Finite Group Databasebeta

← #316176

Subgroups — #316176

Order 512.

436 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 126 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#33213C211yesmaximal, characteristic
2#20829C211yesmaximal, characteristic
3#10633C212yesmaximal, complemented
4#10633C212yesmaximal, complemented
5#9776C211yesmaximal, characteristic, complemented
6#9776C211yesmaximal, characteristic, complemented
7#9194C212yesmaximal
8#9192C212yesmaximal
9#8795C211yesmaximal, characteristic, complemented
10#8795C211yesmaximal, characteristic, complemented
11#8157C211yesmaximal, characteristic
12C2×#498C2211yescharacteristic
13#2921C2211yescharacteristic
14#2921C2211yescharacteristic
15C2×#405C2211yescharacteristic
16#2628C2212yes—
17#1993C2212yes—
18#1943—21no—
19#1941—21no—
20#1849C2211yescharacteristic
21#1849C2211yescharacteristic
22#1846C2211yescharacteristic
23#1837C2211yescharacteristic
24#1837C2211yescharacteristic
25#1811—21no—
26#1810—21no—
27#1809—21no—
28#1807—21no—
29#1778—21no—
30C4×#108—21no—
31#1549C2212yes—
32#1549C2212yes—
33#1548C2212yes—
34#1548C2212yes—
35#1458C2212yes—
36#1452C2211yescharacteristic
37#1452C2211yescharacteristic
38#1286C2212yes—
39#1286C2212yes—
40#1286C2212yes—
41#1286C2212yes—
42#1281C2212yes—
43C22×M16D411yescharacteristic
44C23×C8D411yescharacteristic
45C2×C4⋊Q8C2312yes—
46C2×#117C2311yescharacteristic
47#498—22no—
48#498—22no—
49#433—42no—
50#433—42no—
51#431C2312yes—
52#431C2312yes—
53C2×#108—21no—
54C2×#108—21no—
55C2×#108—21no—
56C2×#108—21no—
57C2×#107—21no—
58C2×#106—21no—
59C2×#106—21no—
60C2×#106—21no—
61#423C2311yescharacteristic
62#423C2311yescharacteristic
63#423C2311yescharacteristic
64#423C2311yescharacteristic
65#407—21no—
66#407—21no—
67#405—22no—
68#405—22no—
69C4×M16—21no—
70C4×M16—21no—
71C2×#97D411yescharacteristic
72C2×#97C2311yescharacteristic
73C2×C4×C8D411yescharacteristic
74C2×C4×C8C2311yescharacteristic
75#400—22no—
76#399—22no—
77#395—22no—
78#393—22no—
79#389—22no—
80#382C2312yes—
81#380—22no—
82#341—21no—
83#341—21no—
84#341—21no—
85#341—21no—
86#340—22no—
87#340—22no—
88#333—42no—
89#333—42no—
90#332—42no—
91#332—42no—
92#324—22no—
93#323—22no—
94#322—22no—
95#322—22no—
96C22×Q8—22no—
97C23×C4C2×D411yescharacteristic
98C2×M16—22no—
99C2×M16—22no—
100C2×M16—21no—
101C2×M16—21no—
102C2×M16—21no—
103C2×M16—21no—
104C2×M16—41no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—41no—
108C22×C8C2×D411yescharacteristic
109C22×C8C2×D411yescharacteristic
110C22×C8C2×D411yescharacteristic
111C22×C8C2×D411yescharacteristic
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116#117—21no—
117#117—21no—
118#117—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4#4112yes—
124C2×C22⋊C4—42no—
125C2×C22⋊C4#4112yes—
126C2×C42C2×D411yescharacteristic
127C2×C42C2411yescharacteristic
128#108—42no—
129#108—42no—
130#108—42no—
131#108—42no—
132#107—42no—
133#106—42no—
134#106—42no—
135#106—42no—
136C4⋊C8—22no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—22no—
140C4⋊C8—22no—
141C4⋊C8—22no—
142C22⋊C8—22no—
143C22⋊C8—22no—
144C22⋊C8—42no—
145C22⋊C8—42no—
146#97—22no—
147#97—21no—
148#97—21no—
149#97—41no—
150#97—41no—
151C4×C8—22no—
152C4×C8—22no—
153C4×C8—22no—
154C4×C8—41no—
155C4×C8—41no—
156C4×C8#4112yes—
157C4×C8#4112yes—
158#95—22no—
159#95—22no—
160#95—42no—
161#95—42no—
162C24#12411yescharacteristic
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—162no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4#12411yescharacteristic
175C22×C4#12412yes—
176C22×C4C22×D411yescharacteristic
177C22×C4C2×#4112yes—
178M16—44no—
179M16—41no—
180M16—41no—
181M16—41no—
182M16—41no—
183M16—41no—
184M16—41no—
185M16—41no—
186M16—41no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—44no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8#12312yes—
203C2×C8#12312yes—
204C2×C8#13712yes—
205C2×C8#13712yes—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—162no—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—162no—
218C42—21no—
219C42—21no—
220C42C2×#4111yescharacteristic
221C42C2×#4111yescharacteristic
222C42H(5,𝔽2)12yes—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—42no—
227C23C2×#12411yescharacteristic
228Q8—162no—
229Q8—162no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4C2×#12311yescharacteristic
247C2×C4#53511yescharacteristic
248C2×C4#55312yes—
249C2×C4#55312yes—
250C2×C4C2×#13711yescharacteristic
251C2×C4C2×#13711yescharacteristic
252C2×C4#57511yescharacteristic
253C2×C4#57511yescharacteristic
254C8—22no—
255C8—22no—
256C8—22no—
257C8—22no—
258C8—82no—
259C8—82no—
260C8—81no—
261C8—81no—
262C8—81no—
263C8—81no—
264C22—22no—
265C22—42no—
266C22—41no—
267C22—42no—
268C22—81no—
269C22#250411yescharacteristic
270C22#313511yescharacteristic
271C22#313911yescharacteristic
272C4—22no—
273C4—22no—
274C4—41no—
275C4—42no—
276C4—162no—
277C4—162no—
278C4#309511yescharacteristic
279C4#309511yescharacteristic
280C4#321611yescharacteristic
281C4#321611yescharacteristic
282C2—21no—
283C2—21no—
284C2—81no—
285C2#2798011yescharacteristic
286C2#2798712yes—
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. #10633C2semidirect
4. #10633C2semidirect
5. #9776C2semidirect
6. #9776C2semidirect
9. #8795C2semidirect
10. #8795C2semidirect

Maximal subgroups

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

subgroupindex
#332132
#208292
#106332
#97762
#91942
#91922
#87952
#81572