Finite Group Databasebeta

← #358276

Subgroups — #358276

Order 512.

1686 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34052C211yesmaximal, characteristic
2C22×#344C211yesmaximal, characteristic
3C2×#1740C211yesmaximal, characteristic
4C2×#1733C211yesmaximal, characteristic
5C2×#1733C211yesmaximal, characteristic
6C2×#1733C211yesmaximal, characteristic
7C22×H(3,ℤ/4)C211yesmaximal, characteristic
8#11263C218yesmaximal, complemented
9C23×M16C2211yescharacteristic
10C23×M16C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16#2317—22no—
17#2317—22no—
18#2316—22no—
19#2315—22no—
20#2296—22no—
21#2296C2214yes—
22#2285—22no—
23C2×#344C418yescomplemented
24C2×#344C2214yes—
25#1740—22no—
26#1740C2214yes—
27C2×#342—22no—
28#1733—24no—
29#1733—22no—
30#1733—22no—
31#1733—22no—
32#1733C2214yes—
33C2×#339—22no—
34C2×#339C2214yes—
35C2×#339C2214yes—
36C2×H(3,ℤ/4)—24no—
37C2×H(3,ℤ/4)C2214yes—
38C24×C4C2311yescharacteristic
39C22×M16—22no—
40C22×M16—22no—
41C22×M16—21no—
42C22×M16—21no—
43C22×M16—21no—
44C22×M16—22no—
45C22×M16—21no—
46C22×M16—21no—
47C22×M16—21no—
48C22×M16—22no—
49C22×M16—21no—
50C22×M16—21no—
51C22×M16—21no—
52C22×M16—21no—
53C22×M16—22no—
54C22×M16—22no—
55C22×M16C2×C414yes—
56C22×M16C2×C414yes—
57C22×M16C2312yes—
58C22×M16C2312yes—
59C23×C8—21no—
60C23×C8—21no—
61C2×#117—21no—
62C2×#117—21no—
63C2×#117—22no—
64C2×#117—22no—
65C2×#117—21no—
66C2×#117—21no—
67C2×#117—21no—
68C2×#117—21no—
69C2×#117—21no—
70C2×#117—21no—
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—21no—
81C2×#117—21no—
82C2×#117—22no—
83C2×#117—21no—
84C2×#117—21no—
85C2×#117—21no—
86C2×#117—21no—
87C2×#117—21no—
88C2×#117—21no—
89C2×#117—21no—
90C2×#117—21no—
91C2×#117—21no—
92C2×#117—22no—
93C2×#117C2×C414yes—
94C2×#117D414yes—
95C2×#117D412yes—
96C2×#117D412yes—
97C2×#117D412yes—
98C2×#117D412yes—
99C2×#117Q812yes—
100C2×#117Q814yes—
101C2×#117C2312yes—
102C2×#117C2312yes—
103C2×#117C2312yes—
104C2×#117C2312yes—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4Q812yes—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4D412yes—
115C22×C42—21no—
116C22×C42—21no—
117C22×C42—21no—
118C22×C42D412yes—
119C22×C42C2312yes—
120#389—28no—
121#389—28no—
122#385—28no—
123#385—28no—
124#384—24no—
125#382—24no—
126#380—28no—
127C4×C4⋊C4—28no—
128#344C2×C4116yescomplemented
129#342—28no—
130#339—28no—
131#339—28no—
132#339—28no—
133H(3,ℤ/4)—28no—
134C25C2×D411yescharacteristic
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—41no—
140C23×C4C22×C412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×Q812yes—
148C23×C4C2×Q811yescharacteristic
149C23×C4C2×Q811yescharacteristic
150C23×C4#4112yes—
151C23×C4#4112yes—
152C23×C4#4111yescharacteristic
153C23×C4#4111yescharacteristic
154C23×C4#4111yescharacteristic
155C23×C4#4111yescharacteristic
156C23×C4#4112yes—
157C23×C4#4112yes—
158C23×C4#4111yescharacteristic
159C23×C4#4111yescharacteristic
160C23×C4#4112yes—
161C23×C4C2411yescharacteristic
162C2×M16—24no—
163C2×M16—24no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—24no—
167C2×M16—24no—
168C2×M16—24no—
169C2×M16—24no—
170C2×M16—24no—
171C2×M16—24no—
172C2×M16—24no—
173C2×M16—24no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—42no—
183C2×M16—42no—
184C2×M16—42no—
185C2×M16—42no—
186C2×M16—42no—
187C2×M16—42no—
188C2×M16—42no—
189C2×M16—41no—
190C2×M16—41no—
191C2×M16—41no—
192C2×M16—41no—
193C2×M16—42no—
194C2×M16—42no—
195C2×M16—42no—
196C2×M16C22×C414yes—
197C2×M16C22×C414yes—
198C22×C8—24no—
199C22×C8—24no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—42no—
203C22×C8—41no—
204C22×C8—41no—
205C22×C8—41no—
206C22×C8—41no—
207C22×C8—42no—
208#117—24no—
209#117—24no—
210#117—24no—
211#117—24no—
212#117—24no—
213#117—22no—
214#117—22no—
215#117—24no—
216#117—24no—
217#117—24no—
218#117—24no—
219#117—24no—
220#117—24no—
221#117—24no—
222#117—22no—
223#117—22no—
224#117—22no—
225#117—22no—
226#117—22no—
227#117—24no—
228#117—22no—
229#117—22no—
230#117—22no—
231#117—22no—
232#117—22no—
233#117—24no—
234#117—22no—
235#117—22no—
236#117—22no—
237#117—22no—
238#117—22no—
239#117—41no—
240#117—41no—
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#117—41no—
250#117—41no—
251#117—41no—
252#117—41no—
253#117—41no—
254#117—41no—
255#117—42no—
256#117—42no—
257#117—42no—
258#117C4⋊C418yes—
259#117C22×C414yes—
260#117C2×D414yes—
261#117C2×Q814yes—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
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—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—216no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—41no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4C2×Q814yes—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—41no—
296C2×C22⋊C4—41no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—41no—
301C2×C22⋊C4—41no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—41no—
305C2×C22⋊C4C4⋊C418yes—
306C2×C22⋊C4C2×D414yes—
307C2×C42—24no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—41no—
324C2×C42—41no—
325C2×C42—41no—
326C2×C42—41no—
327C2×C42—41no—
328C2×C42—42no—
329C2×C42C2×D414yes—
330C2×C42C2×D414yes—
331#95—216no—
332C24—82no—
333C24C2×C4⋊C412yes—
334C24#12011yescharacteristic
335C24#12011yescharacteristic
336C24#12112yes—
337C24C22⋊Q811yescharacteristic
338C24C22⋊Q811yescharacteristic
339C24#12312yes—
340C24#12411yescharacteristic
341C24#12411yescharacteristic
342C24#12412yes—
343C24C22×D411yescharacteristic
344C22×C4—24no—
345C22×C4—28no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—84no—
357C22×C4—84no—
358C22×C4—84no—
359C22×C4—84no—
360C22×C4—84no—
361C22×C4C2×C4⋊C412yes—
362C22×C4#11712yes—
363C22×C4#11712yes—
364C22×C4#11712yes—
365C22×C4#11712yes—
366C22×C4C4×D412yes—
367C22×C4C4×D412yes—
368C22×C4C4×D412yes—
369C22×C4C4×D412yes—
370C22×C4C4×D412yes—
371C22×C4C4×D412yes—
372C22×C4C4×Q812yes—
373C22×C4C4×Q812yes—
374C22×C4#12011yescharacteristic
375C22×C4#12011yescharacteristic
376C22×C4#12112yes—
377C22×C4#12112yes—
378C22×C4#12112yes—
379C22×C4#12112yes—
380C22×C4#12112yes—
381C22×C4C22⋊Q812yes—
382C22×C4C22⋊Q812yes—
383C22×C4C22⋊Q812yes—
384C22×C4C22⋊Q812yes—
385C22×C4C22⋊Q811yescharacteristic
386C22×C4C22⋊Q811yescharacteristic
387C22×C4C22⋊Q811yescharacteristic
388C22×C4C22⋊Q811yescharacteristic
389C22×C4C22⋊Q811yescharacteristic
390C22×C4C22⋊Q811yescharacteristic
391C22×C4C22⋊Q811yescharacteristic
392C22×C4C22⋊Q811yescharacteristic
393C22×C4C22⋊Q811yescharacteristic
394C22×C4C22⋊Q811yescharacteristic
395C22×C4#12312yes—
396C22×C4#12312yes—
397C22×C4#12312yes—
398C22×C4#12312yes—
399C22×C4#12312yes—
400C22×C4#12312yes—
401C22×C4#12312yes—
402C22×C4#12312yes—
403C22×C4#12312yes—
404C22×C4#12411yescharacteristic
405C22×C4#12411yescharacteristic
406C22×C4#12411yescharacteristic
407C22×C4#12411yescharacteristic
408C22×C4#12411yescharacteristic
409C22×C4#12411yescharacteristic
410C22×C4#12411yescharacteristic
411C22×C4#12411yescharacteristic
412C22×C4#12411yescharacteristic
413C22×C4#12411yescharacteristic
414C22×C4#12412yes—
415C22×C4#12512yes—
416C22×C4#12512yes—
417C22×C4#12512yes—
418C22×C4#12512yes—
419C22×C4#12512yes—
420C22×C4#12512yes—
421C22×C4#12512yes—
422C22×C4#12512yes—
423C22×C4#12512yes—
424C22×C4#12512yes—
425C22×C4#12612yes—
426C22×C4#12612yes—
427C22×C4#12612yes—
428C22×C4#12612yes—
429C22×C4#12612yes—
430C22×C4#12612yes—
431C22×C4#12612yes—
432C22×C4#12612yes—
433C22×C4C4⋊Q812yes—
434C22×C4C4⋊Q812yes—
435C22×C4C23×C411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×Q811yescharacteristic
440C22×C4C22×Q811yescharacteristic
441C22×C4C2×#4111yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4C2×#4111yescharacteristic
444C22×C4C2×#4111yescharacteristic
445C22×C4C2×#4111yescharacteristic
446C22×C4C2×#4111yescharacteristic
447C22×C4C2×#4111yescharacteristic
448C22×C4C2×#4111yescharacteristic
449M16—24no—
450M16—24no—
451M16—24no—
452M16—24no—
453M16—42no—
454M16—42no—
455M16—42no—
456M16—42no—
457M16—44no—
458M16—44no—
459M16—44no—
460M16—44no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—44no—
468C2×C8—44no—
469C2×C8—44no—
470C2×C8—44no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—28no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—816no—
487C22⋊C4—24no—
488C22⋊C4—28no—
489C22⋊C4—24no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4C2×C4⋊C418yes—
502C42—24no—
503C42—24no—
504C42—28no—
505C42—24no—
506C42—24no—
507C42—24no—
508C42—24no—
509C42—24no—
510C42—42no—
511C42—42no—
512C42—42no—
513C42—42no—
514C42—42no—
515C42—42no—
516C42—42no—
517C42—44no—
518C42—44no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—82no—
527C23—82no—
528C23—82no—
529C23—82no—
530C23#38212yes—
531C23#38212yes—
532C23#38512yes—
533C23#38512yes—
534C23#38912yes—
535C23#38912yes—
536C23#39311yescharacteristic
537C23#39311yescharacteristic
538C23#39311yescharacteristic
539C23#39311yescharacteristic
540C23#39412yes—
541C23#39412yes—
542C23#39712yes—
543C23#39712yes—
544C23#40012yes—
545C23#40012yes—
546C23C22×C4⋊C411yescharacteristic
547C23C2×#12011yescharacteristic
548C23C2×#12111yescharacteristic
549C23C2×C22⋊Q811yescharacteristic
550C23C2×#12311yescharacteristic
551C23C2×#12411yescharacteristic
552C23C2×#12411yescharacteristic
553C2×C4—28no—
554C2×C4—48no—
555C2×C4—48no—
556C2×C4—48no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—88no—
562C2×C4—84no—
563C2×C4—84no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4—88no—
568C2×C4—88no—
569C2×C4—84no—
570C2×C4#38212yes—
571C2×C4#38212yes—
572C2×C4#38512yes—
573C2×C4#38512yes—
574C2×C4#38712yes—
575C2×C4#38712yes—
576C2×C4#38712yes—
577C2×C4#38712yes—
578C2×C4#38712yes—
579C2×C4#38712yes—
580C2×C4#38712yes—
581C2×C4#38712yes—
582C2×C4#38812yes—
583C2×C4#38812yes—
584C2×C4#38812yes—
585C2×C4#38812yes—
586C2×C4#38812yes—
587C2×C4#38812yes—
588C2×C4#38812yes—
589C2×C4#38812yes—
590C2×C4#38912yes—
591C2×C4#38912yes—
592C2×C4#39311yescharacteristic
593C2×C4#39311yescharacteristic
594C2×C4#39311yescharacteristic
595C2×C4#39311yescharacteristic
596C2×C4#39412yes—
597C2×C4#39412yes—
598C2×C4#39712yes—
599C2×C4#39712yes—
600C2×C4#39812yes—
601C2×C4#39812yes—
602C2×C4#39812yes—
603C2×C4#39812yes—
604C2×C4#39812yes—
605C2×C4#39812yes—
606C2×C4#39812yes—
607C2×C4#39812yes—
608C2×C4#40012yes—
609C2×C4#40012yes—
610C2×C4#40012yes—
611C2×C4#40012yes—
612C2×C4#40012yes—
613C2×C4#40012yes—
614C2×C4#40012yes—
615C2×C4#40012yes—
616C2×C4#40012yes—
617C2×C4#40012yes—
618C2×C4C2×#11711yescharacteristic
619C2×C4C2×#11711yescharacteristic
620C2×C4C2×C4×D411yescharacteristic
621C2×C4C2×C4×D411yescharacteristic
622C2×C4C2×C4×D411yescharacteristic
623C2×C4C2×C4×Q811yescharacteristic
624C2×C4C2×#12111yescharacteristic
625C2×C4C2×#12111yescharacteristic
626C2×C4C2×C22⋊Q811yescharacteristic
627C2×C4C2×C22⋊Q811yescharacteristic
628C2×C4C2×C22⋊Q811yescharacteristic
629C2×C4C2×C22⋊Q811yescharacteristic
630C2×C4C2×#12311yescharacteristic
631C2×C4C2×#12311yescharacteristic
632C2×C4C2×#12311yescharacteristic
633C2×C4C2×#12311yescharacteristic
634C2×C4C2×#12411yescharacteristic
635C2×C4C2×#12411yescharacteristic
636C2×C4C2×#12511yescharacteristic
637C2×C4C2×#12511yescharacteristic
638C2×C4C2×#12511yescharacteristic
639C2×C4C2×#12511yescharacteristic
640C2×C4C2×#12511yescharacteristic
641C2×C4C2×#12611yescharacteristic
642C2×C4C2×#12611yescharacteristic
643C2×C4C2×#12611yescharacteristic
644C2×C4C2×#12611yescharacteristic
645C2×C4C2×C4⋊Q811yescharacteristic
646C8—44no—
647C8—44no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—41no—
656C22—41no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—41no—
661C22—41no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22#144212yes—
672C22#144212yes—
673C22#144212yes—
674C22#144212yes—
675C22C2×#38211yescharacteristic
676C22C2×#38511yescharacteristic
677C22C2×#38911yescharacteristic
678C22C2×#39311yescharacteristic
679C22C2×#39411yescharacteristic
680C22C2×#39711yescharacteristic
681C22C2×#40011yescharacteristic
682C4—48no—
683C4—84no—
684C4—84no—
685C4—84no—
686C4—84no—
687C4#144212yes—
688C4#144212yes—
689C4#144212yes—
690C4#144212yes—
691C4C2×#38711yescharacteristic
692C4C2×#38711yescharacteristic
693C4C2×#38811yescharacteristic
694C4C2×#38811yescharacteristic
695C4C2×#39811yescharacteristic
696C4C2×#39811yescharacteristic
697C4C2×#40011yescharacteristic
698C4C2×#40011yescharacteristic
699C2—21no—
700C2—21no—
701C2—21no—
702C2—21no—
703C2—22no—
704C2—22no—
705C2—22no—
706C2—22no—
707C2—21no—
708C2—21no—
709C2C2×#144211yescharacteristic
710C2#2347212yes—
subgroupG/H
centerC2×C4C2×#124
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#393
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #11263C2semidirect
23. C2×#344C4semidirect
128. #344C2×C4semidirect

Maximal subgroups

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

subgroupindex
#340522
C22×#3442
C2×#17402
C2×#17332
C22×H(3,ℤ/4)2
#112632