Finite Group Databasebeta

← #24120

Subgroups — #24120

Order 256.

675 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 675 conjugacy classes of subgroups — too many to draw. Only its 147 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
2C2×#475C211yesmaximal, characteristic, complemented
3C2×C4×SD16C211yesmaximal, characteristic, complemented
4#2538C211yesmaximal, characteristic, complemented
5#2034C212yesmaximal, complemented
6#2000C212yesmaximal, complemented
7#1990C211yesmaximal, characteristic, complemented
8#1954C211yesmaximal, characteristic, complemented
9#1872C212yesmaximal, complemented
10#1853C212yesmaximal, complemented
11#1802C211yesmaximal, characteristic, complemented
12C22×#41—21no—
13C22×SD16C2211yescharacteristic
14#547—24no—
15#547—24no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×#123—22no—
18C2×C22⋊Q8—21no—
19C2×C22⋊Q8C2212yes—
20C2×#121C2212yes—
21C2×C4×Q8C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C22×C4⋊C4C2212yes—
25#475—24no—
26C4×SD16—24no—
27C2×#106C2212yes—
28C2×#106C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2212yes—
31C2×#103C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2212yes—
34C2×C22⋊C8C2212yes—
35C2×C22⋊C8C2212yes—
36C2×C4×C8C2211yescharacteristic
37#396—22no—
38#390—21no—
39#389—21no—
40#384C2211yescharacteristic
41#380C2212yes—
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45C2×#41—28no—
46C2×#41—44no—
47C2×#41—44no—
48C22×Q8—21no—
49C22×Q8C2311yescharacteristic
50C22×D4—22no—
51C22×D4D411yescharacteristic
52C22×D4C2311yescharacteristic
53C23×C4—22no—
54C23×C4—21no—
55C23×C4C2312yes—
56C2×SD16—22no—
57C2×SD16—22no—
58C2×SD16—22no—
59C22×C8C2311yescharacteristic
60C22×C8C2312yes—
61C22×C8C2311yescharacteristic
62C4⋊Q8—22no—
63C4⋊Q8D414yes—
64#123—216no—
65C22⋊Q8—28no—
66C22⋊Q8—24no—
67C22⋊Q8—42no—
68#121—24no—
69#121—24no—
70C4×Q8—22no—
71C4×Q8—22no—
72C4×D4—22no—
73C4×D4—22no—
74C4×D4—22no—
75C4×D4D414yes—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2312yes—
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2312yes—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4D412yes—
94C2×C42—21no—
95C2×C42C2311yescharacteristic
96#106—22no—
97#106—24no—
98C4⋊C8—22no—
99#103—22no—
100#103—24no—
101#102—22no—
102#102—42no—
103C22⋊C8—24no—
104C22⋊C8—42no—
105C4×C8—22no—
106#95—22no—
107C24—21no—
108C24C2×D412yes—
109#41—432no—
110C2×Q8—24no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—42no—
117C2×Q8—42no—
118C2×Q8#4112yescomplemented
119C2×D4—28no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—44no—
128C2×D4—44no—
129C2×D4SD1614yescomplemented
130C2×D4C2×D412yes—
131C22×C4—28no—
132C22×C4—21no—
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—21no—
143C22×C4—22no—
144C22×C4—44no—
145C22×C4—44no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4C2×D412yes—
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2411yescharacteristic
152SD16—28no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—21no—
159C2×C8—22no—
160C2×C8—21no—
161C2×C8#4112yescomplemented
162C4⋊C4—28no—
163C4⋊C4—24no—
164C4⋊C4—22no—
165C4⋊C4—21no—
166C4⋊C4—22no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—44no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—81no—
175C4⋊C4C2×D412yes—
176C4⋊C4C2×D414yes—
177C22⋊C4—28no—
178C22⋊C4—24no—
179C22⋊C4—24no—
180C22⋊C4—44no—
181C22⋊C4—82no—
182C42—42no—
183C42C2×D412yes—
184C42C2×D412yes—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—22no—
190C23—22no—
191C23—22no—
192C23—41no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23—41no—
198C23C2×SD1614yes—
199C23C22×D411yescharacteristic
200Q8—22no—
201Q8—22no—
202Q8—41no—
203Q8—41no—
204Q8—48no—
205D4—22no—
206D4—22no—
207D4—28no—
208D4—416no—
209D4—41no—
210D4—41no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—24no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—416no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—48no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4#12012yes—
238C2×C4#12012yes—
239C2×C4C2×SD1612yes—
240C2×C4AGL(1,ℤ/8)12yes—
241C2×C4C22×D411yescharacteristic
242C2×C4C22×D411yescharacteristic
243C2×C4C2×#4111yescharacteristic
244C2×C4C2×#4111yescharacteristic
245C2×C4H(5,𝔽2)11yescharacteristic
246C2×C42−1+411yescharacteristic
247C8—22no—
248C8—41no—
249C8—42no—
250C22—24no—
251C22—24no—
252C22—24no—
253C22—42no—
254C22—42no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—81no—
261C22—81no—
262C22—81no—
263C22—81no—
264C22C2×#12011yescharacteristic
265C22#54611yescharacteristic
266C22#54711yescharacteristic
267C22C22×SD1611yescharacteristic
268C22C2×AGL(1,ℤ/8)11yescharacteristic
269C22#57711yescharacteristic
270C22#57811yescharacteristic
271C4—22no—
272C4—42no—
273C4—42no—
274C4—42no—
275C4—44no—
276C4—82no—
277C4—81no—
278C4#45014yes—
279C4#52511yescharacteristic
280C4#53711yescharacteristic
281C4#54811yescharacteristic
282C4#55011yescharacteristic
283C2—44no—
284C2—82no—
285C2#240711yescharacteristic
286C2C2×#45011yescharacteristic
287C2#301311yescharacteristic
288C2#329811yescharacteristic
289C2#333211yescharacteristic
290C2#333511yescharacteristic
291C2#335311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#547C2
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
2. C2×#475C2semidirect
3. C2×C4×SD16C2semidirect
4. #2538C2semidirect
5. #2034C2semidirect
6. #2000C2semidirect
7. #1990C2semidirect
8. #1954C2semidirect
9. #1872C2semidirect
10. #1853C2semidirect
11. #1802C2semidirect
118. C2×Q8#41semidirect
129. C2×D4SD16semidirect
161. 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×#4752
C2×C4×SD162
#25382
#20342
#20002
#19902
#19542
#18722
#18532
#18022