Finite Group Databasebeta

← #196924

Subgroups — #196924

Order 512.

1852 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
1#35084C211yesmaximal, characteristic
2C2×#2275C211yesmaximal, characteristic
3#21740C211yesmaximal, characteristic
4C2×#1508C211yesmaximal, characteristic, complemented
5C22×#328C211yesmaximal, characteristic, complemented
6#9146C212yesmaximal, complemented
7#9145C212yesmaximal, complemented
8#9144C212yesmaximal, complemented
9#9143C212yesmaximal, complemented
10#8020C211yesmaximal, characteristic, complemented
11#8012C211yesmaximal, characteristic, complemented
12C22×C22⋊Q8C2211yescharacteristic
13C22×C4×D4—21no—
14C22×C4×D4C2211yescharacteristic
15C2×#407C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17#2855C2212yes—
18#2848C2212yes—
19#2638—22no—
20#2583—21no—
21#2561—21no—
22#2486C412yescomplemented
23#2483—21no—
24#2415C414yescomplemented
25#2409C412yescomplemented
26#2405—21no—
27#2322—22no—
28#2280C2211yescharacteristic
29#2275—24no—
30#2275—24no—
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4—21no—
34C2×C4×C22⋊C4—21no—
35C2×C4×C22⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4C2211yescharacteristic
37C2×#376C2211yescharacteristic
38C22×#95C2211yescharacteristic
39C22×#95C2211yescharacteristic
40#1870—22no—
41#1869—22no—
42#1817C2212yes—
43#1779C2212yes—
44#1508—24no—
45C2×#328—24no—
46C2×#328C2212yes—
47C2×#328C2212yes—
48C2×#328C2212yes—
49C2×#328C2212yes—
50#1294C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53#1294C2212yes—
54C23×Q8C2×C411yescharacteristic, complemented
55C23×D4D411yescharacteristic
56C24×C4—21no—
57C24×C4—21no—
58C24×C4C2311yescharacteristic
59C24×C4C2311yescharacteristic
60C22×M16—21no—
61C23×C8—21no—
62C2×#123—22no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8C2×C412yescomplemented
70C2×C22⋊Q8C2×C414yescomplemented
71C2×C22⋊Q8C2312yes—
72C2×C4×D4—216no—
73C2×C4×D4—28no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C2×C4×D4C2×C412yes—
79C2×C4×D4C2×C412yes—
80C2×C4×D4D412yes—
81C2×C4×D4D412yes—
82C2×#117C2×C412yescomplemented
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4C2×C411yescharacteristic, complemented
86C22×C4⋊C4D411yescharacteristic
87C22×C22⋊C4—21no—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4D411yescharacteristic
91C22×C22⋊C4D411yescharacteristic
92C22×C42—21no—
93C22×C42—21no—
94C22×C42—21no—
95C22×C42—21no—
96C22×C42C2311yescharacteristic
97#407—22no—
98#407—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—24no—
104C2×C22⋊C8C2312yes—
105C2×C22⋊C8C2312yes—
106#398—22no—
107#397—22no—
108#397—24no—
109#395—22no—
110#394—24no—
111#393—22no—
112#391C2×C414yescomplemented
113#389—24no—
114#389C2×C414yescomplemented
115#388—24no—
116#387—24no—
117#386—22no—
118#385—24no—
119#385—22no—
120#385—22no—
121#385—22no—
122#381—22no—
123#381—21no—
124#381—22no—
125#380C2×C414yescomplemented
126#380C2×C412yescomplemented
127#380C2312yes—
128#379—22no—
129#379—21no—
130C4×C4⋊C4—28no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C22⋊C4—28no—
134C4×C22⋊C4—28no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—22no—
138C4×C22⋊C4—22no—
139C4×C22⋊C4—22no—
140C4×C22⋊C4D414yes—
141#376—22no—
142#376—22no—
143C2×#95—21no—
144C2×#95—21no—
145C2×#95—22no—
146C2×#95—21no—
147C2×#95—22no—
148C2×#95—22no—
149C2×#95D414yes—
150C2×#95C2311yescharacteristic
151C2×#95C2311yescharacteristic
152C2×#95C2311yescharacteristic
153C2×#95C2311yescharacteristic
154#336—24no—
155#328—24no—
156#328—24no—
157#328—24no—
158#328—24no—
159C25C2×D411yescharacteristic
160C25C2×D411yescharacteristic
161C22×Q8—21no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—82no—
165C22×Q8C22×C412yes—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—28no—
169C22×D4C22⋊C412yes—
170C22×D4C22⋊C412yes—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—28no—
176C23×C4—28no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—82no—
186C23×C4—82no—
187C23×C4—82no—
188C23×C4—82no—
189C23×C4—82no—
190C23×C4—82no—
191C23×C4C22×C412yes—
192C23×C4C22×C411yescharacteristic
193C23×C4C22×C411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D412yes—
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D412yes—
202C23×C4C2×D412yes—
203C23×C4#4112yes—
204C23×C4#4112yes—
205C23×C4C2411yescharacteristic
206C2×M16—42no—
207C2×M16—82no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—42no—
211C22×C8—82no—
212#123—48no—
213C22⋊Q8—24no—
214C22⋊Q8—24no—
215C22⋊Q8—22no—
216C22⋊Q8—22no—
217C22⋊Q8—22no—
218C22⋊Q8—22no—
219C22⋊Q8—42no—
220C22⋊Q8—42no—
221C22⋊Q8—42no—
222C22⋊Q8—42no—
223C22⋊Q8—84no—
224C4×D4—264no—
225C4×D4—24no—
226C4×D4—24no—
227C4×D4—24no—
228C4×D4—24no—
229C4×D4—24no—
230C4×D4—24no—
231C4×D4—24no—
232C4×D4—24no—
233#117—82no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4C22⋊C412yes—
248C2×C4⋊C4C22⋊C412yes—
249C2×C4⋊C4C22×C412yes—
250C2×C4⋊C4C22×C414yes—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4C22⋊C412yes—
269C2×C22⋊C4C22⋊C412yes—
270C2×C22⋊C4C22⋊C412yes—
271C2×C22⋊C4C22⋊C412yes—
272C2×C42—24no—
273C2×C42—28no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—82no—
283C2×C42—82no—
284C2×C42—82no—
285C2×C42C22×C412yes—
286C2×C42C2×D412yes—
287C22⋊C8—22no—
288C22⋊C8—22no—
289C22⋊C8—22no—
290C22⋊C8—22no—
291C22⋊C8—42no—
292C22⋊C8—42no—
293C22⋊C8—84no—
294#95—24no—
295#95—24no—
296#95—22no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—22no—
307#95C2×D412yes—
308#95C2×D412yes—
309#95C2×D412yes—
310#95C2×D412yes—
311C24—21no—
312C24—21no—
313C24—22no—
314C24—22no—
315C24—24no—
316C24—24no—
317C24—22no—
318C24—22no—
319C24—42no—
320C24—42no—
321C24C2×C22⋊C411yescharacteristic
322C24C2×C22⋊C412yes—
323C24C2×C22⋊C411yescharacteristic
324C24#12012yes—
325C24#12312yes—
326C24C22×D411yescharacteristic
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—44no—
330C2×Q8—42no—
331C2×Q8—82no—
332C2×Q8—84no—
333C2×Q8—84no—
334C2×D4—216no—
335C2×D4—216no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4#10418yes—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—216no—
360C22×C4—216no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—82no—
389C22×C4—82no—
390C22×C4—82no—
391C22×C4—82no—
392C22×C4—84no—
393C22×C4—82no—
394C22×C4—82no—
395C22×C4—82no—
396C22×C4—82no—
397C22×C4—82no—
398C22×C4—84no—
399C22×C4—84no—
400C22×C4—816no—
401C22×C4—816no—
402C22×C4—84no—
403C22×C4C2×C22⋊C412yes—
404C22×C4C2×C22⋊C412yes—
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C412yes—
408C22×C4C4×D412yes—
409C22×C4C4×D412yes—
410C22×C4C4×D412yes—
411C22×C4C4×D412yes—
412C22×C4#12012yes—
413C22×C4#12112yes—
414C22×C4#12112yes—
415C22×C4#12112yes—
416C22×C4#12112yes—
417C22×C4#12312yes—
418C22×C4C23×C411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4111yescharacteristic
424M16—84no—
425C2×C8—44no—
426C2×C8—44no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—84no—
430C4⋊C4—216no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—44no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—44no—
440C4⋊C4—48no—
441C4⋊C4—84no—
442C22⋊C4—216no—
443C22⋊C4—216no—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—48no—
455C22⋊C4—82no—
456C22⋊C4—84no—
457C42—216no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—82no—
463C42—84no—
464C42—84no—
465C42—84no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—22no—
475C23—24no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—24no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—42no—
485C23—42no—
486C23—48no—
487C23—48no—
488C23—48no—
489C23—48no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—41no—
494C23—41no—
495C23#38612yes—
496C23#38612yes—
497C23C2×#10414yes—
498C23C2×#10414yes—
499C23C22×C22⋊C411yescharacteristic
500C23C2×#12011yescharacteristic
501C23C2×#12311yescharacteristic
502C23#57712yes—
503C23#57812yes—
504Q8—82no—
505Q8—82no—
506Q8—84no—
507D4—216no—
508D4—216no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—48no—
544C2×C4—48no—
545C2×C4—48no—
546C2×C4—42no—
547C2×C4—48no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—816no—
568C2×C4—816no—
569C2×C4—84no—
570C2×C4—84no—
571C2×C4#38612yes—
572C2×C4#38612yes—
573C2×C4C2×#10414yes—
574C2×C4C2×C4×D411yescharacteristic
575C2×C4C2×C4×D411yescharacteristic
576C2×C4C2×#12111yescharacteristic
577C2×C4C2×#12111yescharacteristic
578C8—82no—
579C8—82no—
580C22—24no—
581C22—24no—
582C22—28no—
583C22—22no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—22no—
589C22—28no—
590C22—22no—
591C22—22no—
592C22—24no—
593C22—22no—
594C22—44no—
595C22—44no—
596C22—48no—
597C22—44no—
598C22—44no—
599C22—48no—
600C22—44no—
601C22—44no—
602C22—42no—
603C22—42no—
604C22#189614yes—
605C22C2×#38611yescharacteristic
606C22#288312yes—
607C22#289812yes—
608C22C22×#10412yes—
609C22#300911yescharacteristic
610C22#301111yescharacteristic
611C22#309712yes—
612C4—24no—
613C4—21no—
614C4—21no—
615C4—21no—
616C4—21no—
617C4—22no—
618C4—22no—
619C4—44no—
620C4—44no—
621C4—44no—
622C4—44no—
623C4—42no—
624C4—42no—
625C4—84no—
626C4—82no—
627C4—82no—
628C2—21no—
629C2—21no—
630C2—22no—
631C2—24no—
632C2—44no—
633C2—42no—
634C2#1754814yes—
635C2#2091311yescharacteristic
636C2C2×#189611yescharacteristic
637C2#2098411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1508C2semidirect
5. C22×#328C2semidirect
6. #9146C2semidirect
7. #9145C2semidirect
8. #9144C2semidirect
9. #9143C2semidirect
10. #8020C2semidirect
11. #8012C2semidirect
22. #2486C4semidirect
24. #2415C4semidirect
25. #2409C4semidirect
54. C23×Q8C2×C4semidirect
69. C2×C22⋊Q8C2×C4semidirect
70. C2×C22⋊Q8C2×C4semidirect
82. C2×#117C2×C4semidirect
85. C22×C4⋊C4C2×C4semidirect
112. #391C2×C4semidirect
114. #389C2×C4semidirect
125. #380C2×C4semidirect
126. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
#350842
C2×#22752
#217402
C2×#15082
C22×#3282
#91462
#91452
#91442
#91432
#80202
#80122