Finite Group Databasebeta

← #321694

Subgroups — #321694

Order 512.

725 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 725 conjugacy classes of subgroups — too many to draw. Only its 186 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#33312C211yesmaximal, characteristic, complemented
2#28519C212yesmaximal
3#21631C211yesmaximal, characteristic
4C2×#1696C212yesmaximal, complemented
5C2×#1695C212yesmaximal
6#10279C212yesmaximal, complemented
7#10278C212yesmaximal
8#9702C212yesmaximal, complemented
9#8273C211yesmaximal, characteristic, complemented
10C2×#500C2212yes—
11C2×#500C2212yes—
12C2×#477C2214yes—
13C2×#476C2214yes—
14C2×#446C2211yescharacteristic
15C2×#436C2211yescharacteristic
16#2868C2212yes—
17C4×#124C2211yescharacteristic
18#2096—22no—
19#2093—22no—
20#1984C2211yescharacteristic
21#1959—22no—
22#1954C2211yescharacteristic
23#1923—22no—
24#1847—22no—
25#1845—22no—
26#1845—22no—
27#1831C2211yescharacteristic
28#1767C2212yes—
29#1753—22no—
30C42×C8C2211yescharacteristic
31#1696—28no—
32#1695—28no—
33C2×#334C2212yes—
34C2×#333C2214yes—
35C2×#333C2214yes—
36#1277C2212yes—
37C22×M16—21no—
38C2×C4⋊Q8—24no—
39C2×C4⋊Q8C2314yes—
40C2×#124D411yescharacteristic
41C2×C4×Q8C2311yescharacteristic
42C2×C4×D4C2311yescharacteristic
43#500—24no—
44#500—24no—
45#500—44no—
46#477—28no—
47#477—44no—
48#476—28no—
49#476—44no—
50#446—24no—
51#436—24no—
52C2×#107—24no—
53C2×#107—22no—
54C2×#107—22no—
55C2×#107—24no—
56C2×#106—24no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8C2311yescharacteristic
59C2×C4⋊C8C2311yescharacteristic
60C2×C4⋊C8C2314yes—
61C2×#103—24no—
62C2×#102—24no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—22no—
65C2×#97—21no—
66C2×#97—22no—
67C2×C4×C8D412yes—
68C2×C4×C8D412yes—
69C2×C4×C8C2311yescharacteristic
70C2×C4×C8C2311yescharacteristic
71#391D411yescharacteristic
72#390D411yescharacteristic
73#388—22no—
74#384—22no—
75#384—24no—
76#384—22no—
77#382D411yescharacteristic
78C4×C22⋊C4—22no—
79C43C2311yescharacteristic
80#336—22no—
81#336—22no—
82#336—22no—
83#334D418yes—
84#333—28no—
85#333—28no—
86C22×Q8—24no—
87C22×Q8C2×D411yescharacteristic
88C22×D4C2×D411yescharacteristic
89C23×C4—21no—
90C2×M16—42no—
91C2×M16—44no—
92C22×C8—24no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8C2×D411yescharacteristic
96C22×C8C2×D412yes—
97C22×C8C2×D411yescharacteristic
98C4⋊Q8—24no—
99C4⋊Q8—24no—
100C4⋊Q8—24no—
101C4⋊Q8—24no—
102C4⋊Q8—88no—
103#124—24no—
104C4×Q8—22no—
105C4×Q8—22no—
106C4×D4—22no—
107C4×D4—22no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—44no—
112C2×C4⋊C4—44no—
113C2×C4⋊C4C2×D411yescharacteristic
114C2×C4⋊C4C2×D411yescharacteristic
115C2×C22⋊C4—22no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—21no—
118C2×C42—21no—
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D412yes—
121C2×C42C2×D411yescharacteristic
122C2×C42#4112yes—
123C2×C42C2411yescharacteristic
124#107—48no—
125#107—48no—
126#107—44no—
127#107—44no—
128#106—48no—
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8—24no—
132C4⋊C8—42no—
133C4⋊C8C2×D418yes—
134#103—48no—
135#102—48no—
136C22⋊C8—42no—
137C22⋊C8—44no—
138#97—42no—
139#97—44no—
140C4×C8—24no—
141C4×C8—24no—
142C4×C8—22no—
143C4×C8C2×D414yes—
144#95—22no—
145#95—22no—
146C24—21no—
147C2×Q8—22no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—44no—
151C2×Q8—44no—
152C2×Q8—88no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—22no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—44no—
164C22×C4#12112yes—
165C22×C4#12112yes—
166C22×C4C22×D411yescharacteristic
167C22×C4C22×D411yescharacteristic
168C22×C4C2×#4111yescharacteristic
169M16—84no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—28no—
174C2×C8—44no—
175C2×C8—44no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8#12712yes—
181C2×C8#12712yes—
182C2×C8#13714yes—
183C2×C8#13714yes—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—42no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—44no—
190C4⋊C4—44no—
191C4⋊C4—88no—
192C4⋊C4—88no—
193C22⋊C4—42no—
194C22⋊C4—44no—
195C22⋊C4—44no—
196C42—22no—
197C42—22no—
198C42—22no—
199C42—24no—
200C42—41no—
201C42—41no—
202C42#13714yes—
203C42C22×D412yes—
204C42H(5,𝔽2)12yes—
205C23—42no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—42no—
210C23C2×#12111yescharacteristic
211Q8—44no—
212Q8—164no—
213Q8—164no—
214D4—44no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—42no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—82no—
230C2×C4—84no—
231C2×C4—88no—
232C2×C4#47014yes—
233C2×C4#47214yes—
234C2×C4C2×#12711yescharacteristic
235C2×C4#53511yescharacteristic
236C2×C4D4212yes—
237C2×C4#55012yes—
238C2×C4C2×#13712yes—
239C2×C4C2×#13712yes—
240C2×C4C2×#13712yes—
241C2×C4#57812yes—
242C8—22no—
243C8—24no—
244C8—48no—
245C8—81no—
246C8—81no—
247C8#49714yes—
248C22—81no—
249C22—81no—
250C22—81no—
251C22—82no—
252C22—82no—
253C22#254111yescharacteristic
254C22#303912yes—
255C22C2×#47011yescharacteristic
256C22#305911yescharacteristic
257C22C2×#47212yes—
258C4—22no—
259C4—24no—
260C4—24no—
261C4—81no—
262C4—81no—
263C4—81no—
264C4—164no—
265C4C2×#49711yescharacteristic
266C4#315211yescharacteristic
267C4#321812yes—
268C4#327614yes—
269C2—161no—
270C2#1929712yes—
271C2#1935912yes—
272C2#2844911yescharacteristic
273C2#2848911yescharacteristic
274C2#2849711yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33312C2semidirect
4. C2×#1696C2semidirect
6. #10279C2semidirect
8. #9702C2semidirect
9. #8273C2semidirect

Maximal subgroups

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

subgroupindex
#333122
#285192
#216312
C2×#16962
C2×#16952
#102792
#102782
#97022
#82732