Finite Group Databasebeta

← #18623

Subgroups — #18623

Order 256.

485 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 485 conjugacy classes of subgroups — too many to draw. Only its 131 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#3503C211yesmaximal, characteristic
2#3230C211yesmaximal, characteristic, complemented
3C2×#494C211yesmaximal, characteristic, complemented
4#3079C211yesmaximal, characteristic
5#1633C214yesmaximal, complemented
6#1632C214yesmaximal, complemented
7#1548C211yesmaximal, characteristic, complemented
8#1493C211yesmaximal, characteristic, complemented
9#1476C211yesmaximal, characteristic
10C22×Q16—22no—
11Q82C2212yes—
12#557—21no—
13#554—22no—
14#554C2212yes—
15D4×Q8—22no—
16D4×Q8—21no—
17#544—22no—
18#544—21no—
19#533—21no—
20#533C2211yescharacteristic
21C2×C4⋊Q8—21no—
22C2×C4⋊Q8C2211yescharacteristic
23C2×C4⋊Q8C2211yescharacteristic
24#520—21no—
25C4×#41C2211yescharacteristic
26#500—21no—
27#494—22no—
28#494C2214yes—
29#487—21no—
30#484—22no—
31#477C2212yes—
32#477C2214yes—
33#476C2212yes—
34#475—21no—
35#474—21no—
36#431C2211yescharacteristic
37#428—21no—
38#423C2211yescharacteristic
39#417—21no—
40C2×C4×C8C2211yescharacteristic
41#333C2214yes—
42Q8⋊C8C2214yes—
43Q8⋊C8C2212yes—
44D4⋊C8C2212yes—
45C2×#41D411yescharacteristic
46C22×Q8—22no—
47C22×Q8—21no—
48C22×Q8D412yes—
49C2×Q16—28no—
50C2×Q16—48no—
51C2×M16—21no—
52C22×C8—21no—
53C4⋊Q8—22no—
54C4⋊Q8—22no—
55C4⋊Q8—24no—
56C4⋊Q8—22no—
57C4⋊Q8—22no—
58C4⋊Q8—21no—
59C4⋊Q8—21no—
60C4⋊Q8—21no—
61C4⋊Q8—21no—
62C4⋊Q8—21no—
63C4⋊Q8D412yes—
64C4⋊Q8D411yescharacteristic
65C4⋊Q8D411yescharacteristic
66C4⋊Q8C2312yes—
67C4⋊Q8C2312yes—
68#125—22no—
69#125—24no—
70#125—22no—
71#125—21no—
72C22⋊Q8—22no—
73C22⋊Q8—24no—
74C22⋊Q8—24no—
75C22⋊Q8—22no—
76C22⋊Q8—24no—
77C22⋊Q8—21no—
78C22⋊Q8D412yes—
79C4×Q8—22no—
80C4×Q8—22no—
81C4×Q8—22no—
82C4×Q8—21no—
83C4×Q8D411yescharacteristic
84C4×Q8C2311yescharacteristic
85C4×Q8C2312yes—
86C4×D4—24no—
87C4×D4—22no—
88C4×D4—21no—
89C4×D4—21no—
90C4×D4D411yescharacteristic
91C4×D4C2311yescharacteristic
92#117—22no—
93#117—21no—
94#117—21no—
95#117D411yescharacteristic
96C2×C4⋊C4—22no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C42—21no—
103C2×C42C2311yescharacteristic
104#107—22no—
105#107—24no—
106#106—22no—
107C4⋊C8—21no—
108C4⋊C8C2312yes—
109C4⋊C8C2312yes—
110#103—22no—
111#103—24no—
112#102—22no—
113C22⋊C8—21no—
114C4×C8—21no—
115C4×C8—21no—
116C4×C8C2312yes—
117#41—22no—
118#41Q1614yescomplemented
119C2×Q8—24no—
120C2×Q8—22no—
121C2×Q8—21no—
122C2×Q8—22no—
123C2×Q8—44no—
124C2×Q8—44no—
125C2×Q8—41no—
126C2×Q8—84no—
127C2×Q8C2×D412yes—
128C2×Q8C2×D411yescharacteristic
129C2×Q8C2×D414yes—
130C2×D4—21no—
131C2×D4C2×D411yescharacteristic
132C22×C4—21no—
133C22×C4—24no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—41no—
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141Q16—416no—
142M16—42no—
143C2×C8—24no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—22no—
147C4⋊C4—24no—
148C4⋊C4—24no—
149C4⋊C4—24no—
150C4⋊C4—24no—
151C4⋊C4—22no—
152C4⋊C4—21no—
153C4⋊C4—22no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—21no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4C2×D411yescharacteristic
167C4⋊C4C2×D412yes—
168C4⋊C4C2×D411yescharacteristic
169C22⋊C4—24no—
170C22⋊C4—22no—
171C22⋊C4—21no—
172C22⋊C4—21no—
173C22⋊C4—21no—
174C42—24no—
175C42—21no—
176C42—21no—
177C42—21no—
178C42—22no—
179C42C2×D411yescharacteristic
180C42C2×D411yescharacteristic
181C42C2×D411yescharacteristic
182C42C2411yescharacteristic
183C23—21no—
184C23#12011yescharacteristic
185Q8—21no—
186Q8—24no—
187Q8—44no—
188Q8—44no—
189Q8—44no—
190Q8—42no—
191Q8—84no—
192Q8—82no—
193Q8C2×Q1612yescomplemented
194D4—24no—
195D4—21no—
196D4C2×Q1612yescomplemented
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—24no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—24no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—82no—
210C2×C4—84no—
211C2×C4—82no—
212C2×C4#12011yescharacteristic
213C2×C4#12011yescharacteristic
214C2×C4#12011yescharacteristic
215C2×C4C2×Q1612yes—
216C2×C4#13712yes—
217C2×C4C22×D411yescharacteristic
218C2×C4C22×D411yescharacteristic
219C2×C4C22×D411yescharacteristic
220C8—24no—
221C8—42no—
222C8—42no—
223C22—21no—
224C22—21no—
225C22—22no—
226C22—41no—
227C22—41no—
228C22#45112yes—
229C22C2×#12011yescharacteristic
230C4—22no—
231C4—21no—
232C4—21no—
233C4—41no—
234C4—42no—
235C4—41no—
236C4—44no—
237C4—42no—
238C4—82no—
239C4#45112yes—
240C4#45312yes—
241C4C22×Q1611yescharacteristic
242C4C2×#13711yescharacteristic
243C4#57812yes—
244C2—22no—
245C2—42no—
246C2C2×#45111yescharacteristic
247C2#301111yescharacteristic
248C2C2×#45311yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radicalC2×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3230C2semidirect
3. C2×#494C2semidirect
5. #1633C2semidirect
6. #1632C2semidirect
7. #1548C2semidirect
8. #1493C2semidirect
118. #41Q16semidirect
193. Q8C2×Q16semidirect
196. D4C2×Q16semidirect

Maximal subgroups

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

subgroupindex
#35032
#32302
C2×#4942
#30792
#16332
#16322
#15482
#14932
#14762