Finite Group Databasebeta

← #508709

Subgroups — #508709

Order 512.

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

normal subgroups only

This group has 471 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#33249C211yesmaximal, characteristic
2#21723C211yesmaximal, characteristic
3#13531C211yesmaximal, characteristic
4#13530C211yesmaximal, characteristic
5#13453C212yesmaximal
6#13409C212yesmaximal
7#13366C211yesmaximal, characteristic
8#13366C211yesmaximal, characteristic
9#13133C212yesmaximal
10#13132C212yesmaximal
11#12895C211yesmaximal, characteristic
12C2×#499C2211yescharacteristic
13C2×#499C2211yescharacteristic
14C2×#494C2211yescharacteristic
15C2×#491—21no—
16#2642—21no—
17#2330—21no—
18C2×#367C2212yes—
19C2×#366C2212yes—
20C2×#358C2212yes—
21C2×#358C2212yes—
22C2×#358C2212yes—
23C2×#358C2212yes—
24#1992—21no—
25#1971—21no—
26#1971—21no—
27#1969C2211yescharacteristic
28#1969C2211yescharacteristic
29#1969C2211yescharacteristic
30#1969C2211yescharacteristic
31#1944—21no—
32#1942C2212yes—
33#1845C2211yescharacteristic
34#1776—21no—
35C4×#107C2212yes—
36#1726C2211yescharacteristic
37#1380C2212yes—
38#1380C2212yes—
39#1374C2211yescharacteristic
40#1374C2211yescharacteristic
41#1374C2212yes—
42#1374C2212yes—
43C22×Q16—21no—
44C22×Q16C2311yescharacteristic
45C22×Q16C2311yescharacteristic
46C2×C4⋊Q8—21no—
47C2×#125—21no—
48C2×#125—21no—
49C22×C16C2312yes—
50C22×C16C2312yes—
51#499—24no—
52#499—42no—
53#494—42no—
54#491—44no—
55C2×#107—21no—
56C2×#107C2312yes—
57C2×#107C2311yescharacteristic
58C2×#107C2311yescharacteristic
59C2×#107C2312yes—
60C2×#106—21no—
61C2×#106—21no—
62C2×#103—21no—
63C2×#103—21no—
64C2×#103—21no—
65C2×#103—21no—
66C2×#97D411yescharacteristic
67C2×#97D411yescharacteristic
68C2×#97D411yescharacteristic
69C2×C4×C8C2311yescharacteristic
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2311yescharacteristic
72#400—42no—
73#400—42no—
74#391—21no—
75#391—21no—
76#391—21no—
77#391—21no—
78#389—22no—
79#389—42no—
80#387—42no—
81#384—21no—
82C4×C4⋊C4—22no—
83#376D411yescharacteristic
84#367—42no—
85#366—42no—
86#358—24no—
87#358—24no—
88#358—42no—
89#358—42no—
90#340—22no—
91#340—22no—
92C22×Q8—21no—
93C22×Q8—21no—
94C2×Q16—22no—
95C2×Q16—22no—
96C2×Q16—22no—
97C2×Q16—22no—
98C2×Q16—21no—
99C2×Q16—21no—
100C2×Q16—21no—
101C2×Q16—21no—
102C2×Q16—42no—
103C2×Q16—42no—
104C2×Q16—42no—
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C22×C8C2×D411yescharacteristic
108C22×C8C2411yescharacteristic
109C4⋊Q8—42no—
110C4⋊Q8—42no—
111#125—24no—
112#125—42no—
113#125—42no—
114#125—42no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—42no—
126C2×C42—42no—
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C16—22no—
131C2×C16—22no—
132C2×C16—22no—
133C2×C16—22no—
134C2×C16—22no—
135C2×C16—22no—
136#107—22no—
137#107—22no—
138#107—22no—
139#107—22no—
140#107—22no—
141#107—42no—
142#107#4114yes—
143#106—24no—
144#106—42no—
145#103—42no—
146#103—42no—
147#103—42no—
148#103—42no—
149#97—22no—
150#97—22no—
151#97—41no—
152C4×C8—22no—
153C4×C8—22no—
154C4×C8—21no—
155C4×C8—21no—
156#95—42no—
157#95—42no—
158#95—42no—
159#95—42no—
160#95—42no—
161#95—42no—
162#95D812yes—
163#95D812yes—
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—41no—
176C22×C4—42no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—42no—
182C22×C4C2×D812yes—
183C22×C4C2×D812yes—
184C22×C4C2×D811yescharacteristic
185C22×C4C2×D811yescharacteristic
186C22×C4C22×D411yescharacteristic
187Q16—42no—
188Q16—42no—
189Q16—42no—
190Q16—42no—
191Q16—84no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8AGL(1,ℤ/8)12yes—
200C2×C8#13712yes—
201C2×C8C2×#4112yes—
202C2×C8H(5,𝔽2)12yes—
203C2×C82−1+411yescharacteristic
204C2×C82−1+411yescharacteristic
205C4⋊C4—24no—
206C4⋊C4—48no—
207C4⋊C4—42no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—82no—
215C4⋊C4—82no—
216C4⋊C4—82no—
217C4⋊C4—82no—
218C4⋊C4—162no—
219C4⋊C4—162no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—162no—
227C16—42no—
228C16—42no—
229C23C22×D811yescharacteristic
230Q8—81no—
231Q8—81no—
232Q8—81no—
233Q8—81no—
234Q8—81no—
235Q8—81no—
236Q8—81no—
237Q8—81no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—44no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—82no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4#52511yescharacteristic
265C2×C4#53511yescharacteristic
266C2×C4#53611yescharacteristic
267C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
268C2×C4C2×#13711yescharacteristic
269C2×C4#57511yescharacteristic
270C8—22no—
271C8—22no—
272C8—41no—
273C8#54212yes—
274C8#54212yes—
275C22#310011yescharacteristic
276C22#320511yescharacteristic
277C22#322611yescharacteristic
278C22#341612yes—
279C22#341712yes—
280C4—42no—
281C4—42no—
282C4—41no—
283C4—41no—
284C4—84no—
285C4—161no—
286C4—161no—
287C4—162no—
288C4—161no—
289C4—161no—
290C4#263211yescharacteristic
291C4#311211yescharacteristic
292C4#321511yescharacteristic
293C4#323511yescharacteristic
294C2#3005911yescharacteristic
295C2#3368012yes—
296C2#3378711yescharacteristic
297C2#3378911yescharacteristic
298C2#3380812yes—
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ #1726 ◁ 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
#332492
#217232
#135312
#135302
#134532
#134092
#133662
#131332
#131322
#128952