Finite Group Databasebeta

← #524657

Subgroups — #524657

Order 512.

4590 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#555C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#2293C211yesmaximal, characteristic, complemented
4#14714C211yesmaximal, characteristic, complemented
5#14421C218yesmaximal, complemented
6#14257C212yesmaximal, complemented
7#14161C211yesmaximal, characteristic, complemented
8#14137C212yesmaximal, complemented
9#13970C214yesmaximal, complemented
10#13951C214yesmaximal, complemented
11#13939C212yesmaximal, complemented
12#13906C211yesmaximal, characteristic, complemented
13#13795C212yesmaximal, complemented
14C4×#390C211yesmaximal, characteristic, complemented
15C2×#555—28no—
16C2×#555—28no—
17C2×#555C4116yescomplemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#124C2214yes—
20C22×#121C2212yes—
21C22×#121C2212yes—
22C22×C4×D4C2211yescharacteristic
23C22×C4×D4C2212yes—
24C22×C4×D4C2211yescharacteristic
25C22×#117C2212yes—
26C22×#117C2212yes—
27C22×#117C2212yes—
28C23×C4⋊C4C2212yes—
29D4×C4⋊C4—28no—
30C2×#391C2214yes—
31#2293—28no—
32C2×#390C2211yescharacteristic
33C2×#390C2212yes—
34C2×#390C2211yescharacteristic
35C2×#390C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#388C2214yes—
39C2×#386C2218yes—
40C2×#385C2214yes—
41C2×#384C2211yescharacteristic
42C2×#380C2212yes—
43C2×#380C2214yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2214yes—
48C2×C43C2211yescharacteristic
49#1446C2214yes—
50#1440C2218yes—
51#1439C2218yes—
52#1439C2214yes—
53#1439C2218yes—
54#1438C2218yes—
55#1437C2214yes—
56C4⋊#95C2212yes—
57C4⋊#95C2212yes—
58C4⋊#95C2212yes—
59C4⋊#95C2214yes—
60C4⋊#95C2212yes—
61C4⋊#95C2218yes—
62C4⋊#95C2211yescharacteristic
63C4⋊#95C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2211yescharacteristic
66#1434C2218yes—
67C22⋊#95C2218yes—
68C23×Q8C2312yes—
69C23×D4C2311yescharacteristic
70C23×D4C2311yescharacteristic
71C23×D4C2314yes—
72C24×C4C2312yes—
73C24×C4C2312yes—
74C24×C4C2314yes—
75#555—2128no—
76C2×C4⋊Q8—24no—
77C2×C4⋊Q8—24no—
78C2×C4⋊Q8C2×C418yescomplemented
79C2×#124—216no—
80C2×#124—216no—
81C2×#124C2×C4132yescomplemented
82C2×#121—28no—
83C2×#121—28no—
84C2×#121—28no—
85C2×#121—28no—
86C2×#121C2×C4116yes—
87C2×#121C2×C4116yescomplemented
88C2×C4×D4—28no—
89C2×C4×D4—24no—
90C2×C4×D4—24no—
91C2×C4×D4—48no—
92C2×C4×D4C2×C418yes—
93C2×C4×D4C2×C418yescomplemented
94C2×C4×D4D418yes—
95C2×C4×D4D418yes—
96C2×#117—28no—
97C2×#117—28no—
98C2×#117—28no—
99C2×#117—28no—
100C2×#117—28no—
101C2×#117—28no—
102C2×#117—44no—
103C2×#117C2×C4116yescomplemented
104C2×#117C2×C4116yescomplemented
105C22×C4⋊C4—28no—
106C22×C4⋊C4—28no—
107C22×C4⋊C4—44no—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2318yes—
129C22×C22⋊C4C2312yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2314yes—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#391—48no—
139#390—28no—
140#390—24no—
141#390—24no—
142#390—44no—
143#389—28no—
144#389—28no—
145#388—216no—
146#386—416no—
147#385—216no—
148#384—24no—
149#380—28no—
150#380—48no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—24no—
153C4×C22⋊C4—28no—
154C4×C22⋊C4—48no—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2318yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2318yes—
162C2×#95C2318yes—
163C2×#95C2318yes—
164C2×#95C2314yes—
165C2×#95C2318yes—
166C43—24no—
167C25C2412yes—
168C25C2412yes—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—44no—
172C22×Q8—44no—
173C22×Q8C22×C418yes—
174C22×D4—28no—
175C22×D4—28no—
176C22×D4—22no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—24no—
181C22×D4—22no—
182C22×D4—24no—
183C22×D4—22no—
184C22×D4—48no—
185C22×D4—48no—
186C22×D4C22×C414yes—
187C22×D4C22×C414yes—
188C22×D4C22×C4116yes—
189C22×D4C2×D414yes—
190C22×D4C2×D414yes—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—28no—
198C23×C4—28no—
199C23×C4—28no—
200C23×C4—28no—
201C23×C4—44no—
202C23×C4—48no—
203C23×C4—48no—
204C23×C4C22×C418yes—
205C23×C4C22×C418yes—
206C23×C4C2×D418yes—
207C23×C4C2×D418yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C4⋊Q8—232no—
221#124—2128no—
222#121—264no—
223#121—264no—
224C4×D4—232no—
225C4×D4—432no—
226C4×D4C22⋊C4164yescomplemented
227#117—264no—
228#117—264no—
229#117—432no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—432no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4C22×C414yes—
259C2×C4⋊C4C22×C418yes—
260C2×C4⋊C4C22×C418yes—
261C2×C4⋊C4C22×C414yes—
262C2×C4⋊C4C2×D414yes—
263C2×C4⋊C4C2×D414yes—
264C2×C4⋊C4#4114yes—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—48no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—48no—
286C2×C22⋊C4—48no—
287C2×C22⋊C4—48no—
288C2×C22⋊C4C22×C4116yes—
289C2×C22⋊C4C22×C4116yes—
290C2×C22⋊C4C22×C418yes—
291C2×C22⋊C4C22×C418yes—
292C2×C22⋊C4C2×D418yes—
293C2×C22⋊C4C2×D418yes—
294C2×C42—28no—
295C2×C42—28no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—28no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—42no—
312C2×C42C22×C4116yes—
313C2×C42C22×C414yes—
314C2×C42C2×D414yes—
315C2×C42C2×D414yes—
316C2×C42#4114yes—
317#95—28no—
318#95—24no—
319#95—24no—
320#95—28no—
321#95—216no—
322#95—48no—
323#95—44no—
324#95—44no—
325#95—48no—
326#95—48no—
327#95—48no—
328C24—22no—
329C24—22no—
330C24—24no—
331C24—22no—
332C24—24no—
333C24—22no—
334C24—24no—
335C24—24no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—24no—
342C24—22no—
343C24—24no—
344C24—22no—
345C24—24no—
346C24C23×C414yes—
347C24C23×C414yes—
348C24C22×D414yes—
349C24C22×D414yes—
350C24C2511yescharacteristic
351C2×Q8—216no—
352C2×Q8—416no—
353C2×Q8—88no—
354C2×Q8—84no—
355C2×Q8—84no—
356C2×D4—232no—
357C2×D4—28no—
358C2×D4—28no—
359C2×D4—216no—
360C2×D4—28no—
361C2×D4—28no—
362C2×D4—28no—
363C2×D4—216no—
364C2×D4—432no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—816no—
370C2×D4—88no—
371C2×D4—88no—
372C2×D4C2×C22⋊C4116yes—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—216no—
380C22×C4—216no—
381C22×C4—216no—
382C22×C4—216no—
383C22×C4—216no—
384C22×C4—216no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—28no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—28no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—216no—
410C22×C4—232no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—28no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—28no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—416no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—416no—
432C22×C4—416no—
433C22×C4—816no—
434C22×C4—88no—
435C22×C4—816no—
436C22×C4C2×C22⋊C4132yes—
437C22×C4C23×C414yes—
438C22×C4C23×C418yes—
439C22×C4C23×C414yes—
440C22×C4C23×C412yes—
441C22×C4C23×C412yes—
442C22×C4C23×C412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D414yes—
445C22×C4C22×D414yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C2×#4114yes—
452C22×C4C2×#4114yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4H(5,𝔽2)14yes—
456C22×C4H(5,𝔽2)14yes—
457C22×C4H(5,𝔽2)12yes—
458C22×C42−1+414yes—
459C22×C42−1+412yes—
460C4⋊C4—28no—
461C4⋊C4—216no—
462C4⋊C4—216no—
463C4⋊C4—28no—
464C4⋊C4—416no—
465C4⋊C4—416no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—44no—
470C4⋊C4—84no—
471C4⋊C4C2×C22⋊C4116yes—
472C22⋊C4—232no—
473C22⋊C4—232no—
474C22⋊C4—216no—
475C22⋊C4—416no—
476C22⋊C4—416no—
477C22⋊C4—416no—
478C22⋊C4—416no—
479C22⋊C4—416no—
480C22⋊C4—816no—
481C22⋊C4C2×C22⋊C4132yes—
482C42—232no—
483C42—28no—
484C42—48no—
485C42—48no—
486C42—48no—
487C42—48no—
488C42—44no—
489C42C2×C22⋊C4116yes—
490C23—24no—
491C23—24no—
492C23—28no—
493C23—28no—
494C23—24no—
495C23—24no—
496C23—28no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—28no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—28no—
507C23—24no—
508C23—28no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—42no—
518C23—42no—
519C23—44no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23C22×C22⋊C418yes—
536C23C24×C411yescharacteristic
537C23C23×D411yescharacteristic
538C23C23×D411yescharacteristic
539C23C22×#4111yescharacteristic
540C23C22×#4111yescharacteristic
541C23C2×H(5,𝔽2)12yes—
542C23C2×H(5,𝔽2)11yescharacteristic
543C23C2×H(5,𝔽2)12yes—
544C23C2×2−1+412yes—
545C23C2×2−1+411yescharacteristic
546C23#58512yes—
547Q8—816no—
548D4—232no—
549D4—416no—
550D4—832no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—264no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—216no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—48no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—432no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—48no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—48no—
609C2×C4—832no—
610C2×C4C22×C22⋊C418yes—
611C2×C4C22×C22⋊C414yes—
612C2×C4C22×C22⋊C414yes—
613C2×C4C22×C22⋊C414yes—
614C2×C4C4×#4114yes—
615C2×C4#52018yes—
616C2×C4#52014yes—
617C2×C4#53514yes—
618C2×C4#53614yes—
619C2×C4#53718yes—
620C2×C4#54614yes—
621C2×C4#54614yes—
622C2×C4#54714yes—
623C2×C4#54714yes—
624C22—28no—
625C22—28no—
626C22—28no—
627C22—48no—
628C22—48no—
629C22—44no—
630C22—48no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—48no—
635C22—48no—
636C22—44no—
637C22—48no—
638C22—44no—
639C22—44no—
640C22—84no—
641C22—82no—
642C22—84no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22C23×C22⋊C411yescharacteristic
647C22C2×C4×#4111yescharacteristic
648C22C2×#52012yes—
649C22C2×#52011yescharacteristic
650C22C4×H(5,𝔽2)12yes—
651C22C2×#53511yescharacteristic
652C22C2×#53611yescharacteristic
653C22C2×#53712yes—
654C22C2×#54611yescharacteristic
655C22C2×#54611yescharacteristic
656C22C2×#54711yescharacteristic
657C22C2×#54711yescharacteristic
658C22#348812yes—
659C22#349811yescharacteristic
660C22#350011yescharacteristic
661C22#350612yes—
662C22#351112yes—
663C22#351812yes—
664C22#351912yes—
665C22#354412yes—
666C22#354712yes—
667C22#354712yes—
668C22#354912yes—
669C4—28no—
670C4—24no—
671C4—44no—
672C4—44no—
673C4—44no—
674C4—48no—
675C4—44no—
676C4—84no—
677C4—84no—
678C4—82no—
679C4#231118yes—
680C4#233918yes—
681C2—48no—
682C2—84no—
683C2C2×#231111yescharacteristic
684C2C2×#233911yescharacteristic
685C2#3425811yescharacteristic
686C2#3432112yes—
687C2#3448512yes—
688C2#3892212yes—
689C2#3895311yescharacteristic
690C2#3895811yescharacteristic
691C2#3946912yes—
692C2#3948911yescharacteristic
693C2#3949111yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC22×#555C2
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×D4×C4⋊C4C2semidirect
3. C2×#2293C2semidirect
4. #14714C2semidirect
5. #14421C2semidirect
6. #14257C2semidirect
7. #14161C2semidirect
8. #14137C2semidirect
9. #13970C2semidirect
10. #13951C2semidirect
11. #13939C2semidirect
12. #13906C2semidirect
13. #13795C2semidirect
14. C4×#390C2semidirect
17. C2×#555C4semidirect
78. C2×C4⋊Q8C2×C4semidirect
81. C2×#124C2×C4semidirect
87. C2×#121C2×C4semidirect
93. C2×C4×D4C2×C4semidirect
103. C2×#117C2×C4semidirect
104. C2×#117C2×C4semidirect
226. C4×D4C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5552
C2×D4×C4⋊C42
C2×#22932
#147142
#144212
#142572
#141612
#141372
#139702
#139512
#139392
#139062
#137952
C4×#3902