Finite Group Databasebeta

← #204344

Subgroups — #204344

Order 512.

1127 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3171C211yesmaximal, characteristic
2#21378C211yesmaximal, characteristic, complemented
3#21189C211yesmaximal, characteristic, complemented
4C2×#1565C211yesmaximal, characteristic, complemented
5C2×#1483C211yesmaximal, characteristic, complemented
6C2×#1484C211yesmaximal, characteristic, complemented
7C2×#1455C211yesmaximal, characteristic, complemented
8#9666C211yesmaximal, characteristic, complemented
9#9665C211yesmaximal, characteristic, complemented
10#9663C211yesmaximal, characteristic, complemented
11#9661C211yesmaximal, characteristic, complemented
12#9660C211yesmaximal, characteristic, complemented
13#9659C211yesmaximal, characteristic, complemented
14#9657C211yesmaximal, characteristic, complemented
15#9655C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#529C2211yescharacteristic
18#3171—24no—
19#3171—24no—
20C2×#501C411yescharacteristic
21C2×#496C411yescharacteristic
22C2×#490C411yescharacteristic
23C2×#489C411yescharacteristic
24C2×#469C411yescharacteristic
25C2×#469C411yescharacteristic
26C2×#468C411yescharacteristic
27C2×#468C411yescharacteristic
28C2×C4×SD16C2211yescharacteristic
29C2×C4×SD16C2211yescharacteristic
30C2×C4×D8C2211yescharacteristic
31C2×C4×D8C2211yescharacteristic
32C2×#431C2211yescharacteristic
33C22×C4⋊C8C2211yescharacteristic
34C2×#408—22no—
35C2×C4×M16C2211yescharacteristic
36C2×C4×M16C2211yescharacteristic
37C2×C4×M16C2211yescharacteristic
38#1984C2211yescharacteristic
39#1984C2211yescharacteristic
40#1955—22no—
41#1954C2211yescharacteristic
42#1954C2211yescharacteristic
43#1954C2211yescharacteristic
44#1954C2211yescharacteristic
45#1954C2211yescharacteristic
46#1954C2211yescharacteristic
47#1870—22no—
48#1750—22no—
49#1565—24no—
50C2×#335C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×#334C2211yescharacteristic
54#1484—24no—
55#1483—24no—
56C2×Q8⋊C8C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×D4⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63C2×D4⋊C8C2211yescharacteristic
64#1455—24no—
65C2×#322C2211yescharacteristic
66C2×#322C2211yescharacteristic
67C2×#322C2211yescharacteristic
68C2×C82C2211yescharacteristic
69C22×#41—21no—
70C22×SD16C2×C411yescharacteristic
71C22×SD16C2×C411yescharacteristic
72C22×D8C2×C411yescharacteristic
73C22×D8C2×C411yescharacteristic
74C22×M16—21no—
75C22×M16C2×C411yescharacteristic
76C22×M16C2×C411yescharacteristic
77C22×M16D412yes—
78C23×C8—21no—
79#532—22no—
80#532—22no—
81#532—22no—
82#532—22no—
83C2×C4⋊Q8C2×C411yescharacteristic
84C2×#127C2×C411yescharacteristic
85#529—22no—
86#529—22no—
87#529—22no—
88#529—22no—
89C2×#126—21no—
90C2×#125C2×C411yescharacteristic
91C2×#124—21no—
92C2×#124C2×C411yescharacteristic
93C2×#123—21no—
94C2×C22⋊Q8C2×C411yescharacteristic
95C2×#121—21no—
96C2×#121C2×C411yescharacteristic
97C2×#121C2×C411yescharacteristic
98C2×#121C2×C411yescharacteristic
99C2×C4×Q8C2311yescharacteristic
100C2×C4×D4—21no—
101C2×C4×D4—21no—
102C2×C4×D4C2311yescharacteristic
103C2×C4×D4C2311yescharacteristic
104C2×C4×D4C2311yescharacteristic
105C2×#117—21no—
106C22×C42C2311yescharacteristic
107#501—24no—
108#496—24no—
109#490—24no—
110#489—24no—
111#469—24no—
112#469—24no—
113#468—24no—
114#468—24no—
115C4×SD16—22no—
116C4×SD16—22no—
117C4×SD16—22no—
118C4×SD16—22no—
119C4×D8—22no—
120C4×D8—22no—
121C4×D8—22no—
122C4×D8—22no—
123#431—22no—
124#431—22no—
125C2×#107C2×C411yescharacteristic
126C2×#107C2×C411yescharacteristic
127C2×#106C2×C411yescharacteristic
128C2×#106C2×C411yescharacteristic
129C2×C4⋊C8—22no—
130C2×C4⋊C8—22no—
131C2×C4⋊C8—21no—
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2311yescharacteristic
136C2×#103C2×C411yescharacteristic
137C2×#103C2×C411yescharacteristic
138C2×#102C2×C411yescharacteristic
139C2×#102C2×C411yescharacteristic
140C2×#102C2×C411yescharacteristic
141C2×#102C2×C411yescharacteristic
142C2×#102C2×C411yescharacteristic
143C2×#102C2×C411yescharacteristic
144#408—48no—
145C2×C22⋊C8—21no—
146C2×C22⋊C8—22no—
147C2×C22⋊C8—22no—
148C2×C22⋊C8—22no—
149C2×C22⋊C8—22no—
150C2×C22⋊C8—22no—
151C2×C22⋊C8—22no—
152C4×M16—22no—
153C4×M16—22no—
154C4×M16—22no—
155C4×M16D414yes—
156C4×M16D414yes—
157C4×M16D414yes—
158C2×#97—21no—
159C2×#97C2×C411yescharacteristic
160C2×#97C2×C411yescharacteristic
161C2×#97D412yes—
162C2×C4×C8—21no—
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2311yescharacteristic
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169#336—22no—
170#336—22no—
171#335—22no—
172#335—22no—
173#334—22no—
174#334—22no—
175Q8⋊C8—22no—
176Q8⋊C8—22no—
177D4⋊C8—22no—
178D4⋊C8—22no—
179D4⋊C8—22no—
180D4⋊C8—22no—
181D4⋊C8—22no—
182D4⋊C8—22no—
183#322—22no—
184#322—22no—
185#322—22no—
186C82—22no—
187C2×#41—44no—
188C2×#41—44no—
189C2×#41—42no—
190C2×#41—42no—
191C22×Q8—21no—
192C22×Q8C22×C411yescharacteristic
193C22×D4—21no—
194C22×D4—21no—
195C22×D4—21no—
196C22×D4—21no—
197C22×D4—21no—
198C22×D4C22×C411yescharacteristic
199C22×D4C22×C411yescharacteristic
200C22×D4C22×C411yescharacteristic
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4C22×C411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4#4111yescharacteristic
207C2×SD16—22no—
208C2×SD16—22no—
209C2×SD16—22no—
210C2×SD16—22no—
211C2×SD16—41no—
212C2×SD16—41no—
213C2×D8—22no—
214C2×D8—22no—
215C2×D8—22no—
216C2×D8—22no—
217C2×D8—41no—
218C2×D8—41no—
219C2×M16—24no—
220C2×M16—24no—
221C2×M16—22no—
222C2×M16—22no—
223C2×M16—24no—
224C2×M16—41no—
225C2×M16—41no—
226C2×M16—42no—
227C2×M16—44no—
228C2×M16C22⋊C414yes—
229C2×M16C22⋊C414yes—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—44no—
237C22×C8—42no—
238C22×C8C22×C411yescharacteristic
239C22×C8C22×C411yescharacteristic
240C22×C8C22×C411yescharacteristic
241C22×C8C22×C411yescharacteristic
242C22×C8C2×D412yes—
243C22×C8C2×D412yes—
244C4⋊Q8—22no—
245C4⋊Q8—22no—
246#127—22no—
247#127—22no—
248#126—42no—
249#126—42no—
250#125—22no—
251#125—22no—
252#124—22no—
253#124—22no—
254#124—42no—
255#124—42no—
256#123—42no—
257#123—42no—
258C22⋊Q8—22no—
259C22⋊Q8—22no—
260#121—22no—
261#121—22no—
262#121—22no—
263#121—22no—
264#121—22no—
265#121—22no—
266#121—42no—
267#121—42no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×D4—21no—
273C4×D4—21no—
274C4×D4—21no—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—21no—
282C4×D4—21no—
283C4×D4—21no—
284C4×D4—42no—
285C4×D4—42no—
286C4×D4—42no—
287C4×D4—42no—
288#117—42no—
289#117—42no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4C22×C411yescharacteristic
296C2×C4⋊C4C22×C411yescharacteristic
297C2×C4⋊C4C22×C411yescharacteristic
298C2×C4⋊C4C22×C411yescharacteristic
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C42—21no—
309C2×C42C22×C411yescharacteristic
310C2×C42C2×D412yes—
311C2×C42C2×D412yes—
312C2×C42C2×D412yes—
313C2×C42C2×D411yescharacteristic
314C2×C42#4112yes—
315C2×C42#4111yescharacteristic
316C2×C42C2411yescharacteristic
317#107—22no—
318#107—22no—
319#106—22no—
320#106—22no—
321C4⋊C8—21no—
322C4⋊C8—21no—
323C4⋊C8—21no—
324C4⋊C8—21no—
325C4⋊C8—21no—
326C4⋊C8—21no—
327C4⋊C8—21no—
328C4⋊C8—21no—
329C4⋊C8—42no—
330#103—22no—
331#103—22no—
332#102—22no—
333#102—22no—
334#102—22no—
335#102—22no—
336#102—22no—
337#102—22no—
338C22⋊C8—44no—
339C22⋊C8—44no—
340C22⋊C8—44no—
341C22⋊C8—42no—
342C22⋊C8—44no—
343C22⋊C8—44no—
344C22⋊C8—44no—
345#97—24no—
346#97—24no—
347#97C22⋊C414yes—
348#97C22⋊C414yes—
349C4×C8—21no—
350C4×C8—21no—
351C4×C8—21no—
352C4×C8—21no—
353C4×C8—21no—
354C4×C8—21no—
355C4×C8—42no—
356C4×C8C2×D412yes—
357C4×C8C2×D412yes—
358C4×C8C2×D412yes—
359C4×C8C2×D412yes—
360C4×C8C2×D412yes—
361C4×C8C2×D412yes—
362C24—21no—
363C24—21no—
364C24—21no—
365C24C4×D411yescharacteristic
366#41—84no—
367#41—84no—
368#41—88no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—44no—
376C2×Q8—42no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—82no—
404C2×D4—82no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—42no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4C2×C22⋊C412yes—
430C22×C4C4×D412yes—
431C22×C4C4×D411yescharacteristic
432C22×C4C4×D411yescharacteristic
433C22×C4C4×D411yescharacteristic
434C22×C4#12112yes—
435C22×C4#12112yes—
436C22×C4#12412yes—
437C22×C4#12712yes—
438C22×C4C23×C411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C2×#4111yescharacteristic
441SD16—44no—
442SD16—44no—
443D8—44no—
444D8—44no—
445M16—28no—
446M16—44no—
447M16—48no—
448M16—48no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—42no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—42no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8—84no—
473C2×C8C2×C22⋊C412yes—
474C2×C8C2×C22⋊C412yes—
475C2×C8C2×C22⋊C412yes—
476C2×C8C2×C22⋊C412yes—
477C2×C8#13112yes—
478C2×C8#13112yes—
479C2×C8AGL(1,ℤ/8)14yes—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C42—41no—
507C42—41no—
508C42C2×C22⋊C412yes—
509C42C4×D412yes—
510C42#12112yes—
511C42#12112yes—
512C42#12412yes—
513C42#12712yes—
514C42C22×D411yescharacteristic
515C42C22×D411yescharacteristic
516C42C22×D411yescharacteristic
517C42C2×#4111yescharacteristic
518C23—21no—
519C23—21no—
520C23—22no—
521C23—22no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23#39012yes—
538C23C2×C4×D411yescharacteristic
539Q8—41no—
540Q8—41no—
541Q8—41no—
542Q8—41no—
543Q8—84no—
544D4—41no—
545D4—41no—
546D4—41no—
547D4—41no—
548D4—41no—
549D4—41no—
550D4—41no—
551D4—41no—
552D4—41no—
553D4—41no—
554D4—41no—
555D4—41no—
556D4—84no—
557D4—84no—
558D4—84no—
559D4—84no—
560D4—84no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4#39012yes—
596C2×C4#39012yes—
597C2×C4#39012yes—
598C2×C4C22×C22⋊C411yescharacteristic
599C2×C4C2×C4×D411yescharacteristic
600C2×C4C2×#12111yescharacteristic
601C2×C4C2×#12111yescharacteristic
602C2×C4C2×#12411yescharacteristic
603C2×C4C2×#12711yescharacteristic
604C2×C4C2×#13111yescharacteristic
605C2×C4C2×#13111yescharacteristic
606C2×C4#56811yescharacteristic
607C2×C4#56811yescharacteristic
608C2×C4C2×AGL(1,ℤ/8)12yes—
609C2×C4#57512yes—
610C8—24no—
611C8—22no—
612C8—22no—
613C8—22no—
614C8—24no—
615C8—42no—
616C8—42no—
617C8—81no—
618C8—81no—
619C8—81no—
620C8—81no—
621C8#40814yescomplemented
622C22—22no—
623C22—22no—
624C22—22no—
625C22—42no—
626C22—42no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22C2×#39011yescharacteristic
643C22#293511yescharacteristic
644C22#293511yescharacteristic
645C22C4×AGL(1,ℤ/8)12yes—
646C22#295612yes—
647C4—22no—
648C4—22no—
649C4—21no—
650C4—21no—
651C4—22no—
652C4—81no—
653C4—81no—
654C4—81no—
655C4—81no—
656C4—81no—
657C4C2×#40811yescharacteristic
658C4#287911yescharacteristic
659C4#293511yescharacteristic
660C4#293511yescharacteristic
661C4#305512yes—
662C4#315111yescharacteristic
663C4#315311yescharacteristic
664C2—42no—
665C2—161no—
666C2—161no—
667C2—161no—
668C2#1807612yes—
669C2#1807812yes—
670C2#2118911yescharacteristic
671C2#2123611yescharacteristic
672C2#2124111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#3171C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21378C2semidirect
3. #21189C2semidirect
4. C2×#1565C2semidirect
5. C2×#1483C2semidirect
6. C2×#1484C2semidirect
7. C2×#1455C2semidirect
8. #9666C2semidirect
9. #9665C2semidirect
10. #9663C2semidirect
11. #9661C2semidirect
12. #9660C2semidirect
13. #9659C2semidirect
14. #9657C2semidirect
15. #9655C2semidirect
621. C8#408semidirect

Maximal subgroups

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

subgroupindex
C2×#31712
#213782
#211892
C2×#15652
C2×#14832
C2×#14842
C2×#14552
#96662
#96652
#96632
#96612
#96602
#96592
#96572
#96552