Finite Group Databasebeta

← #18333

Subgroups — #18333

Order 256.

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

normal subgroups only

This group has 475 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#3458C211yesmaximal, characteristic, complemented
2#3215C212yesmaximal, complemented
3#2610C211yesmaximal, characteristic
4#1605C212yesmaximal, complemented
5#1602C212yesmaximal, complemented
6#1599C212yesmaximal, complemented
7#1596C212yesmaximal
8#1506C212yesmaximal, complemented
9#1464C211yesmaximal, characteristic, complemented
10#544—22no—
11#544—22no—
12#544—21no—
13#544C2212yes—
14#543—21no—
15#543C2212yes—
16#533—24no—
17#533—21no—
18#533C2212yes—
19C2×#124C2211yescharacteristic
20C2×#117—21no—
21C2×#117C2211yescharacteristic
22#497—22no—
23#489—22no—
24#464—24no—
25#451C2214yes—
26#450C2214yes—
27#432C2212yes—
28#395C2211yescharacteristic
29#394C2211yescharacteristic
30#393C2211yescharacteristic
31#392C2211yescharacteristic
32#390—21no—
33#388—21no—
34#388—21no—
35#386—21no—
36#376C2211yescharacteristic
37#328C2214yes—
38#328C2214yes—
39#324C2212yes—
40#323C2212yes—
41C22×Q8C2311yescharacteristic
42C22×D4—21no—
43C22×D4C2311yescharacteristic
44C23×C4D411yescharacteristic
45C2×Q16—24no—
46C2×SD16—24no—
47C4⋊Q8—24no—
48C4⋊Q8—21no—
49C4⋊Q8—21no—
50C4⋊Q8—24no—
51C4⋊Q8D412yescomplemented
52C4⋊Q8D412yescomplemented
53#125—24no—
54#125—22no—
55#125—24no—
56#125—22no—
57#124—22no—
58#124—22no—
59C22⋊Q8—24no—
60C22⋊Q8—24no—
61C22⋊Q8—24no—
62C22⋊Q8—22no—
63C22⋊Q8—22no—
64C22⋊Q8—22no—
65C22⋊Q8—24no—
66C22⋊Q8C2314yes—
67C4×Q8—22no—
68C4×Q8—22no—
69C4×Q8—22no—
70C4×Q8D414yescomplemented
71#117—22no—
72#117—22no—
73#117—22no—
74#117—22no—
75#117—22no—
76#117—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4D411yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C22⋊C4—22no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4D411yescharacteristic
90C2×C22⋊C4C2311yescharacteristic
91C2×C42—22no—
92C2×C42—21no—
93C2×C42D411yescharacteristic
94C2×C42C2311yescharacteristic
95C4⋊C8—22no—
96#103—24no—
97#102—24no—
98C22⋊C8C2314yes—
99#97—22no—
100#95—21no—
101#95—21no—
102#95—21no—
103#95C2311yescharacteristic
104#95C2311yescharacteristic
105C24—21no—
106C24C2×D411yescharacteristic
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—81no—
112C2×Q8C2×D414yes—
113C2×D4—22no—
114C2×D4—42no—
115C2×D4—42no—
116C2×D4—81no—
117C2×D4—81no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2411yescharacteristic
133Q16—84no—
134SD16—84no—
135C2×C8—24no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—24no—
140C4⋊C4—24no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—24no—
147C4⋊C4—22no—
148C4⋊C4—24no—
149C4⋊C4—81no—
150C4⋊C4C2×D414yes—
151C22⋊C4—24no—
152C22⋊C4—24no—
153C22⋊C4—22no—
154C22⋊C4—22no—
155C22⋊C4—42no—
156C22⋊C4—41no—
157C22⋊C4—41no—
158C22⋊C4—42no—
159C22⋊C4—42no—
160C22⋊C4—81no—
161C22⋊C4—81no—
162C42—24no—
163C42—24no—
164C42—21no—
165C42—21no—
166C42—42no—
167C42C2×D412yes—
168C42C2×D412yes—
169C23—21no—
170C23—21no—
171C23—41no—
172C23—41no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23—41no—
179C23—41no—
180C23#13712yes—
181C23C22×D411yescharacteristic
182Q8—44no—
183Q8—44no—
184Q8—82no—
185Q8—82no—
186Q8—82no—
187D4—44no—
188D4—84no—
189C2×C4—24no—
190C2×C4—24no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—24no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4#12011yescharacteristic
215C2×C4#12011yescharacteristic
216C2×C4#12012yes—
217C2×C4#13712yes—
218C2×C4C22×D412yes—
219C8—44no—
220C22—21no—
221C22—21no—
222C22—22no—
223C22—42no—
224C22—41no—
225C22—41no—
226C22—81no—
227C22—81no—
228C22—81no—
229C22—81no—
230C22—81no—
231C22—81no—
232C22—81no—
233C22—81no—
234C22C2×#12011yescharacteristic
235C22C2×#13711yescharacteristic
236C22C2×#13711yescharacteristic
237C4—21no—
238C4—21no—
239C4—22no—
240C4—22no—
241C4—22no—
242C4—42no—
243C4—41no—
244C4—44no—
245C4—41no—
246C4—44no—
247C4—81no—
248C4—81no—
249C2—21no—
250C2—21no—
251C2—42no—
252C2—161no—
253C2#300411yescharacteristic
254C2#301811yescharacteristic
255C2#302711yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#392C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3458C2semidirect
2. #3215C2semidirect
4. #1605C2semidirect
5. #1602C2semidirect
6. #1599C2semidirect
8. #1506C2semidirect
9. #1464C2semidirect
51. C4⋊Q8D4semidirect
52. C4⋊Q8D4semidirect
70. C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
#34582
#32152
#26102
#16052
#16022
#15992
#15962
#15062
#14642