Finite Group Databasebeta

← #24660

Subgroups — #24660

Order 256.

438 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 133 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×#462C211yesmaximal, characteristic, complemented
2C2×#432C211yesmaximal, characteristic
3#2857C211yesmaximal, characteristic
4C4×C22⋊Q8C211yesmaximal, characteristic
5#2035C212yesmaximal
6#2001C212yesmaximal
7#1981C211yesmaximal, characteristic, complemented
8#1971C212yesmaximal, complemented
9#1948C211yesmaximal, characteristic, complemented
10#1935C212yesmaximal, complemented
11#1803C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C22⋊Q8C2211yescharacteristic
15C2×C22⋊Q8C2212yes—
16C2×C4×Q8C2211yescharacteristic
17C22×C42C2211yescharacteristic
18#462—24no—
19#432—24no—
20C2×#106C2212yes—
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×C22⋊C8C2212yes—
28C2×#97C2212yes—
29#391C2212yes—
30#391C2211yescharacteristic
31#389—21no—
32#389—21no—
33#389C2211yescharacteristic
34#387—21no—
35#385—21no—
36#384C2211yescharacteristic
37#380—21no—
38#380—21no—
39#380C2211yescharacteristic
40C4×C4⋊C4—21no—
41C4×C4⋊C4C2211yescharacteristic
42C4×C22⋊C4—21no—
43#340C2212yes—
44#340C2212yes—
45C22×Q8C2312yes—
46C22×Q8C2311yescharacteristic
47C23×C4D411yescharacteristic
48C23×C4D411yescharacteristic
49C23×C4C2311yescharacteristic
50C2×Q16—24no—
51C2×Q16—24no—
52C2×Q16—42no—
53C22×C8C2312yes—
54C22×C8C2312yes—
55C4⋊Q8—22no—
56C4⋊Q8—22no—
57C22⋊Q8—22no—
58C22⋊Q8—22no—
59C22⋊Q8—42no—
60C22⋊Q8—42no—
61C4×Q8—22no—
62C4×Q8—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2312yes—
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2311yescharacteristic
75C2×C22⋊C4—21no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—21no—
78C2×C42—22no—
79C2×C42—22no—
80C2×C42—21no—
81C2×C42—21no—
82C2×C42D411yescharacteristic
83C2×C42D411yescharacteristic
84C2×C42C2311yescharacteristic
85C2×C42C2311yescharacteristic
86#106—42no—
87C4⋊C8—22no—
88C4⋊C8D414yes—
89#103—24no—
90#103—24no—
91#103—42no—
92#103—42no—
93C22⋊C8—24no—
94#97—24no—
95#95—21no—
96#95—21no—
97#95—21no—
98#95—21no—
99#95—21no—
100#95—22no—
101#95—22no—
102C24C2×D411yescharacteristic
103C2×Q8—21no—
104C2×Q8—21no—
105C2×Q8—22no—
106C2×Q8—21no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—21no—
111C2×Q8—21no—
112C2×Q8—22no—
113C2×Q8—22no—
114C2×Q8#4112yes—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—41no—
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2411yescharacteristic
139Q16—48no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8C2×D414yes—
146C4⋊C4—24no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—22no—
152C4⋊C4—21no—
153C4⋊C4—22no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—44no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4#4112yes—
162C22⋊C4—24no—
163C22⋊C4—44no—
164C22⋊C4—42no—
165C42—22no—
166C42—22no—
167C42—21no—
168C42—21no—
169C42—42no—
170C42—42no—
171C42C2×D412yes—
172C42#4112yes—
173C23—21no—
174C23—21no—
175C23—21no—
176C23—21no—
177C23—21no—
178C23—21no—
179C23#13712yes—
180C23C22×D411yescharacteristic
181Q8—22no—
182Q8—22no—
183Q8—41no—
184Q8—41no—
185Q8—42no—
186Q8—42no—
187Q8—42no—
188Q8—42no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—44no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—44no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4#12112yes—
225C2×C4#12112yes—
226C2×C4#13712yes—
227C2×C4#13712yes—
228C2×C4C22×D411yescharacteristic
229C2×C4C2×#4111yescharacteristic
230C2×C4C2×#4111yescharacteristic
231C2×C4C2×#4111yescharacteristic
232C2×C4H(5,𝔽2)11yescharacteristic
233C2×C4H(5,𝔽2)11yescharacteristic
234C8—24no—
235C8—42no—
236C22—22no—
237C22—22no—
238C22—22no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—41no—
243C22C2×#12111yescharacteristic
244C22#52511yescharacteristic
245C22#53511yescharacteristic
246C22C2×#13711yescharacteristic
247C22C2×#13711yescharacteristic
248C22C2×#13711yescharacteristic
249C22#57511yescharacteristic
250C4—22no—
251C4—42no—
252C4—41no—
253C4—42no—
254C4—41no—
255C4—41no—
256C4—82no—
257C4—82no—
258C4—81no—
259C4—81no—
260C4#47014yes—
261C4#54611yescharacteristic
262C4#55011yescharacteristic
263C4#55311yescharacteristic
264C4#55511yescharacteristic
265C2—42no—
266C2#241411yescharacteristic
267C2C2×#47011yescharacteristic
268C2#305611yescharacteristic
269C2#311311yescharacteristic
270C2#311611yescharacteristic
271C2#321611yescharacteristic
272C2#321811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#462C2semidirect
7. #1981C2semidirect
8. #1971C2semidirect
9. #1948C2semidirect
10. #1935C2semidirect
11. #1803C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4622
C2×#4322
#28572
C4×C22⋊Q82
#20352
#20012
#19812
#19712
#19482
#19352
#18032