Finite Group Databasebeta

← #363315

Subgroups — #363315

Order 512.

1478 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2318C211yesmaximal, characteristic
2#14349C211yesmaximal, characteristic, complemented
3#13778C211yesmaximal, characteristic, complemented
4#11203C212yesmaximal, complemented
5#11043C212yesmaximal
6#10960C212yesmaximal
7#10912C212yesmaximal, complemented
8#10801C212yesmaximal, complemented
9#10761C212yesmaximal
10C22×C4×D4C2211yescharacteristic
11C22×#99C2214yes—
12#2318—24no—
13#2318—24no—
14C2×#388C411yescharacteristic, complemented
15C2×#388C411yescharacteristic, complemented
16C2×#387C411yescharacteristic, complemented
17C2×#387C411yescharacteristic, complemented
18C2×#386C2211yescharacteristic
19C2×#386C2211yescharacteristic
20C2×#385C2211yescharacteristic
21C2×#385C2211yescharacteristic
22C2×#383C411yescharacteristic, complemented
23C2×#382C411yescharacteristic, complemented
24C2×#381C411yescharacteristic, complemented
25C2×#380C411yescharacteristic, complemented
26C2×#380C2212yes—
27C2×C4×C22⋊C4C2212yes—
28C2×C4×C22⋊C4C2211yescharacteristic
29C22×#95C2211yescharacteristic
30C2×#342C2214yes—
31C2×#342C2214yes—
32C2×#342C2214yes—
33#1443—24no—
34#1436—24no—
35C22⋊#95C2212yes—
36C22⋊#95C2212yes—
37C22⋊#95C2212yes—
38C22⋊#95C2212yes—
39C23×D4C2311yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2311yescharacteristic
42C24×C4C2311yescharacteristic
43C24×C4C2311yescharacteristic
44C2×C4×D4—24no—
45C2×C4×D4—22no—
46C2×C4×D4—22no—
47C2×C4×D4—24no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C22⋊C4C2×C411yescharacteristic
55C22×C22⋊C4C2×C411yescharacteristic
56C22×C22⋊C4C2×C411yescharacteristic
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2314yes—
60C22×C22⋊C4C2311yescharacteristic
61C22×C22⋊C4C2311yescharacteristic
62C22×C22⋊C4C2314yes—
63C22×C22⋊C4C2311yescharacteristic
64C22×C42—22no—
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68C2×#99—28no—
69C2×#99—28no—
70C2×#99—44no—
71#388—24no—
72#388—24no—
73#387—24no—
74#387—24no—
75#386—22no—
76#386—22no—
77#386—22no—
78#386—22no—
79#385—22no—
80#385—22no—
81#385—22no—
82#385—22no—
83#383—24no—
84#382—24no—
85#381—24no—
86#380—28no—
87#380—24no—
88C4×C22⋊C4—22no—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—44no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C411yescharacteristic
105C2×#95C2×C411yescharacteristic
106C2×#95C2×C411yescharacteristic
107C2×#95C2×C411yescharacteristic
108C2×#95C2×C411yescharacteristic
109C2×#95C2×C411yescharacteristic
110C2×#95C2×C411yescharacteristic
111C2×#95C2×C411yescharacteristic
112C2×#95C2×C411yescharacteristic
113C2×#95C2×C411yescharacteristic
114C2×#95C2×C411yescharacteristic
115C2×#95D414yes—
116C2×#95D414yes—
117#342—28no—
118#342—28no—
119#342—44no—
120C25C22×C411yescharacteristic
121C25C2411yescharacteristic
122C22×D4—21no—
123C22×D4—21no—
124C22×D4—22no—
125C22×D4—24no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4#4114yes—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—24no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—81no—
140C23×C4—81no—
141C23×C4—81no—
142C23×C4—81no—
143C23×C4—81no—
144C23×C4—81no—
145C23×C4—81no—
146C23×C4—81no—
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C4×D4—216no—
167C4×D4—216no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4—81no—
179C2×C4⋊C4—81no—
180C2×C4⋊C4—81no—
181C2×C4⋊C4—81no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—81no—
211C2×C22⋊C4—81no—
212C2×C22⋊C4—81no—
213C2×C22⋊C4—81no—
214C2×C22⋊C4#4118yes—
215C2×C22⋊C4#4114yes—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—44no—
222C2×C42—82no—
223C2×C42—82no—
224C2×C42—81no—
225C2×C42—81no—
226#99—416no—
227#95—28no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—44no—
239#95—82no—
240#95—82no—
241#95—82no—
242#95—82no—
243#95—82no—
244#95—82no—
245#95—82no—
246#95—82no—
247#95C22⋊C414yes—
248#95C22⋊C414yes—
249#95C22⋊C414yes—
250#95C22⋊C414yes—
251C24—21no—
252C24—21no—
253C24—22no—
254C24—22no—
255C24—41no—
256C24—41no—
257C24—41no—
258C24—41no—
259C24—41no—
260C24—41no—
261C24—41no—
262C24—41no—
263C24#11712yes—
264C24C23×C411yescharacteristic
265C24C22×D411yescharacteristic
266C24C22×D411yescharacteristic
267C24C2×#4112yes—
268C24C2×#4112yes—
269C24C2×#4114yes—
270C24H(5,𝔽2)12yes—
271C242−1+412yes—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—28no—
275C2×D4—28no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4—44no—
282C2×D4—44no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—28no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—28no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—82no—
367C22×C4—82no—
368C22×C4—84no—
369C22×C4—88no—
370C22×C4—84no—
371C22×C4—84no—
372C22×C4—84no—
373C22×C4—84no—
374C22×C4—84no—
375C22×C4C2×C22⋊C412yes—
376C22×C4C2×C22⋊C412yes—
377C22×C4C2×C22⋊C412yes—
378C22×C4C2×C22⋊C412yes—
379C22×C4C2×C22⋊C412yes—
380C22×C4C2×C22⋊C412yes—
381C22×C4#11712yes—
382C22×C4#11712yes—
383C22×C4#11712yes—
384C4⋊C4—28no—
385C4⋊C4—82no—
386C4⋊C4—82no—
387C4⋊C4—82no—
388C4⋊C4—82no—
389C4⋊C4—88no—
390C22⋊C4—28no—
391C22⋊C4—24no—
392C22⋊C4—24no—
393C22⋊C4—28no—
394C22⋊C4—28no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—42no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—82no—
407C22⋊C4—82no—
408C22⋊C4—82no—
409C22⋊C4—82no—
410C42—28no—
411C42—82no—
412C42—82no—
413C42—88no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—22no—
427C23—24no—
428C23—44no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—44no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—41no—
443C23—42no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—82no—
450C23—82no—
451C23—82no—
452C23—82no—
453C23C22×C22⋊C411yescharacteristic
454C23C2×#11711yescharacteristic
455C23C2×#11711yescharacteristic
456C23C4×#4112yes—
457C23#52012yes—
458C23#52512yes—
459C23#52512yes—
460C23#52512yes—
461C23#53212yes—
462C23#53511yescharacteristic
463C23#53611yescharacteristic
464C23#53712yes—
465C23#55114yes—
466C23#55214yes—
467C23#55314yes—
468C23#55614yes—
469D4—44no—
470D4—44no—
471D4—48no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—48no—
514C2×C4—48no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—48no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—84no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—84no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—84no—
535C2×C4—84no—
536C2×C4—88no—
537C2×C4—84no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4—84no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—84no—
546C2×C4—84no—
547C2×C4—84no—
548C2×C4—84no—
549C2×C4—84no—
550C2×C4—84no—
551C2×C4—84no—
552C22—21no—
553C22—21no—
554C22—22no—
555C22—22no—
556C22—21no—
557C22—22no—
558C22—22no—
559C22—21no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—21no—
567C22—21no—
568C22—22no—
569C22—22no—
570C22—21no—
571C22—21no—
572C22—22no—
573C22—22no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—84no—
582C22—84no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22#228011yescharacteristic
588C22#229711yescharacteristic
589C22#231111yescharacteristic
590C22#234312yes—
591C22#234712yes—
592C22#259812yes—
593C22#259912yes—
594C22#260114yes—
595C22#288311yescharacteristic
596C22#288311yescharacteristic
597C22#288311yescharacteristic
598C22#288411yescharacteristic
599C4—42no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—42no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—42no—
608C4—88no—
609C4—81no—
610C4—81no—
611C4—81no—
612C4—81no—
613C4—164no—
614C2—22no—
615C2—24no—
616C2—22no—
617C2—22no—
618C2—22no—
619C2—82no—
620C2#1393811yescharacteristic
621C2#2072411yescharacteristic
622C2#2072511yescharacteristic
623C2#2142211yescharacteristic
624C2#2142311yescharacteristic
625C2#2187611yescharacteristic
626C2#2187711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#95 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14349C2semidirect
3. #13778C2semidirect
4. #11203C2semidirect
7. #10912C2semidirect
8. #10801C2semidirect
14. C2×#388C4semidirect
15. C2×#388C4semidirect
16. C2×#387C4semidirect
17. C2×#387C4semidirect
22. C2×#383C4semidirect
23. C2×#382C4semidirect
24. C2×#381C4semidirect
25. C2×#380C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23182
#143492
#137782
#112032
#110432
#109602
#109122
#108012
#107612