Finite Group Databasebeta

← #243374

Subgroups — #243374

Order 512.

562 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22113C212yesmaximal
2C4×#500C211yesmaximal, characteristic
3#21341C211yesmaximal, characteristic
4#10650C211yesmaximal, characteristic
5#10618C211yesmaximal, characteristic
6#10589C212yesmaximal
7#10543C211yesmaximal, characteristic
8#10482C212yesmaximal
9#10279C211yesmaximal, characteristic
10C4×#334C211yesmaximal, characteristic
11#8235C212yesmaximal
12C2×#500C2211yescharacteristic
13C2×#477C2212yes—
14C4×C4⋊Q8C2212yes—
15#2295C2211yescharacteristic
16#1990C2212yes—
17#1987C2212yes—
18#1984C2211yescharacteristic
19#1984C2211yescharacteristic
20#1947C2212yes—
21#1942—21no—
22#1942—21no—
23#1942—22no—
24#1939C2211yescharacteristic
25#1939C2211yescharacteristic
26#1933—22no—
27#1928—22no—
28#1921—22no—
29#1916—21no—
30#1916—21no—
31#1845C2211yescharacteristic
32#1803C2212yes—
33C4×#107—21no—
34C4×#107—21no—
35#1768—21no—
36#1767C2211yescharacteristic
37C4×C4⋊C8C2212yes—
38#1746—21no—
39C42×C8C2211yescharacteristic
40C2×#334C2212yes—
41C2×#333C2212yes—
42C2×#333C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45#1277C2211yescharacteristic
46#1277C2211yescharacteristic
47C2×C4⋊Q8C2312yes—
48C2×C4×Q8—22no—
49C2×C4×Q8C2311yescharacteristic
50#500—24no—
51#477—24no—
52#477—24no—
53C2×#107—21no—
54C2×#107—21no—
55C2×#107—22no—
56C2×#107—21no—
57C2×#107—21no—
58C2×#107—22no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8C2311yescharacteristic
62C2×C4⋊C8C2311yescharacteristic
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2312yes—
65C2×#103—22no—
66C2×#103—22no—
67C2×#97—21no—
68C2×C4×C8—22no—
69C2×C4×C8D411yescharacteristic
70C2×C4×C8D411yescharacteristic
71C2×C4×C8D411yescharacteristic
72C2×C4×C8D411yescharacteristic
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2311yescharacteristic
75#391—22no—
76#391D411yescharacteristic
77#391D411yescharacteristic
78#389—22no—
79#389—22no—
80#389D411yescharacteristic
81#389D411yescharacteristic
82#387—21no—
83#387—21no—
84#384C2312yes—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4C2311yescharacteristic
89#376—21no—
90C43C2311yescharacteristic
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#340—22no—
96#340—22no—
97#336—21no—
98#336—21no—
99#336—21no—
100#336—21no—
101#336—21no—
102#336—21no—
103#336—21no—
104#336—22no—
105#334—24no—
106#333—24no—
107#333—24no—
108C22×Q8—22no—
109C22×Q8C2×D411yescharacteristic
110C22×C8—22no—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—22no—
114C22×C8C2×D411yescharacteristic
115C22×C8C2×D411yescharacteristic
116C22×C8C2×D411yescharacteristic
117C22×C8C2×D411yescharacteristic
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8#4114yes—
122C4×Q8—22no—
123C4×Q8—22no—
124C4×Q8—84no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4C2×D411yescharacteristic
135C2×C4⋊C4C2×D411yescharacteristic
136C2×C4⋊C4C2×D411yescharacteristic
137C2×C42—21no—
138C2×C42—21no—
139C2×C42—42no—
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42#4111yescharacteristic
145C2×C42#4111yescharacteristic
146C2×C42C2411yescharacteristic
147#107—28no—
148#107—24no—
149#107—24no—
150#107—42no—
151#107—42no—
152#107—82no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—42no—
159C4⋊C8—42no—
160C4⋊C8#4114yes—
161#103—28no—
162#103—82no—
163#97—42no—
164C4×C8—22no—
165C4×C8—22no—
166C4×C8—22no—
167C4×C8—22no—
168C4×C8—41no—
169C4×C8—82no—
170C4×C8Q1614yes—
171#95—21no—
172#95—21no—
173#95—21no—
174#95—21no—
175#95—21no—
176#95—21no—
177#95—21no—
178#95—42no—
179#95—42no—
180C2×Q8—22no—
181C2×Q8—24no—
182C2×Q8—22no—
183C2×Q8—21no—
184C2×Q8—21no—
185C2×Q8—42no—
186C2×Q8—82no—
187C2×Q8—82no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4#12111yescharacteristic
195C22×C4#12111yescharacteristic
196C22×C4#12111yescharacteristic
197C22×C4#12111yescharacteristic
198C22×C4C22×D411yescharacteristic
199C22×C4C22×D411yescharacteristic
200C22×C4C2×#4111yescharacteristic
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—24no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—41no—
214C2×C8—42no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8C2×Q1612yes—
223C2×C8C2×Q1612yes—
224C2×C8#13512yes—
225C4⋊C4—24no—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—82no—
241C4⋊C4—84no—
242C42—21no—
243C42—21no—
244C42—22no—
245C42—21no—
246C42—21no—
247C42—22no—
248C42—41no—
249C42—41no—
250C42—41no—
251C42—41no—
252C42—41no—
253C42—84no—
254C42C2×Q1612yes—
255C42#13712yes—
256C42C2×#4112yes—
257C42C2×#4112yes—
258C23C2×#12111yescharacteristic
259Q8—42no—
260Q8—42no—
261Q8—84no—
262Q8—84no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—44no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4#46212yes—
293C2×C4#46212yes—
294C2×C4#47212yes—
295C2×C4#52511yescharacteristic
296C2×C4#52512yes—
297C2×C4#52912yes—
298C2×C4#53211yescharacteristic
299C2×C4C22×Q1611yescharacteristic
300C2×C4C2×#13511yescharacteristic
301C2×C4C2×#13711yescharacteristic
302C2×C4#57511yescharacteristic
303C2×C4#57712yes—
304C2×C4#57811yescharacteristic
305C2×C4#57811yescharacteristic
306C8—22no—
307C8—22no—
308C8—22no—
309C8—44no—
310C8—41no—
311C8—81no—
312C8—82no—
313C8—81no—
314C22#239211yescharacteristic
315C22C2×#46211yescharacteristic
316C22#303511yescharacteristic
317C22#305911yescharacteristic
318C22#306011yescharacteristic
319C22C2×#47211yescharacteristic
320C22#306711yescharacteristic
321C4—22no—
322C4—22no—
323C4—22no—
324C4—22no—
325C4—41no—
326C4—81no—
327C4—81no—
328C4—81no—
329C4—81no—
330C4—84no—
331C4—162no—
332C4#310411yescharacteristic
333C4#310811yescharacteristic
334C4#312012yes—
335C4#312112yes—
336C4#317011yescharacteristic
337C4#317211yescharacteristic
338C2#1920911yescharacteristic
339C2#1921011yescharacteristic
340C2#1922011yescharacteristic
341C2#1922711yescharacteristic
342C2#2367911yescharacteristic
343C2#2370411yescharacteristic
344C2#2371611yescharacteristic
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
#221132
C4×#5002
#213412
#106502
#106182
#105892
#105432
#104822
#102792
C4×#3342
#82352