Finite Group Databasebeta

← #6458302

Subgroups — #6458302

Order 512.

1840 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35257C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic
3#27959C214yesmaximal, complemented
4#27957C214yesmaximal, complemented
5#27955C212yesmaximal, complemented
6#27953C212yesmaximal, complemented
7#27951C212yesmaximal, complemented
8#27948C212yesmaximal, complemented
9#23658C211yesmaximal, characteristic, complemented
10#23657C211yesmaximal, characteristic, complemented
11#23656C211yesmaximal, characteristic, complemented
12C2×#2084C211yesmaximal, characteristic, complemented
13#21326C212yesmaximal, complemented
14#20849C212yesmaximal
15#20848C212yesmaximal
16#20802C212yesmaximal, complemented
17#20378C211yesmaximal, characteristic
18C2×#544C2211yescharacteristic
19C2×#520C2212yes—
20C2×C4×#41C2211yescharacteristic
21C2×#428C2212yes—
22C2×#428C2212yes—
23C2×#427C2212yes—
24C22×#106C2212yes—
25C2×#408C2212yes—
26C2×#405C2212yes—
27#2857C2212yes—
28#2851C2212yes—
29#2629C2212yes—
30#2628C2212yes—
31#2454—21no—
32#2453—21no—
33#2452—21no—
34#2451—21no—
35#2450—22no—
36#2449—22no—
37Q8×C4⋊C4—21no—
38Q8×C4⋊C4—21no—
39D4×C4⋊C4—22no—
40D4×C4⋊C4—22no—
41D4×C4⋊C4—21no—
42D4×C4⋊C4—21no—
43#2317C2214yes—
44#2317C2211yescharacteristic
45#2317C2212yes—
46#2311C2211yescharacteristic
47#2311C2211yescharacteristic
48#2296C2211yescharacteristic
49C2×#389C2211yescharacteristic
50#2285C2212yes—
51#2285C2211yescharacteristic
52C2×#384C2212yes—
53C2×#384C2211yescharacteristic
54#2085C2212yes—
55#2085C2214yes—
56#2085C2212yes—
57#2084C2211yescharacteristic
58#2084C2212yes—
59#2084C2212yes—
60#2084C2212yes—
61#2084C2211yescharacteristic
62#2084C2212yes—
63#2084C2212yes—
64#2084C2214yes—
65#2083C2212yes—
66#2083C2212yes—
67#2083C2211yescharacteristic
68#2083C2211yescharacteristic
69#2083C2212yes—
70#1993C2214yes—
71#1992C2214yes—
72#1991C2214yes—
73#1990C2214yes—
74#1854C2214yes—
75#1854C2214yes—
76#1854C2214yes—
77#1854C2214yes—
78#1853C2214yes—
79#1853C2214yes—
80#1853C2214yes—
81#1852C2214yes—
82#1847C2214yes—
83#1847C2214yes—
84#1846C2214yes—
85#1844C2214yes—
86#1750C2214yes—
87#1749C2212yes—
88C4×C22⋊C8C2212yes—
89#1736C2212yes—
90#1736C2212yes—
91#1735C2212yes—
92C2×#340C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C22×M16C2312yes—
97C23×C8C2312yes—
98#544—28no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2312yes—
101C2×#125—21no—
102C2×#125C2312yes—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—22no—
105#520—44no—
106#520—44no—
107C4×#41—28no—
108C2×C4×Q8—22no—
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8D411yescharacteristic
111C2×C4×D4—24no—
112C2×C4×D4—22no—
113C2×C4×D4D412yes—
114C2×C4×D4D412yes—
115C2×C4×D4D411yescharacteristic
116C2×C4×D4D411yescharacteristic
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C2×#117C2312yes—
120C2×#117C2314yes—
121C2×#117C2312yes—
122C2×#117C2311yescharacteristic
123C22×C4⋊C4—21no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2314yes—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134#428—44no—
135#428—44no—
136#427—44no—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#107C2312yes—
140C2×#107C2312yes—
141C2×#106—44no—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×#106C2312yes—
146C2×#106C2314yes—
147C2×#106C2312yes—
148#408Q8116yescomplemented
149C2×C22⋊C8C2314yes—
150C2×C22⋊C8C2314yes—
151#405—28no—
152C2×#97C2314yes—
153C2×C4×C8C2314yes—
154#400—22no—
155#399—22no—
156#398—22no—
157#398—22no—
158#396—22no—
159#396—22no—
160#395—22no—
161#393—22no—
162#391—21no—
163#391—21no—
164#391—22no—
165#390—21no—
166#390—21no—
167#389—24no—
168#389—24no—
169#389—22no—
170#389—22no—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—21no—
175#389—21no—
176#389C2311yescharacteristic
177#389C2311yescharacteristic
178#389C2312yes—
179#389C2311yescharacteristic
180#389C2312yes—
181#389C2311yescharacteristic
182#387—22no—
183#387—22no—
184#386—22no—
185#386—22no—
186#385—22no—
187#385—22no—
188#385—21no—
189#385—21no—
190#385—24no—
191#385—24no—
192#385—24no—
193#385—22no—
194#384—21no—
195#384—21no—
196#384—22no—
197#384—22no—
198#384C2312yes—
199#384C2314yes—
200#384C2312yes—
201#384C2311yescharacteristic
202#384C2312yes—
203#384C2311yescharacteristic
204#382C2312yes—
205#382C2314yes—
206#382C2312yes—
207#380—22no—
208#380—21no—
209#380—22no—
210#380—21no—
211#380—22no—
212#380C2312yes—
213#380C2312yes—
214#380C2312yes—
215#380C2312yes—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C2×#95—21no—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#336C2311yescharacteristic
234#336C2312yes—
235#336C2312yes—
236#336C2311yescharacteristic
237#336C2312yes—
238C2×#41—24no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41C2×Q818yes—
242C22×Q8—22no—
243C22×Q8C2×D411yescharacteristic
244C22×D4C2×D412yes—
245C22×D4C2×D411yescharacteristic
246C23×C4—24no—
247C23×C4—22no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D412yes—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D412yes—
254C23×C4C2412yes—
255C23×C4C2411yescharacteristic
256C2×M16—24no—
257C2×M16—24no—
258C2×M16C2×Q818yes—
259C22×C8—24no—
260C22×C8—24no—
261C22×C8C2×Q818yes—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C22×C8C2414yes—
265C4⋊Q8—28no—
266C4⋊Q8—44no—
267#125—28no—
268#125—44no—
269C22⋊Q8—48no—
270C22⋊Q8—84no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—48no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—28no—
277C4×D4—28no—
278C4×D4—416no—
279C4×D4—48no—
280#117—28no—
281#117—24no—
282#117—24no—
283#117—24no—
284#117—22no—
285#117—22no—
286#117—44no—
287#117—44no—
288#117#4118yes—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—84no—
322C2×C4⋊C4—82no—
323C2×C4⋊C4—82no—
324C2×C4⋊C4—84no—
325C2×C4⋊C4—82no—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2414yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2411yescharacteristic
339C2×C22⋊C4—24no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C42—24no—
351C2×C42—28no—
352C2×C42—24no—
353C2×C42—22no—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D412yes—
359C2×C42C2×D412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362C2×C42C2412yes—
363#107—42no—
364#107—42no—
365#107—44no—
366#107—44no—
367#106—44no—
368#106—44no—
369#106—42no—
370#106—42no—
371#106—44no—
372#106—44no—
373C22⋊C8C2×Q8116yes—
374C22⋊C8C2×Q8116yes—
375#97—28no—
376C4×C8—28no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—24no—
383#95—24no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393C24C22×D412yes—
394C24C22×D411yescharacteristic
395#41—216no—
396#41—216no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—44no—
400C2×Q8—82no—
401C2×Q8—82no—
402C2×Q8C4⋊Q814yes—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4C4⋊Q818yes—
408C2×D4C4⋊Q814yes—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—42no—
430C22×C4—41no—
431C22×C4—42no—
432C22×C4—41no—
433C22×C4—42no—
434C22×C4—48no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4—82no—
440C22×C4—84no—
441C22×C4—81no—
442C22×C4—82no—
443C22×C4—84no—
444C22×C4—81no—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×Q818yes—
454C22×C4C22×Q814yes—
455C22×C4C2×#4114yes—
456C22×C4H(5,𝔽2)12yes—
457C22×C4H(5,𝔽2)12yes—
458C22×C4C2511yescharacteristic
459M16—216no—
460C2×C8—216no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8C22×Q814yes—
468C2×C8C22×Q814yes—
469C2×C8C22×Q818yes—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—42no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—82no—
504C4⋊C4—81no—
505C4⋊C4—81no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4#12314yes—
513C4⋊C4#12314yes—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—48no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C22⋊C4#12318yes—
525C42—24no—
526C42—24no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—48no—
532C42—42no—
533C42—42no—
534C42—44no—
535C42C2×#4118yes—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23C2×#12312yes—
547C23C2×C4⋊Q814yes—
548C23C2×C4⋊Q812yes—
549C23#53512yes—
550C23C23×D411yescharacteristic
551Q8—28no—
552Q8—164no—
553D4—216no—
554D4—28no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—48no—
574C2×C4—48no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—84no—
605C2×C4—162no—
606C2×C4—164no—
607C2×C4—168no—
608C2×C4C2×#12314yes—
609C2×C4C2×#12314yes—
610C2×C4C2×#12312yes—
611C2×C4C2×C4⋊Q814yes—
612C2×C4C2×C4⋊Q812yes—
613C2×C4#53512yes—
614C2×C4#55418yes—
615C2×C4#55418yes—
616C2×C4#57712yes—
617C2×C4#57712yes—
618C2×C4C23×Q812yes—
619C2×C4C22×#4112yes—
620C2×C4C2×H(5,𝔽2)11yescharacteristic
621C2×C4C2×H(5,𝔽2)11yescharacteristic
622C8—24no—
623C8—24no—
624C8—28no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22#250214yes—
639C22C22×#12311yescharacteristic
640C22C22×C4⋊Q811yescharacteristic
641C22C2×#53511yescharacteristic
642C22C2×#57711yescharacteristic
643C22C2×#57711yescharacteristic
644C22#358511yescharacteristic
645C22#358611yescharacteristic
646C4—22no—
647C4—24no—
648C4—22no—
649C4—24no—
650C4—24no—
651C4—41no—
652C4—41no—
653C4—81no—
654C4—82no—
655C4—81no—
656C4—81no—
657C4—84no—
658C4—82no—
659C4—82no—
660C4—81no—
661C4—82no—
662C4#250214yes—
663C4C2×#55412yes—
664C4C2×#55412yes—
665C2—24no—
666C2—44no—
667C2C2×#250211yescharacteristic
668C2#6052211yescharacteristic
669C2#6052511yescharacteristic
670C2#6066511yescharacteristic
671C2#6066811yescharacteristic
672C2#6083311yescharacteristic
673C2#6083611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27959C2semidirect
4. #27957C2semidirect
5. #27955C2semidirect
6. #27953C2semidirect
7. #27951C2semidirect
8. #27948C2semidirect
9. #23658C2semidirect
10. #23657C2semidirect
11. #23656C2semidirect
12. C2×#2084C2semidirect
13. #21326C2semidirect
16. #20802C2semidirect
148. #408Q8semidirect

Maximal subgroups

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

subgroupindex
#352572
#341832
#279592
#279572
#279552
#279532
#279512
#279482
#236582
#236572
#236562
C2×#20842
#213262
#208492
#208482
#208022
#203782