Finite Group Databasebeta

← #351620

Subgroups — #351620

Order 512.

561 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 561 conjugacy classes of subgroups — too many to draw. Only its 150 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#29926C212yesmaximal
2#29876C211yesmaximal, characteristic
3#20548C211yesmaximal, characteristic
4#9731C212yesmaximal, complemented
5#9616C212yesmaximal, complemented
6#9574C212yesmaximal, complemented
7#9573C212yesmaximal, complemented
8#9137C212yesmaximal
9#8599C211yesmaximal, characteristic
10C2×#500C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13#2857C2212yes—
14#2283C2211yescharacteristic
15#2097—22no—
16#2086—22no—
17#2075—22no—
18#2074—22no—
19#2036—22no—
20#2035—22no—
21#1990C2212yes—
22#1981C2212yes—
23#1981C2212yes—
24#1948C2212yes—
25#1947C2212yes—
26#1834C2211yescharacteristic
27#1797C2212yes—
28#1770C2212yes—
29C8×C4⋊C4C2212yes—
30#1768—22no—
31#1753—22no—
32C2×#335C2212yes—
33C2×#332C2212yes—
34C2×#332C2212yes—
35#1296C2212yes—
36#1296C2212yes—
37#1277C2211yescharacteristic
38#1277C2211yescharacteristic
39C22×M16—21no—
40C2×C4⋊Q8C2312yes—
41C2×C22⋊Q8—22no—
42C22×C42C2311yescharacteristic
43#500—24no—
44#431—24no—
45C2×#107—22no—
46C2×#107—22no—
47C2×#106—22no—
48#423—24no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8D412yes—
51C2×C4⋊C8C2312yes—
52C2×C4⋊C8C2312yes—
53C2×C4⋊C8C2312yes—
54C2×#103—22no—
55C2×#103—22no—
56C2×#103—22no—
57C2×#103—22no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8D412yes—
60C2×#97—22no—
61C2×C4×C8—22no—
62C2×C4×C8C2311yescharacteristic
63C2×C4×C8C2311yescharacteristic
64#391—22no—
65#389—22no—
66#389—22no—
67#384C2312yes—
68#383—21no—
69#382D411yescharacteristic
70#382D411yescharacteristic
71#381—21no—
72#380—22no—
73#380—22no—
74#380D412yes—
75C4×C4⋊C4C2312yes—
76C4×C22⋊C4—21no—
77#376—21no—
78#340—22no—
79#340—24no—
80#340—22no—
81#340—22no—
82#336—22no—
83#336—22no—
84#336—22no—
85#335—24no—
86#332—24no—
87#332—24no—
88C22×Q8—22no—
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C23×C4#4111yescharacteristic
92C2×M16—24no—
93C2×M16—82no—
94C22×C8—22no—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8C2×D412yes—
98C22×C8C2×D412yes—
99C4⋊Q8—22no—
100C4⋊Q8—22no—
101C4⋊Q8—22no—
102C4⋊Q8—22no—
103C22⋊Q8—82no—
104C22⋊Q8—82no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4C2×D412yes—
112C2×C4⋊C4C2×D412yes—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—42no—
115C2×C42—22no—
116C2×C42—22no—
117C2×C42—22no—
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42#4111yescharacteristic
121C2×C42C2411yescharacteristic
122#107—44no—
123#107—82no—
124#106—82no—
125C4⋊C8—24no—
126C4⋊C8—24no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8—22no—
132C4⋊C8#4114yes—
133#103—82no—
134#103—82no—
135#103—82no—
136#103—82no—
137C22⋊C8—24no—
138C22⋊C8—82no—
139#97—82no—
140C4×C8—22no—
141C4×C8—82no—
142C4×C8#4114yes—
143#95—22no—
144#95—21no—
145#95—21no—
146#95—21no—
147#95—21no—
148#95—22no—
149#95—42no—
150#95—42no—
151C24#12111yescharacteristic
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8—42no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—21no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4#12111yescharacteristic
173C22×C4#12111yescharacteristic
174C22×C4#12111yescharacteristic
175C22×C4C22×D411yescharacteristic
176C22×C4C22×D411yescharacteristic
177C22×C4C2×#4111yescharacteristic
178M16—84no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—24no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8#12314yes—
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—42no—
206C4⋊C4—82no—
207C4⋊C4—84no—
208C22⋊C4—82no—
209C22⋊C4—84no—
210C42—21no—
211C42—21no—
212C42—22no—
213C42—22no—
214C42—42no—
215C42—42no—
216C42C2×#4112yes—
217C42C2×#4112yes—
218C23—21no—
219C23—22no—
220C23—21no—
221C23—21no—
222C23—22no—
223C23C2×#12111yescharacteristic
224Q8—82no—
225Q8—82no—
226Q8—82no—
227Q8—82no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—41no—
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—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4C2×#12311yescharacteristic
255C2×C4#52511yescharacteristic
256C2×C4#52912yes—
257C2×C4#53212yes—
258C2×C4#57612yes—
259C2×C4#57611yescharacteristic
260C2×C4#57712yes—
261C2×C4#57812yes—
262C2×C4#57811yescharacteristic
263C8—24no—
264C8—42no—
265C8—44no—
266C8—82no—
267C8—82no—
268C22—42no—
269C22—41no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22#285411yescharacteristic
274C22#304612yes—
275C22#305811yescharacteristic
276C22#306111yescharacteristic
277C22#306912yes—
278C4—22no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—82no—
283C4—82no—
284C4—162no—
285C4—162no—
286C4#309712yes—
287C4#312311yescharacteristic
288C4#312711yescharacteristic
289C4#317512yes—
290C4#317612yes—
291C2—81no—
292C2#1954612yes—
293C2#1955012yes—
294C2#3311711yescharacteristic
295C2#3312611yescharacteristic
296C2#3313811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
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
4. #9731C2semidirect
5. #9616C2semidirect
6. #9574C2semidirect
7. #9573C2semidirect

Maximal subgroups

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

subgroupindex
#299262
#298762
#205482
#97312
#96162
#95742
#95732
#91372
#85992