Finite Group Databasebeta

← #358464

Subgroups — #358464

Order 512.

2534 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#428C211yesmaximal, characteristic
2C2×#2296C211yesmaximal, characteristic
3C22×#389C211yesmaximal, characteristic
4C2×#1736C211yesmaximal, characteristic
5C2×#1736C211yesmaximal, characteristic
6C2×#1736C211yesmaximal, characteristic
7C2×#1728C211yesmaximal, characteristic
8#11328C214yesmaximal, complemented
9#11327C214yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C23×M16C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4—21no—
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C42C2211yescharacteristic
18C2×#428—24no—
19C2×#428—24no—
20C2×#428—24no—
21C22×#107C2212yes—
22C22×#106C2212yes—
23#2296—24no—
24#2296—24no—
25C2×#389—22no—
26C2×#389—24no—
27C2×#389—24no—
28C2×#389C2212yes—
29C2×#389C2214yes—
30C2×#389C2212yes—
31C2×#385—21no—
32C2×#385—21no—
33C2×C4×C4⋊C4—22no—
34C22×#95—21no—
35#1736—24no—
36#1736—24no—
37#1736C418yescomplemented
38C2×#340C2212yes—
39C2×#340C2214yes—
40C2×#340C2212yes—
41C2×#340C2214yes—
42#1728—24no—
43C2×#336C2214yes—
44C24×C4—21no—
45C24×C4—21no—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C22×M16—22no—
50C22×M16—22no—
51C22×M16—22no—
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—22no—
55C22×M16—22no—
56C22×M16—22no—
57C22×M16—22no—
58C22×M16—22no—
59C22×M16C2×C414yes—
60C22×M16D414yes—
61C22×M16Q814yes—
62C22×M16Q814yes—
63C23×C8C2312yes—
64C23×C8C2312yes—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117—22no—
72C2×#117—22no—
73C2×#117C2×C414yes—
74C2×#117D414yes—
75C2×#117D414yes—
76C2×#117Q814yes—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—28no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—22no—
90C22×C4⋊C4—22no—
91C22×C4⋊C4—48no—
92C22×C4⋊C4C2×C414yes—
93C22×C4⋊C4D414yes—
94C22×C4⋊C4D414yes—
95C22×C4⋊C4Q814yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C22⋊C4—21no—
102C22×C22⋊C4—21no—
103C22×C42—21no—
104C22×C42—22no—
105C22×C42—22no—
106C22×C42—22no—
107C22×C42—22no—
108C22×C42—22no—
109C22×C42—21no—
110C22×C42—22no—
111C22×C42D412yes—
112C22×C42C2312yes—
113#428—232no—
114C2×#107—24no—
115C2×#107—24no—
116C2×#107—24no—
117C2×#106—24no—
118C2×#106—24no—
119C2×#106—24no—
120#389—216no—
121#389—28no—
122#389—28no—
123#389—24no—
124#389—24no—
125#389—24no—
126#389—24no—
127#385—28no—
128#385—28no—
129C4×C4⋊C4—216no—
130C2×#95—28no—
131C2×#95—24no—
132C2×#95—22no—
133C2×#95—22no—
134#340—28no—
135#340—24no—
136#340—24no—
137#340C2×C4116yescomplemented
138#336—28no—
139C25C2×D411yescharacteristic
140C23×C4—24no—
141C23×C4—22no—
142C23×C4—24no—
143C23×C4—22no—
144C23×C4—24no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—44no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4C22×C412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4C2×D412yes—
174C23×C4C2×Q812yes—
175C23×C4C2×Q812yes—
176C23×C4C2×Q812yes—
177C23×C4C2×Q812yes—
178C23×C4#4112yes—
179C23×C4#4112yes—
180C23×C4#4112yes—
181C23×C4#4112yes—
182C23×C4C2411yescharacteristic
183C2×M16—28no—
184C2×M16—28no—
185C2×M16—28no—
186C2×M16—28no—
187C2×M16—28no—
188C2×M16—28no—
189C2×M16—28no—
190C2×M16—28no—
191C2×M16—28no—
192C2×M16—44no—
193C2×M16—44no—
194C2×M16—44no—
195C2×M16—44no—
196C2×M16C4⋊C4116yescomplemented
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8C22×C414yes—
208C22×C8C2×D414yes—
209C22×C8C2×Q814yes—
210C22×C8C2×Q814yes—
211#117—28no—
212#117—28no—
213#117—28no—
214#117—28no—
215#117—28no—
216#117—44no—
217#117—44no—
218#117C4⋊C4116yescomplemented
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—216no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—432no—
255C2×C4⋊C4—48no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—48no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4C4⋊C4116yescomplemented
264C2×C4⋊C4C22×C414yes—
265C2×C4⋊C4C22×C414yes—
266C2×C4⋊C4C2×D414yes—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D414yes—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4C2×Q814yes—
272C2×C4⋊C4C2×Q814yes—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—42no—
293C2×C42—44no—
294C2×C42—42no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42—44no—
298#107—216no—
299#106—216no—
300#95—216no—
301#95—28no—
302#95—28no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24C4×D412yes—
308C24#12012yes—
309C24#12012yes—
310C24#12112yes—
311C24C22⋊Q812yes—
312C24C22⋊Q812yes—
313C24C4⋊Q812yes—
314C24C22×D411yescharacteristic
315C22×C4—28no—
316C22×C4—28no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—216no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—48no—
340C22×C4—48no—
341C22×C4—48no—
342C22×C4—48no—
343C22×C4—48no—
344C22×C4—416no—
345C22×C4—416no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—48no—
349C22×C4—44no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—48no—
358C22×C4—48no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—44no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—44no—
371C22×C4—48no—
372C22×C4—88no—
373C22×C4—88no—
374C22×C4C2×C4⋊C414yes—
375C22×C4C2×C4⋊C414yes—
376C22×C4C2×C4⋊C414yes—
377C22×C4C4×D414yes—
378C22×C4C4×D414yes—
379C22×C4C4×D412yes—
380C22×C4C4×Q814yes—
381C22×C4#12014yes—
382C22×C4#12012yes—
383C22×C4#12012yes—
384C22×C4#12114yes—
385C22×C4#12114yes—
386C22×C4#12112yes—
387C22×C4C22⋊Q814yes—
388C22×C4C22⋊Q814yes—
389C22×C4C22⋊Q814yes—
390C22×C4C22⋊Q814yes—
391C22×C4C22⋊Q814yes—
392C22×C4C22⋊Q814yes—
393C22×C4C22⋊Q814yes—
394C22×C4C22⋊Q812yes—
395C22×C4C22⋊Q812yes—
396C22×C4#12314yes—
397C22×C4#12314yes—
398C22×C4#12314yes—
399C22×C4#12514yes—
400C22×C4#12514yes—
401C22×C4#12514yes—
402C22×C4#12714yes—
403C22×C4C4⋊Q814yes—
404C22×C4C4⋊Q814yes—
405C22×C4C4⋊Q814yes—
406C22×C4C4⋊Q814yes—
407C22×C4C4⋊Q814yes—
408C22×C4C4⋊Q812yes—
409C22×C4AGL(1,ℤ/8)14yes—
410C22×C4#13714yes—
411C22×C4C23×C411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×Q811yescharacteristic
418C22×C4C22×Q811yescharacteristic
419C22×C4C22×Q811yescharacteristic
420C22×C4C22×Q811yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4C2×#4111yescharacteristic
425M16—232no—
426M16—416no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8C2×C4⋊C418yescomplemented
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—216no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—416no—
455C4⋊C4—416no—
456C4⋊C4—432no—
457C4⋊C4—42no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—42no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—48no—
464C4⋊C4C2×C4⋊C418yes—
465C4⋊C4C2×C4⋊C418yescomplemented
466C22⋊C4—216no—
467C22⋊C4—48no—
468C42—216no—
469C42—42no—
470C42—42no—
471C42—42no—
472C42—42no—
473C42—48no—
474C42—48no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23#38514yes—
492C23#38514yes—
493C23#38914yes—
494C23#39214yes—
495C23#39514yes—
496C23#39514yes—
497C23#39814yes—
498C23C2×C4×D411yescharacteristic
499C23C2×#12011yescharacteristic
500C23C2×#12011yescharacteristic
501C23C2×#12111yescharacteristic
502C23C2×C22⋊Q811yescharacteristic
503C23C2×C22⋊Q811yescharacteristic
504C23C2×C4⋊Q811yescharacteristic
505C23C2×AGL(1,ℤ/8)12yes—
506C23C2×#13712yes—
507C23#57512yes—
508C23#57512yes—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—28no—
514C2×C4—28no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—416no—
518C2×C4—416no—
519C2×C4—432no—
520C2×C4—48no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—48no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—816no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—84no—
551C2×C4—82no—
552C2×C4—84no—
553C2×C4—82no—
554C2×C4#38418yes—
555C2×C4#38514yes—
556C2×C4#38514yes—
557C2×C4#38518yes—
558C2×C4#38914yes—
559C2×C4#38918yes—
560C2×C4#38918yes—
561C2×C4#39214yes—
562C2×C4#39518yes—
563C2×C4#39514yes—
564C2×C4#39514yes—
565C2×C4#39818yes—
566C2×C4#39818yes—
567C2×C4#39814yes—
568C2×C4#39918yes—
569C2×C4C22×C4⋊C411yescharacteristic
570C2×C4C22×C4⋊C411yescharacteristic
571C2×C4C22×C4⋊C411yescharacteristic
572C2×C4C2×C4×D411yescharacteristic
573C2×C4C2×C4×D411yescharacteristic
574C2×C4C2×C4×Q811yescharacteristic
575C2×C4C2×#12011yescharacteristic
576C2×C4C2×#12111yescharacteristic
577C2×C4C2×#12111yescharacteristic
578C2×C4C2×C22⋊Q811yescharacteristic
579C2×C4C2×C22⋊Q811yescharacteristic
580C2×C4C2×C22⋊Q811yescharacteristic
581C2×C4C2×C22⋊Q811yescharacteristic
582C2×C4C2×C22⋊Q811yescharacteristic
583C2×C4C2×C22⋊Q811yescharacteristic
584C2×C4C2×C22⋊Q811yescharacteristic
585C2×C4C2×#12311yescharacteristic
586C2×C4C2×#12311yescharacteristic
587C2×C4C2×#12311yescharacteristic
588C2×C4C2×#12511yescharacteristic
589C2×C4C2×#12511yescharacteristic
590C2×C4C2×#12511yescharacteristic
591C2×C4C2×#12711yescharacteristic
592C2×C4C2×C4⋊Q811yescharacteristic
593C2×C4C2×C4⋊Q811yescharacteristic
594C2×C4C2×C4⋊Q811yescharacteristic
595C2×C4C2×C4⋊Q811yescharacteristic
596C2×C4C2×C4⋊Q811yescharacteristic
597C8—28no—
598C8—44no—
599C22—24no—
600C22—24no—
601C22—28no—
602C22—24no—
603C22—24no—
604C22—28no—
605C22—24no—
606C22—24no—
607C22—28no—
608C22—28no—
609C22#144418yes—
610C22C2×#38511yescharacteristic
611C22C2×#38511yescharacteristic
612C22C2×#38911yescharacteristic
613C22C2×#39211yescharacteristic
614C22C2×#39511yescharacteristic
615C22C2×#39511yescharacteristic
616C22C2×#39811yescharacteristic
617C22C4×AGL(1,ℤ/8)12yes—
618C22C4×#13712yes—
619C22#300311yescharacteristic
620C22#300411yescharacteristic
621C22#300511yescharacteristic
622C22#300511yescharacteristic
623C22#300612yes—
624C22#300712yes—
625C22#305512yes—
626C22#305612yes—
627C22#307712yes—
628C22#307812yes—
629C22#307912yes—
630C22#308012yes—
631C22#316512yes—
632C22#316612yes—
633C4—44no—
634C4—44no—
635C4—48no—
636C4—48no—
637C4—82no—
638C4—84no—
639C4—82no—
640C4#144418yes—
641C4C2×#38411yescharacteristic
642C4C2×#38511yescharacteristic
643C4C2×#38911yescharacteristic
644C4C2×#38911yescharacteristic
645C4C2×#39511yescharacteristic
646C4C2×#39811yescharacteristic
647C4C2×#39811yescharacteristic
648C4C2×#39911yescharacteristic
649C2—28no—
650C2C2×#144411yescharacteristic
651C2#2087311yescharacteristic
652C2#2087411yescharacteristic
653C2#2087512yes—
654C2#2115912yes—
655C2#2339111yescharacteristic
656C2#2339211yescharacteristic
657C2#2355211yescharacteristic
658C2#2355311yescharacteristic
659C2#2355412yes—
660C2#2367012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #11328C2semidirect
9. #11327C2semidirect
37. #1736C4semidirect
137. #340C2×C4semidirect
196. C2×M16C4⋊C4semidirect
218. #117C4⋊C4semidirect
263. C2×C4⋊C4C4⋊C4semidirect
440. C2×C8C2×C4⋊C4semidirect
465. C4⋊C4C2×C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4282
C2×#22962
C22×#3892
C2×#17362
C2×#17282
#113282
#113272