Finite Group Databasebeta

← #125963

Subgroups — #125963

Order 512.

326 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 64 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#32653C211yesmaximal, characteristic
2#13451C211yesmaximal, characteristic
3#13364C211yesmaximal, characteristic
4#10265C211yesmaximal, characteristic
5#7128C211yesmaximal, characteristic
6#7120C211yesmaximal, characteristic
7#7113C211yesmaximal, characteristic
8C2×#500C2211yescharacteristic
9C2×#462—21no—
10#2581—21no—
11C2×#366—21no—
12C2×#358—21no—
13#2097—21no—
14#2085—21no—
15#2066—21no—
16#1984C2211yescharacteristic
17#1969C2211yescharacteristic
18#1942C2211yescharacteristic
19#1878—21no—
20#1863—21no—
21C2×#332—21no—
22#1380—21no—
23#1380—21no—
24#1374C2211yescharacteristic
25#1374C2211yescharacteristic
26#1277C2211yescharacteristic
27C22×Q16—41no—
28C22×Q16D411yescharacteristic
29C2×C4⋊Q8—21no—
30C2×C4×Q8—21no—
31C2×C4×Q8D411yescharacteristic
32C22×C16—21no—
33C22×C16—21no—
34#500—42no—
35#462—42no—
36#462—42no—
37C2×#107—21no—
38C2×#107—21no—
39C2×#107D411yescharacteristic
40C2×#107D411yescharacteristic
41C2×#107D411yescharacteristic
42C2×#106—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8D411yescharacteristic
45C2×#103—21no—
46C2×#103—21no—
47C2×#103—21no—
48C2×#103—41no—
49C2×C4×C8C2311yescharacteristic
50#400—41no—
51#395—41no—
52#391—21no—
53#391—21no—
54#389—21no—
55#389—21no—
56#387—41no—
57#387—41no—
58#382—21no—
59#366—42no—
60#358—42no—
61#340—21no—
62#340—21no—
63#336—21no—
64#336—21no—
65#332—42no—
66C22×Q8—21no—
67C22×Q8—41no—
68C22×Q8D811yescharacteristic
69C2×Q16—21no—
70C2×Q16—41no—
71C2×Q16—41no—
72C2×Q16—44no—
73C2×Q16—82no—
74C2×Q16—82no—
75C2×Q16Q1612yescomplemented
76C22×C8—21no—
77C22×C8—41no—
78C22×C8C2×D411yescharacteristic
79C22×C8C2×D411yescharacteristic
80C4⋊Q8—22no—
81C4⋊Q8—41no—
82C4⋊Q8—41no—
83C4⋊Q8—41no—
84C4×Q8—41no—
85C4×Q8—41no—
86C4×Q8—81no—
87C4×Q8—81no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4D811yescharacteristic
98C2×C42—21no—
99C2×C42—41no—
100C2×C42C2×D411yescharacteristic
101C2×C16—22no—
102C2×C16—41no—
103C2×C16—41no—
104C2×C16—41no—
105C2×C16—41no—
106C2×C16—41no—
107#107—22no—
108#107—22no—
109#107—21no—
110#107—21no—
111#107—41no—
112#107—41no—
113#107—41no—
114#107—41no—
115#106—81no—
116C4⋊C8—22no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119#103—44no—
120#103—41no—
121#103—41no—
122#103—81no—
123#103—81no—
124C4×C8—41no—
125#95—41no—
126#95—41no—
127#95—41no—
128#95—41no—
129#95—41no—
130#95—41no—
131#95—41no—
132#95—41no—
133C2×Q8—22no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—41no—
137C2×Q8—41no—
138C2×Q8—42no—
139C2×Q8—41no—
140C2×Q8—41no—
141C2×Q8—41no—
142C2×Q8—41no—
143C2×Q8—41no—
144C2×Q8—81no—
145C2×Q8—81no—
146C2×Q8—81no—
147C2×Q8—81no—
148C2×Q8—81no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—81no—
156C22×C4#12011yescharacteristic
157C22×C4C2×D811yescharacteristic
158C22×C4AGL(1,ℤ/8)11yescharacteristic
159Q16—42no—
160Q16—81no—
161Q16—88no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—42no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—81no—
171C2×C8—81no—
172C2×C8C2×Q1611yescharacteristic
173C2×C8#13511yescharacteristic
174C2×C8#13711yescharacteristic
175C2×C8#13711yescharacteristic
176C4⋊C4—41no—
177C4⋊C4—42no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—42no—
184C4⋊C4—41no—
185C4⋊C4—81no—
186C4⋊C4—81no—
187C4⋊C4—82no—
188C4⋊C4—82no—
189C4⋊C4—81no—
190C4⋊C4—81no—
191C4⋊C4—81no—
192C42—21no—
193C42—21no—
194C42—41no—
195C42—41no—
196C42—82no—
197C16—42no—
198C16—81no—
199C23#44711yescharacteristic
200Q8—42no—
201Q8—42no—
202Q8—42no—
203Q8—81no—
204Q8—82no—
205Q8—82no—
206Q8—81no—
207Q8—81no—
208Q8—161no—
209Q8—161no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—82no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—161no—
235C2×C4—161no—
236C2×C4#45111yescharacteristic
237C2×C4#45211yescharacteristic
238C2×C4#45311yescharacteristic
239C2×C4#45511yescharacteristic
240C2×C4#45611yescharacteristic
241C2×C4B(4,𝔽2)11yescharacteristic
242C2×C4#50811yescharacteristic
243C2×C4#50911yescharacteristic
244C8—21no—
245C8—21no—
246C8—21no—
247C8—21no—
248C8—82no—
249C8—81no—
250C22#159511yescharacteristic
251C22#164111yescharacteristic
252C22#164911yescharacteristic
253C22#218611yescharacteristic
254C22#219011yescharacteristic
255C22#219511yescharacteristic
256C22#219511yescharacteristic
257C4—21no—
258C4—21no—
259C4—41no—
260C4—81no—
261C4—82no—
262C4—81no—
263C4—161no—
264C4—162no—
265C4—161no—
266C4—161no—
267C4#160211yescharacteristic
268C4#160311yescharacteristic
269C4#163211yescharacteristic
270C4#166111yescharacteristic
271C2#775611yescharacteristic
272C2#1210411yescharacteristic
273C2#1211211yescharacteristic
274C2#1238311yescharacteristic
275C2#1239111yescharacteristic
276C2#1240511yescharacteristic
277C2#1241511yescharacteristic
subgroupG/H
center, socle, prime radicalC23#447
commutator, Frattini, cosocleC2×C4×C8C23
LCSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1984 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
75. C2×Q16Q16semidirect

Maximal subgroups

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

subgroupindex
#326532
#134512
#133642
#102652
#71282
#71202
#71132