Finite Group Databasebeta

← #626073

Subgroups — #626073

Order 512.

3544 conjugacy classes of subgroups.

The subgroup lattice has 781 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2413C212yesmaximal, complemented
2C2×#2618C211yesmaximal, characteristic
3C2×#2618C211yesmaximal, characteristic
4C2×#2316C211yesmaximal, characteristic, complemented
5C22×#389C211yesmaximal, characteristic, complemented
6#15620C214yesmaximal, complemented
7#15618C214yesmaximal, complemented
8#15616C214yesmaximal, complemented
9#15615C214yesmaximal, complemented
10#14927C212yesmaximal, complemented
11#14729C212yesmaximal, complemented
12#14698C211yesmaximal, characteristic
13#14697C211yesmaximal, characteristic, complemented
14#14339C211yesmaximal, characteristic, complemented
15#14135C212yesmaximal, complemented
16C22×#123C2212yes—
17C22×#123C2212yes—
18C22×#121C2211yescharacteristic
19C22×#117C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24C23×C42C2211yescharacteristic
25#2618—28no—
26#2618—28no—
27#2413—216no—
28C2×#398C2212yes—
29C2×#398C2212yes—
30C2×#398C2214yes—
31C2×#396C2212yes—
32C2×#396C2214yes—
33C2×#396C2214yes—
34C2×#396C2212yes—
35C2×#396C2214yes—
36C2×#394C2212yes—
37C2×#394C2212yes—
38#2316—28no—
39C2×#390C2212yes—
40C2×#389—28no—
41C2×#389C2214yes—
42C2×#389C2214yes—
43C2×#389C2214yes—
44C2×#389C2214yes—
45C2×#389C2214yes—
46C2×#389C2212yes—
47C2×#388C2214yes—
48C2×#386C2212yes—
49C2×#386C2212yes—
50C2×#386C2212yes—
51C2×#386C2212yes—
52C2×#385C2212yes—
53C2×#381C2212yes—
54C2×#380C2211yescharacteristic
55C2×#380C2212yes—
56C2×#380C2211yescharacteristic
57C2×#380C2211yescharacteristic
58C22×#95C2211yescharacteristic
59C22×#95C2211yescharacteristic
60#1442C2214yes—
61#1441C2212yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2212yes—
68#1440C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1437C2214yes—
73#1437C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C22⋊#95C2212yes—
79C22⋊#95C2212yes—
80C23×D4C2312yes—
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C2×#123—28no—
90C2×#123—28no—
91C2×#123—28no—
92C2×#123—28no—
93C2×#123—28no—
94C2×#123—44no—
95C2×#121—24no—
96C2×#121—24no—
97C2×#121—24no—
98C2×#117—28no—
99C2×#117—28no—
100C2×#117—44no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4—24no—
109C22×C4⋊C4—44no—
110C22×C4⋊C4D418yes—
111C22×C4⋊C4D418yescomplemented
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2314yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2314yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2314yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2314yes—
141C22×C42—24no—
142C22×C42—44no—
143C22×C42C2312yes—
144C22×C42C2312yes—
145C22×C42C2314yes—
146#398—28no—
147#398—216no—
148#398—28no—
149#396—216no—
150#396—28no—
151#396—28no—
152#396—216no—
153#396—48no—
154#394—28no—
155#394—28no—
156#390—44no—
157#389—216no—
158#389—216no—
159#389—216no—
160#389—216no—
161#389—216no—
162#389—44no—
163#388—216no—
164#386—28no—
165#386—28no—
166#386—28no—
167#386—44no—
168#385—28no—
169#381—28no—
170#380—24no—
171#380—24no—
172#380—28no—
173#380D418yes—
174C2×#95—24no—
175C2×#95—24no—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2312yes—
194C25C2412yes—
195C25C2411yescharacteristic
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—42no—
207C22×D4—42no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—24no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4—42no—
238C23×C4C2×D414yes—
239C23×C4C2×D414yes—
240C23×C4C2×D414yes—
241C23×C4C2×D414yes—
242C23×C4C2×D414yes—
243C23×C4C2×D414yes—
244C23×C4C2×D414yes—
245C23×C4#4114yes—
246C23×C4#4114yes—
247C23×C4C2412yes—
248C23×C4C2414yes—
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2412yes—
252C23×C4C2414yes—
253C23×C4C2412yes—
254C23×C4C2412yes—
255C23×C4C2412yes—
256C23×C4C2412yes—
257C23×C4C2411yescharacteristic
258C23×C4C2411yescharacteristic
259C23×C4C2412yes—
260#123—432no—
261#123—432no—
262#121—416no—
263#117—432no—
264C2×C4⋊C4—232no—
265C2×C4⋊C4—232no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—28no—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—416no—
306C2×C4⋊C4—416no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4C2×D418yes—
314C2×C4⋊C4C2×D418yes—
315C2×C4⋊C4C2×D4116yes—
316C2×C4⋊C4C2×D418yes—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—28no—
329C2×C22⋊C4—28no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—28no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—28no—
338C2×C22⋊C4—28no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—28no—
342C2×C22⋊C4—28no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—44no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—44no—
348C2×C22⋊C4—44no—
349C2×C22⋊C4—44no—
350C2×C22⋊C4C2×D418yes—
351C2×C42—28no—
352C2×C42—28no—
353C2×C42—24no—
354C2×C42—24no—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42—44no—
358C2×C42—42no—
359C2×C42—42no—
360C2×C42—416no—
361#95—28no—
362#95—28no—
363#95—28no—
364#95—24no—
365#95—24no—
366#95—28no—
367#95—28no—
368#95—28no—
369#95—28no—
370#95—24no—
371#95—24no—
372#95—28no—
373#95—28no—
374#95—44no—
375#95—44no—
376#95—44no—
377#95—44no—
378#95C2×D418yes—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—24no—
389C24—22no—
390C24—24no—
391C24—24no—
392C24—22no—
393C24—24no—
394C24C22×D412yes—
395C24C22×D412yes—
396C24C22×D412yes—
397C24C2×#4112yes—
398C24C2×#4112yes—
399C24H(5,𝔽2)12yes—
400C242−1+412yes—
401C24C2511yescharacteristic
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—48no—
406C2×D4—48no—
407C2×D4—48no—
408C2×D4—48no—
409C2×D4—48no—
410C2×D4—44no—
411C2×D4—82no—
412C2×D4—82no—
413C2×D4—84no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—216no—
417C22×C4—28no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—28no—
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—216no—
437C22×C4—28no—
438C22×C4—28no—
439C22×C4—28no—
440C22×C4—28no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—28no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—24no—
481C22×C4—22no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—22no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—48no—
496C22×C4—48no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—48no—
500C22×C4—48no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—44no—
507C22×C4—44no—
508C22×C4—44no—
509C22×C4—44no—
510C22×C4—84no—
511C22×C4—84no—
512C22×C4—84no—
513C22×C4—84no—
514C22×C4#121116yes—
515C22×C4#121116yes—
516C22×C4C22×D414yes—
517C22×C4C22×D414yes—
518C22×C4C22×D418yes—
519C22×C4C22×D412yes—
520C22×C4C22×D414yes—
521C22×C4C22×D412yes—
522C22×C4C22×D412yes—
523C22×C4C22×D414yes—
524C22×C4C22×D412yes—
525C22×C4C22×D414yes—
526C22×C4C2×#4114yes—
527C22×C4C2×#4112yes—
528C22×C4C2×#4112yes—
529C22×C4H(5,𝔽2)14yes—
530C22×C42−1+414yes—
531C22×C42−1+414yes—
532C4⋊C4—216no—
533C4⋊C4—232no—
534C4⋊C4—216no—
535C4⋊C4—48no—
536C4⋊C4—416no—
537C4⋊C4—416no—
538C4⋊C4—416no—
539C4⋊C4—48no—
540C4⋊C4—48no—
541C4⋊C4—44no—
542C4⋊C4—48no—
543C4⋊C4—48no—
544C4⋊C4—416no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4—84no—
548C4⋊C4—88no—
549C22⋊C4—216no—
550C22⋊C4—416no—
551C22⋊C4—48no—
552C22⋊C4—48no—
553C22⋊C4—48no—
554C22⋊C4—48no—
555C22⋊C4—48no—
556C22⋊C4—416no—
557C22⋊C4—84no—
558C22⋊C4—84no—
559C22⋊C4—88no—
560C22⋊C4—88no—
561C42—48no—
562C42—48no—
563C42—416no—
564C23—24no—
565C23—24no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—24no—
575C23—24no—
576C23—24no—
577C23—24no—
578C23—24no—
579C23—24no—
580C23—42no—
581C23—42no—
582C23—44no—
583C23—42no—
584C23—44no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—44no—
601C23—44no—
602C23—44no—
603C23C2×#12114yes—
604C23C2×#12114yes—
605C23#53714yes—
606C23D4214yes—
607C23#54714yes—
608C23#54814yes—
609C23#55314yes—
610C23C23×D411yescharacteristic
611C23C23×D411yescharacteristic
612C23C23×D411yescharacteristic
613C23C22×#4111yescharacteristic
614C23C22×#4111yescharacteristic
615C23C2×H(5,𝔽2)11yescharacteristic
616C23C2×H(5,𝔽2)12yes—
617C23C2×2−1+411yescharacteristic
618C23C2×2−1+412yes—
619C23C2×2−1+412yes—
620C23#58512yes—
621D4—88no—
622D4—816no—
623C2×C4—232no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—232no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—28no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—24no—
653C2×C4—28no—
654C2×C4—28no—
655C2×C4—24no—
656C2×C4—24no—
657C2×C4—416no—
658C2×C4—416no—
659C2×C4—42no—
660C2×C4—416no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—44no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—42no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—44no—
695C2×C4—44no—
696C2×C4—44no—
697C2×C4—44no—
698C2×C4—44no—
699C2×C4—44no—
700C2×C4—44no—
701C2×C4—44no—
702C2×C4—44no—
703C2×C4—88no—
704C2×C4—88no—
705C2×C4C2×#12114yes—
706C2×C4C2×#12118yes—
707C2×C4C2×#12114yes—
708C2×C4#53718yes—
709C22—24no—
710C22—28no—
711C22—24no—
712C22—24no—
713C22—24no—
714C22—24no—
715C22—28no—
716C22—24no—
717C22—28no—
718C22—28no—
719C22—82no—
720C22—84no—
721C22—82no—
722C22—82no—
723C22—82no—
724C22—82no—
725C22—82no—
726C22—84no—
727C22—82no—
728C22—84no—
729C22—84no—
730C22#241618yes—
731C22C22×#12111yescharacteristic
732C22C22×#12111yescharacteristic
733C22C2×#53712yes—
734C22C2×#53711yescharacteristic
735C22#346112yes—
736C22C2×D4211yescharacteristic
737C22C2×#54711yescharacteristic
738C22C2×#54811yescharacteristic
739C22C2×#55311yescharacteristic
740C22#348411yescharacteristic
741C22#348511yescharacteristic
742C22#348611yescharacteristic
743C22#348611yescharacteristic
744C22#349212yes—
745C22#349612yes—
746C22#349712yes—
747C22#349912yes—
748C22#350512yes—
749C22#350912yes—
750C22#353612yes—
751C22#353812yes—
752C22#355912yes—
753C22#356312yes—
754C4—24no—
755C4—24no—
756C4—28no—
757C4—28no—
758C4—44no—
759C4—44no—
760C4—44no—
761C4—48no—
762C4—82no—
763C4—82no—
764C4—82no—
765C4—82no—
766C4—84no—
767C2—28no—
768C2—162no—
769C2C2×#241611yescharacteristic
770C2#3515311yescharacteristic
771C2#3522311yescharacteristic
772C2#3535811yescharacteristic
773C2#3566711yescharacteristic
774C2#3671911yescharacteristic
775C2#3696811yescharacteristic
776C2#3935912yes—
777C2#3942112yes—
778C2#3948712yes—
779C2#3954912yes—
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×#2413C2semidirect
4. C2×#2316C2semidirect
5. C22×#389C2semidirect
6. #15620C2semidirect
7. #15618C2semidirect
8. #15616C2semidirect
9. #15615C2semidirect
10. #14927C2semidirect
11. #14729C2semidirect
13. #14697C2semidirect
14. #14339C2semidirect
15. #14135C2semidirect
111. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24132
C2×#26182
C2×#23162
C22×#3892
#156202
#156182
#156162
#156152
#149272
#147292
#146982
#146972
#143392
#141352