Finite Group Databasebeta

← #7643519

Subgroups — #7643519

Order 512.

6940 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3509C211yesmaximal, characteristic, complemented
2C2×#3508C211yesmaximal, characteristic
3#39164C218yesmaximal, complemented
4#38947C218yesmaximal, complemented
5#38094C218yesmaximal, complemented
6#38093C218yesmaximal, complemented
7#37420C214yesmaximal, complemented
8#37419C214yesmaximal, complemented
9#36639C212yesmaximal, complemented
10#35678C214yesmaximal, complemented
11#35624C214yesmaximal, complemented
12#35322C212yesmaximal, complemented
13#35298C212yesmaximal, complemented
14#35298C212yesmaximal, complemented
15#35289C212yesmaximal, complemented
16#35285C212yesmaximal, complemented
17#34026C211yesmaximal, characteristic, complemented
18#3509—216no—
19#3508—216no—
20C2×#559C2214yes—
21C2×#557C2214yescomplemented
22C2×#556C2214yes—
23C2×#552C2214yes—
24C2×#552C2214yescomplemented
25C2×#550C2214yes—
26C2×#546C2214yes—
27C2×#546C2214yes—
28C2×#541C2212yes—
29C2×#541C2212yescomplemented
30C2×#540C2212yes—
31C2×#540C2212yes—
32C2×#536C2212yes—
33C2×#535C2212yes—
34C2×#533C2211yescharacteristic
35C2×#533C2212yes—
36C2×#533C2212yescomplemented
37C2×#533C2211yescharacteristic, complemented
38C2×#532C2212yes—
39C2×#532C2212yes—
40C22×C4⋊Q8C2211yescharacteristic, complemented
41C22×#127C2211yescharacteristic
42C2×#519C2212yescomplemented
43C2×#518C2212yes—
44C2×C4×#41C2211yescharacteristic
45#2868C2212yescomplemented
46#2862C2212yescomplemented
47#2862C2212yescomplemented
48#2862C2212yescomplemented
49#2861C2218yescomplemented
50#2858C2212yes—
51#2858C2212yes—
52#2858C2212yes—
53#2858C2212yes—
54#2850C2218yescomplemented
55#2850C2218yescomplemented
56#2731C22116yescomplemented
57#2709C22116yescomplemented
58#2707C22132yescomplemented
59#2697C22116yes—
60#2691C22132yescomplemented
61#2684C22116yes—
62#2641C2218yes—
63#2640C2218yescomplemented
64#2621C2218yes—
65#2620C2218yescomplemented
66#2620C2218yescomplemented
67#2566C2218yescomplemented
68#2566C2218yescomplemented
69#2565C2218yescomplemented
70#2565C2218yescomplemented
71#2563C2218yescomplemented
72#2548C2218yes—
73#2548C2218yes—
74#2547C2218yescomplemented
75#2545C2218yescomplemented
76#2545C2218yescomplemented
77#2504C2218yescomplemented
78#2500C2218yescomplemented
79#2500C2218yescomplemented
80#2499C2218yes—
81#2499C2218yes—
82#2480C22132yes—
83#2476C22132yescomplemented
84#2460C22132yescomplemented
85#2458C22132yescomplemented
86#2436C2218yescomplemented
87#2435C2218yes—
88#2435C2218yes—
89#2432C2218yescomplemented
90#2432C2218yescomplemented
91#2327C2218yescomplemented
92#2326C2218yescomplemented
93#2326C2218yescomplemented
94#2311C2218yescomplemented
95#2309C2218yes—
96C4×#124C2218yescomplemented
97C4×#124C2218yescomplemented
98#2298C2212yes—
99#2298C2212yescomplemented
100#2297C2212yes—
101#2297C2218yescomplemented
102#2296C2212yescomplemented
103#2295C2214yescomplemented
104#2293C2214yes—
105#2290C2214yes—
106#2290C2214yescomplemented
107C4×#117C2212yes—
108C4×#117C2212yescomplemented
109C4×#117C2211yescharacteristic, complemented
110C2×C43C2211yescharacteristic, complemented
111C22×#41C2314yes—
112C22×#41C2314yes—
113C22×#41C2311yescharacteristic
114C23×Q8C2312yes—
115C23×D4C2314yes—
116C23×D4C2312yes—
117#559—232no—
118#557—232no—
119#556—232no—
120#552—232no—
121#552—232no—
122#550—232no—
123#546—232no—
124#546—232no—
125#541—216no—
126#541—216no—
127#540—216no—
128#540—216no—
129#536—216no—
130#535—216no—
131#533—28no—
132#533—216no—
133#533—28no—
134#533—216no—
135#532—216no—
136#532—216no—
137C2×C4⋊Q8—28no—
138C2×C4⋊Q8C2314yes—
139C2×C4⋊Q8C2312yes—
140C2×C4⋊Q8C2312yes—
141C2×C4⋊Q8C2312yes—
142C2×C4⋊Q8C2312yes—
143C2×C4⋊Q8C2314yes—
144C2×C4⋊Q8C2318yes—
145C2×#127—28no—
146C2×#127C2312yes—
147C2×#127C2314yes—
148C2×#127C2312yes—
149C2×#127C2314yes—
150C2×#126C2318yes—
151C2×#126C2318yes—
152C2×#125C2312yes—
153C2×#125C2314yes—
154C2×#125C2312yes—
155C2×#125C2314yes—
156C2×#125C2318yes—
157C2×#125C2318yes—
158C2×#124C2314yes—
159C2×#124C2314yes—
160C2×#124C2314yes—
161C2×#124C2314yes—
162C2×#124C2318yes—
163C2×#124C2318yes—
164C2×#124C2318yes—
165C2×#124C2318yes—
166C2×#123C2314yes—
167C2×#123C2314yes—
168C2×#123C2318yes—
169C2×#123C2318yes—
170C2×C22⋊Q8C2314yes—
171C2×C22⋊Q8C2314yes—
172C2×C22⋊Q8C2314yes—
173C2×C22⋊Q8C2314yes—
174C2×C22⋊Q8C2314yes—
175C2×C22⋊Q8C2314yes—
176C2×#121C2314yes—
177C2×#121C2314yes—
178C2×#121C2314yes—
179C2×#121C2314yes—
180C2×#121C2318yes—
181C2×#121C2318yes—
182C2×#120C2314yes—
183C2×#120C2314yes—
184#519—216no—
185#518—216no—
186C4×#41D4116yes—
187C2×C4×Q8C2314yes—
188C2×C4×Q8C2314yes—
189C2×C4×Q8C2314yes—
190C2×C4×Q8C2314yes—
191C2×C4×Q8C2314yes—
192C2×C4×Q8C2318yes—
193C2×C4×Q8C2314yes—
194C2×C4×Q8C2312yes—
195C2×C4×D4C2314yes—
196C2×C4×D4C2314yes—
197C2×C4×D4C2314yes—
198C2×C4×D4C2314yes—
199C2×C4×D4C2318yes—
200C2×C4×D4C2312yes—
201C2×C4×D4C2312yes—
202C2×C4×D4C2312yes—
203C2×#117C2314yes—
204C2×#117C2312yes—
205C2×#117C2312yes—
206C2×#117C2318yes—
207C2×#117C2314yes—
208C2×#117C2314yes—
209C2×#117C2314yes—
210C2×#117C2312yes—
211C2×#117C2311yescharacteristic
212C2×#117C2311yescharacteristic
213C2×#117C2311yescharacteristic
214C22×C4⋊C4C2314yes—
215C22×C22⋊C4C2318yes—
216C22×C42C2312yes—
217C22×C42C2314yes—
218C22×C42C2311yescharacteristic
219C22×C42C2311yescharacteristic
220C22×C42C2311yescharacteristic
221#398C23132yes—
222#397C23132yes—
223#397C23132yes—
224#396C23132yes—
225#396C23132yes—
226#394C23132yes—
227#394C23132yes—
228#394C23132yes—
229#394C23132yes—
230#393C23132yes—
231#391C2318yes—
232#391C2318yes—
233#391C2318yes—
234#391C2318yes—
235#391C2318yes—
236#391C2318yes—
237#391C2314yes—
238#390C2318yes—
239#390C2318yes—
240#390C2318yes—
241#390C2318yes—
242#390C2318yes—
243#390C23132yes—
244#390C2318yes—
245#390C2318yes—
246#390C2318yes—
247#390C2314yes—
248#389C2318yes—
249#388C23132yes—
250#388C23132yes—
251#388C23132yes—
252#388C23132yes—
253#388C2318yes—
254#388C2318yes—
255#387C23132yes—
256#387C23132yes—
257#387C2318yes—
258#387C2318yes—
259#386C23132yes—
260#386C23132yes—
261#386C2314yes—
262#386C2314yes—
263#385C23132yes—
264#385C23132yes—
265#385C2318yes—
266#384C2318yes—
267#384C2318yes—
268#382C2318yes—
269#382C2318yes—
270#382C2318yes—
271#380C23132yes—
272#379C23132yes—
273C4×C4⋊C4C2314yes—
274C4×C4⋊C4C2314yes—
275C4×C4⋊C4C2314yes—
276C4×C4⋊C4C2314yes—
277C4×C4⋊C4C2314yes—
278C4×C22⋊C4C2314yes—
279C4×C22⋊C4C2314yes—
280C4×C22⋊C4C2314yes—
281C4×C22⋊C4C2314yes—
282C4×C22⋊C4C2314yes—
283C4×C22⋊C4C23132yes—
284#376C2312yes—
285#376C2314yes—
286#376C2314yes—
287C43C2312yes—
288C43C2312yes—
289C43C2314yes—
290C43C2312yes—
291C25C2414yes—
292C2×#41—216no—
293C2×#41—216no—
294C2×#41—44no—
295C2×#41—416no—
296C2×#41—416no—
297C2×#41C2×D418yes—
298C22×Q8—28no—
299C22×Q8—48no—
300C22×Q8C2414yes—
301C22×Q8C2414yes—
302C22×Q8C2414yes—
303C22×Q8C2414yes—
304C22×Q8C2411yescharacteristic
305C22×D4—28no—
306C22×D4—216no—
307C22×D4—48no—
308C22×D4—416no—
309C22×D4C2414yes—
310C22×D4C2414yes—
311C22×D4C2414yes—
312C22×D4C2414yes—
313C22×D4C2418yes—
314C22×D4C2418yes—
315C22×D4C2418yes—
316C22×D4C2411yescharacteristic
317C22×D4C2411yescharacteristic
318C22×D4C2411yescharacteristic
319C23×C4C2412yes—
320C23×C4C2412yes—
321C23×C4C2411yescharacteristic
322C23×C4C2411yescharacteristic
323C23×C4C2411yescharacteristic
324C23×C4C2412yes—
325C23×C4C2418yes—
326C23×C4C2418yes—
327C23×C4C2414yes—
328C23×C4C2414yes—
329C4⋊Q8—216no—
330C4⋊Q8—28no—
331C4⋊Q8—28no—
332C4⋊Q8—216no—
333C4⋊Q8—28no—
334C4⋊Q8—28no—
335C4⋊Q8—416no—
336#127—216no—
337#127—28no—
338#127—28no—
339#127—216no—
340#126—232no—
341#126—232no—
342#125—232no—
343#125—232no—
344#125—28no—
345#125—28no—
346#125—216no—
347#125—216no—
348#124—216no—
349#124—216no—
350#124—216no—
351#124—216no—
352#124—416no—
353#124—416no—
354#124—416no—
355#124—416no—
356#123—216no—
357#123—216no—
358#123—232no—
359#123—232no—
360C22⋊Q8—216no—
361C22⋊Q8—216no—
362C22⋊Q8—216no—
363C22⋊Q8—216no—
364C22⋊Q8—216no—
365C22⋊Q8—216no—
366#121—216no—
367#121—216no—
368#121—216no—
369#121—216no—
370#121—232no—
371#121—232no—
372#120—216no—
373#120—216no—
374C4×Q8—232no—
375C4×Q8—216no—
376C4×Q8—216no—
377C4×Q8—216no—
378C4×Q8—216no—
379C4×Q8—216no—
380C4×Q8—216no—
381C4×Q8C2×D4116yes—
382C4×D4—232no—
383C4×D4—216no—
384C4×D4—216no—
385C4×D4—216no—
386C4×D4—216no—
387C4×D4C2×D4116yes—
388C4×D4C2×D4116yes—
389C4×D4C2×D4116yes—
390#117—28no—
391#117—28no—
392#117—216no—
393#117—216no—
394#117—232no—
395#117—48no—
396#117—48no—
397#117—44no—
398#117C2×D418yes—
399#117C2×D418yes—
400#117C2×D418yes—
401C2×C4⋊C4—216no—
402C2×C4⋊C4C2414yes—
403C2×C4⋊C4C2414yes—
404C2×C4⋊C4C2418yes—
405C2×C4⋊C4C2414yes—
406C2×C4⋊C4C2414yes—
407C2×C4⋊C4C2414yes—
408C2×C4⋊C4C2418yes—
409C2×C4⋊C4C2418yes—
410C2×C4⋊C4C2414yes—
411C2×C4⋊C4C2418yes—
412C2×C4⋊C4C2418yes—
413C2×C4⋊C4C2418yes—
414C2×C4⋊C4C2418yes—
415C2×C4⋊C4C2418yes—
416C2×C4⋊C4C2418yes—
417C2×C4⋊C4C2414yes—
418C2×C4⋊C4C24132yes—
419C2×C4⋊C4C2412yes—
420C2×C4⋊C4C2412yes—
421C2×C4⋊C4C2412yes—
422C2×C22⋊C4—232no—
423C2×C22⋊C4C2414yes—
424C2×C22⋊C4C2418yes—
425C2×C22⋊C4C2418yes—
426C2×C22⋊C4C2414yes—
427C2×C22⋊C4C2418yes—
428C2×C22⋊C4C2418yes—
429C2×C22⋊C4C2418yes—
430C2×C22⋊C4C2418yes—
431C2×C22⋊C4C2418yes—
432C2×C22⋊C4C2414yes—
433C2×C22⋊C4C2418yes—
434C2×C22⋊C4C24132yes—
435C2×C22⋊C4C24132yes—
436C2×C22⋊C4C24132yes—
437C2×C22⋊C4C24132yes—
438C2×C22⋊C4C2412yes—
439C2×C22⋊C4C2412yes—
440C2×C22⋊C4C2412yes—
441C2×C42—48no—
442C2×C42—44no—
443C2×C42C2×D418yes—
444C2×C42C2×D418yes—
445C2×C42C2×D418yes—
446C2×C42C2418yes—
447C2×C42C2418yes—
448C2×C42C2414yes—
449C2×C42C2414yes—
450C2×C42C2414yes—
451C2×C42C2414yes—
452C2×C42C2414yes—
453C2×C42C2414yes—
454C2×C42C2414yes—
455C2×C42C2414yes—
456C2×C42C2418yes—
457C2×C42C2414yes—
458C2×C42C2411yescharacteristic
459C2×C42C2411yescharacteristic
460C2×C42C2411yescharacteristic
461C2×C42C2412yes—
462C2×C42C2412yes—
463C2×C42C2412yes—
464C2×C42C2411yescharacteristic
465#95C2418yes—
466#95C2418yes—
467#95C2418yes—
468#95C2418yes—
469#95C2418yes—
470#95C24132yes—
471#95C24132yes—
472#95C24132yes—
473C24—28no—
474C24—48no—
475C24—416no—
476C24C2518yes—
477C24C2511yescharacteristic
478C24C2511yescharacteristic
479C24C2511yescharacteristic
480#41—416no—
481#41—832no—
482#41—832no—
483C2×Q8—28no—
484C2×Q8—28no—
485C2×Q8—28no—
486C2×Q8—28no—
487C2×Q8—216no—
488C2×Q8—216no—
489C2×Q8—216no—
490C2×Q8—48no—
491C2×Q8—42no—
492C2×Q8—432no—
493C2×Q8C22×D414yes—
494C2×D4—216no—
495C2×D4—216no—
496C2×D4—28no—
497C2×D4—28no—
498C2×D4—28no—
499C2×D4—28no—
500C2×D4—216no—
501C2×D4—216no—
502C2×D4—216no—
503C2×D4—216no—
504C2×D4—48no—
505C2×D4—42no—
506C2×D4—42no—
507C2×D4—42no—
508C2×D4—432no—
509C2×D4—416no—
510C2×D4—416no—
511C2×D4—416no—
512C2×D4—832no—
513C2×D4C22×D414yes—
514C2×D4C22×D414yes—
515C2×D4C22×D414yes—
516C22×C4—24no—
517C22×C4—28no—
518C22×C4—28no—
519C22×C4—28no—
520C22×C4—24no—
521C22×C4—24no—
522C22×C4—28no—
523C22×C4—216no—
524C22×C4—216no—
525C22×C4—44no—
526C22×C4—48no—
527C22×C4—48no—
528C22×C4—416no—
529C22×C4—416no—
530C22×C4C22×D418yes—
531C22×C4C22×D414yes—
532C22×C4C22×D414yes—
533C22×C4C22×D418yes—
534C22×C4C22×D418yes—
535C22×C4C22×D414yes—
536C22×C4C2518yes—
537C22×C4C2518yes—
538C22×C4C2518yes—
539C22×C4C2518yes—
540C22×C4C2518yes—
541C22×C4C2511yescharacteristic
542C22×C4C2511yescharacteristic
543C22×C4C2512yes—
544C22×C4C2512yes—
545C22×C4C2512yes—
546C22×C4C2511yescharacteristic
547C22×C4C2511yescharacteristic
548C22×C4C2512yes—
549C4⋊C4—28no—
550C4⋊C4—28no—
551C4⋊C4—216no—
552C4⋊C4—28no—
553C4⋊C4—28no—
554C4⋊C4—28no—
555C4⋊C4—216no—
556C4⋊C4—216no—
557C4⋊C4—28no—
558C4⋊C4—216no—
559C4⋊C4—216no—
560C4⋊C4—216no—
561C4⋊C4—216no—
562C4⋊C4—44no—
563C4⋊C4—48no—
564C4⋊C4—48no—
565C4⋊C4—432no—
566C4⋊C4C22×D418yes—
567C4⋊C4C22×D418yes—
568C4⋊C4C22×D418yes—
569C22⋊C4—28no—
570C22⋊C4—28no—
571C22⋊C4—216no—
572C22⋊C4—216no—
573C22⋊C4—216no—
574C22⋊C4—216no—
575C22⋊C4—216no—
576C22⋊C4—216no—
577C22⋊C4—48no—
578C22⋊C4—48no—
579C22⋊C4—44no—
580C22⋊C4—432no—
581C22⋊C4—432no—
582C22⋊C4—432no—
583C22⋊C4—432no—
584C22⋊C4C22×D418yes—
585C22⋊C4C22×D418yes—
586C22⋊C4C22×D418yes—
587C42—216no—
588C42—216no—
589C42—28no—
590C42—28no—
591C42—28no—
592C42—28no—
593C42—28no—
594C42—28no—
595C42—28no—
596C42—28no—
597C42—44no—
598C42—48no—
599C42C22×D414yes—
600C42C22×D414yes—
601C42C22×D414yes—
602C42C22×D414yes—
603C42C22×D418yes—
604C42C22×D418yes—
605C42C22×D418yes—
606C23—22no—
607C23—24no—
608C23—24no—
609C23—22no—
610C23—24no—
611C23—22no—
612C23—28no—
613C23—216no—
614C23—216no—
615C23—216no—
616C23—416no—
617C23—816no—
618C23—1616no—
619C23C23×D412yes—
620C23C23×D412yes—
621C23C23×D412yes—
622C23C2611yescharacteristic
623Q8—44no—
624Q8—416no—
625Q8—416no—
626Q8—416no—
627Q8—416no—
628D4—44no—
629D4—44no—
630D4—44no—
631D4—416no—
632D4—416no—
633D4—416no—
634D4—416no—
635D4—816no—
636D4—816no—
637D4—816no—
638C2×C4—22no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—28no—
645C2×C4—28no—
646C2×C4—28no—
647C2×C4—216no—
648C2×C4—216no—
649C2×C4—216no—
650C2×C4—28no—
651C2×C4—28no—
652C2×C4—28no—
653C2×C4—28no—
654C2×C4—28no—
655C2×C4—28no—
656C2×C4—28no—
657C2×C4—28no—
658C2×C4—28no—
659C2×C4—216no—
660C2×C4—216no—
661C2×C4—216no—
662C2×C4—416no—
663C2×C4—416no—
664C2×C4—816no—
665C2×C4—816no—
666C2×C4C23×D412yes—
667C2×C4C23×D414yes—
668C2×C4C23×D414yes—
669C2×C4C23×D412yes—
670C2×C4C23×D412yes—
671C2×C4C23×D414yes—
672C2×C4C23×D412yes—
673C2×C4C23×D414yes—
674C2×C4#58514yes—
675C2×C4#58518yes—
676C22—24no—
677C22—24no—
678C22—24no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—44no—
683C22—44no—
684C22—44no—
685C22—48no—
686C22—48no—
687C22—432no—
688C22—48no—
689C22—1616no—
690C22C24×D411yescharacteristic
691C22C2×#58514yes—
692C22C2×#58512yes—
693C4—24no—
694C4—24no—
695C4—24no—
696C4—22no—
697C4—22no—
698C4—22no—
699C4—48no—
700C4—48no—
701C4—48no—
702C4—48no—
703C4—48no—
704C4#344914yes—
705C4#350018yes—
706C2—42no—
707C2—42no—
708C2—42no—
709C2—88no—
710C2C2×#344911yescharacteristic
711C2C2×#350012yes—
712C2#6283914yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
prime radicalC2×#3508C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3509C2semidirect
3. #39164C2semidirect
4. #38947C2semidirect
5. #38094C2semidirect
6. #38093C2semidirect
7. #37420C2semidirect
8. #37419C2semidirect
9. #36639C2semidirect
10. #35678C2semidirect
11. #35624C2semidirect
12. #35322C2semidirect
13. #35298C2semidirect
14. #35298C2semidirect
15. #35289C2semidirect
16. #35285C2semidirect
17. #34026C2semidirect
21. C2×#557C22semidirect
24. C2×#552C22semidirect
29. C2×#541C22semidirect
36. C2×#533C22semidirect
37. C2×#533C22semidirect
40. C22×C4⋊Q8C22semidirect
42. C2×#519C22semidirect
45. #2868C22semidirect
46. #2862C22semidirect
47. #2862C22semidirect
48. #2862C22semidirect
49. #2861C22semidirect
54. #2850C22semidirect
55. #2850C22semidirect
56. #2731C22semidirect
57. #2709C22semidirect
58. #2707C22semidirect
60. #2691C22semidirect
63. #2640C22semidirect
65. #2620C22semidirect
66. #2620C22semidirect
67. #2566C22semidirect
68. #2566C22semidirect
69. #2565C22semidirect
70. #2565C22semidirect
71. #2563C22semidirect
74. #2547C22semidirect
75. #2545C22semidirect
76. #2545C22semidirect
77. #2504C22semidirect
78. #2500C22semidirect
79. #2500C22semidirect
83. #2476C22semidirect
84. #2460C22semidirect
85. #2458C22semidirect
86. #2436C22semidirect
89. #2432C22semidirect
90. #2432C22semidirect
91. #2327C22semidirect
92. #2326C22semidirect
93. #2326C22semidirect
94. #2311C22semidirect
96. C4×#124C22semidirect
97. C4×#124C22semidirect
99. #2298C22semidirect
101. #2297C22semidirect
102. #2296C22semidirect
103. #2295C22semidirect
106. #2290C22semidirect
108. C4×#117C22semidirect
109. C4×#117C22semidirect
110. C2×C43C22semidirect

Maximal subgroups

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

subgroupindex
C2×#35092
C2×#35082
#391642
#389472
#380942
#380932
#374202
#374192
#366392
#356782
#356242
#353222
#352982
#352892
#352852
#340262