Finite Group Databasebeta

← #24128

Subgroups — #24128

Order 256.

700 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 135 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×#547C211yesmaximal, characteristic, complemented
2C2×#476C211yesmaximal, characteristic, complemented
3C2×#442C211yesmaximal, characteristic, complemented
4#2535C211yesmaximal, characteristic, complemented
5#2034C212yesmaximal, complemented
6#1999C212yesmaximal, complemented
7#1993C211yesmaximal, characteristic, complemented
8#1955C211yesmaximal, characteristic, complemented
9#1874C212yesmaximal, complemented
10#1853C212yesmaximal, complemented
11#1804C211yesmaximal, characteristic, complemented
12C22×#41—21no—
13C22×D8C2211yescharacteristic
14#547—24no—
15#547—24no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×#123—22no—
18C2×C22⋊Q8—21no—
19C2×#121C2212yes—
20C2×#121C2212yes—
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C22×C4⋊C4C2212yes—
25#476—24no—
26#442—24no—
27C2×#106C2211yescharacteristic
28C2×#106C2212yes—
29C2×C4⋊C8C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2212yes—
32C2×#102C2211yescharacteristic
33C2×#102C2212yes—
34C2×C22⋊C8C2212yes—
35C2×C22⋊C8C2212yes—
36C2×#97C2211yescharacteristic
37#397—21no—
38#392—21no—
39#386—22no—
40#382C2211yescharacteristic
41#380C2212yes—
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45C2×#41—28no—
46C2×#41—48no—
47C22×Q8—21no—
48C22×D4—22no—
49C22×D4—22no—
50C22×D4D411yescharacteristic
51C22×D4C2311yescharacteristic
52C22×D4C2311yescharacteristic
53C23×C4—22no—
54C23×C4—21no—
55C23×C4—21no—
56C23×C4C2312yes—
57C2×D8—22no—
58C2×D8—22no—
59C2×D8—22no—
60C22×C8C2311yescharacteristic
61C22×C8C2312yes—
62C22×C8C2311yescharacteristic
63C4⋊Q8—22no—
64C4⋊Q8D414yescomplemented
65#123—216no—
66C22⋊Q8—28no—
67#121—24no—
68#121—24no—
69#121—24no—
70#121—42no—
71C4×D4—22no—
72C4×D4—22no—
73C4×D4—22no—
74C4×D4—22no—
75C4×D4—22no—
76C4×D4D414yescomplemented
77C2×C4⋊C4—22no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4D411yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2312yes—
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2312yes—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4D412yes—
92C2×C42C2311yescharacteristic
93#106—22no—
94#106—24no—
95C4⋊C8—22no—
96#102—22no—
97#102—22no—
98#102—24no—
99#102—42no—
100C22⋊C8—24no—
101C22⋊C8—42no—
102#97—22no—
103#95—22no—
104#95—21no—
105#95—21no—
106C24—21no—
107C24—21no—
108C24C2×D412yes—
109#41—432no—
110C2×Q8—24no—
111C2×Q8—44no—
112C2×D4—24no—
113C2×D4—28no—
114C2×D4—22no—
115C2×D4—21no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—44no—
126C2×D4—44no—
127C2×D4—44no—
128C2×D4—44no—
129C2×D4—82no—
130C2×D4C2×D412yes—
131C2×D4#4112yescomplemented
132C22×C4—28no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—24no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—44no—
145C22×C4—44no—
146C22×C4—44no—
147C22×C4—42no—
148C22×C4—82no—
149C22×C4C2×D412yes—
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2411yescharacteristic
153D8—28no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—21no—
160C2×C8—22no—
161C2×C8—21no—
162C2×C8#4112yescomplemented
163C4⋊C4—28no—
164C4⋊C4—24no—
165C4⋊C4—22no—
166C4⋊C4—21no—
167C4⋊C4—22no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4C2×D412yes—
173C4⋊C4C2×D414yes—
174C22⋊C4—28no—
175C22⋊C4—24no—
176C22⋊C4—24no—
177C22⋊C4—44no—
178C22⋊C4—42no—
179C22⋊C4—82no—
180C42C2×D412yes—
181C42C2×D412yes—
182C23—22no—
183C23—22no—
184C23—22no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—22no—
190C23—41no—
191C23—41no—
192C23—41no—
193C23—41no—
194C23—42no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23AGL(1,ℤ/8)14yes—
199C23C22×D411yescharacteristic
200Q8—48no—
201D4—28no—
202D4—22no—
203D4—22no—
204D4—22no—
205D4—22no—
206D4—416no—
207D4—41no—
208D4—41no—
209D4—41no—
210D4—41no—
211D4—88no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—48no—
220C2×C4—416no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—84no—
234C2×C4#12012yes—
235C2×C4#12012yes—
236C2×C4C22×D411yescharacteristic
237C2×C4C22×D411yescharacteristic
238C2×C4C2×#4111yescharacteristic
239C2×C4C2×#4111yescharacteristic
240C2×C4H(5,𝔽2)11yescharacteristic
241C2×C42−1+411yescharacteristic
242C8—22no—
243C8—41no—
244C8—42no—
245C22—24no—
246C22—24no—
247C22—24no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—42no—
256C22—44no—
257C22—82no—
258C22—82no—
259C22—81no—
260C22—81no—
261C22—82no—
262C22C2×#12011yescharacteristic
263C22#54611yescharacteristic
264C22#54711yescharacteristic
265C22C2×AGL(1,ℤ/8)12yes—
266C22#57712yes—
267C4—22no—
268C4—22no—
269C4—42no—
270C4—42no—
271C4—44no—
272C4—81no—
273C4—82no—
274C4#52511yescharacteristic
275C4#53711yescharacteristic
276C4#54811yescharacteristic
277C4#55011yescharacteristic
278C2—44no—
279C2—82no—
280C2—82no—
281C2#240711yescharacteristic
282C2#300311yescharacteristic
283C2#301011yescharacteristic
284C2#330612yes—
285C2#334512yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#547C2semidirect
2. C2×#476C2semidirect
3. C2×#442C2semidirect
4. #2535C2semidirect
5. #2034C2semidirect
6. #1999C2semidirect
7. #1993C2semidirect
8. #1955C2semidirect
9. #1874C2semidirect
10. #1853C2semidirect
11. #1804C2semidirect
64. C4⋊Q8D4semidirect
76. C4×D4D4semidirect
131. C2×D4#41semidirect
162. C2×C8#41semidirect

Maximal subgroups

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

subgroupindex
C2×#5472
C2×#4762
C2×#4422
#25352
#20342
#19992
#19932
#19552
#18742
#18532
#18042