Finite Group Databasebeta

← #6488701

Subgroups — #6488701

Order 512.

1948 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic
2C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
3#33005C214yesmaximal, complemented
4#30129C214yesmaximal, complemented
5#29368C214yesmaximal, complemented
6#29191C212yesmaximal, complemented
7#28614C212yesmaximal, complemented
8#28115C214yesmaximal, complemented
9#27961C211yesmaximal, characteristic, complemented
10#27906C211yesmaximal, characteristic
11#25856C211yesmaximal, characteristic, complemented
12#25718C211yesmaximal, characteristic
13#22268C212yesmaximal, complemented
14#22218C212yesmaximal, complemented
15#20777C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C2×#529—21no—
18C22×C4×Q8C2212yes—
19C2×#492C2212yes—
20C2×#487C2212yes—
21C2×#481C2214yes—
22C2×#448C2214yes—
23#2861—21no—
24#2860—21no—
25#2860C2212yes—
26#2859C2211yescharacteristic
27#2858C2211yescharacteristic
28#2857C2211yescharacteristic
29#2856C2211yescharacteristic
30#2854—21no—
31#2854C2212yes—
32#2850—21no—
33C4×C4⋊Q8—21no—
34C4×C4⋊Q8—21no—
35C4×C4⋊Q8—21no—
36C4×C4⋊Q8—21no—
37C4×C4⋊Q8C2211yescharacteristic
38C4×C4⋊Q8C2212yes—
39C4×C4⋊Q8C2212yes—
40C4×C4⋊Q8C2212yes—
41C4×C4⋊Q8C2212yes—
42C4×#127C2211yescharacteristic
43C4×#124—21no—
44C4×#121—21no—
45C4×#121C2212yes—
46C2×#391C2212yes—
47#2293—21no—
48#2293C2212yes—
49C2×#389—21no—
50C2×#389—21no—
51C2×#384C2211yescharacteristic
52C2×C4×C4⋊C4—21no—
53C2×C4×C4⋊C4—21no—
54C2×C43C2211yescharacteristic
55#2100C2214yes—
56#2094C2214yes—
57#2066C2218yes—
58#2055C2218yes—
59#2043C2214yes—
60#2013C2214yes—
61#1990C2212yes—
62#1957C2212yes—
63#1956C2212yes—
64#1956C2212yes—
65#1927C2218yes—
66#1880C2218yes—
67#1880C2218yes—
68#1877C2218yes—
69#1865C2218yes—
70#1865C2218yes—
71#1844C2212yes—
72#1839C2212yes—
73#1834C2212yes—
74#1831C2212yes—
75#1824C2214yes—
76#1817C2212yes—
77#1797C2212yes—
78#1787C2214yes—
79C4×C4⋊C8C2212yes—
80C4×C22⋊C8C2211yescharacteristic
81C4×C22⋊C8C2211yescharacteristic
82#1744C2212yes—
83C23×Q8C2312yes—
84C22×SD16C2318yes—
85C2×C4⋊Q8—28no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2311yescharacteristic
91C2×#127C2311yescharacteristic
92#529—48no—
93C2×#126—22no—
94C2×#125—21no—
95C2×#124—21no—
96C2×#124—21no—
97C2×#123—22no—
98C2×C22⋊Q8—21no—
99C2×#121—21no—
100C2×#121—21no—
101C2×#121C2312yes—
102C2×C4×Q8—22no—
103C2×C4×Q8—22no—
104C2×C4×Q8—21no—
105C2×C4×Q8—48no—
106C2×C4×Q8C2314yes—
107C2×C4×Q8C2314yes—
108C2×C4×D4—22no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4C2312yes—
112C2×#117—22no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4C2312yes—
119C22×C42—22no—
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#492—28no—
128#487—28no—
129#481—48no—
130#448—48no—
131C2×#106C2318yes—
132C2×C4⋊C8C2314yes—
133C2×#103C2318yes—
134C2×#103C2318yes—
135C2×#103C2318yes—
136C2×#102C2318yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2312yes—
139C2×C4×C8C2312yes—
140C2×C4×C8C2312yes—
141#391—22no—
142#391—22no—
143#391—21no—
144#391—21no—
145#391C2314yes—
146#391C2314yes—
147#390—21no—
148#390—21no—
149#390—21no—
150#390—21no—
151#390—21no—
152#390—21no—
153#390C2312yes—
154#390C2312yes—
155#390C2312yes—
156#390C2312yes—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#386—22no—
182#386—22no—
183#384—21no—
184#384—21no—
185#384C2311yescharacteristic
186#384C2312yes—
187#384C2312yes—
188#384C2311yescharacteristic
189#382—21no—
190#380—21no—
191#380C2312yes—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4C2312yes—
202C4×C22⋊C4—22no—
203C4×C22⋊C4—22no—
204C2×#95—22no—
205C2×#95—22no—
206C43D411yescharacteristic
207C43D411yescharacteristic
208C43D411yescharacteristic
209C43D411yescharacteristic
210C43C2311yescharacteristic
211C43C2311yescharacteristic
212C43C2311yescharacteristic
213C43C2311yescharacteristic
214#340C2318yes—
215#340C2318yes—
216#336C2314yes—
217#336C2314yes—
218#336C2314yes—
219#336C2312yes—
220#336C2312yes—
221C22×Q8—28no—
222C22×Q8—22no—
223C22×Q8—21no—
224C22×Q8—28no—
225C22×Q8—44no—
226C22×Q8C2414yes—
227C22×D4—22no—
228C22×D4—21no—
229C22×D4—21no—
230C22×D4C2412yes—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2411yescharacteristic
241C2×SD16—216no—
242C2×SD16—48no—
243C2×SD16—48no—
244C22×C8C2414yes—
245C22×C8C2414yes—
246C4⋊Q8—28no—
247C4⋊Q8—28no—
248C4⋊Q8—28no—
249C4⋊Q8—24no—
250C4⋊Q8—24no—
251C4⋊Q8#4118yes—
252#127—24no—
253#126—48no—
254#125—44no—
255#124—44no—
256#124—44no—
257#123—48no—
258C22⋊Q8—44no—
259#121—28no—
260#121—44no—
261#121—44no—
262C4×Q8—48no—
263C4×Q8—48no—
264C4×Q8—48no—
265C4×Q8—48no—
266C4×Q8—44no—
267C4×D4—44no—
268C4×D4—44no—
269C4×D4—48no—
270C4×D4—44no—
271#117—48no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—48no—
298C2×C4⋊C4C2414yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—22no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—42no—
320C2×C42—42no—
321C2×C42—42no—
322C2×C42—48no—
323C2×C42SD1618yescomplemented
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352#106—48no—
353C4⋊C8—44no—
354#103—216no—
355#103—48no—
356#103—48no—
357#102—48no—
358C22⋊C8—24no—
359C22⋊C8—24no—
360C4×C8—24no—
361C4×C8—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372C24—22no—
373C24C22×D411yescharacteristic
374C2×Q8—28no—
375C2×Q8—24no—
376C2×Q8—28no—
377C2×Q8—28no—
378C2×Q8—432no—
379C2×Q8—44no—
380C2×Q8—42no—
381C2×Q8—44no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8C2×#4118yes—
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—24no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—42no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—28no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4C2×SD1614yes—
428C22×C4C2×SD1614yes—
429C22×C4C2×SD1614yes—
430C22×C4#13714yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4H(5,𝔽2)14yes—
445C22×C4C2511yescharacteristic
446SD16—816no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C4⋊C4—28no—
454C4⋊C4—28no—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—24no—
458C4⋊C4—28no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—41no—
506C42—41no—
507C42—41no—
508C42—41no—
509C42—41no—
510C42—41no—
511C42—48no—
512C42—48no—
513C42—44no—
514C42C2×SD1614yes—
515C42C2×SD1614yes—
516C42C2×SD1614yes—
517C42C2×SD1614yes—
518C42C2×#4114yes—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23#53512yes—
531C23C22×SD1612yes—
532C23C2×#13712yes—
533C23C23×D411yescharacteristic
534Q8—216no—
535Q8—416no—
536Q8—48no—
537Q8—84no—
538D4—48no—
539D4—88no—
540D4—84no—
541D4—84no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—28no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—416no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—88no—
592C2×C4—88no—
593C2×C4#52512yes—
594C2×C4#52512yes—
595C2×C4#53512yes—
596C2×C4C22×SD1612yes—
597C2×C4C22×SD1612yes—
598C2×C4C22×SD1612yes—
599C2×C4C22×SD1612yes—
600C2×C4C22×SD1612yes—
601C2×C4C22×SD1612yes—
602C2×C4C2×#13712yes—
603C2×C4C2×#13712yes—
604C2×C4#57512yes—
605C2×C4#57512yes—
606C2×C4C22×#4112yes—
607C2×C4C2×H(5,𝔽2)12yes—
608C2×C4#58512yes—
609C8—44no—
610C8—44no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22#320614yes—
625C22C2×#52511yescharacteristic
626C22C2×#53511yescharacteristic
627C22#344911yescharacteristic
628C22C23×SD1611yescharacteristic
629C22C22×#13711yescharacteristic
630C22C2×#57511yescharacteristic
631C22C2×#57511yescharacteristic
632C4—22no—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—41no—
640C4—41no—
641C4—41no—
642C4—41no—
643C4—48no—
644C4—82no—
645C4—82no—
646C4—82no—
647C4—84no—
648C4#310314yes—
649C4#311714yes—
650C4#351314yes—
651C2—24no—
652C2—162no—
653C2#3523311yescharacteristic
654C2C2×#310311yescharacteristic
655C2C2×#311711yescharacteristic
656C2C2×#320611yescharacteristic
657C2#6080111yescharacteristic
658C2#6094111yescharacteristic
659C2#6096611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×C4⋊Q8C2semidirect
3. #33005C2semidirect
4. #30129C2semidirect
5. #29368C2semidirect
6. #29191C2semidirect
7. #28614C2semidirect
8. #28115C2semidirect
9. #27961C2semidirect
11. #25856C2semidirect
13. #22268C2semidirect
14. #22218C2semidirect
15. #20777C2semidirect
323. C2×C42SD16semidirect

Maximal subgroups

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

subgroupindex
#366312
C2×C4×C4⋊Q82
#330052
#301292
#293682
#291912
#286142
#281152
#279612
#279062
#258562
#257182
#222682
#222182
#207772