Finite Group Databasebeta

← #533377

Subgroups — #533377

Order 512.

3918 conjugacy classes of subgroups.

The subgroup lattice has 753 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic
2C2×#2360C212yesmaximal, complemented
3C2×Q8×C22⋊C4C212yesmaximal, complemented
4#14538C212yesmaximal, complemented
5#14401C212yesmaximal
6#14236C212yesmaximal, complemented
7#14220C214yesmaximal, complemented
8#14017C212yesmaximal, complemented
9#14011C214yesmaximal, complemented
10#13993C212yesmaximal, complemented
11#13969C212yesmaximal, complemented
12#13872C211yesmaximal, characteristic
13#13824C212yesmaximal
14C4×#391C212yesmaximal, complemented
15C4×#390C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C4×Q8C2212yes—
19C22×C4×Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2409C4116yescomplemented
23C2×#395C2212yes—
24C2×#393C2212yes—
25#2360—216no—
26Q8×C22⋊C4—216no—
27C2×#391C2214yes—
28C2×#391C2214yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#390C2211yescharacteristic
33C2×#390C2211yescharacteristic
34C2×#389C2214yes—
35C2×#389C2212yes—
36C2×#387C2214yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#385C2214yes—
40C2×#385C2212yes—
41C2×#380C2212yes—
42C2×#380C2214yes—
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2214yes—
48C2×C4×C22⋊C4C2214yes—
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2212yes—
51C2×C43C2211yescharacteristic
52#1444C2214yes—
53#1443C2214yes—
54#1442C2214yes—
55#1441C2212yes—
56#1441C2214yes—
57#1439C2212yes—
58#1439C2214yes—
59#1439C2214yes—
60#1437C2214yes—
61#1437C2214yes—
62#1437C2212yes—
63#1437C2212yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2211yescharacteristic
74C4⋊#95C2211yescharacteristic
75C22⋊#95C2212yes—
76C4×#95C2214yes—
77C4×#95C2214yes—
78C23×Q8—28no—
79C23×Q8—28no—
80C23×Q8—42no—
81C23×Q8—42no—
82C23×Q8C2×C418yescomplemented
83C23×Q8C2312yes—
84C23×Q8C2312yes—
85C23×D4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2312yes—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8C2×C4116yes—
93C2×C4×Q8—28no—
94C2×C4×Q8—28no—
95C2×C4×Q8—28no—
96C2×C4×Q8—28no—
97C2×C4×Q8—44no—
98C2×C4×Q8—44no—
99C2×C4×D4—42no—
100C2×C4×D4—42no—
101C2×C4×D4C2×C418yes—
102C22×C4⋊C4—28no—
103C22×C4⋊C4—42no—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2312yes—
125C22×C42C2312yes—
126C22×C42C2314yes—
127C22×C42C2314yes—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2314yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#395C2×C4116yescomplemented
135#393C2×C4116yes—
136#391—28no—
137#391—216no—
138#391—216no—
139#391C2×C4116yescomplemented
140#391D4116yescomplemented
141#390—24no—
142#390Q818yescomplemented
143#389—28no—
144#389—216no—
145#387—216no—
146#386—28no—
147#386C2×C4116yes—
148#385—216no—
149#385C2×C4116yescomplemented
150#380—28no—
151#380—216no—
152#380—42no—
153#380C2×C418yescomplemented
154C4×C4⋊C4—28no—
155C4×C4⋊C4—42no—
156C4×C22⋊C4—28no—
157C4×C22⋊C4—28no—
158C4×C22⋊C4—216no—
159C4×C22⋊C4—216no—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C43—42no—
183C25C2411yescharacteristic
184C25C2411yescharacteristic
185C22×Q8—264no—
186C22×Q8—24no—
187C22×Q8—28no—
188C22×Q8—24no—
189C22×Q8—28no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—42no—
193C22×Q8—42no—
194C22×Q8—416no—
195C22×Q8—416no—
196C22×Q8—432no—
197C22×Q8—416no—
198C22×Q8C22×C418yes—
199C22×Q8C22×C418yes—
200C22×Q8C2×D418yes—
201C22×Q8#4118yes—
202C22×D4—41no—
203C22×D4—41no—
204C22×D4—41no—
205C22×D4—41no—
206C22×D4C22×C414yes—
207C22×D4C2×Q814yes—
208C22×D4#4114yes—
209C23×C4—24no—
210C23×C4—28no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—28no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—28no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—44no—
225C23×C4—42no—
226C23×C4C22×C414yes—
227C23×C4C22×C414yes—
228C23×C4C22×C418yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2414yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2414yes—
237C23×C4C2412yes—
238C23×C4C2414yes—
239C23×C4C2411yescharacteristic
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C22⋊Q8—264no—
243C4×Q8—432no—
244C4×Q8—432no—
245C4×D4—416no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—416no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—41no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—41no—
271C2×C4⋊C4—41no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4C22×C418yes—
275C2×C4⋊C4C22×C4116yes—
276C2×C4⋊C4C22×C414yes—
277C2×C4⋊C4C2×D418yes—
278C2×C4⋊C4C2×Q814yescomplemented
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—41no—
298C2×C22⋊C4—41no—
299C2×C22⋊C4—41no—
300C2×C22⋊C4—41no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4C22×C4116yes—
304C2×C22⋊C4C22×C414yes—
305C2×C22⋊C4C22×C414yes—
306C2×C22⋊C4C22×C418yes—
307C2×C22⋊C4C2×Q818yes—
308C2×C22⋊C4C2×Q818yescomplemented
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—28no—
314C2×C42—24no—
315C2×C42—28no—
316C2×C42—28no—
317C2×C42—24no—
318C2×C42—28no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—28no—
323C2×C42—44no—
324C2×C42—44no—
325C2×C42—41no—
326C2×C42—42no—
327C2×C42—41no—
328C2×C42—41no—
329C2×C42—42no—
330C2×C42—41no—
331C2×C42—44no—
332C2×C42—44no—
333C2×C42C22×C414yes—
334C2×C42C2×D418yes—
335C2×C42C2×Q814yescomplemented
336#95—24no—
337#95—28no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—28no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—42no—
352#95—42no—
353#95C22×C418yes—
354#95C22×C4116yes—
355#95C22×C4116yes—
356#95C2×D4116yes—
357#95C2×D4116yes—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—21no—
363C24—21no—
364C24—21no—
365C24—21no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—21no—
370C24—21no—
371C24—21no—
372C24—22no—
373C24—21no—
374C24C23×C412yes—
375C24C23×C412yes—
376C24C22×Q812yes—
377C24C22×Q812yes—
378C24C2×#4112yes—
379C24C2×#4112yes—
380C24C2511yescharacteristic
381C2×Q8—216no—
382C2×Q8—232no—
383C2×Q8—216no—
384C2×Q8—216no—
385C2×Q8—216no—
386C2×Q8—216no—
387C2×Q8—216no—
388C2×Q8—48no—
389C2×Q8—48no—
390C2×Q8—416no—
391C2×Q8—48no—
392C2×Q8—4128no—
393C2×Q8C4×D4132yes—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4C4×Q8116yescomplemented
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—216no—
450C22×C4—232no—
451C22×C4—216no—
452C22×C4—28no—
453C22×C4—28no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—416no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—416no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—88no—
481C22×C4—84no—
482C22×C4—84no—
483C22×C4C23×C412yes—
484C22×C4C23×C418yes—
485C22×C4C23×C412yes—
486C22×C4C23×C412yes—
487C22×C4C23×C418yes—
488C22×C4C23×C412yes—
489C22×C4C23×C412yes—
490C22×C4C22×D414yes—
491C22×C4C22×D418yes—
492C22×C4C22×D418yes—
493C22×C4C22×D414yes—
494C22×C4C22×D414yes—
495C22×C4C22×Q814yes—
496C22×C4C22×Q812yes—
497C22×C4C22×Q814yes—
498C22×C4C2×#4118yes—
499C22×C4C2×#4112yes—
500C22×C4C2×#4114yes—
501C22×C4H(5,𝔽2)12yes—
502C22×C4H(5,𝔽2)12yes—
503C22×C42−1+414yes—
504C4⋊C4—216no—
505C4⋊C4—232no—
506C4⋊C4—216no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—48no—
510C4⋊C4—416no—
511C4⋊C4—416no—
512C4⋊C4—44no—
513C4⋊C4—48no—
514C4⋊C4—88no—
515C22⋊C4—232no—
516C22⋊C4—216no—
517C22⋊C4—216no—
518C22⋊C4—48no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—416no—
524C22⋊C4—84no—
525C42—28no—
526C42—216no—
527C42—44no—
528C42—416no—
529C42—416no—
530C42—44no—
531C42—48no—
532C42—48no—
533C42—88no—
534C23—24no—
535C23—24no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—41no—
559C23—41no—
560C23—42no—
561C23—42no—
562C23—41no—
563C23—41no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23C2×C4×Q814yes—
579C23C2×C4×Q814yes—
580C23C24×C411yescharacteristic
581C23C23×D412yes—
582C23C23×Q811yescharacteristic
583C23C22×#4112yes—
584C23C22×#4111yescharacteristic
585C23C2×H(5,𝔽2)11yescharacteristic
586C23C2×H(5,𝔽2)11yescharacteristic
587C23C2×2−1+412yes—
588C23#58511yescharacteristic
589C23#58511yescharacteristic
590C23#58512yes—
591Q8—264no—
592Q8—432no—
593D4—416no—
594C2×C4—28no—
595C2×C4—22no—
596C2×C4—28no—
597C2×C4—22no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—28no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—28no—
619C2×C4—28no—
620C2×C4—24no—
621C2×C4—28no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—28no—
627C2×C4—24no—
628C2×C4—28no—
629C2×C4—24no—
630C2×C4—432no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—42no—
654C2×C4—41no—
655C2×C4—44no—
656C2×C4—416no—
657C2×C4—416no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—42no—
672C2×C4—816no—
673C2×C4C2×C4×D4116yes—
674C2×C4C2×C4×D418yes—
675C2×C4C2×C4×Q814yes—
676C2×C4#51814yes—
677C2×C4D4214yes—
678C2×C4#54618yes—
679C2×C4D4×Q818yes—
680C2×C4#55114yes—
681C22—28no—
682C22—28no—
683C22—24no—
684C22—24no—
685C22—44no—
686C22—44no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—42no—
693C22—42no—
694C22—42no—
695C22—44no—
696C22—44no—
697C22—44no—
698C22—44no—
699C22—42no—
700C22—42no—
701C22—42no—
702C22—42no—
703C22C22×C4×D412yes—
704C22C22×C4×Q811yescharacteristic
705C22C2×#51811yescharacteristic
706C22#342811yescharacteristic
707C22C4×2−1+412yes—
708C22C2×D4211yescharacteristic
709C22C2×#54612yes—
710C22C2×D4×Q812yes—
711C22D4×#4112yes—
712C22C2×#55111yescharacteristic
713C22Q8×#4112yes—
714C22#348212yes—
715C22#349712yes—
716C22#350012yes—
717C22#350111yescharacteristic
718C22#350111yescharacteristic
719C22#350511yescharacteristic
720C22#350511yescharacteristic
721C22#351612yes—
722C22#352412yes—
723C22#354112yes—
724C22#355011yescharacteristic
725C22#355111yescharacteristic
726C4—216no—
727C4—28no—
728C4—24no—
729C4—48no—
730C4—42no—
731C4—42no—
732C4—42no—
733C4—44no—
734C4—44no—
735C4—48no—
736C4—84no—
737C4—82no—
738C4#235818yes—
739C2—44no—
740C2—44no—
741C2C2×#235811yescharacteristic
742C2C4×D4×Q812yes—
743C2#3421211yescharacteristic
744C2#3431212yes—
745C2#3441311yescharacteristic
746C2#5980912yes—
747C2#5981312yes—
748C2#5981711yescharacteristic
749C2#5981811yescharacteristic
750C2#5993311yescharacteristic
751C2#5994611yescharacteristic
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×#2360C2semidirect
3. C2×Q8×C22⋊C4C2semidirect
4. #14538C2semidirect
6. #14236C2semidirect
7. #14220C2semidirect
8. #14017C2semidirect
9. #14011C2semidirect
10. #13993C2semidirect
11. #13969C2semidirect
14. C4×#391C2semidirect
22. #2409C4semidirect
82. C23×Q8C2×C4semidirect
134. #395C2×C4semidirect
139. #391C2×C4semidirect
140. #391D4semidirect
142. #390Q8semidirect
149. #385C2×C4semidirect
153. #380C2×C4semidirect
278. C2×C4⋊C4C2×Q8semidirect
308. C2×C22⋊C4C2×Q8semidirect
335. C2×C42C2×Q8semidirect
400. C2×D4C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
C2×#23602
C2×Q8×C22⋊C42
#145382
#144012
#142362
#142202
#140172
#140112
#139932
#139692
#138722
#138242
C4×#3912
C4×#3902