Finite Group Databasebeta

← #508731

Subgroups — #508731

Order 512.

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

normal subgroups only

This group has 507 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, complemented
2#21696C211yesmaximal, characteristic
3#13530C211yesmaximal, characteristic, complemented
4#13526C211yesmaximal, characteristic
5#13505C212yesmaximal, complemented
6#13483C211yesmaximal, characteristic, complemented
7#13482C211yesmaximal, characteristic
8#13408C212yesmaximal
9#12936C212yesmaximal, complemented
10#12935C212yesmaximal
11#12895C211yesmaximal, characteristic, complemented
12C2×#495C2211yescharacteristic
13C2×#440—21no—
14C2×C4×Q16C2211yescharacteristic
15C2×C4×D8C2211yescharacteristic
16#2647—21no—
17#2326—21no—
18C2×#366C2212yes—
19C2×#366C2212yes—
20C2×#358C2212yes—
21C2×#358C2212yes—
22C2×#357C2212yes—
23C2×#357C2212yes—
24#1973—21no—
25#1970C2211yescharacteristic
26#1969C2211yescharacteristic
27#1959—21no—
28#1944—21no—
29#1943—21no—
30#1942C2212yes—
31#1942C2212yes—
32#1846C2211yescharacteristic
33#1845C2211yescharacteristic
34#1845C2211yescharacteristic
35#1836—21no—
36#1726C2211yescharacteristic
37#1380C2212yes—
38#1380C2212yes—
39#1374C2212yes—
40#1374C2212yes—
41#1374C2211yescharacteristic
42#1374C2211yescharacteristic
43C22×Q16C2311yescharacteristic
44C22×SD16—21no—
45C22×D8C2311yescharacteristic
46C2×#124—21no—
47C2×C4×Q8—21no—
48C2×C4×D4—21no—
49C22×C16C2312yes—
50C22×C16C2312yes—
51#495—22no—
52#495—22no—
53#440—44no—
54C4×Q16—42no—
55C4×D8—42no—
56C2×#107—21no—
57C2×#107—21no—
58C2×#107C2312yes—
59C2×#107C2311yescharacteristic
60C2×#107C2311yescharacteristic
61C2×#107C2312yes—
62C2×#106—21no—
63C2×#103—21no—
64C2×#103—21no—
65C2×#102—21no—
66C2×#102—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#390—21no—
77#389—22no—
78#389—22no—
79#388—42no—
80#385—42no—
81#384—21no—
82#384—21no—
83#382—21no—
84#376D411yescharacteristic
85#366—42no—
86#366—42no—
87#358—24no—
88#358—42no—
89#357—24no—
90#357—42no—
91#340—22no—
92#340—22no—
93C22×Q8—21no—
94C22×D4—21no—
95C23×C4—41no—
96C2×Q16—21no—
97C2×Q16—22no—
98C2×Q16—22no—
99C2×Q16#4112yes—
100C2×SD16—24no—
101C2×SD16—42no—
102C2×SD16—42no—
103C2×D8—21no—
104C2×D8—22no—
105C2×D8—22no—
106C2×D8#4112yes—
107C22×C8C2×D411yescharacteristic
108C22×C8C2×D411yescharacteristic
109C22×C8C2×D411yescharacteristic
110C22×C8C2411yescharacteristic
111#124—24no—
112#124—42no—
113C4×Q8—42no—
114C4×Q8—42no—
115C4×D4—42no—
116C4×D4—42no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—42no—
126C2×C22⋊C4—42no—
127C2×C22⋊C4—41no—
128C2×C22⋊C4—42no—
129C2×C42—41no—
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C16—22no—
134C2×C16—22no—
135C2×C16—22no—
136C2×C16—22no—
137C2×C16—22no—
138C2×C16—22no—
139#107—22no—
140#107—22no—
141#107—22no—
142#107—22no—
143#107—22no—
144#107—22no—
145#107—42no—
146#107—42no—
147#106—42no—
148#103—42no—
149#103—42no—
150#102—42no—
151#102—42no—
152#97—22no—
153#97—22no—
154#97—41no—
155C4×C8—21no—
156C4×C8—21no—
157C4×C8—22no—
158C4×C8—22no—
159#95—42no—
160#95—42no—
161#95—42no—
162#95—42no—
163#95D812yes—
164#95D812yes—
165C24—41no—
166C2×Q8—22no—
167C2×Q8—41no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8—41no—
172C2×D4—22no—
173C2×D4—41no—
174C2×D4—41no—
175C2×D4—41no—
176C2×D4—41no—
177C2×D4—41no—
178C22×C4—41no—
179C22×C4—42no—
180C22×C4—41no—
181C22×C4—42no—
182C22×C4—41no—
183C22×C4—82no—
184C22×C4—162no—
185C22×C4C2×D811yescharacteristic
186C22×C4C2×D812yes—
187C22×C4C2×D812yes—
188C22×C4C2×D811yescharacteristic
189C22×C4C22×D411yescharacteristic
190Q16—24no—
191Q16—42no—
192SD16—48no—
193D8—24no—
194D8—42no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8AGL(1,ℤ/8)12yes—
203C2×C8#13712yes—
204C2×C8C2×#4111yescharacteristic
205C2×C8C2×#4111yescharacteristic
206C2×C8H(5,𝔽2)12yes—
207C2×C82−1+412yes—
208C4⋊C4—42no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—42no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—82no—
217C4⋊C4—162no—
218C4⋊C4—162no—
219C4⋊C4—162no—
220C4⋊C4—162no—
221C22⋊C4—48no—
222C22⋊C4—82no—
223C22⋊C4—162no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—82no—
231C16—42no—
232C16—42no—
233C23—42no—
234C23—82no—
235C23—82no—
236C23—81no—
237C23—81no—
238C23C22×D811yescharacteristic
239Q8—42no—
240Q8—42no—
241Q8—81no—
242Q8—81no—
243D4—42no—
244D4—42no—
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—42no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—82no—
270C2×C4—164no—
271C2×C4#52511yescharacteristic
272C2×C4#53511yescharacteristic
273C2×C4#53611yescharacteristic
274C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
275C2×C4C2×#13711yescharacteristic
276C2×C4#57511yescharacteristic
277C8—22no—
278C8—22no—
279C8—41no—
280C8#53912yes—
281C8#53912yes—
282C22—82no—
283C22—84no—
284C22—162no—
285C22—161no—
286C22—161no—
287C22#310011yescharacteristic
288C22#320511yescharacteristic
289C22#322611yescharacteristic
290C22#341512yes—
291C22#341712yes—
292C4—42no—
293C4—42no—
294C4—41no—
295C4—41no—
296C4—82no—
297C4—162no—
298C4—161no—
299C4—162no—
300C4—161no—
301C4#263311yescharacteristic
302C4#311011yescharacteristic
303C4#321611yescharacteristic
304C4#323411yescharacteristic
305C2—162no—
306C2#3007511yescharacteristic
307C2#3368412yes—
308C2#3378611yescharacteristic
309C2#3378911yescharacteristic
310C2#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
1. #33261C2semidirect
3. #13530C2semidirect
5. #13505C2semidirect
6. #13483C2semidirect
9. #12936C2semidirect
11. #12895C2semidirect

Maximal subgroups

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

subgroupindex
#332612
#216962
#135302
#135262
#135052
#134832
#134822
#134082
#129362
#129352
#128952