Finite Group Databasebeta

← #722616

Subgroups — #722616

Order 512.

3544 conjugacy classes of subgroups.

The subgroup lattice has 754 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic, complemented
2C2×#2628C211yesmaximal, characteristic
3C2×#2396C211yesmaximal, characteristic
4C2×#2323C211yesmaximal, characteristic
5#16603C212yesmaximal, complemented
6#16602C212yesmaximal
7#15847C214yesmaximal, complemented
8#15846C214yesmaximal
9#15218C212yesmaximal, complemented
10#15216C212yesmaximal
11#14876C212yesmaximal, complemented
12#14875C212yesmaximal
13#14161C211yesmaximal, characteristic, complemented
14#14017C212yesmaximal, complemented
15#13969C212yesmaximal
16#13942C211yesmaximal, characteristic, complemented
17#13925C211yesmaximal, characteristic, complemented
18C2×#557—28no—
19C2×#557—28no—
20C2×#557—28no—
21C22×C4⋊Q8C2212yes—
22C22×C4⋊Q8C2211yescharacteristic
23C22×#125C2212yes—
24C22×#125C2214yes—
25C22×C22⋊Q8C2211yescharacteristic
26C22×C4×Q8C2212yes—
27C22×C4×Q8C2212yes—
28C22×C4×Q8C2211yescharacteristic
29C22×C4×Q8C2211yescharacteristic
30C22×#117C2211yescharacteristic
31#2628—28no—
32#2396—28no—
33C2×#400C2211yescharacteristic
34C2×#400C2212yes—
35C2×#399C2211yescharacteristic
36C2×#399C2212yes—
37C2×#398C2212yes—
38C2×#396C2212yes—
39C2×#395C2212yes—
40C2×#395C2211yescharacteristic
41C2×#393C2212yes—
42C2×#393C2211yescharacteristic
43#2323—28no—
44C2×#391C2211yescharacteristic
45C2×#391C2212yes—
46C2×#389C2211yescharacteristic
47C2×#389C2212yes—
48C2×#389C2212yes—
49C2×#388C2212yes—
50C2×#388C2212yes—
51C2×#387C2212yes—
52C2×#382C2211yescharacteristic
53C2×#380C2211yescharacteristic
54C2×#380C2212yes—
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2212yes—
58C2×C4×C22⋊C4C2212yes—
59#1446C2212yes—
60#1446C2212yes—
61#1446C2212yes—
62#1445C2212yes—
63#1444C2214yes—
64#1444C2214yes—
65#1444C2214yes—
66#1442C2214yes—
67#1442C2214yes—
68#1442C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1441C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1437C2214yes—
77#1436C2214yes—
78#1436C2214yes—
79#1436C2214yes—
80#1436C2214yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C4⋊#95C2214yes—
84C4⋊#95C2212yes—
85C4⋊#95C2212yes—
86C4⋊#95C2212yes—
87#1434C2212yes—
88#1434C2212yes—
89C23×Q8C2311yescharacteristic
90C23×Q8C2312yes—
91C23×Q8C2311yescharacteristic
92C24×C4C2311yescharacteristic
93#557—2128no—
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—28no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8D418yes—
100C2×#125—216no—
101C2×#125—28no—
102C2×#125—28no—
103C2×#125—216no—
104C2×#125—216no—
105C2×#125—28no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8Q818yescomplemented
109C2×C4×Q8—28no—
110C2×C4×Q8—28no—
111C2×C4×Q8—28no—
112C2×C4×Q8—28no—
113C2×C4×Q8—28no—
114C2×C4×Q8—28no—
115C2×C4×Q8—24no—
116C2×C4×Q8—24no—
117C2×C4×Q8—24no—
118C2×C4×Q8—24no—
119C2×C4×Q8D418yes—
120C2×C4×Q8D418yes—
121C2×#117—24no—
122C2×#117—24no—
123C2×#117—24no—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2314yes—
127C22×C4⋊C4C2314yes—
128C22×C4⋊C4C2314yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2312yes—
141C22×C4⋊C4C2312yes—
142C22×C4⋊C4C2314yes—
143C22×C4⋊C4C2312yes—
144C22×C22⋊C4C2311yescharacteristic
145C22×C22⋊C4C2311yescharacteristic
146C22×C22⋊C4C2312yes—
147C22×C22⋊C4C2312yes—
148C22×C22⋊C4C2312yes—
149C22×C22⋊C4C2314yes—
150C22×C22⋊C4C2312yes—
151C22×C42C2311yescharacteristic
152C22×C42C2314yes—
153C22×C42C2312yes—
154C22×C42C2312yes—
155C22×C42C2312yes—
156C22×C42C2312yes—
157C22×C42C2311yescharacteristic
158#400—28no—
159#400—24no—
160#399—28no—
161#399—24no—
162#398—28no—
163#396—28no—
164#395—28no—
165#395—24no—
166#393—28no—
167#393—24no—
168#391—24no—
169#391—28no—
170#389—28no—
171#389—24no—
172#389—28no—
173#388—28no—
174#388—28no—
175#387—28no—
176#382—24no—
177#380—24no—
178#380—28no—
179C4×C4⋊C4—24no—
180C4×C4⋊C4—28no—
181C4×C4⋊C4—24no—
182C4×C22⋊C4—28no—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2311yescharacteristic
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2311yescharacteristic
189C2×#95C2314yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2312yes—
197C2×#95C2312yes—
198C2×#95C2312yes—
199C2×#95C2312yes—
200C2×#95C2312yes—
201C2×#95C2314yes—
202C2×#95C2314yes—
203C2×#95C2314yes—
204C2×#95C2314yes—
205C2×#95C2314yes—
206C2×#95C2314yes—
207C2×#95C2314yes—
208C2×#95C2314yes—
209C2×#95C2314yes—
210C25C2411yescharacteristic
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—24no—
215C22×Q8—24no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—42no—
220C22×Q8—42no—
221C22×Q8—44no—
222C22×Q8—44no—
223C22×Q8C2×D414yes—
224C22×Q8C2×D418yes—
225C22×Q8C2×D414yes—
226C22×Q8C2×Q814yes—
227C22×Q8#4114yes—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4C2×Q814yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2414yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2414yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246C23×C4C2412yes—
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2412yes—
250C23×C4C2411yescharacteristic
251C4⋊Q8—264no—
252C4⋊Q8—232no—
253#125—2128no—
254#125—264no—
255C22⋊Q8—232no—
256C4×Q8—264no—
257C4×Q8—264no—
258C4×Q8—232no—
259C4×Q8—232no—
260#117—416no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4C2×D414yes—
313C2×C4⋊C4C2×D418yes—
314C2×C4⋊C4C2×D414yes—
315C2×C4⋊C4C2×D418yes—
316C2×C4⋊C4C2×D418yes—
317C2×C4⋊C4C2×D418yes—
318C2×C4⋊C4C2×Q814yes—
319C2×C4⋊C4C2×Q818yes—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—28no—
333C2×C22⋊C4—42no—
334C2×C22⋊C4—42no—
335C2×C22⋊C4—42no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4C2×Q818yes—
340C2×C42—28no—
341C2×C42—28no—
342C2×C42—28no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—24no—
351C2×C42—24no—
352C2×C42—24no—
353C2×C42—24no—
354C2×C42—44no—
355C2×C42C2×D418yes—
356C2×C42C2×D418yes—
357C2×C42C2×D414yes—
358C2×C42C2×D414yes—
359C2×C42C2×D414yes—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—22no—
365#95—22no—
366#95—24no—
367#95—24no—
368#95—28no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—28no—
375#95—28no—
376#95—28no—
377#95—28no—
378#95—44no—
379#95—44no—
380#95—44no—
381#95—44no—
382#95—44no—
383#95—44no—
384#95—44no—
385#95—44no—
386#95—44no—
387C24—21no—
388C24—21no—
389C24—21no—
390C24—21no—
391C24—22no—
392C24—21no—
393C24—21no—
394C24—22no—
395C24—22no—
396C24—22no—
397C24C22×Q812yes—
398C24C2511yescharacteristic
399C2×Q8—28no—
400C2×Q8—216no—
401C2×Q8—28no—
402C2×Q8—28no—
403C2×Q8—28no—
404C2×Q8—28no—
405C2×Q8—28no—
406C2×Q8—28no—
407C2×Q8—48no—
408C2×Q8—416no—
409C2×Q8—416no—
410C2×Q8—48no—
411C2×Q8—44no—
412C2×Q8—48no—
413C2×Q8—44no—
414C2×Q8—48no—
415C2×Q8C22⋊Q8116yes—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—28no—
463C22×C4—28no—
464C22×C4—28no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—24no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4C22×D414yes—
499C22×C4C22×D418yes—
500C22×C4C22×D414yes—
501C22×C4C22×D414yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D414yes—
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C22×D412yes—
512C22×C4C22×Q814yes—
513C22×C4C22×Q812yes—
514C22×C4C22×Q814yes—
515C22×C4C22×Q812yes—
516C22×C4C2×#4114yes—
517C22×C4C2×#4112yes—
518C22×C42−1+414yes—
519C22×C42−1+412yes—
520C22×C42−1+412yes—
521C4⋊C4—232no—
522C4⋊C4—232no—
523C4⋊C4—232no—
524C4⋊C4—28no—
525C4⋊C4—216no—
526C4⋊C4—216no—
527C4⋊C4—28no—
528C4⋊C4—216no—
529C4⋊C4—216no—
530C4⋊C4—28no—
531C4⋊C4—216no—
532C4⋊C4—48no—
533C4⋊C4—48no—
534C4⋊C4—48no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—48no—
539C4⋊C4—88no—
540C4⋊C4—84no—
541C22⋊C4—216no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—88no—
547C22⋊C4—84no—
548C42—232no—
549C42—216no—
550C42—216no—
551C42—44no—
552C42—48no—
553C42—84no—
554C42#120116yes—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—24no—
559C23—24no—
560C23—41no—
561C23—41no—
562C23—42no—
563C23—41no—
564C23—41no—
565C23—42no—
566C23—41no—
567C23—41no—
568C23—42no—
569C23—42no—
570C23—41no—
571C23—41no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23C23×D411yescharacteristic
579C23C23×D411yescharacteristic
580C23C23×D411yescharacteristic
581C23C23×Q811yescharacteristic
582C23C22×#4111yescharacteristic
583C23C2×2−1+411yescharacteristic
584C23C2×2−1+411yescharacteristic
585C23C2×2−1+412yes—
586C23#58512yes—
587C23#58512yes—
588C23#58512yes—
589Q8—232no—
590Q8—416no—
591Q8—432no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—28no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—28no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—416no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—42no—
668C2×C4—41no—
669C2×C4—42no—
670C2×C4—41no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—44no—
674C2×C4—42no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—42no—
679C2×C4—44no—
680C2×C4—42no—
681C2×C4C2×#12014yes—
682C2×C4C2×#12014yes—
683C2×C4C2×#12014yes—
684C2×C4C2×C22⋊Q818yes—
685C2×C4C2×C22⋊Q814yes—
686C2×C4#53614yes—
687C2×C4#54814yes—
688C2×C4#54814yes—
689C2×C4D4×Q814yes—
690C2×C4D4×Q814yes—
691C22—42no—
692C22—44no—
693C22—42no—
694C22—44no—
695C22—42no—
696C22—82no—
697C22—81no—
698C22—81no—
699C22—81no—
700C22—81no—
701C22—82no—
702C22C22×#12011yescharacteristic
703C22C22×C22⋊Q811yescharacteristic
704C22C2×#53611yescharacteristic
705C22#344912yes—
706C22C2×#54811yescharacteristic
707C22C2×#54811yescharacteristic
708C22C2×D4×Q811yescharacteristic
709C22C2×D4×Q811yescharacteristic
710C22#348512yes—
711C22#348711yescharacteristic
712C22#348711yescharacteristic
713C22#348912yes—
714C22#349712yes—
715C22#349911yescharacteristic
716C22#350011yescharacteristic
717C22#350212yes—
718C22#350412yes—
719C22#351612yes—
720C22#352412yes—
721C22#355512yes—
722C22#355712yes—
723C22#356312yes—
724C22#356512yes—
725C4—24no—
726C4—24no—
727C4—28no—
728C4—44no—
729C4—44no—
730C4—44no—
731C4—48no—
732C4—42no—
733C4—44no—
734C4—44no—
735C4—84no—
736C4—81no—
737C4—81no—
738C4—82no—
739C4—82no—
740C4#240918yes—
741C2—82no—
742C2C2×#240911yescharacteristic
743C2#3514911yescharacteristic
744C2#3566011yescharacteristic
745C2#3584511yescharacteristic
746C2#3604011yescharacteristic
747C2#3911912yes—
748C2#3959112yes—
749C2#4287511yescharacteristic
750C2#4287911yescharacteristic
751C2#4509912yes—
752C2#4510212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#557C2semidirect
5. #16603C2semidirect
7. #15847C2semidirect
9. #15218C2semidirect
11. #14876C2semidirect
13. #14161C2semidirect
14. #14017C2semidirect
16. #13942C2semidirect
17. #13925C2semidirect
108. C2×C22⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5572
C2×#26282
C2×#23962
C2×#23232
#166032
#166022
#158472
#158462
#152182
#152162
#148762
#148752
#141612
#140172
#139692
#139422
#139252