Finite Group Databasebeta

← #309544

Subgroups — #309544

Order 512.

622 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 146 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#25011C212yesmaximal
2#23061C212yesmaximal, complemented
3#10674C211yesmaximal, characteristic, complemented
4#10481C212yesmaximal, complemented
5#10235C211yesmaximal, characteristic
6#9615C212yesmaximal, complemented
7#9541C212yesmaximal, complemented
8#8928C211yesmaximal, characteristic, complemented
9#8885C212yesmaximal, complemented
10C2×#482—22no—
11C2×#476C2212yes—
12#2537C2211yescharacteristic
13#2369C2211yescharacteristic
14#2354C2212yes—
15#2076—22no—
16#2033—22no—
17#1990C2212yes—
18#1956C2211yescharacteristic
19#1954C2212yes—
20#1948C2212yes—
21#1948C2212yes—
22#1934—22no—
23#1926—22no—
24#1920—22no—
25#1854—22no—
26#1845C2212yes—
27#1845C2211yescharacteristic
28#1802C2212yes—
29C4×#106—22no—
30C8×C4⋊C4C2212yes—
31C4×#102C2212yes—
32C2×#335C2212yes—
33C2×#333C2211yescharacteristic
34C2×D4⋊C8C2212yes—
35#1296C2211yescharacteristic
36#1296C2212yes—
37#1279C2212yes—
38#1273C2211yescharacteristic
39C2×C4⋊Q8C2311yescharacteristic
40C2×#121—22no—
41C2×C4×D4C2312yes—
42C2×#117—22no—
43#482—48no—
44#476—24no—
45#476—24no—
46C2×#107—22no—
47C2×#107—22no—
48C2×#107—41no—
49C2×#106—22no—
50C2×#106—22no—
51C2×#106—22no—
52C2×C4⋊C8C2312yes—
53C2×C4⋊C8C2312yes—
54C2×#103—41no—
55C2×#102—22no—
56C2×#102—22no—
57C2×#102D412yes—
58C2×#102D412yes—
59C2×C22⋊C8—22no—
60C2×C22⋊C8—22no—
61C2×C4×C8—22no—
62C2×C4×C8C2311yescharacteristic
63C2×C4×C8C2312yes—
64#394—41no—
65#391—41no—
66#390—22no—
67#389—22no—
68#388—42no—
69#384C2312yes—
70#384C2311yescharacteristic
71#380—22no—
72#380—22no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4C2312yes—
76#340—22no—
77#340—22no—
78#340—22no—
79#340—22no—
80#340—22no—
81#340D412yes—
82#340D412yes—
83#336—22no—
84#335—24no—
85#333—22no—
86D4⋊C8—24no—
87C22×Q8—21no—
88C22×D4—41no—
89C22×D4C2×D412yes—
90C23×C4—22no—
91C22×C8—22no—
92C22×C8—21no—
93C22×C8—22no—
94C22×C8C2×D412yes—
95C22×C8C2×D412yes—
96C4⋊Q8—22no—
97C4⋊Q8—21no—
98C4⋊Q8#4112yes—
99#121—44no—
100#121—82no—
101C4×D4—22no—
102C4×D4—22no—
103C4×D4—22no—
104C4×D4—22no—
105#117—44no—
106#117—82no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4C2×D412yes—
115C2×C4⋊C4C2×D412yes—
116C2×C4⋊C4C2×D412yes—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4—42no—
121C2×C42—22no—
122C2×C42—42no—
123C2×C42—41no—
124C2×C42C2411yescharacteristic
125#107—44no—
126#107—82no—
127#107—82no—
128#106—28no—
129#106—44no—
130#106—44no—
131C4⋊C8—22no—
132C4⋊C8—22no—
133C4⋊C8—22no—
134C4⋊C8#4114yes—
135#103—82no—
136#102—28no—
137#102—24no—
138#102—24no—
139#102—44no—
140C22⋊C8—44no—
141C22⋊C8—44no—
142C4×C8—22no—
143C4×C8—22no—
144C4×C8—22no—
145C4×C8—44no—
146#95—22no—
147#95—42no—
148#95—42no—
149#95—41no—
150C24—22no—
151C2×Q8—22no—
152C2×Q8—41no—
153C2×Q8—81no—
154C2×Q8—81no—
155C2×D4—22no—
156C2×D4—22no—
157C2×D4—22no—
158C2×D4—22no—
159C2×D4—22no—
160C2×D4—22no—
161C2×D4—84no—
162C2×D4—161no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—41no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—41no—
175C22×C4C22×D412yes—
176C22×C4H(5,𝔽2)11yescharacteristic
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—22no—
180C2×C8—24no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8#13514yes—
192C4⋊C4—22no—
193C4⋊C4—24no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—81no—
206C4⋊C4—82no—
207C4⋊C4—161no—
208C4⋊C4AGL(1,ℤ/8)14yes—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—84no—
212C22⋊C4—84no—
213C22⋊C4—162no—
214C42—42no—
215C42—42no—
216C42—84no—
217C42—161no—
218C42C2×#4111yescharacteristic
219C42C2×#4112yes—
220C422−1+411yescharacteristic
221C23—42no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23D4211yescharacteristic
229Q8—82no—
230Q8—161no—
231D4—42no—
232D4—42no—
233D4—42no—
234D4—42no—
235D4—164no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—24no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—81no—
259C2×C4#52512yes—
260C2×C4#53911yescharacteristic
261C2×C4#54712yes—
262C2×C4#55911yescharacteristic
263C2×C4C2×#13512yes—
264C2×C4C2×AGL(1,ℤ/8)12yes—
265C2×C4#57712yes—
266C2×C4#57812yes—
267C8—24no—
268C8—42no—
269C8—42no—
270C8—44no—
271C8—82no—
272C22—82no—
273C22—82no—
274C22—82no—
275C22—82no—
276C22—82no—
277C22—82no—
278C22—82no—
279C22#245511yescharacteristic
280C22#326512yes—
281C22#328312yes—
282C22#328911yescharacteristic
283C22#329211yescharacteristic
284C4—21no—
285C4—21no—
286C4—44no—
287C4—82no—
288C4—82no—
289C4—82no—
290C4—162no—
291C4—161no—
292C4#310612yes—
293C4#312112yes—
294C4#333712yes—
295C4#335312yes—
296C2—162no—
297C2#1986012yes—
298C2#2001911yescharacteristic
299C2#2008411yescharacteristic
300C2#2655212yes—
301C2#2657811yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radical#2537C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23061C2semidirect
3. #10674C2semidirect
4. #10481C2semidirect
6. #9615C2semidirect
7. #9541C2semidirect
8. #8928C2semidirect
9. #8885C2semidirect

Maximal subgroups

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

subgroupindex
#250112
#230612
#106742
#104812
#102352
#96152
#95412
#89282
#88852