Finite Group Databasebeta

← #9716

Subgroups — #9716

Order 256.

325 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 325 conjugacy classes of subgroups — too many to draw. Only its 77 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#1797C211yesmaximal, characteristic, complemented
3C2×#328C212yesmaximal, complemented
4C2×#327C212yesmaximal
5#1434C211yesmaximal, characteristic, complemented
6C2×#124C411yescharacteristic, complemented
7C2×C22⋊Q8C2211yescharacteristic
8C2×#121C2211yescharacteristic
9C22×C42C2211yescharacteristic
10C2×C4⋊C8—21no—
11C2×C22⋊C8C2212yes—
12#391C411yescharacteristic, complemented
13#390C411yescharacteristic, complemented
14#388—21no—
15#388—21no—
16#386—21no—
17#386—21no—
18#382C411yescharacteristic, complemented
19C2×#95—21no—
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—21no—
39C2×C4⋊C4C2×C411yescharacteristic
40C2×C4⋊C4C2×C411yescharacteristic
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C42—42no—
48C2×C42C2×C411yescharacteristic
49C2×C42C2×C411yescharacteristic
50C2×C42D411yescharacteristic
51C2×C42D411yescharacteristic
52C4⋊C8—42no—
53C22⋊C8—22no—
54C22⋊C8—22no—
55#95—21no—
56#95—21no—
57#95—21no—
58#95—21no—
59#95—22no—
60#95—42no—
61#95—42no—
62#95D414yes—
63C24—21no—
64C24C2×D411yescharacteristic
65C2×Q8—22no—
66C2×Q8—41no—
67C2×Q8—41no—
68C2×D4—22no—
69C2×D4—41no—
70C2×D4—41no—
71C2×D4—42no—
72C2×D4—42no—
73C2×D4—41no—
74C2×D4—41no—
75C22×C4—22no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—22no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—24no—
83C22×C4—41no—
84C22×C4—41no—
85C22×C4—41no—
86C22×C4—41no—
87C22×C4—41no—
88C22×C4—42no—
89C22×C4—42no—
90C22×C4—42no—
91C22×C4—42no—
92C22×C4—42no—
93C22×C4—42no—
94C22×C4C22⋊C411yescharacteristic
95C22×C4C22⋊C411yescharacteristic
96C22×C4C22⋊C411yescharacteristic
97C22×C4C22⋊C411yescharacteristic
98C22×C4C22×C411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D412yes—
101C22×C4#4112yes—
102C2×C8—42no—
103C2×C8—42no—
104C2×C8—42no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C22⋊C4—42no—
110C22⋊C4—42no—
111C22⋊C4—42no—
112C22⋊C4—42no—
113C22⋊C4—42no—
114C22⋊C4—41no—
115C22⋊C4—41no—
116C42—22no—
117C42—22no—
118C42—41no—
119C42—41no—
120C23—21no—
121C23—21no—
122C23—21no—
123C23—21no—
124C23—41no—
125C23—41no—
126C23—41no—
127C23—41no—
128C23—41no—
129C23—41no—
130C23—41no—
131C23C2×C22⋊C411yescharacteristic
132C23#12112yes—
133C23#13512yes—
134C23AGL(1,ℤ/8)11yescharacteristic
135C23#13711yescharacteristic
136Q8—82no—
137D4—82no—
138D4—82no—
139D4—82no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—44no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—44no—
161C2×C4—44no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—44no—
172C2×C4—82no—
173C2×C4—82no—
174C2×C4—82no—
175C2×C4—82no—
176C2×C4#9912yes—
177C2×C4C4×D412yes—
178C2×C4#12411yescharacteristic
179C2×C4#12711yescharacteristic
180C8—82no—
181C22—22no—
182C22—22no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—21no—
187C22—22no—
188C22—22no—
189C22—81no—
190C22—81no—
191C22—81no—
192C22—81no—
193C22—81no—
194C22—81no—
195C22—81no—
196C22#39011yescharacteristic
197C22C2×#9911yescharacteristic
198C22#41011yescharacteristic
199C22#41611yescharacteristic
200C22#41711yescharacteristic
201C22#42111yescharacteristic
202C22#42111yescharacteristic
203C22#46312yes—
204C22#46412yes—
205C4—21no—
206C4—22no—
207C4—21no—
208C4—22no—
209C4—41no—
210C4—41no—
211C4—41no—
212C4—84no—
213C4—81no—
214C4—81no—
215C4—81no—
216C2—22no—
217C2—22no—
218C2—161no—
219C2#151211yescharacteristic
220C2#151311yescharacteristic
221C2#151611yescharacteristic
222C2#152111yescharacteristic
223C2#196311yescharacteristic
224C2#196411yescharacteristic
225C2#198311yescharacteristic
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. #1797C2semidirect
3. C2×#328C2semidirect
5. #1434C2semidirect
6. C2×#124C4semidirect
12. #391C4semidirect
13. #390C4semidirect
18. #382C4semidirect

Maximal subgroups

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

subgroupindex
#28502
#17972
C2×#3282
C2×#3272
#14342