Finite Group Databasebeta

← #6432553

Subgroups — #6432553

Order 512.

1860 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34074C211yesmaximal, characteristic, complemented
2C2×#2319C211yesmaximal, characteristic
3#23791C214yesmaximal, complemented
4#23790C214yesmaximal, complemented
5#23787C214yesmaximal, complemented
6#23786C214yesmaximal, complemented
7#23747C211yesmaximal, characteristic, complemented
8#23731C211yesmaximal, characteristic
9#23670C212yesmaximal, complemented
10#23618C212yesmaximal
11#23554C211yesmaximal, characteristic, complemented
12#23454C211yesmaximal, characteristic
13C4×#428C212yesmaximal, complemented
14#20429C211yesmaximal, characteristic, complemented
15C4×#418C212yesmaximal
16C2×#533C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C2×#428C2212yes—
19C2×#428C2212yes—
20C2×#423C2212yes—
21C2×#418C2212yes—
22C2×#418C2212yes—
23C2×C4×M16C2212yes—
24#2331C2212yescomplemented
25#2331C2212yescomplemented
26#2330C2212yescomplemented
27#2330C2212yescomplemented
28#2329—21no—
29#2329—21no—
30#2329—21no—
31#2329—21no—
32#2319—22no—
33#2319—22no—
34#2319C2214yes—
35#2319C2214yes—
36#2296C2212yes—
37#2295—22no—
38#2295—22no—
39C2×#390C2211yescharacteristic
40C2×#390C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#388—21no—
44C2×#388—21no—
45#2285C2211yescharacteristic
46C2×#381—21no—
47C2×#380—21no—
48C4×#117C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×#376C2211yescharacteristic
51#2101C2214yescomplemented
52#2100C2212yescomplemented
53#2099C2212yescomplemented
54#2094C2212yes—
55#2093C2212yes—
56#2092C2214yes—
57#2088C2214yescomplemented
58#2088C2214yescomplemented
59#2087C2214yescomplemented
60#2086C2214yescomplemented
61#2067C2214yes—
62#2067C2214yes—
63#2065C2214yes—
64#2065C2214yes—
65#2059C2214yescomplemented
66#2058C2212yescomplemented
67#2057C2212yescomplemented
68#2025C2214yes—
69#2023C2212yes—
70#2023C2212yes—
71#1776C2214yescomplemented
72#1776C2214yescomplemented
73C4×#107C2214yescomplemented
74C4×#106C2214yescomplemented
75#1768C2214yescomplemented
76#1768C2214yescomplemented
77#1762C2214yes—
78#1762C2214yes—
79C4×#102C2214yes—
80C4×#102C2214yes—
81#1736C2212yes—
82#1736C2212yes—
83#1736C2212yes—
84#1736C2212yes—
85#1728C2211yescharacteristic
86#1728C2211yescharacteristic
87C23×D4C2311yescharacteristic
88C24×C4—21no—
89C22×M16C2312yes—
90C22×M16C2312yes—
91#533—28no—
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2311yescharacteristic
94C2×#125C2312yes—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—21no—
97C2×C4×Q8—21no—
98C2×C4×Q8—21no—
99C2×C4×D4—22no—
100C2×C4×D4—24no—
101C2×C4×D4Q818yescomplemented
102C2×C4×D4C2314yes—
103C2×#117C2312yes—
104C2×#117C2312yes—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C42—21no—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119#428—28no—
120#428—28no—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#106C2314yes—
124C2×#106C2314yes—
125#423—28no—
126C2×C4⋊C8C2314yes—
127C2×C4⋊C8C2314yes—
128#418—28no—
129#418—28no—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C4×M16—28no—
135C2×#97C2314yes—
136C2×C4×C8C2314yes—
137#391—22no—
138#391—22no—
139#391—22no—
140#391—22no—
141#390—21no—
142#390—21no—
143#390—22no—
144#390C2314yes—
145#390C2312yes—
146#390C2312yes—
147#389—21no—
148#389—21no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389C2312yes—
155#389C2312yes—
156#389C2314yes—
157#389C2312yes—
158#389C2312yes—
159#389C2312yes—
160#389C2312yes—
161#388—24no—
162#388—24no—
163#388—24no—
164#388—24no—
165#387—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#385—22no—
174#385—22no—
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#382C2312yes—
178#381—24no—
179#381—24no—
180#380—21no—
181#380—21no—
182#380—24no—
183#380—24no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4C2314yes—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2312yes—
194C4×C22⋊C4—22no—
195C4×C22⋊C4—22no—
196#376—22no—
197#376D412yes—
198#376D412yes—
199#376C2312yes—
200#376C2312yes—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95D411yescharacteristic
205C2×#95D411yescharacteristic
206C2×#95D411yescharacteristic
207C2×#95D411yescharacteristic
208C43—22no—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2312yes—
218#336C2312yes—
219#336C2312yes—
220#336C2312yes—
221C25—21no—
222C22×Q8—21no—
223C22×Q8—21no—
224C22×D4—21no—
225C22×D4—24no—
226C22×D4—24no—
227C22×D4—22no—
228C22×D4C2×Q814yes—
229C22×D4C2412yes—
230C23×C4—21no—
231C23×C4—28no—
232C23×C4—24no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—82no—
236C23×C4—82no—
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2411yescharacteristic
244C2×M16—24no—
245C2×M16—24no—
246C2×M16—24no—
247C2×M16—24no—
248C2×M16—24no—
249C2×M16#4118yes—
250C22×C8C2414yes—
251C22×C8C2414yes—
252C4⋊Q8—24no—
253C4⋊Q8—24no—
254#125—24no—
255#125—24no—
256C22⋊Q8—28no—
257C22⋊Q8—28no—
258C4×Q8—28no—
259C4×Q8—28no—
260C4×D4—216no—
261C4×D4—28no—
262C4×D4C2×Q8116yes—
263#117—24no—
264#117—24no—
265#117—24no—
266#117#4118yes—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—82no—
289C2×C4⋊C4—82no—
290C2×C4⋊C4C2×Q814yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—82no—
309C2×C22⋊C4—82no—
310C2×C22⋊C4—82no—
311C2×C22⋊C4—82no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—82no—
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D412yes—
325C2×C42C2×D412yes—
326C2×C42C2×Q814yes—
327C2×C42#4114yes—
328C2×C42C2411yescharacteristic
329C2×C42C2411yescharacteristic
330C2×C42C2412yes—
331C2×C42C2412yes—
332#107—28no—
333#107—28no—
334#106—28no—
335#106—28no—
336C4⋊C8—28no—
337C4⋊C8—28no—
338#102—28no—
339#102—28no—
340#102—28no—
341#102—28no—
342#97—28no—
343C4×C8—28no—
344#95—24no—
345#95—24no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95C2×D412yes—
359#95C2×D412yes—
360#95C2×D412yes—
361#95C2×D412yes—
362#95C2×D412yes—
363#95C2×D412yes—
364#95C2×D412yes—
365#95C2×D412yes—
366C24—24no—
367C24—22no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24C22×D411yescharacteristic
375C2×Q8—24no—
376C2×Q8—24no—
377C2×Q8—81no—
378C2×Q8—81no—
379C2×Q8—81no—
380C2×Q8—81no—
381C2×D4—24no—
382C2×D4—216no—
383C2×D4—24no—
384C2×D4—24no—
385C2×D4—22no—
386C2×D4—42no—
387C2×D4—48no—
388C2×D4—42no—
389C2×D4C22×Q814yes—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—216no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—84no—
424C22×C4—84no—
425C22×C4—816no—
426C22×C4C22⋊Q814yes—
427C22×C4C22⋊Q814yes—
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×Q812yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4114yes—
441C22×C4C2511yescharacteristic
442M16—216no—
443M16—48no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8C2×#4118yes—
450C4⋊C4—28no—
451C4⋊C4—28no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—82no—
467C4⋊C4—82no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4—82no—
471C4⋊C4—81no—
472C4⋊C4—82no—
473C4⋊C4—81no—
474C4⋊C4—84no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4C22×Q814yes—
478C4⋊C4C2×#4114yes—
479C4⋊C4C2×#4114yes—
480C22⋊C4—28no—
481C22⋊C4—216no—
482C22⋊C4—44no—
483C22⋊C4—84no—
484C22⋊C4—82no—
485C22⋊C4—84no—
486C22⋊C4—82no—
487C22⋊C4—82no—
488C22⋊C4—82no—
489C42—22no—
490C42—22no—
491C42—28no—
492C42—24no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—44no—
498C42—82no—
499C42—82no—
500C42—82no—
501C42—82no—
502C42—84no—
503C42C22⋊Q818yes—
504C42C22×Q814yes—
505C42C2×#4114yes—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—24no—
511C23—48no—
512C23—48no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—48no—
518C23—42no—
519C23—42no—
520C23—84no—
521C23—84no—
522C23—84no—
523C23—84no—
524C23C2×C22⋊Q812yes—
525C23#52512yes—
526C23#53212yes—
527C23C23×D411yescharacteristic
528Q8—84no—
529Q8—84no—
530D4—28no—
531D4—48no—
532D4—44no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—24no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—48no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—816no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4C2×C22⋊Q814yes—
587C2×C4C2×C22⋊Q814yes—
588C2×C4C2×C22⋊Q812yes—
589C2×C4#52512yes—
590C2×C4#52918yes—
591C2×C4#53212yes—
592C2×C4#53318yes—
593C2×C4C23×Q811yescharacteristic
594C2×C4C22×#4111yescharacteristic
595C2×C4C22×#4112yes—
596C2×C4C22×#4112yes—
597C8—28no—
598C8—44no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—816no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22#239614yes—
614C22C22×C22⋊Q811yescharacteristic
615C22C2×#52511yescharacteristic
616C22C2×#53211yescharacteristic
617C22#358511yescharacteristic
618C22#358511yescharacteristic
619C22#358511yescharacteristic
620C22#358611yescharacteristic
621C4—24no—
622C4—24no—
623C4—42no—
624C4—42no—
625C4—44no—
626C4—41no—
627C4—41no—
628C4—44no—
629C4—41no—
630C4—41no—
631C4—44no—
632C4—82no—
633C4—82no—
634C4—82no—
635C4—82no—
636C4#239614yes—
637C4C2×#52912yes—
638C4C2×#53312yes—
639C2—24no—
640C2—84no—
641C2C2×#239611yescharacteristic
642C2#6049011yescharacteristic
643C2#6049211yescharacteristic
644C2#6056711yescharacteristic
645C2#6056811yescharacteristic
646C2#6070311yescharacteristic
647C2#6070411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #34074C2semidirect
3. #23791C2semidirect
4. #23790C2semidirect
5. #23787C2semidirect
6. #23786C2semidirect
7. #23747C2semidirect
9. #23670C2semidirect
11. #23554C2semidirect
13. C4×#428C2semidirect
14. #20429C2semidirect
24. #2331C22semidirect
25. #2331C22semidirect
26. #2330C22semidirect
27. #2330C22semidirect
51. #2101C22semidirect
52. #2100C22semidirect
53. #2099C22semidirect
57. #2088C22semidirect
58. #2088C22semidirect
59. #2087C22semidirect
60. #2086C22semidirect
65. #2059C22semidirect
66. #2058C22semidirect
67. #2057C22semidirect
71. #1776C22semidirect
72. #1776C22semidirect
73. C4×#107C22semidirect
74. C4×#106C22semidirect
75. #1768C22semidirect
76. #1768C22semidirect
101. C2×C4×D4Q8semidirect

Maximal subgroups

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

subgroupindex
#340742
C2×#23192
#237912
#237902
#237872
#237862
#237472
#237312
#236702
#236182
#235542
#234542
C4×#4282
#204292
C4×#4182