Finite Group Databasebeta

← #480958

Subgroups — #480958

Order 512.

825 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36185C211yesmaximal, characteristic
2#17428C212yesmaximal, complemented
3#11574C212yesmaximal, complemented
4#11554C214yesmaximal, complemented
5#11530C211yesmaximal, characteristic, complemented
6#11513C212yesmaximal, complemented
7#11476C211yesmaximal, characteristic, complemented
8#11447C212yesmaximal, complemented
9C2×#556C2211yescharacteristic
10C2×#555C411yescharacteristic
11#2984—22no—
12#2862C411yescharacteristic
13#2850C412yes—
14#2567C2211yescharacteristic
15#2542C411yescharacteristic
16#2521—24no—
17#2500C2212yes—
18#2499C412yes—
19#2427—22no—
20#2362—22no—
21C4×#124C2211yescharacteristic
22#2285C2212yes—
23C42×D4C411yescharacteristic
24#1473—24no—
25#1469—22no—
26#1467C2212yes—
27#1464C2214yes—
28#1461C2212yes—
29#1458C2214yes—
30#1325C2214yes—
31#1325C2214yes—
32#1324C2212yes—
33#1324C2211yescharacteristic
34#1324C2211yescharacteristic
35#1318C2212yes—
36#1318C2212yes—
37#556—41no—
38#556—42no—
39#556—41no—
40#555—42no—
41#555—41no—
42#555—41no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#126D414yes—
46C2×#124—22no—
47C2×#124C2×C411yescharacteristic
48C2×#124D412yes—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8C2×C412yes—
51C2×#121—21no—
52C2×#121C2×C412yes—
53C2×C4×Q8D412yes—
54C2×C4×Q8C2311yescharacteristic
55C2×C4×D4—21no—
56C2×C4×D4C2×C411yescharacteristic
57C2×C4×D4C2×C411yescharacteristic
58C2×#117—21no—
59C2×#117C2×C412yes—
60C2×#117C2312yes—
61C22×C42C2×C412yes—
62#446—44no—
63#446—42no—
64C8×Q8—42no—
65#432—24no—
66#432—22no—
67#431—24no—
68#431—22no—
69#399—22no—
70#397—24no—
71#395—22no—
72#394—24no—
73#391—22no—
74#391—22no—
75#391C2×C412yes—
76#391C2×C411yescharacteristic
77#390C2×C412yes—
78#390C2311yescharacteristic
79#389—22no—
80#388—24no—
81#388—22no—
82#388—24no—
83#387—24no—
84#386—24no—
85#384C2×C412yes—
86#384C2312yes—
87#382C2×C412yes—
88#382C2×C412yes—
89#382C2311yescharacteristic
90#380—24no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—24no—
93C4×C4⋊C4C2×C411yescharacteristic
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4C2×C412yes—
97C43C2×C411yescharacteristic
98#324—22no—
99#324—22no—
100#324C2312yes—
101#324C2314yes—
102#324C2312yes—
103#323—24no—
104#323—24no—
105C22×Q8—21no—
106C22×Q8C22⋊C412yes—
107C22×Q8C22×C411yescharacteristic
108C22×D4—22no—
109C22×D4C22⋊C411yescharacteristic
110C22×D4C22×C411yescharacteristic
111C23×C4—22no—
112C23×C4—21no—
113C23×C4C22×C412yes—
114C4⋊Q8—41no—
115C4⋊Q8—41no—
116C4⋊Q8—41no—
117C4⋊Q8—41no—
118#126—44no—
119#126—44no—
120#124—22no—
121#124—41no—
122#124—42no—
123#124—42no—
124#124—44no—
125#124—44no—
126C22⋊Q8—42no—
127C22⋊Q8—42no—
128C22⋊Q8—48no—
129#121—42no—
130#121—42no—
131#121—42no—
132#121—42no—
133C4×Q8—22no—
134C4×Q8—24no—
135C4×Q8—22no—
136C4×Q8—41no—
137C4×Q8—41no—
138C4×D4—22no—
139C4×D4—21no—
140C4×D4—21no—
141C4×D4—41no—
142C4×D4—41no—
143C4×D4—44no—
144#117—42no—
145#117—42no—
146#117—42no—
147#117—41no—
148#117—41no—
149#117—42no—
150#117—42no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4C22⋊C414yes—
156C2×C4⋊C4C22⋊C412yes—
157C2×C4⋊C4C22⋊C411yescharacteristic
158C2×C4⋊C4C22×C411yescharacteristic
159C2×C4⋊C4C22×C412yes—
160C2×C4⋊C4C22×C412yes—
161C2×C4⋊C4C2×D412yes—
162C2×C4⋊C4C2411yescharacteristic
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4C22⋊C412yes—
171C2×C22⋊C4C22⋊C414yes—
172C2×C22⋊C4C2×D414yes—
173C2×C42—24no—
174C2×C42—22no—
175C2×C42—21no—
176C2×C42—22no—
177C2×C42—21no—
178C2×C42—42no—
179C2×C42C22×C412yes—
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C411yescharacteristic
183C2×C42C2×D414yes—
184C2×C42C2×D412yes—
185C4⋊C8—44no—
186C4⋊C8—42no—
187C4⋊C8—44no—
188C4⋊C8—42no—
189C22⋊C8—22no—
190C22⋊C8—24no—
191C22⋊C8—22no—
192#97—44no—
193#97—42no—
194C4×C8—44no—
195C4×C8—42no—
196#95—21no—
197#95—22no—
198#95—22no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—22no—
203C24—21no—
204C24C2×C22⋊C412yes—
205C2×Q8—21no—
206C2×Q8—21no—
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—44no—
212C2×Q8—42no—
213C2×Q8—82no—
214C2×D4—21no—
215C2×D4—21no—
216C2×D4—21no—
217C2×D4—21no—
218C2×D4—42no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—44no—
222C2×D4—84no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—82no—
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C22⋊C414yes—
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C23×C411yescharacteristic
245C22×C4C22×D412yes—
246C2×C8—42no—
247C2×C8—44no—
248C2×C8—42no—
249C4⋊C4—24no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—48no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—84no—
262C4⋊C4#9912yes—
263C4⋊C4#9911yescharacteristic
264C4⋊C4#9911yescharacteristic
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—44no—
268C22⋊C4—44no—
269C22⋊C4—44no—
270C22⋊C4—44no—
271C22⋊C4—42no—
272C22⋊C4—48no—
273C22⋊C4—44no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—82no—
277C42—21no—
278C42—24no—
279C42—22no—
280C42—24no—
281C42—22no—
282C42—21no—
283C42—21no—
284C42—22no—
285C42—22no—
286C42—44no—
287C42—41no—
288C42—42no—
289C42—42no—
290C42—42no—
291C42—81no—
292C42#13112yes—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—42no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23C22×C22⋊C411yescharacteristic
304Q8—44no—
305Q8—81no—
306Q8—81no—
307Q8—82no—
308Q8—162no—
309D4—42no—
310D4—42no—
311D4—81no—
312D4—81no—
313D4—164no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—22no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—82no—
346C2×C4—88no—
347C2×C4—82no—
348C2×C4C2×#9911yescharacteristic
349C2×C4C2×#9912yes—
350C2×C4C2×#9912yes—
351C2×C4C2×#9911yescharacteristic
352C2×C4C2×#13111yescharacteristic
353C2×C4#56811yescharacteristic
354C8—82no—
355C8—84no—
356C8—82no—
357C22—21no—
358C22—21no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—82no—
363C22—82no—
364C22—82no—
365C22—82no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22#288011yescharacteristic
370C22C22×#9911yescharacteristic
371C22#288911yescharacteristic
372C4—21no—
373C4—21no—
374C4—22no—
375C4—22no—
376C4—21no—
377C4—44no—
378C4—42no—
379C4—42no—
380C4—42no—
381C4—41no—
382C4—44no—
383C4—81no—
384C4—82no—
385C4—81no—
386C4#213212yes—
387C2—21no—
388C2—21no—
389C2—82no—
390C2—161no—
391C2#1741611yescharacteristic
392C2#3336211yescharacteristic
393C2C2×#213211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#555C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#556 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17428C2semidirect
3. #11574C2semidirect
4. #11554C2semidirect
5. #11530C2semidirect
6. #11513C2semidirect
7. #11476C2semidirect
8. #11447C2semidirect

Maximal subgroups

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

subgroupindex
#361852
#174282
#115742
#115542
#115302
#115132
#114762
#114472