Finite Group Databasebeta

← #518697

Subgroups — #518697

Order 512.

3134 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2562C211yesmaximal, characteristic
2C2×#2370C211yesmaximal, characteristic, complemented
3C2×D4×C4⋊C4C211yesmaximal, characteristic
4#14326C212yesmaximal, complemented
5#14283C211yesmaximal, characteristic, complemented
6#14242C212yesmaximal, complemented
7#14190C212yesmaximal
8#14116C214yesmaximal, complemented
9#13994C212yesmaximal, complemented
10#13993C212yesmaximal, complemented
11#13986C214yesmaximal, complemented
12#13911C211yesmaximal, characteristic, complemented
13C4×#394C212yesmaximal
14#13847C211yesmaximal, characteristic
15C22⋊C4×C4⋊C4C212yesmaximal, complemented
16C22⋊C4×C4⋊C4C212yesmaximal, complemented
17#13750C211yesmaximal, characteristic, complemented
18C22×#125C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2212yes—
21C23×C4⋊C4C2212yes—
22#2562C4116yescomplemented
23C2×#400C2212yes—
24C2×#394C2212yes—
25#2370—28no—
26D4×C4⋊C4—28no—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#389C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#388C2214yes—
32C2×#388C2214yes—
33C2×#387C2212yes—
34C2×#387C2212yes—
35C2×#385C2214yes—
36C2×#384C2211yescharacteristic
37C2×#382C2212yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2214yes—
52C2×C4×C22⋊C4C2214yes—
53C2×#376C2211yescharacteristic
54#1445C2211yescharacteristic
55#1445C2211yescharacteristic
56#1443C2214yes—
57#1442C2212yes—
58#1442C2214yes—
59#1442C2212yes—
60#1441C2212yes—
61#1441C2214yes—
62#1441C2212yes—
63#1440C2214yes—
64#1439C2212yes—
65#1439C2212yes—
66#1439C2214yes—
67#1438C2211yescharacteristic
68#1438C2211yescharacteristic
69#1437C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80#1434C2214yes—
81C22⋊#95C2214yes—
82C22⋊#95C2214yes—
83C4×#95C2212yes—
84C4×#95C2214yes—
85C4×#95C2212yes—
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2314yes—
89C2×#125—24no—
90C2×#125—24no—
91C2×#125C2×C418yescomplemented
92C2×C4×D4—24no—
93C2×C4×D4—28no—
94C2×C4×D4—24no—
95C2×C4×D4—48no—
96C2×C4×D4C2×C418yes—
97C22×C4⋊C4—28no—
98C22×C4⋊C4—28no—
99C22×C4⋊C4—28no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2314yes—
129C22×C42C2314yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134#400C2×C4116yescomplemented
135#394C2×C4116yes—
136#389—24no—
137#389—24no—
138#389—28no—
139#389—28no—
140#388—48no—
141#388C2×C4132yescomplemented
142#387—28no—
143#387—28no—
144#385—216no—
145#384—24no—
146#382—44no—
147#380—28no—
148#380C2×C4116yescomplemented
149#380C2×C4116yescomplemented
150C4×C4⋊C4—24no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4—28no—
155C4×C4⋊C4—28no—
156C4×C4⋊C4C2×C418yescomplemented
157C4×C4⋊C4D418yes—
158C4×C22⋊C4—28no—
159C4×C22⋊C4—28no—
160C4×C22⋊C4—48no—
161C4×C22⋊C4—48no—
162#376—24no—
163C2×#95C2314yes—
164C2×#95C2311yescharacteristic
165C2×#95C2311yescharacteristic
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2311yescharacteristic
179C2×#95C2311yescharacteristic
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2312yes—
191C25C2412yes—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4C22×C414yes—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—28no—
204C23×C4—28no—
205C23×C4—28no—
206C23×C4—28no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4C22×C418yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2414yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227#125—232no—
228C4×D4—232no—
229C4×D4—432no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—432no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4C22×C414yes—
267C2×C4⋊C4C22×C414yes—
268C2×C4⋊C4C22×C414yes—
269C2×C4⋊C4C22×C418yes—
270C2×C4⋊C4C22×C418yes—
271C2×C4⋊C4C22×C418yes—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4#4118yes—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4C22×C418yes—
297C2×C22⋊C4C22×C4116yes—
298C2×C22⋊C4C22×C4116yes—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—28no—
309C2×C42—28no—
310C2×C42—44no—
311C2×C42—42no—
312C2×C42—42no—
313C2×C42—42no—
314C2×C42—44no—
315C2×C42—44no—
316C2×C42—44no—
317C2×C42—44no—
318C2×C42—44no—
319C2×C42C22×C418yes—
320C2×C42C22×C414yes—
321C2×C42C2×D418yes—
322C2×C42C2×D414yes—
323C2×C42#4118yes—
324C2×C42#4114yes—
325#95—22no—
326#95—24no—
327#95—22no—
328#95—24no—
329#95—24no—
330#95—28no—
331#95—24no—
332#95—24no—
333#95—22no—
334#95—22no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95C22×C418yes—
350#95C22×C4116yes—
351#95C22×C4116yes—
352#95C2×D418yes—
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—
366C24H(5,𝔽2)14yes—
367C242−1+414yes—
368C24C2511yescharacteristic
369C2×D4—28no—
370C2×D4—28no—
371C2×D4—216no—
372C2×D4—216no—
373C2×D4—28no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—216no—
430C22×C4—216no—
431C22×C4—216no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—416no—
450C22×C4—416no—
451C22×C4—416no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—416no—
457C22×C4—88no—
458C22×C4—88no—
459C22×C4—88no—
460C22×C4C23×C414yes—
461C22×C4C23×C418yes—
462C22×C4C23×C414yes—
463C22×C4C23×C414yes—
464C22×C4C23×C412yes—
465C22×C4C23×C412yes—
466C22×C4C23×C412yes—
467C22×C4C22×D414yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D414yes—
471C22×C4C22×D412yes—
472C22×C4C2×#4114yes—
473C22×C4C2×#4114yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4118yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4114yes—
479C22×C4C2×#4114yes—
480C22×C4H(5,𝔽2)12yes—
481C22×C42−1+412yes—
482C4⋊C4—28no—
483C4⋊C4—216no—
484C4⋊C4—216no—
485C4⋊C4—28no—
486C4⋊C4—28no—
487C4⋊C4—216no—
488C4⋊C4—28no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—416no—
494C4⋊C4—48no—
495C4⋊C4—84no—
496C4⋊C4—84no—
497C4⋊C4#124116yes—
498C22⋊C4—216no—
499C22⋊C4—216no—
500C22⋊C4—232no—
501C22⋊C4—216no—
502C22⋊C4—416no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—416no—
506C42—216no—
507C42—48no—
508C42—48no—
509C42—48no—
510C42—44no—
511C42—44no—
512C42—84no—
513C42—88no—
514C42—88no—
515C42C4×D4116yes—
516C23—24no—
517C23—24no—
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—42no—
535C23—42no—
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#52018yes—
551C23C24×C411yescharacteristic
552C23C23×D411yescharacteristic
553C23C22×#4111yescharacteristic
554C23C22×#4112yes—
555C23C22×#4112yes—
556C23C2×H(5,𝔽2)12yes—
557C23C2×H(5,𝔽2)11yescharacteristic
558C23C2×2−1+412yes—
559C23C2×2−1+411yescharacteristic
560C23#58511yescharacteristic
561C23#58511yescharacteristic
562D4—232no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—432no—
633C2×C4—44no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—44no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—44no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—832no—
651C2×C4C2×C4×D414yes—
652C2×C4C2×C4×D414yes—
653C2×C4C2×C4×D414yes—
654C2×C4C4×#4118yes—
655C2×C4C2×#12418yes—
656C2×C4C2×#12414yes—
657C2×C4#52918yes—
658C2×C4#53214yes—
659C22—28no—
660C22—28no—
661C22—28no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22C22×C4×D411yescharacteristic
675C22C2×C4×#4112yes—
676C22C2×#52012yes—
677C22C4×H(5,𝔽2)11yescharacteristic
678C22C4×2−1+411yescharacteristic
679C22C22×#12411yescharacteristic
680C22C2×#52912yes—
681C22C2×#53211yescharacteristic
682C22#348411yescharacteristic
683C22#348611yescharacteristic
684C22#348811yescharacteristic
685C22#348911yescharacteristic
686C22#349212yes—
687C22#349212yes—
688C22#349212yes—
689C22#349412yes—
690C22#349512yes—
691C22#351111yescharacteristic
692C22#351211yescharacteristic
693C22#351312yes—
694C22#351612yes—
695C22#351912yes—
696C22#352312yes—
697C4—28no—
698C4—24no—
699C4—24no—
700C4—48no—
701C4—44no—
702C4—42no—
703C4—42no—
704C4—44no—
705C4—44no—
706C4—44no—
707C4—44no—
708C4—44no—
709C4—84no—
710C4—81no—
711C4—81no—
712C4C4×#12418yes—
713C2—48no—
714C2C2×C4×#12411yescharacteristic
715C2#3420011yescharacteristic
716C2#3426711yescharacteristic
717C2#3430412yes—
718C2#3430512yes—
719C2#3569912yes—
720C2#3573412yes—
721C2#3590311yescharacteristic
722C2#3591011yescharacteristic
723C2#3592311yescharacteristic
724C2#3592611yescharacteristic
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×#2370C2semidirect
4. #14326C2semidirect
5. #14283C2semidirect
6. #14242C2semidirect
8. #14116C2semidirect
9. #13994C2semidirect
10. #13993C2semidirect
11. #13986C2semidirect
12. #13911C2semidirect
15. C22⋊C4×C4⋊C4C2semidirect
16. C22⋊C4×C4⋊C4C2semidirect
17. #13750C2semidirect
22. #2562C4semidirect
91. C2×#125C2×C4semidirect
134. #400C2×C4semidirect
141. #388C2×C4semidirect
148. #380C2×C4semidirect
149. #380C2×C4semidirect
156. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25622
C2×#23702
C2×D4×C4⋊C42
#143262
#142832
#142422
#141902
#141162
#139942
#139932
#139862
#139112
C4×#3942
#138472
C22⋊C4×C4⋊C42
#137502