Finite Group Databasebeta

← #556772

Subgroups — #556772

Order 512.

6026 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#123C211yesmaximal, characteristic
2C2×#2280C211yesmaximal, characteristic, complemented
3C22×#380C211yesmaximal, characteristic, complemented
4#14658C214yesmaximal, complemented
5#14339C212yesmaximal, complemented
6#14338C212yesmaximal, complemented
7#14287C214yesmaximal, complemented
8#14256C214yesmaximal, complemented
9#14135C214yesmaximal, complemented
10#13923C212yesmaximal, complemented
11#13903C212yesmaximal, complemented
12#13869C212yesmaximal, complemented
13#13867C212yesmaximal
14C24×D4C2211yescharacteristic
15C25×C4C2211yescharacteristic
16C22×#123—28no—
17C22×#123—28no—
18C22×#123C4116yescomplemented
19C22×#123C2212yes—
20C22×#123C2212yes—
21C22×#123C2212yes—
22C22×#123C2212yes—
23C22×#117C2212yescomplemented
24C23×C4⋊C4C2211yescharacteristic
25C23×C4⋊C4C2211yescharacteristic
26C23×C4⋊C4C2212yescomplemented
27C23×C22⋊C4C2211yescharacteristic
28C23×C22⋊C4C2211yescharacteristic
29C23×C22⋊C4C2211yescharacteristic
30C23×C22⋊C4C2212yes—
31C2×#386C2218yes—
32#2280—28no—
33C2×#381C2214yes—
34C2×#381C2212yescomplemented
35C2×#380—28no—
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×#380C2212yes—
39C2×#380C2212yescomplemented
40C2×#380C2212yescomplemented
41C2×#380C2212yescomplemented
42C2×#379C2212yes—
43C2×#379C2214yes—
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yescomplemented
46C22×#95C2212yescomplemented
47#1441C2218yescomplemented
48#1441C2218yescomplemented
49#1441C2218yescomplemented
50#1440C2218yescomplemented
51#1439C2218yescomplemented
52#1439C2218yescomplemented
53#1439C2218yescomplemented
54#1438C2218yes—
55#1437C2218yescomplemented
56#1436C2218yescomplemented
57C22⋊#95C2214yes—
58C22⋊#95C2214yes—
59C22⋊#95C2218yescomplemented
60C26C2311yescharacteristic
61C26C2311yescharacteristic
62C23×D4—24no—
63C23×D4—24no—
64C23×D4—48no—
65C23×D4C2×C418yes—
66C23×D4C2312yes—
67C23×D4C2312yes—
68C24×C4—48no—
69C24×C4C2×C418yes—
70C24×C4D418yes—
71C24×C4D418yes—
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2314yes—
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C2×#123—2128no—
83C2×#123—28no—
84C2×#123—28no—
85C2×#123—28no—
86C2×#123—28no—
87C2×#123—28no—
88C2×#123—28no—
89C2×#123—28no—
90C2×#123—28no—
91C2×#123C2×C4116yes—
92C2×#123C2×C4116yes—
93C2×#123C2×C4116yescomplemented
94C2×#123C2×C4116yescomplemented
95C2×#117—28no—
96C2×#117—48no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—28no—
102C22×C4⋊C4—48no—
103C22×C4⋊C4C2×C418yescomplemented
104C22×C4⋊C4C2×C418yescomplemented
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C22⋊C4—24no—
113C22×C22⋊C4—24no—
114C22×C22⋊C4—24no—
115C22×C22⋊C4—24no—
116C22×C22⋊C4—24no—
117C22×C22⋊C4—24no—
118C22×C22⋊C4—28no—
119C22×C22⋊C4—48no—
120C22×C22⋊C4C2×C418yes—
121C22×C22⋊C4C2×C418yescomplemented
122C22×C22⋊C4C2×C418yescomplemented
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C23116yes—
135C22×C42C2314yes—
136#386—416no—
137#381—28no—
138#381—48no—
139#380—28no—
140#380—28no—
141#380—28no—
142#380—28no—
143#380—28no—
144#380—28no—
145#379—28no—
146#379—48no—
147C4×C22⋊C4—28no—
148C4×C22⋊C4—28no—
149C2×#95—28no—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C23116yes—
155C2×#95C23116yes—
156C2×#95C23116yes—
157C25—24no—
158C25—28no—
159C25—24no—
160C25—22no—
161C25—22no—
162C25—24no—
163C25—24no—
164C25—44no—
165C25—44no—
166C25—44no—
167C25C22×C414yes—
168C25C22×C414yes—
169C25C2×D414yes—
170C25C2×D414yes—
171C25C2412yes—
172C25C2411yescharacteristic
173C25C2411yescharacteristic
174C25C2411yescharacteristic
175C22×D4—232no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—216no—
181C22×D4—48no—
182C22×D4—432no—
183C22×D4—432no—
184C22×D4—432no—
185C22×D4C22×C418yes—
186C22×D4C22×C418yes—
187C23×C4—28no—
188C23×C4—28no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—28no—
198C23×C4—216no—
199C23×C4—432no—
200C23×C4—432no—
201C23×C4—432no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—48no—
209C23×C4—48no—
210C23×C4C22⋊C4164yescomplemented
211C23×C4C22×C414yes—
212C23×C4C22×C414yes—
213C23×C4C22×C414yes—
214C23×C4C22×C414yes—
215C23×C4C22×C414yes—
216C23×C4C22×C414yes—
217C23×C4C22×C414yes—
218C23×C4C22×C414yes—
219C23×C4C22×C414yes—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4C2418yes—
233C23×C4C2418yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241#123—264no—
242#123—264no—
243#123—264no—
244#123—264no—
245#117—432no—
246C2×C4⋊C4—232no—
247C2×C4⋊C4—232no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—48no—
260C2×C4⋊C4—48no—
261C2×C4⋊C4—48no—
262C2×C4⋊C4—432no—
263C2×C4⋊C4C22×C418yes—
264C2×C4⋊C4C22×C418yes—
265C2×C4⋊C4C22×C418yes—
266C2×C4⋊C4C22×C418yes—
267C2×C22⋊C4—232no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—232no—
273C2×C22⋊C4—232no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—216no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—48no—
293C2×C22⋊C4—48no—
294C2×C22⋊C4—432no—
295C2×C22⋊C4—416no—
296C2×C22⋊C4—416no—
297C2×C22⋊C4—416no—
298C2×C22⋊C4C22×C418yes—
299C2×C22⋊C4C22×C418yes—
300C2×C22⋊C4C22×C418yes—
301C2×C22⋊C4C22×C418yes—
302C2×C22⋊C4C22×C418yes—
303C2×C22⋊C4C22×C418yes—
304C2×C42—28no—
305C2×C42—48no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—416no—
311#95—416no—
312#95—416no—
313C24—22no—
314C24—24no—
315C24—22no—
316C24—24no—
317C24—24no—
318C24—216no—
319C24—216no—
320C24—216no—
321C24—216no—
322C24—28no—
323C24—28no—
324C24—216no—
325C24—22no—
326C24—24no—
327C24—22no—
328C24—24no—
329C24—28no—
330C24—22no—
331C24—22no—
332C24—24no—
333C24—28no—
334C24—24no—
335C24—28no—
336C24—216no—
337C24—48no—
338C24—48no—
339C24—48no—
340C24—48no—
341C24—48no—
342C24—432no—
343C24—432no—
344C24—48no—
345C24—48no—
346C24—48no—
347C24—48no—
348C24—44no—
349C24—44no—
350C24—48no—
351C24—816no—
352C24—816no—
353C24C2×C22⋊C4116yes—
354C24C23×C414yes—
355C24C23×C412yes—
356C24C23×C412yes—
357C24C23×C412yes—
358C24C22×D412yes—
359C24C22×D412yes—
360C24C22×D412yes—
361C24C22×D412yes—
362C24C22×D412yes—
363C24C22×D412yes—
364C24H(5,𝔽2)18yes—
365C24H(5,𝔽2)14yes—
366C242−1+414yes—
367C24C2511yescharacteristic
368C2×D4—216no—
369C2×D4—216no—
370C2×D4—232no—
371C2×D4—416no—
372C2×D4—4128no—
373C2×D4—416no—
374C2×D4—416no—
375C2×D4—416no—
376C2×D4—416no—
377C22×C4—216no—
378C22×C4—216no—
379C22×C4—216no—
380C22×C4—216no—
381C22×C4—216no—
382C22×C4—216no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—216no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—24no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—216no—
406C22×C4—216no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—216no—
410C22×C4—216no—
411C22×C4—4128no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—48no—
433C22×C4—416no—
434C22×C4—416no—
435C22×C4—416no—
436C22×C4—832no—
437C22×C4C2×C22⋊C4116yes—
438C22×C4C2×C22⋊C4116yes—
439C22×C4C2×C22⋊C4116yes—
440C22×C4C2×C22⋊C4116yes—
441C22×C4C2×C22⋊C4116yes—
442C22×C4C2×C22⋊C4116yes—
443C22×C4C23×C414yes—
444C22×C4C23×C414yes—
445C22×C4C23×C414yes—
446C22×C4C23×C412yes—
447C22×C4C23×C412yes—
448C22×C4C23×C412yes—
449C22×C4C23×C412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C4⋊C4—216no—
459C4⋊C4—216no—
460C4⋊C4—216no—
461C4⋊C4—216no—
462C4⋊C4—416no—
463C4⋊C4—416no—
464C4⋊C4—416no—
465C22⋊C4—216no—
466C22⋊C4—216no—
467C22⋊C4—216no—
468C22⋊C4—216no—
469C22⋊C4—216no—
470C22⋊C4—216no—
471C22⋊C4—416no—
472C22⋊C4—416no—
473C22⋊C4—416no—
474C22⋊C4—416no—
475C22⋊C4—416no—
476C22⋊C4—832no—
477C42—416no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—216no—
492C23—216no—
493C23—216no—
494C23—216no—
495C23—232no—
496C23—232no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—28no—
509C23—28no—
510C23—28no—
511C23—416no—
512C23—432no—
513C23—432no—
514C23—464no—
515C23—464no—
516C23—416no—
517C23—416no—
518C23—48no—
519C23—48no—
520C23—416no—
521C23—44no—
522C23—44no—
523C23—42no—
524C23—42no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—416no—
529C23—416no—
530C23—832no—
531C23—816no—
532C23—816no—
533C23—832no—
534C23C22×C22⋊C414yes—
535C23C22×C22⋊C414yes—
536C23C22×C22⋊C414yes—
537C23#51818yes—
538C23#52018yes—
539C23#53418yes—
540C23#53418yes—
541C23#53514yes—
542C23#53614yes—
543C23#53718yes—
544C23C24×C411yescharacteristic
545C23C23×D411yescharacteristic
546C23C23×D411yescharacteristic
547C23C2×H(5,𝔽2)14yes—
548C23C2×H(5,𝔽2)12yes—
549C23C2×2−1+412yes—
550C23#58514yes—
551D4—432no—
552D4—432no—
553C2×C4—232no—
554C2×C4—232no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—416no—
574C2×C4—416no—
575C2×C4—416no—
576C2×C4—416no—
577C2×C4—416no—
578C2×C4—416no—
579C2×C4—416no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—416no—
594C2×C4—416no—
595C2×C4—416no—
596C2×C4—416no—
597C2×C4—832no—
598C2×C4C22×C22⋊C414yes—
599C2×C4C22×C22⋊C414yes—
600C2×C4C22×C22⋊C414yes—
601C2×C4C22×C22⋊C414yes—
602C22—28no—
603C22—28no—
604C22—28no—
605C22—28no—
606C22—216no—
607C22—216no—
608C22—216no—
609C22—216no—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—24no—
617C22—464no—
618C22—44no—
619C22—44no—
620C22—48no—
621C22—48no—
622C22—416no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—832no—
627C22—84no—
628C22—84no—
629C22—88no—
630C22#230818yes—
631C22#231118yes—
632C22C23×C22⋊C411yescharacteristic
633C22C2×#51812yes—
634C22C2×#52012yes—
635C22#342812yes—
636C22C2×#53412yes—
637C22C2×#53412yes—
638C22C2×#53511yescharacteristic
639C22C2×#53611yescharacteristic
640C22C2×#53712yes—
641C22#344912yes—
642C22#344912yes—
643C22#353418yes—
644C22#353518yes—
645C4—24no—
646C4—24no—
647C4—24no—
648C4—24no—
649C4—44no—
650C4—44no—
651C4—44no—
652C4—88no—
653C4—88no—
654C2—28no—
655C2—28no—
656C2—44no—
657C2—84no—
658C2C2×#230811yescharacteristic
659C2C2×#231111yescharacteristic
660C2#3404411yescharacteristic
661C2#3437614yes—
662C2#3744012yes—
663C2#3744112yes—
664C2#3745214yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC24×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2280C2semidirect
3. C22×#380C2semidirect
4. #14658C2semidirect
5. #14339C2semidirect
6. #14338C2semidirect
7. #14287C2semidirect
8. #14256C2semidirect
9. #14135C2semidirect
10. #13923C2semidirect
11. #13903C2semidirect
12. #13869C2semidirect
18. C22×#123C4semidirect
23. C22×#117C22semidirect
26. C23×C4⋊C4C22semidirect
34. C2×#381C22semidirect
39. C2×#380C22semidirect
40. C2×#380C22semidirect
41. C2×#380C22semidirect
45. C2×C4×C22⋊C4C22semidirect
46. C22×#95C22semidirect
47. #1441C22semidirect
48. #1441C22semidirect
49. #1441C22semidirect
50. #1440C22semidirect
51. #1439C22semidirect
52. #1439C22semidirect
53. #1439C22semidirect
55. #1437C22semidirect
56. #1436C22semidirect
59. C22⋊#95C22semidirect
93. C2×#123C2×C4semidirect
94. C2×#123C2×C4semidirect
103. C22×C4⋊C4C2×C4semidirect
104. C22×C4⋊C4C2×C4semidirect
121. C22×C22⋊C4C2×C4semidirect
122. C22×C22⋊C4C2×C4semidirect
210. C23×C4C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C23×#1232
C2×#22802
C22×#3802
#146582
#143392
#143382
#142872
#142562
#141352
#139232
#139032
#138692
#138672