Finite Group Databasebeta

← #9519

Subgroups — #9519

Order 256.

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

normal subgroups only

This group has 322 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#3004C211yesmaximal, characteristic
2#2976C211yesmaximal, characteristic
3#1787C211yesmaximal, characteristic, complemented
4#1784C211yesmaximal, characteristic, complemented
5#1513C211yesmaximal, characteristic
6#1305C211yesmaximal, characteristic, complemented
7#1293C211yesmaximal, characteristic, complemented
8C23×Q8C2211yescharacteristic
9C2×#137—21no—
10#525C2211yescharacteristic
11#451—21no—
12#451C412yescomplemented
13#448—21no—
14#448C412yescomplemented
15#436—21no—
16#435—21no—
17#435—21no—
18C8×D4—21no—
19#433—21no—
20#419—21no—
21C2×#103—21no—
22C2×#103—21no—
23C2×#101—21no—
24C2×#101—21no—
25#408—21no—
26#407C2211yescharacteristic
27#407C2211yescharacteristic
28#407C2211yescharacteristic
29C2×C22⋊C8C2211yescharacteristic
30#380C2211yescharacteristic
31#328—21no—
32#327—21no—
33C2×#41D411yescharacteristic
34C22×Q8—21no—
35C22×Q8—21no—
36C22×Q8—21no—
37C22×Q8—21no—
38C22×Q8—21no—
39C22×Q8—42no—
40C22×Q8—42no—
41C22×Q8C2×C412yes—
42C23×C4—21no—
43C23×C4C2311yescharacteristic
44#137—42no—
45#137—42no—
46C2×Q16—22no—
47C2×SD16—22no—
48C2×M16—21no—
49C2×M16—21no—
50C2×M16—21no—
51C2×M16D411yescharacteristic
52C2×M16D411yescharacteristic
53C22×C8—21no—
54C22×C8D411yescharacteristic
55C22×C8D411yescharacteristic
56#123—21no—
57C22⋊Q8C2×C411yescharacteristic
58C22⋊Q8D411yescharacteristic
59#121C2×C411yescharacteristic
60#121D411yescharacteristic
61#120—21no—
62C4×D4—21no—
63C4×D4—21no—
64#117D411yescharacteristic
65C2×C4⋊C4—21no—
66C2×C22⋊C4—21no—
67C4⋊C8—21no—
68C4⋊C8—21no—
69#103—22no—
70#103—22no—
71#103—42no—
72#103—42no—
73#101—42no—
74#101—42no—
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⋊C8—21no—
84C22⋊C8C2×C412yes—
85#97—21no—
86C4×C8—21no—
87#95—21no—
88C24#4111yescharacteristic
89#41—41no—
90#41—41no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—44no—
100C2×Q8—41no—
101C2×Q8—41no—
102C2×Q8—41no—
103C2×Q8—41no—
104C2×Q8—41no—
105C2×Q8—41no—
106C2×Q8—44no—
107C2×Q8—44no—
108C2×Q8—41no—
109C2×Q8—41no—
110C2×Q8C22⋊C411yescharacteristic
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4C22⋊C411yescharacteristic
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4—44no—
120C22×C4—41no—
121C22×C4—41no—
122C22×C4—41no—
123C22×C4C22×C411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128C22×C4#4111yescharacteristic
129Q16—44no—
130SD16—44no—
131M16—41no—
132M16—41no—
133M16—41no—
134M16—41no—
135M16—41no—
136M16—41no—
137M16—81no—
138C2×C8—22no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—81no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4C22⋊C411yescharacteristic
152C4⋊C4C22⋊C411yescharacteristic
153C22⋊C4—21no—
154C22⋊C4—21no—
155C22⋊C4—21no—
156C22⋊C4—21no—
157C22⋊C4—21no—
158C22⋊C4—81no—
159C42—21no—
160C23—21no—
161C23—21no—
162C23—21no—
163C23—41no—
164C23—41no—
165C23C4×D411yescharacteristic
166C23#12111yescharacteristic
167C23#12311yescharacteristic
168Q8—24no—
169Q8—24no—
170Q8—41no—
171Q8—42no—
172Q8—42no—
173Q8—42no—
174Q8—42no—
175Q8—42no—
176Q8—42no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—81no—
181D4—81no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—24no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4C2×C22⋊C411yescharacteristic
207C2×C4C4×D411yescharacteristic
208C2×C4#12011yescharacteristic
209C2×C4#12111yescharacteristic
210C8—42no—
211C8—41no—
212C8—41no—
213C8—41no—
214C8—41no—
215C8—81no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—42no—
225C22—42no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22#38611yescharacteristic
230C22#47111yescharacteristic
231C22#47311yescharacteristic
232C4—21no—
233C4—21no—
234C4—21no—
235C4—44no—
236C4—41no—
237C4—41no—
238C4—41no—
239C4—161no—
240C4B(4,𝔽2)11yescharacteristic
241C4B(4,𝔽2)11yescharacteristic
242C2—21no—
243C2—21no—
244C2—41no—
245C2—41no—
246C2—81no—
247C2#188911yescharacteristic
248C2#190211yescharacteristic
249C2#190811yescharacteristic
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
3. #1787C2semidirect
4. #1784C2semidirect
6. #1305C2semidirect
7. #1293C2semidirect
12. #451C4semidirect
14. #448C4semidirect

Maximal subgroups

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

subgroupindex
#30042
#29762
#17872
#17842
#15132
#13052
#12932