Finite Group Databasebeta

← #509054

Subgroups — #509054

Order 512.

428 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33267C211yesmaximal, characteristic
2#33258C211yesmaximal, characteristic
3#13537C211yesmaximal, characteristic
4#13531C211yesmaximal, characteristic
5#13516C212yesmaximal
6#13505C211yesmaximal, characteristic
7#13481C211yesmaximal, characteristic
8#13475C212yesmaximal
9#13167C212yesmaximal
10#13150C212yesmaximal
11#12653C211yesmaximal, characteristic
12#2863—21no—
13#2859—21no—
14C2×#367C2212yes—
15C2×#366C2212yes—
16C2×C4⋊C16C2212yes—
17C2×#358C2212yes—
18C2×#358C2212yes—
19C2×#346C2212yes—
20#1969C2211yescharacteristic
21#1969C2211yescharacteristic
22#1969C2211yescharacteristic
23#1942—21no—
24#1942—21no—
25#1942C2212yes—
26#1942C2211yescharacteristic
27#1942C2212yes—
28#1942C2211yescharacteristic
29#1942C2211yescharacteristic
30#1933—21no—
31#1933—21no—
32#1933—21no—
33#1845C2211yescharacteristic
34#1845C2211yescharacteristic
35#1834—21no—
36C42×C8C2211yescharacteristic
37#1380C2212yes—
38#1380C2212yes—
39#1380C2212yes—
40#1380C2212yes—
41#1374C2211yescharacteristic
42#1374C2211yescharacteristic
43C22×Q16C2311yescharacteristic
44C2×#125—41no—
45C22×C16C2312yes—
46C22×C16C2312yes—
47C2×#107—21no—
48C2×#107—21no—
49C2×#107C2312yes—
50C2×#107C2312yes—
51C2×#107C2311yescharacteristic
52C2×#107C2311yescharacteristic
53C2×#107C2311yescharacteristic
54C2×#103—21no—
55C2×#103—21no—
56C2×#103—21no—
57C2×C4×C8D411yescharacteristic
58C2×C4×C8D411yescharacteristic
59C2×C4×C8D411yescharacteristic
60C2×C4×C8C2311yescharacteristic
61C2×C4×C8C2311yescharacteristic
62C2×C4×C8C2311yescharacteristic
63#391—21no—
64#391—21no—
65#391—21no—
66#389—22no—
67#389—22no—
68#389—21no—
69#389—21no—
70#389—21no—
71#389—41no—
72#389—41no—
73#387—41no—
74#387—41no—
75#387—41no—
76#384—21no—
77#384—21no—
78#383—41no—
79#382—41no—
80C43D411yescharacteristic
81#367—24no—
82#366—42no—
83C4⋊C16—24no—
84#358—24no—
85#358—42no—
86#346—42no—
87#340—22no—
88#340—22no—
89#340—21no—
90#340—21no—
91#340—21no—
92C22×Q8—21no—
93C2×Q16—22no—
94C2×Q16—21no—
95C2×Q16—22no—
96C2×Q16—21no—
97C22×C8C2×D411yescharacteristic
98C22×C8C2×D411yescharacteristic
99C22×C8C2×D411yescharacteristic
100C22×C8C2411yescharacteristic
101#125—84no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C42D811yescharacteristic
117C2×C42D811yescharacteristic
118C2×C42D811yescharacteristic
119C2×C42D811yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C16—22no—
124C2×C16—22no—
125C2×C16—22no—
126C2×C16—22no—
127C2×C16—22no—
128C2×C16#4114yes—
129#107—22no—
130#107—22no—
131#107—22no—
132#107—22no—
133#107—21no—
134#107—22no—
135#107—22no—
136#107—21no—
137#107—42no—
138#107—42no—
139#103—42no—
140#103—42no—
141#103—42no—
142C4×C8—22no—
143C4×C8—22no—
144C4×C8—22no—
145C4×C8—21no—
146C4×C8—22no—
147C4×C8—21no—
148C4×C8—41no—
149#95—42no—
150#95—42no—
151#95—41no—
152#95—41no—
153#95—41no—
154#95—41no—
155#95—41no—
156#95—41no—
157#95—41no—
158#95—41no—
159#95—41no—
160C2×Q8—41no—
161C2×Q8—41no—
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4C2×D811yescharacteristic
173C22×C4C2×D811yescharacteristic
174C22×C4C2×D811yescharacteristic
175C22×C4C2×D811yescharacteristic
176C22×C4C2×D811yescharacteristic
177C22×C4C2×D811yescharacteristic
178C22×C4C22×D411yescharacteristic
179Q16—42no—
180Q16—42no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8#13512yes—
189C2×C8#13712yes—
190C2×C8C2×#4112yes—
191C2×C8H(5,𝔽2)11yescharacteristic
192C2×C82−1+412yes—
193C2×C82−1+411yescharacteristic
194C4⋊C4—42no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—42no—
204C4⋊C4—82no—
205C4⋊C4—82no—
206C4⋊C4—84no—
207C4⋊C4—84no—
208C4⋊C4—82no—
209C4⋊C4—82no—
210C4⋊C4—82no—
211C4⋊C4—82no—
212C4⋊C4—82no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—22no—
220C42—22no—
221C42—22no—
222C42—41no—
223C16—24no—
224C16—42no—
225C23C22×D811yescharacteristic
226Q8—81no—
227Q8—81no—
228Q8—81no—
229Q8—81no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—82no—
252C2×C4—81no—
253C2×C4—82no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—82no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—82no—
262C2×C4—81no—
263C2×C4#50912yes—
264C2×C4#51012yes—
265C2×C4#51012yes—
266C2×C4#53211yescharacteristic
267C2×C4#53611yescharacteristic
268C2×C4#53711yescharacteristic
269C2×C4C2×#13511yescharacteristic
270C2×C4C2×#13711yescharacteristic
271C2×C4#57511yescharacteristic
272C8—22no—
273C8—22no—
274C8—41no—
275C8#53912yes—
276C8#54112yes—
277C22#316911yescharacteristic
278C22#322611yescharacteristic
279C22#324511yescharacteristic
280C22C2×#50911yescharacteristic
281C22C2×#51011yescharacteristic
282C22C2×#51011yescharacteristic
283C22#341411yescharacteristic
284C4—22no—
285C4—22no—
286C4—22no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—161no—
291C4—162no—
292C4—161no—
293C4—161no—
294C4—161no—
295C4—162no—
296C4#266411yescharacteristic
297C4#317211yescharacteristic
298C4#323211yescharacteristic
299C4#324811yescharacteristic
300C2#3037611yescharacteristic
301C2#3374711yescharacteristic
302C2#3375111yescharacteristic
303C2#3380211yescharacteristic
304C2#3380411yescharacteristic
305C2#3381711yescharacteristic
306C2#3382111yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#332672
#332582
#135372
#135312
#135162
#135052
#134812
#134752
#131672
#131502
#126532