Finite Group Databasebeta

← #24807

Subgroups — #24807

Order 256.

422 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 127 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×#473C211yesmaximal, characteristic, complemented
2#3056C212yesmaximal
3C2×#470C211yesmaximal, characteristic, complemented
4#2930C211yesmaximal, characteristic
5C2×#430C211yesmaximal, characteristic, complemented
6#2064C212yesmaximal, complemented
7#2052C212yesmaximal, complemented
8#1979C211yesmaximal, characteristic, complemented
9#1951C211yesmaximal, characteristic, complemented
10#1901C212yesmaximal, complemented
11#1815C211yesmaximal, characteristic, complemented
12C2×#137C2212yes—
13C22×Q16C2211yescharacteristic
14C2×#131C2211yescharacteristic
15C22×M16C2212yes—
16C22×M16C2211yescharacteristic
17#525C2211yescharacteristic
18C2×C22⋊Q8C2212yes—
19#473—24no—
20#470—22no—
21#470—22no—
22#470—22no—
23#468—22no—
24#467—22no—
25#465—22no—
26#436—22no—
27#435—22no—
28#430—24no—
29C2×#108C2212yes—
30C2×#108C2211yescharacteristic
31C2×#108C2211yescharacteristic
32#428C2212yes—
33C2×#106C2211yescharacteristic
34#424C2211yescharacteristic
35#419C2212yes—
36#417C2212yes—
37C2×#103C2212yes—
38C2×#101C2212yes—
39C2×#101C2212yes—
40#408C2211yescharacteristic
41#407C2211yescharacteristic
42#405C2211yescharacteristic
43#343C2212yes—
44#343C2212yes—
45C2×#41C2311yescharacteristic
46C22×Q8C2312yes—
47C23×C4C2311yescharacteristic
48#137—24no—
49#137—42no—
50C2×Q16—22no—
51C2×Q16—21no—
52C2×Q16—21no—
53C2×Q16—22no—
54C2×Q16—22no—
55C2×SD16—22no—
56#131—22no—
57#131—22no—
58C2×M16—22no—
59C2×M16—22no—
60C2×M16—21no—
61C2×M16—22no—
62C2×M16—24no—
63C2×M16—42no—
64C2×M16D412yes—
65C2×M16D412yes—
66C2×M16C2312yes—
67C2×M16C2311yescharacteristic
68C2×M16C2311yescharacteristic
69C2×M16C2312yes—
70C22×C8—22no—
71C22×C8—21no—
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74#123—21no—
75C22⋊Q8—22no—
76C22⋊Q8—22no—
77C22⋊Q8—22no—
78C22⋊Q8—22no—
79C22⋊Q8D412yes—
80#121D412yes—
81#120—21no—
82C4×D4—22no—
83#117C2311yescharacteristic
84C2×C4⋊C4—22no—
85C2×C4⋊C4C2312yes—
86C2×C22⋊C4—22no—
87#108—24no—
88#108—22no—
89#108D414yescomplemented
90#107—22no—
91#106—21no—
92#106—21no—
93#106—22no—
94C4⋊C8—22no—
95#103—22no—
96#103—24no—
97#103—22no—
98#103—22no—
99#102—22no—
100#101—24no—
101#101—42no—
102C22⋊C8—22no—
103C22⋊C8—22no—
104#97—21no—
105C4×C8—21no—
106C24#4111yescharacteristic
107#41—21no—
108#41—21no—
109#41—21no—
110#41—21no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8C2×D411yescharacteristic
118C2×D4—21no—
119C2×D4—22no—
120C2×D4C2×D411yescharacteristic
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4C2×D412yes—
129C22×C4C2×D412yes—
130C22×C4#4111yescharacteristic
131C22×C4C2411yescharacteristic
132Q16—44no—
133Q16—42no—
134Q16—42no—
135SD16—44no—
136M16—22no—
137M16—24no—
138M16—24no—
139M16—21no—
140M16—22no—
141M16—21no—
142M16—21no—
143M16—22no—
144M16—21no—
145M16—44no—
146M16—42no—
147M16C2×D414yes—
148C2×C8—22no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—22no—
154C2×C8—44no—
155C2×C8—42no—
156C2×C8C2×D412yes—
157C2×C8C2×D411yescharacteristic
158C2×C8C2×D411yescharacteristic
159C2×C8C2×D412yes—
160C2×C8#4112yes—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4C2×D412yes—
167C22⋊C4—22no—
168C22⋊C4—21no—
169C22⋊C4—22no—
170C22⋊C4—42no—
171C22⋊C4—42no—
172C42—21no—
173C23—21no—
174C23—21no—
175C23—21no—
176C23—42no—
177C23#12112yes—
178C23C2×#4111yescharacteristic
179Q8—21no—
180Q8—21no—
181Q8—42no—
182Q8—42no—
183Q8—42no—
184Q8—42no—
185D4—21no—
186D4—21no—
187D4—41no—
188D4—41no—
189D4—82no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—82no—
205C2×C4#12112yes—
206C2×C4C22×D411yescharacteristic
207C2×C4C22×D411yescharacteristic
208C2×C4C22×D411yescharacteristic
209C2×C4C2×#4111yescharacteristic
210C2×C4H(5,𝔽2)11yescharacteristic
211C2×C42−1+411yescharacteristic
212C8—22no—
213C8—24no—
214C8—41no—
215C8—41no—
216C8—42no—
217C8#12112yes—
218C8#12112yes—
219C22—22no—
220C22—22no—
221C22—22no—
222C22—22no—
223C22—41no—
224C22—42no—
225C22—81no—
226C22—81no—
227C22C2×#12111yescharacteristic
228C22#54811yescharacteristic
229C22#55311yescharacteristic
230C4—22no—
231C4—41no—
232C4—42no—
233C4—82no—
234C4—82no—
235C4C2×#12111yescharacteristic
236C4#53711yescharacteristic
237C4D4211yescharacteristic
238C4#54711yescharacteristic
239C2—21no—
240C2—21no—
241C2—42no—
242C2—81no—
243C2#241611yescharacteristic
244C2#306912yes—
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
prime radical#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#473C2semidirect
3. C2×#470C2semidirect
5. C2×#430C2semidirect
6. #2064C2semidirect
7. #2052C2semidirect
8. #1979C2semidirect
9. #1951C2semidirect
10. #1901C2semidirect
11. #1815C2semidirect
89. #108D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4732
#30562
C2×#4702
#29302
C2×#4302
#20642
#20522
#19792
#19512
#19012
#18152