Finite Group Databasebeta

← #343443

Subgroups — #343443

Order 512.

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

normal subgroups only

This group has 580 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#24965C212yesmaximal
3#10655C212yesmaximal, complemented
4#10622C212yesmaximal, complemented
5#10577C211yesmaximal, characteristic, complemented
6#10432C211yesmaximal, characteristic
7#9554C212yesmaximal, complemented
8#8886C212yesmaximal, complemented
9#8815C211yesmaximal, characteristic, complemented
10#2571C2211yescharacteristic
11#2537C2211yescharacteristic
12D4×C4⋊C4C2212yes—
13#2101—22no—
14#2088—22no—
15#2075—22no—
16#2033—22no—
17#1954C2212yes—
18#1948C2212yes—
19#1947C2212yes—
20#1939C2212yes—
21#1874—22no—
22#1864—22no—
23#1844C2211yescharacteristic
24#1844C2211yescharacteristic
25#1834C2212yes—
26#1834C2212yes—
27#1823—22no—
28#1804—22no—
29#1802C2212yes—
30#1802C2212yes—
31C4×#102C2212yes—
32C2×D4⋊C8C2212yes—
33C2×C82C2211yescharacteristic
34#1296C2211yescharacteristic
35#1296C2212yes—
36#1296C2211yescharacteristic
37#1279C2212yes—
38#1277C2212yes—
39C2×#121—22no—
40C2×C4×D4—22no—
41C2×C4×D4C2312yes—
42C22×C4⋊C4—22no—
43C2×#107—22no—
44C2×#106—22no—
45C2×#106—41no—
46C2×#106—41no—
47C2×C4⋊C8—22no—
48C2×C4⋊C8C2312yes—
49C2×C4⋊C8C2312yes—
50C2×#102—22no—
51C2×#102—22no—
52C2×#102D412yes—
53C2×#102D412yes—
54C2×C22⋊C8—22no—
55C2×C22⋊C8—22no—
56C2×C4×C8C2312yes—
57C2×C4×C8C2311yescharacteristic
58#400—41no—
59#394—41no—
60#389—22no—
61#389—22no—
62#385—42no—
63#384C2311yescharacteristic
64#384C2312yes—
65#384C2311yescharacteristic
66#382—22no—
67#380—22no—
68#380—22no—
69C4×C4⋊C4C2312yes—
70#340—22no—
71#340—22no—
72#340—22no—
73#340—22no—
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#340D412yes—
79#340D412yes—
80#336—22no—
81D4⋊C8—24no—
82C82—22no—
83C22×D4—41no—
84C22×D4C2×D412yes—
85C23×C4—22no—
86C23×C4—42no—
87C22×C8—21no—
88C22×C8—22no—
89C22×C8—22no—
90C22×C8C2×D412yes—
91C22×C8C2×D412yes—
92#121—44no—
93#121—82no—
94C4×D4—22no—
95C4×D4—22no—
96C4×D4—22no—
97C4×D4—22no—
98C4×D4—44no—
99C4×D4—82no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—44no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—82no—
111C2×C4⋊C4C2×D412yes—
112C2×C4⋊C4C2×D412yes—
113C2×C4⋊C4C2×D412yes—
114C2×C22⋊C4—22no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—42no—
117C2×C42—22no—
118C2×C42C2411yescharacteristic
119#107—82no—
120#106—82no—
121#106—82no—
122#106—82no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—22no—
127C4⋊C8—82no—
128#102—24no—
129#102—42no—
130#102—44no—
131#102—82no—
132C22⋊C8—82no—
133C22⋊C8—82no—
134C4×C8—22no—
135C4×C8—21no—
136C4×C8—22no—
137C4×C8#4112yes—
138#95—22no—
139#95—41no—
140#95—42no—
141#95—41no—
142#95—42no—
143C24—22no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—84no—
150C2×D4—161no—
151C2×D4#13514yes—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—42no—
164C22×C4—84no—
165C22×C4—84no—
166C22×C4—162no—
167C22×C4C22×D412yes—
168C22×C4H(5,𝔽2)11yescharacteristic
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—42no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—81no—
199C4⋊C4—82no—
200C4⋊C4—81no—
201C4⋊C4—84no—
202C4⋊C4—161no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—84no—
206C22⋊C4—84no—
207C42—42no—
208C42—42no—
209C42C2×#4111yescharacteristic
210C422−1+411yescharacteristic
211C422−1+412yes—
212C23—24no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23D4211yescharacteristic
220D4—24no—
221D4—24no—
222D4—42no—
223D4—42no—
224D4—164no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—81no—
245C2×C4—82no—
246C2×C4—81no—
247C2×C4—168no—
248C2×C4#53612yes—
249C2×C4#53911yescharacteristic
250C2×C4#54712yes—
251C2×C4#56311yescharacteristic
252C2×C4C2×#13512yes—
253C2×C4#57512yes—
254C2×C4#57612yes—
255C2×C4#57712yes—
256C8—24no—
257C8—42no—
258C8—42no—
259C8—82no—
260C8—82no—
261C22—44no—
262C22—44no—
263C22—44no—
264C22—82no—
265C22—82no—
266C22—82no—
267C22—82no—
268C22#268011yescharacteristic
269C22#327512yes—
270C22#328012yes—
271C22#329111yescharacteristic
272C22#329111yescharacteristic
273C4—21no—
274C4—21no—
275C4—82no—
276C4—82no—
277C4—82no—
278C4—161no—
279C4—161no—
280C4—162no—
281C4#323212yes—
282C4#324112yes—
283C4#333712yes—
284C4#334712yes—
285C2—84no—
286C2#1990112yes—
287C2#1998711yescharacteristic
288C2#2029411yescharacteristic
289C2#3108012yes—
290C2#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. #10655C2semidirect
4. #10622C2semidirect
5. #10577C2semidirect
7. #9554C2semidirect
8. #8886C2semidirect
9. #8815C2semidirect

Maximal subgroups

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

subgroupindex
#261072
#249652
#106552
#106222
#105772
#104322
#95542
#88862
#88152