Finite Group Databasebeta

← C2×#32969 #6629101

Subgroups — C2×#32969 #6629101

Order 512.

4610 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×H(5,𝔽2)C211yesmaximal, characteristic
2C2×#3079C212yesmaximal, complemented
3C2×#2542C211yesmaximal, characteristic, complemented
4#32969C2116yesmaximal, direct product factor
5C2×#1859C214yesmaximal, complemented
6C2×#1856C212yesmaximal, complemented
7C2×#1806C212yesmaximal, complemented
8C2×#1800C211yesmaximal, characteristic, complemented
9C2×#1792C212yesmaximal
10C22×H(5,𝔽2)C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic, complemented
12C22×C22⋊Q8—22no—
13C4×H(5,𝔽2)—28no—
14C4×H(5,𝔽2)—24no—
15C4×H(5,𝔽2)C2218yes—
16C2×#520—22no—
17C2×#520C2212yes—
18C2×#518—22no—
19C2×#518—22no—
20C2×#518C2211yescharacteristic
21C2×C4×#41—22no—
22C2×C4×#41C2212yes—
23C22×C4×D4—22no—
24C22×C4×D4—22no—
25C22×C4×D4C2211yescharacteristic
26#3079—28no—
27#3079C22116yescomplemented
28C2×#477—22no—
29C2×#475—22no—
30C2×#476—22no—
31C2×#474—22no—
32C2×#428C2214yescomplemented
33C2×#423C2212yescomplemented
34C22×#104C2214yes—
35C2×#417C2214yes—
36C22×#100C2212yes—
37#2542—24no—
38#2542C2218yescomplemented
39C2×#399—22no—
40C2×#390C2212yes—
41C2×#384C2212yescomplemented
42C2×C4×C4⋊C4C2211yescharacteristic, complemented
43#1859C22132yescomplemented
44#1856C22116yescomplemented
45#1806C22116yescomplemented
46#1800C2218yescomplemented
47#1792C22116yes—
48C2×#343C2212yescomplemented
49C2×#339C2214yescomplemented
50C2×H(5,𝔽2)—24no—
51C2×H(5,𝔽2)—28no—
52C2×H(5,𝔽2)—22no—
53C2×H(5,𝔽2)—24no—
54C2×H(5,𝔽2)Q818yescomplemented
55C2×H(5,𝔽2)C2314yes—
56C22×#41—22no—
57C22×#41C2312yes—
58C23×Q8—21no—
59C23×D4—22no—
60C23×D4—22no—
61C23×D4C2311yescharacteristic
62C24×C4—22no—
63C24×C4D412yes—
64C22×M16C2314yes—
65C2×C4⋊Q8—22no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—24no—
69C2×C4⋊Q8C2314yescomplemented
70C2×C22⋊Q8—216no—
71C2×C22⋊Q8—416no—
72#520—216no—
73#520—24no—
74#520—216no—
75#520—24no—
76#520—24no—
77#520C2318yes—
78#518—216no—
79#518—216no—
80#518—22no—
81#518—216no—
82#518—24no—
83#518—216no—
84#518C2314yes—
85C4×#41—216no—
86C4×#41—216no—
87C4×#41—24no—
88C4×#41—24no—
89C4×#41D418yes—
90C4×#41C2318yes—
91C2×C4×Q8—22no—
92C2×C4×Q8D412yes—
93C2×C4×Q8D412yes—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C2×C4×D4—216no—
98C2×C4×D4—216no—
99C2×C4×D4—24no—
100C2×C4×D4—22no—
101C2×C4×D4—22no—
102C2×C4×D4—24no—
103C2×C4×D4—216no—
104C2×C4×D4—22no—
105C2×C4×D4—216no—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4—24no—
111C2×C4×D4D412yes—
112C2×C4×D4D412yes—
113C2×C4×D4D414yes—
114C2×C4×D4C2314yes—
115C2×#117—24no—
116C2×#117—22no—
117C2×#117—22no—
118C2×#117C2312yes—
119C22×C4⋊C4—22no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—24no—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C22⋊C4—22no—
127C22×C22⋊C4—24no—
128C22×C22⋊C4—22no—
129C22×C22⋊C4—24no—
130C22×C22⋊C4D412yes—
131C22×C42—22no—
132C22×C42—22no—
133C22×C42C2312yes—
134C22×C42C2311yescharacteristic
135#477—216no—
136#476—216no—
137#475—216no—
138#474—216no—
139#428—28no—
140#428C23116yescomplemented
141C2×#107—24no—
142C2×#106—24no—
143#423—24no—
144#423C2318yescomplemented
145C2×C4⋊C8—22no—
146C2×C4⋊C8—22no—
147C2×#104—216no—
148C2×#104C23116yes—
149#417—28no—
150#417C23116yes—
151C2×#103—24no—
152C2×#102—24no—
153C2×#100—28no—
154C2×#100C2318yes—
155#399—216no—
156#390—24no—
157#390C2318yes—
158#384—24no—
159#384C2318yescomplemented
160C4×C4⋊C4—22no—
161C4×C4⋊C4C2314yescomplemented
162C2×#95—21no—
163#343C2318yescomplemented
164#339C23116yescomplemented
165C25—22no—
166C25C2×D412yes—
167H(5,𝔽2)—216no—
168H(5,𝔽2)—232no—
169H(5,𝔽2)—232no—
170H(5,𝔽2)—216no—
171H(5,𝔽2)—216no—
172H(5,𝔽2)—28no—
173H(5,𝔽2)C2×Q8116yescomplemented
174C2×#41—28no—
175C2×#41—216no—
176C2×#41—216no—
177C2×#41—28no—
178C2×#41—22no—
179C2×#41—22no—
180C2×#41—24no—
181C2×#41—24no—
182C2×#41C2×D414yes—
183C2×#41C2×Q818yes—
184C2×#41#4118yes—
185C2×#41C2414yes—
186C22×Q8—24no—
187C22×Q8—48no—
188C22×Q8—42no—
189C22×Q8—41no—
190C22×Q8—41no—
191C22×Q8C2×D412yes—
192C22×D4—24no—
193C22×D4—28no—
194C22×D4—28no—
195C22×D4—216no—
196C22×D4—216no—
197C22×D4—216no—
198C22×D4—22no—
199C22×D4—28no—
200C22×D4—216no—
201C22×D4—28no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—24no—
205C22×D4C2×D412yes—
206C22×D4C2×D412yes—
207C22×D4C2×Q814yes—
208C22×D4#4114yes—
209C22×D4C2412yes—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—216no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—24no—
222C23×C4—22no—
223C23×C4—48no—
224C23×C4—88no—
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D412yes—
227C23×C4C2×D418yes—
228C23×C4C2×D411yescharacteristic
229C23×C4C2411yescharacteristic
230C2×M16—24no—
231C2×M16—28no—
232C2×M16—28no—
233C2×M16C2418yes—
234C22×C8—24no—
235C4⋊Q8—28no—
236C4⋊Q8—28no—
237C4⋊Q8—44no—
238C4⋊Q8—44no—
239C22⋊Q8—464no—
240C4×Q8—28no—
241C4×Q8—28no—
242C4×Q8—24no—
243C4×Q8—24no—
244C4×Q8C2×D418yes—
245C4×Q8C2×D418yes—
246C4×D4—232no—
247C4×D4—216no—
248C4×D4—232no—
249C4×D4—232no—
250C4×D4—28no—
251C4×D4—28no—
252C4×D4—232no—
253C4×D4—28no—
254C4×D4—28no—
255C4×D4—28no—
256C4×D4—28no—
257C4×D4—28no—
258C4×D4—216no—
259C4×D4—28no—
260C4×D4—28no—
261C4×D4—28no—
262C4×D4—24no—
263C4×D4—28no—
264C4×D4—28no—
265C4×D4—24no—
266C4×D4—28no—
267C4×D4—28no—
268C4×D4C2×D418yes—
269C4×D4C2×D418yes—
270#117—28no—
271#117—232no—
272#117—22no—
273#117—22no—
274#117—28no—
275#117—28no—
276#117—28no—
277#117C2×D414yes—
278#117C2414yes—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—216no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—216no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—416no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2414yes—
301C2×C4⋊C4C2412yes—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—216no—
306C2×C22⋊C4—232no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—216no—
314C2×C22⋊C4—416no—
315C2×C22⋊C4C2×D418yes—
316C2×C42—24no—
317C2×C42—216no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—28no—
322C2×C42—28no—
323C2×C42—28no—
324C2×C42—22no—
325C2×C42C2×D412yes—
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2×D414yes—
329C2×C42C2414yes—
330C2×C42C2412yes—
331#107—216no—
332#106—216no—
333C4⋊C8—28no—
334C4⋊C8—28no—
335#104—232no—
336#103—216no—
337#102—216no—
338#100—216no—
339#95—24no—
340C24—28no—
341C24—216no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—24no—
346C24—44no—
347C24—44no—
348C24—84no—
349C24—84no—
350C24C22⋊Q818yes—
351C24C22×D411yescharacteristic
352C24C22×D414yes—
353#41—264no—
354#41—232no—
355#41—232no—
356#41—28no—
357#41—28no—
358#41—28no—
359#41—28no—
360#41—28no—
361#41—24no—
362#41—24no—
363#41C22⋊Q8116yescomplemented
364#41C22×Q818yes—
365#41C2×#4118yes—
366C2×Q8—22no—
367C2×Q8—24no—
368C2×Q8—416no—
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8—84no—
372C2×Q8—88no—
373C2×Q8—84no—
374C2×Q8C22⋊Q818yescomplemented
375C2×Q8C22×D414yes—
376C2×D4—216no—
377C2×D4—232no—
378C2×D4—264no—
379C2×D4—232no—
380C2×D4—264no—
381C2×D4—232no—
382C2×D4—24no—
383C2×D4—28no—
384C2×D4—232no—
385C2×D4—28no—
386C2×D4—24no—
387C2×D4—232no—
388C2×D4—24no—
389C2×D4—22no—
390C2×D4—28no—
391C2×D4—28no—
392C2×D4—24no—
393C2×D4—28no—
394C2×D4—24no—
395C2×D4—28no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4C22⋊Q818yes—
399C2×D4C22⋊Q818yescomplemented
400C2×D4C22×D414yes—
401C2×D4C22×Q814yes—
402C2×D4C2×#4114yes—
403C22×C4—28no—
404C22×C4—216no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—28no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—28no—
417C22×C4—216no—
418C22×C4—28no—
419C22×C4—22no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—24no—
425C22×C4—42no—
426C22×C4—416no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—44no—
432C22×C4—48no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—88no—
436C22×C4—84no—
437C22×C4—84no—
438C22×C4—832no—
439C22×C4—84no—
440C22×C4—88no—
441C22×C4—84no—
442C22×C4#12012yes—
443C22×C4#12012yes—
444C22×C4#12011yescharacteristic
445C22×C4#12012yes—
446C22×C4#12011yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×D414yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×Q812yes—
454C22×C4C2×#4114yes—
455C22×C4C2×#4112yes—
456C22×C4C2511yescharacteristic
457M16—28no—
458M16—28no—
459M16—48no—
460C2×C8—28no—
461C2×C8—48no—
462C4⋊C4—216no—
463C4⋊C4—216no—
464C4⋊C4—28no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—28no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—28no—
471C4⋊C4—432no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4C22×D414yes—
475C4⋊C4C22×D414yes—
476C22⋊C4—216no—
477C22⋊C4—216no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—28no—
481C22⋊C4—28no—
482C22⋊C4—216no—
483C22⋊C4—432no—
484C42—216no—
485C42—28no—
486C42—28no—
487C42—24no—
488C42—24no—
489C42—28no—
490C42—44no—
491C42#12014yes—
492C42#12014yes—
493C42C22×D414yes—
494C42C22×D414yes—
495C23—22no—
496C23—22no—
497C23—232no—
498C23—28no—
499C23—28no—
500C23—24no—
501C23—24no—
502C23—28no—
503C23—24no—
504C23—44no—
505C23—416no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—82no—
512C23—88no—
513C23—82no—
514C23—88no—
515C23—832no—
516C23—88no—
517C23C2×#12011yescharacteristic
518C23C2×#12011yescharacteristic
519C23C2×C22⋊Q812yes—
520C23C2×C22⋊Q818yes—
521C23#52512yes—
522C23C23×D411yescharacteristic
523Q8—28no—
524Q8—24no—
525Q8—88no—
526Q8—88no—
527Q8C2×C22⋊Q818yescomplemented
528D4—264no—
529D4—216no—
530D4—28no—
531D4—28no—
532D4—216no—
533D4—28no—
534D4—28no—
535D4—24no—
536D4—28no—
537D4C2×C22⋊Q818yescomplemented
538C2×C4—232no—
539C2×C4—28no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—216no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—416no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—416no—
561C2×C4—416no—
562C2×C4—44no—
563C2×C4—48no—
564C2×C4—44no—
565C2×C4—88no—
566C2×C4—88no—
567C2×C4—82no—
568C2×C4—88no—
569C2×C4—816no—
570C2×C4—88no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4#45314yes—
574C2×C4C2×#12014yes—
575C2×C4C2×#12012yes—
576C2×C4C2×#12012yes—
577C2×C4C2×#12012yes—
578C2×C4C2×C22⋊Q812yes—
579C2×C4C2×C22⋊Q814yes—
580C2×C4C2×C22⋊Q814yes—
581C2×C4#52512yes—
582C2×C4#52514yes—
583C2×C4#52514yes—
584C2×C4C23×D412yes—
585C2×C4C23×Q811yescharacteristic
586C2×C4C22×#4112yes—
587C2×C4C22×#4111yescharacteristic
588C8—48no—
589C22—24no—
590C22—24no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—28no—
596C22—28no—
597C22—48no—
598C22—416no—
599C22—416no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—84no—
605C22—88no—
606C22—832no—
607C22#284812yes—
608C22#284812yes—
609C22C2×#45312yes—
610C22#302212yes—
611C22C22×#12011yescharacteristic
612C22C22×C22⋊Q811yescharacteristic
613C22C2×#52511yescharacteristic
614C4—28no—
615C4—22no—
616C4—24no—
617C4—48no—
618C4—44no—
619C4—44no—
620C4—44no—
621C4—84no—
622C4#284814yes—
623C4C2×#45314yes—
624C4C22×C22⋊Q812yes—
625C4C2×#52512yes—
626C2—21no—
627C2—21no—
628C2—22no—
629C2—48no—
630C2—48no—
631C2#3296914yesdirect product factor
632C2C2×#284811yescharacteristic
633C2C22×#45311yescharacteristic
634C2C2×#302211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×H(5,𝔽2)C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3079C2semidirect
3. C2×#2542C2semidirect
4. #32969C2direct (N × H)
5. C2×#1859C2semidirect
6. C2×#1856C2semidirect
7. C2×#1806C2semidirect
8. C2×#1800C2semidirect
11. C22×C4⋊Q8C22semidirect
27. #3079C22semidirect
32. C2×#428C22semidirect
33. C2×#423C22semidirect
38. #2542C22semidirect
41. C2×#384C22semidirect
42. C2×C4×C4⋊C4C22semidirect
43. #1859C22semidirect
44. #1856C22semidirect
45. #1806C22semidirect
46. #1800C22semidirect
48. C2×#343C22semidirect
49. C2×#339C22semidirect
54. C2×H(5,𝔽2)Q8semidirect
69. C2×C4⋊Q8C23semidirect
140. #428C23semidirect
144. #423C23semidirect
159. #384C23semidirect
161. C4×C4⋊C4C23semidirect
163. #343C23semidirect
164. #339C23semidirect
173. H(5,𝔽2)C2×Q8semidirect
363. #41C22⋊Q8semidirect
374. C2×Q8C22⋊Q8semidirect
399. C2×D4C22⋊Q8semidirect
527. Q8C2×C22⋊Q8semidirect
537. D4C2×C22⋊Q8semidirect
631. C2#32969direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×H(5,𝔽2)2
C2×#30792
C2×#25422
#329692
C2×#18592
C2×#18562
C2×#18062
C2×#18002
C2×#17922