Finite Group Databasebeta

← #507669

Subgroups — #507669

Order 512.

620 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33172C211yesmaximal, characteristic
2C2×#1942C211yesmaximal, characteristic
3#13537C212yesmaximal, complemented
4#13409C212yesmaximal, complemented
5#13150C214yesmaximal, complemented
6#12993C212yesmaximal
7#12993C212yesmaximal
8#12599C211yesmaximal, characteristic
9C2×#494C2211yescharacteristic
10C2×#467—21no—
11C2×#467C2212yes—
12C22×#107C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15#2857—21no—
16C2×#389—21no—
17C2×#358C2214yes—
18C2×#358C2214yes—
19C2×C22⋊C16C2212yes—
20C2×#346C2212yes—
21#1969—21no—
22#1969C2212yes—
23#1956—21no—
24#1942—21no—
25#1942—21no—
26#1942C2212yes—
27#1942C2212yes—
28#1845C2211yescharacteristic
29C2×#340—21no—
30#1380C2214yes—
31#1380C2214yes—
32#1374C2212yes—
33C22×Q16—21no—
34C22×Q16C2312yes—
35C23×C8D411yescharacteristic
36C23×C8C2311yescharacteristic
37C2×C4⋊Q8—21no—
38C2×C22⋊Q8—22no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—41no—
42C22×C42D411yescharacteristic
43C22×C16C2314yes—
44#494—42no—
45#467—28no—
46#467—44no—
47C2×#107—21no—
48C2×#107—24no—
49C2×#107—21no—
50C2×#107—21no—
51C2×#107—42no—
52C2×#107C2312yes—
53C2×#107C2312yes—
54C2×#107C2312yes—
55C2×#103—22no—
56C2×#103—22no—
57C2×C4×C8—42no—
58C2×C4×C8D411yescharacteristic
59C2×C4×C8D411yescharacteristic
60C2×C4×C8C2311yescharacteristic
61C2×C4×C8C2311yescharacteristic
62#391—22no—
63#389—22no—
64#389—22no—
65#389—41no—
66#389—41no—
67#389—41no—
68#389—41no—
69#384—21no—
70#380—41no—
71#380—41no—
72C2×#95—41no—
73#358—28no—
74#358—44no—
75C22⋊C16—24no—
76#346—42no—
77#340—22no—
78#340—22no—
79C22×Q8—22no—
80C23×C4—41no—
81C23×C4—41no—
82C23×C4D811yescharacteristic
83C23×C4D811yescharacteristic
84C23×C4C2×D411yescharacteristic
85C2×Q16—24no—
86C2×Q16—24no—
87C2×Q16—24no—
88C2×Q16—42no—
89C2×Q16—42no—
90C2×Q16—42no—
91C22×C8—22no—
92C22×C8—22no—
93C22×C8—41no—
94C22×C8—41no—
95C22×C8SD1614yes—
96C22×C8C2×D411yescharacteristic
97C22×C8C2×D411yescharacteristic
98C22×C8C2×D411yescharacteristic
99C22×C8#4114yes—
100C22×C8C2411yescharacteristic
101C4⋊Q8—42no—
102C4⋊Q8—42no—
103C22⋊Q8—44no—
104C22⋊Q8—44no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—84no—
122C2×C22⋊C4—42no—
123C2×C42—41no—
124C2×C42—41no—
125C2×C42D811yescharacteristic
126C2×C42D811yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C16—24no—
130C2×C16—24no—
131C2×C16—24no—
132#107—24no—
133#107—22no—
134#107—24no—
135#107—24no—
136#107—42no—
137#107—42no—
138#107#4114yes—
139#103—44no—
140#103—44no—
141C4×C8—22no—
142C4×C8—22no—
143C4×C8—41no—
144C4×C8—41no—
145#95—42no—
146#95—42no—
147#95—42no—
148#95—42no—
149C24C2×D811yescharacteristic
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—42no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—44no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—82no—
172C22×C4—82no—
173C22×C4—82no—
174C22×C4—82no—
175C22×C4—82no—
176C22×C4#12412yes—
177C22×C4C2×D811yescharacteristic
178C22×C4C2×D811yescharacteristic
179C22×C4C2×D811yescharacteristic
180C22×C4C2×D811yescharacteristic
181C22×C4C2×D811yescharacteristic
182C22×C4C2×SD1612yes—
183C22×C4C22×D411yescharacteristic
184Q16—48no—
185Q16—84no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—28no—
193C2×C8—44no—
194C2×C8#12412yes—
195C2×C8C2×SD1612yes—
196C2×C8C2×SD1612yes—
197C2×C8C2×#4112yes—
198C2×C8C2×#4112yes—
199C2×C8H(5,𝔽2)12yes—
200C4⋊C4—24no—
201C4⋊C4—44no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—82no—
208C4⋊C4—84no—
209C4⋊C4—82no—
210C4⋊C4—82no—
211C4⋊C4—84no—
212C4⋊C4—82no—
213C4⋊C4—84no—
214C4⋊C4—82no—
215C22⋊C4—84no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—41no—
221C42—41no—
222C16—44no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23#48612yes—
228C23#51012yes—
229C23#51012yes—
230C23C22×D811yescharacteristic
231Q8—82no—
232Q8—82no—
233Q8—82no—
234Q8—82no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—44no—
254C2×C4—88no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4—164no—
267C2×C4#48612yes—
268C2×C4#50912yes—
269C2×C4#51012yes—
270C2×C4#52511yescharacteristic
271C2×C4C2×#12411yescharacteristic
272C2×C4#53411yescharacteristic
273C2×C4C22×SD1611yescharacteristic
274C2×C4#57511yescharacteristic
275C2×C4#57511yescharacteristic
276C8—22no—
277C8—22no—
278C8—22no—
279C8#55314yes—
280C22—22no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22#224514yes—
287C22#310011yescharacteristic
288C22C2×#48611yescharacteristic
289C22#318611yescharacteristic
290C22C2×#50911yescharacteristic
291C22C2×#51011yescharacteristic
292C22C2×#51011yescharacteristic
293C22C2×#51011yescharacteristic
294C4—22no—
295C4—22no—
296C4—22no—
297C4—41no—
298C4—41no—
299C4—84no—
300C4—162no—
301C4—162no—
302C4—162no—
303C4#259111yescharacteristic
304C4#310311yescharacteristic
305C4#314211yescharacteristic
306C4#318711yescharacteristic
307C2—24no—
308C2#2932811yescharacteristic
309C2#3367011yescharacteristic
310C2#3367311yescharacteristic
311C2C2×#224511yescharacteristic
312C2#3370711yescharacteristic
313C2#3375911yescharacteristic
314C2#3376111yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #13537C2semidirect
4. #13409C2semidirect
5. #13150C2semidirect

Maximal subgroups

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

subgroupindex
#331722
C2×#19422
#135372
#134092
#131502
#129932
#125992