Finite Group Databasebeta

← #508708

Subgroups — #508708

Order 512.

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

normal subgroups only

This group has 492 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#33247C211yesmaximal, characteristic
2#21723C211yesmaximal, characteristic, complemented
3#13530C211yesmaximal, characteristic, complemented
4#13526C211yesmaximal, characteristic
5#13451C212yesmaximal, complemented
6#13408C212yesmaximal
7#13366C211yesmaximal, characteristic, complemented
8#13365C211yesmaximal, characteristic
9#13133C212yesmaximal, complemented
10#13130C212yesmaximal
11#12895C211yesmaximal, characteristic, complemented
12C2×#500C2211yescharacteristic
13C2×#499C2211yescharacteristic
14C2×#495C2211yescharacteristic
15C2×#489—21no—
16#2640—21no—
17#2330—21no—
18C2×#366C2212yes—
19C2×#366C2212yes—
20C2×#358C2212yes—
21C2×#358C2212yes—
22C2×#357C2212yes—
23C2×#357C2212yes—
24#1992—21no—
25#1973—21no—
26#1973—21no—
27#1970C2211yescharacteristic
28#1970C2211yescharacteristic
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×Q16C2311yescharacteristic
44C22×SD16—21no—
45C22×D8C2311yescharacteristic
46C2×C4⋊Q8—21no—
47C2×#125—21no—
48C2×#124—21no—
49C22×C16C2312yes—
50C22×C16C2312yes—
51#500—42no—
52#499—24no—
53#495—42no—
54#489—44no—
55C2×#107—21no—
56C2×#107—21no—
57C2×#107C2312yes—
58C2×#107C2311yescharacteristic
59C2×#107C2311yescharacteristic
60C2×#107C2312yes—
61C2×#106—21no—
62C2×#103—21no—
63C2×#103—21no—
64C2×#102—21no—
65C2×#102—21no—
66C2×#97D411yescharacteristic
67C2×#97D411yescharacteristic
68C2×#97D411yescharacteristic
69C2×C4×C8C2311yescharacteristic
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2311yescharacteristic
72#397—42no—
73#393—42no—
74#391—21no—
75#391—21no—
76#390—21no—
77#390—21no—
78#389—22no—
79#389—42no—
80#387—42no—
81#384—21no—
82C4×C4⋊C4—22no—
83#376D411yescharacteristic
84#366—42no—
85#366—42no—
86#358—24no—
87#358—42no—
88#357—24no—
89#357—42no—
90#340—22no—
91#340—22no—
92C22×Q8—21no—
93C22×Q8—41no—
94C22×D4—21no—
95C2×Q16—22no—
96C2×Q16—22no—
97C2×Q16—21no—
98C2×Q16—21no—
99C2×SD16—42no—
100C2×SD16—42no—
101C2×SD16—42no—
102C2×D8—22no—
103C2×D8—22no—
104C2×D8—21no—
105C2×D8—21no—
106C22×C8C2×D411yescharacteristic
107C22×C8C2×D411yescharacteristic
108C22×C8C2×D411yescharacteristic
109C22×C8C2411yescharacteristic
110C4⋊Q8—42no—
111C4⋊Q8—42no—
112#125—24no—
113#125—42no—
114#124—42no—
115#124—42no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C22⋊C4—41no—
125C2×C22⋊C4—41no—
126C2×C22⋊C4—42no—
127C2×C22⋊C4—42no—
128C2×C42—42no—
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C16—22no—
133C2×C16—22no—
134C2×C16—22no—
135C2×C16—22no—
136C2×C16—22no—
137C2×C16—22no—
138#107—22no—
139#107—22no—
140#107—22no—
141#107—22no—
142#107—22no—
143#107—42no—
144#107—42no—
145#107#4114yes—
146#106—24no—
147#103—42no—
148#103—42no—
149#102—42no—
150#102—42no—
151#97—22no—
152#97—22no—
153#97—41no—
154C4×C8—22no—
155C4×C8—22no—
156C4×C8—21no—
157C4×C8—21no—
158#95—42no—
159#95—42no—
160#95—42no—
161#95—42no—
162#95D812yes—
163#95D812yes—
164C24—41no—
165C2×Q8—41no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8—82no—
172C2×Q8—162no—
173C2×D4—41no—
174C2×D4—41no—
175C2×D4—41no—
176C2×D4—41no—
177C2×D4—41no—
178C2×D4—41no—
179C22×C4—42no—
180C22×C4—41no—
181C22×C4—42no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4C2×D812yes—
185C22×C4C2×D811yescharacteristic
186C22×C4C2×D812yes—
187C22×C4C2×D811yescharacteristic
188C22×C4C22×D411yescharacteristic
189Q16—42no—
190Q16—42no—
191SD16—84no—
192D8—42no—
193D8—42no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8AGL(1,ℤ/8)12yes—
202C2×C8#13712yes—
203C2×C8C2×#4112yes—
204C2×C8H(5,𝔽2)12yes—
205C2×C82−1+411yescharacteristic
206C2×C82−1+411yescharacteristic
207C4⋊C4—24no—
208C4⋊C4—48no—
209C4⋊C4—42no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—42no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C4⋊C4—82no—
217C4⋊C4—162no—
218C4⋊C4—162no—
219C22⋊C4—82no—
220C22⋊C4—82no—
221C22⋊C4—162no—
222C22⋊C4—162no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—162no—
230C16—42no—
231C16—42no—
232C23—81no—
233C23—82no—
234C23—82no—
235C23—81no—
236C23—81no—
237C23C22×D811yescharacteristic
238Q8—81no—
239Q8—81no—
240Q8—81no—
241Q8—81no—
242Q8—164no—
243D4—81no—
244D4—81no—
245D4—81no—
246D4—81no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—44no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—81no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—82no—
269C2×C4—81no—
270C2×C4#52511yescharacteristic
271C2×C4#53511yescharacteristic
272C2×C4#53611yescharacteristic
273C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
274C2×C4C2×#13711yescharacteristic
275C2×C4#57511yescharacteristic
276C8—22no—
277C8—22no—
278C8—41no—
279C8#54212yes—
280C8#54212yes—
281C22—162no—
282C22—161no—
283C22—162no—
284C22—161no—
285C22—161no—
286C22#310011yescharacteristic
287C22#320511yescharacteristic
288C22#322611yescharacteristic
289C22#341512yes—
290C22#341712yes—
291C4—42no—
292C4—42no—
293C4—41no—
294C4—41no—
295C4—84no—
296C4—161no—
297C4—162no—
298C4—161no—
299C4—161no—
300C4#263211yescharacteristic
301C4#311211yescharacteristic
302C4#321511yescharacteristic
303C4#323511yescharacteristic
304C2—321no—
305C2#3005911yescharacteristic
306C2#3368412yes—
307C2#3378611yescharacteristic
308C2#3378911yescharacteristic
309C2#3380912yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21723C2semidirect
3. #13530C2semidirect
5. #13451C2semidirect
7. #13366C2semidirect
9. #13133C2semidirect
11. #12895C2semidirect

Maximal subgroups

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

subgroupindex
#332472
#217232
#135302
#135262
#134512
#134082
#133662
#133652
#131332
#131302
#128952