Finite Group Databasebeta

← #539190

Subgroups — #539190

Order 512.

4754 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Q8C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic, complemented
3C2×#2331C211yesmaximal, characteristic, complemented
4#14449C214yesmaximal, complemented
5#14362C214yesmaximal, complemented
6#14099C214yesmaximal, complemented
7#14053C212yesmaximal, complemented
8#14053C212yesmaximal, complemented
9#13941C211yesmaximal, characteristic
10#13925C211yesmaximal, characteristic, complemented
11#13922C214yesmaximal, complemented
12#13920C212yesmaximal
13#13850C211yesmaximal, characteristic, complemented
14#13820C212yesmaximal
15#13785C211yesmaximal, characteristic
16C24×Q8C2212yes—
17C2×D4×Q8—28no—
18C2×D4×Q8—28no—
19C2×D4×Q8C4116yescomplemented
20C22×C4⋊Q8C2212yes—
21C22×C4⋊Q8C2211yescharacteristic
22C22×C22⋊Q8C2214yes—
23C22×C22⋊Q8C2212yes—
24C22×C4×Q8C2211yescharacteristic
25C22×C4×D4C2212yes—
26C22×C4×D4C2211yescharacteristic
27#2369—28no—
28#2331—28no—
29C2×#391C2214yes—
30C2×#391C2214yes—
31C2×#391C2214yes—
32C2×#391C2214yes—
33C2×#390C2211yescharacteristic
34C2×#390C2212yes—
35C2×#390C2211yescharacteristic
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#387C2214yes—
40C2×#382C2211yescharacteristic
41C2×#380C2214yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C22⋊C4C2214yes—
48C2×#376C2211yescharacteristic
49#1446C2214yes—
50#1446C2212yes—
51#1446C2214yes—
52#1446C2212yes—
53#1445C2212yes—
54#1445C2212yes—
55#1441C2212yes—
56#1441C2212yes—
57#1441C2214yes—
58#1441C2214yes—
59#1439C2212yes—
60#1439C2214yes—
61#1439C2212yes—
62#1439C2214yes—
63#1437C2214yes—
64#1437C2214yes—
65#1437C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2211yescharacteristic
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C22⋊#95C2214yes—
78C4×#95C2212yes—
79C4×#95C2212yes—
80C23×Q8—28no—
81C23×Q8—28no—
82C23×Q8—48no—
83C23×Q8—48no—
84C23×Q8C2×C4116yes—
85C23×Q8C2311yescharacteristic
86C23×Q8C2314yes—
87C23×Q8C2312yes—
88C23×D4C2311yescharacteristic
89C24×C4C2314yes—
90C24×C4C2312yes—
91D4×Q8—2128no—
92C2×C4⋊Q8—28no—
93C2×C4⋊Q8—28no—
94C2×C4⋊Q8C2×C4116yes—
95C2×C4⋊Q8C2×C418yes—
96C2×C4⋊Q8D418yes—
97C2×C4⋊Q8D418yes—
98C2×C22⋊Q8—28no—
99C2×C22⋊Q8—28no—
100C2×C22⋊Q8—216no—
101C2×C22⋊Q8—216no—
102C2×C22⋊Q8C2×C4116yes—
103C2×C22⋊Q8C2×C4132yes—
104C2×C4×Q8—24no—
105C2×C4×Q8—24no—
106C2×C4×Q8C2×C418yes—
107C2×C4×D4—28no—
108C2×C4×D4—28no—
109C2×C4×D4—24no—
110C2×C4×D4—24no—
111C2×C4×D4C2×C4116yes—
112C2×C4×D4C2×C418yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137C22×C42C2314yes—
138C22×C42C2314yes—
139C22×C42C2311yescharacteristic
140#391—216no—
141#391—216no—
142#391—48no—
143#391—48no—
144#390—28no—
145#390—24no—
146#390Q818yescomplemented
147#389—28no—
148#389—28no—
149#389—28no—
150#387—216no—
151#382—24no—
152#380—48no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—28no—
155C4×C4⋊C4—28no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4—24no—
158C4×C22⋊C4—48no—
159#376—24no—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C25C2412yes—
187C22×Q8—264no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—28no—
191C22×Q8—28no—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8—28no—
195C22×Q8—28no—
196C22×Q8—44no—
197C22×Q8—44no—
198C22×Q8—44no—
199C22×Q8—44no—
200C22×Q8—464no—
201C22×Q8—432no—
202C22×Q8—464no—
203C22×Q8—432no—
204C22×Q8C22×C414yes—
205C22×Q8C22×C4116yes—
206C22×Q8C22×C418yes—
207C22×Q8C2×D418yes—
208C22×Q8C2×D418yes—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4C22×C414yes—
214C22×D4C2×Q814yes—
215C22×D4#4114yes—
216C23×C4—28no—
217C23×C4—28no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—28no—
221C23×C4—28no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4C22×C4116yes—
227C23×C4C22×C418yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2414yes—
239C23×C4C2411yescharacteristic
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C4⋊Q8—264no—
243C4⋊Q8C22⋊C4164yescomplemented
244C22⋊Q8—264no—
245C22⋊Q8—2128no—
246C4×Q8—232no—
247C4×D4—264no—
248C4×D4—232no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4C22×C414yes—
276C2×C4⋊C4C22×C414yes—
277C2×C4⋊C4C22×C418yes—
278C2×C4⋊C4C22×C4116yes—
279C2×C4⋊C4C22×C418yes—
280C2×C4⋊C4C22×C418yes—
281C2×C4⋊C4C2×D418yes—
282C2×C4⋊C4C2×D418yes—
283C2×C4⋊C4C2×D418yes—
284C2×C4⋊C4C2×D418yes—
285C2×C4⋊C4C2×Q814yescomplemented
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4C22×C4116yes—
302C2×C22⋊C4C22×C418yes—
303C2×C22⋊C4C2×Q8116yescomplemented
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—28no—
316C2×C42—28no—
317C2×C42—44no—
318C2×C42—44no—
319C2×C42—44no—
320C2×C42—44no—
321C2×C42—44no—
322C2×C42C22×C418yes—
323C2×C42C22×C414yes—
324C2×C42C2×D414yes—
325C2×C42C2×D414yes—
326C2×C42C2×Q814yescomplemented
327#95—24no—
328#95—24no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—28no—
337#95—24no—
338#95—28no—
339#95—24no—
340#95—28no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—44no—
346#95—42no—
347#95—44no—
348#95—42no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—22no—
359C24—22no—
360C24—24no—
361C24C23×C414yes—
362C24C22×Q814yes—
363C24C2×#4114yes—
364C24C2511yescharacteristic
365C2×Q8—28no—
366C2×Q8—216no—
367C2×Q8—28no—
368C2×Q8—28no—
369C2×Q8—232no—
370C2×Q8—216no—
371C2×Q8—232no—
372C2×Q8—416no—
373C2×Q8—416no—
374C2×Q8—48no—
375C2×Q8—416no—
376C2×Q8—48no—
377C2×Q8—416no—
378C2×Q8—416no—
379C2×Q8—416no—
380C2×Q8—416no—
381C2×Q8—4256no—
382C2×Q8C2×C22⋊C4132yes—
383C2×D4—28no—
384C2×D4—28no—
385C2×D4—28no—
386C2×D4—28no—
387C2×D4—28no—
388C2×D4—28no—
389C2×D4C4×Q8116yes—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
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—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—232no—
429C22×C4—216no—
430C22×C4—232no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—416no—
458C22×C4—416no—
459C22×C4—416no—
460C22×C4—416no—
461C22×C4—416no—
462C22×C4—416no—
463C22×C4—48no—
464C22×C4—416no—
465C22×C4—48no—
466C22×C4C23×C414yes—
467C22×C4C23×C414yes—
468C22×C4C23×C412yes—
469C22×C4C23×C412yes—
470C22×C4C23×C418yes—
471C22×C4C23×C414yes—
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×C4C22×Q814yes—
482C22×C4C22×Q814yes—
483C22×C4C22×Q812yes—
484C22×C4C2×#4112yes—
485C22×C4H(5,𝔽2)14yes—
486C22×C4H(5,𝔽2)14yes—
487C22×C42−1+414yes—
488C22×C42−1+412yes—
489C22×C42−1+412yes—
490C4⋊C4—28no—
491C4⋊C4—28no—
492C4⋊C4—216no—
493C4⋊C4—28no—
494C4⋊C4—232no—
495C4⋊C4—28no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—88no—
500C4⋊C4—88no—
501C4⋊C4C2×C22⋊C4132yes—
502C4⋊C4C2×C22⋊C4132yes—
503C22⋊C4—232no—
504C22⋊C4—216no—
505C22⋊C4—416no—
506C22⋊C4—416no—
507C22⋊C4—88no—
508C22⋊C4—88no—
509C42—216no—
510C42—28no—
511C42—48no—
512C42—48no—
513C42—44no—
514C42—88no—
515C42—88no—
516C42C2×C22⋊C4116yes—
517C23—24no—
518C23—28no—
519C23—24no—
520C23—28no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—28no—
529C23—42no—
530C23—44no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23C2×C4×Q818yes—
540C23C24×C411yescharacteristic
541C23C23×D411yescharacteristic
542C23C23×D411yescharacteristic
543C23C23×Q811yescharacteristic
544C23C22×#4111yescharacteristic
545C23C2×H(5,𝔽2)12yes—
546C23C2×H(5,𝔽2)12yes—
547C23C2×2−1+411yescharacteristic
548C23C2×2−1+411yescharacteristic
549C23C2×2−1+412yes—
550C23#58512yes—
551Q8—232no—
552Q8—432no—
553Q8—464no—
554D4—232no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—464no—
589C2×C4—432no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
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—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
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×C4C22×C22⋊C414yes—
635C2×C4C22×C22⋊C414yes—
636C2×C4C22×C22⋊C4116yes—
637C2×C4C22×C22⋊C414yes—
638C2×C4C2×C4×Q814yes—
639C2×C4#51818yes—
640C2×C4#51914yes—
641C2×C4#54814yes—
642C2×C4#54814yes—
643C2×C4D4×Q814yes—
644C2×C4D4×Q814yes—
645C22—216no—
646C22—28no—
647C22—44no—
648C22—44no—
649C22—48no—
650C22—48no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—48no—
655C22—44no—
656C22C23×C22⋊C411yescharacteristic
657C22C22×C4×Q811yescharacteristic
658C22C2×#51812yes—
659C22C2×#51911yescharacteristic
660C22C4×2−1+412yes—
661C22C2×#54811yescharacteristic
662C22C2×#54811yescharacteristic
663C22C2×D4×Q811yescharacteristic
664C22C2×D4×Q811yescharacteristic
665C22#348411yescharacteristic
666C22#348412yes—
667C22#348411yescharacteristic
668C22#348611yescharacteristic
669C22#348711yescharacteristic
670C22#349712yes—
671C22#350112yes—
672C22#350312yes—
673C22#351412yes—
674C22#352012yes—
675C22#354712yes—
676C22#355012yes—
677C22#355212yes—
678C22#355412yes—
679C4—24no—
680C4—28no—
681C4—216no—
682C4—24no—
683C4—44no—
684C4—44no—
685C4—44no—
686C4—44no—
687C4—44no—
688C4—82no—
689C4—84no—
690C4—82no—
691C4—82no—
692C4Q8×C22⋊C418yes—
693C2—48no—
694C2C2×Q8×C22⋊C411yescharacteristic
695C2#3417111yescharacteristic
696C2#3425711yescharacteristic
697C2#3433412yes—
698C2#3448712yes—
699C2#4487912yes—
700C2#4488312yes—
701C2#4493411yescharacteristic
702C2#4493811yescharacteristic
703C2#4533711yescharacteristic
704C2#4535711yescharacteristic
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×#2369C2semidirect
3. C2×#2331C2semidirect
4. #14449C2semidirect
5. #14362C2semidirect
6. #14099C2semidirect
7. #14053C2semidirect
8. #14053C2semidirect
10. #13925C2semidirect
11. #13922C2semidirect
13. #13850C2semidirect
19. C2×D4×Q8C4semidirect
146. #390Q8semidirect
243. C4⋊Q8C22⋊C4semidirect
285. C2×C4⋊C4C2×Q8semidirect
303. C2×C22⋊C4C2×Q8semidirect
326. C2×C42C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Q82
C2×#23692
C2×#23312
#144492
#143622
#140992
#140532
#139412
#139252
#139222
#139202
#138502
#138202
#137852