Finite Group Databasebeta

← #523518

Subgroups — #523518

Order 512.

2586 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2565C211yesmaximal, characteristic
2C2×#2370C211yesmaximal, characteristic
3C2×Q8×C4⋊C4C211yesmaximal, characteristic
4#14712C211yesmaximal, characteristic
5#14548C214yesmaximal
6#14406C214yesmaximal
7#14283C211yesmaximal, characteristic
8#14277C211yesmaximal, characteristic
9#14211C214yesmaximal
10#14056C214yesmaximal
11#14051C211yesmaximal, characteristic
12#14049C211yesmaximal, characteristic
13#13976C211yesmaximal, characteristic
14#13976C211yesmaximal, characteristic
15#13910C211yesmaximal, characteristic
16#13786C211yesmaximal, characteristic
17#13786C211yesmaximal, characteristic
18C4×#391C211yesmaximal, characteristic
19#13759C211yesmaximal, characteristic
20C22×#125C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24#2565C4116yescomplemented
25C2×#398C2214yes—
26#2370—28no—
27Q8×C4⋊C4—28no—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#391C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2212yes—
34C2×#389C2211yescharacteristic
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#389C2214yes—
40C2×#389C2214yes—
41C2×#387C2212yes—
42C2×#387C2212yes—
43C2×#387C2214yes—
44C2×#387C2214yes—
45C2×#384C2212yes—
46C2×#384C2211yescharacteristic
47C2×#384C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C2×#384C2211yescharacteristic
50C2×#382C2211yescharacteristic
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×#376C2211yescharacteristic
59C2×#376C2211yescharacteristic
60C2×C43C2211yescharacteristic
61#1445C2214yes—
62#1444C2214yes—
63#1444C2214yes—
64#1444C2214yes—
65#1444C2214yes—
66#1442C2214yes—
67#1442C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1436C2212yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2212yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2212yes—
78#1436C2212yes—
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2214yes—
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2211yescharacteristic
84C4⋊#95C2211yescharacteristic
85C4⋊#95C2211yescharacteristic
86C4⋊#95C2211yescharacteristic
87C4⋊#95C2211yescharacteristic
88#1434C2214yes—
89#1434C2214yes—
90#1434C2214yes—
91C4×#95C2214yes—
92C4×#95C2211yescharacteristic
93C4×#95C2211yescharacteristic
94C23×Q8C2311yescharacteristic
95C2×#125—24no—
96C2×#125—24no—
97C2×#125C2×C418yes—
98C2×C4×Q8—24no—
99C2×C4×Q8—24no—
100C2×C4×Q8—24no—
101C2×C4×Q8—24no—
102C2×C4×Q8—24no—
103C2×C4×Q8—24no—
104C2×C4×Q8—42no—
105C2×C4×Q8—42no—
106C2×C4×Q8C2×C418yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
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#391C2×C418yes—
149#391C2×C418yes—
150#391Q818yes—
151#389—28no—
152#389—24no—
153#389—24no—
154#389—24no—
155#389—24no—
156#389—28no—
157#389—28no—
158#389—48no—
159#389—48no—
160#387—28no—
161#387—28no—
162#387—48no—
163#387C2×C4132yes—
164#384—28no—
165#384—24no—
166#384—42no—
167#384—42no—
168#384C2×C418yes—
169#382C2×C418yes—
170C4×C4⋊C4—28no—
171C4×C4⋊C4—24no—
172C4×C4⋊C4—24no—
173C4×C4⋊C4—24no—
174C4×C4⋊C4—28no—
175C4×C4⋊C4C2×C418yes—
176C4×C4⋊C4D418yes—
177#376—24no—
178#376—42no—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2314yes—
194C2×#95C2312yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2312yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2314yes—
202C2×#95C2312yes—
203C2×#95C2314yes—
204C2×#95C2314yes—
205C2×#95C2314yes—
206C2×#95C2312yes—
207C2×#95C2311yescharacteristic
208C2×#95C2311yescharacteristic
209C2×#95C2314yes—
210C43—42no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8C22×C414yes—
216C22×Q8C2×Q814yes—
217C22×Q8#4114yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237#125—232no—
238C4×Q8—232no—
239C4×Q8—416no—
240C4×Q8—416no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4C22×C414yes—
295C2×C4⋊C4C22×C418yes—
296C2×C4⋊C4C22×C414yes—
297C2×C4⋊C4C22×C414yes—
298C2×C4⋊C4C22×C418yes—
299C2×C4⋊C4C22×C418yes—
300C2×C4⋊C4C22×C418yes—
301C2×C4⋊C4C22×C4116yes—
302C2×C4⋊C4C2×D414yes—
303C2×C4⋊C4C2×D414yes—
304C2×C4⋊C4C2×Q814yes—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—28no—
328C2×C42—22no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—42no—
332C2×C42—42no—
333C2×C42—42no—
334C2×C42—42no—
335C2×C42—42no—
336C2×C42—44no—
337C2×C42—44no—
338C2×C42C22×C418yes—
339C2×C42C22×C414yes—
340C2×C42C22×C414yes—
341C2×C42C22×C414yes—
342C2×C42C2×D414yes—
343C2×C42C2×D414yes—
344C2×C42C2×D414yes—
345C2×C42C2×Q814yes—
346C2×C42#4114yes—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—28no—
357#95—24no—
358#95—28no—
359#95—24no—
360#95—24no—
361#95—44no—
362#95—44no—
363#95—42no—
364#95—42no—
365#95—42no—
366#95—44no—
367#95—44no—
368#95—44no—
369#95—44no—
370#95—44no—
371#95C22×C418yes—
372#95C22×C4116yes—
373#95C22×C4116yes—
374#95C22×C4116yes—
375#95C2×D414yes—
376#95C2×D414yes—
377#95C2×Q8116yes—
378C24C2511yescharacteristic
379C2×Q8—28no—
380C2×Q8—28no—
381C2×Q8—28no—
382C2×Q8—28no—
383C2×Q8—28no—
384C2×Q8—28no—
385C2×Q8C4×Q8116yes—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4C23×C414yes—
464C22×C4C23×C414yes—
465C22×C4C23×C414yes—
466C22×C4C23×C414yes—
467C22×C4C23×C412yes—
468C22×C4C23×C412yes—
469C22×C4C23×C414yes—
470C22×C4C23×C414yes—
471C22×C4C23×C412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×Q814yes—
480C22×C4C22×Q814yes—
481C22×C4C22×Q814yes—
482C22×C4C22×Q812yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4114yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4H(5,𝔽2)14yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C42−1+414yes—
491C22×C42−1+414yes—
492C22×C42−1+412yes—
493C4⋊C4—216no—
494C4⋊C4—28no—
495C4⋊C4—216no—
496C4⋊C4—216no—
497C4⋊C4—28no—
498C4⋊C4—28no—
499C4⋊C4—216no—
500C4⋊C4—232no—
501C4⋊C4—28no—
502C4⋊C4—28no—
503C4⋊C4—44no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—84no—
512C4⋊C4—88no—
513C4⋊C4—84no—
514C4⋊C4—84no—
515C4⋊C4—84no—
516C4⋊C4—88no—
517C42—216no—
518C42—28no—
519C42—28no—
520C42—28no—
521C42—28no—
522C42—28no—
523C42—48no—
524C42—48no—
525C42—48no—
526C42—48no—
527C42—48no—
528C42—44no—
529C42—84no—
530C42—84no—
531C42C4×D4116yes—
532C23C24×C411yescharacteristic
533C23C23×D411yescharacteristic
534C23C23×Q811yescharacteristic
535C23C22×#4111yescharacteristic
536C23C22×#4111yescharacteristic
537C23C2×H(5,𝔽2)11yescharacteristic
538C23C2×H(5,𝔽2)12yes—
539C23C2×2−1+411yescharacteristic
540C23C2×2−1+412yes—
541C23C2×2−1+412yes—
542C23#58512yes—
543Q8—232no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—28no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
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—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4C2×C4×D414yes—
640C2×C4C2×C4×D414yes—
641C2×C4C2×C4×D414yes—
642C2×C4C2×C4×Q818yes—
643C2×C4C2×C4×Q814yes—
644C2×C4#52018yes—
645C2×C4#52014yes—
646C2×C4#53614yes—
647C2×C4#54614yes—
648C2×C4D4×Q814yes—
649C2×C4#55214yes—
650C2×C4#55414yes—
651C22C22×C4×D411yescharacteristic
652C22C22×C4×Q811yescharacteristic
653C22C2×#52012yes—
654C22C2×#52011yescharacteristic
655C22C4×2−1+412yes—
656C22C2×#53611yescharacteristic
657C22#344911yescharacteristic
658C22#346112yes—
659C22C2×#54611yescharacteristic
660C22C2×D4×Q811yescharacteristic
661C22C2×#55211yescharacteristic
662C22C2×#55411yescharacteristic
663C22#349811yescharacteristic
664C22#349911yescharacteristic
665C22#350212yes—
666C22#350312yes—
667C22#351112yes—
668C22#351712yes—
669C22#352312yes—
670C22#353812yes—
671C22#354212yes—
672C22#354412yes—
673C22#356312yes—
674C4—24no—
675C4—28no—
676C4—24no—
677C4—28no—
678C4—24no—
679C4—44no—
680C4—44no—
681C4—44no—
682C4—44no—
683C4—44no—
684C4—42no—
685C4—44no—
686C4—42no—
687C4—42no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4#236018yes—
693C2C4×#53611yescharacteristic
694C2C2×#236011yescharacteristic
695C2#3420711yescharacteristic
696C2#3433712yes—
697C2#3446512yes—
698C2#3808111yescharacteristic
699C2#3808911yescharacteristic
700C2#3827512yes—
701C2#3961211yescharacteristic
702C2#3961711yescharacteristic
703C2#3968912yes—
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
24. #2565C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25652
C2×#23702
C2×Q8×C4⋊C42
#147122
#145482
#144062
#142832
#142772
#142112
#140562
#140512
#140492
#139762
#139102
#137862
C4×#3912
#137592