Finite Group Databasebeta

← #352491

Subgroups — #352491

Order 512.

544 conjugacy classes of subgroups.

The subgroup lattice has 321 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33256C211yesmaximal, characteristic
2#29997C212yesmaximal
3#20516C211yesmaximal, characteristic
4#9906C212yesmaximal
5#9618C211yesmaximal, characteristic
6#9617C211yesmaximal, characteristic
7#9616C212yesmaximal
8#8806C211yesmaximal, characteristic
9#8651C211yesmaximal, characteristic
10#8651C211yesmaximal, characteristic
11#8649C212yesmaximal
12C2×#446C2211yescharacteristic
13C2×C8×Q8C2211yescharacteristic
14#2859C2212yes—
15#2295C2211yescharacteristic
16#2088—22no—
17#2086—22no—
18#2068—22no—
19#2066—22no—
20#1948C2212yes—
21#1947C2212yes—
22#1941—21no—
23#1941—21no—
24#1941—21no—
25#1941—21no—
26#1844C2211yescharacteristic
27#1844C2211yescharacteristic
28#1842C2212yes—
29#1834C2212yes—
30#1803C2212yes—
31#1803C2212yes—
32#1771—21no—
33#1770C2211yescharacteristic
34C8×C4⋊C4—21no—
35C8×C4⋊C4C2211yescharacteristic
36#1768—21no—
37#1767C2211yescharacteristic
38#1726—21no—
39C42×C8C2211yescharacteristic
40C2×#335C2212yes—
41C2×#335C2211yescharacteristic
42C2×#335C2211yescharacteristic
43#1296C2212yes—
44#1296C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47C2×#125—22no—
48C2×C4×Q8C2311yescharacteristic
49#446—22no—
50#446Q814yes—
51C8×Q8—22no—
52C8×Q8Q814yes—
53C2×#107—22no—
54C2×#106—21no—
55C2×#106—21no—
56C2×#106—21no—
57C2×#106—21no—
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×#97—21no—
71C2×#97—21no—
72C2×C4×C8—21no—
73C2×C4×C8—21no—
74C2×C4×C8D411yescharacteristic
75C2×C4×C8D411yescharacteristic
76C2×C4×C8D411yescharacteristic
77C2×C4×C8D411yescharacteristic
78C2×C4×C8C2311yescharacteristic
79C2×C4×C8C2311yescharacteristic
80#391D412yes—
81#389—22no—
82#389—22no—
83#389—22no—
84#389—22no—
85#389D412yes—
86#387—22no—
87#384C2312yes—
88#384C2312yes—
89#382—22no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4C2311yescharacteristic
92#376—21no—
93C43C2311yescharacteristic
94#340—22no—
95#340—22no—
96#340—22no—
97#340—22no—
98#340—22no—
99#340—22no—
100#336—21no—
101#336—21no—
102#336—21no—
103#335—24no—
104#335—22no—
105#335—22no—
106C22×Q8C2×D411yescharacteristic
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8C2×D411yescharacteristic
112C22×C8C2×D411yescharacteristic
113C22×C8C2×D411yescharacteristic
114C22×C8C2×D411yescharacteristic
115#125—84no—
116C4×Q8—21no—
117C4×Q8—21no—
118C4×Q8C2×Q812yes—
119C4×Q8C2×Q812yes—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4C2×D412yes—
131C2×C4⋊C4C2×D411yescharacteristic
132C2×C42—21no—
133C2×C42—21no—
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D412yes—
137C2×C42#4112yes—
138C2×C42C2411yescharacteristic
139#107—82no—
140#106—42no—
141#106—42no—
142#106—42no—
143#106—42no—
144#106—82no—
145C4⋊C8—24no—
146C4⋊C8—24no—
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—82no—
154C4⋊C8—82no—
155C4⋊C8C2×Q812yes—
156C4⋊C8C2×Q812yes—
157#103—82no—
158#103—82no—
159#97—24no—
160#97—42no—
161#97—42no—
162C4×C8—24no—
163C4×C8—22no—
164C4×C8—22no—
165C4×C8—41no—
166C4×C8—41no—
167C4×C8—42no—
168C4×C8C2×Q812yes—
169C4×C8C2×Q812yes—
170C4×C8#4112yes—
171C4×C8#4112yes—
172#95—22no—
173#95—22no—
174#95—22no—
175#95—21no—
176#95—42no—
177#95—42no—
178C2×Q8—21no—
179C2×Q8—22no—
180C2×Q8—22no—
181C2×Q8C4⋊Q812yes—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4#12112yes—
188C22×C4#12112yes—
189C22×C4C22×D411yescharacteristic
190C22×C4C22×D411yescharacteristic
191C22×C4C2×#4111yescharacteristic
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8C22⋊Q812yes—
211C2×C8C22⋊Q812yes—
212C2×C8C22⋊Q812yes—
213C2×C8C22⋊Q812yes—
214C2×C8#13512yes—
215C2×C8#13712yes—
216C4⋊C4—21no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—44no—
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—82no—
231C4⋊C4—82no—
232C4⋊C4—84no—
233C4⋊C4—84no—
234C4⋊C4C4⋊Q812yes—
235C42—22no—
236C42—22no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—22no—
242C42—22no—
243C42—24no—
244C42—41no—
245C42—41no—
246C42C22×Q811yescharacteristic
247C42C22×Q811yescharacteristic
248C42C2×#4111yescharacteristic
249C42C2×#4111yescharacteristic
250C23C2×#12111yescharacteristic
251Q8—24no—
252Q8—41no—
253Q8—41no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4#47114yes—
274C2×C4C2×C22⋊Q811yescharacteristic
275C2×C4C2×C22⋊Q811yescharacteristic
276C2×C4C2×C4⋊Q811yescharacteristic
277C2×C4#53211yescharacteristic
278C2×C4#53311yescharacteristic
279C2×C4#53311yescharacteristic
280C2×C4C2×#13511yescharacteristic
281C2×C4C2×#13711yescharacteristic
282C2×C4#57511yescharacteristic
283C2×C4#57511yescharacteristic
284C2×C4#57612yes—
285C2×C4#57812yes—
286C8—22no—
287C8—22no—
288C8—22no—
289C8—42no—
290C8—42no—
291C8—82no—
292C8—82no—
293C8#47914yes—
294C22#285711yescharacteristic
295C22#303511yescharacteristic
296C22#303711yescharacteristic
297C22#306112yes—
298C22C2×#47112yes—
299C4—24no—
300C4—21no—
301C4—21no—
302C4—42no—
303C4—42no—
304C4—42no—
305C4—82no—
306C4—162no—
307C4—162no—
308C4C2×#47911yescharacteristic
309C4#307611yescharacteristic
310C4#307811yescharacteristic
311C4#308011yescharacteristic
312C4#316412yes—
313C4#317611yescharacteristic
314C4#317711yescharacteristic
315C2#1922912yes—
316C2#1935112yes—
317C2#3320011yescharacteristic
318C2#3321611yescharacteristic
319C2#3322011yescharacteristic
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
#332562
#299972
#205162
#99062
#96182
#96172
#96162
#88062
#86512
#86492