Finite Group Databasebeta

← #9038

Subgroups — #9038

Order 256.

234 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 61 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#3091C211yesmaximal, characteristic
2#2976C211yesmaximal, characteristic
3#1826C211yesmaximal, characteristic, complemented
4#1818C211yesmaximal, characteristic, complemented
5#1501C211yesmaximal, characteristic
6#1298C211yesmaximal, characteristic, complemented
7#1282C211yesmaximal, characteristic, complemented
8#525C2211yescharacteristic
9C2×#117C2211yescharacteristic
10#485—21no—
11#484C411yescharacteristic, complemented
12#483C411yescharacteristic, complemented
13#482—21no—
14#481C411yescharacteristic, complemented
15#480C411yescharacteristic, complemented
16#436—21no—
17#435—21no—
18#435—21no—
19C8×D4—21no—
20#433—21no—
21#427—21no—
22#416—21no—
23#408—21no—
24#407C2211yescharacteristic
25C2×C22⋊C8C2211yescharacteristic
26C2×C22⋊C8C2211yescharacteristic
27C2×C22⋊C8C2211yescharacteristic
28C4×C22⋊C4C2211yescharacteristic
29#339—21no—
30#339—21no—
31#336—21no—
32#336—21no—
33#328—21no—
34#327—21no—
35C2×#41D411yescharacteristic
36C23×C4C2311yescharacteristic
37C2×M16—21no—
38C2×M16D411yescharacteristic
39C2×M16D411yescharacteristic
40C22×C8—21no—
41C22×C8—21no—
42C22×C8—21no—
43C22×C8D411yescharacteristic
44C22×C8D411yescharacteristic
45#123—21no—
46C22⋊Q8C2×C411yescharacteristic
47C22⋊Q8D411yescharacteristic
48#121C2×C411yescharacteristic
49#121D411yescharacteristic
50#120—21no—
51C4×D4—21no—
52C4×D4—21no—
53#117—21no—
54#117—21no—
55#117—21no—
56#117—21no—
57#117D411yescharacteristic
58C2×C4⋊C4C2×C411yescharacteristic
59C2×C4⋊C4C2×C411yescharacteristic
60C2×C22⋊C4—21no—
61C2×C22⋊C4—21no—
62C2×C42—21no—
63C2×C42—21no—
64#107—21no—
65#107—21no—
66#106—21no—
67#106—21no—
68C4⋊C8—21no—
69C4⋊C8—21no—
70#103—21no—
71#103—21no—
72#102—21no—
73#102—21no—
74C22⋊C8—21no—
75C22⋊C8—21no—
76C22⋊C8—21no—
77C22⋊C8—21no—
78C22⋊C8—21no—
79C22⋊C8—21no—
80C22⋊C8—21no—
81C22⋊C8—21no—
82C22⋊C8—21no—
83C22⋊C8C2×C411yescharacteristic
84C22⋊C8C2×C411yescharacteristic
85#97—21no—
86C4×C8—21no—
87#95—21no—
88C24#4111yescharacteristic
89#41—41no—
90#41—41no—
91C2×Q8C22⋊C411yescharacteristic
92C2×D4—21no—
93C2×D4—21no—
94C2×D4—21no—
95C2×D4C22⋊C411yescharacteristic
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—41no—
100C22×C4—41no—
101C22×C4—41no—
102C22×C4C22×C411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4#4111yescharacteristic
108M16—41no—
109M16—41no—
110M16—81no—
111C2×C8—21no—
112C2×C8—21no—
113C2×C8—21no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—41no—
118C2×C8—41no—
119C2×C8—41no—
120C2×C8—41no—
121C2×C8—41no—
122C2×C8—41no—
123C2×C8—41no—
124C2×C8—81no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4C22⋊C411yescharacteristic
132C4⋊C4C22⋊C411yescharacteristic
133C22⋊C4—21no—
134C22⋊C4—21no—
135C22⋊C4—21no—
136C22⋊C4—21no—
137C22⋊C4—21no—
138C22⋊C4—22no—
139C22⋊C4—41no—
140C22⋊C4—41no—
141C22⋊C4—41no—
142C42—21no—
143C42—41no—
144C42—41no—
145C42—41no—
146C42—41no—
147C23—21no—
148C23—21no—
149C23—21no—
150C23—41no—
151C23—41no—
152C23C4×D411yescharacteristic
153C23#12111yescharacteristic
154C23#12311yescharacteristic
155Q8—41no—
156D4—41no—
157D4—41no—
158D4—41no—
159D4—81no—
160D4—81no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—81no—
182C2×C4—81no—
183C2×C4C2×C22⋊C411yescharacteristic
184C2×C4C4×D411yescharacteristic
185C2×C4#12011yescharacteristic
186C2×C4#12111yescharacteristic
187C8—41no—
188C8—41no—
189C8—41no—
190C8—41no—
191C8—41no—
192C8—41no—
193C8—81no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—81no—
206C22—81no—
207C22#38611yescharacteristic
208C22#44311yescharacteristic
209C22#44411yescharacteristic
210C4—21no—
211C4—21no—
212C4—21no—
213C4—41no—
214C4—41no—
215C4—41no—
216C4—82no—
217C4—81no—
218C4—81no—
219C4#45811yescharacteristic
220C4#45811yescharacteristic
221C2—21no—
222C2—21no—
223C2—41no—
224C2—41no—
225C2—81no—
226C2#187611yescharacteristic
227C2#188111yescharacteristic
228C2#189011yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
socleC22#386
prime radical#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1826C2semidirect
4. #1818C2semidirect
6. #1298C2semidirect
7. #1282C2semidirect
11. #484C4semidirect
12. #483C4semidirect
14. #481C4semidirect
15. #480C4semidirect

Maximal subgroups

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

subgroupindex
#30912
#29762
#18262
#18182
#15012
#12982
#12822