Finite Group Databasebeta

← #9682

Subgroups — #9682

Order 256.

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

normal subgroups only

This group has 350 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
1#2850C211yesmaximal, characteristic
2#1749C211yesmaximal, characteristic, complemented
3C2×#328C212yesmaximal, complemented
4C2×#327C212yesmaximal
5C4⋊#95C211yesmaximal, characteristic, complemented
6C2×#124C411yescharacteristic, complemented
7C2×C22⋊Q8C2211yescharacteristic
8C2×#121C2211yescharacteristic
9C22×C4⋊C4—21no—
10C22×C42C2211yescharacteristic
11C2×C22⋊C8C2212yes—
12C2×#97—21no—
13#391C411yescharacteristic, complemented
14#390C411yescharacteristic, complemented
15#388—21no—
16#388—21no—
17#386—21no—
18#386—21no—
19#382C411yescharacteristic, complemented
20C2×#95—21no—
21C2×#95C2212yes—
22#336—21no—
23#328—24no—
24#327—24no—
25C22×Q8C2×C411yescharacteristic
26C22×D4—21no—
27C22×D4C2×C411yescharacteristic
28C23×C4—22no—
29C23×C4D411yescharacteristic
30C23×C4D411yescharacteristic
31C23×C4C2311yescharacteristic
32C22×C8—22no—
33#124—42no—
34C22⋊Q8—22no—
35C22⋊Q8—22no—
36#121—22no—
37#121—22no—
38C2×C4⋊C4—24no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—42no—
41C2×C4⋊C4—42no—
42C2×C4⋊C4C2×C411yescharacteristic
43C2×C4⋊C4C2×C411yescharacteristic
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4—21no—
49C2×C22⋊C4—21no—
50C2×C42—42no—
51C2×C42C2×C411yescharacteristic
52C2×C42C2×C411yescharacteristic
53C2×C42D411yescharacteristic
54C2×C42D411yescharacteristic
55C22⋊C8—22no—
56C22⋊C8D414yes—
57#97—42no—
58#95—21no—
59#95—21no—
60#95—21no—
61#95—21no—
62#95—22no—
63#95—22no—
64#95—42no—
65C24—21no—
66C24C2×D411yescharacteristic
67C2×Q8—22no—
68C2×Q8—41no—
69C2×Q8—41no—
70C2×D4—22no—
71C2×D4—41no—
72C2×D4—41no—
73C2×D4—41no—
74C2×D4—41no—
75C2×D4—41no—
76C2×D4—41no—
77C2×D4—41no—
78C2×D4—41no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—22no—
82C22×C4—22no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—24no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—41no—
90C22×C4—41no—
91C22×C4—41no—
92C22×C4—41no—
93C22×C4—42no—
94C22×C4—42no—
95C22×C4—42no—
96C22×C4—42no—
97C22×C4—42no—
98C22×C4—42no—
99C22×C4C22⋊C411yescharacteristic
100C22×C4C22⋊C411yescharacteristic
101C22×C4C22⋊C411yescharacteristic
102C22×C4C22⋊C411yescharacteristic
103C22×C4C22×C411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D412yes—
106C22×C4#4112yes—
107C2×C8—24no—
108C2×C8—42no—
109C2×C8—42no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—48no—
113C4⋊C4—42no—
114C22⋊C4—42no—
115C22⋊C4—42no—
116C22⋊C4—41no—
117C22⋊C4—41no—
118C22⋊C4—42no—
119C22⋊C4—42no—
120C22⋊C4—42no—
121C42—22no—
122C42—22no—
123C42—41no—
124C42—41no—
125C23—21no—
126C23—21no—
127C23—21no—
128C23—21no—
129C23—41no—
130C23—41no—
131C23—41no—
132C23—41no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23C2×C22⋊C411yescharacteristic
137C23#12411yescharacteristic
138C23#12711yescharacteristic
139C23AGL(1,ℤ/8)11yescharacteristic
140C23AGL(1,ℤ/8)11yescharacteristic
141C23#13711yescharacteristic
142C23#13711yescharacteristic
143Q8—82no—
144D4—82no—
145D4—81no—
146D4—81no—
147D4—81no—
148D4—81no—
149C2×C4—24no—
150C2×C4—24no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—44no—
170C2×C4—44no—
171C2×C4—44no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4—82no—
184C2×C4—82no—
185C2×C4#9912yes—
186C2×C4#10412yes—
187C2×C4C4×D412yes—
188C2×C4#12112yes—
189C8—44no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—81no—
203C22—81no—
204C22—81no—
205C22—81no—
206C22—81no—
207C22—81no—
208C22—81no—
209C22#39011yescharacteristic
210C22C2×#9911yescharacteristic
211C22#41011yescharacteristic
212C22#41711yescharacteristic
213C22#41711yescharacteristic
214C22C2×#10411yescharacteristic
215C22#42111yescharacteristic
216C22#48911yescharacteristic
217C22#48911yescharacteristic
218C22#49611yescharacteristic
219C22#49711yescharacteristic
220C4—22no—
221C4—22no—
222C4—22no—
223C4—22no—
224C4—41no—
225C4—41no—
226C4—84no—
227C4—81no—
228C4—81no—
229C4—81no—
230C2—21no—
231C2—21no—
232C2—21no—
233C2—21no—
234C2—161no—
235C2#150411yescharacteristic
236C2#150911yescharacteristic
237C2#151211yescharacteristic
238C2#151311yescharacteristic
239C2#195911yescharacteristic
240C2#196411yescharacteristic
241C2#198011yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
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
2. #1749C2semidirect
3. C2×#328C2semidirect
5. C4⋊#95C2semidirect
6. C2×#124C4semidirect
13. #391C4semidirect
14. #390C4semidirect
19. #382C4semidirect

Maximal subgroups

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

subgroupindex
#28502
#17492
C2×#3282
C2×#3272
C4⋊#952