Finite Group Databasebeta

← #358438

Subgroups — #358438

Order 512.

2242 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#428C211yesmaximal, characteristic
2C22×#427C211yesmaximal, characteristic
3C2×#2317C211yesmaximal, characteristic
4C2×#1736C211yesmaximal, characteristic
5C22×#340C211yesmaximal, characteristic
6C2×#1729C211yesmaximal, characteristic
7#13795C211yesmaximal, characteristic
8#11325C212yesmaximal, complemented
9#11325C212yesmaximal, complemented
10#11324C212yesmaximal, complemented
11#11323C212yesmaximal, complemented
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C2×#428—24no—
20C2×#428—24no—
21C2×#428—24no—
22C2×#427—24no—
23C2×#427—24no—
24C2×#427—24no—
25C22×#107C2211yescharacteristic
26C22×#107C2212yes—
27C22×#107C2211yescharacteristic
28C22×#106C2211yescharacteristic
29C22×#106C2212yes—
30C22×#106C2211yescharacteristic
31#2317—24no—
32#2317—24no—
33C2×#389—22no—
34C2×#385—22no—
35C2×#384C2212yes—
36C2×#382C2212yes—
37#1736—24no—
38C2×#340—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43#1729C418yescomplemented
44C2×#336C2212yes—
45C2×#336C2212yes—
46C4⋊#95—22no—
47C4⋊#95C2212yes—
48C4⋊#95C2212yes—
49C22⋊#95—22no—
50C24×C4—21no—
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C22×M16—22no—
54C22×M16—22no—
55C22×M16—22no—
56C22×M16—22no—
57C22×M16C2×C414yes—
58C22×M16D414yes—
59C22×M16Q814yes—
60C23×C8—22no—
61C23×C8—22no—
62C23×C8—22no—
63C23×C8—22no—
64C23×C8C2×C414yes—
65C23×C8D414yes—
66C23×C8Q814yes—
67C23×C8C2311yescharacteristic
68C23×C8C2312yes—
69C23×C8C2311yescharacteristic
70C2×#117—22no—
71C2×#117—22no—
72C2×#117—22no—
73C2×#117—22no—
74C2×#117—22no—
75C2×#117—22no—
76C2×#117—22no—
77C2×#117—22no—
78C2×#117—22no—
79C2×#117—22no—
80C2×#117—22no—
81C2×#117—22no—
82C2×#117—22no—
83C2×#117—22no—
84C2×#117C2×C414yes—
85C2×#117D414yes—
86C2×#117D414yes—
87C2×#117Q814yes—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—22no—
90C22×C4⋊C4—22no—
91C22×C4⋊C4—22no—
92C22×C4⋊C4—22no—
93C22×C4⋊C4—22no—
94C22×C4⋊C4—22no—
95C22×C4⋊C4—22no—
96C22×C4⋊C4—22no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—22no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4D414yes—
102C22×C4⋊C4D412yes—
103C22×C4⋊C4Q814yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4D412yes—
113C22×C42—21no—
114C22×C42—21no—
115C22×C42C2312yes—
116#428—232no—
117#427—232no—
118C2×#107—22no—
119C2×#107—22no—
120C2×#107—24no—
121C2×#107—22no—
122C2×#107—22no—
123C2×#107—22no—
124C2×#107—22no—
125C2×#107—24no—
126C2×#107—24no—
127C2×#106—22no—
128C2×#106—22no—
129C2×#106—24no—
130C2×#106—22no—
131C2×#106—22no—
132C2×#106—22no—
133C2×#106—22no—
134C2×#106—24no—
135C2×#106—24no—
136#389—216no—
137#385—216no—
138#384—24no—
139#384—24no—
140#382—24no—
141#382—24no—
142C2×#95—24no—
143C2×#95—24no—
144C2×#95—24no—
145#340—24no—
146#340—24no—
147#340—24no—
148#340—24no—
149#336C2×C418yescomplemented
150#336C2×C418yescomplemented
151C25C2×D411yescharacteristic
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—82no—
165C23×C4—82no—
166C23×C4—82no—
167C23×C4—82no—
168C23×C4C22×C412yes—
169C23×C4C2×D414yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4C2×D412yes—
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2×Q812yes—
177C23×C4C2×Q812yes—
178C23×C4C2×Q812yes—
179C23×C4C2×Q812yes—
180C23×C4#4112yes—
181C23×C4#4112yes—
182C23×C4#4112yes—
183C23×C4#4112yes—
184C23×C4C2411yescharacteristic
185C2×M16—28no—
186C2×M16—28no—
187C2×M16—28no—
188C2×M16—28no—
189C2×M16—28no—
190C2×M16—28no—
191C2×M16C4⋊C4116yescomplemented
192C22×C8—28no—
193C22×C8—28no—
194C22×C8—28no—
195C22×C8—28no—
196C22×C8—28no—
197C22×C8—28no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8C4⋊C4116yescomplemented
211C22×C8C22×C412yes—
212C22×C8C22×C412yes—
213C22×C8C22×C414yes—
214C22×C8C2×D412yes—
215C22×C8C2×D412yes—
216C22×C8C2×D414yes—
217C22×C8C2×Q812yes—
218C22×C8C2×Q812yes—
219C22×C8C2×Q814yes—
220#117—28no—
221#117—28no—
222#117—28no—
223#117—28no—
224#117—28no—
225#117—28no—
226#117—28no—
227#117—28no—
228#117—44no—
229#117—44no—
230#117—44no—
231#117—44no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—216no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—84no—
280C2×C4⋊C4—84no—
281C2×C4⋊C4C4⋊C418yes—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2×Q814yes—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4C4⋊C418yes—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—42no—
304C2×C42—42no—
305C2×C42—42no—
306C2×C42—42no—
307C2×C42C22×C414yes—
308C2×C42C2×Q814yes—
309#107—28no—
310#107—28no—
311#107—216no—
312#106—28no—
313#106—28no—
314#106—216no—
315#95—216no—
316#95—84no—
317#95—84no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24C2×C4⋊C412yes—
323C24#12012yes—
324C24#12112yes—
325C24C22⋊Q812yes—
326C24#12312yes—
327C24#12712yes—
328C24C4⋊Q812yes—
329C24C22×D411yescharacteristic
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—44no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—42no—
359C22×C4—44no—
360C22×C4—42no—
361C22×C4—44no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—88no—
369C22×C4—88no—
370C22×C4—88no—
371C22×C4—84no—
372C22×C4—84no—
373C22×C4—84no—
374C22×C4—84no—
375C22×C4—88no—
376C22×C4C2×C4⋊C418yes—
377C22×C4C2×C4⋊C414yes—
378C22×C4C2×C4⋊C414yes—
379C22×C4C2×C4⋊C412yes—
380C22×C4C4×D414yes—
381C22×C4C4×D414yes—
382C22×C4C4×D414yes—
383C22×C4C4×Q814yes—
384C22×C4#12014yes—
385C22×C4#12014yes—
386C22×C4#12012yes—
387C22×C4#12114yes—
388C22×C4#12114yes—
389C22×C4#12112yes—
390C22×C4C22⋊Q814yes—
391C22×C4C22⋊Q814yes—
392C22×C4C22⋊Q814yes—
393C22×C4C22⋊Q814yes—
394C22×C4C22⋊Q814yes—
395C22×C4C22⋊Q814yes—
396C22×C4C22⋊Q814yes—
397C22×C4C22⋊Q814yes—
398C22×C4C22⋊Q812yes—
399C22×C4#12314yes—
400C22×C4#12314yes—
401C22×C4#12312yes—
402C22×C4#12514yes—
403C22×C4#12514yes—
404C22×C4#12514yes—
405C22×C4#12712yes—
406C22×C4C4⋊Q814yes—
407C22×C4C4⋊Q814yes—
408C22×C4C4⋊Q814yes—
409C22×C4C4⋊Q814yes—
410C22×C4C4⋊Q814yes—
411C22×C4C4⋊Q812yes—
412C22×C4C23×C411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×Q811yescharacteristic
419C22×C4C22×Q811yescharacteristic
420C22×C4C22×Q811yescharacteristic
421C22×C4C22×Q811yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C4C2×#4111yescharacteristic
426M16—232no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—22no—
434C2×C8—24no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—232no—
446C2×C8C2×C4⋊C414yes—
447C2×C8C2×C4⋊C414yes—
448C2×C8C2×C4⋊C418yes—
449C4⋊C4—216no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—42no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—42no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—816no—
477C22⋊C4—216no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C42—24no—
481C42—24no—
482C42—24no—
483C42—24no—
484C42—48no—
485C42—48no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23#38414yes—
503C23#38514yes—
504C23#38914yes—
505C23#39214yes—
506C23#39514yes—
507C23#39814yes—
508C23#39914yes—
509C23C22×C4⋊C411yescharacteristic
510C23C2×#12011yescharacteristic
511C23C2×#12111yescharacteristic
512C23C2×C22⋊Q811yescharacteristic
513C23C2×#12311yescharacteristic
514C23C2×#12711yescharacteristic
515C23C2×C4⋊Q811yescharacteristic
516C23#57612yes—
517C23#57712yes—
518C23#57712yes—
519C23#57812yes—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—48no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—48no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—816no—
540C2×C4—816no—
541C2×C4—84no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—84no—
558C2×C4—82no—
559C2×C4#38414yes—
560C2×C4#38514yes—
561C2×C4#38518yes—
562C2×C4#38518yes—
563C2×C4#38914yes—
564C2×C4#38918yes—
565C2×C4#38918yes—
566C2×C4#39214yes—
567C2×C4#39518yes—
568C2×C4#39518yes—
569C2×C4#39514yes—
570C2×C4#39818yes—
571C2×C4#39818yes—
572C2×C4#39814yes—
573C2×C4#39914yes—
574C2×C4C22×C4⋊C411yescharacteristic
575C2×C4C22×C4⋊C411yescharacteristic
576C2×C4C2×C4×D411yescharacteristic
577C2×C4C2×C4×D411yescharacteristic
578C2×C4C2×C4×D411yescharacteristic
579C2×C4C2×C4×Q811yescharacteristic
580C2×C4C2×#12011yescharacteristic
581C2×C4C2×#12011yescharacteristic
582C2×C4C2×#12111yescharacteristic
583C2×C4C2×#12111yescharacteristic
584C2×C4C2×C22⋊Q811yescharacteristic
585C2×C4C2×C22⋊Q811yescharacteristic
586C2×C4C2×C22⋊Q811yescharacteristic
587C2×C4C2×C22⋊Q811yescharacteristic
588C2×C4C2×C22⋊Q811yescharacteristic
589C2×C4C2×C22⋊Q811yescharacteristic
590C2×C4C2×C22⋊Q811yescharacteristic
591C2×C4C2×C22⋊Q811yescharacteristic
592C2×C4C2×#12311yescharacteristic
593C2×C4C2×#12311yescharacteristic
594C2×C4C2×#12511yescharacteristic
595C2×C4C2×#12511yescharacteristic
596C2×C4C2×#12511yescharacteristic
597C2×C4C2×C4⋊Q811yescharacteristic
598C2×C4C2×C4⋊Q811yescharacteristic
599C2×C4C2×C4⋊Q811yescharacteristic
600C2×C4C2×C4⋊Q811yescharacteristic
601C2×C4C2×C4⋊Q811yescharacteristic
602C8—24no—
603C8—24no—
604C8—28no—
605C22—28no—
606C22—24no—
607C22—24no—
608C22—28no—
609C22—28no—
610C22—24no—
611C22—24no—
612C22—28no—
613C22—24no—
614C22—24no—
615C22#144418yes—
616C22C2×#38411yescharacteristic
617C22C2×#38511yescharacteristic
618C22C2×#38911yescharacteristic
619C22C2×#39211yescharacteristic
620C22C2×#39511yescharacteristic
621C22C2×#39811yescharacteristic
622C22C2×#39911yescharacteristic
623C22#291212yes—
624C22#291312yes—
625C22#301212yes—
626C22#301312yes—
627C22#304512yes—
628C22#304612yes—
629C22#308112yes—
630C22#308212yes—
631C22#309612yes—
632C22#309712yes—
633C22#315411yescharacteristic
634C22#315411yescharacteristic
635C22#315511yescharacteristic
636C22#315611yescharacteristic
637C22#316312yes—
638C22#316412yes—
639C4—48no—
640C4—82no—
641C4—82no—
642C4—84no—
643C4—82no—
644C4—82no—
645C4—84no—
646C4#144418yes—
647C4C2×#38511yescharacteristic
648C4C2×#38511yescharacteristic
649C4C2×#38911yescharacteristic
650C4C2×#38911yescharacteristic
651C4C2×#39511yescharacteristic
652C4C2×#39511yescharacteristic
653C4C2×#39811yescharacteristic
654C4C2×#39811yescharacteristic
655C2—28no—
656C2C2×#144411yescharacteristic
657C2#2080211yescharacteristic
658C2#2080311yescharacteristic
659C2#2084912yes—
660C2#2117512yes—
661C2#2342611yescharacteristic
662C2#2342711yescharacteristic
663C2#2355712yes—
664C2#2366212yes—
665C2#2373511yescharacteristic
666C2#2373611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #11325C2semidirect
9. #11325C2semidirect
10. #11324C2semidirect
11. #11323C2semidirect
43. #1729C4semidirect
149. #336C2×C4semidirect
150. #336C2×C4semidirect
191. C2×M16C4⋊C4semidirect
210. C22×C8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4282
C22×#4272
C2×#23172
C2×#17362
C22×#3402
C2×#17292
#137952
#113252
#113242
#113232