Finite Group Databasebeta

← #343630

Subgroups — #343630

Order 512.

602 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 138 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#26107C212yesmaximal, complemented
2#25012C212yesmaximal
3#10654C212yesmaximal, complemented
4#10621C212yesmaximal, complemented
5#10576C211yesmaximal, characteristic, complemented
6#10430C211yesmaximal, characteristic
7#9560C212yesmaximal, complemented
8#8886C212yesmaximal, complemented
9#8819C211yesmaximal, characteristic, complemented
10C2×#481—22no—
11#2571C2211yescharacteristic
12#2537C2211yescharacteristic
13D4×C4⋊C4C2212yes—
14#2101—22no—
15#2088—22no—
16#2077—22no—
17#2034—22no—
18#1954C2212yes—
19#1948C2212yes—
20#1948C2212yes—
21#1939C2212yes—
22#1864—22no—
23#1853—22no—
24#1845C2211yescharacteristic
25#1844C2212yes—
26#1844C2211yescharacteristic
27#1834C2212yes—
28#1804—22no—
29#1802C2212yes—
30#1802C2212yes—
31C4×#102C2212yes—
32C2×D4⋊C8C2212yes—
33C2×#322C2211yescharacteristic
34#1296C2211yescharacteristic
35#1296C2212yes—
36#1296C2211yescharacteristic
37#1279C2212yes—
38#1277C2212yes—
39C2×#121—22no—
40C2×C4×D4—22no—
41C2×C4×D4C2312yes—
42C22×C4⋊C4—22no—
43#481—48no—
44C2×#107—41no—
45C2×#106—22no—
46C2×#106—22no—
47C2×#106—22no—
48C2×#106—22no—
49C2×#106—41no—
50C2×C4⋊C8—22no—
51C2×C4⋊C8C2312yes—
52C2×C4⋊C8C2312yes—
53C2×#102—22no—
54C2×#102—22no—
55C2×#102D412yes—
56C2×#102D412yes—
57C2×C22⋊C8—22no—
58C2×C22⋊C8—22no—
59C2×C4×C8C2312yes—
60C2×C4×C8C2311yescharacteristic
61#400—41no—
62#394—41no—
63#389—22no—
64#389—22no—
65#385—42no—
66#384C2311yescharacteristic
67#384C2312yes—
68#384C2311yescharacteristic
69#382—22no—
70#380—22no—
71#380—22no—
72C4×C4⋊C4C2312yes—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#340—22no—
79#340D412yes—
80#340D412yes—
81#336—22no—
82D4⋊C8—24no—
83#322—22no—
84C22×D4—41no—
85C22×D4C2×D412yes—
86C23×C4—22no—
87C23×C4—42no—
88C22×C8—21no—
89C22×C8—22no—
90C22×C8—22no—
91C22×C8C2×D412yes—
92C22×C8C2×D412yes—
93#121—44no—
94#121—82no—
95C4×D4—22no—
96C4×D4—22no—
97C4×D4—22no—
98C4×D4—22no—
99C4×D4—44no—
100C4×D4—82no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—44no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—82no—
112C2×C4⋊C4C2×D412yes—
113C2×C4⋊C4C2×D412yes—
114C2×C4⋊C4C2×D412yes—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C42—22no—
119C2×C42C2411yescharacteristic
120#107—82no—
121#106—44no—
122#106—44no—
123#106—82no—
124#106—82no—
125#106—82no—
126C4⋊C8—22no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—82no—
131#102—24no—
132#102—42no—
133#102—44no—
134#102—44no—
135C22⋊C8—44no—
136C22⋊C8—44no—
137C4×C8—22no—
138C4×C8—21no—
139C4×C8—22no—
140C4×C8#4112yes—
141#95—22no—
142#95—41no—
143#95—42no—
144#95—41no—
145#95—42no—
146C24—22no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—84no—
153C2×D4—161no—
154C2×D4AGL(1,ℤ/8)14yes—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—42no—
167C22×C4—84no—
168C22×C4—84no—
169C22×C4—162no—
170C22×C4C22×D412yes—
171C22×C4H(5,𝔽2)11yescharacteristic
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—42no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—82no—
202C4⋊C4—81no—
203C4⋊C4—81no—
204C4⋊C4—84no—
205C4⋊C4—161no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—84no—
209C22⋊C4—84no—
210C42—42no—
211C42—42no—
212C42C2×#4111yescharacteristic
213C422−1+411yescharacteristic
214C422−1+412yes—
215C23—24no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23D4211yescharacteristic
223D4—24no—
224D4—24no—
225D4—42no—
226D4—42no—
227D4—164no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—81no—
248C2×C4—82no—
249C2×C4—81no—
250C2×C4—168no—
251C2×C4#53612yes—
252C2×C4#53911yescharacteristic
253C2×C4#54712yes—
254C2×C4#56311yescharacteristic
255C2×C4C2×AGL(1,ℤ/8)12yes—
256C2×C4#57512yes—
257C2×C4#57712yes—
258C2×C4#57712yes—
259C8—24no—
260C8—42no—
261C8—42no—
262C8—82no—
263C8—82no—
264C22—44no—
265C22—44no—
266C22—44no—
267C22—82no—
268C22—82no—
269C22—82no—
270C22—82no—
271C22#268011yescharacteristic
272C22#326512yes—
273C22#326712yes—
274C22#329111yescharacteristic
275C22#329111yescharacteristic
276C4—21no—
277C4—21no—
278C4—82no—
279C4—82no—
280C4—82no—
281C4—161no—
282C4—161no—
283C4—162no—
284C4#323412yes—
285C4#324412yes—
286C4#334512yes—
287C4#335112yes—
288C2—84no—
289C2#2006711yescharacteristic
290C2#2008112yes—
291C2#2014311yescharacteristic
292C2#3109412yes—
293C2#3111211yescharacteristic
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
1. #26107C2semidirect
3. #10654C2semidirect
4. #10621C2semidirect
5. #10576C2semidirect
7. #9560C2semidirect
8. #8886C2semidirect
9. #8819C2semidirect

Maximal subgroups

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

subgroupindex
#261072
#250122
#106542
#106212
#105762
#104302
#95602
#88862
#88192