Finite Group Databasebeta

← #163897

Subgroups — #163897

Order 512.

1164 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34275C211yesmaximal, characteristic
2#22632C212yesmaximal, complemented
3#7601C212yesmaximal, complemented
4#7597C211yesmaximal, characteristic, complemented
5#7587C214yesmaximal, complemented
6#7572C212yesmaximal, complemented
7#7567C212yesmaximal, complemented
8#7563C211yesmaximal, characteristic, complemented
9C2×#535C411yescharacteristic
10C2×#520C411yescharacteristic
11C2×#518C411yescharacteristic
12#2363—22no—
13#2361—22no—
14#2358—24no—
15#2328C412yes—
16#2326C412yes—
17#2317C411yescharacteristic
18C4×#127C2212yes—
19C4×#124C2212yes—
20#2297C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C4×#117C2211yescharacteristic
23#1939—22no—
24#1923—24no—
25#1831C2214yescomplemented
26#1797C2212yes—
27C8×C4⋊C4—22no—
28C4×C22⋊C8C2212yes—
29#1744C2214yescomplemented
30#1272C2211yescharacteristic, complemented
31#1272C2211yescharacteristic, complemented
32#1272C2212yescomplemented
33#1270C2214yes—
34#1270C2214yes—
35#1270C2212yescomplemented
36#1270C2212yescomplemented
37C22×#41C2×C411yescharacteristic
38C23×D4C2×C411yescharacteristic
39#535—24no—
40#535C818yescomplemented
41C2×#127C2×C412yes—
42C2×#124C2×C412yes—
43C2×#121—22no—
44C2×#120—22no—
45#520—44no—
46#518—44no—
47C2×C4×Q8C2×C412yes—
48C2×C4×D4—21no—
49C2×C4×D4—21no—
50C2×C4×D4—22no—
51C2×C4×D4—24no—
52C2×C4×D4C2×C412yes—
53C2×C4×D4C2×C412yes—
54C2×C4×D4C2×C412yes—
55C2×#117—21no—
56C2×#117C2×C411yescharacteristic
57C2×#117C2×C411yescharacteristic
58C2×#117C2×C411yescharacteristic
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C42C2311yescharacteristic
64C2×C4⋊C8—22no—
65C2×C4⋊C8—24no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C4×C8—24no—
69C2×C4×C8—22no—
70#391C2×C411yescharacteristic
71#391C2311yescharacteristic
72#390—24no—
73#390—22no—
74#390C2×C412yes—
75#390C2×C411yescharacteristic
76#390C2311yescharacteristic
77#390C2312yes—
78#389—22no—
79#388—24no—
80#388—24no—
81#386—22no—
82#386—24no—
83#386—24no—
84#385—22no—
85#385—24no—
86#384C2×C412yes—
87#382C2×C412yes—
88C4×C4⋊C4D412yes—
89C4×C4⋊C4D412yes—
90C4×C22⋊C4—24no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4D414yes—
93#376C2×C412yes—
94C43C2312yes—
95#336—24no—
96#336—24no—
97#336C2314yes—
98#336C2312yes—
99#336C2312yes—
100C25—21no—
101C2×#41—22no—
102C2×#41—42no—
103C2×#41—42no—
104C2×#41C2×C814yes—
105C22×Q8C22×C411yescharacteristic
106C22×D4—21no—
107C22×D4—21no—
108C22×D4—24no—
109C22×D4—22no—
110C22×D4—42no—
111C22×D4—42no—
112C22×D4C2×C814yes—
113C22×D4C22×C412yes—
114C22×D4C22×C411yescharacteristic
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4C22×C411yescharacteristic
121C23×C4C2×D412yes—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—24no—
125#127—24no—
126#127C2×C818yes—
127#124—24no—
128#124C2×C818yes—
129#121—216no—
130#120—216no—
131C4×Q8—44no—
132C4×D4—44no—
133C4×D4—44no—
134C4×D4—44no—
135C4×D4—48no—
136C4×D4—44no—
137C4×D4—44no—
138C4×D4—84no—
139C4×D4—84no—
140#117—24no—
141#117—44no—
142#117—42no—
143#117C2×C814yes—
144#117M1614yes—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4C22⋊C411yescharacteristic
151C2×C4⋊C4C22⋊C412yes—
152C2×C4⋊C4C22⋊C411yescharacteristic
153C2×C4⋊C4C22×C412yes—
154C2×C4⋊C4C22×C411yescharacteristic
155C2×C4⋊C4C22×C411yescharacteristic
156C2×C22⋊C4—24no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—48no—
166C2×C22⋊C4—82no—
167C2×C22⋊C4—82no—
168C2×C22⋊C4C22⋊C412yes—
169C2×C22⋊C4C22⋊C412yes—
170C2×C42—22no—
171C2×C42—21no—
172C2×C42—42no—
173C2×C42C22×C412yes—
174C2×C42C22×C411yescharacteristic
175C2×C42C22×C412yes—
176C2×C42C2×D414yes—
177C2×C42C2×D412yes—
178C2×C42C2×D414yes—
179C2×C42C2411yescharacteristic
180C4⋊C8—44no—
181C4⋊C8—84no—
182C22⋊C8—44no—
183C22⋊C8—82no—
184C4×C8—44no—
185C4×C8—48no—
186#95—24no—
187#95—22no—
188#95—22no—
189#95C22⋊C414yes—
190#95C22⋊C414yes—
191C24—22no—
192C24—21no—
193C24—24no—
194C24—42no—
195C24—42no—
196C24—82no—
197C24—82no—
198C24C2×C22⋊C411yescharacteristic
199#41—48no—
200#41—48no—
201C2×Q8—21no—
202C2×Q8—41no—
203C2×Q8—41no—
204C2×Q8C22×C812yes—
205C2×D4—24no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—24no—
209C2×D4—216no—
210C2×D4—42no—
211C2×D4—41no—
212C2×D4—42no—
213C2×D4—41no—
214C2×D4—48no—
215C2×D4—48no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—84no—
221C2×D4—84no—
222C2×D4C22×C814yes—
223C2×D4C22×C812yes—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—44no—
241C22×C4—84no—
242C22×C4—84no—
243C22×C4—84no—
244C22×C4—82no—
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C414yes—
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C22×C812yes—
251C22×C4C2×M1612yes—
252C22×C4C23×C411yescharacteristic
253C22×C4C22×D412yes—
254C2×C8—24no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—44no—
259C2×C8—42no—
260C2×C8—44no—
261C2×C8—44no—
262C2×C8—42no—
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—44no—
268C4⋊C4—42no—
269C4⋊C4—84no—
270C4⋊C4C22⋊C818yes—
271C4⋊C4#10014yes—
272C22⋊C4—216no—
273C22⋊C4—24no—
274C22⋊C4—44no—
275C22⋊C4—42no—
276C22⋊C4—44no—
277C22⋊C4—42no—
278C22⋊C4—44no—
279C22⋊C4—42no—
280C22⋊C4—44no—
281C22⋊C4—44no—
282C22⋊C4—82no—
283C22⋊C4—84no—
284C22⋊C4C22⋊C818yes—
285C42—24no—
286C42—24no—
287C42—24no—
288C42—24no—
289C42—44no—
290C42—41no—
291C42—41no—
292C42—44no—
293C42—42no—
294C42C22×C814yes—
295C42C2×M1614yes—
296C23—21no—
297C23—21no—
298C23—22no—
299C23—22no—
300C23—24no—
301C23—22no—
302C23—48no—
303C23—41no—
304C23—42no—
305C23—41no—
306C23—42no—
307C23—44no—
308C23—42no—
309C23—42no—
310C23—44no—
311C23—88no—
312C23—162no—
313C23—162no—
314C23—164no—
315C23C2×C22⋊C812yes—
316C23C22×C22⋊C411yescharacteristic
317Q8—42no—
318Q8—42no—
319D4—44no—
320D4—42no—
321D4—44no—
322D4—42no—
323D4—48no—
324D4—48no—
325D4—816no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—48no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—164no—
353C2×C4—164no—
354C2×C4—164no—
355C2×C4—162no—
356C2×C4C2×C22⋊C818yes—
357C2×C4C2×C22⋊C812yes—
358C2×C4#41012yes—
359C2×C4C2×#10014yes—
360C2×C4C2×#10012yes—
361C2×C4C23×C811yescharacteristic
362C2×C4C22×M1611yescharacteristic
363C2×C4#57612yes—
364C2×C4#57712yes—
365C8—44no—
366C8—82no—
367C8—84no—
368C22—24no—
369C22—22no—
370C22—41no—
371C22—41no—
372C22—42no—
373C22—44no—
374C22—48no—
375C22—42no—
376C22—44no—
377C22—82no—
378C22—81no—
379C22—81no—
380C22—82no—
381C22—82no—
382C22—84no—
383C22—168no—
384C22C22×C22⋊C811yescharacteristic
385C22C2×#41011yescharacteristic
386C22C22×#10011yescharacteristic
387C22#289712yes—
388C22#290112yes—
389C4—24no—
390C4—28no—
391C4—21no—
392C4—21no—
393C4—22no—
394C4—22no—
395C4—42no—
396C4—42no—
397C4—42no—
398C4—81no—
399C4—81no—
400C4—82no—
401C4#146014yes—
402C2—42no—
403C2—82no—
404C2—84no—
405C2C2×#146011yescharacteristic
406C2#1743812yes—
407C2#1761012yes—
408C2#1768511yescharacteristic
409C2#1778211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
prime radicalC2×#535C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22632C2semidirect
3. #7601C2semidirect
4. #7597C2semidirect
5. #7587C2semidirect
6. #7572C2semidirect
7. #7567C2semidirect
8. #7563C2semidirect
25. #1831C22semidirect
29. #1744C22semidirect
30. #1272C22semidirect
31. #1272C22semidirect
32. #1272C22semidirect
35. #1270C22semidirect
36. #1270C22semidirect
40. #535C8semidirect

Maximal subgroups

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

subgroupindex
#342752
#226322
#76012
#75972
#75872
#75722
#75672
#75632