Finite Group Databasebeta

← D4×#360 #497391

Subgroups — D4×#360 #497391

Order 512.

1148 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#135C211yesmaximal, characteristic
2D4×M32C211yesmaximal, characteristic, complemented
3C22×#360C212yesmaximal, complemented
4D4×#106C211yesmaximal, characteristic, complemented
5#13255C211yesmaximal, characteristic, complemented
6#13210C211yesmaximal, characteristic, complemented
7#13065C212yesmaximal, complemented
8#13001C212yesmaximal, complemented
9#12816C211yesmaximal, characteristic, complemented
10#12735C212yesmaximal, complemented
11C4×#360C211yesmaximal, characteristic, complemented
12C22×#135C2212yes—
13D4×#41—21no—
14C22×M32C2212yescomplemented
15#3287C411yescharacteristic, complemented
16D4×Q16C411yescharacteristic, complemented
17#3285C412yescomplemented
18#3284—21no—
19#3283C411yescharacteristic, complemented
20#3282—22no—
21D4×SD16—21no—
22#3280C412yescomplemented
23D4×D8C411yescharacteristic, complemented
24#3147C2211yescharacteristic
25#3050C2212yes—
26#3035—21no—
27#3002—22no—
28C4×#135C2211yescharacteristic
29C2×C8×D4C2211yescharacteristic
30C22×#106C2212yescomplemented
31D4×C4⋊C4—21no—
32#2169—21no—
33#2168—22no—
34D4×C16—21no—
35#2150C2211yescharacteristic, complemented
36C2×#360—28no—
37C2×#360C2212yes—
38C2×#360C2211yescharacteristic
39C2×#360C2211yescharacteristic
40C2×#360C2212yescomplemented
41C2×#360C2212yescomplemented
42C2×#360C2212yescomplemented
43C2×#360C2212yescomplemented
44#2112C2212yescomplemented
45C4×M32C2211yescharacteristic, complemented
46#1948—21no—
47#1853—22no—
48#1844C2211yescharacteristic, complemented
49#1807C2212yescomplemented
50C4×#106C2211yescharacteristic, complemented
51#1386C2211yescharacteristic, complemented
52#1386C2212yescomplemented
53#1386C2211yescharacteristic, complemented
54C22×#41—22no—
55C2×#135—24no—
56C2×#135—24no—
57C2×#135—22no—
58C2×#135—22no—
59C2×#135—21no—
60C2×#135—42no—
61C2×#135—42no—
62C2×#135C2×C412yescomplemented
63C2×#135C2×C412yescomplemented
64C2×#135C2312yes—
65C2×#135C2311yescharacteristic
66C22×Q16C2×C412yescomplemented
67C22×SD16—22no—
68C22×D8C2×C412yescomplemented
69C23×C8C2312yes—
70#550—21no—
71D4×Q8—21no—
72#548—22no—
73#547—21no—
74#547—41no—
75#546—22no—
76#546—42no—
77D42—21no—
78D42—41no—
79#532—21no—
80#532—41no—
81#525—22no—
82#525—42no—
83C4×#41—21no—
84C2×C4×D4—41no—
85C2×C4×D4—41no—
86C2×C4×D4D411yescharacteristic
87C22×C4⋊C4—22no—
88C2×M32—22no—
89C2×M32—24no—
90C2×M32—42no—
91C2×M32C2311yescharacteristic
92C2×M32C2312yes—
93C2×M32C2311yescharacteristic
94C22×C16—22no—
95#495—21no—
96#495C2×C412yescomplemented
97#494C2×C411yescharacteristic, complemented
98#493C2×C411yescharacteristic, complemented
99#492—21no—
100#467C2×C412yescomplemented
101#467C2×C412yescomplemented
102#466C2×C412yescomplemented
103#466C2×C412yescomplemented
104#465—22no—
105#465—22no—
106#464—22no—
107#463—22no—
108#462—21no—
109#461—21no—
110#460—21no—
111#459—21no—
112#452—22no—
113#452—22no—
114#451—22no—
115#450—22no—
116#449—22no—
117#449—22no—
118#448—22no—
119#447—22no—
120C4×Q16C2×C411yescharacteristic, complemented
121C4×Q16C2×C411yescharacteristic, complemented
122C4×SD16—21no—
123C4×SD16—21no—
124C4×D8C2×C411yescharacteristic, complemented
125C4×D8C2×C411yescharacteristic, complemented
126C8×D4C2×C411yescharacteristic
127C8×D4C2×C412yes—
128C8×D4C2×C411yescharacteristic
129C8×D4D411yescharacteristic
130C8×D4D412yes—
131C8×D4D411yescharacteristic
132#427C2311yescharacteristic
133C2×#106—24no—
134C2×#106—22no—
135C2×#106—21no—
136C2×#106—42no—
137C2×#106C2311yescharacteristic
138C2×#106C2312yes—
139C2×#106C2311yescharacteristic
140C2×C4⋊C8D411yescharacteristic
141#416—21no—
142#416—22no—
143C2×C22⋊C8D412yes—
144C2×C4×C8C2311yescharacteristic
145#389—41no—
146#385—42no—
147#384—21no—
148#380—22no—
149C4×C4⋊C4—21no—
150C4⋊C16—21no—
151C4⋊C16—21no—
152#360—24no—
153#360—22no—
154#360—24no—
155#360—24no—
156#360—42no—
157#360—42no—
158#360D414yesdirect product factor
159C22⋊C16—22no—
160C22⋊C16—22no—
161#346—21no—
162C4×C16—21no—
163#340—22no—
164#340—21no—
165C2×#41—22no—
166C2×#41—22no—
167C2×#41—21no—
168C2×#41—22no—
169C2×#41—44no—
170C2×#41—44no—
171C2×#41—42no—
172C2×#41—42no—
173C2×#41—82no—
174C2×#41—82no—
175C22×Q8—22no—
176C22×D4—22no—
177C22×D4—42no—
178C22×D4—41no—
179C22×D4D811yescharacteristic
180C23×C4—42no—
181C23×C4—42no—
182C23×C4C2×D412yes—
183#135—24no—
184#135—24no—
185#135—22no—
186#135—22no—
187#135—21no—
188#135—21no—
189#135—22no—
190#135—22no—
191#135—42no—
192#135—42no—
193#135—44no—
194#135—44no—
195#135—44no—
196#135—44no—
197#135—81no—
198#135—81no—
199#135C2×D412yes—
200#135#4112yes—
201C2×Q16—24no—
202C2×Q16—22no—
203C2×Q16—21no—
204C2×Q16—42no—
205C2×Q16C22×C412yes—
206C2×Q16C22×C411yescharacteristic
207C2×SD16—22no—
208C2×SD16—22no—
209C2×SD16—22no—
210C2×SD16—21no—
211C2×SD16—22no—
212C2×SD16—44no—
213C2×SD16—44no—
214C2×D8—24no—
215C2×D8—22no—
216C2×D8—21no—
217C2×D8—42no—
218C2×D8C22×C412yes—
219C2×D8C22×C411yescharacteristic
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8C22×C412yes—
225C22×C8C22×C412yes—
226C22×C8C2×D412yes—
227C22×C8C2×D411yescharacteristic
228C22×C8C2×D412yes—
229C22×C8C2×D412yes—
230C22×C8C2411yescharacteristic
231C4⋊Q8—21no—
232C4⋊Q8—41no—
233#127—21no—
234#127—41no—
235#124—22no—
236#124—42no—
237#123—42no—
238#123—42no—
239#123—42no—
240C22⋊Q8—22no—
241C22⋊Q8—22no—
242C22⋊Q8—42no—
243C22⋊Q8—42no—
244#121—22no—
245#121—22no—
246#121—42no—
247#121—41no—
248#121—41no—
249#121—41no—
250#121—41no—
251#121—42no—
252#121—42no—
253#120—42no—
254#120—42no—
255#120—42no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4—41no—
261C4×D4—41no—
262C4×D4—41no—
263C4×D4—42no—
264C4×D4—42no—
265C4×D4—41no—
266C4×D4—42no—
267C4×D4—41no—
268C4×D4—41no—
269C4×D4—84no—
270C4×D4C22⋊C411yescharacteristic
271C4×D4C22⋊C412yes—
272C4×D4C22⋊C411yescharacteristic
273C4×D4SD1611yescharacteristic
274C4×D4SD1612yes—
275C4×D4SD1611yescharacteristic
276#117—21no—
277#117—41no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—41no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—41no—
285C2×C4⋊C4—41no—
286C2×C4⋊C4—82no—
287C2×C4⋊C4D811yescharacteristic
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4D812yes—
291C2×C42—41no—
292C2×C42—41no—
293C2×C42C2×D411yescharacteristic
294M32—21no—
295M32—22no—
296M32—21no—
297M32—21no—
298M32—22no—
299M32—44no—
300M32C2×D412yescomplemented
301C2×C16—21no—
302C2×C16—21no—
303C2×C16—22no—
304C2×C16—44no—
305#107C22×C411yescharacteristic
306#107C22×C411yescharacteristic
307#106—21no—
308#106—21no—
309#106—22no—
310#106—21no—
311#106—21no—
312#106—22no—
313#106—44no—
314#106—81no—
315#106C2×D412yescomplemented
316C4⋊C8C22⋊C411yescharacteristic
317C4⋊C8C22⋊C411yescharacteristic
318C4⋊C8C22⋊C412yes—
319#103—21no—
320#103—21no—
321#103—22no—
322#103—22no—
323#102—21no—
324#102—21no—
325#102—22no—
326#102—22no—
327C22⋊C8C22⋊C412yes—
328C22⋊C8C22⋊C412yes—
329C22⋊C8C22⋊C412yes—
330C22⋊C8C22⋊C412yes—
331C4×C8C22×C411yescharacteristic
332C4×C8C22×C411yescharacteristic
333C4×C8C2×D411yescharacteristic
334C4×C8C2×D411yescharacteristic
335#95—41no—
336#95—42no—
337C24—42no—
338C24C2×D812yes—
339#41—22no—
340#41—22no—
341#41—44no—
342#41—44no—
343#41—41no—
344#41—41no—
345#41—42no—
346#41—42no—
347#41—42no—
348#41—42no—
349#41—84no—
350#41—84no—
351#41—82no—
352#41—82no—
353#41—84no—
354#41—84no—
355C2×Q8—21no—
356C2×Q8—22no—
357C2×Q8—44no—
358C2×Q8—42no—
359C2×Q8—82no—
360C2×D4—24no—
361C2×D4—21no—
362C2×D4—22no—
363C2×D4—44no—
364C2×D4—41no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—41no—
371C2×D4—42no—
372C2×D4—41no—
373C2×D4—82no—
374C2×D4—84no—
375C2×D4—84no—
376C2×D4—84no—
377C2×D4#10212yes—
378C2×D4#10211yescharacteristic
379C2×D4#10211yescharacteristic
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—42no—
394C22×C4—84no—
395C22×C4—84no—
396C22×C4—84no—
397C22×C4—84no—
398C22×C4—162no—
399C22×C4C2×C22⋊C412yes—
400C22×C4C2×C22⋊C412yes—
401C22×C4C2×D812yes—
402C22×C4C2×D811yescharacteristic
403C22×C4C2×D811yescharacteristic
404C22×C4C2×SD1612yes—
405C22×C4C2×SD1612yes—
406C22×C4C22×D411yescharacteristic
407Q16—22no—
408Q16—21no—
409Q16—22no—
410Q16—44no—
411Q16—81no—
412Q16C4×D412yescomplemented
413SD16—22no—
414SD16—22no—
415SD16—44no—
416SD16—44no—
417SD16—42no—
418SD16—42no—
419SD16—82no—
420D8—22no—
421D8—21no—
422D8—22no—
423D8—44no—
424D8—81no—
425D8C4×D412yescomplemented
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8C2×C22⋊C411yescharacteristic
439C2×C8C2×C22⋊C411yescharacteristic
440C2×C8C2×C22⋊C412yes—
441C2×C8C2×C22⋊C412yes—
442C2×C8C23×C411yescharacteristic
443C2×C8C22×D411yescharacteristic
444C2×C8C22×D411yescharacteristic
445C2×C8C2×#4111yescharacteristic
446C2×C8H(5,𝔽2)11yescharacteristic
447C2×C8H(5,𝔽2)11yescharacteristic
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—42no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—42no—
457C4⋊C4—41no—
458C4⋊C4—42no—
459C4⋊C4—41no—
460C4⋊C4—42no—
461C4⋊C4—41no—
462C4⋊C4—84no—
463C4⋊C4—82no—
464C4⋊C4—161no—
465C4⋊C4#10211yescharacteristic
466C4⋊C4#10211yescharacteristic
467C4⋊C4#10212yes—
468C22⋊C4—41no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—41no—
474C22⋊C4—41no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—84no—
478C22⋊C4#10212yes—
479C22⋊C4#10212yes—
480C22⋊C4#10212yes—
481C22⋊C4#10212yes—
482C42—41no—
483C42—41no—
484C42—41no—
485C42—82no—
486C42C2×C22⋊C411yescharacteristic
487C42C2×C22⋊C411yescharacteristic
488C42C2×SD1611yescharacteristic
489C42C2×SD1611yescharacteristic
490C16—22no—
491C16—42no—
492C16—41no—
493C23—24no—
494C23—22no—
495C23—22no—
496C23—42no—
497C23—41no—
498C23—42no—
499C23—84no—
500C23—84no—
501C23—162no—
502C23C2×#10212yes—
503C23C2×#10212yes—
504C23C22×D811yescharacteristic
505Q8—22no—
506Q8—41no—
507Q8—42no—
508Q8—42no—
509Q8—84no—
510D4—24no—
511D4—22no—
512D4—22no—
513D4—42no—
514D4—42no—
515D4—42no—
516D4—41no—
517D4—42no—
518D4—84no—
519D4—84no—
520D4—84no—
521D4—84no—
522D4—82no—
523D4—82no—
524D4—82no—
525D4—161no—
526D4—162no—
527D4#36014yesdirect product factor
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—84no—
548C2×C4—84no—
549C2×C4—84no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—81no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—81no—
559C2×C4—82no—
560C2×C4—81no—
561C2×C4—161no—
562C2×C4—168no—
563C2×C4C2×#10211yescharacteristic
564C2×C4C2×#10211yescharacteristic
565C2×C4C2×#10211yescharacteristic
566C2×C4C2×#10212yes—
567C2×C4C2×#10212yes—
568C2×C4C2×#10211yescharacteristic
569C2×C4C22×C22⋊C411yescharacteristic
570C2×C4D4211yescharacteristic
571C2×C4#54611yescharacteristic
572C2×C4C22×SD1611yescharacteristic
573C2×C4#57611yescharacteristic
574C2×C4#57711yescharacteristic
575C8—22no—
576C8—22no—
577C8—21no—
578C8—21no—
579C8C2×C4×D411yescharacteristic
580C8#51811yescharacteristic
581C8D4211yescharacteristic
582C8#54611yescharacteristic
583C22—21no—
584C22—21no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—44no—
591C22—44no—
592C22—42no—
593C22—81no—
594C22—81no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—168no—
599C22C2×#36014yescomplemented
600C22C22×#10211yescharacteristic
601C22D4×D811yescharacteristic
602C22#329411yescharacteristic
603C4—21no—
604C4—21no—
605C4—21no—
606C4—22no—
607C4—22no—
608C4—42no—
609C4—82no—
610C4—81no—
611C4—81no—
612C4—82no—
613C4—161no—
614C4—162no—
615C4C2×#36012yescomplemented
616C4D4×C22⋊C411yescharacteristic
617C4#289711yescharacteristic
618C4D4×SD1611yescharacteristic
619C4#329511yescharacteristic
620C2—21no—
621C2—21no—
622C2—24no—
623C2—42no—
624C2—82no—
625C2—162no—
626C2D4×#10211yescharacteristic
627C2C22×#36011yescharacteristic
628C2#3338811yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalD4×#135C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. D4×M32C2semidirect
3. C22×#360C2semidirect
4. D4×#106C2semidirect
5. #13255C2semidirect
6. #13210C2semidirect
7. #13065C2semidirect
8. #13001C2semidirect
9. #12816C2semidirect
10. #12735C2semidirect
11. C4×#360C2semidirect
14. C22×M32C22semidirect
15. #3287C4semidirect
16. D4×Q16C4semidirect
17. #3285C4semidirect
19. #3283C4semidirect
22. #3280C4semidirect
23. D4×D8C4semidirect
30. C22×#106C22semidirect
35. #2150C22semidirect
40. C2×#360C22semidirect
41. C2×#360C22semidirect
42. C2×#360C22semidirect
43. C2×#360C22semidirect
44. #2112C22semidirect
45. C4×M32C22semidirect
48. #1844C22semidirect
49. #1807C22semidirect
50. C4×#106C22semidirect
51. #1386C22semidirect
52. #1386C22semidirect
53. #1386C22semidirect
62. C2×#135C2×C4semidirect
63. C2×#135C2×C4semidirect
66. C22×Q16C2×C4semidirect
68. C22×D8C2×C4semidirect
96. #495C2×C4semidirect
97. #494C2×C4semidirect
98. #493C2×C4semidirect
100. #467C2×C4semidirect
101. #467C2×C4semidirect
102. #466C2×C4semidirect
103. #466C2×C4semidirect
120. C4×Q16C2×C4semidirect
121. C4×Q16C2×C4semidirect
124. C4×D8C2×C4semidirect
125. C4×D8C2×C4semidirect
158. #360D4direct (N × H)
300. M32C2×D4semidirect
315. #106C2×D4semidirect
412. Q16C4×D4semidirect
425. D8C4×D4semidirect
527. D4#360direct (N × H)
599. C22C2×#360semidirect
615. C4C2×#360semidirect

Maximal subgroups

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

subgroupindex
D4×#1352
D4×M322
C22×#3602
D4×#1062
#132552
#132102
#130652
#130012
#128162
#127352
C4×#3602