Finite Group Databasebeta

← #523869

Subgroups — #523869

Order 512.

2594 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic
3C2×#2356C211yesmaximal, characteristic
4#14714C211yesmaximal, characteristic
5#14570C214yesmaximal
6#14387C214yesmaximal
7#14274C211yesmaximal, characteristic
8#14274C211yesmaximal, characteristic
9#14179C214yesmaximal
10#14089C214yesmaximal
11#14024C211yesmaximal, characteristic
12#14024C211yesmaximal, characteristic
13#14003C211yesmaximal, characteristic
14#14000C211yesmaximal, characteristic
15#13909C211yesmaximal, characteristic
16#13786C211yesmaximal, characteristic
17#13786C211yesmaximal, characteristic
18C4×#391C211yesmaximal, characteristic
19#13759C211yesmaximal, characteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25C22×C4×Q8C2211yescharacteristic
26#2546C4116yescomplemented
27C2×#398C2214yes—
28#2365—28no—
29#2356—28no—
30C2×#391C2211yescharacteristic
31C2×#391C2211yescharacteristic
32C2×#391C2211yescharacteristic
33C2×#391C2211yescharacteristic
34C2×#391C2211yescharacteristic
35C2×#391C2211yescharacteristic
36C2×#391C2211yescharacteristic
37C2×#391C2211yescharacteristic
38C2×#391C2211yescharacteristic
39C2×#391C2211yescharacteristic
40C2×#391C2211yescharacteristic
41C2×#391C2211yescharacteristic
42C2×#389C2212yes—
43C2×#389C2214yes—
44C2×#387C2212yes—
45C2×#387C2212yes—
46C2×#387C2212yes—
47C2×#387C2212yes—
48C2×#384C2211yescharacteristic
49C2×#383C2212yes—
50C2×#382C2211yescharacteristic
51C2×#382C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2212yes—
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×#376C2212yes—
59C2×#376C2211yescharacteristic
60C2×#376C2211yescharacteristic
61C2×C43C2211yescharacteristic
62#1445C2214yes—
63#1445C2214yes—
64#1445C2214yes—
65#1445C2214yes—
66#1443C2214yes—
67#1442C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1436C2212yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2212yes—
76#1436C2214yes—
77#1436C2214yes—
78#1436C2214yes—
79#1436C2212yes—
80#1436C2214yes—
81#1436C2212yes—
82C4⋊#95C2214yes—
83C4⋊#95C2214yes—
84C4⋊#95C2211yescharacteristic
85C4⋊#95C2211yescharacteristic
86C4⋊#95C2211yescharacteristic
87C4⋊#95C2211yescharacteristic
88#1434C2214yes—
89#1434C2214yes—
90#1434C2214yes—
91C4×#95C2214yes—
92C4×#95C2211yescharacteristic
93C4×#95C2211yescharacteristic
94C23×Q8C2311yescharacteristic
95C23×Q8C2311yescharacteristic
96C23×Q8C2311yescharacteristic
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8—42no—
100C2×C4⋊Q8—42no—
101C2×C4⋊Q8—42no—
102C2×C4⋊Q8—42no—
103C2×C4⋊Q8C2×C418yes—
104C2×C4⋊Q8C2×C418yes—
105C2×C4⋊Q8C2×C418yes—
106C2×C4×Q8—24no—
107C2×C4×Q8—24no—
108C2×C4×Q8—24no—
109C2×C4×Q8—24no—
110C2×C4×Q8—24no—
111C2×C4×Q8—24no—
112C2×C4×Q8—42no—
113C2×C4×Q8—42no—
114C2×C4×Q8C2×C418yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2314yes—
128C22×C4⋊C4C2314yes—
129C22×C4⋊C4C2314yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2312yes—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2312yes—
143C22×C42C2312yes—
144C22×C42C2312yes—
145C22×C42C2311yescharacteristic
146#398C2×C4132yes—
147#391—24no—
148#391—24no—
149#391—24no—
150#391—24no—
151#391—24no—
152#391—24no—
153#391—24no—
154#391—24no—
155#391—42no—
156#391—42no—
157#391C2×C418yes—
158#391C2×C418yes—
159#389—28no—
160#389C2×C4132yes—
161#387—28no—
162#387—28no—
163#387—28no—
164#387—28no—
165#384—42no—
166#383—28no—
167#382—24no—
168#382—42no—
169C4×C4⋊C4—28no—
170C4×C4⋊C4—24no—
171C4×C4⋊C4—24no—
172C4×C4⋊C4—24no—
173C4×C4⋊C4C2×C418yes—
174C4×C4⋊C4D418yes—
175#376—24no—
176#376—28no—
177#376—42no—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C2×#95C2312yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C2×#95C2312yes—
198C2×#95C2312yes—
199C2×#95C2312yes—
200C2×#95C2312yes—
201C2×#95C2314yes—
202C2×#95C2312yes—
203C2×#95C2314yes—
204C2×#95C2312yes—
205C2×#95C2314yes—
206C2×#95C2314yes—
207C2×#95C2314yes—
208C2×#95C2312yes—
209C2×#95C2314yes—
210C2×#95C2312yes—
211C2×#95C2314yes—
212C2×#95C2314yes—
213C2×#95C2311yescharacteristic
214C2×#95C2311yescharacteristic
215C43—42no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—22no—
229C22×Q8—22no—
230C22×Q8—22no—
231C22×Q8—22no—
232C22×Q8—22no—
233C22×Q8—22no—
234C22×Q8C22×C414yes—
235C22×Q8C22×C414yes—
236C22×Q8C22×C414yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253C23×C4C2412yes—
254C23×C4C2411yescharacteristic
255C23×C4C2411yescharacteristic
256C4⋊Q8—232no—
257C4⋊Q8—416no—
258C4⋊Q8—416no—
259C4×Q8—232no—
260C4×Q8—416no—
261C4×Q8—416no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4C22×C414yes—
299C2×C4⋊C4C22×C418yes—
300C2×C4⋊C4C22×C418yes—
301C2×C4⋊C4C22×C418yes—
302C2×C4⋊C4C22×C418yes—
303C2×C4⋊C4C22×C4116yes—
304C2×C4⋊C4C22×C4116yes—
305C2×C4⋊C4C2×D414yes—
306C2×C4⋊C4C2×D414yes—
307C2×C4⋊C4#4114yes—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—28no—
334C2×C42—42no—
335C2×C42—42no—
336C2×C42—42no—
337C2×C42—42no—
338C2×C42—44no—
339C2×C42—42no—
340C2×C42—44no—
341C2×C42C22×C418yes—
342C2×C42C22×C414yes—
343C2×C42C22×C414yes—
344C2×C42C22×C414yes—
345C2×C42C2×D414yes—
346C2×C42C2×D414yes—
347C2×C42C2×D414yes—
348C2×C42#4114yes—
349C2×C42#4114yes—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—28no—
366#95—28no—
367#95—28no—
368#95—28no—
369#95—28no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—44no—
374#95—42no—
375#95—44no—
376#95—44no—
377#95—42no—
378#95—42no—
379#95—44no—
380#95—44no—
381#95—44no—
382#95C22×C418yes—
383#95C22×C4116yes—
384#95C22×C4116yes—
385#95C2×D414yes—
386#95C2×D414yes—
387C24C2511yescharacteristic
388C2×Q8—28no—
389C2×Q8—28no—
390C2×Q8—28no—
391C2×Q8—28no—
392C2×Q8—28no—
393C2×Q8—28no—
394C2×Q8—28no—
395C2×Q8—28no—
396C2×Q8—28no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—44no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
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—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
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—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—22no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4C23×C414yes—
493C22×C4C23×C414yes—
494C22×C4C23×C414yes—
495C22×C4C23×C414yes—
496C22×C4C23×C414yes—
497C22×C4C23×C412yes—
498C22×C4C23×C412yes—
499C22×C4C23×C414yes—
500C22×C4C23×C412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C2×#4114yes—
509C22×C4C2×#4114yes—
510C22×C4C2×#4112yes—
511C22×C4C2×#4112yes—
512C22×C4C2×#4112yes—
513C22×C4C2×#4112yes—
514C22×C4C2×#4112yes—
515C22×C4H(5,𝔽2)12yes—
516C22×C4H(5,𝔽2)12yes—
517C22×C42−1+414yes—
518C22×C42−1+414yes—
519C22×C42−1+414yes—
520C4⋊C4—28no—
521C4⋊C4—216no—
522C4⋊C4—216no—
523C4⋊C4—216no—
524C4⋊C4—216no—
525C4⋊C4—28no—
526C4⋊C4—28no—
527C4⋊C4—416no—
528C4⋊C4—416no—
529C4⋊C4—44no—
530C4⋊C4—48no—
531C4⋊C4—48no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—88no—
536C42—216no—
537C42—28no—
538C42—28no—
539C42—28no—
540C42—28no—
541C42—28no—
542C42—48no—
543C42—48no—
544C42—48no—
545C42—48no—
546C42—48no—
547C42—48no—
548C42—48no—
549C42—44no—
550C42C4×D4116yes—
551C23C24×C411yescharacteristic
552C23C23×D411yescharacteristic
553C23C22×#4111yescharacteristic
554C23C22×#4111yescharacteristic
555C23C22×#4111yescharacteristic
556C23C2×H(5,𝔽2)11yescharacteristic
557C23C2×H(5,𝔽2)11yescharacteristic
558C23C2×2−1+412yes—
559C23C2×2−1+412yes—
560C23C2×2−1+412yes—
561C23#58512yes—
562Q8—416no—
563Q8—416no—
564Q8—416no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—28no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
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—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4C2×C4×D414yes—
666C2×C4C2×C4×D414yes—
667C2×C4C2×C4×D414yes—
668C2×C4C4×#4114yes—
669C2×C4#51814yes—
670C2×C4#51918yes—
671C2×C4#53614yes—
672C2×C4#54614yes—
673C2×C4#55014yes—
674C2×C4#55314yes—
675C2×C4#55514yes—
676C22C22×C4×D411yescharacteristic
677C22C2×C4×#4111yescharacteristic
678C22C2×#51811yescharacteristic
679C22C2×#51912yes—
680C22C4×2−1+412yes—
681C22C2×#53611yescharacteristic
682C22#344911yescharacteristic
683C22#346112yes—
684C22C2×#54611yescharacteristic
685C22C2×#55011yescharacteristic
686C22C2×#55311yescharacteristic
687C22C2×#55511yescharacteristic
688C22#349911yescharacteristic
689C22#349911yescharacteristic
690C22#350712yes—
691C22#350912yes—
692C22#350912yes—
693C22#351512yes—
694C22#352312yes—
695C22#353812yes—
696C22#354112yes—
697C22#355212yes—
698C22#356112yes—
699C4—24no—
700C4—24no—
701C4—28no—
702C4—24no—
703C4—44no—
704C4—44no—
705C4—44no—
706C4—44no—
707C4—44no—
708C4—44no—
709C4—42no—
710C4—42no—
711C4—42no—
712C4—44no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4#236318yes—
718C2C4×#53611yescharacteristic
719C2C2×#236311yescharacteristic
720C2#3420911yescharacteristic
721C2#3432812yes—
722C2#3447512yes—
723C2#3808111yescharacteristic
724C2#3808811yescharacteristic
725C2#3828712yes—
726C2#3974011yescharacteristic
727C2#3974411yescharacteristic
728C2#3981912yes—
subgroupG/H
center, Frattini, socle, prime radical, cosocleC24C25
commutatorC23C24×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
26. #2546C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
C2×#23652
C2×#23562
#147142
#145702
#143872
#142742
#141792
#140892
#140242
#140032
#140002
#139092
#137862
C4×#3912
#137592