Finite Group Databasebeta

← #324862

Subgroups — #324862

Order 512.

534 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30228C211yesmaximal, characteristic
2#29212C211yesmaximal, characteristic
3#28748C211yesmaximal, characteristic
4#22307C211yesmaximal, characteristic
5#10598C212yesmaximal
6#10590C212yesmaximal
7#9897C211yesmaximal, characteristic
8#9705C211yesmaximal, characteristic
9#9566C212yesmaximal
10#9565C212yesmaximal
11#8277C211yesmaximal, characteristic
12C2×#487—21no—
13#2868C2211yescharacteristic
14#2859C2211yescharacteristic
15C4×C4⋊Q8C2211yescharacteristic
16#2099—22no—
17#2099—22no—
18#2059—22no—
19#2043—22no—
20#1993—21no—
21#1990C2211yescharacteristic
22#1990C2211yescharacteristic
23#1990C2211yescharacteristic
24#1990C2211yescharacteristic
25#1984—22no—
26#1981C2212yes—
27#1981C2212yes—
28#1956—21no—
29#1948C2212yes—
30#1947C2212yes—
31#1939C2212yes—
32#1916—22no—
33#1847—21no—
34#1831C2211yescharacteristic
35C8×C4⋊C4C2212yes—
36#1767C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38#1744C2211yescharacteristic
39C2×#334C2212yes—
40C2×#332C2212yes—
41C2×#332C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1277C2212yes—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8D411yescharacteristic
48C2×C4⋊Q8C2311yescharacteristic
49C2×C4⋊Q8C2311yescharacteristic
50C2×#125—21no—
51C2×C4×Q8—21no—
52#487—44no—
53C2×#107—22no—
54C2×#106—22no—
55C2×C4⋊C8—22no—
56C2×C4⋊C8C2312yes—
57C2×C4⋊C8C2312yes—
58C2×C4⋊C8C2312yes—
59C2×#103—22no—
60C2×#103—22no—
61C2×#103—22no—
62C2×#103—22no—
63C2×#97—21no—
64C2×C4×C8—21no—
65C2×C4×C8—22no—
66C2×C4×C8C2311yescharacteristic
67C2×C4×C8C2311yescharacteristic
68#391—21no—
69#389—21no—
70#389—21no—
71#389—21no—
72#389—21no—
73#389—21no—
74#389D412yes—
75#384—21no—
76#384—21no—
77#384—21no—
78#384—21no—
79#384D411yescharacteristic
80#384C2311yescharacteristic
81#384C2311yescharacteristic
82#382—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4C2312yes—
85C43C2311yescharacteristic
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#336—22no—
93#336—22no—
94#336—21no—
95#336—21no—
96#336—21no—
97#336—21no—
98#336D412yes—
99#336D412yes—
100#334—24no—
101#332—24no—
102#332—24no—
103C22×Q8—21no—
104C22×Q8—21no—
105C22×Q8—21no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8C2×D412yes—
110C22×C8C2×D412yes—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—42no—
117C4⋊Q8—42no—
118C4⋊Q8—82no—
119C4⋊Q8SD1614yes—
120#125—82no—
121C4×Q8—82no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4C2×D412yes—
143C2×C4⋊C4C2×D412yes—
144C2×C42—22no—
145C2×C42C2×D411yescharacteristic
146C2×C42C2×D411yescharacteristic
147C2×C42C2×D411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42#4111yescharacteristic
150C2×C42#4111yescharacteristic
151C2×C42C2411yescharacteristic
152#107—82no—
153#106—82no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8—82no—
160C4⋊C8#4114yes—
161#103—44no—
162#103—44no—
163#103—82no—
164#103—82no—
165#97—42no—
166C4×C8—22no—
167C4×C8—22no—
168C4×C8—42no—
169C4×C8—82no—
170#95—22no—
171#95—22no—
172#95—41no—
173#95—41no—
174C2×Q8—24no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—81no—
184C2×Q8—81no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4#12111yescharacteristic
192C22×C4#12111yescharacteristic
193C22×C4#12111yescharacteristic
194C22×C4#12111yescharacteristic
195C22×C4C22×D411yescharacteristic
196C22×C4C22×D411yescharacteristic
197C22×C4C2×#4111yescharacteristic
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—24no—
201C2×C8—22no—
202C2×C8—22no—
203C2×C8—22no—
204C2×C8—22no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C4⋊C4—22no—
214C4⋊C4—24no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—81no—
228C4⋊C4—81no—
229C4⋊C4—82no—
230C4⋊C4—81no—
231C4⋊C4—81no—
232C4⋊C4—82no—
233C4⋊C4—82no—
234C4⋊C4—82no—
235C4⋊C4—82no—
236C4⋊C4—82no—
237C4⋊C4—82no—
238C4⋊C4C2×SD1614yes—
239C42—22no—
240C42—22no—
241C42—22no—
242C42—41no—
243C42—41no—
244C42—42no—
245C42—42no—
246C42C2×SD1612yes—
247C42#13512yes—
248C42C2×#4112yes—
249C422−1+412yes—
250C23C2×#12111yescharacteristic
251Q8—84no—
252Q8—82no—
253Q8—82no—
254Q8—82no—
255Q8—82no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—24no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4#46512yes—
287C2×C4#46512yes—
288C2×C4#53211yescharacteristic
289C2×C4#53611yescharacteristic
290C2×C4#54812yes—
291C2×C4#55612yes—
292C2×C4C22×SD1611yescharacteristic
293C2×C4C2×#13511yescharacteristic
294C2×C4#57612yes—
295C2×C4#57712yes—
296C2×C4#57812yes—
297C8—44no—
298C8—42no—
299C8—42no—
300C8—82no—
301C8—82no—
302C22#254511yescharacteristic
303C22#304211yescharacteristic
304C22#304411yescharacteristic
305C22C2×#46511yescharacteristic
306C22#305811yescharacteristic
307C22#306911yescharacteristic
308C22#306911yescharacteristic
309C4—22no—
310C4—22no—
311C4—22no—
312C4—41no—
313C4—41no—
314C4—44no—
315C4—82no—
316C4—161no—
317C4—161no—
318C4—161no—
319C4—161no—
320C4#317812yes—
321C4#324212yes—
322C4#336112yes—
323C4#336212yes—
324C2#1939011yescharacteristic
325C2#1940211yescharacteristic
326C2#1951411yescharacteristic
327C2#1953011yescharacteristic
328C2#2864011yescharacteristic
329C2#2866611yescharacteristic
330C2#2867311yescharacteristic
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
#302282
#292122
#287482
#223072
#105982
#105902
#98972
#97052
#95662
#95652
#82772