Finite Group Databasebeta

← #481479

Subgroups — #481479

Order 512.

1336 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36009C211yesmaximal, characteristic
2#20354C211yesmaximal, characteristic, complemented
3#17350C211yesmaximal, characteristic, complemented
4#11694C212yesmaximal, complemented
5#11692C214yesmaximal, complemented
6#11691C212yesmaximal, complemented
7C2×#1344C214yesmaximal, complemented
8C2×#547C411yescharacteristic, complemented
9C2×#546—21no—
10C2×#518—21no—
11C2×#518C2211yescharacteristic
12#2972—21no—
13#2621C2211yescharacteristic
14#2617C2212yes—
15#2547C411yescharacteristic, complemented
16#2543C412yescomplemented
17#2535C411yescharacteristic, complemented
18#2486C412yescomplemented
19#2427—21no—
20#2425—22no—
21#2415—21no—
22#2412—22no—
23#2392C2211yescharacteristic
24C2×#397C2212yes—
25#2349C411yescharacteristic, complemented
26#2339—21no—
27#1740—21no—
28#1740C2212yes—
29C2×#342—22no—
30C2×#342C2214yes—
31#1460—21no—
32#1459—22no—
33#1458C2212yes—
34C2×#323C2214yes—
35#1344C2214yes—
36#1344C2218yes—
37#1343C2214yes—
38C22×#41—21no—
39C23×D4D411yescharacteristic
40#547—44no—
41#546—44no—
42#546—44no—
43C2×C4⋊Q8C2×C411yescharacteristic, complemented
44C2×#124—21no—
45C2×#123—22no—
46C2×#123—22no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×#121—22no—
50C2×#121—21no—
51C2×#121C2×C412yescomplemented
52C2×#120—22no—
53#518—28no—
54#518—44no—
55#518—44no—
56C2×C4×D4—21no—
57C2×C4×D4—21no—
58C2×C4×D4—44no—
59C2×C4×D4C2×C412yes—
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4C2×C411yescharacteristic, complemented
63C2×C4×D4D412yes—
64C2×C4×D4D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×#117—22no—
67C2×#117D411yescharacteristic
68C2×#117C2311yescharacteristic
69C22×C4⋊C4C2×C412yescomplemented
70C22×C22⋊C4—24no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4D412yes—
73C22×C22⋊C4C2312yes—
74C8×D4—44no—
75#431—22no—
76C2×C22⋊C8—24no—
77#398—21no—
78#398C2×C412yescomplemented
79#398C2×C411yescharacteristic, complemented
80#397—21no—
81#397—22no—
82#397—22no—
83#397C2×C412yescomplemented
84#397C2×C412yescomplemented
85#397C2×C412yescomplemented
86#397C2312yes—
87#397C2311yescharacteristic
88#396—21no—
89#394—22no—
90#394—22no—
91#394—21no—
92#394C2×C411yescharacteristic, complemented
93#392C2×C412yescomplemented
94#392C2311yescharacteristic
95#391—21no—
96#391—21no—
97#389—21no—
98#389—22no—
99#388—22no—
100#386—22no—
101#386—22no—
102#386—22no—
103#386—22no—
104#385—22no—
105#382C2×C411yescharacteristic, complemented
106#381—22no—
107#380—22no—
108#380—22no—
109#380—21no—
110#380C2×C412yescomplemented
111C4×C4⋊C4C2×C411yescharacteristic, complemented
112C4×C22⋊C4—21no—
113C2×#95—22no—
114C2×#95C2×C412yescomplemented
115#342—22no—
116#342—24no—
117#342C2314yes—
118#324—22no—
119#323—24no—
120#323C2314yes—
121C25C2×D412yes—
122C2×#41—44no—
123C2×#41—44no—
124C2×#41—44no—
125C22×Q8—21no—
126C22×D4—24no—
127C22×D4—24no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4C22⋊C412yes—
132C22×D4C22⋊C411yescharacteristic
133C22×D4C22⋊C411yescharacteristic
134C22×D4D814yes—
135C22×D4SD1614yes—
136C22×D4C22×C411yescharacteristic
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—44no—
141C23×C4C22⋊C412yes—
142C23×C4C22⋊C412yes—
143C23×C4C22×C412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C22×C8—44no—
147C4⋊Q8—42no—
148#124—44no—
149#123—48no—
150#123—48no—
151C22⋊Q8—44no—
152C22⋊Q8—42no—
153C22⋊Q8—42no—
154#121—44no—
155#121—44no—
156#121—48no—
157#120—44no—
158#120—44no—
159C4×D4—24no—
160C4×D4—24no—
161C4×D4—42no—
162C4×D4—42no—
163C4×D4—42no—
164C4×D4—44no—
165C4×D4—44no—
166C4×D4—432no—
167C4×D4—44no—
168C4×D4—42no—
169C4×D4—42no—
170#117—28no—
171#117—42no—
172#117—42no—
173#117—44no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C22⋊C4—216no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—48no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4C22⋊C412yes—
210C2×C22⋊C4C22⋊C412yes—
211C2×C22⋊C4C22×C412yes—
212C2×C22⋊C4C2×D411yescharacteristic
213C2×C22⋊C4C2×D412yes—
214C2×C22⋊C4C2×D412yes—
215C2×C22⋊C4C2411yescharacteristic
216C2×C42—21no—
217C2×C42—42no—
218C2×C42C22⋊C412yes—
219C2×C42C22×C411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C4⋊C8—42no—
222C22⋊C8—24no—
223C22⋊C8—44no—
224C4×C8—42no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—21no—
229#95—22no—
230#95—22no—
231#95—21no—
232#95—21no—
233#95—22no—
234#95C22×C411yescharacteristic
235#95C22×C412yes—
236#95C22×C411yescharacteristic
237C24—24no—
238C24—24no—
239C24—21no—
240C24—82no—
241C24—82no—
242C24C2×C22⋊C412yes—
243C24C2×C22⋊C412yes—
244C24C2×D814yes—
245C24C2×SD1614yes—
246C24C22×D411yescharacteristic
247#41—816no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×D4—24no—
252C2×D4—24no—
253C2×D4—216no—
254C2×D4—216no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—44no—
261C2×D4—44no—
262C2×D4—44no—
263C2×D4—44no—
264C2×D4—81no—
265C2×D4—81no—
266C2×D4—81no—
267C2×D4—82no—
268C2×D4—82no—
269C2×D4#9914yes—
270C2×D4#10218yes—
271C2×D4#10214yes—
272C2×D4#10214yes—
273C2×D4#10414yes—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—416no—
290C22×C4—44no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—164no—
301C22×C4—164no—
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×D812yes—
309C22×C4C2×SD1612yes—
310C22×C4C23×C411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C2×C8—44no—
313C2×C8—164no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—42no—
320C4⋊C4—44no—
321C4⋊C4—42no—
322C4⋊C4—48no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—44no—
326C4⋊C4—81no—
327C4⋊C4—82no—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—28no—
331C22⋊C4—41no—
332C22⋊C4—42no—
333C22⋊C4—41no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—44no—
338C22⋊C4—416no—
339C22⋊C4—44no—
340C22⋊C4—41no—
341C22⋊C4—42no—
342C22⋊C4—41no—
343C22⋊C4—42no—
344C22⋊C4—44no—
345C22⋊C4—44no—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C22⋊C4—42no—
349C22⋊C4—81no—
350C22⋊C4—82no—
351C22⋊C4—81no—
352C42—22no—
353C42—22no—
354C42—42no—
355C42—48no—
356C42—42no—
357C23—21no—
358C23—22no—
359C23—21no—
360C23—22no—
361C23—28no—
362C23—28no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—84no—
377C23—84no—
378C23—84no—
379C23—84no—
380C23—164no—
381C23C2×#9914yes—
382C23C2×#10214yes—
383C23C2×#10214yes—
384C23C2×#10414yes—
385C23C22×C22⋊C411yescharacteristic
386C23C22×D811yescharacteristic
387C23C22×SD1611yescharacteristic
388Q8—84no—
389D4—216no—
390D4—28no—
391D4—82no—
392D4—82no—
393D4—82no—
394D4—88no—
395D4—164no—
396D4#327116yes—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—48no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—84no—
430C2×C4—84no—
431C2×C4—84no—
432C2×C4—84no—
433C2×C4—88no—
434C2×C4—84no—
435C2×C4—162no—
436C2×C4—162no—
437C2×C4—164no—
438C2×C4—1616no—
439C2×C4C2×#9912yes—
440C2×C4C2×#10212yes—
441C2×C4C2×#10212yes—
442C2×C4C2×#10412yes—
443C8—84no—
444C22—21no—
445C22—24no—
446C22—21no—
447C22—24no—
448C22—21no—
449C22—21no—
450C22—24no—
451C22—44no—
452C22—44no—
453C22—41no—
454C22—41no—
455C22—44no—
456C22—41no—
457C22—42no—
458C22—41no—
459C22—42no—
460C22—88no—
461C22—82no—
462C22—81no—
463C22—88no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22C2×#32718yes—
472C22C22×#9911yescharacteristic
473C22C22×#10211yescharacteristic
474C22C22×#10411yescharacteristic
475C4—22no—
476C4—41no—
477C4—42no—
478C4—41no—
479C4—41no—
480C4—42no—
481C4—41no—
482C4—81no—
483C4—82no—
484C4—81no—
485C4—88no—
486C4—81no—
487C4—82no—
488C4C2×#32714yes—
489C2—21no—
490C2—21no—
491C2—28no—
492C2—41no—
493C2—44no—
494C2—41no—
495C2—161no—
496C2C22×#32711yescharacteristic
497C2#3335011yescharacteristic
498C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#36009C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#518 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20354C2semidirect
3. #17350C2semidirect
4. #11694C2semidirect
5. #11692C2semidirect
6. #11691C2semidirect
7. C2×#1344C2semidirect
8. C2×#547C4semidirect
15. #2547C4semidirect
16. #2543C4semidirect
17. #2535C4semidirect
18. #2486C4semidirect
25. #2349C4semidirect
43. C2×C4⋊Q8C2×C4semidirect
51. C2×#121C2×C4semidirect
62. C2×C4×D4C2×C4semidirect
69. C22×C4⋊C4C2×C4semidirect
78. #398C2×C4semidirect
79. #398C2×C4semidirect
83. #397C2×C4semidirect
84. #397C2×C4semidirect
85. #397C2×C4semidirect
92. #394C2×C4semidirect
93. #392C2×C4semidirect
105. #382C2×C4semidirect
110. #380C2×C4semidirect
111. C4×C4⋊C4C2×C4semidirect
114. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
#360092
#203542
#173502
#116942
#116922
#116912
C2×#13442