Finite Group Databasebeta

← #497490

Subgroups — #497490

Order 512.

1543 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36260C211yesmaximal, characteristic
2D4×#99C212yesmaximal, complemented
3#13298C212yesmaximal, complemented
4#13088C214yesmaximal, complemented
5#12856C212yesmaximal, complemented
6#12797C211yesmaximal, characteristic, complemented
7C4×#353C211yesmaximal, characteristic, complemented
8C2×#1393C212yesmaximal, complemented
9C2×#547C411yescharacteristic, complemented
10C2×#546—22no—
11C2×D42C2211yescharacteristic
12C22×#99C2214yes—
13#2858C411yescharacteristic, complemented
14#2856C412yescomplemented
15#2542C2211yescharacteristic
16#2541C411yescharacteristic, complemented
17#2540—22no—
18#2539—24no—
19#2501C412yescomplemented
20#2433C2212yes—
21D4×C22⋊C4—22no—
22C4×#127C2211yescharacteristic
23C2×#390C2212yes—
24C42×D4C411yescharacteristic, complemented
25C2×#353C2212yes—
26C2×#353C2212yes—
27#1964—22no—
28#1898—24no—
29#1799C2212yes—
30#1781C2214yes—
31C4×#99C2212yes—
32#1393C2214yes—
33#1393C2214yes—
34#1393C2211yescharacteristic
35#1393C2212yes—
36#1393C2211yescharacteristic
37C22×#41—21no—
38C23×D4D412yes—
39C23×D4C2312yes—
40#547—41no—
41#547—42no—
42#547—41no—
43#546—416no—
44D42—41no—
45D42—42no—
46D42—41no—
47C2×C4⋊Q8C2×C411yescharacteristic, complemented
48C2×#127C2×C411yescharacteristic
49C2×#127C2×C412yescomplemented
50C2×#127C2311yescharacteristic
51C2×#124—22no—
52C2×#123—24no—
53C2×#123—22no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—21no—
56C2×#121—24no—
57C2×#121—22no—
58C2×#121C2×C412yes—
59C2×#121C2×C412yescomplemented
60C2×#121D412yes—
61C2×#120—24no—
62C2×#120D414yes—
63C2×C4×D4—21no—
64C2×C4×D4—22no—
65C2×C4×D4—42no—
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4C2×C411yescharacteristic, complemented
68C2×C4×D4C2311yescharacteristic
69C22×C4⋊C4C2×C412yescomplemented
70C22×C22⋊C4—24no—
71C22×C22⋊C4—24no—
72C22×C42C2×C412yescomplemented
73C2×#99—28no—
74C2×#99—24no—
75C2×#99—24no—
76C2×#99—44no—
77C2×#99C2314yes—
78C2×#99C2312yes—
79C2×#99C2312yes—
80#399—24no—
81#399—22no—
82#392—22no—
83#392—24no—
84#390—22no—
85#390—24no—
86#390—24no—
87#390—22no—
88#390—42no—
89#390C2×C411yescharacteristic, complemented
90#390C2×C412yescomplemented
91#390C2×C412yescomplemented
92#390C2×C412yescomplemented
93#390C2311yescharacteristic
94#390C2312yes—
95#386—44no—
96#384C2×C411yescharacteristic, complemented
97#384C2×C412yescomplemented
98#380—22no—
99#379—24no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4C2×C411yescharacteristic, complemented
102C4×C22⋊C4—22no—
103C4×C22⋊C4—22no—
104C4×C22⋊C4C2×C412yescomplemented
105C43C2×C411yescharacteristic, complemented
106#353—24no—
107#353—42no—
108#342—22no—
109#342—22no—
110C25C2×D414yes—
111C2×#41—48no—
112C2×#41—41no—
113C2×#41—42no—
114C2×#41—41no—
115C22×Q8—21no—
116C22×D4—22no—
117C22×D4—24no—
118C22×D4—22no—
119C22×D4—24no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—24no—
125C22×D4—42no—
126C22×D4—42no—
127C22×D4—48no—
128C22×D4—82no—
129C22×D4C22⋊C414yes—
130C22×D4C22⋊C412yes—
131C22×D4C22⋊C412yes—
132C22×D4C22⋊C411yescharacteristic
133C22×D4C22×C411yescharacteristic
134C22×D4C22×C412yes—
135C22×D4C2×D414yes—
136C22×D4C2411yescharacteristic
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—44no—
141C23×C4C22×C412yes—
142C23×C4C2×D412yes—
143C4⋊Q8—41no—
144C4⋊Q8—41no—
145#127—22no—
146#127—21no—
147#127—21no—
148#127—41no—
149#127—41no—
150#127—82no—
151#124—48no—
152#123—416no—
153#123—44no—
154#123—44no—
155C22⋊Q8—48no—
156C22⋊Q8—42no—
157C22⋊Q8—42no—
158#121—42no—
159#121—42no—
160#121—48no—
161#121—42no—
162#121—42no—
163#121—42no—
164#121—42no—
165#121—416no—
166#120—44no—
167#120—44no—
168#120—416no—
169C4×D4—28no—
170C4×D4—22no—
171C4×D4—21no—
172C4×D4—21no—
173C4×D4—41no—
174C4×D4—41no—
175C4×D4—48no—
176C4×D4—41no—
177C4×D4—41no—
178C4×D4—88no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C412yes—
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—48no—
204C2×C22⋊C4—416no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—84no—
209C2×C22⋊C4C22⋊C412yes—
210C2×C22⋊C4C22⋊C414yes—
211C2×C22⋊C4C2×D412yes—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—42no—
216C2×C42—82no—
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C412yes—
220C2×C42C22×C411yescharacteristic
221#99—22no—
222#99—22no—
223#99—24no—
224#99—24no—
225#99—22no—
226#99—48no—
227#99—84no—
228#99#4114yescomplemented
229#95—21no—
230#95—22no—
231#95—42no—
232#95—42no—
233C24—24no—
234C24—24no—
235C24—24no—
236C24—24no—
237C24—21no—
238C24—44no—
239C24—48no—
240C24—84no—
241C24C2×C22⋊C412yes—
242C24C2×C22⋊C414yes—
243C24C22×D412yes—
244#41—88no—
245#41—84no—
246#41—84no—
247C2×Q8—44no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—24no—
253C2×D4—24no—
254C2×D4—24no—
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—22no—
258C2×D4—21no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—44no—
264C2×D4—44no—
265C2×D4—44no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—44no—
271C2×D4—42no—
272C2×D4—48no—
273C2×D4—48no—
274C2×D4—48no—
275C2×D4—48no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—84no—
281C2×D4—84no—
282C2×D4—84no—
283C2×D4—82no—
284C2×D4—82no—
285C2×D4—82no—
286C2×D4—82no—
287C2×D4—82no—
288C2×D4—84no—
289C2×D4—84no—
290C2×D4—82no—
291C2×D4—88no—
292C2×D4#9912yes—
293C2×D4#9911yescharacteristic
294C2×D4#9911yescharacteristic
295C2×D4C2×#4112yes—
296C2×D4H(5,𝔽2)11yescharacteristic
297C2×D42−1+411yescharacteristic
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—28no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—42no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—42no—
314C22×C4—48no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—44no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4—88no—
324C22×C4—88no—
325C22×C4—164no—
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C412yes—
330C22×C4C23×C411yescharacteristic
331C4⋊C4—24no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—42no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—48no—
340C4⋊C4—42no—
341C4⋊C4—84no—
342C22⋊C4—28no—
343C22⋊C4—22no—
344C22⋊C4—22no—
345C22⋊C4—24no—
346C22⋊C4—44no—
347C22⋊C4—42no—
348C22⋊C4—44no—
349C22⋊C4—48no—
350C22⋊C4—48no—
351C22⋊C4—48no—
352C22⋊C4—44no—
353C22⋊C4—42no—
354C22⋊C4—48no—
355C22⋊C4—44no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—44no—
359C22⋊C4—44no—
360C22⋊C4—42no—
361C22⋊C4—88no—
362C22⋊C4—88no—
363C22⋊C4—164no—
364C42—24no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—41no—
370C42—42no—
371C42—82no—
372C42—81no—
373C42—84no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—24no—
378C23—22no—
379C23—24no—
380C23—24no—
381C23—24no—
382C23—42no—
383C23—44no—
384C23—44no—
385C23—44no—
386C23—44no—
387C23—42no—
388C23—42no—
389C23—44no—
390C23—44no—
391C23—42no—
392C23—42no—
393C23—48no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—84no—
398C23—84no—
399C23—84no—
400C23—88no—
401C23—88no—
402C23—168no—
403C23C2×#9912yes—
404C23C2×#9912yes—
405C23C22×C22⋊C411yescharacteristic
406C23#54612yes—
407C23#54712yes—
408Q8—82no—
409Q8—82no—
410D4—24no—
411D4—22no—
412D4—42no—
413D4—41no—
414D4—41no—
415D4—48no—
416D4—44no—
417D4—82no—
418D4—82no—
419D4—82no—
420D4—81no—
421D4—81no—
422D4—84no—
423D4—84no—
424D4—84no—
425D4—84no—
426D4—84no—
427D4—84no—
428D4—84no—
429D4—162no—
430D4—168no—
431D4#35314yescomplemented
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—88no—
464C2×C4—84no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—84no—
468C2×C4—82no—
469C2×C4—84no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—84no—
473C2×C4—82no—
474C2×C4—84no—
475C2×C4—1616no—
476C2×C4C2×#9911yescharacteristic
477C2×C4C2×#9911yescharacteristic
478C2×C4C4×#4111yescharacteristic
479C2×C4#52011yescharacteristic
480C22—21no—
481C22—21no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—44no—
490C22—44no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—44no—
496C22—44no—
497C22—44no—
498C22—82no—
499C22—88no—
500C22—88no—
501C22—84no—
502C22—84no—
503C22—82no—
504C22—82no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—164no—
509C22—164no—
510C22C2×#35314yes—
511C22#233911yescharacteristic
512C22C22×#9911yescharacteristic
513C22#288311yescharacteristic
514C4—21no—
515C4—21no—
516C4—21no—
517C4—42no—
518C4—44no—
519C4—42no—
520C4—42no—
521C4—41no—
522C4—42no—
523C4—84no—
524C4—81no—
525C4—82no—
526C4—162no—
527C4—164no—
528C4C2×#35312yes—
529C2—21no—
530C2—21no—
531C2—24no—
532C2—42no—
533C2—42no—
534C2—42no—
535C2—84no—
536C2—161no—
537C2#2247611yescharacteristic
538C2C22×#35311yescharacteristic
539C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radical#36260C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×D42 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. D4×#99C2semidirect
3. #13298C2semidirect
4. #13088C2semidirect
5. #12856C2semidirect
6. #12797C2semidirect
7. C4×#353C2semidirect
8. C2×#1393C2semidirect
9. C2×#547C4semidirect
13. #2858C4semidirect
14. #2856C4semidirect
16. #2541C4semidirect
19. #2501C4semidirect
24. C42×D4C4semidirect
47. C2×C4⋊Q8C2×C4semidirect
49. C2×#127C2×C4semidirect
59. C2×#121C2×C4semidirect
67. C2×C4×D4C2×C4semidirect
69. C22×C4⋊C4C2×C4semidirect
72. C22×C42C2×C4semidirect
89. #390C2×C4semidirect
90. #390C2×C4semidirect
91. #390C2×C4semidirect
92. #390C2×C4semidirect
96. #384C2×C4semidirect
97. #384C2×C4semidirect
101. C4×C4⋊C4C2×C4semidirect
104. C4×C22⋊C4C2×C4semidirect
105. C43C2×C4semidirect
228. #99#41semidirect
431. D4#353semidirect

Maximal subgroups

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

subgroupindex
#362602
D4×#992
#132982
#130882
#128562
#127972
C4×#3532
C2×#13932