Finite Group Databasebeta

← #6433211

Subgroups — #6433211

Order 512.

2209 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3092C211yesmaximal, characteristic
2C22×#484C211yesmaximal, characteristic, complemented
3#35306C211yesmaximal, characteristic
4C2×#2316C211yesmaximal, characteristic, complemented
5#29767C214yesmaximal, complemented
6#29764C214yesmaximal, complemented
7#29761C214yesmaximal, complemented
8#29757C214yesmaximal, complemented
9#23660C212yesmaximal, complemented
10#23615C212yesmaximal, complemented
11#23611C212yesmaximal, complemented
12#23512C212yesmaximal, complemented
13#20911C211yesmaximal, characteristic, complemented
14#20908C211yesmaximal, characteristic
15#20628C211yesmaximal, characteristic, complemented
16C2×#536C2211yescharacteristic
17C2×#525C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#3092—28no—
23C2×#484—28no—
24C2×#484C2212yes—
25C2×#484C2214yes—
26C2×#484C2214yes—
27C2×#481C2212yes—
28C2×#483C2212yes—
29C2×#480C2212yes—
30C2×#428C2212yes—
31C22×#107C2212yes—
32C2×#424C2212yes—
33C2×#417C2212yes—
34C2×#419C2212yes—
35C2×#416C2212yes—
36C22×#103C2212yes—
37C2×#407C2211yescharacteristic
38C22×C22⋊C8C2211yescharacteristic
39#2857C2212yes—
40#2850C2212yes—
41#2621C2212yes—
42#2620C2212yes—
43#2417—22no—
44#2415—22no—
45#2413—22no—
46#2412—22no—
47#2317C2211yescharacteristic
48#2316—22no—
49#2316—22no—
50#2316C2212yes—
51#2316C2214yes—
52#2316C2212yes—
53#2311C2211yescharacteristic
54#2310C2211yescharacteristic
55C2×#389—22no—
56C2×#389C2214yes—
57C2×#380—22no—
58#2085C2218yescomplemented
59#2083C2218yescomplemented
60#2068C2218yescomplemented
61#2068C2218yescomplemented
62#2067C2218yescomplemented
63#2066C2218yescomplemented
64#2043C2218yescomplemented
65#2042C2218yescomplemented
66#1875C2212yes—
67#1874C2212yes—
68#1873C2212yes—
69#1873C2214yes—
70#1872C2212yes—
71#1872C2214yes—
72#1798C2214yes—
73#1797C2214yes—
74#1736C2212yes—
75#1735C2212yes—
76#1729C2212yes—
77C22×#41C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C24×C4—21no—
80C24×C4C2311yescharacteristic
81C22×M16C2312yes—
82C23×C8C2312yes—
83#536—44no—
84C2×C4⋊Q8C2312yes—
85C2×#124C2312yes—
86#525—24no—
87#525—24no—
88C2×#123—21no—
89C2×#123—22no—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8C2311yescharacteristic
96C2×C22⋊Q8C2312yes—
97C2×C22⋊Q8C2311yescharacteristic
98C2×C22⋊Q8C2312yes—
99C2×#121C2311yescharacteristic
100C2×#121C2311yescharacteristic
101C2×#120—21no—
102C2×C4×D4—21no—
103C2×C4×D4—21no—
104C2×#117—21no—
105C2×#117—21no—
106C2×#117—44no—
107C2×#117—44no—
108C2×#117D412yes—
109C2×#117D412yes—
110C2×#117C2311yescharacteristic
111C2×#117C2312yes—
112C2×#117C2312yes—
113C2×#117C2311yescharacteristic
114C2×#117C2311yescharacteristic
115C2×#117C2312yes—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—24no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—24no—
122C22×C4⋊C4D418yes—
123C22×C4⋊C4D412yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4D412yes—
132C22×C42—21no—
133C22×C42C2312yes—
134#484—216no—
135#484—216no—
136#484—28no—
137#483—28no—
138#481—28no—
139#480—28no—
140#428—28no—
141C2×#107—28no—
142C2×#107C2314yes—
143C2×#107C2318yes—
144C2×#106C2314yes—
145#424—44no—
146C2×C4⋊C8C2318yes—
147#419—44no—
148#417—28no—
149#416—44no—
150C2×#103—28no—
151C2×#103C2314yes—
152C2×#103C2318yes—
153C2×#103C2318yes—
154C2×#103C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157#407—24no—
158C2×C22⋊C8—24no—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2312yes—
162C2×C22⋊C8C2312yes—
163#398—24no—
164#397—24no—
165#396—24no—
166#394—24no—
167#391—22no—
168#391C2312yes—
169#391C2312yes—
170#391C2312yes—
171#390C2312yes—
172#389—24no—
173#389—24no—
174#389—28no—
175#389—22no—
176#389—24no—
177#389—24no—
178#389—24no—
179#389C2318yes—
180#389C2318yes—
181#388—24no—
182#386—24no—
183#386—22no—
184#385—22no—
185#384C2312yes—
186#382C2312yes—
187#381—22no—
188#380—24no—
189#380—22no—
190#380—24no—
191#380—24no—
192#380—28no—
193C2×#95—24no—
194#340C2314yes—
195#340C2318yes—
196#340C2314yes—
197#336C2318yes—
198C25C2×D411yescharacteristic
199C2×#41—22no—
200C2×#41—22no—
201C2×#41—24no—
202C2×#41—24no—
203C22×Q8—21no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—24no—
207C22×Q8—24no—
208C22×Q8C2412yes—
209C22×Q8C2411yescharacteristic
210C22×D4—21no—
211C22×D4—21no—
212C22×D4—21no—
213C22×D4C2411yescharacteristic
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—22no—
217C23×C4—28no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—84no—
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4#4114yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C2×M16—24no—
235C2×M16—24no—
236C2×M16—24no—
237C22×C8—24no—
238C22×C8—24no—
239C22×C8—24no—
240C22×C8C2414yes—
241C22×C8C2414yes—
242C4⋊Q8—44no—
243#124—44no—
244#123—48no—
245#123—44no—
246C22⋊Q8—24no—
247C22⋊Q8—24no—
248C22⋊Q8—22no—
249C22⋊Q8—22no—
250C22⋊Q8—24no—
251C22⋊Q8—24no—
252C22⋊Q8—22no—
253C22⋊Q8—22no—
254C22⋊Q8—48no—
255C22⋊Q8—44no—
256C22⋊Q8—44no—
257#121—22no—
258#121—22no—
259#121—22no—
260#121—22no—
261#120—44no—
262C4×D4—44no—
263C4×D4—44no—
264#117—24no—
265#117—24no—
266#117—22no—
267#117—22no—
268#117—24no—
269#117—24no—
270#117—44no—
271#117—44no—
272#117—44no—
273#117—44no—
274#117—44no—
275#117—44no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—28no—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—88no—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D418yes—
304C2×C4⋊C4C2×D418yes—
305C2×C4⋊C4C2×D418yes—
306C2×C4⋊C4C2×D414yes—
307C2×C4⋊C4C2414yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2414yes—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4C2×D412yes—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D414yes—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—44no—
333C2×C42—44no—
334C2×C42C2×D414yes—
335C2×C42C2412yes—
336C2×C42C2412yes—
337#107—216no—
338#107—28no—
339#106—28no—
340C4⋊C8—48no—
341#103—216no—
342#103—28no—
343#103—44no—
344#103—48no—
345#102—28no—
346#102—44no—
347C22⋊C8—24no—
348C22⋊C8—24no—
349C22⋊C8—24no—
350C22⋊C8—24no—
351#95—28no—
352#95—28no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357C24—22no—
358C24—22no—
359C24—24no—
360C24—21no—
361C24—42no—
362C24#12114yes—
363C24C22×D411yescharacteristic
364C24C22×D411yescharacteristic
365C24C22×D411yescharacteristic
366#41—48no—
367#41—48no—
368C2×Q8—22no—
369C2×Q8—21no—
370C2×Q8—22no—
371C2×Q8—21no—
372C2×Q8—24no—
373C2×Q8—22no—
374C2×Q8—24no—
375C2×Q8—22no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—44no—
391C2×D4—82no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—216no—
399C22×C4—216no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—28no—
405C22×C4—24no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—48no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—82no—
429C22×C4—84no—
430C22×C4—84no—
431C22×C4—816no—
432C22×C4#12114yes—
433C22×C4#12114yes—
434C22×C4#12114yes—
435C22×C4C22×D414yes—
436C22×C4C22×D414yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4H(5,𝔽2)14yes—
447C22×C42−1+414yes—
448C22×C4C2511yescharacteristic
449M16—48no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—48no—
457C4⋊C4—28no—
458C4⋊C4—28no—
459C4⋊C4—28no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—84no—
480C4⋊C4—88no—
481C4⋊C4—88no—
482C4⋊C4C22×D418yes—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—44no—
486C22⋊C4—48no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—44no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—84no—
497C42—24no—
498C42—24no—
499C42—44no—
500C42—44no—
501C42—42no—
502C42—44no—
503C23—22no—
504C23—22no—
505C23—21no—
506C23—21no—
507C23—22no—
508C23—22no—
509C23—28no—
510C23—21no—
511C23—21no—
512C23—28no—
513C23—22no—
514C23—44no—
515C23—48no—
516C23—48no—
517C23—42no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—42no—
522C23C2×#12112yes—
523C23C2×#12112yes—
524C23C2×#12112yes—
525C23#53412yes—
526C23#53612yes—
527C23#57814yes—
528C23C23×D411yescharacteristic
529Q8—44no—
530Q8—42no—
531Q8—44no—
532Q8—42no—
533Q8—84no—
534D4—42no—
535D4—42no—
536D4—84no—
537D4—84no—
538D4—84no—
539C2×C4—28no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—416no—
558C2×C4—48no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—88no—
575C2×C4—88no—
576C2×C4—88no—
577C2×C4—88no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4C2×#12112yes—
582C2×C4C2×#12112yes—
583C2×C4C2×#12112yes—
584C2×C4#53412yes—
585C2×C4#53612yes—
586C2×C4#53718yes—
587C2×C4#53918yes—
588C2×C4C23×D411yescharacteristic
589C2×C4C22×#4111yescharacteristic
590C2×C4C2×H(5,𝔽2)12yes—
591C2×C4C2×2−1+412yes—
592C8—44no—
593C8—44no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—416no—
606C22—42no—
607C22—42no—
608C22—81no—
609C22—81no—
610C22—82no—
611C22—82no—
612C22—81no—
613C22#261814yes—
614C22#304414yes—
615C22C22×#12111yescharacteristic
616C22C2×#53411yescharacteristic
617C22C2×#53611yescharacteristic
618C22C2×#57812yes—
619C22#358512yes—
620C4—22no—
621C4—22no—
622C4—48no—
623C4—44no—
624C4—44no—
625C4—81no—
626C4—82no—
627C4—81no—
628C4—84no—
629C4—81no—
630C4—82no—
631C4#261814yes—
632C4C2×#53712yes—
633C4C2×#53912yes—
634C2—24no—
635C2—24no—
636C2—42no—
637C2—161no—
638C2C2×#261811yescharacteristic
639C2C2×#304411yescharacteristic
640C2#6042311yescharacteristic
641C2#6076412yes—
642C2#6087412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
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
2. C22×#484C2semidirect
4. C2×#2316C2semidirect
5. #29767C2semidirect
6. #29764C2semidirect
7. #29761C2semidirect
8. #29757C2semidirect
9. #23660C2semidirect
10. #23615C2semidirect
11. #23611C2semidirect
12. #23512C2semidirect
13. #20911C2semidirect
15. #20628C2semidirect
58. #2085C22semidirect
59. #2083C22semidirect
60. #2068C22semidirect
61. #2068C22semidirect
62. #2067C22semidirect
63. #2066C22semidirect
64. #2043C22semidirect
65. #2042C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30922
C22×#4842
#353062
C2×#23162
#297672
#297642
#297612
#297572
#236602
#236152
#236112
#235122
#209112
#209082
#206282