Finite Group Databasebeta

← #511133

Subgroups — #511133

Order 512.

394 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 118 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#28733C211yesmaximal, characteristic
2#27693C211yesmaximal, characteristic
3#13504C212yesmaximal, complemented
4#13502C212yesmaximal, complemented
5#13458C212yesmaximal, complemented
6#13117C212yesmaximal, complemented
7#12982C211yesmaximal, characteristic
8#12933C212yesmaximal, complemented
9#12833C212yesmaximal, complemented
10#3407C2211yescharacteristic
11C2×#499C2211yescharacteristic
12#3142—21no—
13#2931C2211yescharacteristic
14#2916C2211yescharacteristic
15#2155C2212yes—
16C2×#365C2212yes—
17C2×#365C2212yes—
18#2117C2212yes—
19#2115C2211yescharacteristic
20#2102—21no—
21#2098—21no—
22#2078—21no—
23#2078—21no—
24#1998—21no—
25#1945C2212yes—
26#1884—21no—
27#1868—21no—
28#1815C2212yes—
29#1815C2212yes—
30#1814C2212yes—
31#1812C2211yescharacteristic
32C4×#108C2211yescharacteristic
33#1389C2212yes—
34#1388C2212yes—
35#1386C2212yes—
36#1386C2212yes—
37#1385C2212yes—
38#1376C2212yes—
39#568—21no—
40C2×#131C2311yescharacteristic
41C2×#131C2311yescharacteristic
42#532D411yescharacteristic
43C2×#125—21no—
44#504—24no—
45C2×M32—41no—
46C2×M32C2312yes—
47C2×M32C2312yes—
48#499—41no—
49#499—41no—
50#491—41no—
51#490—41no—
52#488—41no—
53#488—41no—
54#436—41no—
55C8×D4—41no—
56#430—21no—
57#430—41no—
58C2×#108—41no—
59C2×#108C2312yes—
60C2×#108C2312yes—
61#427C2312yes—
62C2×#107—21no—
63C2×#106C2312yes—
64#424—22no—
65#423D411yescharacteristic
66#421—22no—
67#417—22no—
68#417—22no—
69#410—21no—
70#408D411yescharacteristic
71#408D411yescharacteristic
72C4×M16—21no—
73C2×C4×C8C2311yescharacteristic
74#365—24no—
75#365—42no—
76#365—42no—
77#350—24no—
78#349—41no—
79#344—22no—
80#343—22no—
81#343—22no—
82#341—21no—
83#339—22no—
84H(3,ℤ/4)—21no—
85C2×#41C2×D411yescharacteristic
86C2×#41C2×D411yescharacteristic
87#131—22no—
88#131—22no—
89#131—22no—
90#131—22no—
91#131—82no—
92C2×M16—21no—
93C2×M16—21no—
94C2×M16—22no—
95C2×M16—22no—
96C2×M16—42no—
97C2×M16C2×D411yescharacteristic
98C2×M16C2×D411yescharacteristic
99C22×C8—21no—
100C22×C8—21no—
101C22×C8C2×D411yescharacteristic
102C22×C8C2411yescharacteristic
103C4⋊Q8SD1611yescharacteristic
104#127SD1611yescharacteristic
105#125—42no—
106#125—41no—
107#125—41no—
108#124SD1611yescharacteristic
109#124SD1611yescharacteristic
110#121—41no—
111C4×D4—41no—
112#117—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—42no—
115C2×C42C2×D411yescharacteristic
116M32—22no—
117M32—22no—
118M32—22no—
119M32—42no—
120M32#4114yes—
121C2×C16—22no—
122C2×C16—22no—
123C2×C16—42no—
124#108—22no—
125#108—22no—
126#108—22no—
127#108—22no—
128#108—42no—
129#107—22no—
130#107—42no—
131#106—22no—
132#106—22no—
133#106—22no—
134C4⋊C8—41no—
135C4⋊C8—82no—
136#104—82no—
137#103—42no—
138#103—42no—
139#102—42no—
140#102—42no—
141#99—42no—
142C22⋊C8—41no—
143C22⋊C8—41no—
144#97—41no—
145C4×C8—21no—
146C4×C8—21no—
147C4×C8—41no—
148#41—22no—
149#41—22no—
150#41—22no—
151#41—22no—
152C2×Q8C2×SD1611yescharacteristic
153C2×Q8C2×SD1611yescharacteristic
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—41no—
157C2×D4C2×SD1611yescharacteristic
158C2×D4C2×SD1611yescharacteristic
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—42no—
163C22×C4C22×D411yescharacteristic
164M16—22no—
165M16—22no—
166M16—42no—
167M16—42no—
168M16—42no—
169M16—42no—
170M16—42no—
171M16—42no—
172M16—82no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—22no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—82no—
184C2×C8C2×#4112yes—
185C2×C8H(5,𝔽2)11yescharacteristic
186C2×C8H(5,𝔽2)11yescharacteristic
187C2×C82−1+412yes—
188C4⋊C4—41no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—82no—
193C4⋊C4—82no—
194C22⋊C4—41no—
195C22⋊C4—41no—
196C22⋊C4—42no—
197C42—21no—
198C42—42no—
199C42C2×SD1611yescharacteristic
200C42C2×SD1611yescharacteristic
201C16—24no—
202C16—42no—
203C23—21no—
204C23—21no—
205C23#57511yescharacteristic
206Q8—22no—
207Q8—22no—
208D4—22no—
209D4—22no—
210D4—42no—
211D4—42no—
212D4—82no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—81no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—162no—
225C2×C4#53411yescharacteristic
226C2×C4#54712yes—
227C2×C4C22×SD1611yescharacteristic
228C2×C4#57712yes—
229C8—41no—
230C8—41no—
231C8—41no—
232C8—82no—
233C8—82no—
234C8#54212yes—
235C8#55111yescharacteristic
236C8#56311yescharacteristic
237C22—41no—
238C22—41no—
239C22—41no—
240C22—81no—
241C22—81no—
242C22#318711yescharacteristic
243C22#335012yes—
244C4—41no—
245C4—41no—
246C4—41no—
247C4—41no—
248C4—162no—
249C4—162no—
250C4#271711yescharacteristic
251C4#320011yescharacteristic
252C4#333512yes—
253C2—21no—
254C2—22no—
255C2—81no—
256C2—81no—
257C2#3226011yescharacteristic
subgroupG/H
center, socleC2#32260
commutator, Frattini, cosocleC22×C8C24
prime radical#532D4
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22×C4 ◁ #2931 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13504C2semidirect
4. #13502C2semidirect
5. #13458C2semidirect
6. #13117C2semidirect
8. #12933C2semidirect
9. #12833C2semidirect

Maximal subgroups

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

subgroupindex
#287332
#276932
#135042
#135022
#134582
#131172
#129822
#129332
#128332