Finite Group Databasebeta

← #521313

Subgroups — #521313

Order 512.

6174 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#123C211yesmaximal, characteristic
2C2×#2342C212yesmaximal, complemented
3C2×#2308C211yesmaximal, characteristic
4C2×#2280C211yesmaximal, characteristic, complemented
5#14681C211yesmaximal, characteristic, complemented
6#14359C218yesmaximal, complemented
7#14338C211yesmaximal, characteristic, complemented
8#14335C211yesmaximal, characteristic
9#14096C218yesmaximal, complemented
10#13904C212yesmaximal, complemented
11#13870C212yesmaximal, complemented
12#13869C212yesmaximal, complemented
13#13775C211yesmaximal, characteristic, complemented
14C24×D4C2211yescharacteristic
15C25×C4C2211yescharacteristic
16C22×#123—216no—
17C22×#123C4116yescomplemented
18C22×#123C2214yes—
19C22×#123C2214yes—
20C22×#117C2212yes—
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C23×C22⋊C4C2212yes—
24C23×C22⋊C4C2211yescharacteristic
25C23×C22⋊C4C2211yescharacteristic
26C23×C22⋊C4C2211yescharacteristic
27#2342—216no—
28#2308—28no—
29C2×#388C2214yescomplemented
30C2×#388C2214yescomplemented
31C2×#386C2218yes—
32C2×#385C2214yescomplemented
33#2280—28no—
34C2×#381C2212yes—
35C2×#381C2212yes—
36C2×#381C2212yes—
37C2×#381C2212yes—
38C2×#381C2212yes—
39C2×#380C2212yes—
40C2×#380C2214yescomplemented
41C2×#379C2211yescharacteristic
42C2×#379C2211yescharacteristic
43C2×#379C2212yes—
44C2×#379C2211yescharacteristic
45C2×#379C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2214yescomplemented
48C2×C4×C22⋊C4C2214yescomplemented
49#1445C2218yescomplemented
50#1441C2218yescomplemented
51#1440C2218yescomplemented
52#1439C2218yescomplemented
53#1438C2218yes—
54#1437C2218yescomplemented
55#1436C2218yescomplemented
56C4⋊#95C2218yescomplemented
57C22⋊#95C2212yes—
58C22⋊#95C2212yes—
59C22⋊#95C2214yescomplemented
60C22⋊#95C2218yescomplemented
61C22⋊#95C2214yescomplemented
62C26C2311yescharacteristic
63C26C2311yescharacteristic
64C23×D4—24no—
65C23×D4—28no—
66C23×D4—24no—
67C23×D4—44no—
68C23×D4C2×C418yes—
69C23×D4C2314yes—
70C24×C4—24no—
71C24×C4—24no—
72C24×C4—28no—
73C24×C4—44no—
74C24×C4C2×C418yescomplemented
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2312yes—
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2314yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C2×#123—2128no—
84C2×#123—216no—
85C2×#123—216no—
86C2×#123—216no—
87C2×#123—216no—
88C2×#123C2×C4132yescomplemented
89C2×#123C2×C4132yescomplemented
90C2×#117—28no—
91C2×#117—28no—
92C2×#117—44no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—28no—
95C22×C4⋊C4C2×C418yescomplemented
96C22×C4⋊C4C2×C418yescomplemented
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C22⋊C4—28no—
103C22×C22⋊C4—28no—
104C22×C22⋊C4—28no—
105C22×C22⋊C4—28no—
106C22×C22⋊C4—44no—
107C22×C22⋊C4C2×C418yes—
108C22×C22⋊C4C2×C418yes—
109C22×C22⋊C4C2×C418yescomplemented
110C22×C22⋊C4D4116yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C42C2314yes—
127C22×C42C2314yes—
128#388—216no—
129#388—216no—
130#386—232no—
131#385—216no—
132#381—28no—
133#381—28no—
134#381—28no—
135#381—44no—
136#381—44no—
137#380—28no—
138#380—216no—
139#379—24no—
140#379—28no—
141#379—24no—
142#379—24no—
143#379D418yes—
144C4×C22⋊C4—28no—
145C4×C22⋊C4—216no—
146C4×C22⋊C4—216no—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2318yes—
150C2×#95C2318yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2318yes—
154C2×#95C2318yes—
155C2×#95C2318yes—
156C2×#95C2318yes—
157C25—24no—
158C25—24no—
159C25—22no—
160C25—22no—
161C25—24no—
162C25—44no—
163C25—44no—
164C25—44no—
165C25—44no—
166C25C22×C414yes—
167C25C22×C414yes—
168C25C2×D418yes—
169C25C2×D414yes—
170C25#4114yes—
171C25C2412yes—
172C25C2412yes—
173C25C2411yescharacteristic
174C22×D4—232no—
175C22×D4—232no—
176C22×D4—28no—
177C22×D4—28no—
178C22×D4—28no—
179C22×D4—28no—
180C22×D4—416no—
181C22×D4—44no—
182C22×D4—44no—
183C22×D4—432no—
184C22×D4—432no—
185C22×D4C22×C4116yes—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—232no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—232no—
204C23×C4—28no—
205C23×C4—28no—
206C23×C4—28no—
207C23×C4—28no—
208C23×C4—416no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—48no—
215C23×C4—48no—
216C23×C4—432no—
217C23×C4—432no—
218C23×C4C22×C414yes—
219C23×C4C22×C414yes—
220C23×C4C22×C418yes—
221C23×C4C22×C418yes—
222C23×C4C22×C414yes—
223C23×C4C22×C414yes—
224C23×C4C22×C414yes—
225C23×C4C2×D418yes—
226C23×C4C2×D418yes—
227C23×C4C2414yes—
228C23×C4C2414yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237#123—2128no—
238#123—2128no—
239#117—432no—
240C2×C4⋊C4—232no—
241C2×C4⋊C4—232no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—48no—
251C2×C4⋊C4C22×C418yes—
252C2×C4⋊C4C22×C418yes—
253C2×C4⋊C4C22×C4116yes—
254C2×C22⋊C4—232no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—232no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—432no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—48no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—48no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4C22⋊C4164yescomplemented
291C2×C22⋊C4C22×C4116yes—
292C2×C22⋊C4C22×C4116yes—
293C2×C22⋊C4C22×C418yes—
294C2×C22⋊C4C22×C418yes—
295C2×C22⋊C4C2×D418yes—
296C2×C22⋊C4C2×D4116yes—
297C2×C22⋊C4C2×D418yes—
298C2×C22⋊C4C2×D418yes—
299C2×C22⋊C4C2×D418yes—
300C2×C22⋊C4C2×D418yes—
301C2×C42—28no—
302C2×C42—28no—
303C2×C42—28no—
304C2×C42—44no—
305C2×C42—48no—
306#95—216no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—216no—
312#95—48no—
313#95—48no—
314#95—48no—
315#95—48no—
316C24—24no—
317C24—24no—
318C24—28no—
319C24—28no—
320C24—216no—
321C24—216no—
322C24—24no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—24no—
327C24—22no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—28no—
333C24—28no—
334C24—28no—
335C24—28no—
336C24—216no—
337C24—216no—
338C24—48no—
339C24—48no—
340C24—416no—
341C24—416no—
342C24—48no—
343C24—48no—
344C24—416no—
345C24—416no—
346C24—416no—
347C24—416no—
348C24—432no—
349C24—88no—
350C24—88no—
351C24—816no—
352C24C2×C22⋊C4116yes—
353C24C4×D4116yes—
354C24C23×C414yes—
355C24C23×C414yes—
356C24C23×C412yes—
357C24C22×D414yes—
358C24C22×D414yes—
359C24C22×D414yes—
360C24C22×D414yes—
361C24C22×D412yes—
362C24C2×#4114yes—
363C24C2×#4112yes—
364C24H(5,𝔽2)14yes—
365C24C2511yescharacteristic
366C2×D4—232no—
367C2×D4—232no—
368C2×D4—416no—
369C2×D4—4128no—
370C2×D4—416no—
371C2×D4—416no—
372C2×D4—416no—
373C2×D4—416no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—216no—
377C22×C4—216no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—216no—
389C22×C4—216no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—216no—
409C22×C4—216no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—416no—
430C22×C4—4128no—
431C22×C4—416no—
432C22×C4—416no—
433C22×C4—416no—
434C22×C4—416no—
435C22×C4—416no—
436C22×C4—416no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—416no—
442C22×C4—816no—
443C22×C4—88no—
444C22×C4—816no—
445C22×C4C2×C22⋊C4116yes—
446C22×C4C2×C22⋊C4116yes—
447C22×C4C23×C414yes—
448C22×C4C23×C414yes—
449C22×C4C23×C414yes—
450C22×C4C23×C414yes—
451C22×C4C23×C412yes—
452C22×C4C23×C412yes—
453C22×C4C22×D418yes—
454C22×C4C22×D414yes—
455C22×C4C22×D414yes—
456C22×C4C22×D414yes—
457C22×C4C22×D414yes—
458C4⋊C4—216no—
459C4⋊C4—216no—
460C4⋊C4—232no—
461C4⋊C4—416no—
462C4⋊C4—88no—
463C22⋊C4—232no—
464C22⋊C4—216no—
465C22⋊C4—232no—
466C22⋊C4—416no—
467C22⋊C4—416no—
468C22⋊C4—416no—
469C22⋊C4—416no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—88no—
474C22⋊C4—88no—
475C22⋊C4—88no—
476C22⋊C4C2×C22⋊C4132yes—
477C22⋊C4C2×C22⋊C4132yes—
478C42—416no—
479C42—88no—
480C23—24no—
481C23—24no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—216no—
487C23—232no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—28no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—28no—
498C23—28no—
499C23—28no—
500C23—24no—
501C23—28no—
502C23—24no—
503C23—28no—
504C23—24no—
505C23—24no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—28no—
510C23—28no—
511C23—28no—
512C23—232no—
513C23—432no—
514C23—416no—
515C23—416no—
516C23—464no—
517C23—432no—
518C23—416no—
519C23—464no—
520C23—416no—
521C23—48no—
522C23—48no—
523C23—48no—
524C23—48no—
525C23—48no—
526C23—48no—
527C23—48no—
528C23—48no—
529C23—832no—
530C23—816no—
531C23—816no—
532C23—832no—
533C23C22×C22⋊C418yes—
534C23C22×C22⋊C414yes—
535C23C2×C4×D418yes—
536C23C2×C4×D414yes—
537C23#51814yes—
538C23D4218yes—
539C23#54618yes—
540C23C24×C411yescharacteristic
541C23C23×D412yes—
542C23C23×D411yescharacteristic
543C23C22×#4111yescharacteristic
544C23C2×H(5,𝔽2)12yes—
545C23#58514yes—
546C23#58514yes—
547D4—464no—
548C2×C4—232no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—216no—
570C2×C4—416no—
571C2×C4—432no—
572C2×C4—432no—
573C2×C4—416no—
574C2×C4—416no—
575C2×C4—416no—
576C2×C4—44no—
577C2×C4—48no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—48no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—48no—
592C2×C4—44no—
593C2×C4—48no—
594C2×C4—44no—
595C2×C4—48no—
596C2×C4—48no—
597C2×C4—48no—
598C2×C4—48no—
599C2×C4—832no—
600C2×C4C22×C22⋊C418yes—
601C2×C4C22×C22⋊C414yes—
602C2×C4C22×C22⋊C414yes—
603C22—216no—
604C22—28no—
605C22—24no—
606C22—28no—
607C22—216no—
608C22—24no—
609C22—28no—
610C22—28no—
611C22—28no—
612C22—28no—
613C22—28no—
614C22—28no—
615C22—464no—
616C22—48no—
617C22—416no—
618C22—48no—
619C22—44no—
620C22—44no—
621C22—48no—
622C22—44no—
623C22—48no—
624C22—416no—
625C22—48no—
626C22—48no—
627C22—44no—
628C22—832no—
629C22D4×C22⋊C418yes—
630C22C23×C22⋊C411yescharacteristic
631C22C22×C4×D411yescharacteristic
632C22C2×#51811yescharacteristic
633C22#342812yes—
634C22#342812yes—
635C22#344914yes—
636C22C2×D4212yes—
637C22C2×#54612yes—
638C22#348514yes—
639C22#348514yes—
640C22#349114yes—
641C22#353318yes—
642C4—24no—
643C4—24no—
644C4—28no—
645C4—48no—
646C4—44no—
647C4—44no—
648C4—44no—
649C4—84no—
650C4—84no—
651C4—84no—
652C2—28no—
653C2—48no—
654C2—48no—
655C2#3404411yescharacteristic
656C2C2×D4×C22⋊C411yescharacteristic
657C2#3417211yescharacteristic
658C2#3419912yes—
659C2#3435712yes—
660C2#3765414yes—
661C2#3784214yes—
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×#2342C2semidirect
4. C2×#2280C2semidirect
5. #14681C2semidirect
6. #14359C2semidirect
7. #14338C2semidirect
9. #14096C2semidirect
10. #13904C2semidirect
11. #13870C2semidirect
12. #13869C2semidirect
13. #13775C2semidirect
17. C22×#123C4semidirect
29. C2×#388C22semidirect
30. C2×#388C22semidirect
32. C2×#385C22semidirect
40. C2×#380C22semidirect
47. C2×C4×C22⋊C4C22semidirect
48. C2×C4×C22⋊C4C22semidirect
49. #1445C22semidirect
50. #1441C22semidirect
51. #1440C22semidirect
52. #1439C22semidirect
54. #1437C22semidirect
55. #1436C22semidirect
56. C4⋊#95C22semidirect
59. C22⋊#95C22semidirect
60. C22⋊#95C22semidirect
61. C22⋊#95C22semidirect
74. C24×C4C2×C4semidirect
88. C2×#123C2×C4semidirect
89. C2×#123C2×C4semidirect
95. C22×C4⋊C4C2×C4semidirect
96. C22×C4⋊C4C2×C4semidirect
109. C22×C22⋊C4C2×C4semidirect
290. C2×C22⋊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×#23422
C2×#23082
C2×#22802
#146812
#143592
#143382
#143352
#140962
#139042
#138702
#138692
#137752