Finite Group Databasebeta

← #6432846

Subgroups — #6432846

Order 512.

4274 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#536C211yesmaximal, characteristic
2C2×#3095C211yesmaximal, characteristic, complemented
3C22×#451C211yesmaximal, characteristic, complemented
4#35305C211yesmaximal, characteristic
5#29605C218yesmaximal, complemented
6#29599C214yesmaximal, complemented
7#29597C214yesmaximal, complemented
8#23484C214yesmaximal, complemented
9#23389C214yesmaximal, complemented
10#20699C211yesmaximal, characteristic, complemented
11#20599C211yesmaximal, characteristic, complemented
12#20581C211yesmaximal, characteristic
13C23×#41C2211yescharacteristic
14C24×Q8C2211yescharacteristic
15C23×Q16C2212yescomplemented
16C2×#536—22no—
17C2×#536—22no—
18C2×#536—28no—
19C2×#536—28no—
20C2×#536C2214yes—
21C2×#536C2214yes—
22C2×#536C2212yes—
23C22×C4⋊Q8—21no—
24C22×#124—21no—
25C22×#123—22no—
26C22×C22⋊Q8C2211yescharacteristic
27C22×C22⋊Q8C2212yes—
28C22×C22⋊Q8C2212yes—
29C22×#117C2211yescharacteristic
30#3095—28no—
31C2×#484C2212yes—
32C2×#485C2214yescomplemented
33C2×#483C2212yes—
34C2×#451—28no—
35C2×#451C2214yes—
36C2×#451C2214yescomplemented
37C2×#428C2212yescomplemented
38C2×#419C2212yescomplemented
39C2×#419C2214yescomplemented
40C22×#103C2212yescomplemented
41C2×#408C2214yes—
42C2×#407C2211yescharacteristic
43C22×C22⋊C8C2211yescharacteristic
44#2848C2214yes—
45#2616C2214yes—
46#2435—21no—
47#2434—21no—
48#2432—22no—
49#2406—22no—
50#2405—22no—
51#2311C2212yes—
52#2280C2211yescharacteristic
53#2036C22116yescomplemented
54#2035C22116yescomplemented
55#2001C22116yescomplemented
56#2001C22116yescomplemented
57#1825C2214yescomplemented
58#1824C2212yes—
59#1823C2212yes—
60#1787C2214yes—
61#1787C2214yescomplemented
62#1780C2214yes—
63#1779C2214yes—
64#1735C2214yescomplemented
65C22×#41—24no—
66C22×#41—24no—
67C22×#41—48no—
68C22×#41D412yes—
69C22×#41D412yes—
70C22×#41C2312yes—
71C22×#41C2312yes—
72C23×Q8—21no—
73C23×Q8—24no—
74C23×Q8—21no—
75C23×Q8—21no—
76C23×Q8—48no—
77C23×Q8D418yes—
78C23×Q8D411yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×Q8C2312yes—
81C23×Q8C2312yes—
82C23×Q8C2312yes—
83C23×D4D412yes—
84C23×D4D411yescharacteristic
85C24×C4—21no—
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C22×Q16—28no—
89C22×Q16—48no—
90C22×Q16C2318yes—
91C22×M16C2312yes—
92C23×C8C2312yes—
93#536—232no—
94#536—232no—
95#536—216no—
96#536—216no—
97#536—216no—
98#536—216no—
99#536—48no—
100C2×C4⋊Q8—216no—
101C2×C4⋊Q8—24no—
102C2×C4⋊Q8—24no—
103C2×C4⋊Q8—22no—
104C2×#124—216no—
105C2×#124—24no—
106C2×#124—24no—
107C2×#124—22no—
108C2×#123—232no—
109C2×#123—28no—
110C2×#123—28no—
111C2×#123—24no—
112C2×C22⋊Q8—28no—
113C2×C22⋊Q8—28no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—28no—
119C2×C22⋊Q8—28no—
120C2×C22⋊Q8—24no—
121C2×C22⋊Q8—24no—
122C2×C22⋊Q8C2312yes—
123C2×C22⋊Q8C2312yes—
124C2×C22⋊Q8C2318yes—
125C2×C22⋊Q8C2318yes—
126C2×C22⋊Q8C2318yes—
127C2×#117—24no—
128C2×#117—21no—
129C2×#117—21no—
130C2×#117D418yes—
131C2×#117C2312yes—
132C2×#117C2312yes—
133C2×#117C2312yes—
134C22×C4⋊C4—22no—
135C22×C4⋊C4—22no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C22⋊C4—21no—
141C22×C22⋊C4—22no—
142C22×C22⋊C4—22no—
143C22×C22⋊C4—21no—
144C22×C42—21no—
145#485—216no—
146#484—28no—
147#483—28no—
148#451—216no—
149#451—216no—
150#428—28no—
151C2×#107C2314yes—
152C2×#106C2314yes—
153#419—28no—
154#419—48no—
155C2×#103—28no—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2318yes—
159C2×#103C23116yes—
160#408—48no—
161#407—24no—
162C2×C22⋊C8—24no—
163C2×C22⋊C8C2312yes—
164C2×C22⋊C8C2312yes—
165C2×C22⋊C8C2312yes—
166C2×C22⋊C8C2312yes—
167C2×C22⋊C8C23116yes—
168#395—24no—
169#394—28no—
170#394—24no—
171#391—22no—
172#390—22no—
173#386—24no—
174#385—24no—
175#385—28no—
176#381—21no—
177#380C2318yes—
178#380C2312yes—
179#380C2312yes—
180#379—24no—
181#379—21no—
182C4×C22⋊C4—21no—
183C4×C22⋊C4—21no—
184#340C2318yes—
185#340C2318yes—
186C25C2×D411yescharacteristic
187C25C2×D412yes—
188C2×#41—28no—
189C2×#41—28no—
190C2×#41—24no—
191C2×#41—24no—
192C2×#41—24no—
193C2×#41—24no—
194C2×#41—28no—
195C2×#41—28no—
196C2×#41—48no—
197C2×#41—48no—
198C2×#41—44no—
199C2×#41—44no—
200C2×#41—432no—
201C2×#41—432no—
202C22×Q8—28no—
203C22×Q8—24no—
204C22×Q8—28no—
205C22×Q8—28no—
206C22×Q8—24no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—24no—
211C22×Q8—24no—
212C22×Q8—21no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—28no—
216C22×Q8—28no—
217C22×Q8—28no—
218C22×Q8—432no—
219C22×Q8—48no—
220C22×Q8—48no—
221C22×Q8—44no—
222C22×Q8—48no—
223C22×Q8—48no—
224C22×Q8—48no—
225C22×Q8—432no—
226C22×Q8C2×D418yes—
227C22×Q8C2×D418yes—
228C22×Q8C2×D418yes—
229C22×Q8C2×D412yes—
230C22×Q8C2×D412yes—
231C22×Q8C2412yes—
232C22×Q8C2414yes—
233C22×D4—28no—
234C22×D4—24no—
235C22×D4—44no—
236C22×D4—48no—
237C22×D4C2×D414yes—
238C22×D4C2×D412yes—
239C22×D4C2×D414yes—
240C22×D4C2×D412yes—
241C23×C4—28no—
242C23×C4—28no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4—24no—
249C23×C4—24no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—48no—
253C23×C4—48no—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D414yes—
258C23×C4C2×D414yes—
259C23×C4C2×D412yes—
260C23×C4C2×D414yes—
261C23×C4C2414yes—
262C23×C4C2411yescharacteristic
263C23×C4C2411yescharacteristic
264C23×C4C2411yescharacteristic
265C2×Q16—216no—
266C2×Q16—416no—
267C2×Q16—432no—
268C2×M16—24no—
269C2×M16—28no—
270C22×C8—24no—
271C22×C8—28no—
272C22×C8C2414yes—
273C22×C8C2414yes—
274C4⋊Q8—232no—
275C4⋊Q8—232no—
276C4⋊Q8—48no—
277#124—232no—
278#124—232no—
279#124—48no—
280#123—264no—
281#123—264no—
282#123—416no—
283C22⋊Q8—232no—
284C22⋊Q8—232no—
285C22⋊Q8—216no—
286C22⋊Q8—216no—
287C22⋊Q8—216no—
288C22⋊Q8—216no—
289C22⋊Q8—24no—
290C22⋊Q8—24no—
291C22⋊Q8—24no—
292C22⋊Q8—24no—
293C22⋊Q8—416no—
294C22⋊Q8—44no—
295C22⋊Q8—44no—
296#117—28no—
297#117—28no—
298#117—24no—
299#117—24no—
300#117—28no—
301#117C2×D418yes—
302#117C2×D418yes—
303C2×C4⋊C4—216no—
304C2×C4⋊C4—216no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—28no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—28no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4C2×D414yes—
319C2×C4⋊C4C2×D414yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C22⋊C4—216no—
325C2×C22⋊C4—216no—
326C2×C22⋊C4—28no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—28no—
330C2×C22⋊C4—28no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—28no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—28no—
338C2×C22⋊C4—44no—
339C2×C42—28no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343#107—28no—
344#106—28no—
345#103—216no—
346#103—28no—
347#103—28no—
348#103—416no—
349C22⋊C8—24no—
350C22⋊C8—24no—
351C22⋊C8—24no—
352C22⋊C8—24no—
353C22⋊C8—416no—
354#95—22no—
355#95—22no—
356#95—28no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24—28no—
361C24—42no—
362C24—44no—
363C24—44no—
364C24#12014yes—
365C24C22×D414yes—
366C24C22×D411yescharacteristic
367C24C22×D411yescharacteristic
368C24C22×D411yescharacteristic
369#41—432no—
370#41—432no—
371#41—416no—
372#41—416no—
373#41—416no—
374#41—416no—
375C2×Q8—28no—
376C2×Q8—28no—
377C2×Q8—28no—
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8—24no—
381C2×Q8—24no—
382C2×Q8—24no—
383C2×Q8—216no—
384C2×Q8—216no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—28no—
388C2×Q8—28no—
389C2×Q8—28no—
390C2×Q8—44no—
391C2×Q8—416no—
392C2×Q8—416no—
393C2×Q8—432no—
394C2×Q8—48no—
395C2×Q8—48no—
396C2×Q8—432no—
397C2×Q8—432no—
398C2×Q8—416no—
399C2×Q8—416no—
400C2×Q8—416no—
401C2×Q8—42no—
402C2×Q8—48no—
403C2×Q8—44no—
404C2×Q8—416no—
405C2×Q8—48no—
406C2×Q8—48no—
407C2×Q8C22×D418yes—
408C2×D4—28no—
409C2×D4—28no—
410C2×D4—28no—
411C2×D4—28no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—416no—
420C2×D4—432no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—216no—
424C22×C4—24no—
425C22×C4—216no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—28no—
431C22×C4—24no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—28no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—28no—
441C22×C4—28no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—416no—
449C22×C4—416no—
450C22×C4—44no—
451C22×C4—432no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—82no—
460C22×C4#12014yes—
461C22×C4#12014yes—
462C22×C4#12014yes—
463C22×C4C22×D414yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4H(5,𝔽2)18yes—
474C22×C4C2511yescharacteristic
475Q16—432no—
476M16—48no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—28no—
480C2×C8—28no—
481C2×C8—48no—
482C4⋊C4—232no—
483C4⋊C4—232no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—28no—
491C4⋊C4—416no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4C22×D414yes—
496C4⋊C4C22×D414yes—
497C22⋊C4—232no—
498C22⋊C4—232no—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—416no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—82no—
507C22⋊C4—88no—
508C42—28no—
509C42—28no—
510C42—44no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—28no—
517C23—22no—
518C23—28no—
519C23—22no—
520C23—28no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—28no—
525C23—44no—
526C23—416no—
527C23—48no—
528C23—48no—
529C23—48no—
530C23—816no—
531C23C2×#12012yes—
532C23C2×#12012yes—
533C23C2×#12012yes—
534C23#53414yes—
535C23#57814yes—
536C23C23×D411yescharacteristic
537Q8—28no—
538Q8—28no—
539Q8—416no—
540Q8—416no—
541Q8—48no—
542Q8—416no—
543Q8—48no—
544Q8—48no—
545Q8—44no—
546Q8—44no—
547Q8—84no—
548D4—48no—
549D4—416no—
550D4—416no—
551D4—48no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—416no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—416no—
571C2×C4—416no—
572C2×C4—44no—
573C2×C4—416no—
574C2×C4—48no—
575C2×C4—48no—
576C2×C4—44no—
577C2×C4—48no—
578C2×C4—48no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4C2×#12012yes—
589C2×C4C2×#12012yes—
590C2×C4C2×#12012yes—
591C2×C4#53414yes—
592C2×C4#53418yes—
593C2×C4#53518yes—
594C2×C4C23×D411yescharacteristic
595C2×C4C23×D411yescharacteristic
596C2×C4C2×H(5,𝔽2)14yes—
597C8—44no—
598C8—44no—
599C22—28no—
600C22—28no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—416no—
609C22—44no—
610C22—48no—
611C22—48no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—816no—
616C22#261314yes—
617C22#301114yes—
618C22C22×#12011yescharacteristic
619C22C2×#53412yes—
620C22C2×#57812yes—
621C22#358612yes—
622C4—22no—
623C4—22no—
624C4—44no—
625C4—44no—
626C4—44no—
627C4—48no—
628C4—84no—
629C4—82no—
630C4—82no—
631C4#261314yes—
632C4C2×#53412yes—
633C4C2×#53512yes—
634C2—24no—
635C2—24no—
636C2—42no—
637C2—84no—
638C2C2×#261311yescharacteristic
639C2C2×#301111yescharacteristic
640C2#6036811yescharacteristic
641C2#6075914yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3095C2semidirect
3. C22×#451C2semidirect
5. #29605C2semidirect
6. #29599C2semidirect
7. #29597C2semidirect
8. #23484C2semidirect
9. #23389C2semidirect
10. #20699C2semidirect
11. #20599C2semidirect
15. C23×Q16C22semidirect
32. C2×#485C22semidirect
36. C2×#451C22semidirect
37. C2×#428C22semidirect
38. C2×#419C22semidirect
39. C2×#419C22semidirect
40. C22×#103C22semidirect
53. #2036C22semidirect
54. #2035C22semidirect
55. #2001C22semidirect
56. #2001C22semidirect
57. #1825C22semidirect
61. #1787C22semidirect
64. #1735C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5362
C2×#30952
C22×#4512
#353052
#296052
#295992
#295972
#234842
#233892
#206992
#205992
#205812