Finite Group Databasebeta

← #539365

Subgroups — #539365

Order 512.

4042 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic, complemented
3C2×#2331C211yesmaximal, characteristic, complemented
4#14474C214yesmaximal, complemented
5#14449C214yesmaximal, complemented
6#13997C212yesmaximal, complemented
7#13997C212yesmaximal, complemented
8#13969C214yesmaximal, complemented
9#13942C211yesmaximal, characteristic, complemented
10#13907C212yesmaximal, complemented
11#13905C211yesmaximal, characteristic
12#13890C214yesmaximal
13#13847C211yesmaximal, characteristic
14C22⋊C4×C4⋊C4C212yesmaximal, complemented
15#13786C211yesmaximal, characteristic, complemented
16C2×#554—28no—
17C2×#554—28no—
18C2×#554C4116yescomplemented
19C22×C4⋊Q8C2211yescharacteristic
20C22×#125C2212yes—
21C22×C22⋊Q8C2214yes—
22C22×C22⋊Q8C2212yes—
23C22×C4×Q8C2211yescharacteristic
24C22×C4×Q8C2212yes—
25C22×C4×D4C2211yescharacteristic
26C22×C4×D4C2212yes—
27C23×C4⋊C4C2212yes—
28#2365—28no—
29#2331—28no—
30C2×#391C2214yes—
31C2×#391C2212yes—
32C2×#391C2211yescharacteristic
33C2×#391C2211yescharacteristic
34C2×#391C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#387C2214yes—
38C2×#387C2214yes—
39C2×#382C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2214yes—
42C2×#380C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2214yes—
50C2×#376C2211yescharacteristic
51#1445C2212yes—
52#1445C2212yes—
53#1445C2214yes—
54#1445C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2212yes—
58#1441C2212yes—
59#1439C2212yes—
60#1439C2212yes—
61#1439C2214yes—
62#1439C2214yes—
63#1438C2212yes—
64#1438C2212yes—
65#1437C2214yes—
66#1437C2214yes—
67#1437C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2212yes—
73C4⋊#95C2214yes—
74C4⋊#95C2214yes—
75C4⋊#95C2211yescharacteristic
76#1434C2214yes—
77#1434C2214yes—
78C22⋊#95C2214yes—
79C22⋊#95C2214yes—
80C23×Q8C2311yescharacteristic
81C23×Q8C2312yes—
82C23×D4C2311yescharacteristic
83C24×C4C2312yes—
84C24×C4C2314yes—
85#554—2128no—
86C2×C4⋊Q8C2×C418yes—
87C2×C4⋊Q8D418yes—
88C2×C4⋊Q8D418yes—
89C2×#125—28no—
90C2×#125—28no—
91C2×#125C2×C4116yes—
92C2×C22⋊Q8—216no—
93C2×C22⋊Q8—216no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8C2×C4132yes—
97C2×C22⋊Q8C2×C4116yes—
98C2×C4×Q8—28no—
99C2×C4×Q8—24no—
100C2×C4×Q8—24no—
101C2×C4×Q8—48no—
102C2×C4×Q8C2×C418yes—
103C2×C4×D4—28no—
104C2×C4×D4—28no—
105C2×C4×D4—24no—
106C2×C4×D4—24no—
107C2×C4×D4C2×C4116yes—
108C2×C4×D4C2×C418yes—
109C22×C4⋊C4—28no—
110C22×C4⋊C4—28no—
111C22×C4⋊C4C2×C4116yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2314yes—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136C22×C42C2314yes—
137C22×C42C2312yes—
138C22×C42C2311yescharacteristic
139C22×C42C2312yes—
140C22×C42C2314yes—
141C22×C42C2311yescharacteristic
142#391—216no—
143#391—28no—
144#391—24no—
145#391—24no—
146#391—44no—
147#389—28no—
148#389—28no—
149#387—216no—
150#387—216no—
151#382—24no—
152#380—28no—
153#380—48no—
154#380—44no—
155C4×C4⋊C4—28no—
156C4×C4⋊C4—28no—
157C4×C4⋊C4—24no—
158C4×C4⋊C4—24no—
159C4×C22⋊C4—28no—
160C4×C22⋊C4—48no—
161C4×C22⋊C4—44no—
162#376—24no—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C25C2412yes—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—22no—
193C22×Q8—22no—
194C22×Q8—44no—
195C22×Q8—44no—
196C22×Q8C22×C414yes—
197C22×Q8C22×C418yes—
198C22×Q8C2×D418yes—
199C22×Q8C2×D418yes—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4C22×C414yes—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—28no—
212C23×C4—28no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4C22×C418yes—
218C23×C4C22×C4116yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2414yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C4⋊Q8C22⋊C4164yescomplemented
234#125—264no—
235C22⋊Q8—2128no—
236C22⋊Q8—264no—
237C4×Q8—232no—
238C4×Q8—432no—
239C4×D4—264no—
240C4×D4—232no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—264no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4C22×C414yes—
271C2×C4⋊C4C22×C4116yes—
272C2×C4⋊C4C22×C414yes—
273C2×C4⋊C4C22×C418yes—
274C2×C4⋊C4C22×C4116yes—
275C2×C4⋊C4C22×C418yes—
276C2×C4⋊C4C22×C418yes—
277C2×C4⋊C4C2×D418yes—
278C2×C4⋊C4C2×D418yes—
279C2×C4⋊C4C2×D418yes—
280C2×C4⋊C4C2×D418yes—
281C2×C4⋊C4#4114yes—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4C22×C4116yes—
305C2×C22⋊C4C22×C418yes—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—28no—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—22no—
316C2×C42—28no—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—28no—
320C2×C42—44no—
321C2×C42—44no—
322C2×C42—44no—
323C2×C42—44no—
324C2×C42C22×C418yes—
325C2×C42C22×C414yes—
326C2×C42C2×D414yes—
327C2×C42C2×D414yes—
328C2×C42#4114yes—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—28no—
334#95—28no—
335#95—24no—
336#95—24no—
337#95—28no—
338#95—28no—
339#95—24no—
340#95—24no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—44no—
346#95—44no—
347#95—42no—
348#95—42no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24C23×C414yes—
366C242−1+414yes—
367C242−1+414yes—
368C24C2511yescharacteristic
369C2×Q8—28no—
370C2×Q8—28no—
371C2×Q8—28no—
372C2×Q8—48no—
373C2×Q8—416no—
374C2×Q8—48no—
375C2×Q8—416no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8C2×C22⋊C4132yes—
381C2×D4—28no—
382C2×D4—28no—
383C2×D4—216no—
384C2×D4—216no—
385C2×D4—28no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
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—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—216no—
427C22×C4—216no—
428C22×C4—216no—
429C22×C4—232no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—416no—
457C22×C4—416no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—88no—
463C22×C4—88no—
464C22×C4—88no—
465C22×C4—88no—
466C22×C4C23×C414yes—
467C22×C4C23×C414yes—
468C22×C4C23×C414yes—
469C22×C4C23×C412yes—
470C22×C4C23×C412yes—
471C22×C4C23×C418yes—
472C22×C4C23×C412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D418yes—
477C22×C4C22×D418yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C2×#4114yes—
482C22×C4C2×#4114yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4H(5,𝔽2)14yes—
486C22×C4H(5,𝔽2)12yes—
487C22×C42−1+412yes—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—232no—
491C4⋊C4—28no—
492C4⋊C4—216no—
493C4⋊C4—232no—
494C4⋊C4—28no—
495C4⋊C4—416no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—88no—
500C4⋊C4C2×C22⋊C4132yes—
501C4⋊C4C2×C22⋊C4132yes—
502C22⋊C4—216no—
503C22⋊C4—232no—
504C22⋊C4—216no—
505C22⋊C4—48no—
506C22⋊C4—416no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—88no—
510C42—216no—
511C42—28no—
512C42—416no—
513C42—48no—
514C42—48no—
515C42—44no—
516C42—88no—
517C42C2×C22⋊C4116yes—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23#51918yes—
553C23C24×C411yescharacteristic
554C23C23×D411yescharacteristic
555C23C23×D411yescharacteristic
556C23C22×#4111yescharacteristic
557C23C22×#4111yescharacteristic
558C23C2×H(5,𝔽2)11yescharacteristic
559C23C2×H(5,𝔽2)12yes—
560C23C2×2−1+412yes—
561C23C2×2−1+412yes—
562C23C2×2−1+411yescharacteristic
563C23#58512yes—
564Q8—432no—
565Q8—416no—
566D4—232no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—28no—
573C2×C4—22no—
574C2×C4—28no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—24no—
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—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—432no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—832no—
647C2×C4C22×C22⋊C414yes—
648C2×C4C22×C22⋊C414yes—
649C2×C4C22×C22⋊C4116yes—
650C2×C4C22×C22⋊C414yes—
651C2×C4C4×#4114yes—
652C2×C4#52014yes—
653C2×C4#54814yes—
654C2×C4#54814yes—
655C2×C4#55014yes—
656C2×C4#55014yes—
657C22—28no—
658C22—28no—
659C22—28no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22C23×C22⋊C411yescharacteristic
673C22C2×C4×#4111yescharacteristic
674C22C2×#51912yes—
675C22C2×#52011yescharacteristic
676C22C4×H(5,𝔽2)12yes—
677C22C2×#54811yescharacteristic
678C22C2×#54811yescharacteristic
679C22C2×#55011yescharacteristic
680C22C2×#55011yescharacteristic
681C22#348411yescharacteristic
682C22#348511yescharacteristic
683C22#348611yescharacteristic
684C22#348611yescharacteristic
685C22#348612yes—
686C22#349212yes—
687C22#349512yes—
688C22#349612yes—
689C22#351612yes—
690C22#352012yes—
691C22#353612yes—
692C22#354112yes—
693C22#355212yes—
694C22#356312yes—
695C4—24no—
696C4—216no—
697C4—24no—
698C4—44no—
699C4—44no—
700C4—44no—
701C4—44no—
702C4—44no—
703C4—44no—
704C4—82no—
705C4—82no—
706C4—84no—
707C4—82no—
708C4#234118yes—
709C2—48no—
710C2C2×#234111yescharacteristic
711C2#3417311yescharacteristic
712C2#3425611yescharacteristic
713C2#3429912yes—
714C2#3446412yes—
715C2#4510912yes—
716C2#4511212yes—
717C2#4513811yescharacteristic
718C2#4514311yescharacteristic
719C2#4533711yescharacteristic
720C2#4535611yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
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×#2365C2semidirect
3. C2×#2331C2semidirect
4. #14474C2semidirect
5. #14449C2semidirect
6. #13997C2semidirect
7. #13997C2semidirect
8. #13969C2semidirect
9. #13942C2semidirect
10. #13907C2semidirect
14. C22⋊C4×C4⋊C4C2semidirect
15. #13786C2semidirect
18. C2×#554C4semidirect
233. C4⋊Q8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#23652
C2×#23312
#144742
#144492
#139972
#139692
#139422
#139072
#139052
#138902
#138472
C22⋊C4×C4⋊C42
#137862