Finite Group Databasebeta

← #319340

Subgroups — #319340

Order 512.

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

normal subgroups only

This group has 752 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#27892C212yesmaximal
3#25674C212yesmaximal
4#21318C211yesmaximal, characteristic
5#10680C212yesmaximal, complemented
6#10314C212yesmaximal, complemented
7#9834C212yesmaximal, complemented
8#9833C212yesmaximal, complemented
9#8144C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#500C2212yes—
12C2×#498C2211yescharacteristic
13C2×#487C2212yes—
14C22×C4×C8C2211yescharacteristic
15#2857C2212yes—
16#2285C2211yescharacteristic
17#2036—24no—
18#2036—24no—
19#1991—22no—
20#1990C2212yes—
21#1990C2212yes—
22#1990C2212yes—
23#1990C2212yes—
24#1956C2212yes—
25#1956C2212yes—
26#1956C2212yes—
27#1941—22no—
28#1928—24no—
29#1863—24no—
30#1845C2212yes—
31#1844C2211yescharacteristic
32#1834C2212yes—
33#1807—22no—
34C4×C22⋊C8C2212yes—
35#1735—22no—
36C2×#322C2211yescharacteristic
37C2×#322C2211yescharacteristic
38#1273C2212yes—
39C23×C8D412yes—
40#533—24no—
41#533—41no—
42#533—41no—
43C2×C4⋊Q8C2312yes—
44C2×C4⋊Q8C2312yes—
45C2×#125—21no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—22no—
48C2×C4×Q8—22no—
49C2×#117—21no—
50C2×#117—21no—
51C22×C42C2311yescharacteristic
52#500—24no—
53#500—42no—
54#498—41no—
55#498—41no—
56#487—24no—
57#487—42no—
58C2×#107—24no—
59C2×#107—24no—
60C2×#106—22no—
61C2×#106—22no—
62C2×#103—24no—
63C2×#103—24no—
64C2×#103—24no—
65C2×#103—24no—
66C2×C22⋊C8D414yes—
67C2×C4×C8—41no—
68C2×C4×C8—41no—
69C2×C4×C8D412yes—
70C2×C4×C8C2312yes—
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2312yes—
73C2×C4×C8C2311yescharacteristic
74#391—22no—
75#389—22no—
76#389—22no—
77#384C2312yes—
78#384C2312yes—
79#382—21no—
80#380—22no—
81#380—22no—
82#380—22no—
83C4×C4⋊C4—22no—
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
96C22×C8—42no—
97C22×C8—42no—
98C22×C8Q1618yes—
99C22×C8C2×D414yes—
100C22×C8C2×D414yes—
101C22×C8C2×D412yes—
102C22×C8C2×D412yes—
103C4⋊Q8—22no—
104C4⋊Q8—22no—
105C4⋊Q8—22no—
106C4⋊Q8—22no—
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8#4114yes—
111#125—24no—
112#125—42no—
113C22⋊Q8—44no—
114C22⋊Q8—44no—
115C22⋊Q8—84no—
116C4×Q8—44no—
117C4×Q8—44no—
118#117—24no—
119#117—42no—
120#117—44no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
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—42no—
132C2×C4⋊C4—42no—
133C2×C22⋊C4—41no—
134C2×C22⋊C4—41no—
135C2×C22⋊C4—42no—
136C2×C42—22no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—41no—
140C2×C42—41no—
141C2×C42C2×D412yes—
142C2×C42C2×D411yescharacteristic
143C2×C42C2411yescharacteristic
144#107—48no—
145#107—84no—
146#106—44no—
147#106—44no—
148#103—48no—
149#103—48no—
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—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—24no—
195C2×C8—24no—
196C2×C8—24no—
197C2×C8—24no—
198C2×C8—24no—
199C2×C8—24no—
200C2×C8—48no—
201C2×C8C2×Q1614yes—
202C2×C8C2×Q1614yes—
203C4⋊C4—24no—
204C4⋊C4—24no—
205C4⋊C4—48no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—84no—
219C22⋊C4—44no—
220C22⋊C4—82no—
221C22⋊C4—84no—
222C42—21no—
223C42—21no—
224C42—22no—
225C42—22no—
226C42—44no—
227C42—82no—
228C42C2×#4112yes—
229C42H(5,𝔽2)11yescharacteristic
230C42H(5,𝔽2)11yescharacteristic
231C23—22no—
232C23—21no—
233C23—22no—
234C23—21no—
235C23#49412yes—
236C23C2×#12711yescharacteristic
237Q8—48no—
238Q8—84no—
239Q8—84no—
240Q8—84no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—82no—
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#49412yes—
263C2×C4#52511yescharacteristic
264C2×C4#53411yescharacteristic
265C2×C4#54612yes—
266C2×C4#54612yes—
267C2×C4C22×Q1612yes—
268C2×C4#57512yes—
269C2×C4#57514yes—
270C2×C4#57714yes—
271C8—24no—
272C8—24no—
273C8—44no—
274C8—44no—
275C22—22no—
276C22—22no—
277C22—22no—
278C22—42no—
279C22—42no—
280C22#253911yescharacteristic
281C22C2×#49411yescharacteristic
282C22#315012yes—
283C22#315311yescharacteristic
284C22#315412yes—
285C4—22no—
286C4—42no—
287C4—84no—
288C4—162no—
289C4—162no—
290C4—162no—
291C4#310412yes—
292C4#318912yes—
293C4#330714yes—
294C4#332114yes—
295C2—42no—
296C2#1961811yescharacteristic
297C2#1966311yescharacteristic
298C2#1969112yes—
299C2#2830411yescharacteristic
300C2#2835112yes—
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. #10680C2semidirect
6. #10314C2semidirect
7. #9834C2semidirect
8. #9833C2semidirect

Maximal subgroups

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

subgroupindex
#332072
#278922
#256742
#213182
#106802
#103142
#98342
#98332
#81442