Finite Group Databasebeta

← #319337

Subgroups — #319337

Order 512.

768 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 182 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#33207C211yesmaximal, characteristic
2#27878C212yesmaximal
3#25600C212yesmaximal
4#21318C211yesmaximal, characteristic
5#10684C212yesmaximal, complemented
6#10318C212yesmaximal, complemented
7#9835C212yesmaximal, complemented
8#9833C212yesmaximal, complemented
9#8144C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#498C2211yescharacteristic
12C2×#494C2212yes—
13C2×#487C2212yes—
14C22×C4×C8C2211yescharacteristic
15#2857C2212yes—
16#2285C2211yescharacteristic
17#2076—24no—
18#2001—24no—
19#1991—22no—
20#1990C2212yes—
21#1990C2212yes—
22#1990C2212yes—
23#1990C2212yes—
24#1990C2212yes—
25#1990C2212yes—
26#1956C2212yes—
27#1941—22no—
28#1921—24no—
29#1878—24no—
30#1845C2212yes—
31#1844C2211yescharacteristic
32#1834C2212yes—
33#1807—22no—
34C4×C22⋊C8C2212yes—
35#1735—22no—
36C2×#322C2211yescharacteristic
37C2×#322C2211yescharacteristic
38#1273C2212yes—
39C22×Q16—24no—
40C23×C8D412yes—
41#533—24no—
42#533—41no—
43#533—41no—
44C2×C4⋊Q8C2312yes—
45C2×C4⋊Q8C2312yes—
46C2×#125—21no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C4×Q8—22no—
50C2×#117—21no—
51C2×#117—21no—
52C22×C42C2311yescharacteristic
53#498—41no—
54#498—41no—
55#494—24no—
56#494—42no—
57#487—24no—
58#487—42no—
59C2×#107—24no—
60C2×#106—22no—
61C2×#106—22no—
62C2×#103—24no—
63C2×#103—24no—
64C2×C22⋊C8D414yes—
65C2×C4×C8—41no—
66C2×C4×C8—41no—
67C2×C4×C8D412yes—
68C2×C4×C8C2312yes—
69C2×C4×C8C2311yescharacteristic
70C2×C4×C8C2312yes—
71C2×C4×C8C2311yescharacteristic
72#391—22no—
73#389—22no—
74#389—22no—
75#384C2312yes—
76#384C2312yes—
77#382—21no—
78#380—22no—
79#380—22no—
80#380—22no—
81C4×C4⋊C4—22no—
82#340—24no—
83#340—24no—
84#340—24no—
85#340—24no—
86#340—24no—
87#340—24no—
88#340—24no—
89#336D414yes—
90#322—41no—
91#322—41no—
92C22×Q8—22no—
93C22×Q8—22no—
94C23×C4C2×D412yes—
95C23×C4C2×D411yescharacteristic
96C2×Q16—48no—
97C2×Q16—48no—
98C2×Q16—84no—
99C22×C8—42no—
100C22×C8—42no—
101C22×C8Q1618yescomplemented
102C22×C8C2×D414yes—
103C22×C8C2×D414yes—
104C22×C8C2×D412yes—
105C22×C8C2×D412yes—
106C4⋊Q8—22no—
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8#4114yes—
114#125—24no—
115#125—42no—
116C22⋊Q8—44no—
117C22⋊Q8—44no—
118C22⋊Q8—84no—
119C4×Q8—44no—
120C4×Q8—44no—
121#117—24no—
122#117—42no—
123#117—44no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C22⋊C4—41no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—42no—
139C2×C42—22no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—41no—
143C2×C42—41no—
144C2×C42C2×D412yes—
145C2×C42C2×D411yescharacteristic
146C2×C42C2411yescharacteristic
147#107—48no—
148#106—44no—
149#106—44no—
150#103—48no—
151#103—84no—
152C22⋊C8—44no—
153C4×C8—22no—
154C4×C8—22no—
155C4×C8—21no—
156C4×C8—21no—
157C4×C8—22no—
158C4×C8—22no—
159C4×C8—21no—
160C4×C8—21no—
161C4×C8—42no—
162#95—42no—
163#95—42no—
164#95—42no—
165C24#12711yescharacteristic
166C2×Q8—24no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—44no—
171C2×Q8—44no—
172C2×Q8—44no—
173C2×Q8—44no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4#12712yes—
186C22×C4#12711yescharacteristic
187C22×C4C2×Q1614yes—
188C22×C4C22×D411yescharacteristic
189C22×C4C22×D412yes—
190Q16—816no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—24no—
196C2×C8—24no—
197C2×C8—24no—
198C2×C8—24no—
199C2×C8—24no—
200C2×C8—24no—
201C2×C8—48no—
202C2×C8C2×Q1614yes—
203C2×C8C2×Q1614yes—
204C4⋊C4—24no—
205C4⋊C4—24no—
206C4⋊C4—48no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—84no—
220C22⋊C4—44no—
221C22⋊C4—82no—
222C22⋊C4—84no—
223C42—21no—
224C42—21no—
225C42—22no—
226C42—22no—
227C42—44no—
228C42—82no—
229C42C2×#4112yes—
230C42H(5,𝔽2)11yescharacteristic
231C42H(5,𝔽2)11yescharacteristic
232C23—22no—
233C23—21no—
234C23—22no—
235C23—21no—
236C23#49412yes—
237C23C2×#12711yescharacteristic
238Q8—48no—
239Q8—84no—
240Q8—84no—
241Q8—84no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4#49412yes—
264C2×C4#52511yescharacteristic
265C2×C4#53411yescharacteristic
266C2×C4#54612yes—
267C2×C4#54612yes—
268C2×C4C22×Q1612yes—
269C2×C4#57512yes—
270C2×C4#57514yes—
271C2×C4#57714yes—
272C8—24no—
273C8—24no—
274C8—44no—
275C8—44no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—42no—
280C22—42no—
281C22#253911yescharacteristic
282C22C2×#49411yescharacteristic
283C22#315012yes—
284C22#315311yescharacteristic
285C22#315412yes—
286C4—22no—
287C4—42no—
288C4—84no—
289C4—162no—
290C4—162no—
291C4—162no—
292C4#310412yes—
293C4#318912yes—
294C4#331114yes—
295C4#331714yes—
296C2—42no—
297C2#1961811yescharacteristic
298C2#1966311yescharacteristic
299C2#1969212yes—
300C2#2830411yescharacteristic
301C2#2835212yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10684C2semidirect
6. #10318C2semidirect
7. #9835C2semidirect
8. #9833C2semidirect
101. C22×C8Q16semidirect

Maximal subgroups

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

subgroupindex
#332072
#278782
#256002
#213182
#106842
#103182
#98352
#98332
#81442