Finite Group Databasebeta

← #511516

Subgroups — #511516

Order 512.

461 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 461 conjugacy classes of subgroups — too many to draw. Only its 132 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#28737C211yesmaximal, characteristic
2#28676C211yesmaximal, characteristic
3#13477C212yesmaximal
4#13409C212yesmaximal
5#13364C212yesmaximal
6#13264C211yesmaximal, characteristic
7#13150C212yesmaximal
8#13132C212yesmaximal
9#13111C212yesmaximal
10C2×#500C2211yescharacteristic
11C2×#494C2211yescharacteristic
12C2×C8×Q8C2211yescharacteristic
13#2568—21no—
14#2567—21no—
15C2×#367C2212yes—
16C2×C4⋊C16C2211yescharacteristic
17C2×#358C2212yes—
18C2×#358C2212yes—
19C2×#358C2212yes—
20C2×#358C2212yes—
21C2×#346C2211yescharacteristic
22#2087—21no—
23#2087—21no—
24#2056—21no—
25#2056—21no—
26#1987—21no—
27#1981—21no—
28#1969C2212yes—
29#1969C2212yes—
30#1942C2212yes—
31#1942C2212yes—
32C4×#107C2211yescharacteristic
33C4×#107C2211yescharacteristic
34#1380C2212yes—
35#1380C2212yes—
36#1380C2212yes—
37#1374C2212yes—
38#1374C2212yes—
39C22×Q16—21no—
40C22×Q16C2312yes—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4×Q8D411yescharacteristic
44C22×C16C2312yes—
45C22×C16C2312yes—
46#500—41no—
47#500—41no—
48#494—41no—
49#494—41no—
50C8×Q8—41no—
51C8×Q8—41no—
52C2×#107—21no—
53C2×#107C2312yes—
54C2×#107C2312yes—
55C2×#107C2312yes—
56C2×C4⋊C8D411yescharacteristic
57C2×C4⋊C8D411yescharacteristic
58C2×C4⋊C8D411yescharacteristic
59C2×#103—22no—
60C2×#103—22no—
61C2×C4×C8C2311yescharacteristic
62C2×C4×C8C2311yescharacteristic
63C2×C4×C8C2311yescharacteristic
64#400—41no—
65#400—41no—
66#395—41no—
67#395—41no—
68#391—22no—
69#391—22no—
70#391—41no—
71#391—41no—
72#389—22no—
73#389—22no—
74#384—41no—
75#382—41no—
76C4×C4⋊C4—21no—
77C4×C4⋊C4—21no—
78#367—24no—
79C4⋊C16—41no—
80#358—24no—
81#358—24no—
82#358—42no—
83#358—42no—
84#346—41no—
85#340—21no—
86#340—21no—
87#340—22no—
88#340—22no—
89C22×Q8—22no—
90C22×Q8—41no—
91C22×Q8D811yescharacteristic
92C2×Q16—22no—
93C2×Q16—22no—
94C2×Q16—22no—
95C2×Q16—22no—
96C2×Q16—22no—
97C2×Q16—22no—
98C2×Q16—42no—
99C2×Q16—82no—
100C22×C8C2×D411yescharacteristic
101C22×C8C2×D411yescharacteristic
102C22×C8C2×D411yescharacteristic
103C22×C8C2411yescharacteristic
104C4⋊Q8—42no—
105C4⋊Q8—42no—
106C4⋊Q8—42no—
107C4⋊Q8—42no—
108C4×Q8—41no—
109C4×Q8—41no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4D811yescharacteristic
119C2×C4⋊C4D811yescharacteristic
120C2×C4⋊C4D811yescharacteristic
121C2×C42—41no—
122C2×C42—41no—
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D411yescharacteristic
126C2×C16—22no—
127C2×C16—22no—
128C2×C16—22no—
129C2×C16—22no—
130C2×C16—22no—
131C2×C16—22no—
132#107—22no—
133#107—22no—
134#107—22no—
135#107—22no—
136#107—22no—
137#107—42no—
138#107#4114yes—
139C4⋊C8—41no—
140C4⋊C8—41no—
141C4⋊C8—41no—
142#103—82no—
143#103—82no—
144C4×C8—21no—
145C4×C8—21no—
146C4×C8—21no—
147C4×C8—21no—
148C4×C8—21no—
149C4×C8—21no—
150#95—41no—
151#95—41no—
152#95—42no—
153#95—42no—
154#95—42no—
155#95—42no—
156#95—42no—
157#95—42no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8—41no—
161C2×Q8—42no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—82no—
168C2×Q8—162no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4C2×D811yescharacteristic
174C22×C4C2×D811yescharacteristic
175C22×C4C2×D811yescharacteristic
176C22×C4C2×D811yescharacteristic
177C22×C4C2×D811yescharacteristic
178C22×C4C2×D811yescharacteristic
179C22×C4C22×D411yescharacteristic
180Q16—42no—
181Q16—42no—
182Q16—42no—
183Q16—42no—
184Q16—84no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—22no—
189C2×C8—22no—
190C2×C8—22no—
191C2×C8C2×#4112yes—
192C2×C8H(5,𝔽2)11yescharacteristic
193C2×C8H(5,𝔽2)11yescharacteristic
194C2×C82−1+412yes—
195C4⋊C4—24no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—82no—
205C4⋊C4—82no—
206C4⋊C4—82no—
207C4⋊C4—162no—
208C4⋊C4—162no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—82no—
216C42—82no—
217C16—42no—
218C16—42no—
219C23C22×D811yescharacteristic
220Q8—44no—
221Q8—82no—
222Q8—82no—
223Q8—82no—
224Q8—82no—
225Q8—164no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—44no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4#51012yes—
252C2×C4#51012yes—
253C2×C4#53411yescharacteristic
254C2×C4#54812yes—
255C2×C4#57511yescharacteristic
256C2×C4#57712yes—
257C8—41no—
258C8—41no—
259C8—41no—
260C8#54212yes—
261C8#55511yescharacteristic
262C8#56311yescharacteristic
263C22#318611yescharacteristic
264C22#335812yes—
265C22C2×#51011yescharacteristic
266C22C2×#51011yescharacteristic
267C22#341612yes—
268C4—22no—
269C4—22no—
270C4—41no—
271C4—41no—
272C4—41no—
273C4—41no—
274C4—84no—
275C4—162no—
276C4—162no—
277C4—162no—
278C4#273111yescharacteristic
279C4#320011yescharacteristic
280C4#337112yes—
281C2#3262811yescharacteristic
282C2#3376111yescharacteristic
283C2#3376611yescharacteristic
284C2#3393612yes—
285C2#3393912yes—
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ 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
#287372
#286762
#134772
#134092
#133642
#132642
#131502
#131322
#131112