Finite Group Databasebeta

← #544451

Subgroups — #544451

Order 512.

2552 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2396C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#2314C211yesmaximal, characteristic
4#14628C212yesmaximal
5#14556C214yesmaximal
6#14550C214yesmaximal, complemented
7#14340C211yesmaximal, characteristic
8#14326C212yesmaximal, complemented
9#14302C212yesmaximal, complemented
10#14188C212yesmaximal
11#13925C211yesmaximal, characteristic, complemented
12#13911C211yesmaximal, characteristic, complemented
13#13909C212yesmaximal, complemented
14#13884C212yesmaximal, complemented
15#13876C212yesmaximal
16#13875C212yesmaximal
17#13804C211yesmaximal, characteristic, complemented
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C22×#117C2211yescharacteristic
21#2396C4116yescomplemented
22C2×#400C2211yescharacteristic
23C2×#399C2211yescharacteristic
24C2×#398C2212yes—
25C2×#396C2212yes—
26C2×#395C2211yescharacteristic
27C2×#393C2211yescharacteristic
28Q8×C4⋊C4—28no—
29#2314—28no—
30C2×#389C2212yes—
31C2×#389C2214yes—
32C2×#388C2212yes—
33C2×#388C2212yes—
34C2×#388C2214yes—
35C2×#387C2212yes—
36C2×#387C2212yes—
37C2×#387C2214yes—
38C2×#384C2211yescharacteristic
39C2×#384C2212yes—
40C2×#384C2211yescharacteristic
41C2×#382C2211yescharacteristic
42C2×#382C2211yescharacteristic
43C2×#382C2212yes—
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2212yes—
54#1446C2212yes—
55#1445C2212yes—
56#1443C2214yes—
57#1442C2212yes—
58#1442C2214yes—
59#1442C2214yes—
60#1442C2212yes—
61#1441C2212yes—
62#1441C2212yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1440C2214yes—
67#1439C2212yes—
68#1439C2212yes—
69#1439C2214yes—
70#1437C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2214yes—
83#1434C2212yes—
84#1434C2214yes—
85C4×#95C2212yes—
86C23×Q8C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C2×C4×Q8—24no—
89C2×C4×Q8—28no—
90C2×C4×Q8—24no—
91C2×C4×Q8—48no—
92C2×C4×Q8C2×C418yes—
93C2×#117—24no—
94C2×#117—24no—
95C2×#117C2×C418yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C42C2314yes—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2314yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130#400C2×C418yes—
131#399C2×C418yes—
132#398C2×C4116yes—
133#396C2×C4116yes—
134#395C2×C418yes—
135#393C2×C418yes—
136#389—28no—
137#389—216no—
138#388—28no—
139#388—28no—
140#388—48no—
141#387—28no—
142#387—28no—
143#387—48no—
144#384—24no—
145#384—28no—
146#384—24no—
147#382—24no—
148#382—24no—
149#382—44no—
150#380—24no—
151#380—24no—
152#380—44no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4—44no—
156C4×C4⋊C4C2×C4116yes—
157C4×C4⋊C4C2×C418yes—
158C4×C4⋊C4D418yes—
159C4×C22⋊C4C2×C4116yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2311yescharacteristic
184C2×#95C2312yes—
185C2×#95C2311yescharacteristic
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C25C2411yescharacteristic
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8C22×C414yes—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4C22×C414yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C4×Q8—232no—
219C4×Q8—432no—
220#117—232no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4C22×C414yes—
266C2×C4⋊C4C22×C414yes—
267C2×C4⋊C4C22×C418yes—
268C2×C4⋊C4C22×C414yes—
269C2×C4⋊C4C22×C418yes—
270C2×C4⋊C4C22×C418yes—
271C2×C4⋊C4C22×C418yes—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2×D414yes—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4C22×C414yes—
290C2×C22⋊C4C22×C414yes—
291C2×C22⋊C4C22×C418yes—
292C2×C22⋊C4C22×C418yes—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—28no—
300C2×C42—28no—
301C2×C42—24no—
302C2×C42—44no—
303C2×C42—44no—
304C2×C42—44no—
305C2×C42—44no—
306C2×C42—44no—
307C2×C42—44no—
308C2×C42C22×C414yes—
309C2×C42C22×C418yes—
310C2×C42C22×C418yes—
311C2×C42C22×C414yes—
312C2×C42C2×D418yes—
313C2×C42C2×D414yes—
314C2×C42#4114yes—
315#95—24no—
316#95—28no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—28no—
324#95—28no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95C22×C418yes—
338#95C22×C418yes—
339#95C22×C418yes—
340#95C22×C414yes—
341#95C22×C414yes—
342#95C22×C418yes—
343#95C2×D418yes—
344C24—21no—
345C24—21no—
346C24—22no—
347C24—21no—
348C24—22no—
349C24—21no—
350C24—21no—
351C24—21no—
352C24—22no—
353C24—22no—
354C24C23×C412yes—
355C24C2511yescharacteristic
356C2×Q8—28no—
357C2×Q8—28no—
358C2×Q8—216no—
359C2×Q8—28no—
360C2×Q8—216no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
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—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4C23×C414yes—
441C22×C4C23×C412yes—
442C22×C4C23×C412yes—
443C22×C4C23×C412yes—
444C22×C4C23×C414yes—
445C22×C4C23×C414yes—
446C22×C4C23×C412yes—
447C22×C4C23×C412yes—
448C22×C4C23×C414yes—
449C22×C4C23×C412yes—
450C22×C4C22×D414yes—
451C22×C4C22×D414yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4H(5,𝔽2)14yes—
459C22×C4H(5,𝔽2)12yes—
460C22×C42−1+414yes—
461C22×C42−1+412yes—
462C22×C42−1+412yes—
463C22×C42−1+412yes—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—216no—
467C4⋊C4—28no—
468C4⋊C4—216no—
469C4⋊C4—28no—
470C4⋊C4—28no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—416no—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—84no—
480C4⋊C4—84no—
481C4⋊C4—84no—
482C4⋊C4—88no—
483C22⋊C4—28no—
484C22⋊C4—28no—
485C22⋊C4—216no—
486C22⋊C4—48no—
487C22⋊C4—84no—
488C22⋊C4—84no—
489C22⋊C4—88no—
490C42—28no—
491C42—216no—
492C42—216no—
493C42—48no—
494C42—416no—
495C42—84no—
496C42—88no—
497C42C4×D4116yes—
498C23—22no—
499C23—22no—
500C23—24no—
501C23—22no—
502C23—24no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—42no—
510C23—41no—
511C23—41no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23C24×C411yescharacteristic
522C23C23×D411yescharacteristic
523C23C22×#4111yescharacteristic
524C23C2×H(5,𝔽2)11yescharacteristic
525C23C2×H(5,𝔽2)12yes—
526C23C2×2−1+411yescharacteristic
527C23C2×2−1+411yescharacteristic
528C23C2×2−1+411yescharacteristic
529C23C2×2−1+412yes—
530C23#58512yes—
531C23#58512yes—
532Q8—232no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
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—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—416no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4C2×C4×D414yes—
621C2×C4C2×C4×D414yes—
622C2×C4C2×C4×D414yes—
623C2×C4#51914yes—
624C2×C4#52018yes—
625C2×C4#52014yes—
626C2×C4#53614yes—
627C2×C4#53714yes—
628C2×C4#54114yes—
629C2×C4#54214yes—
630C22—42no—
631C22—44no—
632C22—44no—
633C22—42no—
634C22—42no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—82no—
640C22—82no—
641C22C22×C4×D411yescharacteristic
642C22C2×#51911yescharacteristic
643C22C2×#52011yescharacteristic
644C22C2×#52012yes—
645C22#342812yes—
646C22C2×#53611yescharacteristic
647C22C2×#53711yescharacteristic
648C22C2×#54111yescharacteristic
649C22C2×#54211yescharacteristic
650C22#348411yescharacteristic
651C22#348611yescharacteristic
652C22#349212yes—
653C22#350011yescharacteristic
654C22#350011yescharacteristic
655C22#351212yes—
656C22#354612yes—
657C22#354712yes—
658C22#355412yes—
659C22#355412yes—
660C22#355412yes—
661C22#355512yes—
662C22#355512yes—
663C22#355512yes—
664C4—24no—
665C4—28no—
666C4—24no—
667C4—42no—
668C4—44no—
669C4—44no—
670C4—42no—
671C4—42no—
672C4—44no—
673C4—44no—
674C4—44no—
675C4—82no—
676C4—84no—
677C4—82no—
678C4—84no—
679C4#232318yes—
680C2—82no—
681C2C2×#232311yescharacteristic
682C2#3420011yescharacteristic
683C2#3420811yescharacteristic
684C2#3437312yes—
685C2#3438312yes—
686C2#4242911yescharacteristic
687C2#4243111yescharacteristic
688C2#4245912yes—
689C2#4246112yes—
690C2#4279311yescharacteristic
691C2#4279511yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×Q8×C4⋊C4C2semidirect
6. #14550C2semidirect
8. #14326C2semidirect
9. #14302C2semidirect
11. #13925C2semidirect
12. #13911C2semidirect
13. #13909C2semidirect
14. #13884C2semidirect
17. #13804C2semidirect
21. #2396C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23962
C2×Q8×C4⋊C42
C2×#23142
#146282
#145562
#145502
#143402
#143262
#143022
#141882
#139252
#139112
#139092
#138842
#138762
#138752
#138042