Finite Group Databasebeta

← #515593

Subgroups — #515593

Order 512.

2638 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2327C211yesmaximal, characteristic
2C2×#2320C211yesmaximal, characteristic
3C2×#2276C211yesmaximal, characteristic
4#14147C214yesmaximal
5#14144C214yesmaximal
6#14051C211yesmaximal, characteristic
7#14050C211yesmaximal, characteristic
8#14024C211yesmaximal, characteristic
9#14024C211yesmaximal, characteristic
10#14003C211yesmaximal, characteristic
11#14000C211yesmaximal, characteristic
12#13989C214yesmaximal
13#13985C214yesmaximal
14#13978C211yesmaximal, characteristic
15#13976C211yesmaximal, characteristic
16#13779C211yesmaximal, characteristic
17C4×#391C211yesmaximal, characteristic
18C4×#391C211yesmaximal, characteristic
19#13757C211yesmaximal, characteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25#2327C4116yescomplemented
26#2320—28no—
27C2×#391C2211yescharacteristic
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#391C2211yescharacteristic
32C2×#391C2211yescharacteristic
33C2×#391C2211yescharacteristic
34C2×#391C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#387C2212yes—
38C2×#387C2212yes—
39C2×#387C2214yes—
40C2×#387C2214yes—
41C2×#384C2211yescharacteristic
42C2×#383C2212yes—
43C2×#382C2212yes—
44C2×#382C2211yescharacteristic
45C2×#382C2211yescharacteristic
46#2276—28no—
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×C4⋊C4C2211yescharacteristic
54C2×#376C2212yes—
55C2×#376C2211yescharacteristic
56C2×#376C2211yescharacteristic
57C2×C43C2211yescharacteristic
58#1444C2214yes—
59#1443C2214yes—
60#1443C2214yes—
61#1442C2214yes—
62#1442C2214yes—
63#1442C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76C4⋊#95C2211yescharacteristic
77C4⋊#95C2211yescharacteristic
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2211yescharacteristic
82#1434C2212yes—
83#1434C2214yes—
84#1434C2214yes—
85#1434C2212yes—
86#1434C2212yes—
87#1434C2212yes—
88C4×#95C2214yes—
89C4×#95C2211yescharacteristic
90C4×#95C2211yescharacteristic
91C4×#95C2214yes—
92C4×#95C2211yescharacteristic
93C4×#95C2211yescharacteristic
94C23×Q8C2311yescharacteristic
95C23×Q8C2311yescharacteristic
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8C2×C418yes—
99C2×C4×Q8—24no—
100C2×C4×Q8—24no—
101C2×C4×Q8—24no—
102C2×C4×Q8—24no—
103C2×C4×Q8—24no—
104C2×C4×Q8—24no—
105C2×C4×Q8—24no—
106C2×C4×Q8—24no—
107C2×C4×Q8—24no—
108C2×C4×Q8—24no—
109C2×C4×Q8C2×C418yes—
110C2×C4×Q8C2×C418yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138C22×C42C2312yes—
139C22×C42C2312yes—
140C22×C42C2312yes—
141C22×C42C2311yescharacteristic
142#391—24no—
143#391—24no—
144#391—24no—
145#391—24no—
146#391—24no—
147#391—24no—
148#391C2×C418yes—
149#391C2×C418yes—
150#389—24no—
151#389—24no—
152#387—28no—
153#387—28no—
154#387C2×C4132yes—
155#387C2×C4132yes—
156#384—24no—
157#383—28no—
158#382—24no—
159#382—28no—
160#382C2×C418yes—
161C4×C4⋊C4—24no—
162C4×C4⋊C4—24no—
163C4×C4⋊C4—24no—
164C4×C4⋊C4—24no—
165C4×C4⋊C4—28no—
166C4×C4⋊C4—28no—
167C4×C4⋊C4—28no—
168#376—28no—
169#376C2×C418yes—
170#376D418yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C2×#95C2312yes—
195C2×#95C2312yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2312yes—
201C2×#95C2312yes—
202C2×#95C2314yes—
203C2×#95C2314yes—
204C2×#95C2314yes—
205C2×#95C2311yescharacteristic
206C2×#95C2311yescharacteristic
207C2×#95C2311yescharacteristic
208C2×#95C2311yescharacteristic
209C43—24no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8C22×C414yes—
221C22×Q8C22×C414yes—
222C22×Q8#4114yes—
223C22×Q8#4114yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C4⋊Q8—232no—
244C4×Q8—232no—
245C4×Q8—232no—
246C4×Q8—232no—
247C4×Q8—232no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4C22×C414yes—
282C2×C4⋊C4C22×C414yes—
283C2×C4⋊C4C22×C418yes—
284C2×C4⋊C4C22×C418yes—
285C2×C4⋊C4C22×C4116yes—
286C2×C4⋊C4#4114yes—
287C2×C4⋊C4#4114yes—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—42no—
321C2×C42—42no—
322C2×C42—44no—
323C2×C42—44no—
324C2×C42—44no—
325C2×C42—44no—
326C2×C42C22×C418yes—
327C2×C42C22×C418yes—
328C2×C42C22×C414yes—
329C2×C42C22×C414yes—
330C2×C42C22×C414yes—
331C2×C42C2×D414yes—
332C2×C42C2×D414yes—
333C2×C42C2×D414yes—
334C2×C42#4114yes—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—28no—
355#95—28no—
356#95—42no—
357#95—42no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—42no—
362#95—44no—
363#95—42no—
364#95C22×C418yes—
365#95C22×C418yes—
366#95C22×C4116yes—
367#95C22×C4116yes—
368#95C22×C4116yes—
369#95C2×D414yes—
370#95C2×D414yes—
371#95C2×D414yes—
372#95C2×D414yes—
373C24C2511yescharacteristic
374C2×Q8—28no—
375C2×Q8—28no—
376C2×Q8—28no—
377C2×Q8—28no—
378C2×Q8—28no—
379C2×Q8—28no—
380C2×Q8—28no—
381C2×Q8—28no—
382C2×Q8—28no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8#117116yes—
388C22×C4—22no—
389C22×C4—22no—
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—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
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—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4C23×C414yes—
469C22×C4C23×C414yes—
470C22×C4C23×C414yes—
471C22×C4C23×C414yes—
472C22×C4C23×C414yes—
473C22×C4C23×C414yes—
474C22×C4C23×C412yes—
475C22×C4C23×C412yes—
476C22×C4C23×C412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C2×#4114yes—
485C22×C4C2×#4114yes—
486C22×C4C2×#4114yes—
487C22×C4C2×#4114yes—
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C4C2×#4112yes—
493C22×C4C2×#4112yes—
494C22×C4C2×#4112yes—
495C22×C42−1+414yes—
496C22×C42−1+414yes—
497C4⋊C4—216no—
498C4⋊C4—216no—
499C4⋊C4—232no—
500C4⋊C4—216no—
501C4⋊C4—28no—
502C4⋊C4—28no—
503C4⋊C4—216no—
504C4⋊C4—28no—
505C4⋊C4—28no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C4⋊C4—88no—
511C42—216no—
512C42—216no—
513C42—28no—
514C42—216no—
515C42—216no—
516C42—28no—
517C42—28no—
518C42—28no—
519C42—48no—
520C42—48no—
521C42—44no—
522C42—44no—
523C42—84no—
524C42—84no—
525C42—84no—
526C42—84no—
527C42C4×D4116yes—
528C23C24×C411yescharacteristic
529C23C23×D411yescharacteristic
530C23C22×#4111yescharacteristic
531C23C22×#4111yescharacteristic
532C23C22×#4111yescharacteristic
533C23C22×#4111yescharacteristic
534C23C22×#4111yescharacteristic
535C23C2×2−1+412yes—
536C23C2×2−1+412yes—
537C23#58512yes—
538C23#58512yes—
539Q8—232no—
540Q8—416no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
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—44no—
613C2×C4—42no—
614C2×C4—42no—
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—44no—
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—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4C2×#11718yes—
641C2×C4C2×#11714yes—
642C2×C4C2×C4×D414yes—
643C2×C4C2×C4×D414yes—
644C2×C4C2×C4×D414yes—
645C2×C4C4×#4114yes—
646C2×C4#52514yes—
647C2×C4#52514yes—
648C2×C4#52914yes—
649C2×C4#52914yes—
650C22C22×#11711yescharacteristic
651C22C22×C4×D411yescharacteristic
652C22C2×C4×#4111yescharacteristic
653C22C4×2−1+412yes—
654C22C4×2−1+412yes—
655C22C2×#52511yescharacteristic
656C22C2×#52511yescharacteristic
657C22C2×#52911yescharacteristic
658C22C2×#52911yescharacteristic
659C22#349512yes—
660C22#349512yes—
661C22#349911yescharacteristic
662C22#349911yescharacteristic
663C22#350011yescharacteristic
664C22#350011yescharacteristic
665C22#351512yes—
666C22#351612yes—
667C22#351612yes—
668C22#351812yes—
669C22#352112yes—
670C22#352312yes—
671C22#352312yes—
672C22#352412yes—
673C4—28no—
674C4—24no—
675C4—24no—
676C4—42no—
677C4—44no—
678C4—42no—
679C4—44no—
680C4—44no—
681C4—44no—
682C4—42no—
683C4—42no—
684C4—44no—
685C4—44no—
686C4—44no—
687C4—44no—
688C4—44no—
689C4—82no—
690C4—82no—
691C4#229018yes—
692C2C2×#229011yescharacteristic
693C2#3426212yes—
694C2#3427611yescharacteristic
695C2#3427611yescharacteristic
696C2#3428612yes—
697C2#3562211yescharacteristic
698C2#3562311yescharacteristic
699C2#3564512yes—
700C2#3564612yes—
701C2#3566911yescharacteristic
702C2#3567011yescharacteristic
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
25. #2327C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23272
C2×#23202
C2×#22762
#141472
#141442
#140512
#140502
#140242
#140032
#140002
#139892
#139852
#139782
#139762
#137792
C4×#3912
#137572