Finite Group Databasebeta

← #9331

Subgroups — #9331

Order 256.

219 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 219 conjugacy classes of subgroups — too many to draw. Only its 57 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#3196C211yesmaximal, characteristic
2#2978C211yesmaximal, characteristic
3#1516C211yesmaximal, characteristic
4#1511C211yesmaximal, characteristic, complemented
5#1472C211yesmaximal, characteristic, complemented
6#1306C211yesmaximal, characteristic, complemented
7#1293C211yesmaximal, characteristic, complemented
8#525C2211yescharacteristic
9C2×C22⋊Q8C2211yescharacteristic
10#485—21no—
11#484C411yescharacteristic, complemented
12#481C411yescharacteristic, complemented
13#470—21no—
14#468—21no—
15#452—21no—
16#451C411yescharacteristic, complemented
17#448C411yescharacteristic, complemented
18#436—21no—
19#436—21no—
20#435—21no—
21#435—21no—
22#433—21no—
23#408—21no—
24#407C2211yescharacteristic
25#407C2211yescharacteristic
26#407C2211yescharacteristic
27C2×C22⋊C8C2211yescharacteristic
28#381C2211yescharacteristic
29#328—21no—
30#328—21no—
31#328—21no—
32#327—21no—
33#324—21no—
34#324—21no—
35C2×#41D411yescharacteristic
36C22×Q8C2×C411yescharacteristic
37C23×C4C2311yescharacteristic
38C2×Q16—21no—
39C2×SD16—21no—
40C2×M16—21no—
41C2×M16—21no—
42C2×M16—21no—
43C2×M16—21no—
44C22×C8—21no—
45C22×C8—21no—
46#123—21no—
47C22⋊Q8—21no—
48C22⋊Q8—21no—
49C22⋊Q8—21no—
50C22⋊Q8—21no—
51C22⋊Q8C2×C411yescharacteristic
52C22⋊Q8D411yescharacteristic
53#121C2×C411yescharacteristic
54#121D411yescharacteristic
55#120—21no—
56C4×D4—21no—
57C4×D4—21no—
58#117D411yescharacteristic
59C2×C4⋊C4—21no—
60C2×C4⋊C4C2×C411yescharacteristic
61C2×C22⋊C4—21no—
62C2×C22⋊C4—21no—
63#107—21no—
64#106—21no—
65C4⋊C8—21no—
66C4⋊C8—21no—
67#103—21no—
68#103—21no—
69#103—21no—
70#102—21no—
71C22⋊C8—21no—
72C22⋊C8—21no—
73C22⋊C8—21no—
74C22⋊C8—21no—
75C22⋊C8—21no—
76C22⋊C8—21no—
77C22⋊C8—21no—
78C22⋊C8C2×C411yescharacteristic
79C22⋊C8C2×C411yescharacteristic
80C22⋊C8D411yescharacteristic
81C22⋊C8D411yescharacteristic
82C22⋊C8D411yescharacteristic
83C22⋊C8D411yescharacteristic
84#97—21no—
85C4×C8—21no—
86#95—21no—
87#95—21no—
88C24#4111yescharacteristic
89#41—81no—
90C2×Q8—21no—
91C2×Q8—21no—
92C2×Q8—81no—
93C2×Q8C22⋊C411yescharacteristic
94C2×D4—21no—
95C2×D4—21no—
96C2×D4—21no—
97C2×D4C22⋊C411yescharacteristic
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—41no—
102C22×C4—81no—
103C22×C4C22×C411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4#4111yescharacteristic
109Q16—81no—
110SD16—81no—
111M16—81no—
112M16—81no—
113M16—81no—
114M16—81no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—41no—
122C2×C8—81no—
123C2×C8—81no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—41no—
129C4⋊C4—41no—
130C4⋊C4C22⋊C411yescharacteristic
131C4⋊C4C22⋊C411yescharacteristic
132C22⋊C4—21no—
133C22⋊C4—21no—
134C22⋊C4—21no—
135C22⋊C4—21no—
136C22⋊C4—21no—
137C22⋊C4—41no—
138C22⋊C4—41no—
139C22⋊C4—81no—
140C42—21no—
141C23—21no—
142C23—41no—
143C23—41no—
144C23—41no—
145C23C4×D411yescharacteristic
146C23#12111yescharacteristic
147C23#12311yescharacteristic
148Q8—41no—
149Q8—41no—
150Q8—41no—
151Q8—81no—
152D4—41no—
153D4—81no—
154D4—81no—
155D4—81no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—81no—
171C2×C4—81no—
172C2×C4—81no—
173C2×C4—81no—
174C2×C4C2×C22⋊C411yescharacteristic
175C2×C4C4×D411yescharacteristic
176C2×C4#12011yescharacteristic
177C2×C4#12111yescharacteristic
178C8—41no—
179C8—41no—
180C8—41no—
181C8—41no—
182C8—41no—
183C8—41no—
184C8—81no—
185C22—21no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—41no—
195C22—41no—
196C22—41no—
197C22—82no—
198C22#38611yescharacteristic
199C4—21no—
200C4—21no—
201C4—21no—
202C4—21no—
203C4—41no—
204C4—41no—
205C4—41no—
206C4—81no—
207C4—81no—
208C4—161no—
209C2—41no—
210C2—41no—
211C2—41no—
212C2—81no—
213C2#188111yescharacteristic
214C2#190211yescharacteristic
215C2#191311yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radical#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1511C2semidirect
5. #1472C2semidirect
6. #1306C2semidirect
7. #1293C2semidirect
11. #484C4semidirect
12. #481C4semidirect
16. #451C4semidirect
17. #448C4semidirect

Maximal subgroups

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

subgroupindex
#31962
#29782
#15162
#15112
#14722
#13062
#12932