Finite Group Databasebeta

← #6541347

Subgroups — #6541347

Order 512.

2033 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×C4⋊Q8C211yesmaximal, characteristic, complemented
2#34308C211yesmaximal, characteristic
3#29980C211yesmaximal, characteristic, complemented
4#29966C212yesmaximal, complemented
5#29272C211yesmaximal, characteristic, complemented
6#29208C211yesmaximal, characteristic, complemented
7#29157C212yesmaximal, complemented
8#26440C212yesmaximal, complemented
9#26336C214yesmaximal, complemented
10#26040C212yesmaximal, complemented
11#25988C214yesmaximal, complemented
12#24298C212yesmaximal, complemented
13#23774C211yesmaximal, characteristic, complemented
14C2×#2057C212yesmaximal, complemented
15#22696C211yesmaximal, characteristic, complemented
16D4×#107C212yesmaximal, complemented
17#22485C212yesmaximal
18C2×D4×Q8—21no—
19C2×#547C2211yescharacteristic
20C22×C4⋊Q8C2212yes—
21C2×#520—21no—
22C2×#474C2214yes—
23C2×C8×D4C2212yes—
24C22×#107C2214yes—
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2551—21no—
28#2551C2212yes—
29#2546—21no—
30#2440—22no—
31#2440C2214yes—
32#2434—22no—
33#2365C2211yescharacteristic
34#2361C2212yes—
35#2360—21no—
36#2359—22no—
37D4×C4⋊C4—21no—
38D4×C4⋊C4C2211yescharacteristic
39D4×C4⋊C4C2211yescharacteristic
40D4×C4⋊C4C2212yes—
41#2344C2212yes—
42#2339—21no—
43#2323—21no—
44#2322—22no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×#121C2212yes—
48#2293C2211yescharacteristic
49C2×#389C2212yes—
50C42×D4C2211yescharacteristic
51#2099C2211yescharacteristic
52#2099C2212yes—
53#2099C2212yes—
54#2093C2212yes—
55#2093C2211yescharacteristic
56#2086C2214yes—
57#2086C2214yes—
58#2086C2212yes—
59#2067C2212yes—
60#2057C2212yes—
61#2057C2212yes—
62#2057C2212yes—
63#2057C2214yes—
64#2057C2212yes—
65#2057C2214yes—
66#2057C2211yescharacteristic
67#2023C2212yes—
68#2023C2211yescharacteristic
69#1990C2212yes—
70#1988C2212yes—
71#1988C2214yes—
72#1947C2214yes—
73#1947C2212yes—
74#1947C2212yes—
75#1939C2212yes—
76#1918C2214yes—
77#1918C2214yes—
78#1893C2214yes—
79#1874C2214yes—
80#1862C2214yes—
81#1852C2214yes—
82#1852C2214yes—
83#1850C2214yes—
84#1845C2212yes—
85#1831C2211yescharacteristic
86#1823C2214yes—
87#1808C2214yes—
88#1802C2212yes—
89#1797C2212yes—
90C4×#107C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×#102C2212yes—
93C2×#340C2214yes—
94C2×#336C2212yes—
95C22×#41—21no—
96C23×Q8—22no—
97C23×C8C2314yes—
98D4×Q8—48no—
99#547—24no—
100#547Q818yescomplemented
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—48no—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2314yes—
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2312yes—
111C2×#123—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—24no—
115C2×#121C2312yes—
116#520—48no—
117C2×C4×Q8—21no—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×D4—21no—
121C2×C4×D4—22no—
122C2×C4×D4—22no—
123C2×C4×D4—22no—
124C2×C4×D4D411yescharacteristic
125C2×C4×D4D411yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2312yes—
128C2×C4×D4C2311yescharacteristic
129C2×#117—22no—
130C2×#117—21no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4—22no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2314yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C42C2312yes—
138#474—48no—
139C8×D4—28no—
140C2×#107—48no—
141C2×#107C2312yes—
142C2×#107C2314yes—
143C2×#107C2312yes—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×#107C2314yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2314yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152C2×#102C2314yes—
153C2×C22⋊C8C2314yes—
154C2×C4×C8C2312yes—
155#398—24no—
156#395—24no—
157#395—24no—
158#391—21no—
159#391—21no—
160#391—21no—
161#391—22no—
162#390C2311yescharacteristic
163#390C2311yescharacteristic
164#389—21no—
165#389—22no—
166#389—22no—
167#389—21no—
168#389—21no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389C2312yes—
173#389C2312yes—
174#389C2311yescharacteristic
175#389C2312yes—
176#389C2312yes—
177#389C2311yescharacteristic
178#389C2314yes—
179#388—21no—
180#388—22no—
181#388—21no—
182#388—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#386—22no—
187#386—22no—
188#385—22no—
189#385—24no—
190#385—24no—
191#385—22no—
192#385—22no—
193#384—21no—
194#384—21no—
195#384C2311yescharacteristic
196#384C2311yescharacteristic
197#384C2312yes—
198#384C2312yes—
199#381—22no—
200#380—22no—
201#380—21no—
202#380C2312yes—
203#380C2314yes—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—21no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4C2312yes—
218C2×#95—22no—
219C43C2311yescharacteristic
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#336C2311yescharacteristic
232#336C2312yes—
233#336C2311yescharacteristic
234#336C2314yes—
235#336C2312yes—
236C2×#41—28no—
237C2×#41—44no—
238C2×#41—44no—
239C22×Q8—21no—
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8—24no—
243C22×Q8—48no—
244C22×Q8—48no—
245C22×Q8—84no—
246C22×D4—22no—
247C22×D4C2×D411yescharacteristic
248C22×D4C2411yescharacteristic
249C23×C4—22no—
250C23×C4—21no—
251C23×C4—24no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4C2×D412yes—
255C23×C4C2×D412yes—
256C23×C4C2412yes—
257C22×C8—28no—
258C22×C8—28no—
259C22×C8—44no—
260C22×C8C2414yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C4⋊Q8—24no—
264C4⋊Q8—28no—
265C4⋊Q8—22no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8—48no—
269C4⋊Q8—48no—
270C4⋊Q8—48no—
271C4⋊Q8—44no—
272C4⋊Q8—82no—
273C4⋊Q8C2×Q814yescomplemented
274#123—216no—
275C22⋊Q8—28no—
276C22⋊Q8—48no—
277C22⋊Q8—416no—
278#121—24no—
279#121C2×Q818yescomplemented
280C4×Q8—44no—
281C4×Q8—44no—
282C4×Q8—44no—
283C4×D4—24no—
284C4×D4—22no—
285C4×D4—44no—
286C4×D4—44no—
287C4×D4—48no—
288C4×D4—48no—
289C4×D4—48no—
290C4×D4D818yes—
291C4×D4C2×Q814yes—
292C4×D4C2×Q814yescomplemented
293C4×D4#4114yes—
294#117—48no—
295#117—44no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—48no—
323C2×C4⋊C4—48no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×Q818yescomplemented
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C42—22no—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—44no—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D412yes—
358C2×C42C2×D412yes—
359C2×C42C2×D411yescharacteristic
360C2×C42C2412yes—
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—24no—
365#107—24no—
366#107—28no—
367#107—44no—
368#107—44no—
369#107—44no—
370C4⋊C8—24no—
371C4⋊C8—24no—
372C4⋊C8—44no—
373#102—24no—
374#102—24no—
375#102—44no—
376C22⋊C8—28no—
377C4×C8—24no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—21no—
388#95—22no—
389#95—24no—
390#95—24no—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24—21no—
395C24C22×D412yes—
396#41—432no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—48no—
400C2×Q8—42no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—48no—
406C2×Q8—44no—
407C2×Q8—42no—
408C2×Q8—84no—
409C2×Q8—832no—
410C2×D4—28no—
411C2×D4—21no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4C22⋊Q814yes—
417C2×D4C2×D814yes—
418C2×D4#13714yes—
419C2×D4C22×Q812yes—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—28no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—48no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—48no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4—84no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4C2×D818yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×Q814yes—
461C22×C4C2×#4114yes—
462C22×C4C2511yescharacteristic
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—416no—
472C2×C82−1+414yes—
473C4⋊C4—24no—
474C4⋊C4—28no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—81no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4C22⋊Q814yes—
507C4⋊C4C2×D814yes—
508C4⋊C4C22×Q812yes—
509C4⋊C4C22×Q814yes—
510C4⋊C4H(5,𝔽2)14yes—
511C22⋊C4—24no—
512C22⋊C4—24no—
513C22⋊C4—28no—
514C22⋊C4—44no—
515C22⋊C4—48no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—81no—
520C22⋊C4—82no—
521C22⋊C4C22⋊Q818yes—
522C22⋊C4C2×D818yes—
523C42—22no—
524C42—22no—
525C42—24no—
526C42—22no—
527C42—22no—
528C42—42no—
529C42—42no—
530C42—41no—
531C42—44no—
532C42—44no—
533C42—42no—
534C42—44no—
535C42C2×D814yes—
536C42C22×Q812yes—
537C42C2×#4112yes—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23C2×C22⋊Q814yes—
550C23C22×D814yes—
551C23C2×#13714yes—
552C23C23×D411yescharacteristic
553Q8—48no—
554Q8—48no—
555Q8—816no—
556Q8—88no—
557D4—24no—
558D4—416no—
559D4—42no—
560D4#474116yescomplemented
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—48no—
585C2×C4—48no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—416no—
593C2×C4—48no—
594C2×C4—41no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—44no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—816no—
610C2×C4—164no—
611C2×C4—164no—
612C2×C4C2×C22⋊Q812yes—
613C2×C4C2×C22⋊Q814yes—
614C2×C4C2×C22⋊Q812yes—
615C2×C4C22×D812yes—
616C2×C4C22×D814yes—
617C2×C4C22×D812yes—
618C2×C4C22×D812yes—
619C2×C4C2×#13712yes—
620C2×C4#57812yes—
621C2×C4C23×Q811yescharacteristic
622C2×C4C22×#4111yescharacteristic
623C2×C4C2×H(5,𝔽2)12yes—
624C2×C4C2×2−1+412yes—
625C8—24no—
626C8—42no—
627C8—44no—
628C22—24no—
629C22—24no—
630C22—24no—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22C2×#47418yes—
642C22C22×C22⋊Q811yescharacteristic
643C22#348411yescharacteristic
644C22#349911yescharacteristic
645C22C23×D811yescharacteristic
646C22C22×#13711yescharacteristic
647C22C2×#57811yescharacteristic
648C22#358511yescharacteristic
649C4—22no—
650C4—22no—
651C4—22no—
652C4—24no—
653C4—24no—
654C4—22no—
655C4—41no—
656C4—42no—
657C4—44no—
658C4—44no—
659C4—42no—
660C4—81no—
661C4—82no—
662C4—84no—
663C4—81no—
664C4—82no—
665C4C2×#47414yes—
666C4#350312yes—
667C4#352012yes—
668C2—28no—
669C2—82no—
670C2#3573911yescharacteristic
671C2C22×#47411yescharacteristic
672C2#6049811yescharacteristic
673C2#6155911yescharacteristic
674C2#6160211yescharacteristic
675C2#6247111yescharacteristic
676C2#6250411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. D4×C4⋊Q8C2semidirect
3. #29980C2semidirect
4. #29966C2semidirect
5. #29272C2semidirect
6. #29208C2semidirect
7. #29157C2semidirect
8. #26440C2semidirect
9. #26336C2semidirect
10. #26040C2semidirect
11. #25988C2semidirect
12. #24298C2semidirect
13. #23774C2semidirect
14. C2×#2057C2semidirect
15. #22696C2semidirect
16. D4×#107C2semidirect
100. #547Q8semidirect
273. C4⋊Q8C2×Q8semidirect
279. #121C2×Q8semidirect
292. C4×D4C2×Q8semidirect
328. C2×C4⋊C4C2×Q8semidirect
560. D4#474semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊Q82
#343082
#299802
#299662
#292722
#292082
#291572
#264402
#263362
#260402
#259882
#242982
#237742
C2×#20572
#226962
D4×#1072
#224852