Finite Group Databasebeta

← #8367

Subgroups — #8367

Order 256.

569 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 569 conjugacy classes of subgroups — too many to draw. Only its 79 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×#546C211yesmaximal, characteristic
2C4×#100C211yesmaximal, characteristic, complemented
3#1527C212yesmaximal, complemented
4#1490C212yesmaximal, complemented
5#1466C211yesmaximal, characteristic, complemented
6C22×#41C411yescharacteristic
7C23×D4C411yescharacteristic
8#546—24no—
9#546—24no—
10C2×#124C2211yescharacteristic
11C2×#123—21no—
12C2×C22⋊Q8C411yescharacteristic
13C2×#121—21no—
14C2×#121C411yescharacteristic
15C2×#120—21no—
16C2×C4×D4C2211yescharacteristic
17C2×C4×D4C2211yescharacteristic
18C22×C22⋊C4—21no—
19#432C2212yescomplemented
20C2×#100—21no—
21C4×M16C2212yescomplemented
22#343—21no—
23#329—24no—
24D4⋊C8—24no—
25#323—21no—
26#323—21no—
27C25—21no—
28C2×#41—22no—
29C2×#41—22no—
30C2×#41—44no—
31C22×Q8C2×C411yescharacteristic
32C22×D4—21no—
33C22×D4—21no—
34C22×D4—22no—
35C22×D4—22no—
36C22×D4—21no—
37C22×D4—44no—
38C22×D4C2×C411yescharacteristic
39C22×D4C2×C411yescharacteristic
40C22×D4D411yescharacteristic
41C23×C4—21no—
42C23×C4—21no—
43C23×C4C2×C411yescharacteristic
44C23×C4C2×C411yescharacteristic
45C2×M16—22no—
46#124—22no—
47#124D414yes—
48#123—28no—
49C22⋊Q8—22no—
50C22⋊Q8—22no—
51#121—28no—
52#121—22no—
53#121—22no—
54#120—28no—
55C4×D4—22no—
56C4×D4—22no—
57C4×D4—22no—
58C4×D4D414yes—
59C2×C4⋊C4—21no—
60C2×C4⋊C4C2×C411yescharacteristic
61C2×C4⋊C4D411yescharacteristic
62C2×C22⋊C4—28no—
63C2×C22⋊C4—21no—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4D411yescharacteristic
69C2×C22⋊C4D411yescharacteristic
70C2×C42C2311yescharacteristic
71C4⋊C8—22no—
72#100—24no—
73C22⋊C8—22no—
74#97—22no—
75#97—22no—
76C4×C8—22no—
77C24—21no—
78C24—24no—
79C24—21no—
80C24—44no—
81C24—42no—
82C24—42no—
83C24—82no—
84C24C22⋊C411yescharacteristic
85C24C22⋊C411yescharacteristic
86#41—48no—
87#41—44no—
88#41—44no—
89C2×Q8—21no—
90C2×Q8—42no—
91C2×Q8C22⋊C412yes—
92C2×D4—24no—
93C2×D4—24no—
94C2×D4—24no—
95C2×D4—22no—
96C2×D4—24no—
97C2×D4—21no—
98C2×D4—21no—
99C2×D4—22no—
100C2×D4—41no—
101C2×D4—42no—
102C2×D4—48no—
103C2×D4—44no—
104C2×D4—44no—
105C2×D4—44no—
106C2×D4—44no—
107C2×D4—82no—
108C2×D4C22⋊C412yes—
109C2×D4C22⋊C412yes—
110C22×C4—24no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—24no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—44no—
121C22×C4—82no—
122C22×C4C22⋊C411yescharacteristic
123C22×C4C22⋊C411yescharacteristic
124C22×C4C22×C411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127M16—24no—
128M16—42no—
129C2×C8—42no—
130C2×C8—42no—
131C4⋊C4—24no—
132C4⋊C4—21no—
133C4⋊C4—22no—
134C4⋊C4C22⋊C412yes—
135C22⋊C4—24no—
136C22⋊C4—24no—
137C22⋊C4—22no—
138C22⋊C4—24no—
139C22⋊C4—24no—
140C22⋊C4—24no—
141C22⋊C4—22no—
142C42C2×D412yes—
143C42C2×D412yes—
144C23—24no—
145C23—21no—
146C23—22no—
147C23—21no—
148C23—22no—
149C23—21no—
150C23—21no—
151C23—21no—
152C23—41no—
153C23—41no—
154C23—42no—
155C23—42no—
156C23—48no—
157C23—42no—
158C23—48no—
159C23—41no—
160C23—41no—
161C23—41no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23—84no—
166C23—88no—
167C23—88no—
168C23#10412yes—
169C23C2×C22⋊C411yescharacteristic
170Q8—42no—
171Q8—42no—
172D4—24no—
173D4—22no—
174D4—44no—
175D4—42no—
176D4—48no—
177D4—48no—
178D4—42no—
179D4—84no—
180D4#10014yescomplemented
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—48no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—84no—
199C2×C4C2×C22⋊C411yescharacteristic
200C2×C4C2×C22⋊C411yescharacteristic
201C2×C4#12012yes—
202C2×C4#12012yes—
203C8—44no—
204C8—82no—
205C22—21no—
206C22—21no—
207C22—24no—
208C22—22no—
209C22—42no—
210C22—44no—
211C22—44no—
212C22—42no—
213C22—42no—
214C22—41no—
215C22—41no—
216C22—42no—
217C22—41no—
218C22—42no—
219C22—41no—
220C22—42no—
221C22—41no—
222C22—42no—
223C22—81no—
224C22—81no—
225C22—82no—
226C22—82no—
227C22—816no—
228C22#37911yescharacteristic
229C22C2×#10411yescharacteristic
230C22#42111yescharacteristic
231C4—22no—
232C4—21no—
233C4—22no—
234C4—42no—
235C4—41no—
236C4—42no—
237C4—41no—
238C4C2×#10012yes—
239C2—21no—
240C2—21no—
241C2—44no—
242C2—42no—
243C2—81no—
244C2—82no—
245C2#178311yescharacteristic
246C2#178911yescharacteristic
247C2#179511yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC2×#546C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#100C2semidirect
3. #1527C2semidirect
4. #1490C2semidirect
5. #1466C2semidirect
19. #432C22semidirect
21. C4×M16C22semidirect
180. D4#100semidirect

Maximal subgroups

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

subgroupindex
C2×#5462
C4×#1002
#15272
#14902
#14662