Finite Group Databasebeta

← #481690

Subgroups — #481690

Order 512.

1015 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35749C211yesmaximal, characteristic
2#20355C211yesmaximal, characteristic
3#17371C211yesmaximal, characteristic
4#11760C214yesmaximal
5#11756C214yesmaximal
6#11715C212yesmaximal
7#11694C212yesmaximal
8C2×#557C411yescharacteristic, complemented
9C2×#519—21no—
10C2×#519C2211yescharacteristic
11#2983—21no—
12#2634C412yescomplemented
13#2621C2211yescharacteristic
14#2611C2212yes—
15#2599C2212yes—
16#2581C411yescharacteristic, complemented
17#2546C411yescharacteristic, complemented
18#2475—22no—
19#2462—22no—
20#2446—21no—
21#2434—21no—
22#2399C412yescomplemented
23#2396C2211yescharacteristic
24#2365—21no—
25#2362—21no—
26#2356C411yescharacteristic, complemented
27#1740—21no—
28#1740—22no—
29#1740C2212yes—
30#1740C2214yes—
31#1467—22no—
32#1464C2214yes—
33#1461—21no—
34#1458C2212yes—
35#1352C2214yes—
36#1351C2214yes—
37#1344C2218yes—
38C23×Q8D411yescharacteristic
39#557—44no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×#125—22no—
44C2×#125C2×C412yes—
45C2×C22⋊Q8—21no—
46#519—28no—
47#519—44no—
48#519—44no—
49C2×C4×Q8—21no—
50C2×C4×Q8—44no—
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×Q8C2×C412yes—
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8D412yes—
58C2×C4×Q8D411yescharacteristic
59C2×#117—24no—
60C2×#117—22no—
61C2×#117D411yescharacteristic
62C2×#117D412yes—
63C2×#117C2312yes—
64C2×#117C2311yescharacteristic
65#446—44no—
66#432—24no—
67#431—22no—
68#400—22no—
69#400C2×C412yes—
70#400C2×C411yescharacteristic
71#399C2311yescharacteristic
72#398—22no—
73#398—21no—
74#398—21no—
75#398C2×C412yes—
76#398C2×C412yes—
77#398C2×C412yes—
78#397—21no—
79#397C2312yes—
80#396—21no—
81#395—22no—
82#395—22no—
83#395C2×C412yes—
84#395C2×C411yescharacteristic
85#393C2311yescharacteristic
86#391—21no—
87#391—21no—
88#391—21no—
89#391—21no—
90#389—21no—
91#389—22no—
92#389—22no—
93#388—22no—
94#388—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#385—22no—
103#385—22no—
104#383C2×C412yes—
105#382C2×C411yescharacteristic
106C4×C4⋊C4—21no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4C2×C411yescharacteristic
111C4×C22⋊C4—21no—
112#376C2×C412yes—
113#342—24no—
114#342—22no—
115#342C2314yes—
116#324—22no—
117#324—24no—
118#323C2314yes—
119C22×Q8—24no—
120C22×Q8—24no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8C22⋊C412yes—
125C22×Q8C22⋊C411yescharacteristic
126C22×Q8C22⋊C411yescharacteristic
127C22×Q8C22×C411yescharacteristic
128C23×C4C2×D412yes—
129C23×C4C2×D411yescharacteristic
130C4⋊Q8—42no—
131C4⋊Q8—44no—
132C4⋊Q8—44no—
133#125—44no—
134#125—48no—
135C22⋊Q8—42no—
136C22⋊Q8—42no—
137C4×Q8—24no—
138C4×Q8—24no—
139C4×Q8—44no—
140C4×Q8—42no—
141C4×Q8—42no—
142C4×Q8—44no—
143C4×Q8—44no—
144C4×Q8—42no—
145C4×Q8—44no—
146C4×Q8—432no—
147#117—216no—
148#117—28no—
149#117—42no—
150#117—44no—
151#117—42no—
152#117—44no—
153#117—48no—
154#117—44no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4C22⋊C411yescharacteristic
171C2×C4⋊C4C22⋊C412yes—
172C2×C4⋊C4C22⋊C411yescharacteristic
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22×C411yescharacteristic
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C411yescharacteristic
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C2×D412yes—
179C2×C4⋊C4C2×D411yescharacteristic
180C2×C22⋊C4—24no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4C2×D412yes—
185C2×C22⋊C4C2×D411yescharacteristic
186C2×C22⋊C4C2411yescharacteristic
187C2×C42—22no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—22no—
191C2×C42—44no—
192C2×C42—42no—
193C2×C42C22⋊C412yes—
194C2×C42C22⋊C412yes—
195C2×C42C22⋊C412yes—
196C2×C42C22×C412yes—
197C2×C42C22×C411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D412yes—
200C4⋊C8—42no—
201C4⋊C8—44no—
202C22⋊C8—24no—
203#97—44no—
204C4×C8—42no—
205#95—22no—
206#95—21no—
207#95—21no—
208#95—22no—
209#95—22no—
210#95—21no—
211#95—22no—
212#95—22no—
213#95—21no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—21no—
219#95—22no—
220#95C22×C411yescharacteristic
221#95C22×C412yes—
222#95C22×C411yescharacteristic
223C24C22×D411yescharacteristic
224C2×Q8—216no—
225C2×Q8—216no—
226C2×Q8—24no—
227C2×Q8—22no—
228C2×Q8—24no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—24no—
232C2×Q8—22no—
233C2×Q8—42no—
234C2×Q8—41no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8#9914yes—
239C2×Q8#10414yes—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—44no—
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4AGL(1,ℤ/8)12yes—
262C22×C4#13712yes—
263C22×C4C23×C411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C2×C8—44no—
266C4⋊C4—24no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—24no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—416no—
284C4⋊C4—48no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—42no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C22⋊C4—28no—
292C22⋊C4—42no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—44no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C42—22no—
301C42—24no—
302C42—24no—
303C42—22no—
304C42—42no—
305C42—42no—
306C42—42no—
307C42—44no—
308C42—44no—
309C42—416no—
310C42—48no—
311C42—42no—
312C42—42no—
313C23—21no—
314C23—21no—
315C23—41no—
316C23—41no—
317C23C22×C22⋊C411yescharacteristic
318C23C2×AGL(1,ℤ/8)11yescharacteristic
319C23C2×#13711yescharacteristic
320Q8—216no—
321Q8—28no—
322Q8—28no—
323Q8—42no—
324Q8—42no—
325Q8—84no—
326C2×C4—28no—
327C2×C4—28no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—48no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—84no—
368C2×C4—84no—
369C2×C4C2×#9914yes—
370C2×C4C2×#9912yes—
371C2×C4#41712yes—
372C2×C4#41712yes—
373C2×C4C2×#10414yes—
374C2×C4C2×#10412yes—
375C8—84no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—81no—
385C22C22×#9911yescharacteristic
386C22C2×#41711yescharacteristic
387C22C22×#10411yescharacteristic
388C4—24no—
389C4—22no—
390C4—24no—
391C4—42no—
392C4—41no—
393C4—42no—
394C4—42no—
395C4—41no—
396C4—41no—
397C4—42no—
398C4—41no—
399C4—81no—
400C4—82no—
401C4—88no—
402C4—81no—
403C4—82no—
404C4#150414yes—
405C2—21no—
406C2—21no—
407C2—41no—
408C2—41no—
409C2C2×#150411yescharacteristic
410C2#3334911yescharacteristic
411C2#3336011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×#519 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. C2×#557C4semidirect
12. #2634C4semidirect
16. #2581C4semidirect
17. #2546C4semidirect
22. #2399C4semidirect
26. #2356C4semidirect

Maximal subgroups

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

subgroupindex
#357492
#203552
#173712
#117602
#117562
#117152
#116942