Finite Group Databasebeta

← #674775

Subgroups — #674775

Order 512.

3530 conjugacy classes of subgroups.

The subgroup lattice has 785 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2536C211yesmaximal, characteristic, complemented
2C2×#2535C211yesmaximal, characteristic
3C2×#2426C212yesmaximal, complemented
4C2×#2351C211yesmaximal, characteristic, complemented
5C2×C4×#124C211yesmaximal, characteristic, complemented
6#15621C214yesmaximal, complemented
7#15617C214yesmaximal, complemented
8#15220C212yesmaximal, complemented
9#15126C212yesmaximal, complemented
10#14922C214yesmaximal, complemented
11#14707C211yesmaximal, characteristic, complemented
12#14201C214yesmaximal, complemented
13#13969C212yesmaximal, complemented
14#13941C211yesmaximal, characteristic, complemented
15C4×#390C211yesmaximal, characteristic, complemented
16C22×#124C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×#121C2212yes—
19C22×C4×Q8C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23#2536—28no—
24#2535—28no—
25#2426—216no—
26C2×#398C2212yes—
27C2×#397C2212yes—
28C2×#396C2214yes—
29C2×#394C2212yes—
30C2×#394C2214yes—
31C2×#394C2214yes—
32C2×#392C2212yes—
33#2351—28no—
34C4×#124—28no—
35C2×#391C2211yescharacteristic
36C2×#390C2212yes—
37C2×#390C2214yes—
38C2×#390C2211yescharacteristic
39C2×#390C2212yes—
40C2×#389C2214yes—
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#388C2214yes—
44C2×#388C2214yes—
45C2×#388C2214yes—
46C2×#386C2214yes—
47C2×#382C2212yes—
48C2×#382C2211yescharacteristic
49C2×#382C2211yescharacteristic
50C2×#382C2211yescharacteristic
51C2×#380C2212yes—
52C2×#380C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C22⋊C4C2214yes—
56C2×C4×C22⋊C4C2212yes—
57C2×C4×C22⋊C4C2212yes—
58C2×C43C2211yescharacteristic
59#1443C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1440C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1438C2214yes—
69#1437C2214yes—
70#1437C2212yes—
71#1437C2212yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C22⋊#95C2212yes—
79C22⋊#95C2212yes—
80C23×Q8C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C23×D4C2314yes—
84C24×C4C2312yes—
85C24×C4C2312yes—
86C2×#124—24no—
87C2×#124—24no—
88C2×#124D418yescomplemented
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×#121—28no—
93C2×#121—28no—
94C2×#121—44no—
95C2×C4×Q8—28no—
96C2×C4×Q8—28no—
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×C4×Q8—44no—
100C2×C4×Q8D418yescomplemented
101C2×C4×D4—24no—
102C2×C4×D4—24no—
103C2×C4×D4—24no—
104C2×C4×D4—24no—
105C2×C4×D4—24no—
106C2×C4×D4—24no—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2314yes—
135C22×C42C2312yes—
136C22×C42C2312yes—
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139C22×C42C2311yescharacteristic
140#398—28no—
141#397—28no—
142#396—216no—
143#394—28no—
144#394—216no—
145#394—216no—
146#392—28no—
147#391—24no—
148#390—28no—
149#390—24no—
150#390—216no—
151#390—28no—
152#389—28no—
153#389—28no—
154#389—216no—
155#388—216no—
156#388—216no—
157#388—216no—
158#386—216no—
159#382—24no—
160#382—24no—
161#382—28no—
162#382D418yes—
163#380—28no—
164#380—28no—
165C4×C4⋊C4—24no—
166C4×C4⋊C4—28no—
167C4×C22⋊C4—28no—
168C4×C22⋊C4—216no—
169C4×C22⋊C4—44no—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2311yescharacteristic
178C2×#95C2311yescharacteristic
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C43—24no—
191C25C2412yes—
192C25C2412yes—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8C2×D414yes—
198C22×Q8C2×D414yes—
199C22×Q8#4114yes—
200C22×D4—24no—
201C22×D4—28no—
202C22×D4—28no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—28no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—42no—
214C22×D4—44no—
215C22×D4—44no—
216C22×D4—44no—
217C22×D4—44no—
218C22×D4C2×D414yes—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2414yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244#124—232no—
245C22⋊Q8—432no—
246#121—432no—
247C4×Q8—232no—
248C4×Q8—432no—
249C4×D4—416no—
250C4×D4—416no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4C2×D418yes—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×D418yes—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—42no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4—42no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4C2×D4116yes—
325C2×C22⋊C4#4118yes—
326C2×C42—24no—
327C2×C42—22no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—24no—
331C2×C42—28no—
332C2×C42—24no—
333C2×C42—28no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—44no—
342C2×C42—42no—
343C2×C42—42no—
344C2×C42—42no—
345C2×C42C2×D418yes—
346C2×C42C2×D414yes—
347C2×C42C2×D414yes—
348C2×C42C2×D414yes—
349C2×C42#4114yes—
350#95—28no—
351#95—24no—
352#95—28no—
353#95—24no—
354#95—22no—
355#95—22no—
356#95—28no—
357#95—28no—
358#95—24no—
359#95—28no—
360#95—24no—
361#95—28no—
362#95—28no—
363#95—44no—
364#95—44no—
365#95—44no—
366#95—42no—
367#95—42no—
368#95C2×D418yes—
369#95C2×D418yes—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—24no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—24no—
389C24—24no—
390C24—24no—
391C24—22no—
392C24—22no—
393C24—22no—
394C24C22×D414yes—
395C24C2×#4114yes—
396C24C2511yescharacteristic
397C2×Q8—28no—
398C2×Q8—28no—
399C2×Q8—28no—
400C2×Q8—28no—
401C2×Q8—28no—
402C2×Q8—28no—
403C2×Q8#121116yes—
404C2×D4—28no—
405C2×D4—28no—
406C2×D4—28no—
407C2×D4—28no—
408C2×D4—48no—
409C2×D4—48no—
410C2×D4—44no—
411C2×D4—416no—
412C2×D4—416no—
413C2×D4—416no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—88no—
420C2×D4—88no—
421C2×D4—88no—
422C2×D4—88no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—416no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—48no—
494C22×C4—88no—
495C22×C4—84no—
496C22×C4—84no—
497C22×C4C22×D414yes—
498C22×C4C22×D414yes—
499C22×C4C22×D418yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D414yes—
504C22×C4C22×D412yes—
505C22×C4C22×D414yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C2×#4114yes—
510C22×C4C2×#4114yes—
511C22×C4C2×#4112yes—
512C22×C4C2×#4112yes—
513C22×C4C2×#4114yes—
514C22×C4C2×#4112yes—
515C22×C4C2×#4112yes—
516C22×C4H(5,𝔽2)12yes—
517C22×C42−1+412yes—
518C4⋊C4—216no—
519C4⋊C4—28no—
520C4⋊C4—216no—
521C4⋊C4—48no—
522C4⋊C4—416no—
523C4⋊C4—416no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—48no—
527C4⋊C4—416no—
528C4⋊C4—88no—
529C22⋊C4—232no—
530C22⋊C4—48no—
531C22⋊C4—416no—
532C22⋊C4—48no—
533C22⋊C4—416no—
534C22⋊C4—416no—
535C22⋊C4—48no—
536C22⋊C4—416no—
537C22⋊C4—48no—
538C22⋊C4—48no—
539C22⋊C4—48no—
540C22⋊C4—416no—
541C22⋊C4—88no—
542C22⋊C4—88no—
543C42—216no—
544C42—28no—
545C42—48no—
546C42—416no—
547C42—48no—
548C42—44no—
549C42—48no—
550C42#121116yes—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—28no—
558C23—24no—
559C23—24no—
560C23—28no—
561C23—24no—
562C23—24no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—44no—
577C23—44no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—44no—
583C23—42no—
584C23—44no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—44no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—44no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—44no—
600C23—42no—
601C23—42no—
602C23—44no—
603C23—42no—
604C23—44no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—44no—
609C23C23×D411yescharacteristic
610C23C23×D411yescharacteristic
611C23C23×D411yescharacteristic
612C23C22×#4111yescharacteristic
613C23C22×#4111yescharacteristic
614C23C22×#4112yes—
615C23C2×H(5,𝔽2)11yescharacteristic
616C23C2×2−1+411yescharacteristic
617C23#58512yes—
618C23#58512yes—
619C23#58512yes—
620Q8—232no—
621D4—416no—
622D4—416no—
623D4—832no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—24no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—28no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—28no—
645C2×C4—24no—
646C2×C4—28no—
647C2×C4—28no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—42no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—44no—
696C2×C4—42no—
697C2×C4—816no—
698C2×C4—816no—
699C2×C4C2×#12114yes—
700C2×C4C2×#12114yes—
701C2×C4C2×#12118yes—
702C2×C4C2×#12114yes—
703C2×C4C2×#12114yes—
704C2×C4#53714yes—
705C2×C4D4214yes—
706C2×C4#54714yes—
707C2×C4#54814yes—
708C2×C4#55314yes—
709C22—44no—
710C22—44no—
711C22—44no—
712C22—48no—
713C22—44no—
714C22—44no—
715C22—44no—
716C22—48no—
717C22—44no—
718C22—44no—
719C22—44no—
720C22—44no—
721C22—82no—
722C22—84no—
723C22—82no—
724C22—82no—
725C22—82no—
726C22—82no—
727C22—82no—
728C22—82no—
729C22—82no—
730C22—82no—
731C22—82no—
732C22—82no—
733C22—84no—
734C22—82no—
735C22C22×#12111yescharacteristic
736C22C22×#12111yescharacteristic
737C22#344411yescharacteristic
738C22C2×#53711yescharacteristic
739C22#345511yescharacteristic
740C22C2×D4211yescharacteristic
741C22C2×#54711yescharacteristic
742C22C2×#54811yescharacteristic
743C22D4×#4112yes—
744C22D4×#4112yes—
745C22D4×#4112yes—
746C22C2×#55311yescharacteristic
747C22#348012yes—
748C22#348712yes—
749C22#350012yes—
750C22#350012yes—
751C22#350512yes—
752C22#351011yescharacteristic
753C22#351011yescharacteristic
754C22#351512yes—
755C22#352412yes—
756C22#355112yes—
757C22#356512yes—
758C4—24no—
759C4—28no—
760C4—24no—
761C4—44no—
762C4—44no—
763C4—48no—
764C4—44no—
765C4—44no—
766C4—84no—
767C4—82no—
768C4—82no—
769C4—82no—
770C4#241618yes—
771C2—84no—
772C2—84no—
773C2C2×#241611yescharacteristic
774C2#3474311yescharacteristic
775C2#3567711yescharacteristic
776C2#3613511yescharacteristic
777C2#3643012yes—
778C2#3653911yescharacteristic
779C2#3918811yescharacteristic
780C2#3937411yescharacteristic
781C2#3988312yes—
782C2#5965812yes—
783C2#5981812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2535C2
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×#2536C2semidirect
3. C2×#2426C2semidirect
4. C2×#2351C2semidirect
5. C2×C4×#124C2semidirect
6. #15621C2semidirect
7. #15617C2semidirect
8. #15220C2semidirect
9. #15126C2semidirect
10. #14922C2semidirect
11. #14707C2semidirect
12. #14201C2semidirect
13. #13969C2semidirect
14. #13941C2semidirect
15. C4×#390C2semidirect
88. C2×#124D4semidirect
100. C2×C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25362
C2×#25352
C2×#24262
C2×#23512
C2×C4×#1242
#156212
#156172
#152202
#151262
#149222
#147072
#142012
#139692
#139412
C4×#3902