Finite Group Databasebeta

← #415576

Subgroups — #415576

Order 512.

1428 conjugacy classes of subgroups; only the 254 normal subgroups are shown.

normal subgroups only

This group has 1428 conjugacy classes of subgroups — too many to draw. Only its 254 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
2#13923C213yesmaximal
3#11390C211yesmaximal, characteristic
4#11390C211yesmaximal, characteristic
5#11287C213yesmaximal
6#11287C213yesmaximal
7#10846C213yesmaximal
8C2×#543C2211yescharacteristic
9C23×C4⋊C4C2211yescharacteristic
10#2329—26no—
11#2315C2213yes—
12C4×C22⋊Q8—26no—
13C2×#380C2213yes—
14C2×C4×C22⋊C4C2213yes—
15C2×#342C2213yes—
16C2×#342C2213yes—
17C2×#342C2216yes—
18#1441—26no—
19#1436—26no—
20C22⋊#95C2213yes—
21C22⋊#95C2213yes—
22C22⋊#95C2213yes—
23C22⋊#95C2213yes—
24C24×C4C2311yescharacteristic
25C24×C4C2311yescharacteristic
26C24×C4C2311yescharacteristic
27#543—26no—
28#543Q814yes—
29C2×C22⋊Q8—26no—
30C2×C4×Q8—22no—
31C22×C4⋊C4—23no—
32C22×C4⋊C4—23no—
33C22×C4⋊C4—26no—
34C22×C4⋊C4—23no—
35C22×C4⋊C4Q814yescomplemented
36C22×C22⋊C4C2313yes—
37C22×C22⋊C4C2313yes—
38C22×C22⋊C4C2313yes—
39C22×C22⋊C4C2313yes—
40C22×C42D416yes—
41#391—26no—
42#389—26no—
43#387—212no—
44#385—212no—
45#380—26no—
46#380—212no—
47C4×C4⋊C4—26no—
48C4×C22⋊C4—26no—
49C4×C22⋊C4—26no—
50C2×#95—26no—
51C2×#95—26no—
52C2×#95—23no—
53C2×#95—26no—
54C2×#95—26no—
55C2×#95D416yes—
56#342—26no—
57#342—26no—
58#342—46no—
59C25C2411yescharacteristic
60C22×Q8—22no—
61C23×C4—23no—
62C23×C4—23no—
63C23×C4—23no—
64C23×C4—23no—
65C23×C4—23no—
66C23×C4—23no—
67C23×C4—44no—
68C23×C4—82no—
69C23×C4—82no—
70C23×C4C2×D416yes—
71C23×C4C2×D416yes—
72C23×C4C2×D413yes—
73C23×C4C2×D413yes—
74C23×C4C2×Q812yes—
75C23×C4C2×Q812yes—
76C23×C4C2×Q812yes—
77C23×C4#4116yes—
78C22⋊Q8—224no—
79C22⋊Q8—224no—
80C4×Q8—84no—
81C2×C4⋊C4—26no—
82C2×C4⋊C4—212no—
83C2×C4⋊C4—212no—
84C2×C4⋊C4—26no—
85C2×C4⋊C4—26no—
86C2×C4⋊C4—43no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—46no—
89C2×C4⋊C4—46no—
90C2×C4⋊C4—46no—
91C2×C4⋊C4—86no—
92C2×C22⋊C4—26no—
93C2×C22⋊C4—26no—
94C2×C22⋊C4—26no—
95C2×C22⋊C4—23no—
96C2×C22⋊C4—23no—
97C2×C22⋊C4—26no—
98C2×C22⋊C4—26no—
99C2×C22⋊C4—26no—
100C2×C22⋊C4—26no—
101C2×C22⋊C4—26no—
102C2×C22⋊C4—23no—
103C2×C22⋊C4—23no—
104C2×C22⋊C4—23no—
105C2×C22⋊C4—23no—
106C2×C22⋊C4C2×Q816yes—
107C2×C22⋊C4#4116yes—
108C2×C42—26no—
109C2×C42—26no—
110C2×C42—86no—
111#95—212no—
112#95—212no—
113#95—412no—
114#95—412no—
115#95—46no—
116#95—86no—
117#95—86no—
118C24—42no—
119C24—42no—
120C24C22×D413yes—
121C24C22×Q811yescharacteristic
122C24C22×Q811yescharacteristic
123C24C2×#4113yes—
124C24C2×#4113yes—
125C24H(5,𝔽2)11yescharacteristic
126C242−1+413yes—
127C2×Q8—28no—
128C2×Q8—82no—
129C2×Q8—82no—
130C22×C4—212no—
131C22×C4—26no—
132C22×C4—26no—
133C22×C4—26no—
134C22×C4—26no—
135C22×C4—26no—
136C22×C4—26no—
137C22×C4—26no—
138C22×C4—44no—
139C22×C4—46no—
140C22×C4—46no—
141C22×C4—412no—
142C22×C4—46no—
143C22×C4—41no—
144C22×C4—43no—
145C22×C4—46no—
146C22×C4—46no—
147C22×C4—46no—
148C22×C4—46no—
149C22×C4—46no—
150C22×C4—46no—
151C22×C4—41no—
152C22×C4—43no—
153C22×C4—46no—
154C22×C4—46no—
155C22×C4—46no—
156C22×C4—46no—
157C22×C4—43no—
158C22×C4—43no—
159C22×C4—43no—
160C22×C4—43no—
161C22×C4—43no—
162C22×C4—43no—
163C22×C4—82no—
164C22×C4—82no—
165C22×C4—86no—
166C22×C4—86no—
167C22×C4#12013yes—
168C22×C4#12011yescharacteristic
169C22×C4C22⋊Q8112yes—
170C22×C4C22⋊Q816yes—
171C22×C4C22⋊Q816yes—
172C22×C4#12416yes—
173C22×C4#12416yes—
174C4⋊C4—224no—
175C4⋊C4—412no—
176C4⋊C4—44no—
177C4⋊C4—86no—
178C4⋊C4—86no—
179C4⋊C4—812no—
180C22⋊C4—212no—
181C22⋊C4—212no—
182C22⋊C4—46no—
183C22⋊C4—46no—
184C22⋊C4—46no—
185C22⋊C4—46no—
186C22⋊C4—46no—
187C22⋊C4—46no—
188C42—412no—
189C42—86no—
190C23—26no—
191C23—26no—
192C23—41no—
193C23—43no—
194C23—41no—
195C23—43no—
196C23—42no—
197C23—46no—
198C23—42no—
199C23—46no—
200C23C2×#12011yescharacteristic
201C23C2×C22⋊Q813yes—
202C23C2×C22⋊Q813yes—
203C23#52513yes—
204C23C2×#12413yes—
205C23#53613yes—
206C23#53713yes—
207C23#55616yes—
208C23#55713yes—
209C23#55713yes—
210C23Q8211yescharacteristic
211C23#55913yes—
212Q8—88no—
213C2×C4—22no—
214C2×C4—26no—
215C2×C4—212no—
216C2×C4—212no—
217C2×C4—46no—
218C2×C4—46no—
219C2×C4—46no—
220C2×C4—46no—
221C2×C4—46no—
222C2×C4—412no—
223C2×C4—42no—
224C2×C4—46no—
225C2×C4—44no—
226C2×C4—46no—
227C2×C4—412no—
228C2×C4—412no—
229C2×C4—46no—
230C2×C4—46no—
231C2×C4—46no—
232C2×C4—42no—
233C2×C4—46no—
234C2×C4—46no—
235C2×C4—86no—
236C2×C4—86no—
237C2×C4—82no—
238C2×C4—86no—
239C2×C4—83no—
240C2×C4—83no—
241C2×C4—86no—
242C2×C4—86no—
243C2×C4—83no—
244C2×C4—83no—
245C2×C4—86no—
246C2×C4—86no—
247C2×C4—83no—
248C2×C4—83no—
249C2×C4#39316yes—
250C2×C4#39312yes—
251C22—26no—
252C22—26no—
253C22—23no—
254C22—23no—
255C22—23no—
256C22—23no—
257C22—26no—
258C22—26no—
259C22—46no—
260C22—43no—
261C22—43no—
262C22—41no—
263C22—43no—
264C22C2×#39311yescharacteristic
265C22#256813yes—
266C22#258113yes—
267C22#258213yes—
268C22#259213yes—
269C22#261611yescharacteristic
270C22#284811yescharacteristic
271C22#302512yes—
272C22#302612yes—
273C4—22no—
274C4—26no—
275C4—44no—
276C4—412no—
277C4—86no—
278C4—82no—
279C4—163no—
280C4—163no—
281C2—23no—
282C2—23no—
283C2—23no—
284C2—23no—
285C2#1520111yescharacteristic
286C2#2346711yescharacteristic
287C2#2346811yescharacteristic
288C2#2971112yes—
289C2#3297712yes—
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
35. C22×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C4×#5432
#139232
#113902
#112872
#108462