Finite Group Databasebeta

← #213809

Subgroups — #213809

Order 512.

895 conjugacy classes of subgroups.

The subgroup lattice has 363 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35624C211yesmaximal, characteristic
2#17391C212yesmaximal, complemented
3#9106C212yesmaximal, complemented
4#9102C212yesmaximal, complemented
5#8964C212yesmaximal, complemented
6#8960C212yesmaximal, complemented
7#8733C211yesmaximal, characteristic, complemented
8#8732C211yesmaximal, characteristic, complemented
9#8322C212yesmaximal
10C2×#541C2211yescharacteristic
11C2×#518C2211yescharacteristic
12#2983C2212yescomplemented
13#2982C2212yescomplemented
14#2566C412yes—
15#2547C412yes—
16#2499C412yes—
17#2480—24no—
18#2460—24no—
19#2435C412yes—
20#2326C2212yes—
21C4×#124C2212yes—
22#2298C2211yescharacteristic
23#1478C2212yes—
24#1478C2212yes—
25#1478C2212yes—
26#1478C2212yes—
27#1473—24no—
28#1470—24no—
29#1461C2212yescomplemented
30#1461C2212yescomplemented
31#1458C2212yes—
32#1458C2212yes—
33#1276C2211yescharacteristic, complemented
34#1276C2212yescomplemented
35#1276C2211yescharacteristic, complemented
36#1274C2211yescharacteristic, complemented
37#1274C2211yescharacteristic, complemented
38#1274C2212yescomplemented
39C23×D4C2×C411yescharacteristic
40#541—41no—
41#541—42no—
42#541—41no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#126D414yes—
45C2×#125C2×C411yescharacteristic
46C2×#125D414yes—
47C2×#124C2×C412yes—
48C2×#124C2×C412yes—
49C2×C22⋊Q8C2×C412yes—
50#518—42no—
51#518—41no—
52#518—41no—
53C2×C4×Q8C2312yes—
54C2×C4×D4—22no—
55C2×C4×D4C2×C412yes—
56C2×C4×D4C2312yes—
57C2×#117C2×C411yescharacteristic
58C2×#117C2311yescharacteristic
59C22×C22⋊C4—22no—
60#446—42no—
61#446—42no—
62C8×Q8—42no—
63C8×Q8—42no—
64#432—24no—
65#431—24no—
66#431C2312yes—
67#431C2312yes—
68#431C2312yes—
69#431C2312yes—
70#397—24no—
71#397—24no—
72#396—24no—
73#394—24no—
74#391C2×C412yes—
75#391C2×C412yes—
76#390C2×C412yes—
77#390C2×C412yes—
78#388—24no—
79#388—22no—
80#388—24no—
81#387—24no—
82#387—22no—
83#385—24no—
84#385—24no—
85#384C2×C412yes—
86#382C2×C412yes—
87#380—24no—
88#379—24no—
89C4×C4⋊C4C2×C412yes—
90C4×C22⋊C4—24no—
91C4×C22⋊C4C2×C412yes—
92#376C2311yescharacteristic
93C43C2311yescharacteristic
94D4⋊C8—42no—
95D4⋊C8—42no—
96D4⋊C8—42no—
97D4⋊C8—42no—
98#324—22no—
99#324—22no—
100#324—24no—
101#324—24no—
102#323—22no—
103#323—22no—
104C25—21no—
105C22×Q8C22×C412yes—
106C22×D4—24no—
107C22×D4—21no—
108C22×D4—41no—
109C22×D4—41no—
110C22×D4—42no—
111C22×D4C22×C412yes—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4C22×C411yescharacteristic
115C4⋊Q8—41no—
116C4⋊Q8—41no—
117#126—44no—
118#126—44no—
119#125—41no—
120#125—41no—
121#125—44no—
122#125—44no—
123#124—24no—
124#124—24no—
125#124—42no—
126#124—42no—
127C22⋊Q8—42no—
128C22⋊Q8—42no—
129C4×Q8—22no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×D4—22no—
134C4×D4—22no—
135C4×D4—22no—
136C4×D4—22no—
137C4×D4—42no—
138C4×D4—42no—
139C4×D4—44no—
140C4×D4—44no—
141#117—21no—
142#117—21no—
143#117—22no—
144#117—41no—
145#117—41no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4C22⋊C414yes—
149C2×C4⋊C4C22⋊C414yes—
150C2×C4⋊C4C22⋊C414yes—
151C2×C4⋊C4C22×C412yes—
152C2×C4⋊C4C22×C412yes—
153C2×C4⋊C4C2×D412yes—
154C2×C4⋊C4C2×D414yes—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—44no—
164C2×C22⋊C4—44no—
165C2×C22⋊C4C22⋊C414yes—
166C2×C22⋊C4C2×D412yes—
167C2×C42—21no—
168C2×C42—22no—
169C2×C42C22×C412yes—
170C2×C42C22×C412yes—
171C2×C42C22×C411yescharacteristic
172C2×C42C2×D414yes—
173C2×C42C2411yescharacteristic
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—44no—
179C4⋊C8—44no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184#97—44no—
185C4×C8—22no—
186C4×C8—22no—
187C4×C8—22no—
188C4×C8—22no—
189C4×C8—44no—
190#95—24no—
191#95—24no—
192#95—22no—
193#95—22no—
194#95—24no—
195#95—22no—
196C24—22no—
197C24—44no—
198C24—42no—
199C24—84no—
200C24C2×C22⋊C411yescharacteristic
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—22no—
212C2×D4—44no—
213C2×D4—42no—
214C2×D4—44no—
215C2×D4—48no—
216C2×D4—44no—
217C2×D4#13114yes—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—44no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—84no—
229C22×C4—84no—
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C2×C22⋊C414yes—
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C414yes—
234C22×C4C23×C411yescharacteristic
235C22×C4C22×D412yes—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—84no—
252C4⋊C4—82no—
253C22⋊C4—22no—
254C22⋊C4—22no—
255C22⋊C4—44no—
256C22⋊C4—48no—
257C22⋊C4—48no—
258C22⋊C4—44no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—84no—
263C22⋊C4—84no—
264C22⋊C4—82no—
265C42—21no—
266C42—21no—
267C42—24no—
268C42—24no—
269C42—22no—
270C42—22no—
271C42—22no—
272C42—22no—
273C42—42no—
274C42—82no—
275C42H(5,𝔽2)11yescharacteristic
276C42H(5,𝔽2)11yescharacteristic
277C42H(5,𝔽2)12yes—
278C23—21no—
279C23—24no—
280C23—22no—
281C23—44no—
282C23—44no—
283C23—42no—
284C23—48no—
285C23—42no—
286C23—42no—
287C23—168no—
288C23—164no—
289C23C22×C22⋊C411yescharacteristic
290Q8—42no—
291Q8—42no—
292Q8—42no—
293Q8—42no—
294D4—24no—
295D4—24no—
296D4—48no—
297D4—42no—
298D4—42no—
299C2×C4—22no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—21no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—44no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—162no—
321C2×C4—162no—
322C2×C4—164no—
323C2×C4#51811yescharacteristic
324C2×C4#51811yescharacteristic
325C2×C4#53412yes—
326C2×C4#53412yes—
327C2×C4C2×#13112yes—
328C2×C4#56812yes—
329C8—82no—
330C8—82no—
331C8—82no—
332C8—82no—
333C22—22no—
334C22—48no—
335C22—42no—
336C22—41no—
337C22—44no—
338C22—82no—
339C22—84no—
340C22—82no—
341C22—82no—
342C22—168no—
343C22#230811yescharacteristic
344C22#288012yes—
345C4—22no—
346C4—22no—
347C4—44no—
348C4—42no—
349C4—41no—
350C4—44no—
351C4—82no—
352C4—82no—
353C4—82no—
354C4#297612yes—
355C4#297711yescharacteristic
356C4#297811yescharacteristic
357C2—22no—
358C2—84no—
359C2—81no—
360C2#2174411yescharacteristic
361C2#2178512yes—
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17391C2semidirect
3. #9106C2semidirect
4. #9102C2semidirect
5. #8964C2semidirect
6. #8960C2semidirect
7. #8733C2semidirect
8. #8732C2semidirect
12. #2983C22semidirect
13. #2982C22semidirect
29. #1461C22semidirect
30. #1461C22semidirect
33. #1276C22semidirect
34. #1276C22semidirect
35. #1276C22semidirect
36. #1274C22semidirect
37. #1274C22semidirect
38. #1274C22semidirect

Maximal subgroups

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

subgroupindex
#356242
#173912
#91062
#91022
#89642
#89602
#87332
#87322
#83222