Finite Group Databasebeta

← #331544

Subgroups — #331544

Order 512.

650 conjugacy classes of subgroups.

The subgroup lattice has 319 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33316C211yesmaximal, characteristic
2#33289C211yesmaximal, characteristic
3#33256C211yesmaximal, characteristic
4#33232C211yesmaximal, characteristic
5#30305C211yesmaximal, characteristic
6#30228C211yesmaximal, characteristic
7#10686C212yesmaximal
8#10685C212yesmaximal
9#10684C212yesmaximal
10#10684C212yesmaximal
11#8132C211yesmaximal, characteristic
12C2×#501—22no—
13C2×#491—22no—
14#2868C2211yescharacteristic
15#2868C2211yescharacteristic
16#2859C2211yescharacteristic
17#2859C2211yescharacteristic
18#2099—22no—
19#2099—22no—
20#2084—22no—
21#2058—22no—
22#1990—22no—
23#1990C2212yes—
24#1990C2212yes—
25#1990C2212yes—
26#1990C2212yes—
27#1960—22no—
28#1956C2212yes—
29#1956C2212yes—
30#1941—22no—
31#1941—22no—
32#1847—22no—
33#1845C2212yes—
34#1844C2212yes—
35#1844C2212yes—
36#1844C2211yescharacteristic
37#1844C2211yescharacteristic
38#1834—22no—
39#1834C2212yes—
40#1834C2211yescharacteristic
41#1834C2211yescharacteristic
42#1744C2211yescharacteristic
43C4×#97C2211yescharacteristic
44C42×C8C2211yescharacteristic
45#1273C2212yes—
46#1273C2212yes—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8C2312yes—
53C2×#125—21no—
54C2×#125—21no—
55#501—48no—
56#491—48no—
57C2×#107—24no—
58C2×#106—24no—
59C2×#106—22no—
60C2×#106—22no—
61C2×#106—24no—
62C2×#103—24no—
63C2×#103—24no—
64C2×#97D412yes—
65C2×#97D412yes—
66C2×C4×C8D412yes—
67C2×C4×C8D412yes—
68C2×C4×C8C2311yescharacteristic
69C2×C4×C8C2312yes—
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2312yes—
72#389—21no—
73#389—21no—
74#389—21no—
75#389—21no—
76#389—21no—
77#389—21no—
78#389—21no—
79#389—21no—
80#384—21no—
81#384—21no—
82#384—21no—
83#384—21no—
84#384—21no—
85#384—21no—
86#384—21no—
87#384C2311yescharacteristic
88#384C2311yescharacteristic
89#384C2311yescharacteristic
90#384C2311yescharacteristic
91#384C2312yes—
92#382—21no—
93#382—21no—
94C43C2311yescharacteristic
95#340—24no—
96#340—24no—
97#340—24no—
98#340—22no—
99#340—22no—
100#340—24no—
101#340—24no—
102#336D412yes—
103#336D412yes—
104C22×Q8—22no—
105C22×Q8—41no—
106C22×C8C2×D414yes—
107C22×C8C2×D412yes—
108C22×C8C2×D414yes—
109C22×C8C2×D412yes—
110C4⋊Q8—24no—
111C4⋊Q8—24no—
112C4⋊Q8—44no—
113C4⋊Q8—44no—
114C4⋊Q8—44no—
115C4⋊Q8—44no—
116C4⋊Q8—82no—
117#125—44no—
118#125—44no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
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—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C42C2×D411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2411yescharacteristic
154#107—48no—
155#106—44no—
156#106—48no—
157#106—44no—
158#106—84no—
159#103—48no—
160#103—48no—
161#97—24no—
162#97SD1618yes—
163C4×C8—24no—
164C4×C8—22no—
165C4×C8—22no—
166C4×C8—22no—
167C4×C8—22no—
168C4×C8—22no—
169C4×C8—22no—
170C4×C8—42no—
171#95—41no—
172#95—41no—
173#95—41no—
174#95—41no—
175C2×Q8—44no—
176C2×Q8—44no—
177C2×Q8—44no—
178C2×Q8—82no—
179C2×Q8—82no—
180C2×Q8—82no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4#12711yescharacteristic
188C22×C4#12711yescharacteristic
189C22×C4#12711yescharacteristic
190C22×C4#12711yescharacteristic
191C22×C4C22×D411yescharacteristic
192C22×C4C22×D411yescharacteristic
193C22×C4C22×D411yescharacteristic
194C2×C8—24no—
195C2×C8—22no—
196C2×C8—24no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—24no—
200C2×C8—22no—
201C2×C8—22no—
202C2×C8—24no—
203C2×C8—24no—
204C2×C8C2×SD1618yes—
205C2×C8#13714yes—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—44no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—44no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—44no—
219C4⋊C4—42no—
220C4⋊C4—44no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—82no—
225C4⋊C4—82no—
226C4⋊C4—82no—
227C4⋊C4—82no—
228C4⋊C4—82no—
229C4⋊C4—82no—
230C4⋊C4—82no—
231C4⋊C4—82no—
232C4⋊C4—82no—
233C4⋊C4—82no—
234C4⋊C4—82no—
235C42—22no—
236C42—22no—
237C42—22no—
238C42—22no—
239C42C2×SD1614yes—
240C42AGL(1,ℤ/8)14yes—
241C422−1+412yes—
242C422−1+412yes—
243C23C2×#12711yescharacteristic
244Q8—88no—
245Q8—164no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—82no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—82no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—82no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—82no—
272C2×C4#49212yes—
273C2×C4#49212yes—
274C2×C4#49612yes—
275C2×C4#49612yes—
276C2×C4#53612yes—
277C2×C4#53611yescharacteristic
278C2×C4#53611yescharacteristic
279C2×C4#53612yes—
280C2×C4C22×SD1612yes—
281C2×C4C2×AGL(1,ℤ/8)12yes—
282C2×C4C2×#13712yes—
283C2×C4#57512yes—
284C2×C4#57712yes—
285C2×C4#57812yes—
286C8—28no—
287C8—24no—
288C8—42no—
289C8—44no—
290C22#265811yescharacteristic
291C22C2×#49211yescharacteristic
292C22C2×#49611yescharacteristic
293C22#315211yescharacteristic
294C22#315311yescharacteristic
295C22#315411yescharacteristic
296C22#315611yescharacteristic
297C4—22no—
298C4—22no—
299C4—22no—
300C4—22no—
301C4—162no—
302C4—161no—
303C4—161no—
304C4—161no—
305C4—161no—
306C4—162no—
307C4#322714yes—
308C4#323612yes—
309C4#323712yes—
310C4#324214yes—
311C2#1962211yescharacteristic
312C2#1962411yescharacteristic
313C2#1967511yescharacteristic
314C2#1967811yescharacteristic
315C2#3027711yescharacteristic
316C2#3028411yescharacteristic
317C2#3028811yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#127
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
#333162
#332892
#332562
#332322
#303052
#302282
#106862
#106852
#106842
#81322