Finite Group Databasebeta

← #222912

Subgroups — #222912

Order 512.

1156 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2372C211yesmaximal, characteristic
2#22555C212yesmaximal, complemented
3#9878C212yesmaximal, complemented
4#9084C212yesmaximal, complemented
5#9083C212yesmaximal
6#8700C211yesmaximal, characteristic, complemented
7#8699C211yesmaximal, characteristic, complemented
8#8559C211yesmaximal, characteristic, complemented
9C4×#323C211yesmaximal, characteristic, complemented
10#7911C211yesmaximal, characteristic, complemented
11#7910C211yesmaximal, characteristic, complemented
12C22×#124C411yescharacteristic
13C2×#432C2212yes—
14C2×#407C2212yes—
15#2372—22no—
16#2372—22no—
17#2372—22no—
18#2372—22no—
19C2×#391C411yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#388C411yescharacteristic
22C2×#388C411yescharacteristic
23C2×#388C411yescharacteristic
24C2×#388C2211yescharacteristic
25C2×#386C411yescharacteristic
26C2×#386C2211yescharacteristic
27C2×C4×C22⋊C4C411yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31C2×#376C411yescharacteristic
32C2×#376C2211yescharacteristic
33#1924—22no—
34#1923—22no—
35#1851—22no—
36#1817C2212yes—
37#1797C2212yes—
38#1779C2212yes—
39#1753—22no—
40C4×C22⋊C8C2212yes—
41C2×#323C2212yes—
42C2×#323C2212yes—
43#1280C2211yescharacteristic
44#1280C2211yescharacteristic
45#1280C2212yes—
46#1280C2212yes—
47#1280C2212yes—
48#1280C2212yes—
49#1280C2211yescharacteristic
50#1280C2211yescharacteristic
51C23×Q8C2×C411yescharacteristic
52C23×D4C2×C411yescharacteristic
53C24×C4C2×C411yescharacteristic
54C24×C4C2311yescharacteristic
55C22×M16—22no—
56C2×#124—24no—
57C2×#124—22no—
58C2×#124—22no—
59C2×#124—24no—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C411yescharacteristic
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C411yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2×C411yescharacteristic
70C22×C22⋊C4C2×C411yescharacteristic
71C22×C22⋊C4C2311yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79#432—24no—
80#432—42no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83#407—24no—
84#407—42no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8C2312yes—
89C2×C22⋊C8C2312yes—
90C2×C22⋊C8C2312yes—
91C2×C22⋊C8C2312yes—
92C2×#97—22no—
93C2×C4×C8—22no—
94#391—22no—
95#391—22no—
96#391—22no—
97#391—22no—
98#388—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#386—22no—
107#386—22no—
108#386—22no—
109#386—22no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—21no—
115C4×C22⋊C4—21no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4D412yes—
119C4×C22⋊C4D412yes—
120C4×C22⋊C4D412yes—
121C4×C22⋊C4D412yes—
122#376—22no—
123#376—22no—
124#376—21no—
125#376—21no—
126#376—21no—
127#376—21no—
128C2×#95C2×C411yescharacteristic
129C2×#95C2×C411yescharacteristic
130C2×#95C2×C411yescharacteristic
131C2×#95C2×C411yescharacteristic
132C2×#95C2×C411yescharacteristic
133C2×#95C2×C411yescharacteristic
134C2×#95C2×C411yescharacteristic
135C2×#95C2×C411yescharacteristic
136C2×#95C2×C411yescharacteristic
137C2×#95C2311yescharacteristic
138C2×#95C2311yescharacteristic
139C2×#95C2311yescharacteristic
140#336—22no—
141#336—22no—
142#336—22no—
143#336—22no—
144#323—24no—
145#323—42no—
146C25C22×C411yescharacteristic
147C25C22×C411yescharacteristic
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—81no—
151C22×Q8—81no—
152C22×Q8M1614yescomplemented
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—81no—
157C22×D4—81no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—42no—
165C23×C4—41no—
166C23×C4—42no—
167C23×C4—81no—
168C23×C4—81no—
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C411yescharacteristic
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C2×D412yes—
182C23×C4C2×D412yes—
183C23×C4C2411yescharacteristic
184C2×M16—44no—
185C2×M16—44no—
186C2×M16—82no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191#124—216no—
192#124—216no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—81no—
196C2×C4⋊C4—81no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—81no—
222C2×C22⋊C4—81no—
223C2×C22⋊C4C22⋊C412yes—
224C2×C22⋊C4C22⋊C412yes—
225C2×C22⋊C4C22⋊C412yes—
226C2×C22⋊C4C22⋊C412yes—
227C2×C22⋊C4M1614yescomplemented
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—22no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—41no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—81no—
248C2×C42—81no—
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C4⋊C8—44no—
254C4⋊C8—82no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257C22⋊C8—22no—
258C22⋊C8—22no—
259C22⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8—22no—
262C22⋊C8—44no—
263C22⋊C8—44no—
264C22⋊C8—82no—
265C22⋊C8#4114yes—
266#97—44no—
267C4×C8—82no—
268#95—22no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—22no—
274#95—21no—
275#95—21no—
276#95—22no—
277#95—21no—
278#95—21no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95C22⋊C411yescharacteristic
283#95C22⋊C411yescharacteristic
284#95C22⋊C411yescharacteristic
285#95C22⋊C411yescharacteristic
286#95C22⋊C412yes—
287#95C22⋊C412yes—
288C24—21no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—41no—
301C24—41no—
302C24—41no—
303C24—41no—
304C24—41no—
305C24—41no—
306C24—41no—
307C24—41no—
308C24C2×C22⋊C412yes—
309C24C2×M1612yes—
310C24C23×C411yescharacteristic
311C2×Q8—28no—
312C2×Q8—82no—
313C2×Q8—82no—
314C2×Q8—84no—
315C2×Q8—84no—
316C2×D4—28no—
317C2×D4—82no—
318C2×D4—82no—
319C2×D4—84no—
320C2×D4—84no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—81no—
387C22×C4—81no—
388C22×C4—84no—
389C22×C4—82no—
390C22×C4—82no—
391C22×C4—84no—
392C22×C4—84no—
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C412yes—
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C412yes—
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×M1612yes—
402C22×C4C2×M1612yes—
403C22×C4C2×M1612yes—
404C22×C4C2×M1612yes—
405C22×C4C2×M1612yes—
406C22×C4#13112yes—
407C22×C4C22×D412yes—
408C22×C4C2×#4112yes—
409C22×C4H(5,𝔽2)11yescharacteristic
410C22×C42−1+411yescharacteristic
411M16—88no—
412C2×C8—24no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C4⋊C4—82no—
425C4⋊C4—82no—
426C22⋊C4—24no—
427C22⋊C4—24no—
428C22⋊C4—28no—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—28no—
432C22⋊C4—24no—
433C22⋊C4—42no—
434C22⋊C4—44no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—82no—
440C22⋊C4—82no—
441C42—24no—
442C42—24no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—42no—
448C42—42no—
449C42—42no—
450C42—42no—
451C42—42no—
452C42—42no—
453C42—42no—
454C42—42no—
455C42—81no—
456C42—81no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—81no—
492C23—81no—
493C23—81no—
494C23—81no—
495C23—81no—
496C23—81no—
497C23—81no—
498C23—81no—
499C23—81no—
500C23—81no—
501C23—81no—
502C23—81no—
503C23—81no—
504C23—81no—
505C23—81no—
506C23—81no—
507C23C22×C22⋊C411yescharacteristic
508C23C4×#4111yescharacteristic
509C23#52011yescharacteristic
510C23C22×M1611yescharacteristic
511C23C22×M1611yescharacteristic
512C23C2×#13111yescharacteristic
513C23#56811yescharacteristic
514Q8—88no—
515D4—88no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—84no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—84no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4#40714yes—
596C2×C4#41012yes—
597C2×C4#41312yes—
598C2×C4#54612yes—
599C2×C4#54712yes—
600C8—44no—
601C8—82no—
602C8—82no—
603C8—82no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22#233911yescharacteristic
639C22C2×#40711yescharacteristic
640C22#288011yescharacteristic
641C22C2×#41011yescharacteristic
642C22#288311yescharacteristic
643C22C2×#41311yescharacteristic
644C22#288811yescharacteristic
645C22#296511yescharacteristic
646C22#297111yescharacteristic
647C22#298911yescharacteristic
648C22#299111yescharacteristic
649C4—22no—
650C4—22no—
651C4—22no—
652C4—22no—
653C4—24no—
654C4—41no—
655C4—42no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—81no—
664C4—81no—
665C4—81no—
666C2—21no—
667C2—21no—
668C2—21no—
669C2—21no—
670C2—82no—
671C2—82no—
672C2#1736611yescharacteristic
673C2#1736711yescharacteristic
674C2#1741811yescharacteristic
675C2#1742211yescharacteristic
676C2#2247311yescharacteristic
677C2#2247711yescharacteristic
678C2#2248311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22555C2semidirect
3. #9878C2semidirect
4. #9084C2semidirect
6. #8700C2semidirect
7. #8699C2semidirect
8. #8559C2semidirect
9. C4×#323C2semidirect
10. #7911C2semidirect
11. #7910C2semidirect
152. C22×Q8M16semidirect
227. C2×C22⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#23722
#225552
#98782
#90842
#90832
#87002
#86992
#85592
C4×#3232
#79112
#79102