Finite Group Databasebeta

← #316204

Subgroups — #316204

Order 512.

415 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 415 conjugacy classes of subgroups — too many to draw. Only its 122 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#29881C211yesmaximal, characteristic
2#21995C211yesmaximal, characteristic
3#10636C212yesmaximal, complemented
4#10633C212yesmaximal, complemented
5#9776C211yesmaximal, characteristic, complemented
6#9767C211yesmaximal, characteristic, complemented
7#9193C212yesmaximal
8#9192C212yesmaximal
9#8794C211yesmaximal, characteristic, complemented
10#8792C211yesmaximal, characteristic, complemented
11#8158C211yesmaximal, characteristic
12C2×#498C2211yescharacteristic
13#2921C2211yescharacteristic
14C2×#433C2211yescharacteristic
15#2905C2211yescharacteristic
16#2628C2212yes—
17#2088—22no—
18#2077—22no—
19#1993C2212yes—
20#1950—22no—
21#1855—22no—
22#1849C2211yescharacteristic
23#1848C2211yescharacteristic
24#1838C2211yescharacteristic
25#1837C2211yescharacteristic
26#1835C2211yescharacteristic
27#1549C2212yes—
28#1549C2212yes—
29#1548C2212yes—
30#1548C2212yes—
31#1458C2212yes—
32#1452C2211yescharacteristic
33#1450C2211yescharacteristic
34#1286C2212yes—
35#1286C2212yes—
36#1286C2212yes—
37#1286C2212yes—
38#1281C2212yes—
39C22×M16—21no—
40C2×C4⋊Q8C2312yes—
41C2×#117C2311yescharacteristic
42#498—21no—
43#498—21no—
44#498—22no—
45#433—21no—
46#433—21no—
47#433—22no—
48#433—42no—
49#431C2312yes—
50#431C2312yes—
51C2×#108—22no—
52C2×#108—22no—
53C2×#106—22no—
54#424—42no—
55#423—21no—
56#423C2311yescharacteristic
57#423C2311yescharacteristic
58#423C2311yescharacteristic
59#423C2311yescharacteristic
60C2×C4⋊C8D412yes—
61#407—21no—
62C2×C22⋊C8D412yes—
63C4×M16—21no—
64C2×#97C2311yescharacteristic
65C2×C4×C8C2311yescharacteristic
66#400—22no—
67#399—22no—
68#395—22no—
69#393—22no—
70#389—22no—
71#382C2312yes—
72#380—22no—
73#341—22no—
74#341—22no—
75#340—22no—
76#340—24no—
77#333—42no—
78#333—42no—
79#332—42no—
80#332—42no—
81#324—22no—
82#323—22no—
83#322—21no—
84#322—22no—
85C82—21no—
86C22×Q8—22no—
87C23×C4C2×D411yescharacteristic
88C2×M16—21no—
89C2×M16—21no—
90C2×M16—21no—
91C2×M16—21no—
92C2×M16—41no—
93C2×M16—82no—
94C22×C8—41no—
95C22×C8C2×D412yes—
96C22×C8C2×D412yes—
97C4⋊Q8—22no—
98C4⋊Q8—22no—
99C4⋊Q8—22no—
100C4⋊Q8—22no—
101#117—21no—
102#117—21no—
103#117—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4#4112yes—
109C2×C22⋊C4—42no—
110C2×C22⋊C4#4112yes—
111C2×C42C2×D411yescharacteristic
112C2×C42C2411yescharacteristic
113#108—82no—
114#108—82no—
115#106—42no—
116#106—42no—
117C4⋊C8—22no—
118C4⋊C8—22no—
119C4⋊C8—22no—
120C4⋊C8—22no—
121C4⋊C8—22no—
122C4⋊C8—22no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125C4⋊C8—42no—
126C22⋊C8—22no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8—42no—
131#97—21no—
132#97—21no—
133#97—41no—
134C4×C8—22no—
135C4×C8—22no—
136C4×C8—41no—
137C4×C8#4111yescharacteristic
138C4×C8#4111yescharacteristic
139C4×C8#4112yes—
140#95—22no—
141#95—22no—
142#95—42no—
143#95—42no—
144C24#12411yescharacteristic
145C2×Q8—42no—
146C2×Q8—42no—
147C2×Q8—162no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4#12411yescharacteristic
155C22×C4#12412yes—
156C22×C4C22×D411yescharacteristic
157C22×C4C2×#4112yes—
158M16—42no—
159M16—42no—
160M16—42no—
161M16—42no—
162M16—82no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—82no—
175C2×C8#12312yes—
176C2×C8#12312yes—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—162no—
186C22⋊C4—22no—
187C22⋊C4—22no—
188C22⋊C4—162no—
189C42—21no—
190C42—21no—
191C42C2×#4111yescharacteristic
192C42C2×#4111yescharacteristic
193C42H(5,𝔽2)12yes—
194C23—21no—
195C23—22no—
196C23—42no—
197C23C2×#12411yescharacteristic
198Q8—162no—
199Q8—162no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4C2×#12311yescharacteristic
215C2×C4#53511yescharacteristic
216C2×C4#55312yes—
217C2×C4#55312yes—
218C2×C4#57712yes—
219C2×C4#57812yes—
220C8—22no—
221C8—22no—
222C8—42no—
223C8—82no—
224C8—82no—
225C8—81no—
226C8—81no—
227C8—81no—
228C8—81no—
229C22—22no—
230C22—41no—
231C22—42no—
232C22—82no—
233C22#250411yescharacteristic
234C22#313712yes—
235C4—22no—
236C4—22no—
237C4—41no—
238C4—42no—
239C4—162no—
240C4—162no—
241C4#309712yes—
242C4#322211yescharacteristic
243C4#322311yescharacteristic
244C2—22no—
245C2—81no—
246C2#2798511yescharacteristic
247C2#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. #10636C2semidirect
4. #10633C2semidirect
5. #9776C2semidirect
6. #9767C2semidirect
9. #8794C2semidirect
10. #8792C2semidirect

Maximal subgroups

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

subgroupindex
#298812
#219952
#106362
#106332
#97762
#97672
#91932
#91922
#87942
#87922
#81582