Finite Group Databasebeta

← #351617

Subgroups — #351617

Order 512.

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

normal subgroups only

This group has 569 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#29941C212yesmaximal
2#29808C211yesmaximal, characteristic
3#20548C211yesmaximal, characteristic
4#9733C212yesmaximal, complemented
5#9616C212yesmaximal, complemented
6#9570C212yesmaximal, complemented
7#9569C212yesmaximal, complemented
8#9139C212yesmaximal
9#8599C211yesmaximal, characteristic
10C2×#494C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13#2857C2212yes—
14#2283C2211yescharacteristic
15#2101—22no—
16#2085—22no—
17#2077—22no—
18#2076—22no—
19#2056—22no—
20#2001—22no—
21#1988C2212yes—
22#1988C2212yes—
23#1956C2212yes—
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×Q16—22no—
40C22×M16—21no—
41C2×C4⋊Q8C2312yes—
42C2×C22⋊Q8—22no—
43C22×C42C2311yescharacteristic
44#494—24no—
45#431—24no—
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×C22⋊C8—22no—
57C2×C22⋊C8D412yes—
58C2×#97—22no—
59C2×C4×C8—22no—
60C2×C4×C8C2311yescharacteristic
61C2×C4×C8C2311yescharacteristic
62#391—22no—
63#389—22no—
64#389—22no—
65#384C2312yes—
66#383—21no—
67#382D411yescharacteristic
68#382D411yescharacteristic
69#381—21no—
70#380—22no—
71#380—22no—
72#380D412yes—
73C4×C4⋊C4C2312yes—
74C4×C22⋊C4—21no—
75#376—21no—
76#340—22no—
77#340—22no—
78#340—22no—
79#340—22no—
80#340—22no—
81#340—22no—
82#340—22no—
83#336—22no—
84#336—22no—
85#336—22no—
86#335—24no—
87#332—24no—
88#332—24no—
89C22×Q8—22no—
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4#4111yescharacteristic
93C2×Q16—44no—
94C2×Q16—82no—
95C2×Q16—82no—
96C2×M16—24no—
97C2×M16—82no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8—22no—
101C22×C8C2×D412yes—
102C22×C8C2×D412yes—
103C4⋊Q8—22no—
104C4⋊Q8—22no—
105C4⋊Q8—22no—
106C4⋊Q8—22no—
107C22⋊Q8—82no—
108C22⋊Q8—82no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4C2×D412yes—
116C2×C4⋊C4C2×D412yes—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—42no—
119C2×C42—22no—
120C2×C42—22no—
121C2×C42—22no—
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D411yescharacteristic
124C2×C42#4111yescharacteristic
125C2×C42C2411yescharacteristic
126#107—82no—
127#106—82no—
128C4⋊C8—24no—
129C4⋊C8—24no—
130C4⋊C8—22no—
131C4⋊C8—22no—
132C4⋊C8—22no—
133C4⋊C8—22no—
134C4⋊C8—22no—
135C4⋊C8#4114yes—
136#103—82no—
137#103—82no—
138C22⋊C8—24no—
139C22⋊C8—82no—
140#97—82no—
141C4×C8—22no—
142C4×C8—82no—
143C4×C8#4114yes—
144#95—22no—
145#95—21no—
146#95—21no—
147#95—21no—
148#95—21no—
149#95—22no—
150#95—42no—
151#95—42no—
152C24#12111yescharacteristic
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—42no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4#12111yescharacteristic
174C22×C4#12111yescharacteristic
175C22×C4#12111yescharacteristic
176C22×C4C22×D411yescharacteristic
177C22×C4C22×D411yescharacteristic
178C22×C4C2×#4111yescharacteristic
179Q16—88no—
180M16—84no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—24no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8#12314yes—
196C4⋊C4—22no—
197C4⋊C4—24no—
198C4⋊C4—22no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—82no—
208C4⋊C4—84no—
209C22⋊C4—82no—
210C22⋊C4—84no—
211C42—21no—
212C42—21no—
213C42—22no—
214C42—22no—
215C42—42no—
216C42—42no—
217C42C2×#4112yes—
218C42C2×#4112yes—
219C23—21no—
220C23—22no—
221C23—21no—
222C23—21no—
223C23—22no—
224C23C2×#12111yescharacteristic
225Q8—82no—
226Q8—82no—
227Q8—82no—
228Q8—82no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
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—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4C2×#12311yescharacteristic
256C2×C4#52511yescharacteristic
257C2×C4#52912yes—
258C2×C4#53212yes—
259C2×C4#57612yes—
260C2×C4#57611yescharacteristic
261C2×C4#57712yes—
262C2×C4#57812yes—
263C2×C4#57811yescharacteristic
264C8—24no—
265C8—42no—
266C8—44no—
267C8—82no—
268C8—82no—
269C22—42no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22#285411yescharacteristic
275C22#304512yes—
276C22#305911yescharacteristic
277C22#306111yescharacteristic
278C22#306912yes—
279C4—22no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—82no—
284C4—82no—
285C4—162no—
286C4—162no—
287C4#309612yes—
288C4#312211yescharacteristic
289C4#312611yescharacteristic
290C4#317512yes—
291C4#317712yes—
292C2—81no—
293C2#1953912yes—
294C2#1954312yes—
295C2#3311611yescharacteristic
296C2#3312611yescharacteristic
297C2#3313911yescharacteristic
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. #9733C2semidirect
5. #9616C2semidirect
6. #9570C2semidirect
7. #9569C2semidirect

Maximal subgroups

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

subgroupindex
#299412
#298082
#205482
#97332
#96162
#95702
#95692
#91392
#85992