Finite Group Databasebeta

← #508728

Subgroups — #508728

Order 512.

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

normal subgroups only

This group has 483 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#33261C211yesmaximal, characteristic
2#21698C211yesmaximal, characteristic
3#13531C211yesmaximal, characteristic
4#13530C211yesmaximal, characteristic
5#13507C212yesmaximal
6#13483C211yesmaximal, characteristic
7#13483C211yesmaximal, characteristic
8#13409C212yesmaximal
9#12938C212yesmaximal
10#12936C212yesmaximal
11#12895C211yesmaximal, characteristic
12C2×#494C2211yescharacteristic
13C2×#441—21no—
14C2×C4×Q16C2211yescharacteristic
15C2×C4×Q16C2211yescharacteristic
16#2647—21no—
17#2327—21no—
18C2×#367C2212yes—
19C2×#366C2212yes—
20C2×#358C2212yes—
21C2×#358C2212yes—
22C2×#358C2212yes—
23C2×#358C2212yes—
24#1971—21no—
25#1969C2211yescharacteristic
26#1969C2211yescharacteristic
27#1960—21no—
28#1944—21no—
29#1943—21no—
30#1942C2212yes—
31#1942C2212yes—
32#1846C2211yescharacteristic
33#1846C2211yescharacteristic
34#1845C2211yescharacteristic
35#1836—21no—
36#1726C2211yescharacteristic
37#1380C2212yes—
38#1380C2212yes—
39#1374C2212yes—
40#1374C2212yes—
41#1374C2211yescharacteristic
42#1374C2211yescharacteristic
43C22×Q16—21no—
44C22×Q16C2311yescharacteristic
45C22×Q16C2311yescharacteristic
46C2×C4⋊Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49C22×C16C2312yes—
50C22×C16C2312yes—
51#494—22no—
52#494—22no—
53#441—44no—
54C4×Q16—42no—
55C4×Q16—42no—
56C2×#107—21no—
57C2×#107C2312yes—
58C2×#107C2312yes—
59C2×#107C2311yescharacteristic
60C2×#107C2311yescharacteristic
61C2×#106—21no—
62C2×#106—21no—
63C2×#103—21no—
64C2×#103—21no—
65C2×#103—21no—
66C2×#103—21no—
67C2×#97D411yescharacteristic
68C2×#97D411yescharacteristic
69C2×#97D411yescharacteristic
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73#400—42no—
74#398—42no—
75#391—21no—
76#391—21no—
77#389—22no—
78#389—22no—
79#387—42no—
80#387—42no—
81#384—21no—
82#382—21no—
83#382—21no—
84#376D411yescharacteristic
85#367—42no—
86#366—42no—
87#358—24no—
88#358—24no—
89#358—42no—
90#358—42no—
91#340—22no—
92#340—22no—
93C22×Q8—21no—
94C22×Q8—21no—
95C2×Q16—24no—
96C2×Q16—21no—
97C2×Q16—21no—
98C2×Q16—22no—
99C2×Q16—22no—
100C2×Q16—22no—
101C2×Q16—22no—
102C2×Q16—42no—
103C2×Q16—42no—
104C2×Q16#4112yes—
105C2×Q16#4112yes—
106C22×C8C2×D411yescharacteristic
107C22×C8C2×D411yescharacteristic
108C22×C8C2×D411yescharacteristic
109C22×C8C2411yescharacteristic
110C4⋊Q8—24no—
111C4⋊Q8—42no—
112C4×Q8—42no—
113C4×Q8—42no—
114C4×Q8—42no—
115C4×Q8—42no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—42no—
126C2×C42—41no—
127C2×C42—41no—
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C16—22no—
132C2×C16—22no—
133C2×C16—22no—
134C2×C16—22no—
135C2×C16—22no—
136C2×C16—22no—
137#107—22no—
138#107—22no—
139#107—22no—
140#107—22no—
141#107—22no—
142#107—22no—
143#107—42no—
144#106—42no—
145#106—42no—
146#103—42no—
147#103—42no—
148#103—42no—
149#103—42no—
150#97—22no—
151#97—22no—
152#97—41no—
153C4×C8—21no—
154C4×C8—21no—
155C4×C8—22no—
156C4×C8—22no—
157#95—42no—
158#95—42no—
159#95—42no—
160#95—42no—
161#95—42no—
162#95—42no—
163#95D812yes—
164#95D812yes—
165C2×Q8—22no—
166C2×Q8—22no—
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—
176C2×Q8—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4C2×D812yes—
184C22×C4C2×D811yescharacteristic
185C22×C4C2×D812yes—
186C22×C4C2×D811yescharacteristic
187C22×C4C22×D411yescharacteristic
188Q16—24no—
189Q16—24no—
190Q16—48no—
191Q16—42no—
192Q16—42no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8AGL(1,ℤ/8)12yes—
201C2×C8#13712yes—
202C2×C8C2×#4111yescharacteristic
203C2×C8C2×#4111yescharacteristic
204C2×C8H(5,𝔽2)12yes—
205C2×C82−1+412yes—
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—42no—
215C4⋊C4—82no—
216C4⋊C4—82no—
217C4⋊C4—162no—
218C4⋊C4—162no—
219C4⋊C4—162no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—82no—
227C42—82no—
228C16—42no—
229C16—42no—
230C23C22×D811yescharacteristic
231Q8—42no—
232Q8—42no—
233Q8—42no—
234Q8—42no—
235Q8—81no—
236Q8—81no—
237Q8—81no—
238Q8—81no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—82no—
265C2×C4#52511yescharacteristic
266C2×C4#53511yescharacteristic
267C2×C4#53611yescharacteristic
268C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
269C2×C4C2×#13711yescharacteristic
270C2×C4#57511yescharacteristic
271C8—22no—
272C8—22no—
273C8—41no—
274C8#53912yes—
275C8#53912yes—
276C22#310011yescharacteristic
277C22#320511yescharacteristic
278C22#322611yescharacteristic
279C22#341612yes—
280C22#341712yes—
281C4—42no—
282C4—42no—
283C4—41no—
284C4—41no—
285C4—82no—
286C4—82no—
287C4—162no—
288C4—162no—
289C4—161no—
290C4—161no—
291C4#263311yescharacteristic
292C4#311011yescharacteristic
293C4#321611yescharacteristic
294C4#323411yescharacteristic
295C2#3007511yescharacteristic
296C2#3368112yes—
297C2#3378711yescharacteristic
298C2#3378911yescharacteristic
299C2#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
#332612
#216982
#135312
#135302
#135072
#134832
#134092
#129382
#129362
#128952