Finite Group Databasebeta

← #413855

Subgroups — #413855

Order 512.

1480 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 1480 conjugacy classes of subgroups — too many to draw. Only its 270 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
1C4×#543C211yesmaximal, characteristic
2C2×#2062C212yesmaximal
3#14149C213yesmaximal
4#11222C216yesmaximal
5#10846C213yesmaximal
6C2×#543C2211yescharacteristic
7#2329—26no—
8#2315C2213yes—
9C4×C22⋊Q8—26no—
10C2×#385C2216yes—
11C2×#380C2213yes—
12C2×C4×C22⋊C4C2213yes—
13C22×#95C2211yescharacteristic
14#2062—28no—
15C2×#342C2216yes—
16C2×#342C2216yes—
17#1441—26no—
18#1436—26no—
19C22⋊#95C2216yes—
20C24×C4C2312yes—
21C24×C4C2311yescharacteristic
22#543—26no—
23#543D414yes—
24C2×C22⋊Q8—26no—
25C2×C4×Q8—22no—
26C22×C4⋊C4—23no—
27C22×C4⋊C4—26no—
28C22×C22⋊C4C2313yes—
29C22×C22⋊C4C2316yes—
30C22×C22⋊C4C2313yes—
31C22×C42D416yes—
32#391—26no—
33#389—26no—
34#387—212no—
35#385—212no—
36#385—212no—
37#385—212no—
38#380—26no—
39#380—26no—
40#380Q8112yescomplemented
41C4×C4⋊C4—26no—
42C4×C22⋊C4—26no—
43C4×C22⋊C4—26no—
44C2×#95—23no—
45C2×#95—26no—
46C2×#95—23no—
47C2×#95—26no—
48C2×#95—23no—
49C2×#95—26no—
50C2×#95D416yes—
51#342—212no—
52#342—46no—
53C25C2411yescharacteristic
54C22×Q8—22no—
55C23×C4—26no—
56C23×C4—26no—
57C23×C4—22no—
58C23×C4—26no—
59C23×C4—44no—
60C23×C4—82no—
61C23×C4—82no—
62C23×C4C2×D416yes—
63C23×C4C2×D416yes—
64C23×C4C2×D413yes—
65C23×C4C2×D413yes—
66C23×C4C2×Q816yes—
67C23×C4#4112yes—
68C22⋊Q8—224no—
69C22⋊Q8—224no—
70C4×Q8—84no—
71C2×C4⋊C4—26no—
72C2×C4⋊C4—26no—
73C2×C4⋊C4—26no—
74C2×C4⋊C4—212no—
75C2×C4⋊C4—412no—
76C2×C4⋊C4—412no—
77C2×C4⋊C4—86no—
78C2×C4⋊C4—86no—
79C2×C22⋊C4—26no—
80C2×C22⋊C4—26no—
81C2×C22⋊C4—26no—
82C2×C22⋊C4—26no—
83C2×C22⋊C4—26no—
84C2×C22⋊C4—26no—
85C2×C22⋊C4—26no—
86C2×C22⋊C4—26no—
87C2×C22⋊C4—23no—
88C2×C22⋊C4—23no—
89C2×C22⋊C4—26no—
90C2×C22⋊C4C2×D416yes—
91C2×C22⋊C4C2×Q8112yes—
92C2×C22⋊C4#4116yes—
93C2×C42—26no—
94C2×C42—26no—
95C2×C42—86no—
96#95—212no—
97#95—212no—
98#95—212no—
99#95—412no—
100#95—43no—
101#95—41no—
102#95—412no—
103#95—86no—
104C24—42no—
105C24—42no—
106C24C22×D413yes—
107C24C22×D411yescharacteristic
108C24C22×Q813yes—
109C24C2×#4113yes—
110C24C2×#4111yescharacteristic
111C24H(5,𝔽2)11yescharacteristic
112C242−1+413yes—
113C2×Q8—28no—
114C2×Q8—82no—
115C2×Q8—82no—
116C22×C4—26no—
117C22×C4—26no—
118C22×C4—212no—
119C22×C4—26no—
120C22×C4—212no—
121C22×C4—26no—
122C22×C4—26no—
123C22×C4—46no—
124C22×C4—46no—
125C22×C4—46no—
126C22×C4—46no—
127C22×C4—46no—
128C22×C4—46no—
129C22×C4—46no—
130C22×C4—46no—
131C22×C4—46no—
132C22×C4—46no—
133C22×C4—46no—
134C22×C4—46no—
135C22×C4—46no—
136C22×C4—42no—
137C22×C4—46no—
138C22×C4—46no—
139C22×C4—46no—
140C22×C4—46no—
141C22×C4—86no—
142C22×C4—82no—
143C22×C4—84no—
144C22×C4—812no—
145C22×C4#12013yes—
146C22×C4#12011yescharacteristic
147C22×C4#121112yes—
148C22×C4C22⋊Q816yes—
149C22×C4C22⋊Q816yes—
150C22×C4C4⋊Q816yes—
151C22×C4C4⋊Q816yes—
152C4⋊C4—224no—
153C4⋊C4—86no—
154C4⋊C4—812no—
155C4⋊C4—86no—
156C4⋊C4—824no—
157C22⋊C4—212no—
158C22⋊C4—212no—
159C22⋊C4—212no—
160C22⋊C4—212no—
161C22⋊C4—46no—
162C22⋊C4—46no—
163C22⋊C4—46no—
164C22⋊C4—46no—
165C42—412no—
166C42—86no—
167C23—26no—
168C23—26no—
169C23—42no—
170C23—43no—
171C23—41no—
172C23—46no—
173C23—43no—
174C23—41no—
175C23—42no—
176C23—46no—
177C23C2×#12011yescharacteristic
178C23C2×#12113yes—
179C23C2×C22⋊Q813yes—
180C23#52513yes—
181C23C2×C4⋊Q813yes—
182C23#53613yes—
183C23#53713yes—
184C23D4×Q813yes—
185C23#55011yescharacteristic
186C23#55413yes—
187C23#55613yes—
188C23#55716yes—
189Q8—88no—
190C2×C4—22no—
191C2×C4—26no—
192C2×C4—212no—
193C2×C4—46no—
194C2×C4—46no—
195C2×C4—46no—
196C2×C4—46no—
197C2×C4—46no—
198C2×C4—46no—
199C2×C4—412no—
200C2×C4—412no—
201C2×C4—412no—
202C2×C4—412no—
203C2×C4—412no—
204C2×C4—412no—
205C2×C4—46no—
206C2×C4—46no—
207C2×C4—44no—
208C2×C4—46no—
209C2×C4—412no—
210C2×C4—86no—
211C2×C4—86no—
212C2×C4—86no—
213C2×C4—86no—
214C2×C4—86no—
215C2×C4—86no—
216C2×C4—86no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—86no—
220C2×C4#39516yes—
221C2×C4#39512yes—
222C22—23no—
223C22—23no—
224C22—23no—
225C22—23no—
226C22—26no—
227C22—26no—
228C22—26no—
229C22—26no—
230C22—43no—
231C22—41no—
232C22—43no—
233C22—43no—
234C22—46no—
235C22C2×#39511yescharacteristic
236C22#253813yes—
237C22#254613yes—
238C22#258113yes—
239C22#258413yes—
240C22#261611yescharacteristic
241C22#284711yescharacteristic
242C22#302512yes—
243C22#302612yes—
244C4—22no—
245C4—26no—
246C4—412no—
247C4—812no—
248C4—86no—
249C4—84no—
250C2—23no—
251C2—23no—
252C2—23no—
253C2—23no—
254C2#1515811yescharacteristic
255C2#2356411yescharacteristic
256C2#2356511yescharacteristic
257C2#2971112yes—
258C2#3292812yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, prime radical, cosocleC25C24
LCSG ▷ C25 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
40. #380Q8semidirect

Maximal subgroups

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

subgroupindex
C4×#5432
C2×#20622
#141492
#112222
#108462