23—22no—
153C23—41no—
154C23—41no—
155C23—42no—
156C23—42no—
157C23C22×D411yescharacteristic
158Q8—22no—
159Q8—22no—
160Q8—22no—
161Q8—84no—
162Q8C22⋊Q814yes—
163D4—22no—
164D4—22no—
165D4—41no—
166D4—41no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—84no—
189C2×C4#12112yes—
190C2×C4#12112yes—
191C2×C4C22×D411yescharacteristic
192C2×C4C22×Q811yescharacteristic
193C2×C4C2×#4111yescharacteristic
194C2×C4C2×#4111yescharacteristic
195C2×C4H(5,𝔽2)11yescharacteristic
196C2×C42−1+411yescharacteristic
197C8—22no—
198C8—41no—
199C8—42no—
200C22—44no—
201C22—42no—
202C22—81no—
203C22—81no—
204C22—82no—
205C22C2×#12111yescharacteristic
206C22#54611yescharacteristic
207C22D4×Q811yescharacteristic
208C22#57512yes—
209C22#57712yes—
210C4—24no—
211C4—22no—
212C4—22no—
213C4—42no—
214C4—42no—
215C4—42no—
216C4—81no—
217C4—82no—
218C4—82no—
219C4C2×C22⋊Q811yescharacteristic
220C4#53711yescharacteristic
221C4#55211yescharacteristic
222C4#55411yescharacteristic
223C2—82no—
224C2#241711yescharacteristic
225C2#304011yescharacteristic
226C2#304311yescharacteristic
227C2#332112yes—
228C2#338812yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
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×#475C2semidirect
4. #2351C2semidirect
5. #2085C2semidirect
7. #1993C2semidirect
8. #1940C2semidirect
10. #1805C2semidirect
49. C2×SD16Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#4752
C2×#4402
#25632
#23512
#20852
#20652
#19932
#19402
#18802
#18052
#18042