Finite Group Databasebeta

← #24342

Subgroups — #24342

Order 256.

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

normal subgroups only

This group has 624 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×D4×Q8C211yesmaximal, characteristic
2C2×#477C211yesmaximal, characteristic, complemented
3C2×#440C211yesmaximal, characteristic
4#2581C211yesmaximal, characteristic, complemented
5#2057C212yesmaximal, complemented
6#2024C212yesmaximal, complemented
7#1993C211yesmaximal, characteristic, complemented
8#1985C211yesmaximal, characteristic, complemented
9#1877C212yesmaximal
10#1862C212yesmaximal
11#1804C211yesmaximal, characteristic
12C23×Q8—21no—
13C22×SD16C2211yescharacteristic
14D4×Q8—24no—
15D4×Q8—24no—
16C2×C4⋊Q8C2212yes—
17C2×C4⋊Q8C2211yescharacteristic
18C2×C22⋊Q8—22no—
19C2×C22⋊Q8—21no—
20C2×C4×Q8C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic
22C2×C4×D4C2212yes—
23C2×C4×D4C2211yescharacteristic
24#477—24no—
25#440—24no—
26C2×#107C2212yes—
27C2×#107C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×#103C2212yes—
30C2×#103C2211yescharacteristic
31C2×#102C2212yes—
32C2×#102C2211yescharacteristic
33C2×#97C2211yescharacteristic
34#400—21no—
35#395—21no—
36#391C2212yes—
37#389C2212yes—
38#387—22no—
39#382C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43#336C2212yes—
44#336C2212yes—
45C22×Q8—28no—
46C22×Q8—21no—
47C22×Q8—22no—
48C22×Q8—48no—
49C22×Q8D411yescharacteristic
50C22×Q8C2311yescharacteristic
51C22×D4C2311yescharacteristic
52C23×C4—22no—
53C23×C4—21no—
54C2×SD16—22no—
55C2×SD16—22no—
56C2×SD16Q814yescomplemented
57C22×C8C2311yescharacteristic
58C22×C8C2312yes—
59C22×C8C2311yescharacteristic
60C4⋊Q8—24no—
61C4⋊Q8—24no—
62C4⋊Q8—22no—
63C4⋊Q8D414yes—
64C22⋊Q8—28no—
65C22⋊Q8—216no—
66C4×Q8—22no—
67C4×Q8—22no—
68C4×Q8—22no—
69C4×Q8—22no—
70C4×D4—24no—
71C4×D4—24no—
72C4×D4—22no—
73C4×D4D414yes—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4D412yes—
79C2×C4⋊C4D411yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2312yes—
82C2×C4⋊C4C2312yes—
83C2×C4⋊C4C2311yescharacteristic
84C2×C22⋊C4—22no—
85C2×C22⋊C4—21no—
86C2×C42—21no—
87C2×C42C2312yes—
88C2×C42C2311yescharacteristic
89#107—22no—
90#107—24no—
91C4⋊C8—22no—
92#103—22no—
93#103—24no—
94#102—22no—
95#102—24no—
96#97—22no—
97#95—21no—
98#95—21no—
99#95—22no—
100#95—22no—
101#95—22no—
102C24—21no—
103C2×Q8—28no—
104C2×Q8—22no—
105C2×Q8—24no—
106C2×Q8—24no—
107C2×Q8—21no—
108C2×Q8—21no—
109C2×Q8—21no—
110C2×Q8—21no—
111C2×Q8—21no—
112C2×Q8—44no—
113C2×Q8—44no—
114C2×Q8—432no—
115C2×Q8—44no—
116C2×Q8C2×Q812yescomplemented
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—21no—
120C2×D4C2×D412yes—
121C2×D4C2×Q812yes—
122C2×D4#4112yes—
123C22×C4—28no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—21no—
127C22×C4—24no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—44no—
131C22×C4—44no—
132C22×C4—44no—
133C22×C4C2×D412yes—
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D412yes—
137C22×C4C2411yescharacteristic
138SD16—28no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—22no—
147C2×C8C2×Q812yescomplemented
148C4⋊C4—28no—
149C4⋊C4—24no—
150C4⋊C4—24no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—42no—
159C4⋊C4—44no—
160C4⋊C4C2×D414yes—
161C4⋊C4C2×D412yes—
162C22⋊C4—28no—
163C22⋊C4—24no—
164C42—22no—
165C42—22no—
166C42—42no—
167C42C2×D412yes—
168C42C2×D412yes—
169C23—22no—
170C23—22no—
171C23—22no—
172C23—42no—
173C23—42no—
174C23C22×D411yescharacteristic
175Q8—28no—
176Q8—22no—
177Q8—22no—
178Q8—48no—
179Q8—416no—
180Q8—41no—
181Q8—41no—
182D4—22no—
183D4—22no—
184D4—22no—
185D4C22⋊Q814yes—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—24no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—48no—
196C2×C4—416no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4#12012yes—
208C2×C4#12012yes—
209C2×C4AGL(1,ℤ/8)14yes—
210C2×C4C22×D411yescharacteristic
211C2×C4C22×D411yescharacteristic
212C2×C4C22×Q811yescharacteristic
213C2×C4C2×#4111yescharacteristic
214C2×C42−1+411yescharacteristic
215C2×C42−1+411yescharacteristic
216C8—22no—
217C8—41no—
218C8—42no—
219C22—24no—
220C22—44no—
221C22—44no—
222C22—42no—
223C22—42no—
224C22C2×#12011yescharacteristic
225C22#54811yescharacteristic
226C22D4×Q811yescharacteristic
227C22C2×AGL(1,ℤ/8)12yes—
228C22#57812yes—
229C4—22no—
230C4—24no—
231C4—22no—
232C4—42no—
233C4—44no—
234C4—42no—
235C4—42no—
236C4—82no—
237C4—81no—
238C4C2×C22⋊Q811yescharacteristic
239C4#53611yescharacteristic
240C4#54811yescharacteristic
241C4D4×Q811yescharacteristic
242C2—44no—
243C2#240911yescharacteristic
244C2#300311yescharacteristic
245C2#301111yescharacteristic
246C2#336712yes—
247C2#338612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#477C2semidirect
4. #2581C2semidirect
5. #2057C2semidirect
6. #2024C2semidirect
7. #1993C2semidirect
8. #1985C2semidirect
56. C2×SD16Q8semidirect
116. C2×Q8C2×Q8semidirect
147. C2×C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Q82
C2×#4772
C2×#4402
#25812
#20572
#20242
#19932
#19852
#18772
#18622
#18042