Finite Group Databasebeta

← #512500

Subgroups — #512500

Order 512.

504 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 166 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#33257C211yesmaximal, characteristic
2#33242C211yesmaximal, characteristic
3#13283C211yesmaximal, characteristic
4#13283C211yesmaximal, characteristic
5#13207C211yesmaximal, characteristic
6#13207C211yesmaximal, characteristic
7#13150C214yesmaximal
8#13148C214yesmaximal
9C4×#346C211yesmaximal, characteristic
10C2×#500C2211yescharacteristic
11C2×#494C2211yescharacteristic
12C2×#487—21no—
13#2864—21no—
14#2862—21no—
15C2×#358C2214yes—
16C2×#358C2214yes—
17C2×#346C2212yes—
18C2×#346C2211yescharacteristic
19C2×#346C2211yescharacteristic
20C2×C4×C16C2212yes—
21#1969—21no—
22#1969—21no—
23#1969C2212yes—
24#1969C2212yes—
25#1942C2212yes—
26#1942C2212yes—
27#1941—21no—
28#1941—21no—
29#1846C2211yescharacteristic
30#1846C2211yescharacteristic
31#1835—21no—
32C42×C8C2211yescharacteristic
33#1380C2214yes—
34#1380C2214yes—
35C22×Q16—21no—
36C22×Q16C2312yes—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C22×C16C2312yes—
40C22×C16C2312yes—
41#500—22no—
42#500—22no—
43#494—22no—
44#494—22no—
45#487—44no—
46C2×#107—21no—
47C2×#107C2312yes—
48C2×#107C2312yes—
49C2×#107C2312yes—
50C2×#106—21no—
51C2×#106—21no—
52C2×#103—22no—
53C2×#103—22no—
54C2×C4×C8D411yescharacteristic
55C2×C4×C8D411yescharacteristic
56C2×C4×C8D411yescharacteristic
57C2×C4×C8C2311yescharacteristic
58C2×C4×C8C2311yescharacteristic
59C2×C4×C8C2311yescharacteristic
60#391—22no—
61#391—22no—
62#391—42no—
63#389—22no—
64#389—22no—
65#387—42no—
66#387—42no—
67#382—21no—
68#382—21no—
69#382—42no—
70C43D411yescharacteristic
71#358—28no—
72#358—28no—
73#346—24no—
74#346—22no—
75#346—22no—
76C4×C16—24no—
77#340—22no—
78#340—22no—
79C22×Q8—22no—
80C22×Q8—41no—
81C2×Q16—24no—
82C2×Q16—24no—
83C2×Q16—22no—
84C2×Q16—22no—
85C2×Q16—42no—
86C2×Q16—42no—
87C2×Q16—42no—
88C22×C8C2×D411yescharacteristic
89C22×C8C2×D411yescharacteristic
90C22×C8C2×D411yescharacteristic
91C22×C8C2411yescharacteristic
92C4⋊Q8—42no—
93C4⋊Q8—42no—
94C4⋊Q8—42no—
95C4⋊Q8—42no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C42D812yes—
105C2×C42D812yes—
106C2×C42C2×D411yescharacteristic
107C2×C42C2×D411yescharacteristic
108C2×C42C2×D411yescharacteristic
109C2×C16—22no—
110C2×C16—22no—
111C2×C16—22no—
112C2×C16—22no—
113C2×C16#4114yes—
114C2×C16#4114yes—
115#107—24no—
116#107—24no—
117#107—22no—
118#107—22no—
119#107—42no—
120#106—42no—
121#106—42no—
122#103—44no—
123#103—44no—
124C4×C8—22no—
125C4×C8—22no—
126C4×C8—22no—
127C4×C8—22no—
128C4×C8SD1614yes—
129C4×C8#4112yes—
130C4×C8#4112yes—
131#95—41no—
132#95—41no—
133#95—41no—
134#95—41no—
135#95—44no—
136#95—44no—
137#95—42no—
138#95—42no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8—42no—
145C2×Q8—84no—
146C2×Q8—82no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4C2×D812yes—
152C22×C4C2×D811yescharacteristic
153C22×C4C2×D811yescharacteristic
154C22×C4C2×D812yes—
155C22×C4C22×D411yescharacteristic
156Q16—44no—
157Q16—44no—
158Q16—84no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8#12412yes—
164C2×C8C2×SD1612yes—
165C2×C8C2×SD1612yes—
166C2×C8#13512yes—
167C2×C8#13512yes—
168C2×C8C2×#4111yescharacteristic
169C2×C8C2×#4111yescharacteristic
170C2×C8C2×#4112yes—
171C2×C8C2×#4112yes—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—84no—
179C4⋊C4—82no—
180C4⋊C4—82no—
181C4⋊C4—82no—
182C4⋊C4—84no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—24no—
188C42—42no—
189C42#12412yes—
190C42C2×SD1612yes—
191C16—24no—
192C16—24no—
193C23C22×D811yescharacteristic
194Q8—82no—
195Q8—82no—
196Q8—82no—
197Q8—82no—
198Q8—164no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—84no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4#48612yes—
218C2×C4#48612yes—
219C2×C4#51014yes—
220C2×C4C2×#12411yescharacteristic
221C2×C4#53211yescharacteristic
222C2×C4#53211yescharacteristic
223C2×C4C22×SD1611yescharacteristic
224C2×C4C2×#13511yescharacteristic
225C2×C4C2×#13511yescharacteristic
226C8—22no—
227C8—22no—
228C8#48814yes—
229C8#52912yes—
230C8#52912yes—
231C22C2×#48611yescharacteristic
232C22#316911yescharacteristic
233C22#316911yescharacteristic
234C22C2×#51012yes—
235C22#341412yes—
236C4—22no—
237C4—22no—
238C4—24no—
239C4—42no—
240C4—162no—
241C4—162no—
242C4—162no—
243C4—162no—
244C4#286111yescharacteristic
245C4C2×#48811yescharacteristic
246C4#316711yescharacteristic
247C4#316711yescharacteristic
248C2#3327711yescharacteristic
249C2#3370511yescharacteristic
250C2#3370611yescharacteristic
251C2#3373812yes—
252C2#3374712yes—
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#332572
#332422
#132832
#132072
#131502
#131482
C4×#3462