Finite Group Databasebeta

← #719395

Subgroups — #719395

Order 512.

3448 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2501C211yesmaximal, characteristic
2C2×#2413C212yesmaximal, complemented
3C2×#2618C211yesmaximal, characteristic
4C22×#398C211yesmaximal, characteristic, complemented
5C2×#2344C211yesmaximal, characteristic, complemented
6#16550C214yesmaximal, complemented
7#15661C214yesmaximal, complemented
8#15629C214yesmaximal, complemented
9#15069C212yesmaximal, complemented
10#14912C212yesmaximal, complemented
11#14788C214yesmaximal, complemented
12#14152C211yesmaximal, characteristic
13#14149C211yesmaximal, characteristic, complemented
14#14135C212yesmaximal, complemented
15#13776C211yesmaximal, characteristic, complemented
16C22×#123C2212yes—
17C22×#123C2212yes—
18C22×#123C2212yes—
19C22×#121C2211yescharacteristic
20C22×#117C2212yes—
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2212yes—
23C23×C4⋊C4C2211yescharacteristic
24#2618—28no—
25#2501—28no—
26#2413—216no—
27C2×#399C2212yes—
28C2×#399C2212yes—
29C2×#398—28no—
30C2×#398C2212yes—
31C2×#398C2214yes—
32C2×#398C2214yes—
33C2×#398C2212yes—
34C2×#398C2212yes—
35C2×#396C2214yes—
36C2×#396C2214yes—
37C2×#394C2212yes—
38#2344—28no—
39C2×#390C2212yes—
40C2×#389C2214yes—
41C2×#389C2212yes—
42C2×#388C2214yes—
43C2×#387C2214yes—
44C2×#386C2212yes—
45C2×#386C2212yes—
46C2×#386C2212yes—
47C2×#385C2212yes—
48C2×#384C2212yes—
49C2×#381C2212yes—
50C2×#381C2211yescharacteristic
51C2×#381C2211yescharacteristic
52C2×#380C2211yescharacteristic
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C22⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2211yescharacteristic
56C22×#95C2212yes—
57C22×#95C2211yescharacteristic
58#1445C2212yes—
59#1444C2214yes—
60#1444C2214yes—
61#1442C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1440C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1438C2212yes—
71#1437C2214yes—
72#1437C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77C22⋊#95C2211yescharacteristic
78C22⋊#95C2212yes—
79C22⋊#95C2211yescharacteristic
80C23×D4C2312yes—
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2312yes—
84C24×C4C2312yes—
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C2×#123—28no—
88C2×#123—28no—
89C2×#123—28no—
90C2×#123—28no—
91C2×#123—28no—
92C2×#123—28no—
93C2×#123—44no—
94C2×#123—44no—
95C2×#123—44no—
96C2×#121—24no—
97C2×#121—24no—
98C2×#121—24no—
99C2×#117—28no—
100C2×#117—28no—
101C2×#117—44no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—28no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—28no—
108C22×C4⋊C4—24no—
109C22×C4⋊C4—44no—
110C22×C4⋊C4D418yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2311yescharacteristic
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2311yescharacteristic
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2311yescharacteristic
138C22×C22⋊C4C2311yescharacteristic
139C22×C22⋊C4C2314yes—
140C22×C22⋊C4C2314yes—
141C22×C42C2314yes—
142C22×C42C2312yes—
143C22×C42C2312yes—
144#399—28no—
145#399—28no—
146#398—216no—
147#398—28no—
148#398—28no—
149#398—28no—
150#398—216no—
151#396—216no—
152#396—216no—
153#394—28no—
154#390—28no—
155#389—28no—
156#389—216no—
157#388—216no—
158#387—216no—
159#386—28no—
160#386—28no—
161#386—44no—
162#385—28no—
163#384—28no—
164#381—28no—
165#381—24no—
166#381D418yescomplemented
167#380D418yes—
168C4×C4⋊C4—28no—
169C4×C22⋊C4—24no—
170C4×C22⋊C4—24no—
171C2×#95—24no—
172C2×#95—28no—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C25C2412yes—
193C25C2411yescharacteristic
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—44no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4—28no—
219C23×C4—22no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4C2×D414yes—
228C23×C4C2×D418yes—
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4#4114yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2414yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2414yes—
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245#123—432no—
246#123—432no—
247#123—432no—
248#121—416no—
249#117—432no—
250C2×C4⋊C4—232no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
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—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—432no—
295C2×C4⋊C4—416no—
296C2×C4⋊C4C2×D418yes—
297C2×C4⋊C4C2×D414yes—
298C2×C4⋊C4C2×D414yes—
299C2×C4⋊C4C2×D418yes—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—28no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—28no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—28no—
326C2×C22⋊C4—28no—
327C2×C22⋊C4—28no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—42no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—42no—
332C2×C22⋊C4—42no—
333C2×C22⋊C4—42no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4C2×D414yes—
339C2×C22⋊C4C2×D418yes—
340C2×C22⋊C4C2×D414yes—
341C2×C22⋊C4#4114yes—
342C2×C42—28no—
343C2×C42—28no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—44no—
347C2×C42—44no—
348C2×C42—44no—
349#95—24no—
350#95—24no—
351#95—28no—
352#95—28no—
353#95—24no—
354#95—24no—
355#95—28no—
356#95—28no—
357#95—28no—
358#95—28no—
359#95—28no—
360#95—28no—
361#95—28no—
362#95—44no—
363#95—44no—
364#95—44no—
365#95—44no—
366#95C2×D418yes—
367#95C2×D4116yes—
368C24—24no—
369C24—24no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—24no—
378C24—24no—
379C24—24no—
380C24—24no—
381C24C22×D412yes—
382C24C22×D412yes—
383C24C22×D412yes—
384C24C2×#4112yes—
385C24C2×#4112yes—
386C24H(5,𝔽2)12yes—
387C24H(5,𝔽2)12yes—
388C24C2511yescharacteristic
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—48no—
397C2×D4—44no—
398C2×D4—84no—
399C2×D4—84no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—216no—
403C22×C4—216no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—216no—
407C22×C4—28no—
408C22×C4—216no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
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—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—416no—
472C22×C4—416no—
473C22×C4—48no—
474C22×C4—44no—
475C22×C4—48no—
476C22×C4—44no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—44no—
481C22×C4—48no—
482C22×C4—44no—
483C22×C4—48no—
484C22×C4—44no—
485C22×C4—84no—
486C22×C4—84no—
487C22×C4#121116yes—
488C22×C4C22×D418yes—
489C22×C4C22×D414yes—
490C22×C4C22×D414yes—
491C22×C4C22×D414yes—
492C22×C4C22×D414yes—
493C22×C4C22×D414yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C2×#4114yes—
499C22×C4C2×#4112yes—
500C22×C4C2×#4112yes—
501C22×C42−1+418yes—
502C4⋊C4—28no—
503C4⋊C4—28no—
504C4⋊C4—216no—
505C4⋊C4—48no—
506C4⋊C4—416no—
507C4⋊C4—416no—
508C4⋊C4—416no—
509C4⋊C4—416no—
510C4⋊C4—416no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—48no—
516C4⋊C4—48no—
517C4⋊C4—44no—
518C22⋊C4—28no—
519C22⋊C4—216no—
520C22⋊C4—28no—
521C22⋊C4—28no—
522C22⋊C4—416no—
523C22⋊C4—416no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—44no—
529C22⋊C4—48no—
530C22⋊C4—88no—
531C22⋊C4—88no—
532C42—416no—
533C42—48no—
534C42—48no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—28no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—28no—
543C23—28no—
544C23—28no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—44no—
555C23—42no—
556C23—42no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23C2×#12114yes—
569C23#53214yes—
570C23#54014yes—
571C23D4214yes—
572C23D4214yes—
573C23#54614yes—
574C23#55014yes—
575C23C23×D411yescharacteristic
576C23C23×D411yescharacteristic
577C23C23×D411yescharacteristic
578C23C22×#4111yescharacteristic
579C23C22×#4111yescharacteristic
580C23C2×H(5,𝔽2)11yescharacteristic
581C23C2×H(5,𝔽2)11yescharacteristic
582C23C2×2−1+414yes—
583C23#58514yes—
584D4—88no—
585D4—816no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—28no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—416no—
614C2×C4—432no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—432no—
622C2×C4—416no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
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—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—88no—
664C2×C4C2×#12114yes—
665C2×C4C2×#12114yes—
666C22—24no—
667C22—28no—
668C22—24no—
669C22—28no—
670C22—28no—
671C22—28no—
672C22—28no—
673C22—28no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—84no—
682C22—84no—
683C22—84no—
684C22—84no—
685C22#242818yes—
686C22C22×#12111yescharacteristic
687C22C2×#53211yescharacteristic
688C22#344912yes—
689C22C2×#54011yescharacteristic
690C22C2×D4211yescharacteristic
691C22C2×D4211yescharacteristic
692C22C2×#54611yescharacteristic
693C22C2×#55011yescharacteristic
694C22#348512yes—
695C22#348612yes—
696C22#348914yes—
697C22#349112yes—
698C22#349912yes—
699C22#350912yes—
700C22#351514yes—
701C22#353514yes—
702C22#355914yes—
703C4—24no—
704C4—24no—
705C4—44no—
706C4—48no—
707C4—44no—
708C4—44no—
709C4—44no—
710C4—44no—
711C4—44no—
712C4—48no—
713C4—82no—
714C4—82no—
715C4—82no—
716C2—28no—
717C2—162no—
718C2C2×#242811yescharacteristic
719C2#3508911yescharacteristic
720C2#3566711yescharacteristic
721C2#3602412yes—
722C2#3765312yes—
723C2#3802912yes—
724C2#4139512yes—
725C2#4349712yes—
726C2#4444712yes—
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
2. C2×#2413C2semidirect
4. C22×#398C2semidirect
5. C2×#2344C2semidirect
6. #16550C2semidirect
7. #15661C2semidirect
8. #15629C2semidirect
9. #15069C2semidirect
10. #14912C2semidirect
11. #14788C2semidirect
13. #14149C2semidirect
14. #14135C2semidirect
15. #13776C2semidirect
166. #381D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25012
C2×#24132
C2×#26182
C22×#3982
C2×#23442
#165502
#156612
#156292
#150692
#149122
#147882
#141522
#141492
#141352
#137762