Finite Group Databasebeta

← #316177

Subgroups — #316177

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#33215C211yesmaximal, characteristic
2#20829C211yesmaximal, characteristic
3#10633C212yesmaximal, complemented
4#10633C212yesmaximal, complemented
5#9778C212yesmaximal, complemented
6#9194C212yesmaximal
7#9192C212yesmaximal
8#8795C212yesmaximal, complemented
9#8157C211yesmaximal, characteristic
10C2×#500C2211yescharacteristic
11#2921C2212yes—
12C2×#405C2211yescharacteristic
13#2628C2212yes—
14#1993C2212yes—
15#1944—21no—
16#1942—21no—
17#1849C2211yescharacteristic
18#1849C2211yescharacteristic
19#1846C2211yescharacteristic
20#1837C2212yes—
21#1811—21no—
22#1810—21no—
23#1809—21no—
24#1808—21no—
25#1778—21no—
26C4×#108—21no—
27#1549C2212yes—
28#1549C2212yes—
29#1548C2212yes—
30#1548C2212yes—
31#1458C2211yescharacteristic
32#1458C2211yescharacteristic
33#1452C2212yes—
34#1286C2212yes—
35#1286C2212yes—
36#1286C2212yes—
37#1286C2212yes—
38#1281C2212yes—
39C22×M16D411yescharacteristic
40C23×C8D411yescharacteristic
41C2×C4⋊Q8C2312yes—
42C2×#117C2311yescharacteristic
43#500—24no—
44#433—42no—
45#433—42no—
46#431C2312yes—
47#431C2312yes—
48C2×#108—21no—
49C2×#108—21no—
50C2×#108—21no—
51C2×#108—21no—
52C2×#107—21no—
53C2×#107—21no—
54C2×#107—21no—
55C2×#106—21no—
56#423C2312yes—
57#423C2312yes—
58#407—22no—
59#405—24no—
60C4×M16—22no—
61C2×#97D411yescharacteristic
62C2×#97C2311yescharacteristic
63C2×C4×C8D411yescharacteristic
64C2×C4×C8C2311yescharacteristic
65#400—22no—
66#399—22no—
67#395—22no—
68#393—22no—
69#389—22no—
70#382C2312yes—
71#380—22no—
72#341—22no—
73#341—22no—
74#340—22no—
75#340—22no—
76#333—42no—
77#333—42no—
78#332—42no—
79#332—42no—
80#324—21no—
81#324—21no—
82#323—21no—
83#323—21no—
84#322—24no—
85C22×Q8—22no—
86C23×C4C2×D411yescharacteristic
87C2×M16—22no—
88C2×M16—22no—
89C2×M16—22no—
90C2×M16—22no—
91C2×M16—41no—
92C22×C8—22no—
93C22×C8—22no—
94C22×C8—41no—
95C22×C8C2×D411yescharacteristic
96C22×C8C2×D411yescharacteristic
97C22×C8C2×D411yescharacteristic
98C22×C8C2×D411yescharacteristic
99C4⋊Q8—22no—
100C4⋊Q8—22no—
101C4⋊Q8—22no—
102C4⋊Q8—22no—
103#117—22no—
104#117—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4#4111yescharacteristic
110C2×C4⋊C4#4111yescharacteristic
111C2×C22⋊C4—42no—
112C2×C22⋊C4#4111yescharacteristic
113C2×C22⋊C4#4111yescharacteristic
114C2×C42C2×D411yescharacteristic
115C2×C42C2411yescharacteristic
116#108—42no—
117#108—42no—
118#108—42no—
119#108—42no—
120#107—42no—
121#107—42no—
122#107—42no—
123#106—42no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—44no—
133#97—22no—
134#97—22no—
135#97—42no—
136C4×C8—22no—
137C4×C8—22no—
138C4×C8—22no—
139C4×C8—42no—
140C4×C8#4114yes—
141#95—22no—
142#95—22no—
143#95—42no—
144#95—42no—
145C24#12411yescharacteristic
146C2×Q8—42no—
147C2×Q8—42no—
148C2×Q8—162no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4#12411yescharacteristic
158C22×C4#12411yescharacteristic
159C22×C4#12411yescharacteristic
160C22×C4C22×D411yescharacteristic
161C22×C4C2×#4111yescharacteristic
162C22×C4C2×#4111yescharacteristic
163M16—44no—
164M16—42no—
165M16—42no—
166M16—42no—
167M16—42no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—44no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8#12312yes—
182C2×C8#12312yes—
183C2×C8#13712yes—
184C2×C8#13712yes—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—162no—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C22⋊C4—162no—
197C42—21no—
198C42—21no—
199C42C2×#4112yes—
200C42H(5,𝔽2)12yes—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—41no—
205C23—41no—
206C23C2×#12411yescharacteristic
207Q8—162no—
208Q8—162no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4C2×#12311yescharacteristic
229C2×C4#53511yescharacteristic
230C2×C4#55312yes—
231C2×C4#55312yes—
232C2×C4C2×#13711yescharacteristic
233C2×C4C2×#13711yescharacteristic
234C2×C4#57511yescharacteristic
235C2×C4#57511yescharacteristic
236C8—22no—
237C8—22no—
238C8—22no—
239C8—22no—
240C8—82no—
241C8—82no—
242C8—82no—
243C8—82no—
244C22—21no—
245C22—21no—
246C22—42no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22—81no—
251C22#250411yescharacteristic
252C22#313511yescharacteristic
253C22#313911yescharacteristic
254C4—22no—
255C4—22no—
256C4—41no—
257C4—41no—
258C4—41no—
259C4—162no—
260C4—162no—
261C4#309511yescharacteristic
262C4#309511yescharacteristic
263C4#321611yescharacteristic
264C4#321611yescharacteristic
265C2—21no—
266C2—21no—
267C2—81no—
268C2#2798011yescharacteristic
269C2#2798811yescharacteristic
270C2#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. #10633C2semidirect
4. #10633C2semidirect
5. #9778C2semidirect
8. #8795C2semidirect

Maximal subgroups

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

subgroupindex
#332152
#208292
#106332
#97782
#91942
#91922
#87952
#81572