Finite Group Databasebeta

← #325392

Subgroups — #325392

Order 512.

544 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30305C211yesmaximal, characteristic
2#29150C211yesmaximal, characteristic
3#28717C211yesmaximal, characteristic
4#22238C211yesmaximal, characteristic
5#10626C212yesmaximal
6#10620C212yesmaximal
7#9905C211yesmaximal, characteristic
8#9897C211yesmaximal, characteristic
9#8647C212yesmaximal
10#8643C212yesmaximal
11#8277C211yesmaximal, characteristic
12C2×#487—21no—
13C2×C8×Q8C2212yes—
14#2868C2211yescharacteristic
15#2859C2211yescharacteristic
16C4×C4⋊Q8C2211yescharacteristic
17#2084—22no—
18#2072—22no—
19#2058—22no—
20#2026—22no—
21#1990—21no—
22#1984C2212yes—
23#1960—21no—
24#1916—22no—
25#1916—22no—
26#1844C2211yescharacteristic
27#1844C2211yescharacteristic
28#1836—21no—
29#1834C2211yescharacteristic
30#1834C2211yescharacteristic
31#1831C2211yescharacteristic
32#1803C2212yes—
33#1803C2212yes—
34#1803C2212yes—
35#1803C2212yes—
36#1767C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38#1744C2211yescharacteristic
39C2×#334C2212yes—
40#1296C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1277C2212yes—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8D411yescharacteristic
49C2×#125—21no—
50C2×C4×Q8C2312yes—
51#487—44no—
52C8×Q8—24no—
53C8×Q8—24no—
54C2×#106—22no—
55C2×#106—22no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8C2312yes—
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2312yes—
60C2×#103—22no—
61C2×#103—22no—
62C2×#103—22no—
63C2×#103—22no—
64C2×#97—21no—
65C2×C4×C8—22no—
66C2×C4×C8—21no—
67C2×C4×C8C2311yescharacteristic
68C2×C4×C8C2311yescharacteristic
69#391D412yes—
70#389—21no—
71#389—21no—
72#389—21no—
73#389—21no—
74#389—21no—
75#389—21no—
76#384—21no—
77#384—21no—
78#384—21no—
79#384D411yescharacteristic
80#384C2311yescharacteristic
81#384C2311yescharacteristic
82#384C2311yescharacteristic
83#384C2311yescharacteristic
84#382—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C43C2311yescharacteristic
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#336—24no—
95#336—21no—
96#336—21no—
97#336—21no—
98#336—21no—
99#336D412yes—
100#336D412yes—
101#334—24no—
102C22×Q8—41no—
103C22×Q8C2×D412yes—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8C2×D412yes—
108C22×C8C2×D412yes—
109C4⋊Q8—22no—
110C4⋊Q8—42no—
111C4⋊Q8—42no—
112C4⋊Q8—82no—
113C4⋊Q8—82no—
114C4⋊Q8SD1614yes—
115#125—82no—
116C4×Q8—22no—
117C4×Q8—22no—
118C4×Q8—22no—
119C4×Q8—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—22no—
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—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
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×C42—22no—
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D411yescharacteristic
146C2×C42C2×D411yescharacteristic
147C2×C42C2×D411yescharacteristic
148C2×C42#4111yescharacteristic
149C2×C42#4111yescharacteristic
150C2×C42C2411yescharacteristic
151#106—82no—
152#106—82no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8—44no—
160#103—44no—
161#103—44no—
162#103—82no—
163#103—82no—
164#97—42no—
165C4×C8—22no—
166C4×C8—44no—
167C4×C8—42no—
168C4×C8#4114yes—
169#95—22no—
170#95—22no—
171#95—41no—
172#95—41no—
173C2×Q8—22no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—22no—
177C2×Q8—22no—
178C2×Q8—82no—
179C2×Q8—82no—
180C2×Q8—82no—
181C2×Q8C2×SD1614yes—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4#12111yescharacteristic
189C22×C4#12111yescharacteristic
190C22×C4#12111yescharacteristic
191C22×C4#12111yescharacteristic
192C22×C4C22×D411yescharacteristic
193C22×C4C22×D411yescharacteristic
194C22×C4C2×#4111yescharacteristic
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8#12414yes—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—81no—
227C4⋊C4—81no—
228C4⋊C4—81no—
229C4⋊C4—81no—
230C4⋊C4—81no—
231C4⋊C4—81no—
232C4⋊C4—82no—
233C4⋊C4—82no—
234C4⋊C4—82no—
235C4⋊C4—82no—
236C4⋊C4—82no—
237C42—22no—
238C42—22no—
239C42—22no—
240C42—41no—
241C42—41no—
242C42—42no—
243C42—42no—
244C42C2×SD1612yes—
245C42#13512yes—
246C42C2×#4112yes—
247C422−1+412yes—
248C23C2×#12111yescharacteristic
249Q8—24no—
250Q8—24no—
251Q8—42no—
252Q8—42no—
253Q8—164no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—24no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—81no—
273C2×C4—81no—
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#46512yes—
285C2×C4#46512yes—
286C2×C4C2×#12411yescharacteristic
287C2×C4#53611yescharacteristic
288C2×C4#54712yes—
289C2×C4#55612yes—
290C2×C4C22×SD1611yescharacteristic
291C2×C4C2×#13511yescharacteristic
292C2×C4#57712yes—
293C2×C4#57712yes—
294C2×C4#57812yes—
295C8—24no—
296C8—42no—
297C8—82no—
298C8—82no—
299C8—82no—
300C22#254711yescharacteristic
301C22#304311yescharacteristic
302C22#304411yescharacteristic
303C22C2×#46511yescharacteristic
304C22#305711yescharacteristic
305C22#306811yescharacteristic
306C22#306911yescharacteristic
307C4—22no—
308C4—22no—
309C4—22no—
310C4—44no—
311C4—41no—
312C4—41no—
313C4—82no—
314C4—161no—
315C4—161no—
316C4—161no—
317C4—161no—
318C4#313712yes—
319C4#324212yes—
320C4#333512yes—
321C4#334012yes—
322C2#1939611yescharacteristic
323C2#1940211yescharacteristic
324C2#1952111yescharacteristic
325C2#1952911yescharacteristic
326C2#2870611yescharacteristic
327C2#2872911yescharacteristic
328C2#2873511yescharacteristic
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
#303052
#291502
#287172
#222382
#106262
#106202
#99052
#98972
#86472
#86432
#82772