Finite Group Databasebeta

← #507749

Subgroups — #507749

Order 512.

490 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#500C211yesmaximal, characteristic
2#21276C211yesmaximal, characteristic
3#13505C211yesmaximal, characteristic
4#13481C211yesmaximal, characteristic
5#13452C211yesmaximal, characteristic
6#13364C211yesmaximal, characteristic
7#13132C212yesmaximal
8#13111C212yesmaximal
9#13109C212yesmaximal
10#12936C212yesmaximal
11#12894C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×C4×Q16C2211yescharacteristic
14C4×C4⋊Q8—21no—
15#2295—21no—
16C2×#367C2212yes—
17C2×#366C2212yes—
18C2×#358C2212yes—
19C2×#358C2212yes—
20#1990—21no—
21#1969C2211yescharacteristic
22#1969C2211yescharacteristic
23#1942—21no—
24#1942C2211yescharacteristic
25#1942C2211yescharacteristic
26#1942C2212yes—
27#1933—21no—
28#1933—21no—
29#1845C2211yescharacteristic
30C4×#107—21no—
31C4×#107C2211yescharacteristic
32C4×#107C2212yes—
33C4×#103—21no—
34C42×C8C2211yescharacteristic
35#1380C2212yes—
36#1380C2212yes—
37#1380C2212yes—
38#1380C2212yes—
39#1374C2211yescharacteristic
40#1374C2212yes—
41#1374C2212yes—
42#1374C2211yescharacteristic
43C22×Q16C2311yescharacteristic
44C2×C4⋊Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—41no—
47C22×C16C2312yes—
48C22×C16C2312yes—
49#500—24no—
50C4×Q16—22no—
51C4×Q16—22no—
52C2×#107—21no—
53C2×#107—21no—
54C2×#107C2312yes—
55C2×#107C2311yescharacteristic
56C2×#107C2311yescharacteristic
57C2×#107C2312yes—
58C2×#107C2311yescharacteristic
59C2×#103—21no—
60C2×#103—21no—
61C2×#103—21no—
62C2×C4×C8D411yescharacteristic
63C2×C4×C8D411yescharacteristic
64C2×C4×C8D411yescharacteristic
65C2×C4×C8C2311yescharacteristic
66C2×C4×C8C2311yescharacteristic
67C2×C4×C8C2311yescharacteristic
68#391—21no—
69#391—21no—
70#391—41no—
71#389—22no—
72#389—21no—
73#389—21no—
74#389—41no—
75#387—41no—
76#387—41no—
77#384—21no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4—41no—
81C4×C4⋊C4—41no—
82#376—41no—
83C43D411yescharacteristic
84#367—24no—
85#366—24no—
86#358—24no—
87#358—24no—
88#340—21no—
89#340—21no—
90#340—21no—
91#340—22no—
92#340—22no—
93C22×Q8—21no—
94C22×Q8—41no—
95C2×Q16—22no—
96C2×Q16—21no—
97C2×Q16—22no—
98C2×Q16#4112yes—
99C22×C8C2×D411yescharacteristic
100C22×C8C2×D411yescharacteristic
101C22×C8C2×D411yescharacteristic
102C22×C8C2411yescharacteristic
103C4⋊Q8—24no—
104C4⋊Q8—42no—
105C4×Q8—24no—
106C4×Q8—42no—
107C4×Q8—84no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C42—41no—
121C2×C42—42no—
122C2×C42—41no—
123C2×C42D811yescharacteristic
124C2×C42D811yescharacteristic
125C2×C42D811yescharacteristic
126C2×C42D811yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C16—22no—
131C2×C16—22no—
132C2×C16—22no—
133C2×C16—22no—
134C2×C16—22no—
135C2×C16—22no—
136#107—24no—
137#107—21no—
138#107—22no—
139#107—22no—
140#107—22no—
141#107—22no—
142#107—22no—
143#107—42no—
144#107#4114yes—
145#107#4112yes—
146#103—24no—
147#103—42no—
148#103—42no—
149C4×C8—22no—
150C4×C8—21no—
151C4×C8—21no—
152C4×C8—22no—
153C4×C8—22no—
154C4×C8—41no—
155C4×C8—41no—
156#95—41no—
157#95—41no—
158#95—41no—
159#95—41no—
160#95—41no—
161#95—42no—
162#95—41no—
163#95—41no—
164#95—42no—
165C2×Q8—22no—
166C2×Q8—44no—
167C2×Q8—41no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8—41no—
172C2×Q8—82no—
173C2×Q8—82no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—42no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—42no—
180C22×C4C2×D811yescharacteristic
181C22×C4C2×D811yescharacteristic
182C22×C4C2×D811yescharacteristic
183C22×C4C2×D811yescharacteristic
184C22×C4C2×D811yescharacteristic
185C22×C4C2×D811yescharacteristic
186C22×C4C22×D411yescharacteristic
187Q16—24no—
188Q16—42no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8#13712yes—
198C2×C8C2×#4111yescharacteristic
199C2×C8C2×#4112yes—
200C2×C8C2×#4111yescharacteristic
201C2×C82−1+412yes—
202C4⋊C4—22no—
203C4⋊C4—24no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—41no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—82no—
215C4⋊C4—84no—
216C4⋊C4—82no—
217C4⋊C4—82no—
218C4⋊C4—82no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—22no—
226C42—22no—
227C42—22no—
228C42—22no—
229C42—44no—
230C42—84no—
231C42—82no—
232C16—42no—
233C16—42no—
234C23C22×D811yescharacteristic
235Q8—42no—
236Q8—42no—
237Q8—88no—
238Q8—81no—
239Q8—81no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—82no—
260C2×C4—81no—
261C2×C4—82no—
262C2×C4—81no—
263C2×C4—82no—
264C2×C4—81no—
265C2×C4—82no—
266C2×C4—81no—
267C2×C4—82no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4#50812yes—
273C2×C4#50912yes—
274C2×C4#51012yes—
275C2×C4#51012yes—
276C2×C4#52511yescharacteristic
277C2×C4#52511yescharacteristic
278C2×C4#53611yescharacteristic
279C2×C4C2×#13711yescharacteristic
280C2×C4#57511yescharacteristic
281C2×C4#57511yescharacteristic
282C8—22no—
283C8—41no—
284C8—41no—
285C8#55212yes—
286C8#55612yes—
287C22#310011yescharacteristic
288C22#310011yescharacteristic
289C22#322611yescharacteristic
290C22C2×#50811yescharacteristic
291C22C2×#50911yescharacteristic
292C22C2×#51011yescharacteristic
293C22C2×#51011yescharacteristic
294C4—22no—
295C4—22no—
296C4—22no—
297C4—22no—
298C4—41no—
299C4—41no—
300C4—82no—
301C4—84no—
302C4—82no—
303C4—161no—
304C4—162no—
305C4—161no—
306C4#259911yescharacteristic
307C4#311611yescharacteristic
308C4#311711yescharacteristic
309C4#324011yescharacteristic
310C2#2938811yescharacteristic
311C2#3366611yescharacteristic
312C2#3366811yescharacteristic
313C2#3367011yescharacteristic
314C2#3367311yescharacteristic
315C2#3379711yescharacteristic
316C2#3380411yescharacteristic
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
C4×#5002
#212762
#135052
#134812
#134522
#133642
#131322
#131112
#131092
#129362
#128942