Finite Group Databasebeta

← #6629678

Subgroups — #6629678

Order 512.

2400 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3095C211yesmaximal, characteristic, complemented
2C2×#3094C211yesmaximal, characteristic
3C2×#2620C211yesmaximal, characteristic
4#33996C211yesmaximal, characteristic, complemented
5#33121C214yesmaximal, complemented
6#33120C214yesmaximal, complemented
7#33115C218yesmaximal, complemented
8#21261C212yesmaximal, complemented
9#21171C212yesmaximal, complemented
10#21061C212yesmaximal, complemented
11#21060C212yesmaximal, complemented
12#20912C211yesmaximal, characteristic
13#20911C211yesmaximal, characteristic, complemented
14#20627C211yesmaximal, characteristic, complemented
15C22×#124C2211yescharacteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21#3095—28no—
22#3094—28no—
23C2×#484C2212yescomplemented
24C2×#481C2212yes—
25C2×#485C2214yescomplemented
26C2×#482C2214yes—
27C2×#483C2212yescomplemented
28C2×#480C2212yes—
29C2×#428C2212yes—
30C2×#428C2212yes—
31C2×#423C2212yes—
32C2×#419C2212yes—
33C2×#419C2212yes—
34C2×#418C2212yes—
35C2×#418C2212yes—
36C2×#407C2211yescharacteristic
37C2×#407C2211yescharacteristic
38#2620—22no—
39#2620—22no—
40#2620C2214yes—
41#2620C2214yes—
42C2×#396—22no—
43C2×#394—22no—
44C2×#391C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#384C2211yescharacteristic
47#2274—22no—
48#2274C2214yescomplemented
49#2274C2212yescomplemented
50#2274C2212yescomplemented
51#2273—22no—
52C4×#117—22no—
53C4×#117—22no—
54C4×#117C2212yes—
55C4×#117C2212yes—
56#1983C2218yescomplemented
57#1983C2218yescomplemented
58#1948C2218yescomplemented
59#1947C2218yescomplemented
60#1928C2218yescomplemented
61#1927C2218yescomplemented
62#1926C2218yescomplemented
63#1925C2218yescomplemented
64#1875C2214yes—
65#1874C2214yes—
66#1873C2214yescomplemented
67#1872C2214yescomplemented
68#1798C2214yescomplemented
69#1798C2214yescomplemented
70#1736C2212yes—
71#1736C2212yes—
72#1728C2212yes—
73C23×Q8C2311yescharacteristic
74C23×D4—21no—
75C23×D4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C22×M16C2312yes—
78C22×M16C2312yes—
79C2×#124—22no—
80C2×#124—28no—
81C2×#124C2314yes—
82C2×C22⋊Q8—21no—
83C2×C22⋊Q8—21no—
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8C2312yes—
87C2×C22⋊Q8C2312yes—
88C2×#121—21no—
89C2×#121—21no—
90C2×#121—24no—
91C2×#121—24no—
92C2×#121C2312yes—
93C2×#121C2312yes—
94C2×#117—21no—
95C2×#117—21no—
96C2×#117—21no—
97C2×#117—21no—
98C2×#117—24no—
99C2×#117—28no—
100C2×#117—28no—
101C2×#117D418yes—
102C2×#117D412yes—
103C2×#117D412yes—
104C2×#117C2312yes—
105C2×#117C2312yes—
106C2×#117C2312yes—
107C2×#117C2312yes—
108C2×#117C2312yes—
109C2×#117C2312yes—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4D412yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—22no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—22no—
124C22×C22⋊C4D412yes—
125C22×C42—21no—
126C22×C42—21no—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#485—216no—
130#484—28no—
131#483—28no—
132#482—216no—
133#481—28no—
134#480—28no—
135#428—28no—
136#428—28no—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141#423—28no—
142C2×C4⋊C8C2318yes—
143#419—28no—
144#419—28no—
145#418—28no—
146#418—28no—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2318yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2318yes—
153#407—24no—
154#407—24no—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2312yes—
159#396—28no—
160#396—28no—
161#394—28no—
162#394—28no—
163#391—22no—
164#391C2314yes—
165#390—22no—
166#390C2314yes—
167#384—22no—
168#384C2314yes—
169C4×C4⋊C4—24no—
170C4×C4⋊C4—24no—
171C4×C4⋊C4—24no—
172C4×C4⋊C4—24no—
173C4×C4⋊C4C2314yes—
174C4×C4⋊C4C2314yes—
175C4×C4⋊C4C2314yes—
176C4×C4⋊C4C2314yes—
177C4×C22⋊C4—24no—
178C4×C22⋊C4—24no—
179C4×C22⋊C4—24no—
180C4×C22⋊C4—24no—
181C4×C22⋊C4—24no—
182C4×C22⋊C4—24no—
183#376—22no—
184#376—22no—
185C2×#95—22no—
186C43—22no—
187C43—22no—
188#340C2318yes—
189#340C2318yes—
190#336C2318yes—
191C25—21no—
192C25C2×D411yescharacteristic
193C22×Q8—21no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8C2412yes—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—21no—
203C22×D4—22no—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—44no—
207C22×D4—84no—
208C22×D4C2412yes—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D414yes—
221C23×C4C2×D412yes—
222C23×C4C2×D412yes—
223C23×C4#4114yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C2×M16—24no—
228C2×M16—24no—
229C2×M16—24no—
230C2×M16—24no—
231C2×M16—24no—
232C2×M16—24no—
233C22×C8C2414yes—
234C22×C8C2414yes—
235#124—216no—
236#124—48no—
237C22⋊Q8—24no—
238C22⋊Q8—24no—
239C22⋊Q8—24no—
240C22⋊Q8—24no—
241C22⋊Q8—44no—
242C22⋊Q8—44no—
243#121—24no—
244#121—24no—
245#121—24no—
246#121—24no—
247#121—44no—
248#121—44no—
249#117—28no—
250#117—28no—
251#117—28no—
252#117—28no—
253#117—24no—
254#117—24no—
255#117—24no—
256#117—28no—
257#117—28no—
258#117—28no—
259#117—28no—
260#117—28no—
261#117C2×D418yes—
262#117C2×D418yes—
263#117#4118yes—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—84no—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4C2×D414yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2414yes—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—48no—
307C2×C22⋊C4—84no—
308C2×C22⋊C4C2×D414yes—
309C2×C22⋊C4C2×D414yes—
310C2×C42—28no—
311C2×C42—28no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42C2×D412yes—
322C2×C42C2×D412yes—
323C2×C42#4118yes—
324C2×C42C2412yes—
325C2×C42C2412yes—
326#107—28no—
327#107—28no—
328#106—28no—
329#106—28no—
330C4⋊C8—216no—
331#103—28no—
332#103—28no—
333#103—216no—
334#102—28no—
335#102—28no—
336#102—216no—
337C22⋊C8—24no—
338C22⋊C8—24no—
339C22⋊C8—24no—
340C22⋊C8—24no—
341#95—28no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—42no—
350C24—42no—
351C24—44no—
352C24—44no—
353C24—42no—
354C24—42no—
355C24—42no—
356C24#12114yes—
357C24C22×D411yescharacteristic
358C24C22×D411yescharacteristic
359C24C22×D411yescharacteristic
360C2×Q8—22no—
361C2×Q8—24no—
362C2×Q8—22no—
363C2×Q8—24no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—48no—
368C2×Q8—48no—
369C2×D4—22no—
370C2×D4—24no—
371C2×D4—22no—
372C2×D4—24no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—84no—
381C2×D4—84no—
382C2×D4—816no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—216no—
390C22×C4—216no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—82no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4—82no—
420C22×C4#12114yes—
421C22×C4#12114yes—
422C22×C4#12114yes—
423C22×C4#12318yes—
424C22×C4C22×D414yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D414yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D412yes—
432C22×C4C2×#4112yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4114yes—
436C22×C4C2×#4114yes—
437C22×C4C2511yescharacteristic
438M16—48no—
439M16—48no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C4⋊C4—28no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—28no—
450C4⋊C4—28no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—88no—
463C4⋊C4C22×D414yes—
464C4⋊C4C22×D414yes—
465C4⋊C4C2×#4118yes—
466C22⋊C4—28no—
467C22⋊C4—28no—
468C22⋊C4—28no—
469C22⋊C4—28no—
470C22⋊C4—28no—
471C22⋊C4—28no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—416no—
477C22⋊C4—88no—
478C42—28no—
479C42—28no—
480C42—28no—
481C42—24no—
482C42—28no—
483C42—24no—
484C42—84no—
485C42—84no—
486C42—84no—
487C42—84no—
488C42#12318yes—
489C42C2×#4118yes—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—28no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—84no—
509C23—84no—
510C23—84no—
511C23—84no—
512C23—88no—
513C23—88no—
514C23—84no—
515C23C2×#12112yes—
516C23C2×#12112yes—
517C23C2×#12112yes—
518C23C2×#12312yes—
519C23#52512yes—
520C23C23×D411yescharacteristic
521Q8—44no—
522Q8—44no—
523Q8—84no—
524D4—44no—
525D4—44no—
526D4—88no—
527D4—88no—
528D4—84no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—48no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4—88no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4C2×#12112yes—
568C2×C4C2×#12112yes—
569C2×C4C2×#12112yes—
570C2×C4C2×#12318yes—
571C2×C4C2×#12312yes—
572C2×C4#52512yes—
573C2×C4#52918yes—
574C2×C4#53218yes—
575C2×C4C23×D411yescharacteristic
576C2×C4C22×#4111yescharacteristic
577C2×C4C22×#4112yes—
578C2×C4C22×#4112yes—
579C8—44no—
580C8—44no—
581C22—22no—
582C22—22no—
583C22—24no—
584C22—24no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—88no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—84no—
600C22—84no—
601C22—168no—
602C22#285414yes—
603C22C22×#12111yescharacteristic
604C22C22×#12311yescharacteristic
605C22C2×#52511yescharacteristic
606C22#358512yes—
607C22#358612yes—
608C4—22no—
609C4—22no—
610C4—28no—
611C4—44no—
612C4—44no—
613C4—44no—
614C4—48no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4#285414yes—
619C4C2×#52912yes—
620C4C2×#53212yes—
621C2—24no—
622C2—42no—
623C2—42no—
624C2—162no—
625C2C2×#285411yescharacteristic
626C2#6042611yescharacteristic
627C2#6042711yescharacteristic
628C2#6052012yes—
629C2#6056712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3095C2semidirect
4. #33996C2semidirect
5. #33121C2semidirect
6. #33120C2semidirect
7. #33115C2semidirect
8. #21261C2semidirect
9. #21171C2semidirect
10. #21061C2semidirect
11. #21060C2semidirect
13. #20911C2semidirect
14. #20627C2semidirect
23. C2×#484C22semidirect
25. C2×#485C22semidirect
27. C2×#483C22semidirect
48. #2274C22semidirect
49. #2274C22semidirect
50. #2274C22semidirect
56. #1983C22semidirect
57. #1983C22semidirect
58. #1948C22semidirect
59. #1947C22semidirect
60. #1928C22semidirect
61. #1927C22semidirect
62. #1926C22semidirect
63. #1925C22semidirect
66. #1873C22semidirect
67. #1872C22semidirect
68. #1798C22semidirect
69. #1798C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30952
C2×#30942
C2×#26202
#339962
#331212
#331202
#331152
#212612
#211712
#210612
#210602
#209122
#209112
#206272