Finite Group Databasebeta

← #481555

Subgroups — #481555

Order 512.

956 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36011C211yesmaximal, characteristic
2#20393C211yesmaximal, characteristic, complemented
3#17353C211yesmaximal, characteristic, complemented
4#11729C214yesmaximal, complemented
5#11727C214yesmaximal, complemented
6#11718C212yesmaximal, complemented
7#11691C212yesmaximal, complemented
8C2×#555C411yescharacteristic, complemented
9C2×#520C2211yescharacteristic
10#2982—21no—
11C2×#410—21no—
12#2635C2212yes—
13#2628C2211yescharacteristic
14#2591C412yescomplemented
15#2568C411yescharacteristic, complemented
16#2536C411yescharacteristic, complemented
17#2484C412yescomplemented
18#2474—22no—
19#2459—22no—
20#2420—21no—
21#2407—21no—
22#2395C2212yes—
23#2392C2211yescharacteristic
24#2362—21no—
25#2357C411yescharacteristic, complemented
26#2352—21no—
27#2274—21no—
28C4×#99—22no—
29C4×#99C2214yes—
30#1740C2212yes—
31#1460—21no—
32#1460C2214yes—
33#1458—22no—
34#1458C2212yes—
35#1352C2218yes—
36#1344C2214yes—
37#1343C2214yes—
38C22×#41D411yescharacteristic
39#555—44no—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×#124—22no—
42C2×C22⋊Q8—21no—
43C2×#121—21no—
44C2×#121C2×C412yes—
45#520—44no—
46C2×C4×Q8C2×C411yescharacteristic
47C2×C4×Q8D411yescharacteristic
48C2×C4×D4—21no—
49C2×C4×D4C2×C411yescharacteristic
50C2×C4×D4C2×C411yescharacteristic
51C2×C4×D4D412yes—
52C2×C4×D4D411yescharacteristic
53C2×C4×D4C2312yes—
54C2×#117—21no—
55C2×#117—22no—
56C2×#117C2×C412yes—
57C2×#117D412yes—
58C2×#117C2311yescharacteristic
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4D411yescharacteristic
61C8×Q8—44no—
62#431—24no—
63#431—22no—
64#410—28no—
65C2×#99—24no—
66#400C2×C412yes—
67#400C2×C412yes—
68#400C2×C412yes—
69#399C2312yes—
70#398C2×C411yescharacteristic
71#398C2×C412yes—
72#397—22no—
73#397C2311yescharacteristic
74#396—21no—
75#395—21no—
76#394—22no—
77#394—22no—
78#394—21no—
79#394C2×C411yescharacteristic
80#394C2×C412yes—
81#393—21no—
82#392—22no—
83#392C2311yescharacteristic
84#391—21no—
85#391—22no—
86#390—21no—
87#389—21no—
88#388—22no—
89#388—22no—
90#388—22no—
91#388—22no—
92#388—22no—
93#387—22no—
94#387—22no—
95#386—22no—
96#386—22no—
97#386—22no—
98#385—22no—
99#382—21no—
100#382C2×C411yescharacteristic
101#381—22no—
102#381C2×C412yes—
103#380—21no—
104#380—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—44no—
108C4×C4⋊C4C2×C411yescharacteristic
109C4×C22⋊C4—22no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—24no—
112C4×C22⋊C4C2×C412yes—
113#342—22no—
114#342C2314yes—
115#324—22no—
116#323—24no—
117#323C2314yes—
118C2×#41—24no—
119C2×#41—24no—
120C2×#41—42no—
121C22×Q8—21no—
122C22×Q8C22⋊C411yescharacteristic
123C22×D4—22no—
124C22×D4—21no—
125C22×D4C22⋊C411yescharacteristic
126C22×D4C22×C411yescharacteristic
127C22×D4C2×D412yes—
128C23×C4—21no—
129C23×C4C22⋊C412yes—
130C23×C4C22⋊C412yes—
131C23×C4C22×C412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C4⋊Q8—42no—
135#124—48no—
136C22⋊Q8—44no—
137#121—44no—
138#121—44no—
139C4×Q8—24no—
140C4×Q8—42no—
141C4×D4—24no—
142C4×D4—42no—
143C4×D4—44no—
144C4×D4—42no—
145C4×D4—44no—
146C4×D4—44no—
147#117—28no—
148#117—42no—
149#117—44no—
150#117—44no—
151#117—44no—
152#117—44no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22⋊C411yescharacteristic
165C2×C4⋊C4C22⋊C411yescharacteristic
166C2×C4⋊C4C22×C411yescharacteristic
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C22×C411yescharacteristic
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C2×D411yescharacteristic
171C2×C22⋊C4—24no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4C22⋊C412yes—
185C2×C22⋊C4C2×D411yescharacteristic
186C2×C22⋊C4C2×D412yes—
187C2×C22⋊C4C2411yescharacteristic
188C2×C42—22no—
189C2×C42—21no—
190C2×C42—22no—
191C2×C42—42no—
192C2×C42—44no—
193C2×C42C22⋊C412yes—
194C2×C42C22×C411yescharacteristic
195C2×C42C2×D412yes—
196C2×C42C2×D411yescharacteristic
197C4⋊C8—44no—
198C4⋊C8—42no—
199#99—216no—
200C22⋊C8—24no—
201C4×C8—44no—
202C4×C8—42no—
203#95—22no—
204#95—21no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—21no—
212#95—22no—
213#95—21no—
214#95—44no—
215#95C22×C411yescharacteristic
216#95C22×C412yes—
217#95C22×C411yescharacteristic
218C24—21no—
219C24C2×C22⋊C412yes—
220C24C22×D411yescharacteristic
221#41—48no—
222#41—48no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—42no—
226C2×Q8—82no—
227C2×Q8#10414yes—
228C2×D4—24no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—24no—
232C2×D4—22no—
233C2×D4—42no—
234C2×D4—41no—
235C2×D4—42no—
236C2×D4—44no—
237C2×D4—81no—
238C2×D4—81no—
239C2×D4—81no—
240C2×D4—82no—
241C2×D4—82no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4#13514yes—
270C22×C4C23×C411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C2×C8—44no—
273C4⋊C4—24no—
274C4⋊C4—22no—
275C4⋊C4—24no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—42no—
281C4⋊C4—48no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—44no—
286C4⋊C4—82no—
287C4⋊C4—81no—
288C4⋊C4#9914yes—
289C22⋊C4—24no—
290C22⋊C4—28no—
291C22⋊C4—42no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—42no—
295C22⋊C4—44no—
296C22⋊C4—44no—
297C22⋊C4—42no—
298C22⋊C4—44no—
299C22⋊C4—42no—
300C22⋊C4—44no—
301C22⋊C4—81no—
302C22⋊C4—81no—
303C22⋊C4—82no—
304C22⋊C4—82no—
305C22⋊C4—82no—
306C22⋊C4—82no—
307C42—24no—
308C42—22no—
309C42—22no—
310C42—42no—
311C42—48no—
312C42—44no—
313C42—42no—
314C42—82no—
315C42—164no—
316C23—21no—
317C23—21no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—41no—
322C23—41no—
323C23—42no—
324C23—42no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23C22×C22⋊C411yescharacteristic
333C23C2×#13512yes—
334Q8—28no—
335Q8—162no—
336D4—48no—
337D4—44no—
338D4—82no—
339D4—84no—
340D4—164no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—42no—
353C2×C4—48no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—48no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—84no—
376C2×C4—84no—
377C2×C4—84no—
378C2×C4—84no—
379C2×C4—84no—
380C2×C4—162no—
381C2×C4C2×#9914yes—
382C2×C4C2×#9912yes—
383C2×C4#41612yes—
384C2×C4#41612yes—
385C2×C4C2×#10414yes—
386C2×C4C2×#10412yes—
387C8—84no—
388C22—21no—
389C22—21no—
390C22—22no—
391C22—42no—
392C22—41no—
393C22—41no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—81no—
398C22—82no—
399C22—82no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22C22×#9911yescharacteristic
408C22C2×#41611yescharacteristic
409C22C22×#10411yescharacteristic
410C4—24no—
411C4—22no—
412C4—24no—
413C4—41no—
414C4—42no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—81no—
419C4—82no—
420C4—81no—
421C4—88no—
422C4—82no—
423C4—81no—
424C4#150114yes—
425C2—21no—
426C2—21no—
427C2—42no—
428C2—82no—
429C2—161no—
430C2C2×#150111yescharacteristic
431C2#3335011yescharacteristic
432C2#3336211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#36011C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20393C2semidirect
3. #17353C2semidirect
4. #11729C2semidirect
5. #11727C2semidirect
6. #11718C2semidirect
7. #11691C2semidirect
8. C2×#555C4semidirect
14. #2591C4semidirect
15. #2568C4semidirect
16. #2536C4semidirect
17. #2484C4semidirect
25. #2357C4semidirect

Maximal subgroups

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

subgroupindex
#360112
#203932
#173532
#117292
#117272
#117182
#116912