Finite Group Databasebeta

← #554196

Subgroups — #554196

Order 512.

3310 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2591C211yesmaximal, characteristic
2C2×#2325C211yesmaximal, characteristic
3C22×#389C211yesmaximal, characteristic, complemented
4#14696C212yesmaximal, complemented
5#14580C214yesmaximal, complemented
6#14518C214yesmaximal, complemented
7#14434C214yesmaximal, complemented
8#14405C214yesmaximal, complemented
9#14259C212yesmaximal, complemented
10#14137C211yesmaximal, characteristic
11#14015C212yesmaximal, complemented
12#14015C212yesmaximal, complemented
13#13923C211yesmaximal, characteristic, complemented
14#13880C211yesmaximal, characteristic, complemented
15#13796C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C22×C4×D4C2212yes—
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2591C4116yescomplemented
24C2×#400C2212yes—
25C2×#394C2212yes—
26#2325—28no—
27C2×#390C2212yes—
28C2×#389—28no—
29C2×#389C2214yes—
30C2×#389C2214yes—
31C2×#388C2212yes—
32C2×#388C2214yes—
33C2×#388C2214yes—
34C2×#387C2212yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#386C2212yes—
40C2×#386C2212yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#385C2212yes—
44C2×#385C2214yes—
45C2×#385C2214yes—
46C2×#384C2212yes—
47C2×#380C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50C22×#95C2211yescharacteristic
51C22×#95C2211yescharacteristic
52#1444C2214yes—
53#1444C2214yes—
54#1444C2214yes—
55#1443C2214yes—
56#1442C2214yes—
57#1441C2212yes—
58#1441C2212yes—
59#1440C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1437C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71#1434C2214yes—
72#1434C2214yes—
73#1434C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C23×D4C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C2×#121—24no—
89C2×#121—24no—
90C2×#121C2×C418yes—
91C2×C4×D4—28no—
92C2×C4×D4—44no—
93C2×C4×D4—44no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—44no—
100C22×C4⋊C4—44no—
101C22×C4⋊C4—44no—
102C22×C4⋊C4C2×C418yescomplemented
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C42—24no—
127C22×C42—44no—
128C22×C42C2314yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132#400C2×C4116yescomplemented
133#394C2×C4116yescomplemented
134#390—44no—
135#389—216no—
136#389—216no—
137#388—28no—
138#388—48no—
139#388C2×C4132yescomplemented
140#387—28no—
141#386—28no—
142#386—28no—
143#386—44no—
144#386—44no—
145#386C2×C4116yescomplemented
146#386C2×C4116yescomplemented
147#385—28no—
148#385—48no—
149#385—44no—
150#385—44no—
151#385—48no—
152#384—44no—
153#380—24no—
154C4×C22⋊C4—24no—
155C4×C22⋊C4C2×C418yescomplemented
156C2×#95—24no—
157C2×#95D418yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C25C2412yes—
181C25C2411yescharacteristic
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—42no—
191C22×D4—42no—
192C22×D4C22×C418yes—
193C22×D4C22×C414yes—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4C22×C414yes—
224C23×C4C22×C414yes—
225C23×C4C22×C414yes—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2×D414yes—
229C23×C4#4114yes—
230C23×C4C2412yes—
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2414yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2412yes—
243#121—232no—
244C4×D4—432no—
245C2×C4⋊C4—232no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—416no—
271C2×C4⋊C4—416no—
272C2×C4⋊C4—416no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4C22×C418yes—
281C2×C4⋊C4C22×C418yes—
282C2×C4⋊C4C22×C414yes—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4C22×C418yes—
311C2×C22⋊C4C22×C418yes—
312C2×C22⋊C4C22×C4116yes—
313C2×C22⋊C4C22×C4116yes—
314C2×C42—24no—
315C2×C42—28no—
316C2×C42—28no—
317C2×C42—24no—
318C2×C42—42no—
319C2×C42—42no—
320C2×C42—42no—
321C2×C42—44no—
322C2×C42—42no—
323C2×C42—416no—
324C2×C42—42no—
325C2×C42—42no—
326C2×C42C22×C418yes—
327#95—28no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—28no—
333#95—28no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95C22×C418yes—
345#95C22×C4116yes—
346#95C22×C4116yes—
347#95C2×D418yes—
348#95C2×D418yes—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—24no—
357C24—22no—
358C24—22no—
359C24—24no—
360C24—22no—
361C24—22no—
362C24—24no—
363C24C23×C414yes—
364C24C23×C412yes—
365C24C22×D412yes—
366C24C2×#4112yes—
367C24H(5,𝔽2)14yes—
368C242−1+414yes—
369C24C2511yescharacteristic
370C2×D4—216no—
371C2×D4—216no—
372C2×D4—216no—
373C2×D4—28no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4—82no—
380C2×D4—82no—
381C2×D4—84no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—216no—
385C22×C4—28no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—216no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—28no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—48no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—416no—
458C22×C4—48no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—48no—
463C22×C4—416no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—84no—
469C22×C4—84no—
470C22×C4—88no—
471C22×C4—88no—
472C22×C4—88no—
473C22×C4—88no—
474C22×C4C4×D4116yes—
475C22×C4C23×C414yes—
476C22×C4C23×C418yes—
477C22×C4C23×C414yes—
478C22×C4C23×C412yes—
479C22×C4C23×C412yes—
480C22×C4C23×C414yes—
481C22×C4C22×D414yes—
482C22×C4C22×D414yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4H(5,𝔽2)14yes—
488C22×C4H(5,𝔽2)14yes—
489C22×C42−1+414yes—
490C4⋊C4—216no—
491C4⋊C4—216no—
492C4⋊C4—28no—
493C4⋊C4—416no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—416no—
497C4⋊C4—48no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—88no—
501C4⋊C4—88no—
502C22⋊C4—216no—
503C22⋊C4—216no—
504C22⋊C4—232no—
505C22⋊C4—416no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4—88no—
512C22⋊C4—88no—
513C42—48no—
514C42—416no—
515C42—48no—
516C42—84no—
517C23—24no—
518C23—24no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—24no—
524C23—24no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—28no—
529C23—24no—
530C23—28no—
531C23—24no—
532C23—24no—
533C23—28no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23C2×C4×D414yes—
553C23#52018yes—
554C23#53514yes—
555C23#53614yes—
556C23#54218yes—
557C23C24×C411yescharacteristic
558C23C23×D411yescharacteristic
559C23C22×#4111yescharacteristic
560C23C2×H(5,𝔽2)12yes—
561C23C2×H(5,𝔽2)12yes—
562C23C2×H(5,𝔽2)12yes—
563C23C2×2−1+412yes—
564C23C2×2−1+412yes—
565C23#58512yes—
566D4—432no—
567D4—88no—
568C2×C4—232no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—28no—
599C2×C4—416no—
600C2×C4—416no—
601C2×C4—416no—
602C2×C4—416no—
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—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—48no—
618C2×C4—44no—
619C2×C4—44no—
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—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—88no—
643C2×C4—816no—
644C2×C4—88no—
645C2×C4C2×C4×D414yes—
646C2×C4C2×C4×D414yes—
647C2×C4#52018yes—
648C22—24no—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—216no—
654C22—24no—
655C22—216no—
656C22—44no—
657C22—44no—
658C22—48no—
659C22—48no—
660C22—44no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—84no—
666C22—84no—
667C22#232618yes—
668C22C22×C4×D411yescharacteristic
669C22C2×#52012yes—
670C22C2×#52012yes—
671C22C4×H(5,𝔽2)12yes—
672C22C2×#53511yescharacteristic
673C22C2×#53611yescharacteristic
674C22C2×#54212yes—
675C22#348411yescharacteristic
676C22#348411yescharacteristic
677C22#348511yescharacteristic
678C22#348611yescharacteristic
679C22#349212yes—
680C22#350512yes—
681C22#353714yes—
682C22#354114yes—
683C22#355214yes—
684C22#355914yes—
685C4—28no—
686C4—24no—
687C4—24no—
688C4—44no—
689C4—44no—
690C4—44no—
691C4—44no—
692C4—48no—
693C4—44no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4—84no—
698C2—28no—
699C2—84no—
700C2C2×#232611yescharacteristic
701C2#3420011yescharacteristic
702C2#3426411yescharacteristic
703C2#3446114yes—
704C2#3671611yescharacteristic
705C2#3671711yescharacteristic
706C2#3682614yes—
707C2#3695811yescharacteristic
708C2#3723311yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC22×#121C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C22×#389C2semidirect
4. #14696C2semidirect
5. #14580C2semidirect
6. #14518C2semidirect
7. #14434C2semidirect
8. #14405C2semidirect
9. #14259C2semidirect
11. #14015C2semidirect
12. #14015C2semidirect
13. #13923C2semidirect
14. #13880C2semidirect
15. #13796C2semidirect
23. #2591C4semidirect
102. C22×C4⋊C4C2×C4semidirect
132. #400C2×C4semidirect
133. #394C2×C4semidirect
139. #388C2×C4semidirect
145. #386C2×C4semidirect
146. #386C2×C4semidirect
155. C4×C22⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25912
C2×#23252
C22×#3892
#146962
#145802
#145182
#144342
#144052
#142592
#141372
#140152
#139232
#138802
#137962