Finite Group Databasebeta

← #363282

Subgroups — #363282

Order 512.

2738 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#556C211yesmaximal, characteristic
2C2×#1984C211yesmaximal, characteristic, complemented
3C2×#1981C211yesmaximal, characteristic, complemented
4C2×#1956C211yesmaximal, characteristic, complemented
5C2×#1802C211yesmaximal, characteristic
6C2×C4×#102C211yesmaximal, characteristic
7#13926C211yesmaximal, characteristic, complemented
8#11009C214yesmaximal
9#10789C214yesmaximal, complemented
10C2×#556—24no—
11C2×#556—24no—
12C2×#556—24no—
13C2×#556—24no—
14C2×#556—24no—
15C2×#556C418yescomplemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#126—22no—
18C22×#124C2212yes—
19C22×C22⋊Q8—21no—
20C22×C4×Q8C2212yes—
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C22×#117—21no—
24C22×C4⋊C8C2211yescharacteristic
25C22×#103C2214yes—
26C22×#103C2214yes—
27C22×#102C2212yes—
28C22×#102C2212yes—
29C22×C4×C8C2211yescharacteristic
30C2×#391—21no—
31C2×#384C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1984—24no—
34#1981—24no—
35#1956—24no—
36#1802—24no—
37C4×#102—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2214yes—
41#1441—22no—
42C4⋊#95—21no—
43C4⋊#95C2214yes—
44C23×Q8D411yescharacteristic
45C23×Q8C2312yes—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C23×C8C2312yes—
49C23×C8C2312yes—
50#556—232no—
51#556—232no—
52#556—232no—
53#556—232no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8C2×C414yes—
58C2×C4⋊Q8D414yes—
59C2×C4⋊Q8D414yes—
60C2×#126—216no—
61C2×#126—216no—
62C2×#126—216no—
63C2×#124—24no—
64C2×#124—24no—
65C2×#124—24no—
66C2×#124—24no—
67C2×#124—24no—
68C2×#124C2×C418yes—
69C2×C22⋊Q8—28no—
70C2×C22⋊Q8—28no—
71C2×C22⋊Q8—28no—
72C2×C4×Q8—24no—
73C2×C4×Q8—24no—
74C2×C4×Q8—24no—
75C2×C4×Q8—24no—
76C2×C4×Q8—24no—
77C2×C4×Q8—22no—
78C2×C4×Q8—22no—
79C2×C4×Q8—22no—
80C2×C4×Q8—22no—
81C2×C4×Q8C2×C418yes—
82C2×C4×Q8C2×C414yes—
83C2×C4×Q8D414yes—
84C2×C4×D4—22no—
85C2×C4×D4—22no—
86C2×C4×D4—22no—
87C2×C4×D4C2×C414yes—
88C2×C4×D4D414yes—
89C2×C4×D4D414yes—
90C2×#117—28no—
91C2×#117—28no—
92C2×#117—28no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—22no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4D411yescharacteristic
99C22×C4⋊C4D412yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4—22no—
104C22×C22⋊C4—21no—
105C22×C22⋊C4—22no—
106C22×C42—22no—
107C22×C42—21no—
108C22×C42C2312yes—
109C22×C42C2311yescharacteristic
110C2×C4⋊C8—22no—
111C2×C4⋊C8—22no—
112C2×C4⋊C8—22no—
113C2×#103—28no—
114C2×#103—28no—
115C2×#103—28no—
116C2×#103—28no—
117C2×#103—44no—
118C2×#103—44no—
119C2×#102—24no—
120C2×#102—24no—
121C2×#102—24no—
122C2×#102—24no—
123C2×#102—44no—
124C2×C4×C8—22no—
125C2×C4×C8—22no—
126C2×C4×C8—22no—
127#391—44no—
128#384—22no—
129#384—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C2×#95—24no—
133C2×#95—21no—
134C2×#95—24no—
135#340—24no—
136#340—24no—
137#336—28no—
138C25—21no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—24no—
149C22×Q8—24no—
150C22×Q8—24no—
151C22×Q8—24no—
152C22×Q8C22⋊C414yes—
153C22×Q8C22×C414yes—
154C22×Q8C2×D414yes—
155C22×Q8C2×D414yes—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4C22×C412yes—
164C22×D4C2×D412yes—
165C22×D4C2×D412yes—
166C23×C4—22no—
167C23×C4—21no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4C2×D412yes—
178C23×C4C2×D412yes—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2411yescharacteristic
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—24no—
192C22×C8—24no—
193C22×C8#4114yes—
194C4⋊Q8—28no—
195C4⋊Q8—28no—
196C4⋊Q8—28no—
197C4⋊Q8C22⋊C4116yes—
198#126—2128no—
199#124—216no—
200#124—216no—
201#124—216no—
202#124—216no—
203C22⋊Q8—264no—
204C4×Q8—216no—
205C4×Q8—216no—
206C4×Q8—216no—
207C4×Q8—216no—
208C4×Q8—28no—
209C4×Q8—28no—
210C4×Q8—28no—
211C4×Q8—28no—
212C4×D4—28no—
213C4×D4—28no—
214C4×D4—28no—
215C4×D4C22⋊C4116yes—
216#117—264no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—82no—
247C2×C4⋊C4C22⋊C414yes—
248C2×C4⋊C4C22⋊C418yes—
249C2×C4⋊C4C22×C412yes—
250C2×C4⋊C4C2×D412yes—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4#4114yes—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C42—28no—
263C2×C42—28no—
264C2×C42—28no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42C22×C414yes—
273C2×C42C22×C412yes—
274C2×C42C2×D414yes—
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C4⋊C8—44no—
281#103—416no—
282#103—416no—
283#102—216no—
284#102—48no—
285C4×C8—44no—
286#95—48no—
287#95—42no—
288#95—84no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—42no—
293C24—42no—
294C24—41no—
295C24—41no—
296C24—41no—
297C24—41no—
298C24—42no—
299C24—42no—
300C24C22×D411yescharacteristic
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—28no—
305C2×Q8—28no—
306C2×Q8—28no—
307C2×Q8—28no—
308C2×Q8—28no—
309C2×Q8—28no—
310C2×Q8—28no—
311C2×Q8—28no—
312C2×Q8—28no—
313C2×Q8—44no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—44no—
321C2×Q8—44no—
322C2×Q8—44no—
323C2×Q8—44no—
324C2×Q8—44no—
325C2×Q8C2×C22⋊C418yes—
326C2×D4—22no—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—24no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—24no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4C2×C22⋊C414yes—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—216no—
363C22×C4—24no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—48no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—44no—
387C22×C4C2×C22⋊C414yes—
388C22×C4C2×C22⋊C414yes—
389C22×C4C2×C22⋊C412yes—
390C22×C4C2×C22⋊C412yes—
391C22×C4#12012yes—
392C22×C4#12012yes—
393C22×C4#12012yes—
394C22×C4#12012yes—
395C22×C4#13512yes—
396C22×C4#13512yes—
397C22×C4AGL(1,ℤ/8)12yes—
398C22×C4#13712yes—
399C22×C4C23×C411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C2×#4112yes—
406C22×C4C2×#4112yes—
407C22×C4H(5,𝔽2)12yes—
408C22×C42−1+412yes—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—44no—
420C2×C8—44no—
421C4⋊C4—232no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—216no—
426C4⋊C4—216no—
427C4⋊C4—28no—
428C4⋊C4—216no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—44no—
436C4⋊C4—84no—
437C4⋊C4—84no—
438C4⋊C4C2×C22⋊C414yes—
439C22⋊C4—232no—
440C22⋊C4—232no—
441C22⋊C4—216no—
442C42—232no—
443C42—24no—
444C42—24no—
445C42—24no—
446C42—24no—
447C42—84no—
448C42C2×C22⋊C414yes—
449C42C2×C22⋊C414yes—
450C42#12014yes—
451C42#12014yes—
452C23—24no—
453C23—44no—
454C23—42no—
455C23—42no—
456C23—44no—
457C23—44no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—82no—
468C23—82no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23C22×C22⋊C411yescharacteristic
476C23C2×#12011yescharacteristic
477C23C2×#12011yescharacteristic
478C23#54812yes—
479C23#55012yes—
480C23C2×#13511yescharacteristic
481C23C2×#13511yescharacteristic
482C23C2×AGL(1,ℤ/8)11yescharacteristic
483C23C2×#13711yescharacteristic
484C23#57712yes—
485C23#57812yes—
486Q8—216no—
487Q8—44no—
488Q8—48no—
489Q8—44no—
490Q8—44no—
491Q8—44no—
492Q8—416no—
493D4—24no—
494D4—24no—
495D4—44no—
496D4—44no—
497D4—42no—
498D4—42no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—28no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—24no—
510C2×C4—48no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—48no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—432no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—84no—
536C2×C4—84no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4#37914yes—
541C2×C4#37914yes—
542C2×C4#41614yes—
543C2×C4#41714yes—
544C2×C4#44914yes—
545C2×C4#44914yes—
546C2×C4#45214yes—
547C2×C4#45214yes—
548C2×C4C22×C22⋊C411yescharacteristic
549C2×C4C22×C22⋊C411yescharacteristic
550C2×C4C4×#4112yes—
551C2×C4#52012yes—
552C2×C4C2×#12011yescharacteristic
553C2×C4C2×#12011yescharacteristic
554C2×C4#52512yes—
555C2×C4#53211yescharacteristic
556C2×C4#53211yescharacteristic
557C2×C4#53511yescharacteristic
558C2×C4#53611yescharacteristic
559C2×C4#53712yes—
560C2×C4#54612yes—
561C2×C4#54612yes—
562C2×C4#54712yes—
563C2×C4#54712yes—
564C2×C4#54812yes—
565C2×C4#55012yes—
566C8—44no—
567C8—82no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—84no—
572C22—84no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—84no—
578C22—84no—
579C22C2×#37911yescharacteristic
580C22#234112yes—
581C22#240712yes—
582C22#243111yescharacteristic
583C22#243211yescharacteristic
584C22C2×#41611yescharacteristic
585C22C2×#41711yescharacteristic
586C22#289812yes—
587C22C2×#44911yescharacteristic
588C22C2×#44911yescharacteristic
589C22C2×#45211yescharacteristic
590C22C2×#45211yescharacteristic
591C22#301011yescharacteristic
592C22#301111yescharacteristic
593C22#301312yes—
594C22#336012yes—
595C22#336412yes—
596C22#336712yes—
597C22#336912yes—
598C22#339312yes—
599C22#339412yes—
600C22#339712yes—
601C22#339812yes—
602C4—24no—
603C4—48no—
604C4—44no—
605C4—44no—
606C4—44no—
607C4—84no—
608C4—162no—
609C4#178818yes—
610C4#229711yescharacteristic
611C4#231111yescharacteristic
612C4#233912yes—
613C4#240712yes—
614C4#243111yescharacteristic
615C4#243211yescharacteristic
616C2—84no—
617C2#1393711yescharacteristic
618C2C2×#178811yescharacteristic
619C2#2060311yescharacteristic
620C2#2254112yes—
621C2#2254412yes—
622C2#2418512yes—
623C2#2418912yes—
624C2#2562811yescharacteristic
625C2#2563211yescharacteristic
626C2#2564611yescharacteristic
627C2#2565011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1984C2semidirect
3. C2×#1981C2semidirect
4. C2×#1956C2semidirect
7. #13926C2semidirect
9. #10789C2semidirect
15. C2×#556C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5562
C2×#19842
C2×#19812
C2×#19562
C2×#18022
C2×C4×#1022
#139262
#110092
#107892