Finite Group Databasebeta

← #497525

Subgroups — #497525

Order 512.

1060 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#38094C211yesmaximal, characteristic
2#22405C212yesmaximal
3#13299C212yesmaximal, complemented
4#13091C212yesmaximal, complemented
5#13090C212yesmaximal, complemented
6#12858C211yesmaximal, characteristic, complemented
7#12857C211yesmaximal, characteristic, complemented
8#12800C211yesmaximal, characteristic, complemented
9#12627C211yesmaximal, characteristic, complemented
10#12582C211yesmaximal, characteristic, complemented
11#12581C211yesmaximal, characteristic, complemented
12C2×#556C411yescharacteristic, complemented
13C2×#546C2211yescharacteristic
14C2×#410C2212yes—
15C22×#99C2212yes—
16#2850C411yescharacteristic, complemented
17#2691—22no—
18#2684—22no—
19#2640C411yescharacteristic, complemented
20#2621C411yescharacteristic, complemented
21#2563C2211yescharacteristic
22#2545C411yescharacteristic, complemented
23#2500C411yescharacteristic, complemented
24#2499C411yescharacteristic, complemented
25#2480—22no—
26#2458—22no—
27#2435C2211yescharacteristic
28#2432C2211yescharacteristic
29#2342—22no—
30#2326C2211yescharacteristic
31#2309C2211yescharacteristic
32#2297C2211yescharacteristic
33#2290C411yescharacteristic, complemented
34C2×#354C2212yes—
35C2×#354C2212yes—
36#1964—22no—
37#1899—22no—
38#1898—22no—
39#1799C2212yes—
40#1782C2212yes—
41#1781C2212yes—
42C4×#99C2212yes—
43#1393C2211yescharacteristic
44#1393C2211yescharacteristic
45#1393C2212yes—
46#1393C2212yes—
47#1393C2212yes—
48#1393C2212yes—
49#1393C2211yescharacteristic
50#1393C2211yescharacteristic
51C22×#41C2311yescharacteristic
52C23×D4C2311yescharacteristic
53#556—42no—
54#556—42no—
55#546—42no—
56#546—42no—
57C2×C4⋊Q8C2×C411yescharacteristic, complemented
58C2×C4⋊Q8C2×C411yescharacteristic, complemented
59C2×#126—21no—
60C2×#126—21no—
61C2×#124—21no—
62C2×#124C2×C411yescharacteristic
63C2×#124C2×C411yescharacteristic
64C2×#124C2311yescharacteristic
65C2×#123D412yes—
66C2×C22⋊Q8C2×C411yescharacteristic
67C2×C22⋊Q8C2×C411yescharacteristic
68C2×C22⋊Q8C2×C411yescharacteristic, complemented
69C2×#121C2×C411yescharacteristic
70C2×#121D412yes—
71C2×#120D412yes—
72C2×C4×Q8—21no—
73C2×C4×Q8C2×C411yescharacteristic, complemented
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C2×#117—22no—
77C2×#117C2×C411yescharacteristic
78C2×#117C2×C411yescharacteristic
79C2×#117C2×C411yescharacteristic, complemented
80C22×C22⋊C4—22no—
81C22×C22⋊C4D412yes—
82C22×C42C2×C411yescharacteristic, complemented
83#410—24no—
84#410—42no—
85C2×#99—24no—
86C2×#99—22no—
87C2×#99—22no—
88C2×#99—22no—
89C2×#99—22no—
90C2×#99—42no—
91C2×#99C2312yes—
92C2×#99C2312yes—
93C2×#99C2312yes—
94C2×#99C2312yes—
95#398—22no—
96#397—22no—
97#397—22no—
98#397—22no—
99#396—22no—
100#394—22no—
101#394—22no—
102#394—22no—
103#393—22no—
104#393—22no—
105#391C2×C411yescharacteristic, complemented
106#391C2×C411yescharacteristic, complemented
107#391C2×C411yescharacteristic, complemented
108#391C2×C411yescharacteristic, complemented
109#390—22no—
110#390C2×C411yescharacteristic
111#390C2×C411yescharacteristic
112#390C2×C411yescharacteristic
113#390C2311yescharacteristic
114#390C2311yescharacteristic
115#390C2311yescharacteristic
116#388—22no—
117#388—22no—
118#388—22no—
119#388—21no—
120#388—22no—
121#388—42no—
122#388—42no—
123#387—22no—
124#387—21no—
125#386—22no—
126#386—21no—
127#386—22no—
128#385—22no—
129#385—22no—
130#385—21no—
131#385—42no—
132#384C2×C411yescharacteristic, complemented
133#382C2×C411yescharacteristic, complemented
134#382C2×C411yescharacteristic, complemented
135#381—22no—
136#380—22no—
137#380—22no—
138#379—22no—
139#379—22no—
140#379—22no—
141C4×C4⋊C4C2×C411yescharacteristic, complemented
142C4×C22⋊C4—22no—
143C4×C22⋊C4—22no—
144C4×C22⋊C4—42no—
145C4×C22⋊C4C2×C411yescharacteristic
146C4×C22⋊C4C2×C411yescharacteristic, complemented
147#376C2×C411yescharacteristic, complemented
148#354—24no—
149#354—42no—
150#342—22no—
151#342—22no—
152C25C2×D412yes—
153C2×#41—22no—
154C2×#41—21no—
155C2×#41—21no—
156C2×#41—41no—
157C2×#41—41no—
158C2×#41—81no—
159C22×Q8—21no—
160C22×Q8C22×C411yescharacteristic
161C22×Q8C22×C411yescharacteristic
162C22×D4—22no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—42no—
166C22×D4—81no—
167C22×D4C22⋊C411yescharacteristic
168C22×D4C22⋊C412yes—
169C22×D4C22×C411yescharacteristic
170C22×D4C2×D412yes—
171C22×D4C2×D412yes—
172C22×D4C2411yescharacteristic
173C23×C4—21no—
174C23×C4—42no—
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C2×D412yes—
178C23×C4C2×D412yes—
179C4⋊Q8—42no—
180C4⋊Q8—81no—
181#126—44no—
182#126—42no—
183#126—42no—
184#124—22no—
185#124—21no—
186#124—21no—
187#124—42no—
188#124—44no—
189#124—81no—
190#123—42no—
191#123—42no—
192C22⋊Q8—42no—
193C22⋊Q8—41no—
194C22⋊Q8—41no—
195C22⋊Q8—42no—
196#121—41no—
197#121—41no—
198#121—42no—
199#121—42no—
200#120—42no—
201#120—42no—
202C4×Q8—42no—
203C4×Q8—42no—
204C4×Q8—42no—
205C4×D4—22no—
206C4×D4—22no—
207C4×D4—42no—
208#117—42no—
209#117—41no—
210#117—41no—
211#117—44no—
212#117—81no—
213#117—82no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4C22⋊C411yescharacteristic
225C2×C4⋊C4C22⋊C412yes—
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—82no—
258C2×C22⋊C4C22⋊C412yes—
259C2×C22⋊C4C22⋊C412yes—
260C2×C22⋊C4C22⋊C412yes—
261C2×C22⋊C4C22⋊C411yescharacteristic
262C2×C22⋊C4C22⋊C412yes—
263C2×C22⋊C4C22⋊C411yescharacteristic
264C2×C22⋊C4C2×D412yes—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42—81no—
272C2×C42C22×C411yescharacteristic
273C2×C42C22×C411yescharacteristic
274C2×C42C22×C411yescharacteristic
275C2×C42C22×C411yescharacteristic
276C2×C42C22×C411yescharacteristic
277#99—22no—
278#99—22no—
279#99—22no—
280#99—22no—
281#99—22no—
282#99—22no—
283#99—22no—
284#99—44no—
285#99—44no—
286#99—82no—
287#99—82no—
288#99#4114yes—
289#95—21no—
290#95—21no—
291#95—22no—
292#95—22no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—42no—
301#95—42no—
302#95—42no—
303#95—42no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—42no—
309C24—82no—
310C24—82no—
311C24C2×C22⋊C412yes—
312C24C2×C22⋊C411yescharacteristic
313C24C2×C22⋊C411yescharacteristic
314C24C22×D412yes—
315#41—81no—
316#41—82no—
317#41—81no—
318#41—82no—
319#41—81no—
320#41—81no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—42no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—82no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—42no—
347C2×D4—82no—
348C2×D4—82no—
349C2×D4—82no—
350C2×D4—82no—
351C2×D4—81no—
352C2×D4—82no—
353C2×D4C2×#4112yes—
354C2×D4H(5,𝔽2)11yescharacteristic
355C2×D42−1+411yescharacteristic
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4—81no—
387C22×C4—82no—
388C22×C4—84no—
389C22×C4—84no—
390C22×C4—162no—
391C22×C4C2×C22⋊C412yes—
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C412yes—
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C23×C411yescharacteristic
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—42no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—42no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—81no—
411C4⋊C4—82no—
412C4⋊C4—81no—
413C4⋊C4—82no—
414C4⋊C4—84no—
415C4⋊C4—162no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—24no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—81no—
440C22⋊C4—82no—
441C22⋊C4—81no—
442C22⋊C4—82no—
443C22⋊C4—82no—
444C22⋊C4—82no—
445C22⋊C4—82no—
446C22⋊C4—82no—
447C22⋊C4—82no—
448C22⋊C4—81no—
449C22⋊C4—84no—
450C22⋊C4—84no—
451C22⋊C4—162no—
452C22⋊C4#9912yes—
453C22⋊C4#9912yes—
454C42—22no—
455C42—21no—
456C42—21no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42—42no—
461C42—42no—
462C42—41no—
463C42—41no—
464C42—81no—
465C42—84no—
466C42—162no—
467C23—22no—
468C23—21no—
469C23—22no—
470C23—21no—
471C23—22no—
472C23—21no—
473C23—21no—
474C23—22no—
475C23—22no—
476C23—21no—
477C23—41no—
478C23—41no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—44no—
486C23—42no—
487C23—41no—
488C23—42no—
489C23—41no—
490C23—82no—
491C23—82no—
492C23—82no—
493C23—162no—
494C23—162no—
495C23—164no—
496C23C2×#9912yes—
497C23C22×C22⋊C411yescharacteristic
498C23#54612yes—
499C23#54712yes—
500Q8—81no—
501Q8—81no—
502Q8—81no—
503Q8—81no—
504Q8—84no—
505D4—42no—
506D4—41no—
507D4—41no—
508D4—81no—
509D4—81no—
510D4—82no—
511D4—82no—
512D4—82no—
513D4—82no—
514D4—82no—
515D4—162no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—44no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—84no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—168no—
578C2×C4C2×#9912yes—
579C2×C4C2×#9911yescharacteristic
580C2×C4C2×#9911yescharacteristic
581C2×C4C4×#4111yescharacteristic
582C2×C4#52011yescharacteristic
583C22—21no—
584C22—21no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—41no—
595C22—42no—
596C22—41no—
597C22—42no—
598C22—41no—
599C22—42no—
600C22—82no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—82no—
606C22—82no—
607C22—84no—
608C22—84no—
609C22—162no—
610C22—162no—
611C22#233911yescharacteristic
612C22C22×#9911yescharacteristic
613C22#288311yescharacteristic
614C4—22no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—21no—
619C4—41no—
620C4—42no—
621C4—41no—
622C4—42no—
623C4—81no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—84no—
628C4—81no—
629C4—81no—
630C4—81no—
631C4—81no—
632C4—162no—
633C4—162no—
634C4—162no—
635C2—21no—
636C2—21no—
637C2—42no—
638C2—42no—
639C2—42no—
640C2—81no—
641C2—82no—
642C2#2247611yescharacteristic
643C2#3335811yescharacteristic
644C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC2×#546C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#546 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13299C2semidirect
4. #13091C2semidirect
5. #13090C2semidirect
6. #12858C2semidirect
7. #12857C2semidirect
8. #12800C2semidirect
9. #12627C2semidirect
10. #12582C2semidirect
11. #12581C2semidirect
12. C2×#556C4semidirect
16. #2850C4semidirect
19. #2640C4semidirect
20. #2621C4semidirect
22. #2545C4semidirect
23. #2500C4semidirect
24. #2499C4semidirect
33. #2290C4semidirect
57. C2×C4⋊Q8C2×C4semidirect
58. C2×C4⋊Q8C2×C4semidirect
68. C2×C22⋊Q8C2×C4semidirect
73. C2×C4×Q8C2×C4semidirect
79. C2×#117C2×C4semidirect
82. C22×C42C2×C4semidirect
105. #391C2×C4semidirect
106. #391C2×C4semidirect
107. #391C2×C4semidirect
108. #391C2×C4semidirect
132. #384C2×C4semidirect
133. #382C2×C4semidirect
134. #382C2×C4semidirect
141. C4×C4⋊C4C2×C4semidirect
146. C4×C22⋊C4C2×C4semidirect
147. #376C2×C4semidirect

Maximal subgroups

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

subgroupindex
#380942
#224052
#132992
#130912
#130902
#128582
#128572
#128002
#126272
#125822
#125812