Finite Group Databasebeta

← #24798

Subgroups — #24798

Order 256.

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

normal subgroups only

This group has 423 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#3059C211yesmaximal, characteristic, complemented
3#3046C212yesmaximal
4C2×#443C211yesmaximal, characteristic
5#2913C211yesmaximal, characteristic, complemented
6#2064C212yesmaximal, complemented
7#2038C212yesmaximal, complemented
8#1969C211yesmaximal, characteristic, complemented
9#1945C211yesmaximal, characteristic, complemented
10#1882C212yesmaximal, complemented
11#1816C211yesmaximal, characteristic, complemented
12C2×#137C2212yes—
13C2×#135C2211yescharacteristic
14C22×Q16C2211yescharacteristic
15C2×#131C2211yescharacteristic
16C2×#131C2211yescharacteristic
17#536C2212yes—
18#520C2212yes—
19#473—24no—
20#470—22no—
21#467—21no—
22#467—21no—
23#464—22no—
24#463—22no—
25#462—22no—
26#461—22no—
27#443—24no—
28C2×#108C2211yescharacteristic
29C2×#108C2211yescharacteristic
30#428—22no—
31#428C2212yes—
32#427—21no—
33#427C2211yescharacteristic
34C2×#107—21no—
35C2×#107C2211yescharacteristic
36#424C2212yes—
37C2×#104C2211yescharacteristic
38C2×#104C2211yescharacteristic
39#419C2212yes—
40#417C2212yes—
41C2×#103C2212yes—
42C2×#101C2212yes—
43C2×C4×C8C2211yescharacteristic
44#391C2212yes—
45#343C2212yes—
46#339C2212yes—
47C2×#41C2311yescharacteristic
48C2×#41C2311yescharacteristic
49C22×Q8C2312yes—
50#137—24no—
51#137—24no—
52#135—22no—
53#135—22no—
54C2×Q16—22no—
55C2×Q16—21no—
56C2×Q16—21no—
57C2×Q16—22no—
58C2×Q16—22no—
59C2×Q16D411yescharacteristic
60C2×SD16—22no—
61C2×SD16D411yescharacteristic
62C2×SD16D411yescharacteristic
63C2×D8D411yescharacteristic
64#131—22no—
65#131—22no—
66#131—22no—
67#131D414yes—
68C2×M16—21no—
69C2×M16—21no—
70C2×M16C2311yescharacteristic
71C2×M16C2312yes—
72C2×M16C2311yescharacteristic
73C22×C8—21no—
74C22×C8—21no—
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C4⋊Q8—22no—
78#124—22no—
79#123—42no—
80C22⋊Q8—22no—
81C22⋊Q8—22no—
82C4×Q8—22no—
83C4×D4—22no—
84C4×D4—42no—
85#117—22no—
86#117C2312yes—
87C2×C4⋊C4—22no—
88C2×C4⋊C4C2312yes—
89C2×C42C2311yescharacteristic
90#108—22no—
91#108—22no—
92#107—21no—
93#107—21no—
94#107—22no—
95#107D411yescharacteristic
96#107D411yescharacteristic
97#106—22no—
98#106D412yes—
99C4⋊C8—22no—
100C4⋊C8—22no—
101#104—22no—
102#104—22no—
103#103—22no—
104#103—22no—
105#103—22no—
106#103—22no—
107#103—22no—
108#102—22no—
109#101—24no—
110C4×C8—22no—
111#95—42no—
112#41—21no—
113#41—21no—
114#41—21no—
115#41—21no—
116#41—21no—
117#41—21no—
118#41C2×D412yes—
119#41#4112yes—
120C2×Q8—22no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8C2×D411yescharacteristic
127C2×Q8C2×D411yescharacteristic
128C2×D4—21no—
129C2×D4—21no—
130C2×D4C2×D411yescharacteristic
131C2×D4C2×D411yescharacteristic
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—42no—
138C22×C4C2411yescharacteristic
139Q16—22no—
140Q16—44no—
141Q16—42no—
142Q16—42no—
143SD16—22no—
144SD16—22no—
145SD16—44no—
146D8—22no—
147M16—21no—
148M16—24no—
149M16—22no—
150M16—21no—
151M16—21no—
152M16—22no—
153M16—42no—
154M16C2×D412yes—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—24no—
159C2×C8—22no—
160C2×C8—21no—
161C2×C8—42no—
162C2×C8C2×D411yescharacteristic
163C2×C8C2×D411yescharacteristic
164C2×C8C2×D412yes—
165C2×C8C2×D411yescharacteristic
166C2×C8C2×D411yescharacteristic
167C2×C8#4112yes—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4C2×D412yes—
174C4⋊C4C2×D412yes—
175C22⋊C4—22no—
176C22⋊C4—42no—
177C22⋊C4—42no—
178C22⋊C4—42no—
179C42—22no—
180C42—21no—
181C42—21no—
182C42—42no—
183C23—21no—
184C23—21no—
185C23H(5,𝔽2)11yescharacteristic
186Q8—21no—
187Q8—21no—
188Q8—21no—
189Q8—42no—
190Q8—42no—
191Q8—42no—
192Q8—42no—
193Q8#12112yes—
194D4—22no—
195D4—22no—
196D4—21no—
197D4—21no—
198D4—21no—
199D4—41no—
200D4—41no—
201D4#12112yes—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—82no—
218C2×C4—162no—
219C2×C4C22×D411yescharacteristic
220C2×C4C22×D411yescharacteristic
221C2×C4C22×D411yescharacteristic
222C2×C4C2×#4111yescharacteristic
223C2×C4C2×#4111yescharacteristic
224C2×C42−1+411yescharacteristic
225C8—22no—
226C8—21no—
227C8—21no—
228C8—41no—
229C8—42no—
230C8#12112yes—
231C8#12112yes—
232C22—21no—
233C22—21no—
234C22—22no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22#53711yescharacteristic
241C22D4211yescharacteristic
242C22#55311yescharacteristic
243C4—22no—
244C4—42no—
245C4—41no—
246C4—42no—
247C4—42no—
248C4—82no—
249C4—82no—
250C4C2×#12111yescharacteristic
251C4C2×#12111yescharacteristic
252C4#54711yescharacteristic
253C4#54811yescharacteristic
254C2—21no—
255C2—21no—
256C2—42no—
257C2—81no—
258C2#241611yescharacteristic
259C2#329011yescharacteristic
260C2#329311yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#135C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#473C2semidirect
2. #3059C2semidirect
5. #2913C2semidirect
6. #2064C2semidirect
7. #2038C2semidirect
8. #1969C2semidirect
9. #1945C2semidirect
10. #1882C2semidirect
11. #1816C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4732
#30592
#30462
C2×#4432
#29132
#20642
#20382
#19692
#19452
#18822
#18162