Finite Group Databasebeta

← #481621

Subgroups — #481621

Order 512.

1106 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35737C211yesmaximal, characteristic
2#20393C211yesmaximal, characteristic, complemented
3#17370C211yesmaximal, characteristic, complemented
4#11777C214yesmaximal, complemented
5#11774C214yesmaximal, complemented
6#11694C212yesmaximal, complemented
7#11691C212yesmaximal, complemented
8C2×D4×Q8C411yescharacteristic, complemented
9C2×#548—21no—
10C2×#520C2211yescharacteristic
11#2975—21no—
12C2×#410—21no—
13#2621C412yescomplemented
14#2621C2211yescharacteristic
15#2547C411yescharacteristic, complemented
16#2547C2212yes—
17#2535C411yescharacteristic, complemented
18#2535C2212yes—
19#2427—22no—
20#2427—21no—
21#2417—21no—
22#2417—22no—
23#2392C412yescomplemented
24#2392C2211yescharacteristic
25#2351C411yescharacteristic, complemented
26Q8×C22⋊C4—21no—
27#2274—21no—
28C4×#99—22no—
29C4×#99C2214yes—
30#1740C2212yes—
31#1465C2214yes—
32#1463—22no—
33#1461—21no—
34#1458C2212yes—
35#1344C2214yes—
36#1344C2218yes—
37#1343C2214yes—
38C22×#41D411yescharacteristic
39C23×Q8—21no—
40D4×Q8—44no—
41#548—48no—
42C2×C4⋊Q8C2×C412yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#124—21no—
45C2×#124—22no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—21no—
50C2×#121—21no—
51C2×#121—22no—
52#520—44no—
53C2×C4×Q8—21no—
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8D411yescharacteristic
56C2×C4×D4—22no—
57C2×C4×D4—21no—
58C2×C4×D4C2×C412yes—
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4D412yes—
62C2×C4×D4D411yescharacteristic
63C2×C4×D4C2312yes—
64C2×#117—22no—
65C2×#117C2×C412yes—
66C2×#117C2311yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D412yes—
69#436—44no—
70#431—22no—
71#410—28no—
72C2×#99—24no—
73#407—24no—
74#398—22no—
75#398—21no—
76#398C2×C411yescharacteristic
77#398C2×C412yes—
78#397—22no—
79#397—21no—
80#397C2×C412yes—
81#397C2×C412yes—
82#397C2×C412yes—
83#397C2312yes—
84#397C2311yescharacteristic
85#396—22no—
86#396—21no—
87#394—22no—
88#394—21no—
89#394C2×C411yescharacteristic
90#392C2×C412yes—
91#392C2311yescharacteristic
92#391—21no—
93#391—22no—
94#391—22no—
95#391—21no—
96#389—22no—
97#389—21no—
98#386—22no—
99#386—22no—
100#386—22no—
101#382C2×C412yes—
102#382C2×C411yescharacteristic
103#380—22no—
104#380—22no—
105#380—22no—
106#380—21no—
107#380—22no—
108C4×C4⋊C4—44no—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4C2×C411yescharacteristic
111C4×C22⋊C4—22no—
112C4×C22⋊C4—21no—
113C4×C22⋊C4—24no—
114#342—22no—
115#342C2314yes—
116#324—22no—
117#324—24no—
118#323C2314yes—
119C2×#41—24no—
120C2×#41—24no—
121C2×#41—42no—
122C22×Q8—22no—
123C22×Q8—21no—
124C22×Q8—44no—
125C22×Q8—44no—
126C22×Q8—44no—
127C22×Q8C22⋊C411yescharacteristic
128C22×D4—22no—
129C22×D4—21no—
130C22×D4C22⋊C411yescharacteristic
131C22×D4C22×C411yescharacteristic
132C22×D4C2×D412yes—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4C22⋊C412yes—
136C23×C4C22⋊C412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C2×M16—44no—
141C4⋊Q8—42no—
142C4⋊Q8—44no—
143#124—48no—
144#124—44no—
145C22⋊Q8—48no—
146C22⋊Q8—48no—
147C22⋊Q8—44no—
148C22⋊Q8—44no—
149#121—44no—
150#121—48no—
151C4×Q8—24no—
152C4×Q8—42no—
153C4×Q8—44no—
154C4×D4—24no—
155C4×D4—42no—
156C4×D4—44no—
157C4×D4—42no—
158C4×D4—44no—
159C4×D4—44no—
160C4×D4—48no—
161C4×D4—44no—
162#117—28no—
163#117—42no—
164#117—44no—
165#117—44no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22⋊C411yescharacteristic
181C2×C4⋊C4C22×C412yes—
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C4⋊C4C22×C411yescharacteristic
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C4⋊C4C2×D412yes—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4C22⋊C412yes—
199C2×C22⋊C4C22×C412yes—
200C2×C22⋊C4C2×D411yescharacteristic
201C2×C22⋊C4C2411yescharacteristic
202C2×C42—22no—
203C2×C42—21no—
204C2×C42—42no—
205C2×C42—44no—
206C2×C42C22⋊C412yes—
207C2×C42C22×C412yes—
208C2×C42C22×C411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C4⋊C8—42no—
211#99—216no—
212C22⋊C8—24no—
213C22⋊C8—44no—
214C4×C8—42no—
215#95—22no—
216#95—22no—
217#95—21no—
218#95—22no—
219#95—22no—
220#95—21no—
221#95—22no—
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—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—44no—
236C2×Q8—44no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—44no—
240C2×Q8—42no—
241C2×Q8—816no—
242C2×Q8#10414yes—
243C2×D4—24no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—24no—
247C2×D4—22no—
248C2×D4—42no—
249C2×D4—41no—
250C2×D4—42no—
251C2×D4—44no—
252C2×D4—44no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—82no—
256C2×D4—81no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—44no—
276C22×C4—42no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4AGL(1,ℤ/8)14yes—
288C22×C4C23×C411yescharacteristic
289C22×C4C22×D411yescharacteristic
290M16—164no—
291C2×C8—44no—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—24no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—42no—
302C4⋊C4—48no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—82no—
308C4⋊C4—81no—
309C4⋊C4#9914yes—
310C22⋊C4—24no—
311C22⋊C4—28no—
312C22⋊C4—42no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—44no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C22⋊C4—44no—
321C22⋊C4—42no—
322C22⋊C4—44no—
323C22⋊C4—44no—
324C22⋊C4—42no—
325C22⋊C4—82no—
326C22⋊C4—81no—
327C42—22no—
328C42—24no—
329C42—22no—
330C42—42no—
331C42—44no—
332C42—48no—
333C42—42no—
334C42—164no—
335C23—21no—
336C23—21no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—42no—
342C23—41no—
343C23—42no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23C22×C22⋊C411yescharacteristic
352C23C2×AGL(1,ℤ/8)12yes—
353Q8—28no—
354Q8—88no—
355Q8—84no—
356D4—48no—
357D4—42no—
358D4—42no—
359D4—44no—
360D4—84no—
361D4—88no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—42no—
374C2×C4—48no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—48no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—84no—
397C2×C4—84no—
398C2×C4—88no—
399C2×C4—84no—
400C2×C4—84no—
401C2×C4—84no—
402C2×C4—162no—
403C2×C4C2×#9914yes—
404C2×C4C2×#9912yes—
405C2×C4#41812yes—
406C2×C4#41812yes—
407C2×C4C2×#10414yes—
408C2×C4C2×#10412yes—
409C8—84no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—42no—
414C22—42no—
415C22—44no—
416C22—42no—
417C22—41no—
418C22—41no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—42no—
423C22—81no—
424C22—82no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22C22×#9911yescharacteristic
430C22C2×#41811yescharacteristic
431C22C22×#10411yescharacteristic
432C4—24no—
433C4—22no—
434C4—24no—
435C4—41no—
436C4—41no—
437C4—41no—
438C4—42no—
439C4—41no—
440C4—81no—
441C4—82no—
442C4—81no—
443C4—88no—
444C4—82no—
445C4—81no—
446C4#150714yes—
447C2—21no—
448C2—21no—
449C2—42no—
450C2—82no—
451C2—82no—
452C2C2×#150711yescharacteristic
453C2#3334911yescharacteristic
454C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#35737C2
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. #17370C2semidirect
4. #11777C2semidirect
5. #11774C2semidirect
6. #11694C2semidirect
7. #11691C2semidirect
8. C2×D4×Q8C4semidirect
13. #2621C4semidirect
15. #2547C4semidirect
17. #2535C4semidirect
23. #2392C4semidirect
25. #2351C4semidirect

Maximal subgroups

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

subgroupindex
#357372
#203932
#173702
#117772
#117742
#116942
#116912