Finite Group Databasebeta

← #128805

Subgroups — #128805

Order 512.

408 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 100 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#21539C211yesmaximal, characteristic
2#12607C211yesmaximal, characteristic
3C2×#1342C211yesmaximal, characteristic
4C2×#1340C211yesmaximal, characteristic
5#10813C211yesmaximal, characteristic
6#7255C211yesmaximal, characteristic
7#7254C211yesmaximal, characteristic
8C2×#470C2211yescharacteristic
9C2×#441C411yescharacteristic, complemented
10C2×C4×M16C2211yescharacteristic
11C4×C22⋊Q8—21no—
12C2×#349C2211yescharacteristic
13C2×#349C2211yescharacteristic
14C2×#347—21no—
15#1969C411yescharacteristic, complemented
16#1941C411yescharacteristic, complemented
17#1933—21no—
18#1894—21no—
19#1809C2211yescharacteristic
20#1776C411yescharacteristic, complemented
21#1763—21no—
22C2×#344—21no—
23C2×#339—21no—
24C2×#338C2211yescharacteristic
25C2×H(3,ℤ/4)C2211yescharacteristic
26#1376—21no—
27#1342C414yescomplemented
28#1340C414yescomplemented
29C22×Q16C2×C411yescharacteristic
30C22×M16—21no—
31C22×M16D411yescharacteristic
32C22×M16C2311yescharacteristic
33C2×C22⋊Q8—21no—
34C2×C4×Q8—21no—
35C2×#117—21no—
36C22×C42D411yescharacteristic
37C2×M32—21no—
38C2×M32—21no—
39#470—21no—
40#470—21no—
41#470C2×C412yes—
42#470C2×C412yes—
43#441—42no—
44C2×#107C2×C411yescharacteristic
45C2×#106C2×C411yescharacteristic
46C2×#106C2×C411yescharacteristic
47C2×#103—21no—
48C2×#103—21no—
49C2×#103—21no—
50C4×M16—42no—
51C2×#97C2×C411yescharacteristic
52C2×#97D411yescharacteristic
53C2×C4×C8C2×C411yescharacteristic
54C2×C4×C8D411yescharacteristic
55#391—21no—
56#389—21no—
57#387—41no—
58#385—41no—
59#380—21no—
60C4×C4⋊C4—21no—
61C4×C4⋊C4—41no—
62C4×C22⋊C4—41no—
63#349—21no—
64#349—21no—
65#349C2×C412yes—
66#349C2×C412yes—
67#347—42no—
68#344—42no—
69#340—21no—
70#340—21no—
71#340—21no—
72#339—42no—
73#338—21no—
74#338C2×C412yes—
75H(3,ℤ/4)—21no—
76H(3,ℤ/4)C2×C412yes—
77C22×Q8—21no—
78C23×C4—21no—
79C23×C4C2×D411yescharacteristic
80C2×Q16—21no—
81C2×Q16—41no—
82C2×Q16—41no—
83C2×Q16C4212yes—
84C2×M16—21no—
85C2×M16—21no—
86C2×M16—22no—
87C2×M16—41no—
88C2×M16—41no—
89C2×M16—41no—
90C2×M16—41no—
91C2×M16—41no—
92C2×M16—41no—
93C2×M16—41no—
94C2×M16—41no—
95C2×M16—41no—
96C2×M16C22×C411yescharacteristic
97C2×M16C22×C411yescharacteristic
98C2×M16#4111yescharacteristic
99C2×M16#4111yescharacteristic
100C22×C8—41no—
101C22×C8C22⋊C411yescharacteristic
102C22×C8C22⋊C411yescharacteristic
103C22×C8C22×C411yescharacteristic
104C22×C8C2×D411yescharacteristic
105C22⋊Q8—22no—
106C22⋊Q8—22no—
107C22⋊Q8—41no—
108C22⋊Q8—41no—
109C4×Q8—42no—
110C4×Q8—42no—
111#117—22no—
112#117—41no—
113#117—41no—
114#117—41no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C22⋊C4—41no—
123C2×C22⋊C4—41no—
124C2×C22⋊C4—41no—
125C2×C42—22no—
126C2×C42—21no—
127C2×C42—41no—
128C2×C42—41no—
129C2×C42—41no—
130C2×C42—41no—
131C2×C42C22⋊C411yescharacteristic
132C2×C42C22⋊C411yescharacteristic
133M32—22no—
134M32—22no—
135M32—41no—
136M32—41no—
137C2×C16—41no—
138C2×C16—41no—
139#107—41no—
140#106—21no—
141#106—41no—
142#106C4212yes—
143#103—22no—
144#103—22no—
145#103—42no—
146#103—42no—
147#97—22no—
148#97—22no—
149C4×C8—41no—
150C4×C8—41no—
151#95—41no—
152#95—41no—
153#95—41no—
154#95—41no—
155#95—41no—
156C24AGL(1,ℤ/8)11yescharacteristic
157C2×Q8—22no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8—41no—
161C2×Q8—41no—
162C2×Q8—41no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4C2×C22⋊C411yescharacteristic
178C22×C4C4×D411yescharacteristic
179C22×C4C4×D411yescharacteristic
180C22×C4AGL(1,ℤ/8)11yescharacteristic
181C22×C4AGL(1,ℤ/8)11yescharacteristic
182C22×C4#13711yescharacteristic
183Q16—42no—
184Q16—42no—
185M16—41no—
186M16—41no—
187M16—41no—
188M16—41no—
189M16—42no—
190M16—42no—
191M16—42no—
192M16—42no—
193M16—81no—
194M16—81no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—42no—
206C2×C8—81no—
207C2×C8—81no—
208C2×C8C2×C4211yescharacteristic
209C2×C8#11711yescharacteristic
210C2×C8C4×D411yescharacteristic
211C2×C8C4×D411yescharacteristic
212C4⋊C4—22no—
213C4⋊C4—42no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—82no—
222C4⋊C4—81no—
223C4⋊C4—82no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—81no—
228C22⋊C4—82no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—42no—
233C42—42no—
234C42—81no—
235C42—82no—
236C42—82no—
237C42#9912yes—
238C16—42no—
239C16—42no—
240C23—21no—
241C23—21no—
242C23—41no—
243C23—41no—
244C23#41811yescharacteristic
245C23#44011yescharacteristic
246C23#44211yescharacteristic
247Q8—42no—
248Q8—42no—
249Q8—81no—
250Q8—81no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—84no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—82no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—161no—
282C2×C4—161no—
283C2×C4C4×C22⋊C411yescharacteristic
284C2×C4C2×#9911yescharacteristic
285C2×C4#41011yescharacteristic
286C2×C4#41711yescharacteristic
287C2×C4#42111yescharacteristic
288C2×C4#42111yescharacteristic
289C2×C4#44011yescharacteristic
290C2×C4#44211yescharacteristic
291C8—21no—
292C8—21no—
293C8—21no—
294C8—21no—
295C8—41no—
296C8—41no—
297C8—82no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—81no—
307C22#151011yescharacteristic
308C22#151411yescharacteristic
309C22#176211yescharacteristic
310C4—21no—
311C4—21no—
312C4—42no—
313C4—42no—
314C4—82no—
315C4—82no—
316C4—161no—
317C4—162no—
318C4—161no—
319C4#151211yescharacteristic
320C4#151311yescharacteristic
321C4C4×#9911yescharacteristic
322C4#175911yescharacteristic
323C2—21no—
324C2—21no—
325C2—41no—
326C2—41no—
327C2#821611yescharacteristic
328C2#1183611yescharacteristic
329C2#1183811yescharacteristic
subgroupG/H
center, socleC22#1514
commutatorC2×C8C2×C42
Frattini, cosocleC22×M16C23
prime radicalC24AGL(1,ℤ/8)
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. C2×#441C4semidirect
15. #1969C4semidirect
16. #1941C4semidirect
20. #1776C4semidirect
27. #1342C4semidirect
28. #1340C4semidirect

Maximal subgroups

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

subgroupindex
#215392
#126072
C2×#13422
C2×#13402
#108132
#72552
#72542