Finite Group Databasebeta

← #481623

Subgroups — #481623

Order 512.

1101 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36020C211yesmaximal, characteristic
2#20393C211yesmaximal, characteristic, complemented
3#17372C211yesmaximal, characteristic, complemented
4#11775C214yesmaximal, complemented
5#11773C214yesmaximal, complemented
6#11694C212yesmaximal, complemented
7#11691C212yesmaximal, complemented
8C2×#547—21no—
9C2×#547C411yescharacteristic, complemented
10C2×#520C2211yescharacteristic
11#2983—21no—
12C2×#410—21no—
13#2621C2211yescharacteristic
14#2618C412yescomplemented
15#2546C411yescharacteristic, complemented
16#2543C2212yes—
17#2536C411yescharacteristic, complemented
18#2486C2212yes—
19#2429—21no—
20#2425—22no—
21#2417—21no—
22#2413—22no—
23#2392C2211yescharacteristic
24C2×#398C412yescomplemented
25#2349C411yescharacteristic, complemented
26#2339—21no—
27#2274—21no—
28C4×#99—22no—
29C4×#99C2214yes—
30#1740C2212yes—
31#1466C2214yes—
32#1464—22no—
33#1460—21no—
34#1458C2212yes—
35#1344C2214yes—
36#1344C2218yes—
37#1343C2214yes—
38C22×#41—21no—
39C22×#41D411yescharacteristic
40#547—44no—
41#547—48no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#123—22no—
45C2×#123—22no—
46C2×#123—22no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×#121—22no—
51C2×#121C2×C412yes—
52#520—44no—
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×Q8D411yescharacteristic
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4C2×C411yescharacteristic
58C2×C4×D4C2×C411yescharacteristic
59C2×C4×D4D412yes—
60C2×C4×D4D411yescharacteristic
61C2×C4×D4C2312yes—
62C2×#117—22no—
63C2×#117D412yes—
64C2×#117C2311yescharacteristic
65C22×C4⋊C4—22no—
66C22×C4⋊C4C2×C412yes—
67C22×C4⋊C4C2×C412yes—
68C22×C4⋊C4D411yescharacteristic
69#446—44no—
70#432—24no—
71#431—22no—
72#410—28no—
73C2×#99—24no—
74#398—22no—
75#398—22no—
76#398—21no—
77#398C2×C412yes—
78#398C2×C412yes—
79#398C2×C411yescharacteristic
80#398C2×C412yes—
81#398C2×C412yes—
82#397—21no—
83#397C2312yes—
84#397C2311yescharacteristic
85#396—22no—
86#396—22no—
87#396—21no—
88#394—21no—
89#394C2×C411yescharacteristic
90#394C2×C412yes—
91#392C2311yescharacteristic
92#391—21no—
93#391—21no—
94#389—21no—
95#389—22no—
96#389—22no—
97#388—22no—
98#388—22no—
99#386—22no—
100#386—22no—
101#386—22no—
102#385—22no—
103#382C2×C411yescharacteristic
104#381—22no—
105#380—22no—
106#380—21no—
107#380C2×C412yes—
108C4×C4⋊C4—44no—
109C4×C4⋊C4C2×C411yescharacteristic
110C4×C22⋊C4—21no—
111C4×C22⋊C4—24no—
112C2×#95—22no—
113C2×#95C2×C412yes—
114#342—22no—
115#342C2314yes—
116#324—22no—
117#323—24no—
118#323C2314yes—
119C2×#41—24no—
120C2×#41—24no—
121C2×#41—42no—
122C2×#41—44no—
123C2×#41—44no—
124C2×#41—44no—
125C22×Q8—21no—
126C22×Q8C22⋊C411yescharacteristic
127C22×D4—22no—
128C22×D4—21no—
129C22×D4C22⋊C411yescharacteristic
130C22×D4C22×C411yescharacteristic
131C22×D4C2×D412yes—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4C22⋊C412yes—
136C23×C4C22⋊C412yes—
137C23×C4C22×C412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C4⋊Q8—42no—
141C4⋊Q8—44no—
142#123—48no—
143#123—48no—
144#123—48no—
145C22⋊Q8—44no—
146C22⋊Q8—44no—
147C22⋊Q8—44no—
148#121—48no—
149#121—44no—
150C4×Q8—24no—
151C4×Q8—42no—
152C4×D4—24no—
153C4×D4—42no—
154C4×D4—44no—
155C4×D4—42no—
156C4×D4—44no—
157C4×D4—44no—
158C4×D4—44no—
159#117—28no—
160#117—42no—
161#117—44no—
162#117—44no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—48no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22⋊C411yescharacteristic
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C411yescharacteristic
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C2×D411yescharacteristic
187C2×C22⋊C4—24no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4C22⋊C412yes—
199C2×C22⋊C4C2×D411yescharacteristic
200C2×C22⋊C4C2×D412yes—
201C2×C22⋊C4C2411yescharacteristic
202C2×C42—21no—
203C2×C42—42no—
204C2×C42—44no—
205C2×C42C22⋊C412yes—
206C2×C42C22×C411yescharacteristic
207C2×C42C2×D412yes—
208C2×C42C2×D411yescharacteristic
209C4⋊C8—42no—
210C4⋊C8—44no—
211#99—216no—
212C22⋊C8—24no—
213#97—44no—
214C4×C8—42no—
215#95—21no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—21no—
222#95—21no—
223#95—22no—
224#95—44no—
225#95C22×C411yescharacteristic
226#95C22×C412yes—
227#95C22×C411yescharacteristic
228C24—21no—
229C24C2×C22⋊C412yes—
230C24C22×D411yescharacteristic
231#41—48no—
232#41—48no—
233#41—816no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—42no—
245C2×D4—41no—
246C2×D4—42no—
247C2×D4—44no—
248C2×D4—44no—
249C2×D4—42no—
250C2×D4—44no—
251C2×D4—81no—
252C2×D4—81no—
253C2×D4#10414yes—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4AGL(1,ℤ/8)14yes—
288C22×C4C23×C411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C2×C8—44no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—24no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—48no—
305C4⋊C4—44no—
306C4⋊C4—42no—
307C4⋊C4—44no—
308C4⋊C4—42no—
309C4⋊C4—82no—
310C4⋊C4#9914yes—
311C22⋊C4—24no—
312C22⋊C4—28no—
313C22⋊C4—42no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C22⋊C4—44no—
321C22⋊C4—42no—
322C22⋊C4—44no—
323C22⋊C4—44no—
324C22⋊C4—42no—
325C22⋊C4—81no—
326C22⋊C4—82no—
327C42—22no—
328C42—24no—
329C42—22no—
330C42—42no—
331C42—48no—
332C42—42no—
333C42—164no—
334C23—22no—
335C23—21no—
336C23—22no—
337C23—21no—
338C23—42no—
339C23—42no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23C2×#10414yes—
350C23C22×C22⋊C411yescharacteristic
351C23C2×AGL(1,ℤ/8)12yes—
352Q8—44no—
353Q8—84no—
354D4—28no—
355D4—48no—
356D4—84no—
357D4—82no—
358D4—88no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—42no—
373C2×C4—48no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—48no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—84no—
395C2×C4—84no—
396C2×C4—88no—
397C2×C4—88no—
398C2×C4—84no—
399C2×C4—84no—
400C2×C4—84no—
401C2×C4—162no—
402C2×C4C2×#9914yes—
403C2×C4C2×#9912yes—
404C2×C4#41812yes—
405C2×C4#41812yes—
406C2×C4C2×#10412yes—
407C8—84no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—24no—
412C22—42no—
413C22—44no—
414C22—44no—
415C22—41no—
416C22—42no—
417C22—41no—
418C22—42no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22C22×#9911yescharacteristic
428C22C2×#41811yescharacteristic
429C22C22×#10411yescharacteristic
430C4—24no—
431C4—22no—
432C4—42no—
433C4—41no—
434C4—42no—
435C4—41no—
436C4—41no—
437C4—41no—
438C4—42no—
439C4—81no—
440C4—82no—
441C4—88no—
442C4—81no—
443C4—82no—
444C4#150714yes—
445C2—21no—
446C2—21no—
447C2—42no—
448C2—44no—
449C2—161no—
450C2C2×#150711yescharacteristic
451C2#3335011yescharacteristic
452C2#3336011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#36020C2
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. #17372C2semidirect
4. #11775C2semidirect
5. #11773C2semidirect
6. #11694C2semidirect
7. #11691C2semidirect
9. C2×#547C4semidirect
14. #2618C4semidirect
15. #2546C4semidirect
17. #2536C4semidirect
24. C2×#398C4semidirect
25. #2349C4semidirect

Maximal subgroups

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

subgroupindex
#360202
#203932
#173722
#117752
#117732
#116942
#116912