Finite Group Databasebeta

← #343424

Subgroups — #343424

Order 512.

658 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 158 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#26040C212yesmaximal, complemented
2#25010C212yesmaximal
3#10649C212yesmaximal, complemented
4#10615C212yesmaximal, complemented
5#10573C211yesmaximal, characteristic, complemented
6#10430C211yesmaximal, characteristic
7#9554C212yesmaximal, complemented
8#8882C212yesmaximal, complemented
9#8818C211yesmaximal, characteristic, complemented
10C2×#480—22no—
11C2×#474—22no—
12#2551C2211yescharacteristic
13#2537C2211yescharacteristic
14D4×C4⋊C4C2212yes—
15#2086—22no—
16#2075—22no—
17#2057—22no—
18#2033—22no—
19#1954C2212yes—
20#1947C2212yes—
21#1947C2212yes—
22#1939C2212yes—
23#1862—22no—
24#1852—22no—
25#1845C2212yes—
26#1845C2212yes—
27#1845C2211yescharacteristic
28#1845C2211yescharacteristic
29#1802C2212yes—
30#1802C2212yes—
31C4×#102C2212yes—
32C2×D4⋊C8C2212yes—
33C2×C82C2211yescharacteristic
34#1296C2211yescharacteristic
35#1296C2212yes—
36#1296C2211yescharacteristic
37#1279C2212yes—
38#1277C2212yes—
39C2×C4⋊Q8—22no—
40C2×#121—22no—
41C2×C4×D4—22no—
42C2×C4×D4C2312yes—
43C22×C4⋊C4—22no—
44#480—48no—
45#474—48no—
46C2×#107—22no—
47C2×#107—22no—
48C2×#107—22no—
49C2×#107—22no—
50C2×#107—22no—
51C2×#107—22no—
52C2×#107—41no—
53C2×#107—41no—
54C2×C4⋊C8—22no—
55C2×C4⋊C8C2312yes—
56C2×C4⋊C8C2312yes—
57C2×#102—22no—
58C2×#102—22no—
59C2×#102D412yes—
60C2×#102D412yes—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×C4×C8C2312yes—
64C2×C4×C8C2311yescharacteristic
65#395—41no—
66#394—41no—
67#389—22no—
68#389—22no—
69#385—42no—
70#384C2311yescharacteristic
71#384C2312yes—
72#384C2311yescharacteristic
73#380—22no—
74#380—22no—
75C4×C4⋊C4C2312yes—
76#340—22no—
77#340—22no—
78#340—22no—
79#340—22no—
80#340D412yes—
81#340D412yes—
82#336—22no—
83D4⋊C8—24no—
84C82—22no—
85C22×Q8—41no—
86C22×D4—41no—
87C22×D4C2×D412yes—
88C23×C4—22no—
89C23×C4—42no—
90C22×C8—21no—
91C22×C8—22no—
92C22×C8—22no—
93C22×C8C2×D412yes—
94C22×C8C2×D412yes—
95C4⋊Q8—44no—
96C4⋊Q8—82no—
97#121—44no—
98#121—82no—
99C4×D4—22no—
100C4×D4—22no—
101C4×D4—22no—
102C4×D4—22no—
103C4×D4—44no—
104C4×D4—82no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—44no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—82no—
116C2×C4⋊C4C2×D412yes—
117C2×C4⋊C4C2×D412yes—
118C2×C4⋊C4C2×D412yes—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—42no—
121C2×C22⋊C4—42no—
122C2×C42—22no—
123C2×C42C2411yescharacteristic
124#107—44no—
125#107—44no—
126#107—44no—
127#107—44no—
128#107—82no—
129#107—82no—
130#107—82no—
131#107—82no—
132C4⋊C8—22no—
133C4⋊C8—22no—
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—44no—
137#102—24no—
138#102—44no—
139#102—44no—
140#102D818yes—
141C22⋊C8—44no—
142C22⋊C8—44no—
143C4×C8—22no—
144C4×C8—21no—
145C4×C8—22no—
146C4×C8#4112yes—
147#95—22no—
148#95—41no—
149#95—42no—
150#95—41no—
151C24—22no—
152C2×Q8—84no—
153C2×Q8—161no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—22no—
157C2×D4—22no—
158C2×D4—22no—
159C2×D4—84no—
160C2×D4—161no—
161C2×D4C2×D814yes—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—41no—
172C22×C4—42no—
173C22×C4—41no—
174C22×C4—84no—
175C22×C4—84no—
176C22×C4—162no—
177C22×C4C22×D412yes—
178C22×C4H(5,𝔽2)11yescharacteristic
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8C2×D814yes—
193C2×C8#13714yes—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—22no—
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—84no—
208C4⋊C4—84no—
209C4⋊C4—82no—
210C4⋊C4—81no—
211C4⋊C4—81no—
212C4⋊C4C2×D814yes—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—84no—
216C22⋊C4—84no—
217C42—42no—
218C42—42no—
219C42C2×#4111yescharacteristic
220C422−1+411yescharacteristic
221C422−1+412yes—
222C23—24no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—42no—
228C23—42no—
229C23D4211yescharacteristic
230Q8—164no—
231D4—24no—
232D4—24no—
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—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—168no—
259C2×C4#53612yes—
260C2×C4#53911yescharacteristic
261C2×C4#54712yes—
262C2×C4#56311yescharacteristic
263C2×C4C22×D812yes—
264C2×C4C2×#13712yes—
265C2×C4#57612yes—
266C2×C4#57812yes—
267C8—24no—
268C8—24no—
269C8—24no—
270C8—82no—
271C8—82no—
272C22—44no—
273C22—44no—
274C22—44no—
275C22—82no—
276C22—82no—
277C22—82no—
278C22—82no—
279C22#268011yescharacteristic
280C22#327612yes—
281C22D4×D812yes—
282C22#329211yescharacteristic
283C22#329311yescharacteristic
284C4—21no—
285C4—21no—
286C4—44no—
287C4—82no—
288C4—82no—
289C4—162no—
290C4—161no—
291C4—161no—
292C4#322612yes—
293C4#324312yes—
294C4#333412yes—
295C4#334612yes—
296C2—84no—
297C2#1988712yes—
298C2D8211yescharacteristic
299C2#2032411yescharacteristic
300C2#3107812yes—
301C2#3111011yescharacteristic
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. #26040C2semidirect
3. #10649C2semidirect
4. #10615C2semidirect
5. #10573C2semidirect
7. #9554C2semidirect
8. #8882C2semidirect
9. #8818C2semidirect

Maximal subgroups

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

subgroupindex
#260402
#250102
#106492
#106152
#105732
#104302
#95542
#88822
#88182