Finite Group Databasebeta

← #480952

Subgroups — #480952

Order 512.

836 conjugacy classes of subgroups.

The subgroup lattice has 397 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36185C211yesmaximal, characteristic
2#17418C212yesmaximal
3#11570C212yesmaximal, complemented
4#11552C214yesmaximal, complemented
5#11531C211yesmaximal, characteristic, complemented
6#11513C212yesmaximal, complemented
7#11477C211yesmaximal, characteristic, complemented
8#11447C212yesmaximal, complemented
9C2×#556C411yescharacteristic
10C2×#555C2211yescharacteristic
11#2924—22no—
12#2862C411yescharacteristic
13#2850C412yes—
14#2567C411yescharacteristic
15#2542C2211yescharacteristic
16#2521—24no—
17#2500C412yes—
18#2499C2212yes—
19#2427—22no—
20#2362—22no—
21C4×#124C2211yescharacteristic
22#2285C2212yes—
23C42×D4C411yescharacteristic
24#1471—22no—
25#1469—24no—
26#1465C2212yes—
27#1464C2214yes—
28#1459C2212yes—
29#1458C2214yes—
30#1325C2214yes—
31#1325C2214yes—
32#1324C2212yes—
33#1324C2211yescharacteristic
34#1324C2211yescharacteristic
35#1319C2212yes—
36#1319C2212yes—
37#556—42no—
38#556—41no—
39#556—41no—
40#555—41no—
41#555—41no—
42#555—42no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#126—22no—
46C2×#124—21no—
47C2×#124C2×C411yescharacteristic
48C2×#124D414yes—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8C2×C412yes—
51C2×#121C2×C412yes—
52C2×#121D412yes—
53C2×C4×Q8—21no—
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×D4—21no—
56C2×C4×D4C2×C411yescharacteristic
57C2×C4×D4C2311yescharacteristic
58C2×#117C2×C412yes—
59C2×#117D412yes—
60C2×#117C2312yes—
61C22×C42C2×C412yes—
62#433—44no—
63#432—24no—
64#431—24no—
65#424—42no—
66#407—22no—
67C2×C22⋊C8—22no—
68#405—42no—
69#399—22no—
70#397—24no—
71#395—22no—
72#394—24no—
73#391—22no—
74#391—22no—
75#391C2×C412yes—
76#391C2311yescharacteristic
77#390C2×C412yes—
78#390C2×C411yescharacteristic
79#389—22no—
80#388—24no—
81#388—24no—
82#388—22no—
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#324C2312yes—
99#324C2314yes—
100#324C2312yes—
101#323—24no—
102#323—22no—
103#323—24no—
104#323—22no—
105C22×Q8—21no—
106C22×Q8C22⋊C412yes—
107C22×Q8C22×C411yescharacteristic
108C22×D4C22⋊C411yescharacteristic
109C22×D4C22⋊C414yes—
110C22×D4C22×C411yescharacteristic
111C23×C4—22no—
112C23×C4C22×C412yes—
113C23×C4C2×D412yes—
114C2×M16—42no—
115C22×C8—42no—
116C4⋊Q8—41no—
117C4⋊Q8—41no—
118C4⋊Q8—41no—
119C4⋊Q8—41no—
120#126—44no—
121#126—44no—
122#124—22no—
123#124—44no—
124#124—44no—
125#124—41no—
126#124—42no—
127#124—42no—
128C22⋊Q8—48no—
129C22⋊Q8—42no—
130C22⋊Q8—42no—
131#121—42no—
132#121—42no—
133#121—42no—
134#121—42no—
135C4×Q8—41no—
136C4×Q8—41no—
137C4×Q8—42no—
138C4×Q8—41no—
139C4×Q8—41no—
140C4×D4—22no—
141C4×D4—21no—
142C4×D4—21no—
143C4×D4—41no—
144C4×D4—41no—
145C4×D4—44no—
146#117—22no—
147#117—22no—
148#117—24no—
149#117—42no—
150#117—42no—
151#117—42no—
152#117—42no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4C22⋊C412yes—
160C2×C4⋊C4C22⋊C411yescharacteristic
161C2×C4⋊C4C22×C411yescharacteristic
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C2411yescharacteristic
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4C22⋊C414yes—
172C2×C22⋊C4C22⋊C412yes—
173C2×C22⋊C4C2×D414yes—
174C2×C22⋊C4C2×D412yes—
175C2×C42—24no—
176C2×C42—22no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—22no—
180C2×C42—21no—
181C2×C42—42no—
182C2×C42C22×C412yes—
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C411yescharacteristic
185C2×C42C22×C411yescharacteristic
186C2×C42C2×D414yes—
187C4⋊C8—44no—
188C4⋊C8—44no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—24no—
192C22⋊C8—42no—
193C22⋊C8—42no—
194#97—44no—
195C4×C8—44no—
196#95—21no—
197#95—22no—
198#95—24no—
199#95—24no—
200#95—24no—
201#95—22no—
202#95—22no—
203C24C2×C22⋊C412yes—
204C24C2×C22⋊C412yes—
205C2×Q8—21no—
206C2×Q8—21no—
207C2×Q8—42no—
208C2×Q8—44no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8—41no—
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—22no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—42no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—82no—
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C411yescharacteristic
242C22×C4C2×C22⋊C414yes—
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C23×C411yescharacteristic
245C22×C4C22×D412yes—
246M16—84no—
247C2×C8—42no—
248C2×C8—44no—
249C2×C8—42no—
250C2×C8—84no—
251C4⋊C4—24no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—48no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—84no—
264C4⋊C4#9912yes—
265C4⋊C4#9911yescharacteristic
266C4⋊C4#9911yescharacteristic
267C22⋊C4—24no—
268C22⋊C4—24no—
269C22⋊C4—22no—
270C22⋊C4—22no—
271C22⋊C4—44no—
272C22⋊C4—42no—
273C22⋊C4—44no—
274C22⋊C4—44no—
275C22⋊C4—48no—
276C22⋊C4—44no—
277C22⋊C4—44no—
278C22⋊C4—82no—
279C42—21no—
280C42—24no—
281C42—24no—
282C42—21no—
283C42—21no—
284C42—22no—
285C42—22no—
286C42—44no—
287C42—41no—
288C42—42no—
289C42—42no—
290C42—42no—
291C42—41no—
292C42—41no—
293C42—81no—
294C42#13112yes—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23C22×C22⋊C411yescharacteristic
306Q8—81no—
307Q8—81no—
308Q8—82no—
309Q8—82no—
310Q8—162no—
311D4—42no—
312D4—42no—
313D4—81no—
314D4—81no—
315D4—164no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—24no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—24no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—44no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—82no—
349C2×C4—88no—
350C2×C4C2×#9912yes—
351C2×C4C2×#9912yes—
352C2×C4C2×#9911yescharacteristic
353C2×C4C2×#9911yescharacteristic
354C2×C4C2×#13111yescharacteristic
355C2×C4#56811yescharacteristic
356C8—82no—
357C8—82no—
358C8—84no—
359C22—21no—
360C22—21no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—82no—
368C22—82no—
369C22—82no—
370C22—82no—
371C22#288011yescharacteristic
372C22C22×#9911yescharacteristic
373C22#288911yescharacteristic
374C4—21no—
375C4—21no—
376C4—22no—
377C4—22no—
378C4—21no—
379C4—44no—
380C4—42no—
381C4—42no—
382C4—44no—
383C4—41no—
384C4—81no—
385C4—81no—
386C4—82no—
387C4—81no—
388C4#212612yes—
389C2—21no—
390C2—21no—
391C2—82no—
392C2—82no—
393C2#1741611yescharacteristic
394C2C2×#212611yescharacteristic
395C2#3337111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#555C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#555 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11570C2semidirect
4. #11552C2semidirect
5. #11531C2semidirect
6. #11513C2semidirect
7. #11477C2semidirect
8. #11447C2semidirect

Maximal subgroups

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

subgroupindex
#361852
#174182
#115702
#115522
#115312
#115132
#114772
#114472