Finite Group Databasebeta

← #18632

Subgroups — #18632

Order 256.

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

normal subgroups only

This group has 469 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#3229C211yesmaximal, characteristic, complemented
3C2×#487C211yesmaximal, characteristic, complemented
4#3039C211yesmaximal, characteristic
5#1645C214yesmaximal, complemented
6#1643C214yesmaximal, complemented
7#1548C211yesmaximal, characteristic, complemented
8#1493C211yesmaximal, characteristic, complemented
9#1476C211yesmaximal, characteristic
10C2×#137—21no—
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#498—21no—
27#487—22no—
28#487—21no—
29#487C2214yes—
30#483—22no—
31#475C2214yes—
32#462—21no—
33#462C2212yes—
34#461—21no—
35#461C2212yes—
36#431C2211yescharacteristic
37#423C2211yescharacteristic
38#419—21no—
39C2×#103—22no—
40C2×C4×C8C2211yescharacteristic
41#333C2214yes—
42Q8⋊C8C2214yes—
43Q8⋊C8C2212yes—
44D4⋊C8C2212yes—
45C2×#41D411yescharacteristic
46C22×Q8—22no—
47C22×Q8—21no—
48C22×Q8D412yes—
49#137—44no—
50C2×Q16—22no—
51C2×SD16—22no—
52C2×M16—21no—
53C22×C8—21no—
54C4⋊Q8—22no—
55C4⋊Q8—22no—
56C4⋊Q8—24no—
57C4⋊Q8—22no—
58C4⋊Q8—22no—
59C4⋊Q8—21no—
60C4⋊Q8—21no—
61C4⋊Q8—21no—
62C4⋊Q8—21no—
63C4⋊Q8—21no—
64C4⋊Q8D412yes—
65C4⋊Q8D411yescharacteristic
66C4⋊Q8D411yescharacteristic
67C4⋊Q8C2312yes—
68C4⋊Q8C2312yes—
69#125—22no—
70#125—24no—
71#125—22no—
72#125—21no—
73C22⋊Q8—22no—
74C22⋊Q8—24no—
75C22⋊Q8—24no—
76C22⋊Q8—22no—
77C22⋊Q8—24no—
78C22⋊Q8—21no—
79C22⋊Q8D412yes—
80C4×Q8—22no—
81C4×Q8—22no—
82C4×Q8—22no—
83C4×Q8—21no—
84C4×Q8D411yescharacteristic
85C4×Q8C2311yescharacteristic
86C4×Q8C2312yes—
87C4×D4—24no—
88C4×D4—22no—
89C4×D4—21no—
90C4×D4—21no—
91C4×D4D411yescharacteristic
92C4×D4C2311yescharacteristic
93#117—22no—
94#117—21no—
95#117—21no—
96#117D411yescharacteristic
97C2×C4⋊C4—22no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C42—21no—
104C2×C42C2311yescharacteristic
105#106—24no—
106C4⋊C8—21no—
107C4⋊C8C2312yes—
108C4⋊C8C2312yes—
109#103—28no—
110#103—24no—
111#103—22no—
112#103—22no—
113C22⋊C8—21no—
114C4×C8—21no—
115C4×C8—21no—
116C4×C8C2312yes—
117#41—22no—
118#41SD1614yes—
119C2×Q8—24no—
120C2×Q8—22no—
121C2×Q8—21no—
122C2×Q8—22no—
123C2×Q8—44no—
124C2×Q8—41no—
125C2×Q8—42no—
126C2×Q8—42no—
127C2×Q8—84no—
128C2×Q8C2×D412yes—
129C2×Q8C2×D411yescharacteristic
130C2×Q8C2×D414yes—
131C2×D4—21no—
132C2×D4C2×D411yescharacteristic
133C22×C4—21no—
134C22×C4—24no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—41no—
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142Q16—44no—
143SD16—44no—
144M16—42no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—42no—
149C4⋊C4—24no—
150C4⋊C4—24no—
151C4⋊C4—24no—
152C4⋊C4—24no—
153C4⋊C4—22no—
154C4⋊C4—21no—
155C4⋊C4—22no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—21no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4C2×D411yescharacteristic
169C4⋊C4C2×D411yescharacteristic
170C4⋊C4C2×D412yes—
171C22⋊C4—24no—
172C22⋊C4—22no—
173C22⋊C4—21no—
174C22⋊C4—21no—
175C22⋊C4—21no—
176C42—24no—
177C42—21no—
178C42—21no—
179C42—21no—
180C42—22no—
181C42C2×D411yescharacteristic
182C42C2×D411yescharacteristic
183C42C2×D411yescharacteristic
184C42C2411yescharacteristic
185C23—21no—
186C23#12011yescharacteristic
187Q8—21no—
188Q8—24no—
189Q8—44no—
190Q8—44no—
191Q8—42no—
192Q8—42no—
193Q8—42no—
194Q8—84no—
195Q8—82no—
196Q8C2×SD1612yes—
197D4—24no—
198D4—21no—
199D4C2×SD1612yes—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—24no—
208C2×C4—22no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—82no—
213C2×C4—84no—
214C2×C4—82no—
215C2×C4#12011yescharacteristic
216C2×C4#12011yescharacteristic
217C2×C4#12011yescharacteristic
218C2×C4C2×SD1612yes—
219C2×C4#13712yes—
220C2×C4C22×D411yescharacteristic
221C2×C4C22×D411yescharacteristic
222C2×C4C22×D411yescharacteristic
223C8—42no—
224C8—42no—
225C8—42no—
226C22—21no—
227C22—21no—
228C22—22no—
229C22—41no—
230C22—41no—
231C22#44812yes—
232C22C2×#12011yescharacteristic
233C4—22no—
234C4—21no—
235C4—21no—
236C4—41no—
237C4—42no—
238C4—41no—
239C4—44no—
240C4—42no—
241C4—82no—
242C4#44812yes—
243C4#45512yes—
244C4C22×SD1611yescharacteristic
245C4C2×#13711yescharacteristic
246C4#57812yes—
247C2—22no—
248C2—42no—
249C2C2×#44811yescharacteristic
250C2#301111yescharacteristic
251C2C2×#45511yescharacteristic
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. #3229C2semidirect
3. C2×#487C2semidirect
5. #1645C2semidirect
6. #1643C2semidirect
7. #1548C2semidirect
8. #1493C2semidirect

Maximal subgroups

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

subgroupindex
#35032
#32292
C2×#4872
#30392
#16452
#16432
#15482
#14932
#14762