Finite Group Databasebeta

← #6432563

Subgroups — #6432563

Order 512.

1862 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34255C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#23787C214yesmaximal, complemented
4#23786C214yesmaximal, complemented
5#23781C212yesmaximal, complemented
6#23780C212yesmaximal, complemented
7#23775C212yesmaximal, complemented
8#23774C212yesmaximal, complemented
9#23744C211yesmaximal, characteristic, complemented
10#23731C211yesmaximal, characteristic
11#23671C212yesmaximal, complemented
12#23614C212yesmaximal
13#23552C211yesmaximal, characteristic, complemented
14C2×#2023C211yesmaximal, characteristic
15#20453C212yesmaximal, complemented
16#20430C211yesmaximal, characteristic, complemented
17#20416C212yesmaximal
18C22×C4⋊Q8C2211yescharacteristic
19C2×#518C2211yescharacteristic
20C2×#428C2212yes—
21C2×#428C2212yes—
22C2×#424C2212yes—
23C2×#418C2212yes—
24C22×#102C2212yes—
25C2×#405C2212yes—
26#2369—21no—
27#2369—21no—
28#2365—21no—
29#2365—21no—
30#2357—21no—
31#2357—21no—
32#2354—21no—
33#2354—21no—
34#2345—22no—
35#2342—22no—
36#2342—22no—
37Q8×C22⋊C4—22no—
38#2331C2214yescomplemented
39#2319C2212yes—
40#2319C2212yes—
41#2317C2211yescharacteristic
42#2316C2212yes—
43#2309C2211yescharacteristic
44C4×C4⋊Q8C2212yescomplemented
45C4×C4⋊Q8C2212yescomplemented
46#2296C2211yescharacteristic
47#2293C2212yes—
48#2293C2212yes—
49C2×#390C2211yescharacteristic
50#2285C2211yescharacteristic
51#2274C2212yes—
52#2274C2211yescharacteristic
53C4×#117C2211yescharacteristic
54#2101C2212yescomplemented
55#2101C2212yescomplemented
56#2100C2211yescharacteristic, complemented
57#2100C2211yescharacteristic, complemented
58#2099C2211yescharacteristic, complemented
59#2099C2211yescharacteristic, complemented
60#2094C2212yes—
61#2093C2212yes—
62#2092C2214yes—
63#2088C2214yescomplemented
64#2088C2214yescomplemented
65#2087C2214yescomplemented
66#2086C2214yescomplemented
67#2067C2214yes—
68#2067C2214yes—
69#2065C2214yes—
70#2065C2214yes—
71#2058C2212yescomplemented
72#2058C2212yescomplemented
73#2057C2212yescomplemented
74#2057C2212yescomplemented
75#2023C2212yes—
76#2023C2212yes—
77#2023C2214yes—
78#1776C2214yescomplemented
79#1776C2214yescomplemented
80C4×#107C2214yescomplemented
81C4×#106C2214yescomplemented
82#1768C2214yescomplemented
83#1767C2212yescomplemented
84C4×C4⋊C8C2212yescomplemented
85#1762C2214yes—
86#1762C2214yes—
87C4×#102C2214yes—
88C4×#102C2214yes—
89#1736C2212yes—
90#1735C2212yes—
91#1735C2212yes—
92C2×#340C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C23×Q8—21no—
96C23×D4C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99C2×C4⋊Q8—28no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104#518—24no—
105#518Q818yescomplemented
106C2×C4×Q8—22no—
107C2×C4×Q8—22no—
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4C2312yes—
111C2×C4×D4C2312yes—
112C2×#117—22no—
113C2×#117—22no—
114C2×#117C2311yescharacteristic
115C2×#117C2312yes—
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C22×C4⋊C4—21no—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2312yes—
122C22×C22⋊C4—22no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C42C2311yescharacteristic
126#428—28no—
127#428—28no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#106C2314yes—
131C2×#106C2314yes—
132#424—28no—
133C2×C4⋊C8C2314yes—
134C2×C4⋊C8C2314yes—
135#418—28no—
136C2×#102—28no—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141#405—28no—
142C2×#97C2314yes—
143C2×C4×C8C2314yes—
144#391—22no—
145#391—22no—
146#391—24no—
147#391—24no—
148#390—21no—
149#390—21no—
150#390—21no—
151#390—21no—
152#390C2312yes—
153#390C2312yes—
154#390C2312yes—
155#390C2312yes—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389C2314yes—
161#389C2314yes—
162#389C2311yescharacteristic
163#389C2311yescharacteristic
164#389C2311yescharacteristic
165#389C2311yescharacteristic
166#389C2312yes—
167#389C2312yes—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#387—24no—
177#387—24no—
178#385—21no—
179#385—21no—
180#385—22no—
181#384—21no—
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#382—21no—
185#382C2312yes—
186#381—22no—
187#381—22no—
188#380—21no—
189#380—21no—
190#380—24no—
191#380—22no—
192#380—22no—
193#379—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C22⋊C4—24no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4D412yes—
218#376C2312yes—
219C43C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2312yes—
234#336C2311yescharacteristic
235#336C2311yescharacteristic
236#336C2312yes—
237C25—21no—
238C22×Q8—28no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—82no—
242C22×Q8—82no—
243C22×D4—21no—
244C22×D4—24no—
245C22×D4—24no—
246C22×D4—22no—
247C22×D4C2×Q814yes—
248C22×D4C2412yes—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—21no—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2411yescharacteristic
256C2×M16—24no—
257C2×M16—24no—
258C2×M16#4118yes—
259C22×C8—24no—
260C22×C8—24no—
261C22×C8—24no—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C22×C8C2414yes—
265C4⋊Q8—28no—
266C4⋊Q8—28no—
267C4⋊Q8—28no—
268C4⋊Q8—28no—
269C4×Q8—84no—
270C4×Q8—84no—
271C4×D4—28no—
272C4×D4—28no—
273C4×D4—24no—
274C4×D4—24no—
275C4×D4C2×Q818yescomplemented
276C4×D4C2×Q818yescomplemented
277#117—22no—
278#117—24no—
279#117—24no—
280#117—24no—
281#117—24no—
282#117—84no—
283#117—84no—
284#117C2×Q814yes—
285#117C2×Q814yes—
286#117#4114yes—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4#4118yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2414yes—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—216no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—82no—
334C2×C22⋊C4—82no—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C42—21no—
338C2×C42—24no—
339C2×C42—22no—
340C2×C42—21no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2412yes—
355#107—28no—
356#107—28no—
357#106—28no—
358#106—28no—
359C4⋊C8—28no—
360C4⋊C8—28no—
361#102—28no—
362#102—28no—
363#102—28no—
364#102—28no—
365#97—28no—
366C4×C8—28no—
367#95—21no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—24no—
379#95—21no—
380#95—21no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24—22no—
386C24—24no—
387C24—42no—
388C24—42no—
389C24—81no—
390C24—81no—
391C24—81no—
392C24—81no—
393C24C22×D411yescharacteristic
394C2×Q8—28no—
395C2×Q8—28no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—82no—
399C2×Q8—82no—
400C2×Q8—816no—
401C2×D4—24no—
402C2×D4—216no—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—42no—
407C2×D4—48no—
408C2×D4—42no—
409C2×D4C22×Q814yes—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—28no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—42no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4—82no—
434C22×C4—81no—
435C22×C4—81no—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×Q812yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4114yes—
448C22×C4C2×#4114yes—
449C22×C4C2511yescharacteristic
450M16—216no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—48no—
460C2×C8C2×#4118yes—
461C4⋊C4—28no—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—84no—
488C4⋊C4—84no—
489C4⋊C4C22⋊Q814yes—
490C4⋊C4C22⋊Q814yes—
491C4⋊C4C22×Q812yes—
492C4⋊C4C22×Q812yes—
493C4⋊C4C2×#4118yes—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—28no—
497C22⋊C4—28no—
498C22⋊C4—42no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C22⋊C4—82no—
506C22⋊C4—82no—
507C22⋊C4—82no—
508C22⋊C4C22⋊Q818yes—
509C42—24no—
510C42—22no—
511C42—22no—
512C42—24no—
513C42—24no—
514C42—24no—
515C42—22no—
516C42—22no—
517C42—42no—
518C42—42no—
519C42—42no—
520C42—44no—
521C42—82no—
522C42—82no—
523C42—84no—
524C42—84no—
525C42C22×Q814yes—
526C42C2×#4114yes—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—24no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—48no—
539C23—84no—
540C23—84no—
541C23—162no—
542C23—162no—
543C23—162no—
544C23—162no—
545C23C2×C22⋊Q812yes—
546C23#52512yes—
547C23#53212yes—
548C23C23×D411yescharacteristic
549Q8—416no—
550Q8—88no—
551D4—28no—
552D4—48no—
553D4—44no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—28no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—88no—
601C2×C4—88no—
602C2×C4—88no—
603C2×C4—162no—
604C2×C4C2×C22⋊Q814yes—
605C2×C4C2×C22⋊Q814yes—
606C2×C4C2×C22⋊Q812yes—
607C2×C4#52512yes—
608C2×C4#52918yes—
609C2×C4#53212yes—
610C2×C4#53318yes—
611C2×C4#57612yes—
612C2×C4#57712yes—
613C2×C4C23×Q811yescharacteristic
614C2×C4C22×#4111yescharacteristic
615C2×C4C22×#4112yes—
616C2×C4C22×#4112yes—
617C8—28no—
618C8—42no—
619C8—42no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—168no—
634C22#239614yes—
635C22C22×C22⋊Q811yescharacteristic
636C22C2×#52511yescharacteristic
637C22C2×#53211yescharacteristic
638C22C2×#57611yescharacteristic
639C22C2×#57711yescharacteristic
640C22#358511yescharacteristic
641C22#358611yescharacteristic
642C4—24no—
643C4—24no—
644C4—22no—
645C4—22no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—41no—
651C4—48no—
652C4—41no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—81no—
657C4—81no—
658C4#239614yes—
659C4C2×#52912yes—
660C4C2×#53312yes—
661C2—24no—
662C2—84no—
663C2C2×#239611yescharacteristic
664C2#6049411yescharacteristic
665C2#6049511yescharacteristic
666C2#6057811yescharacteristic
667C2#6058211yescharacteristic
668C2#6072211yescharacteristic
669C2#6072711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34216C2semidirect
3. #23787C2semidirect
4. #23786C2semidirect
5. #23781C2semidirect
6. #23780C2semidirect
7. #23775C2semidirect
8. #23774C2semidirect
9. #23744C2semidirect
11. #23671C2semidirect
13. #23552C2semidirect
15. #20453C2semidirect
16. #20430C2semidirect
38. #2331C22semidirect
44. C4×C4⋊Q8C22semidirect
45. C4×C4⋊Q8C22semidirect
54. #2101C22semidirect
55. #2101C22semidirect
56. #2100C22semidirect
57. #2100C22semidirect
58. #2099C22semidirect
59. #2099C22semidirect
63. #2088C22semidirect
64. #2088C22semidirect
65. #2087C22semidirect
66. #2086C22semidirect
71. #2058C22semidirect
72. #2058C22semidirect
73. #2057C22semidirect
74. #2057C22semidirect
78. #1776C22semidirect
79. #1776C22semidirect
80. C4×#107C22semidirect
81. C4×#106C22semidirect
82. #1768C22semidirect
83. #1767C22semidirect
84. C4×C4⋊C8C22semidirect
105. #518Q8semidirect
275. C4×D4C2×Q8semidirect
276. C4×D4C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#342552
#342162
#237872
#237862
#237812
#237802
#237752
#237742
#237442
#237312
#236712
#236142
#235522
C2×#20232
#204532
#204302
#204162