Finite Group Databasebeta

← #622144

Subgroups — #622144

Order 512.

3336 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2853C212yesmaximal, complemented
2C2×#2623C212yesmaximal, complemented
3C2×#2618C211yesmaximal, characteristic
4C22×#389C211yesmaximal, characteristic, complemented
5#15808C214yesmaximal, complemented
6#15804C214yesmaximal, complemented
7#15765C214yesmaximal, complemented
8#15696C214yesmaximal, complemented
9#14946C212yesmaximal, complemented
10#14777C212yesmaximal, complemented
11#14152C211yesmaximal, characteristic
12#14137C211yesmaximal, characteristic
13#13965C211yesmaximal, characteristic, complemented
14#13880C211yesmaximal, characteristic, complemented
15#13776C211yesmaximal, characteristic, complemented
16C22×#123C2212yes—
17C22×#121C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2853—216no—
24#2623—216no—
25#2618—28no—
26C2×#400C2214yes—
27C2×#399C2214yes—
28C2×#399C2214yes—
29C2×#398C2214yes—
30C2×#398C2212yes—
31C2×#397C2214yes—
32C2×#396C2212yes—
33C2×#396C2212yes—
34C2×#394C2212yes—
35C2×#390C2212yes—
36C2×#389—28no—
37C2×#389C2214yes—
38C2×#389C2214yes—
39C2×#388C2214yes—
40C2×#388C2214yes—
41C2×#387C2214yes—
42C2×#387C2214yes—
43C2×#386C2212yes—
44C2×#386C2212yes—
45C2×#386C2212yes—
46C2×#386C2212yes—
47C2×#386C2212yes—
48C2×#385C2212yes—
49C2×#385C2212yes—
50C2×#385C2212yes—
51C2×#385C2212yes—
52C2×#384C2212yes—
53C2×#381C2212yes—
54C2×#380C2211yescharacteristic
55C2×C4×C22⋊C4C2211yescharacteristic
56C2×C4×C22⋊C4C2211yescharacteristic
57C22×#95C2211yescharacteristic
58C22×#95C2211yescharacteristic
59#1445C2212yes—
60#1445C2214yes—
61#1445C2214yes—
62#1445C2214yes—
63#1444C2214yes—
64#1444C2212yes—
65#1444C2212yes—
66#1444C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1438C2212yes—
72#1437C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2214yes—
78C22⋊#95C2212yes—
79C23×D4C2312yes—
80C23×D4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C2×#123—28no—
89C2×#123—28no—
90C2×#123—28no—
91C2×#121—24no—
92C2×#121—24no—
93C2×#121—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—28no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—44no—
101C22×C4⋊C4—44no—
102C22×C4⋊C4D418yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C42—28no—
131C22×C42—24no—
132C22×C42C2314yes—
133C22×C42C2312yes—
134C22×C42C2312yes—
135#400—216no—
136#399—216no—
137#399—48no—
138#398—28no—
139#398—216no—
140#397—216no—
141#396—28no—
142#396—28no—
143#394—28no—
144#390—44no—
145#389—216no—
146#389—216no—
147#388—216no—
148#388—216no—
149#387—216no—
150#387—216no—
151#386—28no—
152#386—28no—
153#386—28no—
154#386—28no—
155#386—28no—
156#385—28no—
157#385—28no—
158#385—28no—
159#385—28no—
160#384—44no—
161#381D4116yescomplemented
162#380—24no—
163C4×C22⋊C4—24no—
164C4×C22⋊C4—24no—
165C2×#95—24no—
166C2×#95—24no—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C25C2412yes—
188C25C2411yescharacteristic
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—42no—
200C22×D4—42no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4C2×D414yes—
227C23×C4C2×D418yes—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4#4114yes—
231C23×C4#4114yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2414yes—
238C23×C4C2414yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245#123—432no—
246#121—416no—
247C2×C4⋊C4—232no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—416no—
284C2×C4⋊C4—416no—
285C2×C4⋊C4—416no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4C2×D418yes—
293C2×C4⋊C4C2×D418yes—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4C2×D418yes—
327C2×C22⋊C4C2×D418yes—
328C2×C42—28no—
329C2×C42—28no—
330C2×C42—28no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—42no—
334C2×C42—42no—
335C2×C42—416no—
336C2×C42—42no—
337C2×C42—42no—
338#95—28no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95—24no—
349#95—24no—
350#95—28no—
351#95—28no—
352#95—24no—
353#95—24no—
354#95—44no—
355#95—44no—
356#95C2×D4116yes—
357#95C2×D4116yes—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—24no—
368C24—24no—
369C24—22no—
370C24—22no—
371C24—24no—
372C24—24no—
373C24C22×D412yes—
374C24C22×D414yes—
375C24C2×#4112yes—
376C24C2×#4112yes—
377C242−1+414yes—
378C24C2511yescharacteristic
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—48no—
385C2×D4—48no—
386C2×D4—48no—
387C2×D4—44no—
388C2×D4—82no—
389C2×D4—82no—
390C2×D4—84no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—216no—
394C22×C4—216no—
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—28no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—416no—
464C22×C4—416no—
465C22×C4—48no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—48no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—88no—
478C22×C4—88no—
479C22×C4#123116yes—
480C22×C4#127116yes—
481C22×C4C22×D418yes—
482C22×C4C22×D418yes—
483C22×C4C22×D414yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D414yes—
487C22×C4C22×D414yes—
488C22×C4C22×D414yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C4C2×#4112yes—
493C22×C4H(5,𝔽2)14yes—
494C22×C42−1+414yes—
495C4⋊C4—216no—
496C4⋊C4—216no—
497C4⋊C4—416no—
498C4⋊C4—416no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—416no—
502C4⋊C4—44no—
503C4⋊C4—416no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—88no—
509C4⋊C4—88no—
510C22⋊C4—216no—
511C22⋊C4—216no—
512C22⋊C4—416no—
513C22⋊C4—416no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—84no—
519C22⋊C4—84no—
520C22⋊C4—88no—
521C22⋊C4—88no—
522C42—48no—
523C42—416no—
524C42—84no—
525C23—24no—
526C23—24no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—28no—
532C23—28no—
533C23—24no—
534C23—24no—
535C23—28no—
536C23—28no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—44no—
541C23—44no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23C2×#12314yes—
561C23C2×#12714yes—
562C23#53614yes—
563C23#54718yes—
564C23#54718yes—
565C23C23×D411yescharacteristic
566C23C23×D412yes—
567C23C22×#4111yescharacteristic
568C23C22×#4111yescharacteristic
569C23C2×H(5,𝔽2)12yes—
570C23C2×2−1+412yes—
571C23C2×2−1+412yes—
572C23#58512yes—
573C23#58512yes—
574D4—88no—
575D4—816no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—232no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—28no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—416no—
605C2×C4—416no—
606C2×C4—416no—
607C2×C4—416no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—88no—
652C2×C4—88no—
653C2×C4C2×#12314yes—
654C2×C4C2×#12314yes—
655C2×C4C2×#12718yes—
656C22—24no—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—216no—
661C22—216no—
662C22—24no—
663C22—24no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—84no—
669C22—84no—
670C22—84no—
671C22—84no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22#250118yes—
676C22C22×#12311yescharacteristic
677C22C22×#12711yescharacteristic
678C22C2×#53611yescharacteristic
679C22#345912yes—
680C22#346112yes—
681C22C2×#54712yes—
682C22C2×#54712yes—
683C22#348411yescharacteristic
684C22#348511yescharacteristic
685C22#348611yescharacteristic
686C22#348711yescharacteristic
687C22#348814yes—
688C22#348914yes—
689C22#350512yes—
690C22#350712yes—
691C22#355214yes—
692C22#355314yes—
693C4—24no—
694C4—24no—
695C4—28no—
696C4—44no—
697C4—48no—
698C4—48no—
699C4—44no—
700C4—44no—
701C4—82no—
702C4—82no—
703C4—82no—
704C4—82no—
705C4—82no—
706C2—28no—
707C2—162no—
708C2C2×#250111yescharacteristic
709C2#3498311yescharacteristic
710C2#3518211yescharacteristic
711C2#3556511yescharacteristic
712C2#3563411yescharacteristic
713C2#3712011yescharacteristic
714C2#3723311yescharacteristic
715C2#3779014yes—
716C2#3779114yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2853C2semidirect
2. C2×#2623C2semidirect
4. C22×#389C2semidirect
5. #15808C2semidirect
6. #15804C2semidirect
7. #15765C2semidirect
8. #15696C2semidirect
9. #14946C2semidirect
10. #14777C2semidirect
13. #13965C2semidirect
14. #13880C2semidirect
15. #13776C2semidirect
161. #381D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28532
C2×#26232
C2×#26182
C22×#3892
#158082
#158042
#157652
#156962
#149462
#147772
#141522
#141372
#139652
#138802
#137762