Finite Group Databasebeta

← #321689

Subgroups — #321689

Order 512.

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

normal subgroups only

This group has 829 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#33302C211yesmaximal, characteristic, complemented
2#28520C212yesmaximal
3#21631C211yesmaximal, characteristic
4C2×#1690C212yesmaximal, complemented
5C2×#1689C212yesmaximal
6#10276C212yesmaximal, complemented
7#10274C212yesmaximal
8#9703C212yesmaximal, complemented
9#8273C211yesmaximal, characteristic, complemented
10C2×#494—22no—
11C2×#494—22no—
12C2×#494C2212yes—
13C2×#494C2212yes—
14C2×#475C2214yes—
15C2×#474C2214yes—
16C2×#446C2211yescharacteristic
17C2×#436C2211yescharacteristic
18#2868C2212yes—
19C4×#124C2211yescharacteristic
20#2095—22no—
21#2094—22no—
22#1984C2211yescharacteristic
23#1959—22no—
24#1954C2211yescharacteristic
25#1923—22no—
26#1847—22no—
27#1831C2211yescharacteristic
28#1767C2212yes—
29#1753—22no—
30C42×C8C2211yescharacteristic
31#1690—28no—
32#1689—28no—
33C2×#334C2212yes—
34C2×#332C2214yes—
35C2×#332C2214yes—
36#1277C2212yes—
37C22×Q16—22no—
38C22×Q16—24no—
39C22×Q16—22no—
40C22×M16—21no—
41C2×C4⋊Q8—24no—
42C2×C4⋊Q8C2314yes—
43C2×#124D411yescharacteristic
44C2×C4×Q8C2311yescharacteristic
45C2×C4×D4C2311yescharacteristic
46#494—24no—
47#494—24no—
48#494—44no—
49#494—48no—
50#494—48no—
51#475—28no—
52#475—44no—
53#474—28no—
54#474—44no—
55#446—24no—
56#436—24no—
57C2×#107—24no—
58C2×#106—24no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8C2311yescharacteristic
61C2×C4⋊C8C2311yescharacteristic
62C2×C4⋊C8C2314yes—
63C2×#103—24no—
64C2×#102—24no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—22no—
67C2×#97—21no—
68C2×#97—22no—
69C2×C4×C8D412yes—
70C2×C4×C8D412yes—
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73#391D411yescharacteristic
74#390D411yescharacteristic
75#388—22no—
76#384—22no—
77#384—22no—
78#384—24no—
79#382D411yescharacteristic
80C4×C22⋊C4—22no—
81C43C2311yescharacteristic
82#336—22no—
83#336—22no—
84#336—22no—
85#334D418yes—
86#332—28no—
87#332—28no—
88C22×Q8—24no—
89C22×Q8C2×D411yescharacteristic
90C22×D4C2×D411yescharacteristic
91C23×C4—21no—
92C2×Q16—48no—
93C2×Q16—48no—
94C2×Q16—48no—
95C2×Q16—44no—
96C2×Q16—44no—
97C2×Q16—48no—
98C2×Q16—48no—
99C2×M16—42no—
100C2×M16—44no—
101C22×C8—24no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8C2×D411yescharacteristic
105C22×C8C2×D412yes—
106C22×C8C2×D411yescharacteristic
107C4⋊Q8—24no—
108C4⋊Q8—24no—
109C4⋊Q8—24no—
110C4⋊Q8—24no—
111C4⋊Q8—48no—
112C4⋊Q8—48no—
113#124—24no—
114C4×Q8—22no—
115C4×Q8—22no—
116C4×D4—22no—
117C4×D4—22no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—44no—
122C2×C4⋊C4—44no—
123C2×C4⋊C4C2×D411yescharacteristic
124C2×C4⋊C4C2×D411yescharacteristic
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—21no—
128C2×C42—21no—
129C2×C42C2×D412yes—
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42#4112yes—
133C2×C42C2411yescharacteristic
134#107—48no—
135#106—48no—
136C4⋊C8—22no—
137C4⋊C8—22no—
138C4⋊C8—24no—
139C4⋊C8—42no—
140C4⋊C8C2×D418yes—
141#103—48no—
142#102—48no—
143C22⋊C8—42no—
144C22⋊C8—44no—
145#97—42no—
146#97—44no—
147C4×C8—24no—
148C4×C8—24no—
149C4×C8—22no—
150C4×C8C2×D414yes—
151#95—22no—
152#95—22no—
153C24—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—22no—
157C2×Q8—44no—
158C2×Q8—44no—
159C2×Q8—48no—
160C2×Q8—48no—
161C2×D4—22no—
162C2×D4—22no—
163C2×D4—22no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—44no—
172C22×C4#12112yes—
173C22×C4#12112yes—
174C22×C4C22×D411yescharacteristic
175C22×C4C22×D411yescharacteristic
176C22×C4C2×#4111yescharacteristic
177Q16—88no—
178Q16—816no—
179Q16—88no—
180M16—84no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—28no—
185C2×C8—44no—
186C2×C8—44no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8#12712yes—
192C2×C8#12712yes—
193C2×C8AGL(1,ℤ/8)14yes—
194C2×C8AGL(1,ℤ/8)14yes—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—42no—
198C4⋊C4—44no—
199C4⋊C4—44no—
200C4⋊C4—44no—
201C4⋊C4—44no—
202C4⋊C4—88no—
203C4⋊C4—88no—
204C22⋊C4—42no—
205C22⋊C4—44no—
206C22⋊C4—44no—
207C42—22no—
208C42—22no—
209C42—22no—
210C42—24no—
211C42—41no—
212C42—41no—
213C42#13714yes—
214C42C22×D412yes—
215C42H(5,𝔽2)12yes—
216C23—42no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—42no—
221C23C2×#12111yescharacteristic
222Q8—44no—
223Q8—84no—
224Q8—84no—
225Q8—88no—
226D4—44no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—82no—
242C2×C4—88no—
243C2×C4—84no—
244C2×C4#47014yes—
245C2×C4#47314yes—
246C2×C4C2×#12711yescharacteristic
247C2×C4#53511yescharacteristic
248C2×C4D4212yes—
249C2×C4#55012yes—
250C2×C4C2×AGL(1,ℤ/8)12yes—
251C2×C4C2×AGL(1,ℤ/8)12yes—
252C2×C4C2×#13712yes—
253C2×C4#57812yes—
254C8—22no—
255C8—24no—
256C8—48no—
257C8—81no—
258C8—81no—
259C8#49614yes—
260C22—81no—
261C22—81no—
262C22—81no—
263C22—82no—
264C22—82no—
265C22#254111yescharacteristic
266C22#303812yes—
267C22C2×#47011yescharacteristic
268C22#305911yescharacteristic
269C22C2×#47312yes—
270C4—22no—
271C4—24no—
272C4—24no—
273C4—81no—
274C4—81no—
275C4—81no—
276C4—164no—
277C4C2×#49611yescharacteristic
278C4#315111yescharacteristic
279C4#321712yes—
280C4#327614yes—
281C2—161no—
282C2#1929312yes—
283C2#1935412yes—
284C2#2844811yescharacteristic
285C2#2848911yescharacteristic
286C2#2849811yescharacteristic
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. #33302C2semidirect
4. C2×#1690C2semidirect
6. #10276C2semidirect
8. #9703C2semidirect
9. #8273C2semidirect

Maximal subgroups

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

subgroupindex
#333022
#285202
#216312
C2×#16902
C2×#16892
#102762
#102742
#97032
#82732