Finite Group Databasebeta

← #18352

Subgroups — #18352

Order 256.

498 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 115 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#3501C211yesmaximal, characteristic, complemented
2#3193C212yesmaximal, complemented
3#2635C211yesmaximal, characteristic
4#1617C212yesmaximal, complemented
5#1614C212yesmaximal, complemented
6#1599C212yesmaximal, complemented
7#1596C212yesmaximal, complemented
8#1511C212yesmaximal, complemented
9#1465C211yesmaximal, characteristic, complemented
10#554—24no—
11#554—22no—
12D4×Q8—22no—
13#544C2212yes—
14#543—21no—
15#543C2212yes—
16#533—21no—
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8C2212yes—
19C2×#121C2211yescharacteristic
20#518—21no—
21C2×C4×D4C2211yescharacteristic
22#484C2214yes—
23#480—22no—
24#468—24no—
25#450C2214yes—
26#448—22no—
27#407C2212yes—
28#400C2211yescharacteristic
29#399C2211yescharacteristic
30#394—21no—
31#394C2211yescharacteristic
32#392—21no—
33#392C2211yescharacteristic
34#388—21no—
35#386—21no—
36#381C2211yescharacteristic
37#328C2214yes—
38#328C2214yes—
39#324C2212yes—
40#323C2212yes—
41C22×Q8—21no—
42C22×Q8D412yescomplemented
43C22×D4—21no—
44C22×D4—21no—
45C22×D4D411yescharacteristic
46C22×D4C2311yescharacteristic
47C23×C4D411yescharacteristic
48C23×C4C2311yescharacteristic
49C2×SD16—24no—
50C2×M16—22no—
51C4⋊Q8—21no—
52C4⋊Q8—21no—
53C4⋊Q8—24no—
54#125—22no—
55#125—24no—
56C22⋊Q8—22no—
57C22⋊Q8—22no—
58C22⋊Q8—24no—
59C22⋊Q8—24no—
60C22⋊Q8—24no—
61C22⋊Q8—22no—
62C22⋊Q8—22no—
63C22⋊Q8—22no—
64C22⋊Q8—22no—
65C22⋊Q8—24no—
66C22⋊Q8D414yescomplemented
67C22⋊Q8C2314yes—
68#121—22no—
69#121—22no—
70C4×Q8—22no—
71C4×Q8—22no—
72C4×D4—24no—
73C4×D4—24no—
74C4×D4—22no—
75C4×D4—22no—
76#117—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4D412yescomplemented
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4D411yescharacteristic
93C2×C22⋊C4C2311yescharacteristic
94C2×C42D411yescharacteristic
95#107—24no—
96#103—24no—
97#102—24no—
98C22⋊C8—22no—
99C22⋊C8C2314yes—
100#95—21no—
101#95—21no—
102#95—21no—
103#95C2311yescharacteristic
104#95C2311yescharacteristic
105C24—21no—
106C24C2×D411yescharacteristic
107C24C2×D411yescharacteristic
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—82no—
112C2×Q8—82no—
113C2×Q8C2×D414yes—
114C2×D4—22no—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—42no—
119C2×D4—42no—
120C2×D4—81no—
121C2×D4—81no—
122C2×D4—81no—
123C2×D4—81no—
124C2×D4—81no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—81no—
135C22×C4—81no—
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D412yes—
141C22×C4C2×D412yes—
142C22×C4C2411yescharacteristic
143SD16—84no—
144M16—82no—
145C2×C8—24no—
146C4⋊C4—22no—
147C4⋊C4—24no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—24no—
151C4⋊C4—22no—
152C4⋊C4—24no—
153C4⋊C4—22no—
154C4⋊C4—24no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—81no—
158C4⋊C4C2×D414yes—
159C22⋊C4—24no—
160C22⋊C4—24no—
161C22⋊C4—22no—
162C22⋊C4—24no—
163C22⋊C4—22no—
164C22⋊C4—41no—
165C22⋊C4—41no—
166C22⋊C4—42no—
167C22⋊C4—81no—
168C22⋊C4—81no—
169C22⋊C4—81no—
170C22⋊C4—81no—
171C42—24no—
172C42—21no—
173C42—21no—
174C23—21no—
175C23—21no—
176C23—21no—
177C23—21no—
178C23—41no—
179C23—41no—
180C23—41no—
181C23—41no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23#12012yes—
190C23#13712yes—
191C23C22×D411yescharacteristic
192Q8—44no—
193Q8—44no—
194Q8—82no—
195Q8—82no—
196D4—28no—
197D4—44no—
198D4—84no—
199D4—84no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—24no—
204C2×C4—24no—
205C2×C4—21no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—81no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4#12011yescharacteristic
228C2×C4#12011yescharacteristic
229C2×C4#13712yes—
230C2×C4C22×D412yes—
231C8—44no—
232C22—21no—
233C22—21no—
234C22—22no—
235C22—22no—
236C22—22no—
237C22—22no—
238C22—42no—
239C22—41no—
240C22—41no—
241C22—81no—
242C22—81no—
243C22—81no—
244C22—81no—
245C22—81no—
246C22—81no—
247C22—81no—
248C22—81no—
249C22—81no—
250C22—81no—
251C22C2×#12011yescharacteristic
252C22C2×#13711yescharacteristic
253C22C2×#13711yescharacteristic
254C4—22no—
255C4—22no—
256C4—21no—
257C4—21no—
258C4—41no—
259C4—44no—
260C4—41no—
261C4—44no—
262C4—81no—
263C4—81no—
264C4—81no—
265C2—21no—
266C2—21no—
267C2—42no—
268C2—42no—
269C2—161no—
270C2#300411yescharacteristic
271C2#302111yescharacteristic
272C2#302611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#2635C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3501C2semidirect
2. #3193C2semidirect
4. #1617C2semidirect
5. #1614C2semidirect
6. #1599C2semidirect
7. #1596C2semidirect
8. #1511C2semidirect
9. #1465C2semidirect
42. C22×Q8D4semidirect
66. C22⋊Q8D4semidirect
81. C2×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
#35012
#31932
#26352
#16172
#16142
#15992
#15962
#15112
#14652