Finite Group Databasebeta

← #9239

Subgroups — #9239

Order 256.

343 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 343 conjugacy classes of subgroups — too many to draw. Only its 65 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
1C2×#452C211yesmaximal, characteristic
2#2880C211yesmaximal, characteristic
3#1789C211yesmaximal, characteristic
4#1508C211yesmaximal, characteristic, complemented
5#1472C211yesmaximal, characteristic, complemented
6#1293C211yesmaximal, characteristic, complemented
7#1282C211yesmaximal, characteristic, complemented
8C22×#41C2211yescharacteristic
9C22×Q16C411yescharacteristic, complemented
10C22×SD16C411yescharacteristic, complemented
11C2×#131—21no—
12C2×C22⋊Q8C2211yescharacteristic
13#452—21no—
14#452—21no—
15#452—21no—
16#452—21no—
17C2×#104—21no—
18C2×#104—21no—
19C2×#103C411yescharacteristic, complemented
20C2×#102C411yescharacteristic, complemented
21#408—21no—
22#408—22no—
23#407—21no—
24#407C2211yescharacteristic
25#407C2211yescharacteristic
26C2×C22⋊C8—21no—
27C2×C22⋊C8C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29C4×C22⋊C4C2211yescharacteristic
30#328—22no—
31#324—21no—
32#324—21no—
33C2×#41—21no—
34C2×#41—21no—
35C2×#41—42no—
36C2×#41—42no—
37C2×#41D411yescharacteristic
38C2×#41D411yescharacteristic
39C2×#41D411yescharacteristic
40C2×#41D411yescharacteristic
41C22×Q8C2×C411yescharacteristic
42C22×Q8C2×C411yescharacteristic
43C22×D4—21no—
44C22×D4C2×C411yescharacteristic
45C23×C4—21no—
46C23×C4C2311yescharacteristic
47C2×Q16—21no—
48C2×Q16—21no—
49C2×Q16—42no—
50C2×SD16—21no—
51C2×SD16—21no—
52C2×SD16—42no—
53#131—42no—
54#131—42no—
55C2×M16—22no—
56C2×M16—21no—
57C2×M16—21no—
58C22×C8—22no—
59C22×C8—21no—
60C22×C8C2×C411yescharacteristic
61C22×C8C2×C411yescharacteristic
62C22⋊Q8—21no—
63C22⋊Q8—21no—
64C22⋊Q8—21no—
65C22⋊Q8—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4C2×C411yescharacteristic
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C42—21no—
71#104—42no—
72#104—42no—
73#103—21no—
74#103—21no—
75#102—21no—
76#102—21no—
77C22⋊C8—22no—
78C22⋊C8—22no—
79C22⋊C8—21no—
80C22⋊C8—21no—
81C22⋊C8—21no—
82C22⋊C8—21no—
83C22⋊C8—21no—
84C22⋊C8—21no—
85C22⋊C8D411yescharacteristic
86C22⋊C8D411yescharacteristic
87C22⋊C8D412yes—
88#95—21no—
89C24—21no—
90C24#4111yescharacteristic
91#41—22no—
92#41—22no—
93#41—22no—
94#41—22no—
95#41—41no—
96#41—41no—
97#41—41no—
98#41—41no—
99#41—44no—
100#41—44no—
101C2×Q8—21no—
102C2×Q8—22no—
103C2×Q8—21no—
104C2×Q8—21no—
105C2×Q8—41no—
106C2×Q8—41no—
107C2×Q8—41no—
108C2×Q8C22⋊C411yescharacteristic
109C2×Q8C22⋊C411yescharacteristic
110C2×D4—21no—
111C2×D4—22no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—21no—
116C2×D4—41no—
117C2×D4—44no—
118C2×D4C22⋊C411yescharacteristic
119C2×D4C22⋊C411yescharacteristic
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—44no—
130C22×C4—41no—
131C22×C4C22×C411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4#4111yescharacteristic
137Q16—42no—
138Q16—42no—
139SD16—42no—
140SD16—42no—
141M16—44no—
142M16—41no—
143M16—41no—
144M16—41no—
145M16—41no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—22no—
150C2×C8—21no—
151C2×C8—44no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—41no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C22⋊C4—22no—
162C22⋊C4—41no—
163C22⋊C4—41no—
164C22⋊C4—41no—
165C42—41no—
166C42—41no—
167C23—21no—
168C23—21no—
169C23—21no—
170C23—21no—
171C23—41no—
172C23—41no—
173C23—42no—
174C23—41no—
175C23—41no—
176C23—81no—
177C23C4×D411yescharacteristic
178C23#12111yescharacteristic
179C23#12311yescharacteristic
180Q8—21no—
181Q8—21no—
182Q8—21no—
183Q8—42no—
184Q8—41no—
185Q8—41no—
186Q8—81no—
187Q8#9912yes—
188D4—24no—
189D4—21no—
190D4—21no—
191D4—21no—
192D4—42no—
193D4—42no—
194D4—42no—
195D4—42no—
196D4#9912yes—
197C2×C4—24no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4C2×C22⋊C411yescharacteristic
222C2×C4C4×D411yescharacteristic
223C2×C4#12011yescharacteristic
224C2×C4#12111yescharacteristic
225C8—41no—
226C8—42no—
227C8—41no—
228C8—41no—
229C8—41no—
230C8—81no—
231C22—21no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—22no—
238C22—21no—
239C22—21no—
240C22—41no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—81no—
246C22—81no—
247C22—82no—
248C22—82no—
249C22#38611yescharacteristic
250C22#44311yescharacteristic
251C22#44411yescharacteristic
252C4—21no—
253C4—21no—
254C4—22no—
255C4—21no—
256C4—41no—
257C4—82no—
258C4—81no—
259C4—81no—
260C4C2×#9911yescharacteristic
261C4C2×#9911yescharacteristic
262C2—21no—
263C2—21no—
264C2—41no—
265C2—41no—
266C2—42no—
267C2—81no—
268C2#187611yescharacteristic
269C2#189811yescharacteristic
270C2#190211yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC22×#41C22
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. #1508C2semidirect
5. #1472C2semidirect
6. #1293C2semidirect
7. #1282C2semidirect
9. C22×Q16C4semidirect
10. C22×SD16C4semidirect
19. C2×#103C4semidirect
20. C2×#102C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4522
#28802
#17892
#15082
#14722
#12932
#12822