Finite Group Databasebeta

← #517680

Subgroups — #517680

Order 512.

3230 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2621C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic, complemented
3C2×#2360C212yesmaximal
4C2×#2274C211yesmaximal, characteristic
5#14313C212yesmaximal
6#14226C212yesmaximal, complemented
7#14223C212yesmaximal, complemented
8#14208C214yesmaximal, complemented
9#14202C214yesmaximal
10#14155C211yesmaximal, characteristic, complemented
11#14150C211yesmaximal, characteristic, complemented
12#14142C212yesmaximal, complemented
13#14138C212yesmaximal
14C4×#397C212yesmaximal
15#13852C211yesmaximal, characteristic
16#13817C212yesmaximal, complemented
17#13804C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2212yes—
20C22×C4×Q8C2212yes—
21C22×#117C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24#2621C4116yescomplemented
25C2×#398C2212yes—
26C2×#398C2212yes—
27C2×#397C2212yes—
28C2×#397C2212yes—
29#2365—28no—
30#2360—216no—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#389C2212yes—
36C2×#389C2214yes—
37C2×#389C2212yes—
38C2×#387C2214yes—
39C2×#387C2214yes—
40C2×#385C2214yes—
41C2×#382C2211yescharacteristic
42#2274—28no—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2212yes—
54C2×C4×C22⋊C4C2214yes—
55C2×C4×C22⋊C4C2212yes—
56C2×C4×C22⋊C4C2212yes—
57#1445C2212yes—
58#1445C2212yes—
59#1444C2212yes—
60#1443C2214yes—
61#1442C2212yes—
62#1442C2214yes—
63#1441C2212yes—
64#1441C2211yescharacteristic
65#1441C2211yescharacteristic
66#1440C2214yes—
67#1440C2214yes—
68#1439C2211yescharacteristic
69#1439C2212yes—
70#1439C2211yescharacteristic
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2214yes—
81#1434C2214yes—
82#1434C2214yes—
83#1434C2212yes—
84C4×#95C2212yes—
85C4×#95C2214yes—
86C4×#95C2212yes—
87C23×Q8C2312yes—
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8C2×C418yescomplemented
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8C2×C4116yes—
96C2×C4×Q8—28no—
97C2×C4×Q8—28no—
98C2×C4×Q8—28no—
99C2×#117—24no—
100C2×#117—24no—
101C2×#117—24no—
102C2×#117—24no—
103C2×#117—24no—
104C2×#117C2×C418yes—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4D418yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C42C2312yes—
131C22×C42C2314yes—
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136C22×C42C2314yes—
137C22×C42C2312yes—
138C22×C42C2312yes—
139C22×C42C2312yes—
140C22×C42C2311yescharacteristic
141#398C2×C4116yescomplemented
142#398C2×C4116yescomplemented
143#397C2×C4116yes—
144#397C2×C4116yes—
145#391—28no—
146#391—28no—
147#391—28no—
148#391C2×C4116yescomplemented
149#389—28no—
150#389—216no—
151#389—28no—
152#387—216no—
153#387—216no—
154#385—216no—
155#382C2×C418yescomplemented
156C4×C4⋊C4—28no—
157C4×C4⋊C4—24no—
158C4×C4⋊C4—24no—
159C4×C4⋊C4—24no—
160C4×C4⋊C4—28no—
161C4×C4⋊C4—28no—
162C4×C4⋊C4—24no—
163C4×C4⋊C4—24no—
164C4×C4⋊C4—28no—
165C4×C4⋊C4D418yes—
166C4×C22⋊C4—28no—
167C4×C22⋊C4—216no—
168C4×C22⋊C4—28no—
169C4×C22⋊C4—28no—
170C2×#95C2312yes—
171C2×#95C2311yescharacteristic
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2311yescharacteristic
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2312yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2312yes—
198C2×#95C2312yes—
199C25C2411yescharacteristic
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—24no—
204C22×Q8—24no—
205C22×Q8—24no—
206C22×Q8C22×C418yes—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4C22×C414yes—
216C23×C4C2×D414yes—
217C23×C4C2×D418yes—
218C23×C4#4114yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2414yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C4⋊Q8—232no—
236C22⋊Q8—264no—
237C4×Q8—432no—
238#117—232no—
239#117—232no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—232no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4C22×C414yes—
275C2×C4⋊C4C22×C414yes—
276C2×C4⋊C4C22×C418yes—
277C2×C4⋊C4C22×C4116yes—
278C2×C4⋊C4C22×C418yes—
279C2×C4⋊C4C22×C418yes—
280C2×C4⋊C4C2×D418yes—
281C2×C4⋊C4C2×D414yes—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4#4114yes—
286C2×C4⋊C4#4114yes—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4C22×C414yes—
303C2×C22⋊C4C22×C414yes—
304C2×C22⋊C4C22×C4116yes—
305C2×C42—24no—
306C2×C42—22no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42—28no—
311C2×C42—24no—
312C2×C42—28no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—28no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—28no—
323C2×C42—44no—
324C2×C42—44no—
325C2×C42—44no—
326C2×C42—44no—
327C2×C42—44no—
328C2×C42—44no—
329C2×C42C22×C414yes—
330C2×C42C22×C414yes—
331C2×C42C2×D418yes—
332C2×C42C2×D414yes—
333C2×C42#4114yes—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—28no—
339#95—24no—
340#95—28no—
341#95—28no—
342#95—24no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—22no—
347#95—22no—
348#95—28no—
349#95—28no—
350#95—24no—
351#95—28no—
352#95—28no—
353#95—24no—
354#95—42no—
355#95—44no—
356#95—44no—
357#95—42no—
358#95C22×C418yes—
359#95C22×C418yes—
360#95C22×C4116yes—
361#95C22×C4116yes—
362#95C2×D418yes—
363C24—21no—
364C24—21no—
365C24—21no—
366C24—21no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24C23×C412yes—
372C24C22×D412yes—
373C24C2×#4112yes—
374C24C2511yescharacteristic
375C2×Q8—216no—
376C2×Q8—216no—
377C2×Q8—216no—
378C2×Q8—48no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—48no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—216no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—416no—
458C22×C4—84no—
459C22×C4—84no—
460C22×C4—88no—
461C22×C4C4×D4116yes—
462C22×C4C23×C412yes—
463C22×C4C23×C412yes—
464C22×C4C23×C418yes—
465C22×C4C23×C418yes—
466C22×C4C23×C412yes—
467C22×C4C23×C412yes—
468C22×C4C23×C412yes—
469C22×C4C23×C412yes—
470C22×C4C22×D414yes—
471C22×C4C22×D414yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D414yes—
476C22×C4C22×D414yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C2×#4114yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4114yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C42−1+414yes—
489C22×C42−1+414yes—
490C4⋊C4—28no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—216no—
494C4⋊C4—232no—
495C4⋊C4—216no—
496C4⋊C4—216no—
497C4⋊C4—216no—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—28no—
501C4⋊C4—48no—
502C4⋊C4—44no—
503C4⋊C4—416no—
504C4⋊C4—88no—
505C4⋊C4#121116yes—
506C22⋊C4—216no—
507C22⋊C4—216no—
508C22⋊C4—232no—
509C22⋊C4—28no—
510C22⋊C4—28no—
511C22⋊C4—48no—
512C42—216no—
513C42—28no—
514C42—48no—
515C42—416no—
516C42—48no—
517C42—48no—
518C42—88no—
519C42—84no—
520C42—84no—
521C42—84no—
522C42C4×D4116yes—
523C23—22no—
524C23—22no—
525C23—24no—
526C23—24no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—41no—
536C23—41no—
537C23—42no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23C2×C4×D414yes—
546C23C24×C411yescharacteristic
547C23C23×D411yescharacteristic
548C23C23×D411yescharacteristic
549C23C22×#4111yescharacteristic
550C23C22×#4111yescharacteristic
551C23C22×#4111yescharacteristic
552C23C22×#4111yescharacteristic
553C23C2×2−1+412yes—
554C23C2×2−1+412yes—
555C23#58512yes—
556C23#58512yes—
557Q8—432no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—232no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—816no—
645C2×C4C2×C4×D414yes—
646C2×C4C2×C4×D414yes—
647C2×C4C2×C4×D414yes—
648C2×C4C2×C4×D414yes—
649C2×C4C2×C4×D414yes—
650C2×C4C4×#4114yes—
651C2×C4C2×#12118yes—
652C2×C4C2×#12114yes—
653C2×C4#52514yes—
654C2×C4#52914yes—
655C2×C4#53214yes—
656C22—28no—
657C22—24no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—44no—
663C22—44no—
664C22—42no—
665C22—42no—
666C22—44no—
667C22C22×C4×D411yescharacteristic
668C22C22×C4×D411yescharacteristic
669C22C2×C4×#4111yescharacteristic
670C22C4×2−1+412yes—
671C22C4×2−1+412yes—
672C22C22×#12111yescharacteristic
673C22C2×#52511yescharacteristic
674C22C2×#52911yescharacteristic
675C22C2×#53211yescharacteristic
676C22#348511yescharacteristic
677C22#348512yes—
678C22#348611yescharacteristic
679C22#348612yes—
680C22#349512yes—
681C22#349911yescharacteristic
682C22#350011yescharacteristic
683C22#350712yes—
684C22#350712yes—
685C22#350712yes—
686C22#351512yes—
687C22#351612yes—
688C22#352112yes—
689C22#352312yes—
690C4—28no—
691C4—24no—
692C4—24no—
693C4—24no—
694C4—44no—
695C4—44no—
696C4—44no—
697C4—44no—
698C4—44no—
699C4—42no—
700C4—48no—
701C4—42no—
702C4—48no—
703C4—82no—
704C4—82no—
705C4C4×#12118yes—
706C2—44no—
707C2C2×C4×#12111yescharacteristic
708C2#3426611yescharacteristic
709C2#3426612yes—
710C2#3427611yescharacteristic
711C2#3428612yes—
712C2#3570011yescharacteristic
713C2#3571211yescharacteristic
714C2#3577312yes—
715C2#3578912yes—
716C2#3584611yescharacteristic
717C2#3600911yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
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×#2365C2semidirect
6. #14226C2semidirect
7. #14223C2semidirect
8. #14208C2semidirect
10. #14155C2semidirect
11. #14150C2semidirect
12. #14142C2semidirect
16. #13817C2semidirect
17. #13804C2semidirect
24. #2621C4semidirect
92. C2×C4⋊Q8C2×C4semidirect
141. #398C2×C4semidirect
142. #398C2×C4semidirect
148. #391C2×C4semidirect
155. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26212
C2×#23652
C2×#23602
C2×#22742
#143132
#142262
#142232
#142082
#142022
#141552
#141502
#141422
#141382
C4×#3972
#138522
#138172
#138042