Finite Group Databasebeta

← #508593

Subgroups — #508593

Order 512.

489 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33215C211yesmaximal, characteristic
2#21156C211yesmaximal, characteristic
3#13539C212yesmaximal, complemented
4#13515C212yesmaximal, complemented
5#13514C212yesmaximal, complemented
6#13451C212yesmaximal, complemented
7#12919C212yesmaximal
8#12868C212yesmaximal
9#12784C211yesmaximal, characteristic
10C2×#500C2211yescharacteristic
11C2×#427C2211yescharacteristic
12C22×#107C2211yescharacteristic
13C2×#405C2211yescharacteristic
14#2628—21no—
15#2317—21no—
16C2×#366C2212yes—
17C2×#366C2212yes—
18C2×C4⋊C16C2211yescharacteristic
19C2×C4⋊C16C2211yescharacteristic
20C2×C22⋊C16C2212yes—
21#1993—21no—
22#1944—21no—
23#1944—21no—
24#1944—21no—
25#1942C2212yes—
26#1942C2212yes—
27#1942C2212yes—
28#1846C2211yescharacteristic
29#1809—21no—
30#1808C2212yes—
31#1736—21no—
32#1380C2212yes—
33#1380C2212yes—
34#1380C2212yes—
35#1380C2212yes—
36#1380C2212yes—
37#1380C2212yes—
38#1374C2212yes—
39C22×M16D411yescharacteristic
40C23×C8C2311yescharacteristic
41C2×C4⋊Q8—21no—
42C2×#117—21no—
43C2×#117D411yescharacteristic
44C22×C4⋊C4—21no—
45C22×C16C2312yes—
46C22×C16C2312yes—
47#500—41no—
48#500—41no—
49#427—22no—
50#427Q814yescomplemented
51C2×#107—21no—
52C2×#107—21no—
53C2×#107—41no—
54C2×#107—41no—
55C2×#107C2312yes—
56C2×#107C2312yes—
57C2×#107C2312yes—
58C2×#107C2312yes—
59C2×#106—21no—
60C2×#106—21no—
61#405—41no—
62#405—41no—
63C2×#97D411yescharacteristic
64C2×#97D411yescharacteristic
65C2×C4×C8C2311yescharacteristic
66C2×C4×C8C2311yescharacteristic
67#400—41no—
68#399—41no—
69#395—41no—
70#393—41no—
71#389—22no—
72#389—22no—
73#389—22no—
74#385—41no—
75#385—41no—
76#384—41no—
77#382—21no—
78#382—41no—
79#380—22no—
80#366—42no—
81#366—42no—
82C4⋊C16—41no—
83C4⋊C16—41no—
84C22⋊C16—24no—
85#340—22no—
86#340—22no—
87#340—22no—
88#340—22no—
89C22×Q8—41no—
90C23×C4—41no—
91C23×C4C2×D411yescharacteristic
92C2×M16—22no—
93C2×M16—22no—
94C2×M16—41no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—21no—
99C22×C8C2×D411yescharacteristic
100C22×C8C2×D411yescharacteristic
101C22×C8C2×D411yescharacteristic
102C22×C8C2×Q812yes—
103C22×C8#4112yes—
104C22×C8C2411yescharacteristic
105C4⋊Q8—42no—
106C4⋊Q8—42no—
107#117—24no—
108#117—41no—
109#117—41no—
110#117—42no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4D811yescharacteristic
123C2×C4⋊C4D811yescharacteristic
124C2×C22⋊C4—41no—
125C2×C22⋊C4—42no—
126C2×C22⋊C4D811yescharacteristic
127C2×C22⋊C4D811yescharacteristic
128C2×C42—41no—
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C16—22no—
132C2×C16—22no—
133C2×C16—22no—
134C2×C16—22no—
135C2×C16—22no—
136C2×C16—22no—
137#107—22no—
138#107—22no—
139#107—22no—
140#107—22no—
141#107—22no—
142#107—22no—
143#107—22no—
144#107—42no—
145#107—42no—
146#107C2×Q814yes—
147#106—24no—
148#106—42no—
149#97—41no—
150#97—41no—
151C4×C8—21no—
152C4×C8—21no—
153C4×C8—21no—
154C4×C8—21no—
155#95—42no—
156#95—41no—
157#95—41no—
158#95—42no—
159#95—42no—
160#95—42no—
161C24C2×D811yescharacteristic
162C2×Q8—82no—
163C2×Q8—162no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—82no—
173C22×C4—162no—
174C22×C4C22⋊Q812yes—
175C22×C4C2×D811yescharacteristic
176C22×C4C2×D811yescharacteristic
177C22×C4C2×D811yescharacteristic
178C22×C4C2×D811yescharacteristic
179C22×C4C2×D811yescharacteristic
180C22×C4#13712yes—
181C22×C4C22×D411yescharacteristic
182M16—44no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—22no—
190C2×C8—22no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8C22⋊Q812yes—
195C2×C8#13712yes—
196C2×C8C22×Q811yescharacteristic
197C2×C8C2×#4111yescharacteristic
198C2×C8H(5,𝔽2)11yescharacteristic
199C2×C8H(5,𝔽2)11yescharacteristic
200C2×C82−1+411yescharacteristic
201C2×C82−1+411yescharacteristic
202C4⋊C4—24no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—82no—
213C4⋊C4—82no—
214C4⋊C4—162no—
215C4⋊C4—162no—
216C4⋊C4—162no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—44no—
220C22⋊C4—162no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—44no—
226C16—42no—
227C16—42no—
228C23—22no—
229C23—22no—
230C23—21no—
231C23—21no—
232C23#47412yes—
233C23C22×D811yescharacteristic
234Q8—164no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—44no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—164no—
262C2×C4#47412yes—
263C2×C4C2×C22⋊Q811yescharacteristic
264C2×C4#53411yescharacteristic
265C2×C4#53611yescharacteristic
266C2×C4C2×#13711yescharacteristic
267C2×C4C2×#13711yescharacteristic
268C2×C4#57511yescharacteristic
269C8—22no—
270C8—22no—
271C8—41no—
272C8#53911yescharacteristic
273C8#53911yescharacteristic
274C8#54411yescharacteristic
275C8#54411yescharacteristic
276C22—22no—
277C22—24no—
278C22—42no—
279C22—41no—
280C22—41no—
281C22C2×#47411yescharacteristic
282C22#318611yescharacteristic
283C22#322611yescharacteristic
284C22#341512yes—
285C22#341712yes—
286C4—22no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—84no—
292C4—162no—
293C4—162no—
294C4—162no—
295C4#262711yescharacteristic
296C4#307811yescharacteristic
297C4#319511yescharacteristic
298C4#323611yescharacteristic
299C2—42no—
300C2#2996611yescharacteristic
301C2#3363312yes—
302C2#3376511yescharacteristic
303C2#3376811yescharacteristic
304C2#3380912yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #13539C2semidirect
4. #13515C2semidirect
5. #13514C2semidirect
6. #13451C2semidirect
50. #427Q8semidirect

Maximal subgroups

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

subgroupindex
#332152
#211562
#135392
#135152
#135142
#134512
#129192
#128682
#127842