Finite Group Databasebeta

← #358486

Subgroups — #358486

Order 512.

2126 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#428C211yesmaximal, characteristic
2#34052C211yesmaximal, characteristic
3C2×#2317C211yesmaximal, characteristic
4C2×#1742C211yesmaximal, characteristic
5C2×#1736C211yesmaximal, characteristic
6C2×#1733C211yesmaximal, characteristic
7C2×#1733C211yesmaximal, characteristic
8#11340C214yesmaximal, complemented
9#11339C214yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C23×M16C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×#428—24no—
18C2×#428—22no—
19C2×#428—24no—
20C2×#428C2214yes—
21C22×#107—21no—
22C22×#106—21no—
23#2317—24no—
24#2317—24no—
25#2317—21no—
26#2317—21no—
27#2317C2212yes—
28#2317C2212yes—
29#2316—22no—
30#2315—22no—
31#2296—22no—
32#2296—22no—
33C2×#389—22no—
34C2×#385—22no—
35#2285—22no—
36C2×#384C2212yes—
37C2×#382C2212yes—
38#1742—24no—
39C2×#343C2212yes—
40C2×#343C2212yes—
41#1736—22no—
42#1736C2214yes—
43C2×#340—22no—
44#1733—24no—
45#1733C418yescomplemented
46C2×#339C2214yes—
47C2×#339C2214yes—
48C24×C4—21no—
49C24×C4C2311yescharacteristic
50C22×M16—22no—
51C22×M16—21no—
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—21no—
55C22×M16—22no—
56C22×M16—22no—
57C22×M16—22no—
58C22×M16—22no—
59C22×M16—22no—
60C22×M16C2×C414yes—
61C22×M16D414yes—
62C22×M16Q814yes—
63C22×M16Q814yes—
64C22×M16C2312yes—
65C22×M16C2312yes—
66C23×C8—21no—
67C23×C8—21no—
68C2×#117—22no—
69C2×#117—21no—
70C2×#117—21no—
71C2×#117—22no—
72C2×#117—22no—
73C2×#117—22no—
74C2×#117—22no—
75C2×#117—21no—
76C2×#117—21no—
77C2×#117—22no—
78C2×#117—22no—
79C2×#117—21no—
80C2×#117—22no—
81C2×#117—21no—
82C2×#117—21no—
83C2×#117—21no—
84C2×#117—22no—
85C2×#117—22no—
86C2×#117—22no—
87C2×#117—22no—
88C2×#117—22no—
89C2×#117C2×C414yes—
90C2×#117C2×C414yes—
91C2×#117D412yes—
92C2×#117D412yes—
93C2×#117D414yes—
94C2×#117Q814yes—
95C2×#117Q814yes—
96C2×#117C2312yes—
97C2×#117C2312yes—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—22no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—22no—
105C22×C4⋊C4—22no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4D411yescharacteristic
110C22×C4⋊C4D414yes—
111C22×C4⋊C4D412yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4D412yes—
119C22×C42—21no—
120C22×C42—21no—
121C22×C42D412yes—
122C22×C42C2312yes—
123#428—216no—
124#428—28no—
125#428—28no—
126C2×#107—28no—
127C2×#107—24no—
128C2×#106—28no—
129C2×#106—24no—
130#389—24no—
131#389—24no—
132#389—216no—
133#389—24no—
134#389—24no—
135#385—24no—
136#385—24no—
137#385—216no—
138#385—24no—
139#385—24no—
140#384—24no—
141#384—22no—
142#384—22no—
143#384—24no—
144#382—24no—
145#382—22no—
146#382—22no—
147#382—24no—
148#380—28no—
149C4×C4⋊C4—28no—
150C2×#95—24no—
151#343—24no—
152#343—24no—
153#340—28no—
154#339—28no—
155#339C2×C4116yescomplemented
156C25C2×D411yescharacteristic
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—24no—
163C23×C4—41no—
164C23×C4—41no—
165C23×C4—42no—
166C23×C4—84no—
167C23×C4C22×C412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D412yes—
178C23×C4C2×Q812yes—
179C23×C4C2×Q812yes—
180C23×C4C2×Q812yes—
181C23×C4C2×Q812yes—
182C23×C4#4112yes—
183C23×C4#4112yes—
184C23×C4#4112yes—
185C23×C4#4112yes—
186C23×C4C2411yescharacteristic
187C2×M16—24no—
188C2×M16—28no—
189C2×M16—24no—
190C2×M16—24no—
191C2×M16—28no—
192C2×M16—28no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—22no—
196C2×M16—24no—
197C2×M16—22no—
198C2×M16—28no—
199C2×M16—28no—
200C2×M16—22no—
201C2×M16—22no—
202C2×M16—22no—
203C2×M16—22no—
204C2×M16—42no—
205C2×M16—44no—
206C2×M16—44no—
207C2×M16—42no—
208C2×M16C4⋊C4116yescomplemented
209C2×M16C22×C414yes—
210C2×M16C2×D414yes—
211C2×M16C2×Q814yes—
212C2×M16C2×Q814yes—
213C22×C8—24no—
214C22×C8—24no—
215C22×C8—24no—
216C22×C8—22no—
217C22×C8—24no—
218C22×C8—22no—
219C22×C8—42no—
220C22×C8—42no—
221#117—24no—
222#117—28no—
223#117—24no—
224#117—24no—
225#117—28no—
226#117—24no—
227#117—28no—
228#117—24no—
229#117—22no—
230#117—22no—
231#117—24no—
232#117—24no—
233#117—22no—
234#117—22no—
235#117—22no—
236#117—22no—
237#117—42no—
238#117—42no—
239#117—42no—
240#117—42no—
241#117—42no—
242#117—42no—
243#117—42no—
244#117—42no—
245#117—42no—
246#117—42no—
247#117—42no—
248#117—42no—
249#117C4⋊C418yescomplemented
250#117C4⋊C4116yescomplemented
251#117C22×C414yes—
252#117C2×Q814yes—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—216no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—41no—
290C2×C4⋊C4—41no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—88no—
297C2×C4⋊C4C4⋊C418yescomplemented
298C2×C4⋊C4C2×D414yes—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D414yes—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—42no—
316C2×C22⋊C4—42no—
317C2×C42—24no—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—42no—
331C2×C42—42no—
332C2×C42C22×C414yes—
333C2×C42C2×D414yes—
334C2×C42C2×Q814yes—
335#107—216no—
336#106—216no—
337#95—216no—
338#95—28no—
339#95—28no—
340C24—42no—
341C24—42no—
342C24C4×D412yes—
343C24#12011yescharacteristic
344C24#12011yescharacteristic
345C24#12012yes—
346C24#12112yes—
347C24C22⋊Q812yes—
348C24C22⋊Q812yes—
349C24C4⋊Q812yes—
350C24C22×D411yescharacteristic
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—28no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—48no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—82no—
377C22×C4—82no—
378C22×C4—82no—
379C22×C4—82no—
380C22×C4—88no—
381C22×C4—84no—
382C22×C4—84no—
383C22×C4—88no—
384C22×C4C2×C4⋊C414yes—
385C22×C4C2×C4⋊C412yes—
386C22×C4C2×C4⋊C414yes—
387C22×C4C2×C4⋊C412yes—
388C22×C4C4×D412yes—
389C22×C4C4×D412yes—
390C22×C4C4×D414yes—
391C22×C4C4×D412yes—
392C22×C4C4×Q814yes—
393C22×C4#12012yes—
394C22×C4#12012yes—
395C22×C4#12012yes—
396C22×C4#12011yescharacteristic
397C22×C4#12011yescharacteristic
398C22×C4#12112yes—
399C22×C4#12112yes—
400C22×C4#12112yes—
401C22×C4#12112yes—
402C22×C4#12112yes—
403C22×C4C22⋊Q812yes—
404C22×C4C22⋊Q812yes—
405C22×C4C22⋊Q814yes—
406C22×C4C22⋊Q812yes—
407C22×C4C22⋊Q812yes—
408C22×C4C22⋊Q814yes—
409C22×C4C22⋊Q812yes—
410C22×C4C22⋊Q812yes—
411C22×C4C22⋊Q814yes—
412C22×C4C22⋊Q814yes—
413C22×C4C22⋊Q812yes—
414C22×C4C22⋊Q812yes—
415C22×C4#12314yes—
416C22×C4#12312yes—
417C22×C4#12314yes—
418C22×C4#12312yes—
419C22×C4#12514yes—
420C22×C4#12514yes—
421C22×C4#12514yes—
422C22×C4#12712yes—
423C22×C4#12712yes—
424C22×C4C4⋊Q814yes—
425C22×C4C4⋊Q812yes—
426C22×C4C4⋊Q812yes—
427C22×C4C4⋊Q814yes—
428C22×C4C4⋊Q812yes—
429C22×C4C4⋊Q812yes—
430C22×C4C4⋊Q814yes—
431C22×C4C4⋊Q812yes—
432C22×C4C23×C411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×Q811yescharacteristic
439C22×C4C22×Q811yescharacteristic
440C22×C4C22×Q811yescharacteristic
441C22×C4C22×Q811yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4C2×#4111yescharacteristic
444C22×C4C2×#4111yescharacteristic
445C22×C4C2×#4111yescharacteristic
446M16—24no—
447M16—24no—
448M16—24no—
449M16—216no—
450M16—24no—
451M16—24no—
452M16—42no—
453M16—42no—
454M16—48no—
455M16C2×C4⋊C418yes—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—216no—
460C2×C8—24no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—48no—
464C4⋊C4—24no—
465C4⋊C4—216no—
466C4⋊C4—28no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—42no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—42no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—88no—
488C4⋊C4—88no—
489C4⋊C4—816no—
490C4⋊C4C2×C4⋊C418yes—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—28no—
494C22⋊C4—216no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C42—24no—
501C42—24no—
502C42—24no—
503C42—24no—
504C42—24no—
505C42—28no—
506C42—42no—
507C42—42no—
508C42—44no—
509C42—44no—
510C42—44no—
511C42C2×C4⋊C418yes—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—42no—
519C23—42no—
520C23—44no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23#38512yes—
527C23#38512yes—
528C23#38514yes—
529C23#38914yes—
530C23#39212yes—
531C23#39212yes—
532C23#39512yes—
533C23#39512yes—
534C23#39514yes—
535C23#39814yes—
536C23C2×C4×D411yescharacteristic
537C23C2×#12011yescharacteristic
538C23C2×#12011yescharacteristic
539C23C2×#12111yescharacteristic
540C23C2×C22⋊Q811yescharacteristic
541C23C2×C22⋊Q811yescharacteristic
542C23C2×C4⋊Q811yescharacteristic
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—44no—
546C2×C4—48no—
547C2×C4—44no—
548C2×C4—48no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—48no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—88no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4—88no—
568C2×C4—88no—
569C2×C4—84no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4#38414yes—
573C2×C4#38414yes—
574C2×C4#38512yes—
575C2×C4#38514yes—
576C2×C4#38512yes—
577C2×C4#38514yes—
578C2×C4#38514yes—
579C2×C4#38914yes—
580C2×C4#38914yes—
581C2×C4#38914yes—
582C2×C4#38914yes—
583C2×C4#38914yes—
584C2×C4#39212yes—
585C2×C4#39212yes—
586C2×C4#39514yes—
587C2×C4#39514yes—
588C2×C4#39514yes—
589C2×C4#39512yes—
590C2×C4#39512yes—
591C2×C4#39814yes—
592C2×C4#39814yes—
593C2×C4#39814yes—
594C2×C4#39814yes—
595C2×C4#39814yes—
596C2×C4#39914yes—
597C2×C4#39914yes—
598C2×C4C22×C4⋊C411yescharacteristic
599C2×C4C22×C4⋊C411yescharacteristic
600C2×C4C22×C4⋊C411yescharacteristic
601C2×C4C2×C4×D411yescharacteristic
602C2×C4C2×C4×D411yescharacteristic
603C2×C4C2×C4×Q811yescharacteristic
604C2×C4C2×#12011yescharacteristic
605C2×C4C2×#12111yescharacteristic
606C2×C4C2×#12111yescharacteristic
607C2×C4C2×C22⋊Q811yescharacteristic
608C2×C4C2×C22⋊Q811yescharacteristic
609C2×C4C2×C22⋊Q811yescharacteristic
610C2×C4C2×C22⋊Q811yescharacteristic
611C2×C4C2×C22⋊Q811yescharacteristic
612C2×C4C2×C22⋊Q811yescharacteristic
613C2×C4C2×C22⋊Q811yescharacteristic
614C2×C4C2×#12311yescharacteristic
615C2×C4C2×#12311yescharacteristic
616C2×C4C2×#12311yescharacteristic
617C2×C4C2×#12511yescharacteristic
618C2×C4C2×#12511yescharacteristic
619C2×C4C2×#12511yescharacteristic
620C2×C4C2×#12711yescharacteristic
621C2×C4C2×C4⋊Q811yescharacteristic
622C2×C4C2×C4⋊Q811yescharacteristic
623C2×C4C2×C4⋊Q811yescharacteristic
624C2×C4C2×C4⋊Q811yescharacteristic
625C2×C4C2×C4⋊Q811yescharacteristic
626C8—28no—
627C8—44no—
628C22—22no—
629C22—22no—
630C22—24no—
631C22—24no—
632C22—21no—
633C22—21no—
634C22—22no—
635C22—22no—
636C22—21no—
637C22—21no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—24no—
642C22—22no—
643C22—24no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22#144414yes—
649C22#144414yes—
650C22C2×#38511yescharacteristic
651C22C2×#38511yescharacteristic
652C22C2×#38911yescharacteristic
653C22C2×#39211yescharacteristic
654C22C2×#39511yescharacteristic
655C22C2×#39511yescharacteristic
656C22C2×#39811yescharacteristic
657C22#302112yes—
658C22#302212yes—
659C4—48no—
660C4—48no—
661C4—82no—
662C4—82no—
663C4—84no—
664C4—82no—
665C4—82no—
666C4#144414yes—
667C4#144414yes—
668C4C2×#38411yescharacteristic
669C4C2×#38511yescharacteristic
670C4C2×#38911yescharacteristic
671C4C2×#38911yescharacteristic
672C4C2×#39511yescharacteristic
673C4C2×#39811yescharacteristic
674C4C2×#39811yescharacteristic
675C4C2×#39911yescharacteristic
676C2—21no—
677C2—21no—
678C2—21no—
679C2—24no—
680C2—24no—
681C2—21no—
682C2C2×#144411yescharacteristic
683C2#2086512yes—
684C2#2340011yescharacteristic
685C2#2340111yescharacteristic
686C2#2356212yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #11340C2semidirect
9. #11339C2semidirect
45. #1733C4semidirect
155. #339C2×C4semidirect
208. C2×M16C4⋊C4semidirect
249. #117C4⋊C4semidirect
250. #117C4⋊C4semidirect
297. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4282
#340522
C2×#23172
C2×#17422
C2×#17362
C2×#17332
#113402
#113392