Finite Group Databasebeta

← #188464

Subgroups — #188464

Order 512.

1132 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
1#34243C211yesmaximal, characteristic
2#21953C211yesmaximal, characteristic, complemented
3C2×#1509C211yesmaximal, characteristic
4C2×#1500C211yesmaximal, characteristic
5#13783C211yesmaximal, characteristic, complemented
6#8578C212yesmaximal, complemented
7#8576C212yesmaximal, complemented
8#8573C212yesmaximal
9#8571C212yesmaximal
10#8022C211yesmaximal, characteristic, complemented
11#8014C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2370C412yescomplemented
16#2367C412yescomplemented
17#2359—22no—
18#2355C411yescharacteristic, complemented
19#2354C411yescharacteristic, complemented
20#2352—21no—
21Q8×C4⋊C4C411yescharacteristic, complemented
22D4×C4⋊C4—21no—
23D4×C4⋊C4C411yescharacteristic, complemented
24#2334—22no—
25#2314—21no—
26C4×C22⋊Q8C2212yes—
27C4×#121C2212yes—
28C2×#389—21no—
29#2285C2211yescharacteristic
30#2274—21no—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×#376C2211yescharacteristic
34#1940—22no—
35#1939—21no—
36#1939—21no—
37#1797C2212yes—
38#1797C2212yes—
39#1509—24no—
40#1500—24no—
41C2×#328C2211yescharacteristic
42C2×#328C2211yescharacteristic
43C2×#328C2212yes—
44C2×#327C2211yescharacteristic
45C2×#327C2211yescharacteristic
46C2×#327C2212yes—
47#1436—22no—
48C4⋊#95—21no—
49C4⋊#95C2212yes—
50#1434C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53#1294C2212yes—
54#1294C2212yes—
55#529—22no—
56#529—22no—
57#529D414yes—
58#529Q814yes—
59C2×#126—21no—
60C2×#125C2×C411yescharacteristic
61C2×#124C2×C411yescharacteristic
62C2×#123—21no—
63C2×C22⋊Q8C2311yescharacteristic
64C2×#121C2311yescharacteristic
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8C2×C412yes—
67C2×C4×D4—21no—
68C2×C4×D4—22no—
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C412yes—
71C2×#117—21no—
72C2×#117—21no—
73C2×#117—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C42—21no—
80C22×C42—21no—
81C22×C42C2312yes—
82C22×C42C2311yescharacteristic
83#432—22no—
84#432—22no—
85#431—22no—
86#431—22no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C22⋊C8C2311yescharacteristic
92C2×C22⋊C8C2311yescharacteristic
93C2×C22⋊C8C2312yes—
94C2×#97—21no—
95C2×C4×C8—21no—
96#391C2×C412yes—
97#390C2×C412yes—
98#389—28no—
99#389—22no—
100#389—22no—
101#389C2×C412yes—
102#389C2×C412yes—
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#387—22no—
108#387—22no—
109#387—22no—
110#387—22no—
111#386—22no—
112#385—22no—
113#385—22no—
114#385—22no—
115#384—21no—
116#384C2×C411yescharacteristic
117#384C2×C411yescharacteristic
118#382—21no—
119#382—21no—
120#380—21no—
121#380—21no—
122#380—21no—
123#380C2312yes—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4C2×C412yes—
132C4×C4⋊C4C2×C412yes—
133C4×C4⋊C4C2×C412yes—
134C4×C4⋊C4C2×C411yescharacteristic
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C4⋊C4C2×C412yes—
137C4×C4⋊C4D411yescharacteristic
138C4×C4⋊C4D412yes—
139C4×C4⋊C4D411yescharacteristic
140C4×C22⋊C4—22no—
141C4×C22⋊C4—22no—
142C4×C22⋊C4—22no—
143#376—22no—
144#376—22no—
145C2×#95—22no—
146C2×#95—21no—
147C2×#95—22no—
148C2×#95—22no—
149C2×#95D412yes—
150C2×#95C2311yescharacteristic
151C2×#95C2312yes—
152C2×#95C2311yescharacteristic
153#336—22no—
154#336—22no—
155#328—22no—
156#328—22no—
157#328—24no—
158#327—22no—
159#327—22no—
160#327—24no—
161C22×Q8C22×C411yescharacteristic
162C22×D4—21no—
163C22×D4C22×C411yescharacteristic
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4C2411yescharacteristic
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—22no—
177#126—24no—
178#126—24no—
179#125—22no—
180#125C4⋊C414yescomplemented
181#124—22no—
182#124C4⋊C414yescomplemented
183#123—24no—
184#123—24no—
185C22⋊Q8—21no—
186C22⋊Q8—21no—
187C22⋊Q8C2×D412yes—
188C22⋊Q8C2×Q812yes—
189#121—21no—
190#121—21no—
191#121C2×D412yes—
192#121C2×Q812yes—
193C4×Q8—22no—
194C4×Q8—82no—
195C4×Q8C4⋊C414yescomplemented
196C4×D4—24no—
197C4×D4—24no—
198C4×D4—22no—
199C4×D4—82no—
200C4×D4—82no—
201C4×D4—82no—
202C4×D4C4⋊C414yescomplemented
203#117—24no—
204#117—24no—
205#117—44no—
206#117—81no—
207#117—81no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4—82no—
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22⋊C411yescharacteristic
230C2×C4⋊C4C22⋊C411yescharacteristic
231C2×C4⋊C4C22⋊C412yes—
232C2×C4⋊C4C22⋊C412yes—
233C2×C4⋊C4C22⋊C411yescharacteristic
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C22×C412yes—
237C2×C4⋊C4C22×C412yes—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C42—21no—
248C2×C42—24no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—82no—
257C2×C42—81no—
258C2×C42—81no—
259C2×C42C22×C412yes—
260C2×C42C22×C412yes—
261C2×C42C22×C411yescharacteristic
262C2×C42C22×C411yescharacteristic
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×Q812yes—
269C2×C42#4112yes—
270C2×C42#4112yes—
271C4⋊C8—42no—
272C4⋊C8—42no—
273C4⋊C8—82no—
274C4⋊C8—82no—
275C22⋊C8—21no—
276C22⋊C8—21no—
277C22⋊C8—22no—
278C22⋊C8—21no—
279C22⋊C8—21no—
280C22⋊C8—22no—
281#97—42no—
282C4×C8—42no—
283#95—28no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—21no—
293#95—21no—
294#95—22no—
295#95—21no—
296#95—21no—
297#95—22no—
298#95—42no—
299#95—82no—
300#95—82no—
301#95C22⋊C412yes—
302#95C22⋊C412yes—
303#95C22⋊C412yes—
304#95C22⋊C412yes—
305C24—21no—
306C24C22×D411yescharacteristic
307C2×Q8—21no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8C2×C4⋊C412yes—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—21no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4C2×C4⋊C412yes—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—44no—
367C22×C4—82no—
368C22×C4—82no—
369C22×C4C2×C22⋊C412yes—
370C22×C4C2×C22⋊C412yes—
371C22×C4C2×C22⋊C412yes—
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C412yes—
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4#12112yes—
378C22×C4#12112yes—
379C22×C4C22⋊Q812yes—
380C22×C4C22⋊Q812yes—
381C22×C4C23×C411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×Q811yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—42no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—42no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—42no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—48no—
411C4⋊C4—44no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—81no—
415C4⋊C4—81no—
416C4⋊C4—82no—
417C4⋊C4—82no—
418C4⋊C4—88no—
419C4⋊C4#10414yes—
420C4⋊C4C2×C4⋊C412yes—
421C4⋊C4C2×C4⋊C412yes—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—24no—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—44no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—82no—
433C22⋊C4—82no—
434C42—24no—
435C42—42no—
436C42—42no—
437C42—42no—
438C42—42no—
439C42—48no—
440C42—44no—
441C42—82no—
442C42—82no—
443C42—82no—
444C42—84no—
445C42C2×C4⋊C412yes—
446C42C2×C4⋊C412yes—
447C42#11712yes—
448C42#11712yes—
449C42#12112yes—
450C42#12112yes—
451C42C22⋊Q812yes—
452C42C22⋊Q812yes—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—22no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23C22×C22⋊C411yescharacteristic
465C23C2×#12111yescharacteristic
466C23C2×C22⋊Q811yescharacteristic
467C23#57611yescharacteristic
468C23#57711yescharacteristic
469C23#57711yescharacteristic
470C23#57811yescharacteristic
471Q8—42no—
472Q8—42no—
473D4—42no—
474D4—42no—
475D4—42no—
476D4—42no—
477D4—42no—
478D4—42no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—48no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—81no—
540C2×C4—82no—
541C2×C4—81no—
542C2×C4—82no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—82no—
547C2×C4—81no—
548C2×C4—82no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—164no—
552C2×C4#38012yes—
553C2×C4#38012yes—
554C2×C4#38012yes—
555C2×C4#38012yes—
556C2×C4C2×#10414yes—
557C2×C4C2×#10412yes—
558C2×C4#42112yes—
559C2×C4C22×C4⋊C411yescharacteristic
560C2×C4C2×#11711yescharacteristic
561C2×C4C2×#12111yescharacteristic
562C2×C4C2×C22⋊Q811yescharacteristic
563C8—81no—
564C8—81no—
565C8—82no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22C2×#38011yescharacteristic
586C22#288311yescharacteristic
587C22#288411yescharacteristic
588C22#289711yescharacteristic
589C22#289811yescharacteristic
590C22C22×#10411yescharacteristic
591C22C2×#42111yescharacteristic
592C22#304511yescharacteristic
593C22#304611yescharacteristic
594C22#308311yescharacteristic
595C22#308411yescharacteristic
596C4—24no—
597C4—21no—
598C4—21no—
599C4—21no—
600C4—21no—
601C4—22no—
602C4—22no—
603C4—22no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—82no—
611C4—84no—
612C4—82no—
613C4—82no—
614C4—82no—
615C4#180614yes—
616C2—21no—
617C2—21no—
618C2—21no—
619C2—21no—
620C2—82no—
621C2#1754711yescharacteristic
622C2#1754911yescharacteristic
623C2#1760011yescharacteristic
624C2#1760211yescharacteristic
625C2#2063411yescharacteristic
626C2#2065211yescharacteristic
627C2C2×#180611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21953C2semidirect
5. #13783C2semidirect
6. #8578C2semidirect
7. #8576C2semidirect
10. #8022C2semidirect
11. #8014C2semidirect
15. #2370C4semidirect
16. #2367C4semidirect
18. #2355C4semidirect
19. #2354C4semidirect
21. Q8×C4⋊C4C4semidirect
23. D4×C4⋊C4C4semidirect
180. #125C4⋊C4semidirect
182. #124C4⋊C4semidirect
195. C4×Q8C4⋊C4semidirect
202. C4×D4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342432
#219532
C2×#15092
C2×#15002
#137832
#85782
#85762
#85732
#85712
#80222
#80142