Finite Group Databasebeta

← #361114

Subgroups — #361114

Order 512.

1680 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2290C211yesmaximal, characteristic
2C2×#1917C212yesmaximal, complemented
3C2×#1798C211yesmaximal, characteristic, complemented
4C2×#1749C211yesmaximal, characteristic, complemented
5#11042C212yesmaximal, complemented
6#10964C212yesmaximal
7#10922C212yesmaximal, complemented
8#10796C211yesmaximal, characteristic, complemented
9#10796C211yesmaximal, characteristic, complemented
10#10713C211yesmaximal, characteristic, complemented
11#10712C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2212yes—
20C22×#97C2212yes—
21#2290—28no—
22C2×#388C412yes—
23C2×#387C412yes—
24C2×#386C412yes—
25C2×#385C412yes—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29C2×#376C2211yescharacteristic
30#1917—28no—
31#1798—24no—
32#1749—24no—
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42C2×#336C2211yescharacteristic
43C23×D4C2×C411yescharacteristic
44C24×C4C2×C412yes—
45C24×C4C2311yescharacteristic
46C24×C4C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4×D4—24no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—24no—
55C2×#117—22no—
56C2×#117—24no—
57C2×#117—22no—
58C2×#117—42no—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2×C412yes—
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2×C412yes—
65C22×C22⋊C4C2×C412yes—
66C22×C22⋊C4C2×C412yes—
67C22×C42—22no—
68C22×C42—24no—
69C22×C42—22no—
70C22×C42—42no—
71C22×C42C2×C412yes—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×C4⋊C8—24no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—42no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—42no—
89C2×C22⋊C8—42no—
90C2×C22⋊C8—42no—
91C2×C22⋊C8—42no—
92C2×#97—24no—
93C2×#97—24no—
94C2×#97—42no—
95#388—28no—
96#387—28no—
97#386—28no—
98#385—28no—
99C4×C4⋊C4D418yes—
100C4×C22⋊C4—24no—
101C4×C22⋊C4—24no—
102#376—24no—
103C2×#95C2×C412yes—
104C2×#95C2×C412yes—
105C2×#95C2×C412yes—
106C2×#95C2×C412yes—
107C2×#95C2×C412yes—
108#336—24no—
109#336—22no—
110#336—24no—
111#336—24no—
112#336—24no—
113#336—24no—
114#336—22no—
115#336—22no—
116#336—42no—
117#336D414yescomplemented
118C25C22×C411yescharacteristic
119C25C22×C411yescharacteristic
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—24no—
123C22×D4—22no—
124C22×D4—42no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4C22×C412yes—
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C412yes—
152C23×C4C22×C412yes—
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C412yes—
157C23×C4#4114yes—
158C23×C4#4112yes—
159C23×C4#4112yes—
160C23×C4C2411yescharacteristic
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8C2×D414yes—
188C22×C8#4114yes—
189C4×D4—216no—
190C4×D4—216no—
191#117—48no—
192#117—48no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4C22⋊C418yes—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C42—22no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—22no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—48no—
235C2×C42—48no—
236C2×C42—42no—
237C2×C42C2×D414yes—
238C2×C42C2×D414yes—
239C2×C42C2×D414yes—
240C2×C42C2×D412yes—
241C2×C42#4112yes—
242C4⋊C8—48no—
243C22⋊C8—216no—
244C22⋊C8—48no—
245C22⋊C8—48no—
246C22⋊C8—48no—
247#97—48no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95C22⋊C418yes—
253C24—21no—
254C24—21no—
255C24—21no—
256C24—21no—
257C24—21no—
258C24—21no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—21no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24#11712yes—
273C24#11712yes—
274C24#13114yes—
275C24C23×C411yescharacteristic
276C2×D4—28no—
277C2×D4—48no—
278C2×D4—48no—
279C2×D4—44no—
280C2×D4—44no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—48no—
341C22×C4—48no—
342C22×C4—48no—
343C22×C4—48no—
344C22×C4—48no—
345C22×C4—48no—
346C22×C4—48no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4C2×C22⋊C412yes—
366C22×C4C2×C22⋊C412yes—
367C22×C4C2×C22⋊C414yes—
368C22×C4C2×C22⋊C414yes—
369C22×C4#11712yes—
370C22×C4#11712yes—
371C22×C4C4×D414yes—
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D412yes—
374C22×C4C2×#4112yes—
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4112yes—
378C22×C4C2×#4111yescharacteristic
379C22×C4H(5,𝔽2)11yescharacteristic
380C22×C4H(5,𝔽2)12yes—
381C22×C42−1+411yescharacteristic
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C4⋊C4—28no—
411C4⋊C4—48no—
412C4⋊C4—48no—
413C22⋊C4—28no—
414C22⋊C4—28no—
415C22⋊C4—48no—
416C22⋊C4—48no—
417C22⋊C4—48no—
418C42—24no—
419C42—24no—
420C42—24no—
421C42—24no—
422C42—24no—
423C42—24no—
424C42—24no—
425C42—24no—
426C42—48no—
427C42—42no—
428C42—42no—
429C23—22no—
430C23—24no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—24no—
444C23—24no—
445C23—22no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—42no—
460C23—42no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—41no—
474C23#43314yes—
475C23C22×C22⋊C411yescharacteristic
476C23C2×#11711yescharacteristic
477C23C2×#11711yescharacteristic
478C23C2×C4×D411yescharacteristic
479C23C4×#4111yescharacteristic
480C23#51811yescharacteristic
481C23#52011yescharacteristic
482C23C2×#13112yes—
483C23#56812yes—
484C23#56812yes—
485C23#56812yes—
486D4—416no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—432no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4#52512yes—
549C2×C4#52512yes—
550C2×C4#52512yes—
551C2×C4#52912yes—
552C2×C4#53812yes—
553C2×C4#53912yes—
554C2×C4D4212yes—
555C2×C4#54611yescharacteristic
556C2×C4#54612yes—
557C2×C4#54812yes—
558C2×C4#54811yescharacteristic
559C2×C4#55012yes—
560C2×C4#55112yes—
561C2×C4#55211yescharacteristic
562C2×C4#55312yes—
563C2×C4#55311yescharacteristic
564C8—44no—
565C8—44no—
566C8—82no—
567C8—82no—
568C22—24no—
569C22—24no—
570C22—24no—
571C22—44no—
572C22—44no—
573C22—42no—
574C22—42no—
575C22—44no—
576C22—44no—
577C22—42no—
578C22—42no—
579C22—44no—
580C22—42no—
581C22—44no—
582C22—42no—
583C22—42no—
584C22—82no—
585C22—81no—
586C22—81no—
587C22—82no—
588C22—81no—
589C22—81no—
590C22#228011yescharacteristic
591C22#229011yescharacteristic
592C22#231811yescharacteristic
593C22D4×C22⋊C411yescharacteristic
594C22#234111yescharacteristic
595C22#235211yescharacteristic
596C22#235311yescharacteristic
597C22#287912yes—
598C22#288012yes—
599C22C2×#43311yescharacteristic
600C22#292111yescharacteristic
601C22#292312yes—
602C22#292311yescharacteristic
603C22#292311yescharacteristic
604C22#292412yes—
605C22#293012yes—
606C22#293312yes—
607C22#296712yes—
608C22#296812yes—
609C22#297612yes—
610C22#297812yes—
611C22#299212yes—
612C22#299512yes—
613C4—42no—
614C4—42no—
615C4—44no—
616C4—44no—
617C4—44no—
618C4—42no—
619C4—42no—
620C4—82no—
621C4#245612yes—
622C4#246012yes—
623C4#246312yes—
624C4#246712yes—
625C2—44no—
626C2—82no—
627C2#1389011yescharacteristic
628C2#2071611yescharacteristic
629C2#2071911yescharacteristic
630C2#2110411yescharacteristic
631C2#2110511yescharacteristic
632C2#2200111yescharacteristic
633C2#2200311yescharacteristic
634C2#2240012yes—
635C2#2253312yes—
636C2#2275312yes—
637C2#2279412yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutator, socleC24C23×C4
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1917C2semidirect
3. C2×#1798C2semidirect
4. C2×#1749C2semidirect
5. #11042C2semidirect
7. #10922C2semidirect
8. #10796C2semidirect
9. #10796C2semidirect
10. #10713C2semidirect
11. #10712C2semidirect
117. #336D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22902
C2×#19172
C2×#17982
C2×#17492
#110422
#109642
#109222
#107962
#107132
#107122