Finite Group Databasebeta

← #316186

Subgroups — #316186

Order 512.

451 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 451 conjugacy classes of subgroups — too many to draw. Only its 130 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#33215C211yesmaximal, characteristic
2#20768C211yesmaximal, characteristic
3#10636C212yesmaximal, complemented
4#10636C212yesmaximal, complemented
5#9778C212yesmaximal, complemented
6#9193C211yesmaximal, characteristic
7#9193C211yesmaximal, characteristic
8#9191C211yesmaximal, characteristic
9#9191C211yesmaximal, characteristic
10#8795C212yesmaximal, complemented
11#8157C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13#2905C2211yescharacteristic
14#2905C2211yescharacteristic
15C2×#405C2211yescharacteristic
16#2628C2211yescharacteristic
17#2628C2211yescharacteristic
18#1993C2211yescharacteristic
19#1993C2211yescharacteristic
20#1944—21no—
21#1942—21no—
22#1846C2211yescharacteristic
23#1838—21no—
24#1838C2212yes—
25#1837—21no—
26#1837C2212yes—
27#1809—21no—
28#1808—21no—
29#1738—21no—
30C2×#341—21no—
31#1549C2212yes—
32#1549C2212yes—
33#1548C2212yes—
34#1548C2212yes—
35#1458C2211yescharacteristic
36#1458C2211yescharacteristic
37#1452C2212yes—
38#1286C2212yes—
39#1286C2212yes—
40#1286C2212yes—
41#1286C2212yes—
42#1281C2212yes—
43C22×M16—21no—
44C22×M16—21no—
45C22×M16D411yescharacteristic
46C23×C8D411yescharacteristic
47C2×C4⋊Q8C2311yescharacteristic
48C2×C4⋊Q8C2311yescharacteristic
49C2×#117C2311yescharacteristic
50#500—24no—
51#431C2312yes—
52#431C2312yes—
53C2×#107—21no—
54C2×#107—21no—
55C2×#107—21no—
56C2×#106—21no—
57#424—42no—
58#424—42no—
59#423—21no—
60#423—21no—
61#423C2312yes—
62#423C2312yes—
63#405—24no—
64C2×#97D411yescharacteristic
65C2×#97C2311yescharacteristic
66C2×C4×C8D411yescharacteristic
67C2×C4×C8C2311yescharacteristic
68#400—21no—
69#400—21no—
70#399—21no—
71#399—21no—
72#395—21no—
73#395—21no—
74#393—21no—
75#393—21no—
76#389—21no—
77#389—21no—
78#382C2311yescharacteristic
79#382C2311yescharacteristic
80#380—21no—
81#380—21no—
82#341—22no—
83#341—22no—
84#341—22no—
85#341—22no—
86#340—21no—
87#340—21no—
88#340—21no—
89#340—21no—
90#333—42no—
91#333—42no—
92#332—42no—
93#332—42no—
94#324—21no—
95#324—21no—
96#323—21no—
97#323—21no—
98#322—24no—
99C22×Q8—21no—
100C22×Q8—21no—
101C23×C4C2×D411yescharacteristic
102C2×M16—22no—
103C2×M16—22no—
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—41no—
107C2×M16—41no—
108C2×M16—41no—
109C2×M16—42no—
110C2×M16—42no—
111C2×M16—42no—
112C2×M16—42no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—41no—
116C22×C8—41no—
117C22×C8—41no—
118C22×C8C2×D411yescharacteristic
119C22×C8C2×D411yescharacteristic
120C22×C8C2×D411yescharacteristic
121C22×C8C2×D411yescharacteristic
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126#117—22no—
127#117—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4#4111yescharacteristic
137C2×C4⋊C4#4111yescharacteristic
138C2×C22⋊C4—41no—
139C2×C22⋊C4—41no—
140C2×C22⋊C4#4111yescharacteristic
141C2×C22⋊C4#4111yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2411yescharacteristic
144#107—42no—
145#107—42no—
146#107—42no—
147#106—42no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—42no—
155C4⋊C8—42no—
156C22⋊C8—22no—
157C22⋊C8—22no—
158#97—22no—
159#97—22no—
160C4×C8—22no—
161C4×C8—22no—
162C4×C8—22no—
163C4×C8#4114yes—
164#95—21no—
165#95—21no—
166#95—21no—
167#95—21no—
168#95—41no—
169#95—41no—
170#95—41no—
171#95—41no—
172C24#12411yescharacteristic
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—161no—
176C2×Q8—161no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4#12411yescharacteristic
186C22×C4#12411yescharacteristic
187C22×C4#12411yescharacteristic
188C22×C4#13712yes—
189C22×C4#13712yes—
190C22×C4C22×D411yescharacteristic
191C22×C4C2×#4111yescharacteristic
192C22×C4C2×#4111yescharacteristic
193M16—44no—
194M16—42no—
195M16—42no—
196M16—42no—
197M16—42no—
198M16—82no—
199M16—82no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—44no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—82no—
216C2×C8—82no—
217C2×C8#12312yes—
218C2×C8#12312yes—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—161no—
232C4⋊C4—161no—
233C22⋊C4—22no—
234C22⋊C4—22no—
235C22⋊C4—161no—
236C22⋊C4—161no—
237C42—21no—
238C42—21no—
239C42C2×#4112yes—
240C42H(5,𝔽2)12yes—
241C23—21no—
242C23—21no—
243C23—41no—
244C23—41no—
245C23#49112yes—
246C23C2×#12411yescharacteristic
247Q8—162no—
248Q8—162no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4#49112yes—
274C2×C4C2×#12311yescharacteristic
275C2×C4#53511yescharacteristic
276C2×C4#55312yes—
277C2×C4#55312yes—
278C2×C4C2×#13711yescharacteristic
279C2×C4C2×#13711yescharacteristic
280C2×C4#57511yescharacteristic
281C2×C4#57511yescharacteristic
282C8—22no—
283C8—22no—
284C8—41no—
285C8—41no—
286C8—82no—
287C8—82no—
288C8—82no—
289C8—82no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—42no—
294C22—41no—
295C22—41no—
296C22—81no—
297C22#250411yescharacteristic
298C22C2×#49111yescharacteristic
299C22#314211yescharacteristic
300C4—22no—
301C4—22no—
302C4—22no—
303C4—41no—
304C4—41no—
305C4—161no—
306C4—161no—
307C4—161no—
308C4—161no—
309C4#309511yescharacteristic
310C4#309511yescharacteristic
311C4#321511yescharacteristic
312C4#321511yescharacteristic
313C2—21no—
314C2—21no—
315C2—42no—
316C2#2798311yescharacteristic
317C2#2798811yescharacteristic
318C2#2798911yescharacteristic
subgroupG/H
center, socleC22#2504
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#124
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #10636C2semidirect
4. #10636C2semidirect
5. #9778C2semidirect
10. #8795C2semidirect

Maximal subgroups

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

subgroupindex
#332152
#207682
#106362
#97782
#91932
#91912
#87952
#81572