Finite Group Databasebeta

← #308914

Subgroups — #308914

Order 512.

512 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28828C211yesmaximal, characteristic
2#26507C212yesmaximal
3#23158C211yesmaximal, characteristic
4#10601C212yesmaximal
5#10352C212yesmaximal
6#9823C211yesmaximal, characteristic
7#9730C211yesmaximal, characteristic
8#9615C211yesmaximal, characteristic
9#9594C211yesmaximal, characteristic
10#8928C212yesmaximal
11C8×#107C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13#2571C2212yes—
14#2369C2212yes—
15#2101—22no—
16#2086—22no—
17#2085—22no—
18#1990C2212yes—
19#1987C2212yes—
20#1987C2212yes—
21#1956C2212yes—
22#1949—22no—
23#1948C2212yes—
24#1948C2212yes—
25#1942—21no—
26#1942—21no—
27#1939C2212yes—
28#1921—22no—
29#1920—22no—
30#1846—21no—
31#1845C2211yescharacteristic
32C4×#107—21no—
33C4×#107C2211yescharacteristic
34C4×#107C2211yescharacteristic
35C8×C4⋊C4C2211yescharacteristic
36C8×C4⋊C4C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×#335C2211yescharacteristic
39C2×#333C2212yes—
40C2×C82C2211yescharacteristic
41#1296C2212yes—
42#1279C2211yescharacteristic
43#1279C2211yescharacteristic
44#1279C2212yes—
45C2×C4⋊Q8C2312yes—
46#500—22no—
47#500Q814yes—
48C2×#107—21no—
49C2×#107—21no—
50C2×#107—21no—
51C2×#107—21no—
52C2×#107D411yescharacteristic
53C2×#107D411yescharacteristic
54C2×#107D411yescharacteristic
55C2×#107D411yescharacteristic
56C2×#106—22no—
57C2×#106—22no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2311yescharacteristic
61C2×C4⋊C8C2311yescharacteristic
62C2×#103—42no—
63C2×C4×C8—21no—
64C2×C4×C8—21no—
65C2×C4×C8C2311yescharacteristic
66C2×C4×C8C2311yescharacteristic
67C2×C4×C8C2311yescharacteristic
68#400—42no—
69#391—42no—
70#389—22no—
71#389—22no—
72#389—22no—
73#389—22no—
74#384C2312yes—
75#382—22no—
76#382—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4C2311yescharacteristic
80C4×C4⋊C4C2312yes—
81C4×C4⋊C4C2311yescharacteristic
82#340—22no—
83#340—22no—
84#340—22no—
85#340—22no—
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340D412yes—
91#340D411yescharacteristic
92#340D411yescharacteristic
93#336—21no—
94#336—22no—
95#336—21no—
96#335—22no—
97#335—22no—
98#333—24no—
99C82—22no—
100C22×Q8—22no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—22no—
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C22×C8C2×D411yescharacteristic
108C22×C8C2×D411yescharacteristic
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112C4⋊Q8C2×Q814yes—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4C2×D411yescharacteristic
121C2×C4⋊C4C2×D412yes—
122C2×C4⋊C4C2×D411yescharacteristic
123C2×C4⋊C4C2×D411yescharacteristic
124C2×C4⋊C4C2×D412yes—
125C2×C4⋊C4C2×D411yescharacteristic
126C2×C42—21no—
127C2×C42—22no—
128C2×C42—21no—
129C2×C42—42no—
130C2×C42C2411yescharacteristic
131#107—24no—
132#107—24no—
133#107—22no—
134#107—22no—
135#107—22no—
136#107—22no—
137#107—42no—
138#107—81no—
139#106—44no—
140#106—82no—
141C4⋊C8—22no—
142C4⋊C8—22no—
143C4⋊C8—21no—
144C4⋊C8—21no—
145C4⋊C8—21no—
146C4⋊C8—44no—
147C4⋊C8#4112yes—
148#103—84no—
149C4×C8—21no—
150C4×C8—21no—
151C4×C8—21no—
152C4×C8—21no—
153C4×C8—42no—
154C4×C8—42no—
155C4×C8C2×Q812yes—
156C4×C8#4112yes—
157#95—21no—
158#95—22no—
159#95—21no—
160#95—42no—
161#95—42no—
162#95—42no—
163#95—42no—
164#95—42no—
165C2×Q8—24no—
166C2×Q8—42no—
167C2×Q8—82no—
168C2×Q8—82no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4C22×D411yescharacteristic
178C22×C4C22×D411yescharacteristic
179C22×C4H(5,𝔽2)11yescharacteristic
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—22no—
185C2×C8—21no—
186C2×C8—22no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—42no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—41no—
199C2×C8C22⋊Q812yes—
200C2×C8C22⋊Q812yes—
201C2×C8#13512yes—
202C2×C8#13512yes—
203C2×C8#13712yes—
204C4⋊C4—21no—
205C4⋊C4—24no—
206C4⋊C4—24no—
207C4⋊C4—22no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—82no—
224C4⋊C4—162no—
225C4⋊C4—162no—
226C4⋊C4#13512yes—
227C42—41no—
228C42—41no—
229C42—42no—
230C42—41no—
231C42—41no—
232C42—42no—
233C42—162no—
234C42C22×Q811yescharacteristic
235C42C2×#4111yescharacteristic
236C42C2×#4111yescharacteristic
237C422−1+411yescharacteristic
238C23D4211yescharacteristic
239Q8—84no—
240Q8—162no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—44no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—82no—
267C2×C4C2×C22⋊Q811yescharacteristic
268C2×C4#53211yescharacteristic
269C2×C4#54211yescharacteristic
270C2×C4#54811yescharacteristic
271C2×C4D4×Q811yescharacteristic
272C2×C4#55411yescharacteristic
273C2×C4C2×#13511yescharacteristic
274C2×C4C2×#13511yescharacteristic
275C2×C4C2×#13511yescharacteristic
276C2×C4C2×#13711yescharacteristic
277C2×C4#57711yescharacteristic
278C2×C4#57811yescharacteristic
279C2×C4#57811yescharacteristic
280C2×C4#57811yescharacteristic
281C8—22no—
282C8—22no—
283C8—42no—
284C8—42no—
285C8—41no—
286C8—81no—
287C8—82no—
288C8#47914yes—
289C22#245211yescharacteristic
290C22#327811yescharacteristic
291C22#328311yescharacteristic
292C22#328311yescharacteristic
293C22#328511yescharacteristic
294C22#329011yescharacteristic
295C22#329211yescharacteristic
296C4—21no—
297C4—21no—
298C4—42no—
299C4—81no—
300C4—81no—
301C4—81no—
302C4—82no—
303C4—84no—
304C4—82no—
305C4—162no—
306C4C2×#47911yescharacteristic
307C4#308211yescharacteristic
308C4#316811yescharacteristic
309C4#317711yescharacteristic
310C4#336011yescharacteristic
311C4#337411yescharacteristic
312C4#338311yescharacteristic
313C4#338311yescharacteristic
314C2#1994211yescharacteristic
315C2#1995911yescharacteristic
316C2#2001111yescharacteristic
317C2#2001911yescharacteristic
318C2#2646111yescharacteristic
319C2#2648311yescharacteristic
320C2#2650111yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#288282
#265072
#231582
#106012
#103522
#98232
#97302
#96152
#95942
#89282
C8×#1072