Finite Group Databasebeta

← #4309137

Subgroups — #4309137

Order 512.

4622 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2847C212yesmaximal, complemented
2C2×#2619C211yesmaximal, characteristic, complemented
3C2×#2280C211yesmaximal, characteristic, complemented
4#16660C214yesmaximal, complemented
5#16512C218yesmaximal, complemented
6#16249C214yesmaximal, complemented
7#15233C212yesmaximal, complemented
8#15219C212yesmaximal, complemented
9#15130C212yesmaximal, complemented
10#15124C212yesmaximal, complemented
11#14340C211yesmaximal, characteristic, complemented
12#14257C212yesmaximal, complemented
13C25×C4C2211yescharacteristic
14C22×#127C2211yescharacteristic, complemented
15C22×#121C2212yes—
16C22×#121C2212yes—
17C22×#121C2212yes—
18C22×#120C2214yes—
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21#2847—216no—
22#2619—28no—
23C2×#394C2218yes—
24C2×#394C2214yescomplemented
25C2×#392C2214yes—
26C2×#392C2218yes—
27C2×#390C2212yescomplemented
28C2×#390C2212yescomplemented
29C2×#390C2212yescomplemented
30C2×#386C2214yes—
31C2×#386C2214yes—
32C2×#386C2214yes—
33C2×#384C2211yescharacteristic, complemented
34C2×#382C2211yescharacteristic, complemented
35C2×#382C2211yescharacteristic, complemented
36#2280—28no—
37C2×#381C2212yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2214yes—
42C2×#379C2212yes—
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2212yes—
45#1443C2218yescomplemented
46#1441C2214yescomplemented
47#1440C2218yes—
48#1440C2218yescomplemented
49#1439C2214yes—
50#1439C2218yescomplemented
51#1439C2218yescomplemented
52#1438C2218yes—
53#1438C2218yes—
54#1438C2218yescomplemented
55C4⋊#95C2212yescomplemented
56C4⋊#95C2212yescomplemented
57C4⋊#95C2212yescomplemented
58C26C2311yescharacteristic
59C23×D4C2312yes—
60C23×D4C2314yes—
61C23×D4C2314yes—
62C23×D4C2314yes—
63C23×D4C2314yes—
64C24×C4—24no—
65C24×C4—24no—
66C24×C4—42no—
67C24×C4—42no—
68C24×C4—42no—
69C24×C4—42no—
70C24×C4D418yescomplemented
71C24×C4C2312yes—
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C24×C4C2311yescharacteristic
75C2×#127—24no—
76C2×#127—42no—
77C2×#127—42no—
78C2×#121—28no—
79C2×#121—28no—
80C2×#121—28no—
81C2×#121—28no—
82C2×#121—28no—
83C2×#121—44no—
84C2×#121—44no—
85C2×#121—44no—
86C2×#121—44no—
87C2×#120—216no—
88C2×#120—416no—
89C2×#117—24no—
90C2×#117—42no—
91C2×#117—42no—
92C2×#117—42no—
93C2×#117—42no—
94C2×#117D418yescomplemented
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2318yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2318yes—
112C22×C22⋊C4C2318yes—
113C22×C22⋊C4C2312yes—
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117#394—216no—
118#394—416no—
119#392—216no—
120#392—416no—
121#390—28no—
122#390—44no—
123#390—44no—
124#386—216no—
125#386—216no—
126#386—216no—
127#384—42no—
128#382—24no—
129#382—42no—
130#381—28no—
131#380—28no—
132#380—28no—
133#380—28no—
134#380—216no—
135#379—28no—
136C4×C22⋊C4—28no—
137C4×C22⋊C4—28no—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2318yes—
145C2×#95C2318yes—
146C2×#95C2318yes—
147C2×#95C2314yes—
148C25—22no—
149C25—22no—
150C25—28no—
151C25—41no—
152C25—41no—
153C25—41no—
154C25—41no—
155C25—44no—
156C25C2×D414yes—
157C25C2414yes—
158C25C2411yescharacteristic
159C25C2412yes—
160C22×D4—28no—
161C22×D4—28no—
162C22×D4—28no—
163C22×D4—28no—
164C22×D4—28no—
165C22×D4—216no—
166C22×D4—216no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—28no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—48no—
176C22×D4—48no—
177C22×D4—44no—
178C22×D4—44no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4—44no—
182C22×D4—44no—
183C22×D4—48no—
184C22×D4—48no—
185C22×D4—48no—
186C22×D4—48no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—232no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—416no—
203C23×C4—416no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—41no—
209C23×C4—41no—
210C23×C4—41no—
211C23×C4—41no—
212C23×C4—416no—
213C23×C4—416no—
214C23×C4—416no—
215C23×C4—416no—
216C23×C4C2×D418yes—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2×D418yes—
221C23×C4#4114yes—
222C23×C4C2414yes—
223C23×C4C2414yes—
224C23×C4C2414yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231#127—416no—
232#121—432no—
233#121—432no—
234#121—432no—
235#120—464no—
236#117—416no—
237#117—416no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—48no—
254C2×C4⋊C4C2×D418yes—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—216no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—216no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—48no—
277C2×C22⋊C4—48no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—48no—
283C2×C22⋊C4—48no—
284C2×C22⋊C4—48no—
285C2×C22⋊C4—48no—
286C2×C22⋊C4—48no—
287C2×C22⋊C4—48no—
288C2×C22⋊C4—48no—
289C2×C22⋊C4—48no—
290C2×C22⋊C4C2×D418yes—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42C2×D414yes—
299C2×C42C2×D414yes—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—48no—
307#95—44no—
308#95—48no—
309#95—48no—
310C24—22no—
311C24—22no—
312C24—21no—
313C24—21no—
314C24—21no—
315C24—21no—
316C24—24no—
317C24—24no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—28no—
326C24—216no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—28no—
333C24—24no—
334C24—24no—
335C24—28no—
336C24—28no—
337C24—28no—
338C24—44no—
339C24—44no—
340C24—48no—
341C24—416no—
342C24—416no—
343C24—48no—
344C24—48no—
345C24—44no—
346C24—44no—
347C24—44no—
348C24—44no—
349C24—416no—
350C24—416no—
351C24—48no—
352C24—48no—
353C24C22×D412yes—
354C24C22×D412yes—
355C24C22×D414yes—
356C24C2×#4112yes—
357C24H(5,𝔽2)18yes—
358C24C2511yescharacteristic
359C2×D4—416no—
360C2×D4—416no—
361C2×D4—416no—
362C2×D4—416no—
363C2×D4—416no—
364C2×D4—416no—
365C2×D4—416no—
366C2×D4—416no—
367C2×D4—416no—
368C2×D4—416no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4—48no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—816no—
377C2×D4—816no—
378C2×D4—816no—
379C2×D4—816no—
380C2×D4—816no—
381C2×D4—816no—
382C2×D4—816no—
383C2×D4—88no—
384C2×D4—88no—
385C2×D4—816no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—216no—
412C22×C4—216no—
413C22×C4—216no—
414C22×C4—216no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—28no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—28no—
435C22×C4—28no—
436C22×C4—24no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—4128no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4#121116yes—
455C22×C4C22×D414yes—
456C22×C4C22×D414yes—
457C22×C4C22×D414yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C4H(5,𝔽2)12yes—
466C22×C4H(5,𝔽2)12yes—
467C22×C4H(5,𝔽2)12yes—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—88no—
474C22⋊C4—48no—
475C22⋊C4—416no—
476C22⋊C4—416no—
477C22⋊C4—416no—
478C22⋊C4—416no—
479C22⋊C4—48no—
480C22⋊C4—416no—
481C22⋊C4—416no—
482C22⋊C4—416no—
483C22⋊C4—416no—
484C22⋊C4—816no—
485C22⋊C4—816no—
486C22⋊C4—816no—
487C42—28no—
488C42—28no—
489C42—48no—
490C23—24no—
491C23—22no—
492C23—24no—
493C23—24no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—28no—
510C23—28no—
511C23—28no—
512C23—416no—
513C23—432no—
514C23—416no—
515C23—432no—
516C23—416no—
517C23—432no—
518C23—432no—
519C23—432no—
520C23—432no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—44no—
526C23—44no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—44no—
531C23—42no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—48no—
549C23—48no—
550C23—48no—
551C23—48no—
552C23—48no—
553C23—48no—
554C23—48no—
555C23—48no—
556C23—48no—
557C23—48no—
558C23—48no—
559C23—48no—
560C23C2×#12114yes—
561C23#53518yes—
562C23C23×D411yescharacteristic
563C23C23×D411yescharacteristic
564C23C22×#4111yescharacteristic
565C23C2×H(5,𝔽2)11yescharacteristic
566C23C2×H(5,𝔽2)11yescharacteristic
567C23C2×H(5,𝔽2)11yescharacteristic
568C23C2×H(5,𝔽2)11yescharacteristic
569C23C2×H(5,𝔽2)14yes—
570C23#58514yes—
571D4—832no—
572D4—832no—
573D4—832no—
574D4—832no—
575D4—816no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—232no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—432no—
591C2×C4—416no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—432no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—48no—
616C2×C4—48no—
617C2×C4—44no—
618C2×C4—48no—
619C2×C4—48no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—48no—
623C2×C4—44no—
624C2×C4—48no—
625C2×C4C2×#12114yes—
626C2×C4C2×#12114yes—
627C2×C4#53414yes—
628C2×C4#53414yes—
629C2×C4#53414yes—
630C2×C4#53514yes—
631C2×C4#53814yes—
632C2×C4#53814yes—
633C22—28no—
634C22—28no—
635C22—28no—
636C22—28no—
637C22—24no—
638C22—464no—
639C22—44no—
640C22—44no—
641C22—42no—
642C22—44no—
643C22—42no—
644C22—44no—
645C22—48no—
646C22—44no—
647C22—48no—
648C22—44no—
649C22—48no—
650C22—44no—
651C22—48no—
652C22—44no—
653C22—44no—
654C22—84no—
655C22—84no—
656C22—84no—
657C22—84no—
658C22—88no—
659C22—88no—
660C22—88no—
661C22—88no—
662C22—84no—
663C22—84no—
664C22—84no—
665C22C22×#12111yescharacteristic
666C22C2×#53411yescharacteristic
667C22C2×#53411yescharacteristic
668C22C2×#53411yescharacteristic
669C22C2×#53512yes—
670C22C2×#53511yescharacteristic
671C22#344912yes—
672C22C2×#53811yescharacteristic
673C22C2×#53811yescharacteristic
674C22#348814yes—
675C22#349414yes—
676C22#353712yes—
677C22#353712yes—
678C22#353912yes—
679C22#354012yes—
680C22#354214yes—
681C22#354214yes—
682C4—24no—
683C4—44no—
684C4—44no—
685C4—44no—
686C4—84no—
687C4—84no—
688C4—84no—
689C4—84no—
690C4#261718yes—
691C2—48no—
692C2—44no—
693C2—164no—
694C2C2×#261711yescharacteristic
695C2#3525212yes—
696C2#3726912yes—
697C2#3727012yes—
698C2#3728312yes—
699C2#3728412yes—
700C2#3744812yes—
701C2#3745011yescharacteristic
702C2#3745511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2847C2semidirect
2. C2×#2619C2semidirect
3. C2×#2280C2semidirect
4. #16660C2semidirect
5. #16512C2semidirect
6. #16249C2semidirect
7. #15233C2semidirect
8. #15219C2semidirect
9. #15130C2semidirect
10. #15124C2semidirect
11. #14340C2semidirect
12. #14257C2semidirect
14. C22×#127C22semidirect
24. C2×#394C22semidirect
27. C2×#390C22semidirect
28. C2×#390C22semidirect
29. C2×#390C22semidirect
33. C2×#384C22semidirect
34. C2×#382C22semidirect
35. C2×#382C22semidirect
45. #1443C22semidirect
46. #1441C22semidirect
48. #1440C22semidirect
50. #1439C22semidirect
51. #1439C22semidirect
54. #1438C22semidirect
55. C4⋊#95C22semidirect
56. C4⋊#95C22semidirect
57. C4⋊#95C22semidirect
70. C24×C4D4semidirect
94. C2×#117D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28472
C2×#26192
C2×#22802
#166602
#165122
#162492
#152332
#152192
#151302
#151242
#143402
#142572