Finite Group Databasebeta

← #330781

Subgroups — #330781

Order 512.

504 conjugacy classes of subgroups.

The subgroup lattice has 337 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33311C211yesmaximal, characteristic
2#29996C212yesmaximal
3#21341C211yesmaximal, characteristic
4#10654C212yesmaximal
5#10652C211yesmaximal, characteristic
6#10651C211yesmaximal, characteristic
7#10620C212yesmaximal
8#10618C211yesmaximal, characteristic
9#10617C211yesmaximal, characteristic
10#9904C212yesmaximal
11#8784C211yesmaximal, characteristic
12C2×#499—21no—
13C2×#499—21no—
14#2859C2212yes—
15#2295C2211yescharacteristic
16#2088—22no—
17#2088—22no—
18#2066—22no—
19#2066—22no—
20#1984C2211yescharacteristic
21#1984C2211yescharacteristic
22#1948C2212yes—
23#1948C2212yes—
24#1939C2211yescharacteristic
25#1939C2211yescharacteristic
26#1916—21no—
27#1916—21no—
28#1846—21no—
29#1846—21no—
30#1845C2211yescharacteristic
31#1845C2211yescharacteristic
32#1844C2212yes—
33#1842C2212yes—
34#1803C2212yes—
35#1803C2212yes—
36#1768—21no—
37#1767C2211yescharacteristic
38#1746—21no—
39C42×C8C2211yescharacteristic
40#1296C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1277C2211yescharacteristic
45#1277C2211yescharacteristic
46#1277C2212yes—
47C2×#125—22no—
48C2×C4×Q8C2311yescharacteristic
49#499—44no—
50#499—44no—
51C2×#107—21no—
52C2×#107—21no—
53C2×#107—21no—
54C2×#107—21no—
55C2×#106—22no—
56C2×#106—22no—
57C2×#106—22no—
58C2×#106—22no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2311yescharacteristic
66C2×C4⋊C8C2311yescharacteristic
67C2×#103—22no—
68C2×#103—22no—
69C2×#97—21no—
70C2×C4×C8D411yescharacteristic
71C2×C4×C8D411yescharacteristic
72C2×C4×C8D411yescharacteristic
73C2×C4×C8D411yescharacteristic
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2311yescharacteristic
76#391D411yescharacteristic
77#391D411yescharacteristic
78#389—22no—
79#389—22no—
80#389—22no—
81#389—22no—
82#389D411yescharacteristic
83#389D411yescharacteristic
84#387—21no—
85#387—21no—
86#384C2312yes—
87#384C2312yes—
88#382—22no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4C2311yescharacteristic
91#376—21no—
92C43C2311yescharacteristic
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—
104C22×Q8C2×D411yescharacteristic
105C22×C8—21no—
106C22×C8—21no—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8C2×D411yescharacteristic
110C22×C8C2×D411yescharacteristic
111C22×C8C2×D411yescharacteristic
112C22×C8C2×D411yescharacteristic
113#125—44no—
114#125—44no—
115C4×Q8—22no—
116C4×Q8—22no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4C2×D411yescharacteristic
129C2×C4⋊C4C2×D411yescharacteristic
130C2×C4⋊C4C2×D411yescharacteristic
131C2×C42—21no—
132C2×C42—21no—
133C2×C42C2×D411yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42#4111yescharacteristic
138C2×C42#4111yescharacteristic
139C2×C42C2411yescharacteristic
140#107—42no—
141#107—42no—
142#107—42no—
143#107—42no—
144#106—44no—
145#106—44no—
146#106—82no—
147#106—82no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—42no—
155C4⋊C8—42no—
156C4⋊C8—82no—
157C4⋊C8—82no—
158#103—82no—
159#103—82no—
160#97—42no—
161C4×C8—22no—
162C4×C8—22no—
163C4×C8—22no—
164C4×C8—22no—
165C4×C8—22no—
166C4×C8SD1614yes—
167#95—21no—
168#95—21no—
169#95—21no—
170#95—21no—
171#95—21no—
172#95—21no—
173#95—21no—
174#95—42no—
175#95—42no—
176C2×Q8—22no—
177C2×Q8—21no—
178C2×Q8—22no—
179C2×Q8—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4#12111yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4#12111yescharacteristic
189C22×C4#12111yescharacteristic
190C22×C4C22×D411yescharacteristic
191C22×C4C22×D411yescharacteristic
192C22×C4C2×#4111yescharacteristic
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8C2×SD1612yes—
214C2×C8C2×SD1612yes—
215C2×C8#13712yes—
216C2×C8#13712yes—
217C4⋊C4—22no—
218C4⋊C4—21no—
219C4⋊C4—22no—
220C4⋊C4—21no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—82no—
233C4⋊C4—82no—
234C4⋊C4—84no—
235C4⋊C4—84no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—22no—
240C42—21no—
241C42—41no—
242C42—41no—
243C42—41no—
244C42—41no—
245C42—41no—
246C42—41no—
247C42C2×SD1612yes—
248C42AGL(1,ℤ/8)12yes—
249C422−1+412yes—
250C422−1+412yes—
251C23C2×#12111yescharacteristic
252Q8—42no—
253Q8—42no—
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—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4#46012yes—
285C2×C4#46012yes—
286C2×C4#53611yescharacteristic
287C2×C4#53611yescharacteristic
288C2×C4#53612yes—
289C2×C4#54112yes—
290C2×C4C22×SD1611yescharacteristic
291C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
292C2×C4C2×#13711yescharacteristic
293C2×C4C2×#13711yescharacteristic
294C2×C4#57611yescharacteristic
295C2×C4#57712yes—
296C2×C4#57811yescharacteristic
297C8—22no—
298C8—22no—
299C8—22no—
300C8—22no—
301C8—81no—
302C8—81no—
303C8—82no—
304C8—82no—
305C22#262111yescharacteristic
306C22C2×#46011yescharacteristic
307C22#303911yescharacteristic
308C22#306011yescharacteristic
309C22#306111yescharacteristic
310C22#306711yescharacteristic
311C22#306711yescharacteristic
312C4—22no—
313C4—22no—
314C4—22no—
315C4—41no—
316C4—41no—
317C4—81no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4—162no—
322C4—162no—
323C4#322911yescharacteristic
324C4#322911yescharacteristic
325C4#323911yescharacteristic
326C4#323911yescharacteristic
327C4#324112yes—
328C4#324212yes—
329C2#1920511yescharacteristic
330C2#1920611yescharacteristic
331C2#1936011yescharacteristic
332C2#1936111yescharacteristic
333C2#2986611yescharacteristic
334C2#2988211yescharacteristic
335C2#2988811yescharacteristic
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
#333112
#299962
#213412
#106542
#106522
#106512
#106202
#106182
#106172
#99042
#87842