Finite Group Databasebeta

← #616160

Subgroups — #616160

Order 512.

4146 conjugacy classes of subgroups.

The subgroup lattice has 739 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic, complemented
2C2×#2857C211yesmaximal, characteristic, complemented
3C2×#2417C212yesmaximal
4C22×#384C211yesmaximal, characteristic, complemented
5C2×C42×D4C211yesmaximal, characteristic
6#15184C212yesmaximal, complemented
7#15182C212yesmaximal, complemented
8#15133C214yesmaximal, complemented
9#15131C214yesmaximal, complemented
10#14714C211yesmaximal, characteristic, complemented
11#14712C211yesmaximal, characteristic, complemented
12#14686C211yesmaximal, characteristic, complemented
13#14139C212yesmaximal, complemented
14#13907C214yesmaximal, complemented
15#13872C214yesmaximal
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2212yes—
21C23×C4⋊C4C2214yes—
22C23×C4⋊C4C2212yes—
23C23×C42C2211yescharacteristic
24C23×C42C2211yescharacteristic
25#2857—28no—
26#2857—28no—
27#2417—216no—
28C2×#398C2214yes—
29C2×#396C2214yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#389C2214yes—
35C2×#389C2214yes—
36C2×#389C2212yes—
37C2×#389C2214yes—
38C2×#386C2214yes—
39C2×#384—28no—
40C2×#384C2211yescharacteristic
41C2×#384C2214yes—
42C2×#384C2212yes—
43C2×#384C2211yescharacteristic
44C2×#384C2211yescharacteristic
45C2×#380C2214yes—
46C2×#380C2214yes—
47C2×#380C2212yes—
48C2×#380C2214yes—
49C2×#380C2214yes—
50C42×D4—28no—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2212yes—
54C2×C4×C22⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2211yescharacteristic
56C2×C4×C22⋊C4C2212yes—
57C2×C43C2211yescharacteristic
58#1441C2218yes—
59#1441C2218yes—
60#1441C2218yes—
61#1441C2218yes—
62#1439C2218yes—
63C4⋊#95C2214yes—
64C4⋊#95C2214yes—
65C4⋊#95C2214yes—
66C4⋊#95C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C22⋊#95C2218yes—
70C23×Q8C2312yes—
71C23×D4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C24×C4C2314yes—
75C24×C4C2311yescharacteristic
76C24×C4C2312yes—
77C2×C4⋊Q8—24no—
78C2×C4⋊Q8—24no—
79C2×C4⋊Q8—24no—
80C2×C22⋊Q8—28no—
81C2×C22⋊Q8—28no—
82C2×C22⋊Q8—28no—
83C2×C22⋊Q8—28no—
84C2×C22⋊Q8—28no—
85C2×C22⋊Q8—44no—
86C2×C4×D4—24no—
87C2×C4×D4—28no—
88C2×C4×D4—24no—
89C2×C4×D4—28no—
90C2×C4×D4D418yes—
91C2×C4×D4Q8116yescomplemented
92C22×C4⋊C4—28no—
93C22×C4⋊C4—216no—
94C22×C4⋊C4—28no—
95C22×C4⋊C4—216no—
96C22×C4⋊C4—44no—
97C22×C4⋊C4—48no—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2318yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C42—24no—
120C22×C42—24no—
121C22×C42—24no—
122C22×C42—24no—
123C22×C42—24no—
124C22×C42D418yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134#398—216no—
135#396—216no—
136#391—28no—
137#391—28no—
138#391—28no—
139#391—44no—
140#389—216no—
141#389—216no—
142#389—28no—
143#389—48no—
144#386—216no—
145#384—28no—
146#384—216no—
147#384—24no—
148#384—24no—
149#384D418yescomplemented
150#380—216no—
151#380—28no—
152#380—216no—
153#380—216no—
154#380—48no—
155C4×C4⋊C4—28no—
156C4×C4⋊C4—24no—
157C4×C22⋊C4—24no—
158C4×C22⋊C4—28no—
159C4×C22⋊C4—28no—
160C4×C22⋊C4—24no—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2318yes—
166C2×#95C2318yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2318yes—
171C2×#95C2318yes—
172C2×#95C2318yes—
173C43—24no—
174C25C2411yescharacteristic
175C25C2411yescharacteristic
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—24no—
186C22×D4C2×D414yes—
187C22×D4C2×Q818yes—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—24no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—28no—
209C23×C4—216no—
210C23×C4—28no—
211C23×C4—48no—
212C23×C4—44no—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4C2×D418yes—
217C23×C4C2×Q818yes—
218C23×C4C2×Q818yes—
219C23×C4#4114yes—
220C23×C4#4118yes—
221C23×C4C2414yes—
222C23×C4C2414yes—
223C23×C4C2414yes—
224C23×C4C2411yescharacteristic
225C23×C4C2414yes—
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C4⋊Q8—416no—
235C22⋊Q8—432no—
236C22⋊Q8—432no—
237C4×D4—232no—
238C4×D4—264no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—216no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—216no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—432no—
273C2×C4⋊C4—464no—
274C2×C4⋊C4—48no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—48no—
278C2×C4⋊C4C2×D418yes—
279C2×C4⋊C4C2×D4116yes—
280C2×C4⋊C4C2×D414yes—
281C2×C4⋊C4C2×Q818yes—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—48no—
298C2×C22⋊C4C2×D414yes—
299C2×C22⋊C4C2×D414yes—
300C2×C22⋊C4C2×Q818yes—
301C2×C22⋊C4C2×Q818yes—
302C2×C42—22no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—22no—
307C2×C42—24no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—232no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—416no—
324C2×C42C2×D414yes—
325C2×C42C2×D418yes—
326C2×C42C2×D414yes—
327C2×C42C2×D414yes—
328C2×C42C2×D414yes—
329C2×C42C2×Q818yes—
330C2×C42#4114yes—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—28no—
335#95—216no—
336#95—216no—
337#95—28no—
338#95—28no—
339#95—28no—
340#95—48no—
341#95—48no—
342#95—48no—
343C24—21no—
344C24—21no—
345C24—21no—
346C24—22no—
347C24—21no—
348C24—22no—
349C24—21no—
350C24—21no—
351C24—21no—
352C24—22no—
353C24—21no—
354C24—21no—
355C24—22no—
356C24—21no—
357C24—22no—
358C24C22×D412yes—
359C24C22×D412yes—
360C24C22×D412yes—
361C24C22×Q814yes—
362C24C22×Q814yes—
363C24C2×#4112yes—
364C24C2×#4114yes—
365C24C2511yescharacteristic
366C2×Q8—48no—
367C2×Q8—48no—
368C2×Q8—48no—
369C2×Q8—48no—
370C2×Q8—48no—
371C2×Q8—48no—
372C2×Q8—48no—
373C2×D4—28no—
374C2×D4—216no—
375C2×D4—28no—
376C2×D4—216no—
377C2×D4C4⋊Q8116yescomplemented
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—28no—
390C22×C4—216no—
391C22×C4—28no—
392C22×C4—216no—
393C22×C4—28no—
394C22×C4—216no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—216no—
398C22×C4—216no—
399C22×C4—28no—
400C22×C4—216no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—28no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—28no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—28no—
428C22×C4—216no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—216no—
432C22×C4—216no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—28no—
443C22×C4—24no—
444C22×C4—28no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—28no—
448C22×C4—28no—
449C22×C4—24no—
450C22×C4—416no—
451C22×C4—432no—
452C22×C4—44no—
453C22×C4—48no—
454C22×C4—44no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—44no—
458C22×C4—48no—
459C22×C4—44no—
460C22×C4—416no—
461C22×C4—88no—
462C22×C4—816no—
463C22×C4#121116yes—
464C22×C4C22⋊Q8132yescomplemented
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D418yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D414yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D414yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D414yes—
477C22×C4C22×Q814yes—
478C22×C4C22×Q814yes—
479C22×C4C22×Q814yes—
480C22×C4C22×Q814yes—
481C22×C4C22×Q814yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4114yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4114yes—
487C22×C4C2×#4114yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C4H(5,𝔽2)14yes—
490C22×C42−1+412yes—
491C22×C42−1+414yes—
492C4⋊C4—28no—
493C4⋊C4—216no—
494C4⋊C4—216no—
495C4⋊C4—232no—
496C4⋊C4—416no—
497C4⋊C4—432no—
498C4⋊C4—416no—
499C4⋊C4—48no—
500C4⋊C4—416no—
501C4⋊C4—416no—
502C4⋊C4—416no—
503C4⋊C4—48no—
504C4⋊C4—416no—
505C4⋊C4—88no—
506C22⋊C4—28no—
507C22⋊C4—216no—
508C22⋊C4—28no—
509C22⋊C4—216no—
510C22⋊C4—416no—
511C22⋊C4—416no—
512C22⋊C4—88no—
513C42—28no—
514C42—216no—
515C42—28no—
516C42—216no—
517C42—44no—
518C42—48no—
519C42—44no—
520C42—48no—
521C42#121116yes—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—22no—
526C23—24no—
527C23—22no—
528C23—22no—
529C23—24no—
530C23—24no—
531C23—22no—
532C23—24no—
533C23—22no—
534C23—22no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—22no—
541C23—24no—
542C23—22no—
543C23—24no—
544C23—22no—
545C23—22no—
546C23—24no—
547C23—24no—
548C23—22no—
549C23—22no—
550C23—24no—
551C23—24no—
552C23—41no—
553C23—41no—
554C23—42no—
555C23—42no—
556C23—41no—
557C23—42no—
558C23—41no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23C2×#12114yes—
563C23C2×C22⋊Q818yes—
564C23C2×C4⋊Q814yes—
565C23C2×C4⋊Q814yes—
566C23#53214yes—
567C23#53318yes—
568C23C23×D411yescharacteristic
569C23C23×D411yescharacteristic
570C23C23×D411yescharacteristic
571C23C23×Q812yes—
572C23C22×#4111yescharacteristic
573C23C22×#4111yescharacteristic
574C23C22×#4112yes—
575C23C2×H(5,𝔽2)11yescharacteristic
576C23C2×H(5,𝔽2)12yes—
577C23C2×2−1+411yescharacteristic
578C23C2×2−1+412yes—
579Q8—816no—
580D4—232no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—232no—
598C2×C4—264no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—28no—
604C2×C4—22no—
605C2×C4—216no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—48no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—416no—
631C2×C4—432no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—48no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—48no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—48no—
648C2×C4—48no—
649C2×C4—832no—
650C2×C4C2×#12114yes—
651C2×C4C2×#12114yes—
652C2×C4C2×#12114yes—
653C2×C4C2×#12118yes—
654C2×C4C2×C22⋊Q818yes—
655C2×C4C2×C22⋊Q818yes—
656C2×C4C2×C4⋊Q814yes—
657C2×C4#53714yes—
658C2×C4#53718yes—
659C2×C4D4214yes—
660C2×C4#54618yes—
661C2×C4#54714yes—
662C2×C4#54814yes—
663C2×C4D4×Q818yes—
664C2×C4#55218yes—
665C2×C4#55314yes—
666C2×C4#55418yes—
667C22—24no—
668C22—24no—
669C22—28no—
670C22—24no—
671C22—24no—
672C22—28no—
673C22—24no—
674C22—28no—
675C22—24no—
676C22—28no—
677C22—24no—
678C22—24no—
679C22—24no—
680C22—42no—
681C22—44no—
682C22—42no—
683C22—44no—
684C22—42no—
685C22—44no—
686C22—42no—
687C22—44no—
688C22#285718yescomplemented
689C22C22×#12111yescharacteristic
690C22C22×#12111yescharacteristic
691C22C22×C22⋊Q812yes—
692C22C22×C4⋊Q811yescharacteristic
693C22C2×#53211yescharacteristic
694C22C2×#53312yes—
695C22C2×#53711yescharacteristic
696C22C2×#53712yes—
697C22C2×D4211yescharacteristic
698C22C2×#54612yes—
699C22C2×#54711yescharacteristic
700C22C2×#54811yescharacteristic
701C22C2×D4×Q812yes—
702C22C2×#55212yes—
703C22C2×#55311yescharacteristic
704C22C2×#55412yes—
705C22#349811yescharacteristic
706C22#349911yescharacteristic
707C22#350312yes—
708C22#350312yes—
709C22#350812yes—
710C22#350912yes—
711C22#351112yes—
712C4—24no—
713C4—24no—
714C4—28no—
715C4—48no—
716C4—42no—
717C4—44no—
718C4—42no—
719C4—44no—
720C4—84no—
721C4—84no—
722C4—84no—
723C4#241618yes—
724C4#2417116yescomplemented
725C2—28no—
726C2—44no—
727C2C2×#241611yescharacteristic
728C2C2×#241712yes—
729C2#3527311yescharacteristic
730C2#3528012yes—
731C2D4×C4⋊Q811yescharacteristic
732C2#3612111yescharacteristic
733C2#3612912yes—
734C2#3618011yescharacteristic
735C2#3618111yescharacteristic
736C2#3618912yes—
737C2C2×#285711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2857C2semidirect
2. C2×#2857C2semidirect
4. C22×#384C2semidirect
6. #15184C2semidirect
7. #15182C2semidirect
8. #15133C2semidirect
9. #15131C2semidirect
10. #14714C2semidirect
11. #14712C2semidirect
12. #14686C2semidirect
13. #14139C2semidirect
14. #13907C2semidirect
91. C2×C4×D4Q8semidirect
149. #384D4semidirect
377. C2×D4C4⋊Q8semidirect
464. C22×C4C22⋊Q8semidirect
688. C22#2857semidirect
724. C4#2417semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#24172
C22×#3842
C2×C42×D42
#151842
#151822
#151332
#151312
#147142
#147122
#146862
#141392
#139072
#138722