Finite Group Databasebeta

← #24697

Subgroups — #24697

Order 256.

390 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 123 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
2C2×#440C211yesmaximal, characteristic
3#2568C211yesmaximal, characteristic, complemented
4#2357C211yesmaximal, characteristic
5#2073C212yesmaximal
6#2055C212yesmaximal
7#1993C211yesmaximal, characteristic, complemented
8#1949C212yesmaximal, complemented
9#1940C211yesmaximal, characteristic, complemented
10#1927C212yesmaximal, complemented
11#1805C211yesmaximal, characteristic, complemented
12C22×SD16C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15C2×C4×D4C2211yescharacteristic
16#476—24no—
17#440—24no—
18C2×#107C2211yescharacteristic
19C2×#106C2212yes—
20C2×C4⋊C8C2212yes—
21C2×C4⋊C8C2211yescharacteristic
22C2×#103C2212yes—
23C2×#103C2211yescharacteristic
24C2×#102C2212yes—
25C2×#102C2211yescharacteristic
26C2×#97C2211yescharacteristic
27#400—21no—
28#400—21no—
29#391—22no—
30#391C2212yes—
31#390C2212yes—
32#388—22no—
33#382C2212yes—
34#382C2211yescharacteristic
35#381—21no—
36C4×C4⋊C4C2212yes—
37C4×C4⋊C4C2211yescharacteristic
38C4×C22⋊C4—21no—
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×SD16—22no—
50C22×C8C2311yescharacteristic
51C22×C8C2312yes—
52C22×C8C2311yescharacteristic
53C4⋊Q8—22no—
54C4⋊Q8—22no—
55C4×Q8—22no—
56C4×Q8—22no—
57C4×D4—22no—
58C4×D4D414yes—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4D412yes—
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2311yescharacteristic
66C2×C22⋊C4—21no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69C2×C42—21no—
70C2×C42—22no—
71C2×C42C2312yes—
72C2×C42C2311yescharacteristic
73#107—22no—
74#106—24no—
75C4⋊C8—24no—
76C4⋊C8—22no—
77#103—24no—
78#103—22no—
79#102—22no—
80#102—24no—
81#97—22no—
82#95—21no—
83#95—22no—
84#95—21no—
85#95—22no—
86#95—22no—
87#95—22no—
88#95—22no—
89#95D411yescharacteristic
90#95D411yescharacteristic
91C24—21no—
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—44no—
98C2×Q8—42no—
99C2×Q8#4112yes—
100C2×D4—21no—
101C2×D4—21no—
102C2×D4—21no—
103C2×D4—21no—
104C2×D4—21no—
105C2×D4C2×D412yes—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—24no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—82no—
113C22×C4C2×D412yes—
114C22×C4C2×D412yes—
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2411yescharacteristic
118SD16—28no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8#4112yes—
128C4⋊C4—21no—
129C4⋊C4—22no—
130C4⋊C4—21no—
131C4⋊C4—22no—
132C4⋊C4—21no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4C2×D412yes—
139C22⋊C4—24no—
140C22⋊C4—44no—
141C22⋊C4—82no—
142C42—22no—
143C42—22no—
144C42—42no—
145C42—82no—
146C42C2×D412yes—
147C42#4112yes—
148C23—22no—
149C23—41no—
150C23—41no—
151C23—42no—
152C23—42no—
153C23C22×D411yescharacteristic
154Q8—22no—
155Q8—22no—
156Q8—41no—
157Q8—41no—
158Q8—84no—
159D4—22no—
160D4—22no—
161D4—41no—
162D4—41no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—44no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—84no—
184C2×C4#12112yes—
185C2×C4#12112yes—
186C2×C4C22×D411yescharacteristic
187C2×C4C2×#4111yescharacteristic
188C2×C4C2×#4111yescharacteristic
189C2×C4C2×#4111yescharacteristic
190C2×C42−1+411yescharacteristic
191C2×C42−1+411yescharacteristic
192C8—22no—
193C8—42no—
194C8—41no—
195C22—42no—
196C22—42no—
197C22—42no—
198C22—82no—
199C22—82no—
200C22C2×#12111yescharacteristic
201C22#54711yescharacteristic
202C22#54811yescharacteristic
203C22#57512yes—
204C22#57712yes—
205C4—22no—
206C4—22no—
207C4—42no—
208C4—42no—
209C4—42no—
210C4—42no—
211C4—82no—
212C4—82no—
213C4—81no—
214C4#52511yescharacteristic
215C4#53611yescharacteristic
216C4#55211yescharacteristic
217C4#55611yescharacteristic
218C2—82no—
219C2#241511yescharacteristic
220C2#304011yescharacteristic
221C2#304211yescharacteristic
222C2#335012yes—
223C2#337112yes—
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
3. #2568C2semidirect
7. #1993C2semidirect
8. #1949C2semidirect
9. #1940C2semidirect
10. #1927C2semidirect
11. #1805C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4762
C2×#4402
#25682
#23572
#20732
#20552
#19932
#19492
#19402
#19272
#18052