Finite Group Databasebeta

← #324780

Subgroups — #324780

Order 512.

608 conjugacy classes of subgroups.

The subgroup lattice has 321 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33289C211yesmaximal, characteristic
2#29206C211yesmaximal, characteristic
3#28748C211yesmaximal, characteristic
4C8×C4⋊Q8C211yesmaximal, characteristic
5#10598C214yesmaximal
6#9896C211yesmaximal, characteristic
7#9694C211yesmaximal, characteristic
8#9565C214yesmaximal
9C4×#334C211yesmaximal, characteristic
10C2×#487—21no—
11C2×C8×Q8—21no—
12#2868C2211yescharacteristic
13#2868C2211yescharacteristic
14C4×C4⋊Q8C2211yescharacteristic
15#2058—22no—
16#2043—22no—
17#1990—21no—
18#1990—21no—
19#1990—21no—
20#1990C2212yes—
21#1990C2212yes—
22#1984—21no—
23#1981C2214yes—
24#1956—21no—
25#1948C2214yes—
26#1939—21no—
27#1939C2212yes—
28#1844—21no—
29#1842C2211yescharacteristic
30#1834—21no—
31#1834—21no—
32C8×C4⋊C4—21no—
33C8×C4⋊C4C2212yes—
34C4×C4⋊C8C2211yescharacteristic
35C4×C4⋊C8C2211yescharacteristic
36C42×C8C2211yescharacteristic
37C2×#334C2212yes—
38C2×#332C2214yes—
39#1296C2214yes—
40#1277C2212yes—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8D411yescharacteristic
47C2×C4⋊Q8C2312yes—
48C2×C4×Q8—21no—
49#487—44no—
50C8×Q8—44no—
51C2×#106—24no—
52C2×C4⋊C8—22no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8C2312yes—
56C2×C4⋊C8C2312yes—
57C2×C4⋊C8C2312yes—
58C2×#103—24no—
59C2×#103—24no—
60C2×C4×C8—22no—
61C2×C4×C8—21no—
62C2×C4×C8—21no—
63C2×C4×C8D411yescharacteristic
64C2×C4×C8D411yescharacteristic
65C2×C4×C8D411yescharacteristic
66C2×C4×C8D411yescharacteristic
67C2×C4×C8C2311yescharacteristic
68C2×C4×C8C2311yescharacteristic
69#391—21no—
70#389—21no—
71#389D412yes—
72#384—21no—
73#384—21no—
74#384—21no—
75#384—21no—
76#384—21no—
77#384—21no—
78#384—21no—
79#384D411yescharacteristic
80#384C2312yes—
81C4×C4⋊C4—21no—
82C4×C4⋊C4C2312yes—
83C43C2311yescharacteristic
84#340—22no—
85#340—22no—
86#340—22no—
87#340—22no—
88#340—24no—
89#336—22no—
90#336—22no—
91#336—21no—
92#336—21no—
93#334—24no—
94#332—28no—
95C22×Q8—21no—
96C22×Q8—22no—
97C22×Q8—41no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8—22no—
101C22×C8C2×D411yescharacteristic
102C22×C8C2×D411yescharacteristic
103C22×C8C2×D411yescharacteristic
104C22×C8C2×D411yescharacteristic
105C4⋊Q8—22no—
106C4⋊Q8—24no—
107C4⋊Q8—24no—
108C4⋊Q8—42no—
109C4⋊Q8—42no—
110C4⋊Q8—42no—
111C4⋊Q8—42no—
112C4⋊Q8—82no—
113C4⋊Q8—82no—
114C4⋊Q8—82no—
115C4⋊Q8SD1614yes—
116C4×Q8—42no—
117C4×Q8—42no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4C2×D412yes—
135C2×C4⋊C4C2×D412yes—
136C2×C42—22no—
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42#4111yescharacteristic
142C2×C42#4111yescharacteristic
143C2×C42C2411yescharacteristic
144#106—84no—
145C4⋊C8—24no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—44no—
150C4⋊C8—42no—
151C4⋊C8—42no—
152C4⋊C8#4114yes—
153#103—48no—
154#103—84no—
155C4×C8—22no—
156C4×C8—22no—
157C4×C8—22no—
158C4×C8—22no—
159C4×C8—22no—
160C4×C8—44no—
161C4×C8—42no—
162C4×C8—42no—
163C4×C8SD1614yes—
164#95—22no—
165#95—22no—
166C2×Q8—24no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—44no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—44no—
173C2×Q8—82no—
174C2×Q8—82no—
175C2×Q8—82no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4#12111yescharacteristic
181C22×C4#12111yescharacteristic
182C22×C4#12111yescharacteristic
183C22×C4#12111yescharacteristic
184C22×C4C22×D411yescharacteristic
185C22×C4C22×D411yescharacteristic
186C22×C4C2×#4111yescharacteristic
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—24no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8C2×SD1612yes—
207C2×C8C2×SD1612yes—
208C4⋊C4—22no—
209C4⋊C4—24no—
210C4⋊C4—24no—
211C4⋊C4—44no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—44no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—44no—
220C4⋊C4—82no—
221C4⋊C4—82no—
222C4⋊C4—82no—
223C4⋊C4—82no—
224C4⋊C4—82no—
225C4⋊C4—82no—
226C4⋊C4—82no—
227C4⋊C4—82no—
228C4⋊C4C2×SD1614yes—
229C42—22no—
230C42—22no—
231C42—22no—
232C42—42no—
233C42—42no—
234C42C2×SD1612yes—
235C42C2×SD1612yes—
236C42#13512yes—
237C42#13512yes—
238C42#13712yes—
239C42#13712yes—
240C42C2×#4112yes—
241C422−1+412yes—
242C23C2×#12111yescharacteristic
243Q8—48no—
244Q8—84no—
245Q8—84no—
246Q8—164no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—24no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4#46112yes—
266C2×C4#46112yes—
267C2×C4#46412yes—
268C2×C4#46412yes—
269C2×C4#46512yes—
270C2×C4#46512yes—
271C2×C4#47112yes—
272C2×C4#47312yes—
273C2×C4#53211yescharacteristic
274C2×C4#53611yescharacteristic
275C2×C4#54812yes—
276C2×C4#55612yes—
277C2×C4C22×SD1611yescharacteristic
278C2×C4C22×SD1611yescharacteristic
279C2×C4C2×#13511yescharacteristic
280C2×C4C2×#13511yescharacteristic
281C2×C4C2×#13711yescharacteristic
282C2×C4C2×#13711yescharacteristic
283C2×C4#57612yes—
284C8—22no—
285C8—22no—
286C8—44no—
287C8—41no—
288C8—41no—
289C8—82no—
290C8—82no—
291C22#254511yescharacteristic
292C22C2×#46111yescharacteristic
293C22C2×#46411yescharacteristic
294C22C2×#46511yescharacteristic
295C22#305811yescharacteristic
296C22C2×#47111yescharacteristic
297C22C2×#47311yescharacteristic
298C4—22no—
299C4—22no—
300C4—22no—
301C4—44no—
302C4—82no—
303C4—162no—
304C4—162no—
305C4#167114yes—
306C4#168014yes—
307C4#316711yescharacteristic
308C4#318511yescharacteristic
309C4#322911yescharacteristic
310C4#323111yescharacteristic
311C4#336112yes—
312C4#336212yes—
313C2C2×#167111yescharacteristic
314C2C2×#168011yescharacteristic
315C2#1919211yescharacteristic
316C2#1921411yescharacteristic
317C2#2864011yescharacteristic
318C2#2865311yescharacteristic
319C2#2866911yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ 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
#332892
#292062
#287482
C8×C4⋊Q82
#105982
#98962
#96942
#95652
C4×#3342