Finite Group Databasebeta

← #4310241

Subgroups — #4310241

Order 512.

2128 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2644C211yesmaximal, characteristic
2C2×#2647C211yesmaximal, characteristic, complemented
3#16550C214yesmaximal, complemented
4#16532C214yesmaximal, complemented
5#16469C214yesmaximal, complemented
6#16434C214yesmaximal, complemented
7#15231C211yesmaximal, characteristic, complemented
8#15224C211yesmaximal, characteristic
9#15222C211yesmaximal, characteristic
10#15197C211yesmaximal, characteristic, complemented
11#15196C211yesmaximal, characteristic, complemented
12#15155C211yesmaximal, characteristic
13#15151C211yesmaximal, characteristic, complemented
14#15134C211yesmaximal, characteristic
15#14712C211yesmaximal, characteristic, complemented
16#14700C211yesmaximal, characteristic
17#14276C211yesmaximal, characteristic, complemented
18#14260C211yesmaximal, characteristic
19#13759C211yesmaximal, characteristic, complemented
20#2647—28no—
21#2644—28no—
22C2×#400C2212yes—
23C2×#400C2212yes—
24C2×#399C2214yes—
25C2×#399C2214yes—
26C2×#398C2212yes—
27C2×#398C2212yes—
28C2×#398C2214yes—
29C2×#397C2212yes—
30C2×#397C2212yes—
31C2×#397C2212yes—
32C2×#396C2212yes—
33C2×#396C2214yes—
34C2×#390C2211yescharacteristic
35C2×#390C2211yescharacteristic
36C2×#390C2211yescharacteristic
37C2×#390C2211yescharacteristic
38C2×#390C2211yescharacteristic
39C2×#390C2211yescharacteristic
40C2×#390C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#389C2214yes—
49C2×#388C2214yes—
50C2×#384C2211yescharacteristic
51C2×#384C2211yescharacteristic
52C2×#384C2211yescharacteristic
53C2×#384C2211yescharacteristic
54C2×#382C2211yescharacteristic
55C2×#382C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×#376C2211yescharacteristic
59C2×#376C2211yescharacteristic
60C2×C43C2211yescharacteristic
61#1445C2214yes—
62#1444C2214yes—
63#1442C2214yes—
64#1442C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1440C2214yes—
71#1440C2214yes—
72#1440C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2214yes—
77#1437C2214yes—
78#1436C2212yes—
79#1436C2212yes—
80#1436C2212yes—
81#1436C2212yes—
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2211yescharacteristic
84C4⋊#95C2211yescharacteristic
85C4⋊#95C2211yescharacteristic
86C4⋊#95C2214yes—
87C4⋊#95C2211yescharacteristic
88C4⋊#95C2211yescharacteristic
89C4⋊#95C2211yescharacteristic
90C4⋊#95C2211yescharacteristic
91C4×#95C2211yescharacteristic
92C4×#95C2211yescharacteristic
93C23×D4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134#400—28no—
135#400—28no—
136#399—48no—
137#399—48no—
138#398—28no—
139#398—28no—
140#398—48no—
141#397—28no—
142#397—28no—
143#397—28no—
144#396—28no—
145#396—48no—
146#390—24no—
147#390—24no—
148#390—24no—
149#390—24no—
150#390—24no—
151#390—42no—
152#390—42no—
153#389—24no—
154#389—24no—
155#389—24no—
156#389—24no—
157#389—24no—
158#389—24no—
159#389—24no—
160#389—48no—
161#388—48no—
162#384—24no—
163#384—24no—
164#384—24no—
165#384—42no—
166#382—24no—
167#382—42no—
168C4×C4⋊C4—24no—
169C4×C4⋊C4—24no—
170#376—42no—
171#376D418yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C2×#95C2314yes—
198C2×#95C2311yescharacteristic
199C2×#95C2311yescharacteristic
200C2×#95C2311yescharacteristic
201C2×#95C2311yescharacteristic
202C43—42no—
203C25C2412yes—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
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—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—42no—
327C2×C42—42no—
328C2×C42—42no—
329C2×C42C2×D414yes—
330C2×C42C2×D414yes—
331C2×C42C2×D414yes—
332C2×C42#4114yes—
333C2×C42#4114yes—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—28no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—44no—
350#95—42no—
351#95—42no—
352#95—42no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95C2×D414yes—
361#95C2×D414yes—
362#95C2×D414yes—
363#95C2×D414yes—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—24no—
368C24—24no—
369C24—22no—
370C24—24no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—24no—
375C24C2511yescharacteristic
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—44no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
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—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—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C2×#4114yes—
473C22×C4C2×#4114yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4H(5,𝔽2)14yes—
477C22×C4H(5,𝔽2)14yes—
478C22×C4H(5,𝔽2)12yes—
479C22×C42−1+414yes—
480C22×C42−1+412yes—
481C22×C42−1+412yes—
482C22×C42−1+412yes—
483C4⋊C4—48no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—416no—
493C4⋊C4—88no—
494C4⋊C4—84no—
495C4⋊C4—84no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C4⋊C4—84no—
502C4⋊C4—84no—
503C4⋊C4—88no—
504C22⋊C4—416no—
505C22⋊C4—84no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4—88no—
512C22⋊C4—88no—
513C22⋊C4—88no—
514C42—28no—
515C42—28no—
516C42—28no—
517C42—28no—
518C42—28no—
519C42—48no—
520C42—48no—
521C42—44no—
522C42—44no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—44no—
529C23—42no—
530C23—42no—
531C23—44no—
532C23—44no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23C23×D411yescharacteristic
547C23C22×#4111yescharacteristic
548C23C22×#4111yescharacteristic
549C23C2×H(5,𝔽2)11yescharacteristic
550C23C2×H(5,𝔽2)12yes—
551C23C2×H(5,𝔽2)12yes—
552C23C2×2−1+411yescharacteristic
553C23C2×2−1+411yescharacteristic
554C23C2×2−1+411yescharacteristic
555C23C2×2−1+412yes—
556C23#58512yes—
557D4—88no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
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—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
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—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4#53214yes—
651C2×C4#53614yes—
652C2×C4#53714yes—
653C2×C4#53914yes—
654C2×C4#53914yes—
655C2×C4#54018yes—
656C2×C4#54114yes—
657C2×C4#54114yes—
658C22—82no—
659C22—84no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—84no—
666C22—84no—
667C22—82no—
668C22—84no—
669C22C2×#53211yescharacteristic
670C22C2×#53611yescharacteristic
671C22C2×#53711yescharacteristic
672C22C2×#53911yescharacteristic
673C22C2×#53911yescharacteristic
674C22C2×#54012yes—
675C22C2×#54111yescharacteristic
676C22C2×#54111yescharacteristic
677C22#346112yes—
678C22#349512yes—
679C22#349811yescharacteristic
680C22#349811yescharacteristic
681C22#349911yescharacteristic
682C22#350011yescharacteristic
683C22#350812yes—
684C22#353612yes—
685C22#354212yes—
686C22#354412yes—
687C22#354712yes—
688C22#355012yes—
689C22#355212yes—
690C22#355412yes—
691C22#355912yes—
692C4—28no—
693C4—42no—
694C4—42no—
695C4—42no—
696C4—42no—
697C4—42no—
698C4—44no—
699C4—44no—
700C4—42no—
701C4—82no—
702C4—82no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4#266418yes—
710C2—162no—
711C2C2×#266411yescharacteristic
712C2#3527811yescharacteristic
713C2#3531811yescharacteristic
714C2#3729911yescharacteristic
715C2#3730111yescharacteristic
716C2#3730311yescharacteristic
717C2#3730411yescharacteristic
718C2#3733012yes—
719C2#3733712yes—
720C2#3750212yes—
721C2#3762812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2647C2semidirect
3. #16550C2semidirect
4. #16532C2semidirect
5. #16469C2semidirect
6. #16434C2semidirect
7. #15231C2semidirect
10. #15197C2semidirect
11. #15196C2semidirect
13. #15151C2semidirect
15. #14712C2semidirect
17. #14276C2semidirect
19. #13759C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26442
C2×#26472
#165502
#165322
#164692
#164342
#152312
#152242
#152222
#151972
#151962
#151552
#151512
#151342
#147122
#147002
#142762
#142602
#137592