Finite Group Databasebeta

← #517708

Subgroups — #517708

Order 512.

3362 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#398C211yesmaximal, characteristic
2C2×#2344C211yesmaximal, characteristic
3C2×#2344C212yesmaximal
4C2×#2274C211yesmaximal, characteristic
5#14339C211yesmaximal, characteristic
6#14314C212yesmaximal, complemented
7#14226C212yesmaximal, complemented
8#14223C212yesmaximal, complemented
9#14208C214yesmaximal, complemented
10#14203C214yesmaximal, complemented
11#13908C211yesmaximal, characteristic
12#13904C211yesmaximal, characteristic
13#13874C212yesmaximal
14#13870C212yesmaximal
15C4×#398C212yesmaximal, complemented
16#13802C211yesmaximal, characteristic
17#13799C212yesmaximal
18C22×#117C2211yescharacteristic
19C22×#117C2212yes—
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2212yes—
23C23×C4⋊C4C2211yescharacteristic
24C2×#398C4116yescomplemented
25C2×#398C2211yescharacteristic
26C2×#398C2212yes—
27C2×#398C2212yes—
28C2×#398C2211yescharacteristic
29C2×#398C2211yescharacteristic
30C2×#398C2211yescharacteristic
31#2344—28no—
32#2344—216no—
33C2×#389C2214yes—
34C2×#389C2212yes—
35C2×#387C2214yes—
36C2×#387C2214yes—
37C2×#387C2214yes—
38C2×#381C2212yes—
39C2×#381C2212yes—
40C2×#381C2212yes—
41C2×#381C2212yes—
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2212yes—
45#2274—28no—
46C2×C4×C4⋊C4C2214yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2212yes—
54C2×C4×C22⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2211yescharacteristic
56C2×C4×C22⋊C4C2211yescharacteristic
57C22×#95C2212yes—
58C22×#95C2211yescharacteristic
59#1445C2212yes—
60#1445C2212yes—
61#1444C2212yes—
62#1443C2214yes—
63#1442C2214yes—
64#1442C2212yes—
65#1441C2212yes—
66#1441C2212yes—
67#1441C2212yes—
68#1441C2212yes—
69#1441C2214yes—
70#1441C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2214yes—
79#1434C2214yes—
80#1434C2214yes—
81#1434C2212yes—
82C22⋊#95C2212yes—
83C22⋊#95C2212yes—
84C22⋊#95C2212yes—
85C4×#95C2214yes—
86C4×#95C2212yes—
87C24×C4C2312yes—
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C2×#117—24no—
93C2×#117—24no—
94C2×#117—24no—
95C2×#117—28no—
96C2×#117—28no—
97C2×#117—24no—
98C2×#117—24no—
99C2×#117—28no—
100C2×#117D418yes—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—28no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—28no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4C2×C418yes—
108C22×C4⋊C4C2×C4116yes—
109C22×C4⋊C4C2×C418yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2311yescharacteristic
130C22×C22⋊C4C2311yescharacteristic
131C22×C22⋊C4C2311yescharacteristic
132C22×C22⋊C4C2311yescharacteristic
133C22×C42C2314yes—
134C22×C42C2312yes—
135C22×C42C2314yes—
136C22×C42C2312yes—
137C22×C42C2312yes—
138C22×C42C2312yes—
139C22×C42C2312yes—
140C22×C42C2312yes—
141#398C2×C418yes—
142#398C2×C4116yes—
143#398C2×C4116yes—
144#398C2×C418yes—
145#398C2×C418yes—
146#398C2×C418yes—
147#389—216no—
148#389—28no—
149#387—216no—
150#387—216no—
151#387—216no—
152#381—28no—
153#381—28no—
154#381—28no—
155#381—28no—
156#380—24no—
157#380—24no—
158#380—28no—
159C4×C4⋊C4—28no—
160C4×C4⋊C4—216no—
161C4×C4⋊C4—28no—
162C4×C4⋊C4—28no—
163C4×C4⋊C4—28no—
164C4×C4⋊C4—28no—
165C4×C22⋊C4—24no—
166C4×C22⋊C4—28no—
167C4×C22⋊C4—24no—
168C4×C22⋊C4—24no—
169C4×C22⋊C4D418yes—
170C2×#95C2×C4116yes—
171C2×#95C2×C418yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2312yes—
194C2×#95C2311yescharacteristic
195C2×#95C2312yes—
196C2×#95C2311yescharacteristic
197C2×#95C2312yes—
198C2×#95C2312yes—
199C2×#95C2312yes—
200C2×#95C2312yes—
201C25C2411yescharacteristic
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—28no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4C22×C418yes—
221C23×C4C22×C418yes—
222C23×C4C22×C414yes—
223C23×C4C22×C414yes—
224C23×C4C22×C414yes—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4#4114yes—
228C23×C4#4114yes—
229C23×C4C2414yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2412yes—
244C23×C4C2412yes—
245#117—232no—
246#117—232no—
247#117—432no—
248C2×C4⋊C4—232no—
249C2×C4⋊C4—264no—
250C2×C4⋊C4—232no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4C22×C418yes—
289C2×C4⋊C4C22×C418yes—
290C2×C4⋊C4C22×C414yes—
291C2×C4⋊C4C22×C414yes—
292C2×C4⋊C4C22×C414yes—
293C2×C4⋊C4C22×C414yes—
294C2×C4⋊C4C22×C418yes—
295C2×C4⋊C4C22×C414yes—
296C2×C4⋊C4C22×C414yes—
297C2×C4⋊C4C22×C418yes—
298C2×C4⋊C4C22×C418yes—
299C2×C4⋊C4C2×D418yes—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4C2×D414yes—
313C2×C22⋊C4C2×D414yes—
314C2×C22⋊C4C2×D414yes—
315C2×C22⋊C4C2×D414yes—
316C2×C22⋊C4#4114yes—
317C2×C22⋊C4#4114yes—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—28no—
324C2×C42—28no—
325C2×C42—28no—
326C2×C42—28no—
327C2×C42—28no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—44no—
332C2×C42—44no—
333C2×C42—44no—
334C2×C42—44no—
335C2×C42—44no—
336C2×C42C2×D418yes—
337C2×C42C2×D418yes—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—24no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—24no—
351#95—28no—
352#95—28no—
353#95—28no—
354#95—28no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95C22×C418yes—
360#95C22×C418yes—
361#95C22×C414yes—
362#95C22×C418yes—
363#95C22×C418yes—
364#95C22×C418yes—
365#95C22×C414yes—
366#95C2×D418yes—
367C24—24no—
368C24—24no—
369C24C23×C412yes—
370C24C22×D412yes—
371C24C22×D412yes—
372C24C2×#4112yes—
373C24C2×#4112yes—
374C24C2×#4112yes—
375C24C2×#4112yes—
376C24C2511yescharacteristic
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—216no—
399C22×C4—216no—
400C22×C4—216no—
401C22×C4—216no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
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—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—28no—
437C22×C4—216no—
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—24no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—416no—
458C22×C4—416no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—84no—
464C22×C4—84no—
465C22×C4C4×D4116yes—
466C22×C4C4×D4116yes—
467C22×C4C23×C414yes—
468C22×C4C23×C412yes—
469C22×C4C23×C412yes—
470C22×C4C23×C414yes—
471C22×C4C23×C414yes—
472C22×C4C23×C412yes—
473C22×C4C23×C412yes—
474C22×C4C23×C412yes—
475C22×C4C23×C412yes—
476C22×C4C23×C414yes—
477C22×C4C22×D414yes—
478C22×C4C22×D414yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D414yes—
484C22×C4C22×D414yes—
485C22×C4C2×#4114yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4114yes—
491C22×C42−1+418yes—
492C4⋊C4—216no—
493C4⋊C4—216no—
494C4⋊C4—28no—
495C4⋊C4—28no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—216no—
499C4⋊C4—28no—
500C4⋊C4—216no—
501C4⋊C4—216no—
502C4⋊C4—216no—
503C4⋊C4—28no—
504C4⋊C4—216no—
505C4⋊C4—416no—
506C4⋊C4—48no—
507C4⋊C4—416no—
508C4⋊C4—84no—
509C22⋊C4—28no—
510C22⋊C4—28no—
511C22⋊C4—28no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4#121116yes—
515C42—216no—
516C42—216no—
517C42—48no—
518C42—416no—
519C42—416no—
520C42—84no—
521C42—84no—
522C42—84no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—28no—
527C23—24no—
528C23—28no—
529C23—24no—
530C23—24no—
531C23—28no—
532C23—28no—
533C23C2×C4×D414yes—
534C23C2×C4×D414yes—
535C23C4×#4114yes—
536C23C2×#12114yes—
537C23#52514yes—
538C23#52914yes—
539C23#53214yes—
540C23C24×C411yescharacteristic
541C23C23×D411yescharacteristic
542C23C23×D411yescharacteristic
543C23C22×#4111yescharacteristic
544C23C22×#4111yescharacteristic
545C23C22×#4111yescharacteristic
546C23C22×#4111yescharacteristic
547C23C2×2−1+414yes—
548C23#58514yes—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—232no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—416no—
597C2×C4—432no—
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—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—432no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—88no—
636C2×C4C2×C4×D414yes—
637C2×C4C2×C4×D414yes—
638C2×C4C2×C4×D414yes—
639C2×C4C2×C4×D414yes—
640C2×C4#51918yes—
641C2×C4C2×#12118yes—
642C22—24no—
643C22—28no—
644C22—24no—
645C22—28no—
646C22—28no—
647C22—28no—
648C22—28no—
649C22—28no—
650C22C4×#12118yes—
651C22C22×C4×D411yescharacteristic
652C22C22×C4×D411yescharacteristic
653C22C2×C4×#4111yescharacteristic
654C22C2×#51912yes—
655C22#342812yes—
656C22C22×#12111yescharacteristic
657C22C2×#52511yescharacteristic
658C22C2×#52911yescharacteristic
659C22C2×#53211yescharacteristic
660C22#348512yes—
661C22#348914yes—
662C22#349112yes—
663C22#349514yes—
664C22#349912yes—
665C22#350912yes—
666C22#351514yes—
667C22#351514yes—
668C4—24no—
669C4—24no—
670C4—28no—
671C4—28no—
672C4—24no—
673C4—24no—
674C4—42no—
675C4—42no—
676C4—44no—
677C4—44no—
678C4—44no—
679C4—44no—
680C4—44no—
681C4—44no—
682C4—84no—
683C4—84no—
684C2—28no—
685C2C2×C4×#12111yescharacteristic
686C2#3419911yescharacteristic
687C2#3420312yes—
688C2#3420911yescharacteristic
689C2#3423912yes—
690C2#3602012yes—
691C2#3602112yes—
692C2#3602312yes—
693C2#3602412yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #14314C2semidirect
7. #14226C2semidirect
8. #14223C2semidirect
9. #14208C2semidirect
10. #14203C2semidirect
15. C4×#398C2semidirect
24. C2×#398C4semidirect

Maximal subgroups

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

subgroupindex
C22×#3982
C2×#23442
C2×#22742
#143392
#143142
#142262
#142232
#142082
#142032
#139082
#139042
#138742
#138702
C4×#3982
#138022
#137992