Finite Group Databasebeta

← #874903

Subgroups — #874903

Order 512.

2222 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2C2×#2584C211yesmaximal, characteristic
3#16553C214yesmaximal
4#15852C214yesmaximal
5#15627C214yesmaximal
6#15234C211yesmaximal, characteristic
7#15230C211yesmaximal, characteristic
8#15228C211yesmaximal, characteristic
9#15228C211yesmaximal, characteristic
10#15196C211yesmaximal, characteristic
11#15143C211yesmaximal, characteristic
12#14712C211yesmaximal, characteristic
13#14707C211yesmaximal, characteristic
14#14406C214yesmaximal
15#14051C211yesmaximal, characteristic
16#14050C211yesmaximal, characteristic
17#14049C211yesmaximal, characteristic
18#14049C211yesmaximal, characteristic
19C4×#391C211yesmaximal, characteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×#125C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23#2584—28no—
24#2567—28no—
25C2×#398C2212yes—
26C2×#395C2212yes—
27C2×#395C2212yes—
28C2×#395C2212yes—
29C2×#395C2214yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#389C2212yes—
33C2×#389C2211yescharacteristic
34C2×#389C2212yes—
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2214yes—
43C2×#387C2214yes—
44C2×#387C2214yes—
45C2×#384C2212yes—
46C2×#384C2211yescharacteristic
47C2×#384C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C2×#382C2211yescharacteristic
50C2×#382C2211yescharacteristic
51C2×#382C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C43C2211yescharacteristic
56#1446C2214yes—
57#1445C2214yes—
58#1445C2214yes—
59#1444C2214yes—
60#1444C2214yes—
61#1444C2214yes—
62#1444C2214yes—
63#1443C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2212yes—
74#1436C2214yes—
75#1436C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2214yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2211yescharacteristic
84C4⋊#95C2211yescharacteristic
85C4⋊#95C2211yescharacteristic
86C4⋊#95C2211yescharacteristic
87C4⋊#95C2211yescharacteristic
88#1434C2214yes—
89C4×#95C2211yescharacteristic
90C4×#95C2211yescharacteristic
91C4×#95C2211yescharacteristic
92C4×#95C2211yescharacteristic
93C23×Q8C2311yescharacteristic
94C23×Q8C2312yes—
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8—42no—
97C2×C4⋊Q8—42no—
98C2×#125—24no—
99C2×#125—42no—
100C2×#125—42no—
101C2×C4×Q8—42no—
102C2×C4×Q8—42no—
103C2×C4×Q8D418yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2314yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2312yes—
140C22×C42C2312yes—
141#398—28no—
142#395—28no—
143#395—28no—
144#395—28no—
145#395—48no—
146#391—28no—
147#391—28no—
148#389—28no—
149#389—28no—
150#389—24no—
151#389—24no—
152#389—24no—
153#389—24no—
154#389—24no—
155#389—24no—
156#389—24no—
157#389—24no—
158#389—48no—
159#387—216no—
160#387—48no—
161#384—28no—
162#384—42no—
163#384—42no—
164#384Q818yescomplemented
165#382—42no—
166#382—42no—
167#382Q818yescomplemented
168C4×C4⋊C4—24no—
169C4×C4⋊C4—24no—
170C4×C4⋊C4—42no—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2312yes—
200C2×#95C2314yes—
201C2×#95C2314yes—
202C2×#95C2312yes—
203C43—42no—
204C22×Q8—24no—
205C22×Q8—24no—
206C22×Q8—24no—
207C22×Q8—41no—
208C22×Q8—41no—
209C22×Q8—42no—
210C22×Q8—42no—
211C22×Q8—44no—
212C22×Q8—44no—
213C22×Q8C2×D414yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C4⋊Q8—416no—
234#125—416no—
235C4×Q8—416no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—41no—
275C2×C4⋊C4—41no—
276C2×C4⋊C4—41no—
277C2×C4⋊C4—41no—
278C2×C4⋊C4—41no—
279C2×C4⋊C4—41no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
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⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4C2×D414yes—
299C2×C4⋊C4C2×D414yes—
300C2×C4⋊C4C2×D414yes—
301C2×C4⋊C4C2×Q814yes—
302C2×C4⋊C4C2×Q814yes—
303C2×C4⋊C4C2×Q814yes—
304C2×C4⋊C4C2×Q814yes—
305C2×C4⋊C4C2×Q8116yes—
306C2×C42—22no—
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—24no—
317C2×C42—28no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—42no—
329C2×C42—42no—
330C2×C42—42no—
331C2×C42—44no—
332C2×C42C2×D414yes—
333C2×C42C2×D414yes—
334C2×C42C2×D414yes—
335C2×C42C2×Q814yes—
336C2×C42C2×Q814yes—
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—24no—
346#95—24no—
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—24no—
356#95—42no—
357#95—42no—
358#95—44no—
359#95—42no—
360#95—44no—
361#95—44no—
362#95—44no—
363#95—44no—
364#95—44no—
365#95—44no—
366#95—44no—
367#95C2×Q8116yes—
368#95#4118yes—
369C24C2511yescharacteristic
370C2×Q8—28no—
371C2×Q8—28no—
372C2×Q8—44no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×Q8—84no—
377C2×Q8—84no—
378C2×Q8—88no—
379C2×Q8—88no—
380C22×C4—22no—
381C22×C4—22no—
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—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
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—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
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—24no—
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—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×Q814yes—
469C22×C4C22×Q814yes—
470C22×C4C22×Q814yes—
471C22×C4C22×Q814yes—
472C22×C4C22×Q814yes—
473C22×C4C22×Q814yes—
474C22×C4C22×Q812yes—
475C22×C4C22×Q812yes—
476C22×C4C2×#4114yes—
477C22×C4C2×#4114yes—
478C22×C4C2×#4114yes—
479C22×C4H(5,𝔽2)14yes—
480C22×C4H(5,𝔽2)12yes—
481C22×C4H(5,𝔽2)12yes—
482C22×C42−1+414yes—
483C22×C42−1+412yes—
484C22×C42−1+412yes—
485C4⋊C4—28no—
486C4⋊C4—28no—
487C4⋊C4—28no—
488C4⋊C4—28no—
489C4⋊C4—232no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—48no—
503C4⋊C4—88no—
504C4⋊C4—84no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C4⋊C4—88no—
511C42—28no—
512C42—28no—
513C42—28no—
514C42—28no—
515C42—28no—
516C42—48no—
517C42—48no—
518C42—48no—
519C42—44no—
520C42—44no—
521C42—84no—
522C23C23×D411yescharacteristic
523C23C23×Q811yescharacteristic
524C23C23×Q811yescharacteristic
525C23C22×#4112yes—
526C23C2×H(5,𝔽2)11yescharacteristic
527C23C2×H(5,𝔽2)11yescharacteristic
528C23C2×H(5,𝔽2)12yes—
529C23C2×2−1+412yes—
530C23C2×2−1+411yescharacteristic
531C23C2×2−1+411yescharacteristic
532C23#58512yes—
533Q8—416no—
534Q8—816no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
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—22no—
550C2×C4—22no—
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—28no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
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—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4#53214yes—
631C2×C4#53414yes—
632C2×C4#53614yes—
633C2×C4#54414yes—
634C2×C4#54414yes—
635C2×C4D4×Q814yes—
636C2×C4D4×Q814yes—
637C2×C4#55418yes—
638C2×C4Q8214yes—
639C2×C4#56314yes—
640C22C2×#53211yescharacteristic
641C22C2×#53411yescharacteristic
642C22C2×#53611yescharacteristic
643C22C2×#54411yescharacteristic
644C22C2×#54411yescharacteristic
645C22C2×D4×Q811yescharacteristic
646C22C2×D4×Q811yescharacteristic
647C22C2×#55412yes—
648C22C2×Q8211yescharacteristic
649C22Q8×#4112yes—
650C22#349811yescharacteristic
651C22#350011yescharacteristic
652C22#350212yes—
653C22#350312yes—
654C22#351112yes—
655C22#351512yes—
656C22#352012yes—
657C22#352012yes—
658C22C2×#56311yescharacteristic
659C22#354212yes—
660C22#354412yes—
661C22#354612yes—
662C22#355512yes—
663C4—24no—
664C4—24no—
665C4—28no—
666C4—42no—
667C4—44no—
668C4—44no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—44no—
673C4—44no—
674C4—44no—
675C4—44no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4#273418yes—
682C2#3508511yescharacteristic
683C2#3655911yescharacteristic
684C2#3656211yescharacteristic
685C2#3730011yescharacteristic
686C2#3896312yes—
687C2#3961711yescharacteristic
688C2#3961911yescharacteristic
689C2#3998312yes—
690C2#4617412yes—
691C2C2×#273411yescharacteristic
692C2#5945912yes—
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
164. #384Q8semidirect
167. #382Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#25842
#165532
#158522
#156272
#152342
#152302
#152282
#151962
#151432
#147122
#147072
#144062
#140512
#140502
#140492
C4×#3912