Finite Group Databasebeta

← #481556

Subgroups — #481556

Order 512.

966 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36019C211yesmaximal, characteristic
2#20393C211yesmaximal, characteristic, complemented
3#17351C211yesmaximal, characteristic, complemented
4#11730C214yesmaximal, complemented
5#11728C214yesmaximal, complemented
6#11717C212yesmaximal, complemented
7#11691C212yesmaximal, complemented
8C2×#556C411yescharacteristic, complemented
9C2×#520C2211yescharacteristic
10#2972—21no—
11C2×#410—21no—
12#2631C2212yes—
13#2628C2211yescharacteristic
14#2610C412yescomplemented
15#2599C412yescomplemented
16#2563C411yescharacteristic, complemented
17#2535C411yescharacteristic, complemented
18#2481—22no—
19#2467—22no—
20#2421—21no—
21#2407—21no—
22#2398C2212yes—
23#2392C2211yescharacteristic
24#2363—21no—
25#2356C411yescharacteristic, complemented
26#2352—21no—
27#2274—21no—
28C4×#99—22no—
29C4×#99C2214yes—
30#1740C2212yes—
31#1461—21no—
32#1459C2214yes—
33#1458C2212yes—
34C2×#324—22no—
35#1351C2218yes—
36#1344C2214yes—
37#1343C2214yes—
38C22×#41D411yescharacteristic
39#556—44no—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×#126—22no—
42C2×#124C2×C412yes—
43C2×C22⋊Q8—21no—
44C2×#121—21no—
45#520—44no—
46C2×C4×Q8C2×C412yes—
47C2×C4×Q8C2×C411yescharacteristic
48C2×C4×Q8D411yescharacteristic
49C2×C4×D4—21no—
50C2×C4×D4C2×C411yescharacteristic
51C2×C4×D4C2×C411yescharacteristic
52C2×C4×D4D412yes—
53C2×C4×D4D411yescharacteristic
54C2×C4×D4C2312yes—
55C2×#117—21no—
56C2×#117—22no—
57C2×#117C2×C412yes—
58C2×#117C2311yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D412yes—
61C8×D4—44no—
62#431—22no—
63#410—28no—
64C2×#99—24no—
65C2×C22⋊C8—24no—
66#400—21no—
67#400—22no—
68#399—21no—
69#398C2×C411yescharacteristic
70#397—22no—
71#397C2×C412yes—
72#397C2311yescharacteristic
73#396—21no—
74#395C2×C412yes—
75#394—22no—
76#394—21no—
77#394C2×C411yescharacteristic
78#393C2×C412yes—
79#393C2×C412yes—
80#393C2312yes—
81#392—22no—
82#392C2×C412yes—
83#392C2311yescharacteristic
84#391—21no—
85#390—21no—
86#390—22no—
87#389—21no—
88#388—22no—
89#388—22no—
90#388—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#386—22no—
95#386—22no—
96#386—22no—
97#386—22no—
98#385—22no—
99#383C2×C412yes—
100#382—21no—
101#382C2×C411yescharacteristic
102#380—22no—
103#380—21no—
104#380—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—44no—
107C4×C4⋊C4C2×C411yescharacteristic
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—24no—
112#376C2×C412yes—
113#342—22no—
114#342C2314yes—
115#324—22no—
116#324—24no—
117#323C2314yes—
118C2×#41—24no—
119C2×#41—24no—
120C2×#41—42no—
121C22×Q8C22⋊C411yescharacteristic
122C22×Q8C22×C412yes—
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×C4C2×D411yescharacteristic
132C23×C4C2×D412yes—
133C23×C4C2×D412yes—
134C22×C8—44no—
135C4⋊Q8—42no—
136#126—48no—
137#124—44no—
138C22⋊Q8—44no—
139#121—44no—
140C4×Q8—24no—
141C4×Q8—42no—
142C4×Q8—44no—
143C4×D4—24no—
144C4×D4—42no—
145C4×D4—44no—
146C4×D4—42no—
147C4×D4—44no—
148C4×D4—44no—
149#117—28no—
150#117—42no—
151#117—44no—
152#117—44no—
153#117—44no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4C22⋊C412yes—
165C2×C4⋊C4C22⋊C411yescharacteristic
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C411yescharacteristic
168C2×C4⋊C4C22×C411yescharacteristic
169C2×C4⋊C4C22×C411yescharacteristic
170C2×C4⋊C4C2×D411yescharacteristic
171C2×C4⋊C4C2×D412yes—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4C22⋊C412yes—
186C2×C22⋊C4C22×C412yes—
187C2×C22⋊C4C2×D411yescharacteristic
188C2×C22⋊C4C2411yescharacteristic
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—42no—
193C2×C42—44no—
194C2×C42C22⋊C412yes—
195C2×C42C22×C412yes—
196C2×C42C22×C411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C4⋊C8—42no—
199#99—216no—
200C22⋊C8—24no—
201C22⋊C8—44no—
202C4×C8—42no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—21no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—21no—
211#95—22no—
212#95—21no—
213#95—22no—
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—22no—
226C2×Q8—42no—
227C2×Q8—82no—
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—42no—
233C2×D4—41no—
234C2×D4—42no—
235C2×D4—44no—
236C2×D4—81no—
237C2×D4—81no—
238C2×D4—81no—
239C2×D4—82no—
240C2×D4—82no—
241C2×D4#10414yes—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
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⋊C412yes—
263C22×C4C2×C22⋊C411yescharacteristic
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—
273C2×C8—164no—
274C4⋊C4—24no—
275C4⋊C4—22no—
276C4⋊C4—24no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—44no—
282C4⋊C4—42no—
283C4⋊C4—48no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—82no—
288C4⋊C4—81no—
289C4⋊C4#9914yes—
290C22⋊C4—24no—
291C22⋊C4—28no—
292C22⋊C4—42no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—42no—
296C22⋊C4—44no—
297C22⋊C4—44no—
298C22⋊C4—42no—
299C22⋊C4—44no—
300C22⋊C4—44no—
301C22⋊C4—42no—
302C22⋊C4—81no—
303C22⋊C4—82no—
304C22⋊C4—81no—
305C22⋊C4—82no—
306C22⋊C4—82no—
307C22⋊C4—82no—
308C42—22no—
309C42—22no—
310C42—24no—
311C42—42no—
312C42—48no—
313C42—44no—
314C42—42no—
315C42—82no—
316C42—164no—
317C23—22no—
318C23—21no—
319C23—22no—
320C23—21no—
321C23—42no—
322C23—41no—
323C23—42no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23C2×#10414yes—
333C23C22×C22⋊C411yescharacteristic
334C23C2×#13512yes—
335Q8—44no—
336Q8—84no—
337D4—28no—
338D4—48no—
339D4—82no—
340D4—84no—
341D4—164no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—42no—
355C2×C4—48no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—48no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—84no—
378C2×C4—84no—
379C2×C4—84no—
380C2×C4—84no—
381C2×C4—84no—
382C2×C4—162no—
383C2×C4C2×#9914yes—
384C2×C4C2×#9912yes—
385C2×C4#41612yes—
386C2×C4#41612yes—
387C2×C4C2×#10412yes—
388C8—84no—
389C22—21no—
390C22—21no—
391C22—22no—
392C22—24no—
393C22—42no—
394C22—44no—
395C22—41no—
396C22—42no—
397C22—41no—
398C22—42no—
399C22—81no—
400C22—82no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22C22×#9911yescharacteristic
409C22C2×#41611yescharacteristic
410C22C22×#10411yescharacteristic
411C4—24no—
412C4—22no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—42no—
417C4—41no—
418C4—42no—
419C4—81no—
420C4—81no—
421C4—82no—
422C4—88no—
423C4—82no—
424C4—81no—
425C4#150114yes—
426C2—21no—
427C2—21no—
428C2—42no—
429C2—44no—
430C2—161no—
431C2C2×#150111yescharacteristic
432C2#3335111yescharacteristic
433C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#36019C2
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. #17351C2semidirect
4. #11730C2semidirect
5. #11728C2semidirect
6. #11717C2semidirect
7. #11691C2semidirect
8. C2×#556C4semidirect
14. #2610C4semidirect
15. #2599C4semidirect
16. #2563C4semidirect
17. #2535C4semidirect
25. #2356C4semidirect

Maximal subgroups

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

subgroupindex
#360192
#203932
#173512
#117302
#117282
#117172
#116912