Finite Group Databasebeta

← #504298

Subgroups — #504298

Order 512.

428 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28828C211yesmaximal, characteristic
2#23158C211yesmaximal, characteristic
3#13539C212yesmaximal
4#13516C212yesmaximal
5#13282C211yesmaximal, characteristic
6#13276C211yesmaximal, characteristic
7#13178C212yesmaximal
8#13176C211yesmaximal, characteristic
9#13159C211yesmaximal, characteristic
10#13109C212yesmaximal
11C4×#367C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13#2571—21no—
14#2369—21no—
15C2×#367C2211yescharacteristic
16C2×#367C2211yescharacteristic
17C2×#366C2212yes—
18C2×C4⋊C16C2211yescharacteristic
19C2×C4⋊C16C2212yes—
20C2×C4×C16C2211yescharacteristic
21#2085—22no—
22#1987—21no—
23#1948—21no—
24#1942C2212yes—
25#1942C2212yes—
26#1921—22no—
27#1846C2212yes—
28#1845C2211yescharacteristic
29C4×#107C2212yes—
30C4×#107C2211yescharacteristic
31C4×#107C2211yescharacteristic
32C8×C4⋊C4C2211yescharacteristic
33#1380C2211yescharacteristic
34#1380C2212yes—
35#1380C2212yes—
36#1380C2212yes—
37#1380C2212yes—
38#1380C2212yes—
39#1380C2211yescharacteristic
40#1374C2212yes—
41C2×C4⋊Q8—21no—
42C22×C16C2311yescharacteristic
43C22×C16C2312yes—
44C22×C16C2311yescharacteristic
45#500—22no—
46#500Q814yes—
47C2×#107—21no—
48C2×#107—21no—
49C2×#107C2312yes—
50C2×#107C2311yescharacteristic
51C2×#107C2311yescharacteristic
52C2×#107C2311yescharacteristic
53C2×#107C2312yes—
54C2×#107C2311yescharacteristic
55C2×#106—22no—
56C2×C4⋊C8D411yescharacteristic
57C2×C4×C8C2312yes—
58C2×C4×C8C2311yescharacteristic
59#400—42no—
60#391—42no—
61#389—22no—
62#389—22no—
63#389—41no—
64#389—41no—
65#384—21no—
66#382—22no—
67#382—41no—
68C4×C4⋊C4—21no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—21no—
71C4×C4⋊C4—41no—
72C4×C4⋊C4D411yescharacteristic
73#367—22no—
74#367—22no—
75#366—24no—
76C4⋊C16—24no—
77C4⋊C16—22no—
78C4×C16—22no—
79#340—21no—
80#340—22no—
81#340—21no—
82#340—22no—
83#340—22no—
84#336D412yes—
85C22×Q8—41no—
86C22×C8C2×D412yes—
87C22×C8C2×D411yescharacteristic
88C22×C8C2411yescharacteristic
89C4⋊Q8—24no—
90C4⋊Q8—42no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4D811yescharacteristic
103C2×C4⋊C4D811yescharacteristic
104C2×C42—41no—
105C2×C42—41no—
106C2×C42—42no—
107C2×C42C2×D412yes—
108C2×C42C2×D411yescharacteristic
109C2×C16—21no—
110C2×C16—21no—
111C2×C16—21no—
112C2×C16—22no—
113C2×C16—22no—
114C2×C16—22no—
115C2×C16—21no—
116C2×C16—21no—
117C2×C16#4112yes—
118#107—24no—
119#107—22no—
120#107—21no—
121#107—22no—
122#107—21no—
123#107—21no—
124#107—21no—
125#107—21no—
126#107—21no—
127#107—22no—
128#107—21no—
129#107—81no—
130#107C2×Q814yes—
131#107#4112yes—
132#106—82no—
133C4⋊C8—22no—
134C4×C8—22no—
135C4×C8—22no—
136C4×C8C2×Q812yes—
137C4×C8#4112yes—
138#95—42no—
139#95—41no—
140#95—41no—
141#95—42no—
142#95—42no—
143#95—42no—
144#95—42no—
145#95D812yes—
146C2×Q8—44no—
147C2×Q8—161no—
148C2×Q8—161no—
149C22×C4—41no—
150C22×C4—42no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—42no—
155C22×C4C2×D812yes—
156C22×C4C2×D812yes—
157C22×C4C2×D811yescharacteristic
158C22×C4C2×D811yescharacteristic
159C22×C4C22×D411yescharacteristic
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8C22⋊Q812yes—
166C2×C8C22×Q811yescharacteristic
167C2×C8C2×#4111yescharacteristic
168C2×C8C2×#4111yescharacteristic
169C2×C8C2×#4111yescharacteristic
170C2×C82−1+411yescharacteristic
171C2×C82−1+411yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—24no—
175C4⋊C4—22no—
176C4⋊C4—42no—
177C4⋊C4—41no—
178C4⋊C4—44no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—42no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—42no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—82no—
188C4⋊C4—161no—
189C4⋊C4—162no—
190C4⋊C4—162no—
191C42—22no—
192C42—22no—
193C42—82no—
194C42—82no—
195C42—162no—
196C42C22⋊Q812yes—
197C42#13712yes—
198C16—22no—
199C16—42no—
200C16—41no—
201C23C22×D811yescharacteristic
202Q8—164no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—44no—
213C2×C4—81no—
214C2×C4—81no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—82no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—82no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—82no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4#47412yes—
230C2×C4#47412yes—
231C2×C4#50812yes—
232C2×C4#51012yes—
233C2×C4C2×C22⋊Q811yescharacteristic
234C2×C4#54711yescharacteristic
235C2×C4#54811yescharacteristic
236C2×C4C2×#13711yescharacteristic
237C2×C4#57711yescharacteristic
238C2×C4#57811yescharacteristic
239C8—22no—
240C8—42no—
241C8#52911yescharacteristic
242C8#54111yescharacteristic
243C8#55711yescharacteristic
244C8#55711yescharacteristic
245C22C2×#47411yescharacteristic
246C22#333411yescharacteristic
247C22#335811yescharacteristic
248C22C2×#50811yescharacteristic
249C22C2×#51011yescharacteristic
250C22#341511yescharacteristic
251C22#341611yescharacteristic
252C4—22no—
253C4—42no—
254C4—42no—
255C4—84no—
256C4—82no—
257C4—161no—
258C4—161no—
259C4—162no—
260C4—161no—
261C4#222914yes—
262C4#245311yescharacteristic
263C4#308211yescharacteristic
264C4#335611yescharacteristic
265C4#337411yescharacteristic
266C2#2650611yescharacteristic
267C2C2×#222911yescharacteristic
268C2#3363111yescharacteristic
269C2#3390711yescharacteristic
270C2#3392411yescharacteristic
271C2#3392811yescharacteristic
272C2#3393611yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#288282
#231582
#135392
#135162
#132822
#132762
#131782
#131762
#131592
#131092
C4×#3672