Finite Group Databasebeta

← #8572

Subgroups — #8572

Order 256.

329 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 329 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C4×C22⋊Q8C211yesmaximal, characteristic
2#1797C211yesmaximal, characteristic
3C2×#328C212yesmaximal
4C4⋊#95C211yesmaximal, characteristic
5#1294C212yesmaximal
6C2×C22⋊Q8C2211yescharacteristic
7C2×C4×Q8C411yescharacteristic, complemented
8C22×C4⋊C4—21no—
9C22×C42C2211yescharacteristic
10C2×C4⋊C8—21no—
11C2×C22⋊C8C2212yes—
12#391C411yescharacteristic, complemented
13#389C411yescharacteristic, complemented
14#387—21no—
15#385—21no—
16#380C2211yescharacteristic
17C4×C4⋊C4—21no—
18C4×C4⋊C4C411yescharacteristic, complemented
19C4×C22⋊C4—21no—
20C2×#95—21no—
21C2×#95C2212yes—
22#336—21no—
23#328—24no—
24C22×Q8C2×C411yescharacteristic
25C23×C4—22no—
26C23×C4D411yescharacteristic
27C23×C4D411yescharacteristic
28C23×C4C2311yescharacteristic
29C22×C8—22no—
30C22⋊Q8—21no—
31C22⋊Q8—21no—
32C22⋊Q8D412yes—
33C22⋊Q8Q812yes—
34C4×Q8—42no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—24no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—42no—
40C2×C4⋊C4—42no—
41C2×C4⋊C4C2×C411yescharacteristic
42C2×C4⋊C4C2×C411yescharacteristic
43C2×C4⋊C4C2×C411yescharacteristic
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C42—21no—
47C2×C42—21no—
48C2×C42—42no—
49C2×C42C2×C411yescharacteristic
50C2×C42C2×C411yescharacteristic
51C2×C42D411yescharacteristic
52C2×C42D411yescharacteristic
53C4⋊C8—42no—
54C22⋊C8—22no—
55C22⋊C8—22no—
56#95—21no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95—21no—
61#95—22no—
62#95—22no—
63#95—42no—
64C24C2×D411yescharacteristic
65C2×Q8—21no—
66C2×Q8—41no—
67C2×Q8—41no—
68C2×Q8C4⋊C412yescomplemented
69C22×C4—22no—
70C22×C4—22no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—24no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—41no—
79C22×C4—41no—
80C22×C4—41no—
81C22×C4—41no—
82C22×C4—42no—
83C22×C4—42no—
84C22×C4—42no—
85C22×C4—42no—
86C22×C4—42no—
87C22×C4—42no—
88C22×C4C22⋊C411yescharacteristic
89C22×C4C22⋊C411yescharacteristic
90C22×C4C22⋊C411yescharacteristic
91C22×C4C22⋊C411yescharacteristic
92C22×C4SD1612yes—
93C22×C4Q1612yes—
94C22×C4C22×C411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C22×C4C2×Q811yescharacteristic
98C22×C4#4111yescharacteristic
99C22×C4#4111yescharacteristic
100C2×C8—42no—
101C2×C8—42no—
102C2×C8—42no—
103C4⋊C4—21no—
104C4⋊C4—22no—
105C4⋊C4—22no—
106C4⋊C4—41no—
107C4⋊C4—41no—
108C4⋊C4—42no—
109C4⋊C4—48no—
110C4⋊C4—42no—
111C4⋊C4C4⋊C412yescomplemented
112C22⋊C4—22no—
113C22⋊C4—22no—
114C22⋊C4—41no—
115C22⋊C4—41no—
116C42—22no—
117C42—22no—
118C42—41no—
119C42—41no—
120C42—42no—
121C42—42no—
122C23—21no—
123C23—21no—
124C23—21no—
125C23—21no—
126C23C2×C22⋊C411yescharacteristic
127C23#12111yescharacteristic
128C23C22⋊Q811yescharacteristic
129C23C2×SD1611yescharacteristic
130C23C2×Q1611yescharacteristic
131C23AGL(1,ℤ/8)11yescharacteristic
132C23#13711yescharacteristic
133Q8—42no—
134Q8—42no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—24no—
142C2×C4—24no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—44no—
162C2×C4—44no—
163C2×C4—44no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—82no—
171C2×C4—82no—
172C2×C4—82no—
173C2×C4—82no—
174C2×C4#9912yes—
175C2×C4#10312yes—
176C2×C4#10312yes—
177C2×C4#10412yes—
178C2×C4C2×C4⋊C411yescharacteristic
179C2×C4#11711yescharacteristic
180C2×C4#12111yescharacteristic
181C2×C4C22⋊Q811yescharacteristic
182C8—82no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—21no—
195C22#38011yescharacteristic
196C22C2×#9911yescharacteristic
197C22#41011yescharacteristic
198C22C2×#10311yescharacteristic
199C22#41711yescharacteristic
200C22C2×#10411yescharacteristic
201C22#42111yescharacteristic
202C22#46011yescharacteristic
203C22#46211yescharacteristic
204C22#47511yescharacteristic
205C22#47711yescharacteristic
206C4—21no—
207C4—21no—
208C4—21no—
209C4—21no—
210C4—22no—
211C4—42no—
212C4—42no—
213C4—41no—
214C4—41no—
215C4—84no—
216C4—81no—
217C4—81no—
218C4#32814yescomplemented
219C2—21no—
220C2—21no—
221C2—21no—
222C2—21no—
223C2C2×#32811yescharacteristic
224C2#150911yescharacteristic
225C2#151311yescharacteristic
226C2#151811yescharacteristic
227C2#179911yescharacteristic
228C2#180311yescharacteristic
229C2#180611yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C2×C4×Q8C4semidirect
12. #391C4semidirect
13. #389C4semidirect
18. C4×C4⋊C4C4semidirect
68. C2×Q8C4⋊C4semidirect
111. C4⋊C4C4⋊C4semidirect
218. C4#328semidirect

Maximal subgroups

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

subgroupindex
C4×C22⋊Q82
#17972
C2×#3282
C4⋊#952
#12942