Finite Group Databasebeta

← #33046

Subgroups — #33046

Order 256.

415 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 415 conjugacy classes of subgroups — too many to draw. Only its 131 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×#487C211yesmaximal, characteristic, complemented
2C2×#405C211yesmaximal, characteristic
3#2628C211yesmaximal, characteristic
4#2621C211yesmaximal, characteristic
5#1993C211yesmaximal, characteristic, complemented
6#1960C211yesmaximal, characteristic, complemented
7#1935C212yesmaximal, complemented
8#1933C212yesmaximal, complemented
9#1894C212yesmaximal
10#1892C212yesmaximal
11#1835C211yesmaximal, characteristic, complemented
12C22×M16C2211yescharacteristic
13C23×C8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×C22⋊Q8C2212yes—
17C2×#117C2211yescharacteristic
18#487—24no—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23#405—24no—
24C2×#97C2211yescharacteristic
25C2×#97C2211yescharacteristic
26C2×C4×C8C2211yescharacteristic
27C2×C4×C8C2211yescharacteristic
28#400—21no—
29#399—21no—
30#398—21no—
31#398—21no—
32#397—21no—
33#397—21no—
34#395—21no—
35#393—21no—
36#391C2212yes—
37#389C2212yes—
38#382C2211yescharacteristic
39#382C2211yescharacteristic
40#380C2212yes—
41#340C2212yes—
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45C22×Q8—21no—
46C22×Q8C2312yes—
47C23×C4C2311yescharacteristic
48C2×M16—22no—
49C2×M16—22no—
50C2×M16—22no—
51C22×C8—22no—
52C22×C8—22no—
53C22×C8—22no—
54C22×C8C2311yescharacteristic
55C22×C8C2311yescharacteristic
56C22×C8C2311yescharacteristic
57C22×C8C2311yescharacteristic
58C4⋊Q8—22no—
59C4⋊Q8—22no—
60C4⋊Q8—22no—
61C4⋊Q8—22no—
62C22⋊Q8—24no—
63C22⋊Q8—24no—
64#117—22no—
65#117—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4D411yescharacteristic
72C2×C4⋊C4D411yescharacteristic
73C2×C4⋊C4C2312yes—
74C2×C4⋊C4C2312yes—
75C2×C4⋊C4C2312yes—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4D411yescharacteristic
79C2×C22⋊C4D411yescharacteristic
80C2×C42C2311yescharacteristic
81C2×C42C2311yescharacteristic
82#103—24no—
83#103—24no—
84#103—24no—
85#103—24no—
86#97—22no—
87#97—22no—
88C4×C8—22no—
89C4×C8D414yes—
90#95—22no—
91#95—21no—
92#95—21no—
93#95—21no—
94#95—21no—
95#95—22no—
96#95—22no—
97#95—22no—
98C24C2×D411yescharacteristic
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—42no—
106C2×Q8—82no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4#4112yes—
121C22×C4#4112yes—
122C22×C4C2411yescharacteristic
123M16—28no—
124C2×C8—28no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8C2×D412yes—
134C2×C8C2×D412yes—
135C2×C8#4112yes—
136C2×C8#4112yes—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C4⋊C4—44no—
149C4⋊C4—82no—
150C22⋊C4—22no—
151C22⋊C4—22no—
152C22⋊C4—44no—
153C22⋊C4—82no—
154C42—21no—
155C42—21no—
156C42C2×D412yes—
157C42#4112yes—
158C23—21no—
159C23—21no—
160C23—22no—
161C23—22no—
162C23#12412yes—
163C23C22×D411yescharacteristic
164Q8—42no—
165Q8—42no—
166Q8—42no—
167Q8—42no—
168Q8—84no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4#12112yes—
194C2×C4#12112yes—
195C2×C4#12412yes—
196C2×C4C22×D411yescharacteristic
197C2×C4C2×#4111yescharacteristic
198C2×C4C2×#4111yescharacteristic
199C2×C4C2×#4111yescharacteristic
200C2×C4C2×#4111yescharacteristic
201C2×C4C2×#4111yescharacteristic
202C8—22no—
203C8—22no—
204C8—22no—
205C8—22no—
206C22—22no—
207C22—24no—
208C22—42no—
209C22—41no—
210C22—41no—
211C22C2×#12111yescharacteristic
212C22C2×#12411yescharacteristic
213C22#53211yescharacteristic
214C22#57712yes—
215C22#57812yes—
216C4—22no—
217C4—22no—
218C4—22no—
219C4—41no—
220C4—41no—
221C4—82no—
222C4—82no—
223C4—82no—
224C4—82no—
225C4#52511yescharacteristic
226C4#52511yescharacteristic
227C4#52911yescharacteristic
228C4#52911yescharacteristic
229C2—42no—
230C2#285011yescharacteristic
231C2#305711yescharacteristic
232C2#305911yescharacteristic
233C2#313712yes—
234C2#317812yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#487C2semidirect
5. #1993C2semidirect
6. #1960C2semidirect
7. #1935C2semidirect
8. #1933C2semidirect
11. #1835C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4872
C2×#4052
#26282
#26212
#19932
#19602
#19352
#19332
#18942
#18922
#18352