Finite Group Databasebeta

← #25923

Subgroups — #25923

Order 256.

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

normal subgroups only

This group has 507 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#3139C211yesmaximal, characteristic, complemented
3#3039C211yesmaximal, characteristic
4#2862C211yesmaximal, characteristic, complemented
5#2028C214yesmaximal, complemented
6#1997C211yesmaximal, characteristic, complemented
7#1994C211yesmaximal, characteristic, complemented
8#1882C214yesmaximal
9C4×#104C211yesmaximal, characteristic
10C2×#137C2212yes—
11Q82—21no—
12#557—21no—
13#554—21no—
14#554—22no—
15D4×Q8—22no—
16D4×Q8—21no—
17#544—22no—
18#544C2212yes—
19#533—21no—
20#533C2211yescharacteristic
21C2×C4⋊Q8—21no—
22C2×C4⋊Q8C2211yescharacteristic
23C2×C4⋊Q8C2211yescharacteristic
24#520C2212yes—
25C4×#41C2211yescharacteristic
26#491—22no—
27#487—21no—
28#487—21no—
29#462—21no—
30#462—21no—
31#461—21no—
32#461—21no—
33#423C2211yescharacteristic
34C2×#104C2212yes—
35#419C2214yes—
36#419C2212yes—
37C4×M16C2211yescharacteristic
38#391—22no—
39#391C2214yes—
40#382—22no—
41C43C2211yescharacteristic
42#343C2214yes—
43#339C2214yes—
44#339C2212yes—
45C2×#41C2311yescharacteristic
46C22×Q8—22no—
47C22×Q8D412yes—
48C22×Q8C2312yes—
49#137—28no—
50C2×Q16—22no—
51C2×SD16—22no—
52C2×M16C2312yes—
53C2×M16C2312yes—
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⋊Q8—21no—
65C4⋊Q8—21no—
66C4⋊Q8—21no—
67C4⋊Q8D411yescharacteristic
68C4⋊Q8D411yescharacteristic
69#125—24no—
70#125—22no—
71#125—22no—
72#125D412yes—
73C22⋊Q8—22no—
74C22⋊Q8—22no—
75C22⋊Q8—24no—
76C22⋊Q8—24no—
77C22⋊Q8—21no—
78C22⋊Q8—21no—
79C22⋊Q8—24no—
80C4×Q8—22no—
81C4×Q8—22no—
82C4×Q8—22no—
83C4×Q8—21no—
84C4×Q8—21no—
85C4×Q8D411yescharacteristic
86C4×Q8D411yescharacteristic
87C4×D4—22no—
88C4×D4—21no—
89C4×D4—21no—
90C4×D4—24no—
91C4×D4D411yescharacteristic
92C4×D4D411yescharacteristic
93#117—22no—
94#117—21no—
95#117C2311yescharacteristic
96#117C2312yes—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4D412yes—
102C2×C4⋊C4C2312yes—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42C2312yes—
107C2×C42C2311yescharacteristic
108C4⋊C8—21no—
109C4⋊C8—21no—
110#104—24no—
111#103—22no—
112#103—22no—
113#103—24no—
114#103—24no—
115#97—21no—
116C4×C8—21no—
117#95—28no—
118#41—22no—
119#41#4114yes—
120C2×Q8—24no—
121C2×Q8—21no—
122C2×Q8—22no—
123C2×Q8—21no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—24no—
128C2×Q8—24no—
129C2×Q8—44no—
130C2×Q8—84no—
131C2×Q8C2×D411yescharacteristic
132C2×D4—21no—
133C2×D4C2×D411yescharacteristic
134C22×C4—21no—
135C22×C4—22no—
136C22×C4—24no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D414yes—
143C22×C4C2411yescharacteristic
144Q16—44no—
145SD16—44no—
146M16—22no—
147M16—24no—
148M16—22no—
149C2×C8—22no—
150C2×C8—22no—
151C4⋊C4—24no—
152C4⋊C4—24no—
153C4⋊C4—24no—
154C4⋊C4—24no—
155C4⋊C4—21no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4C2×D412yes—
170C4⋊C4C2×D412yes—
171C4⋊C4C2×D411yescharacteristic
172C4⋊C4C2×D411yescharacteristic
173C22⋊C4—24no—
174C22⋊C4—22no—
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C42—24no—
179C42—24no—
180C42—21no—
181C42—21no—
182C42—21no—
183C42—22no—
184C42—22no—
185C42—22no—
186C42—42no—
187C42C2×D411yescharacteristic
188C42C2×D411yescharacteristic
189C42C2×D411yescharacteristic
190C42C2×D411yescharacteristic
191C23—21no—
192C23C22×D411yescharacteristic
193Q8—21no—
194Q8—44no—
195Q8—42no—
196Q8—42no—
197Q8—42no—
198Q8—42no—
199Q8—44no—
200Q8—82no—
201Q8—84no—
202Q8#12412yes—
203D4—24no—
204D4—21no—
205D4#12412yes—
206C2×C4—21no—
207C2×C4—24no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—24no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—24no—
215C2×C4—22no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—44no—
221C2×C4—42no—
222C2×C4—44no—
223C2×C4—84no—
224C2×C4#12011yescharacteristic
225C2×C4#12011yescharacteristic
226C2×C4#12011yescharacteristic
227C2×C4#12011yescharacteristic
228C2×C4C22×D411yescharacteristic
229C2×C4C22×D411yescharacteristic
230C2×C4C2×#4112yes—
231C2×C42−1+412yes—
232C8—42no—
233C8—42no—
234C22—21no—
235C22—21no—
236C22—22no—
237C22—41no—
238C22—41no—
239C22C2×#12011yescharacteristic
240C22#54812yes—
241C4—22no—
242C4—21no—
243C4—21no—
244C4—44no—
245C4—41no—
246C4—44no—
247C4—42no—
248C4—41no—
249C4—42no—
250C4—82no—
251C4#45512yes—
252C4#45612yes—
253C4C2×#12411yescharacteristic
254C4#53611yescharacteristic
255C4#54812yes—
256C2—22no—
257C2—42no—
258C2#243611yescharacteristic
259C2C2×#45511yescharacteristic
260C2C2×#45611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3139C2semidirect
4. #2862C2semidirect
5. #2028C2semidirect
6. #1997C2semidirect
7. #1994C2semidirect

Maximal subgroups

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

subgroupindex
#35032
#31392
#30392
#28622
#20282
#19972
#19942
#18822
C4×#1042