Finite Group Databasebeta

← #379579

Subgroups — #379579

Order 512.

984 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23451C211yesmaximal, characteristic
2#21301C211yesmaximal, characteristic, complemented
3#20384C211yesmaximal, characteristic
4#11386C211yesmaximal, characteristic, complemented
5#11269C211yesmaximal, characteristic, complemented
6#11053C211yesmaximal, characteristic
7#10930C211yesmaximal, characteristic
8#10929C211yesmaximal, characteristic
9#10819C211yesmaximal, characteristic, complemented
10#10818C211yesmaximal, characteristic, complemented
11#10809C211yesmaximal, characteristic, complemented
12#10809C211yesmaximal, characteristic, complemented
13#10791C211yesmaximal, characteristic
14#10791C211yesmaximal, characteristic
15#10696C211yesmaximal, characteristic, complemented
16C22×#131C2211yescharacteristic
17C2×#533C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C22×#104C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C2×#416C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#408C2211yescharacteristic
24C2×#405C2211yescharacteristic
25C22×#97C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28#2297C2211yescharacteristic
29#2026C412yescomplemented
30#2025C412yescomplemented
31#2024C412yescomplemented
32#2023C412yescomplemented
33#1985—22no—
34#1985—22no—
35#1753—22no—
36#1752—22no—
37C2×#341C2211yescharacteristic
38C2×#341C2211yescharacteristic
39C2×#341C2211yescharacteristic
40C2×#341C2211yescharacteristic
41#1735C2211yescharacteristic
42#1735C2211yescharacteristic
43C2×#339C2211yescharacteristic
44C2×#339C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C2×#339C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49#1729C2211yescharacteristic
50#1729C2211yescharacteristic
51#1729C2211yescharacteristic
52#1729C2211yescharacteristic
53#1729C2211yescharacteristic
54#1728C2211yescharacteristic
55C2×#336C2211yescharacteristic
56C2×#336C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C4×#95C2211yescharacteristic
59C22×#41C2311yescharacteristic
60C2×#131—22no—
61C2×#131—22no—
62C2×#131—22no—
63C2×#131—22no—
64C2×#131—41no—
65C2×#131—41no—
66C22×M16—21no—
67C22×M16C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C23×C8—21no—
72C23×C8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76#533—41no—
77#533—41no—
78#533—41no—
79#533—41no—
80C2×C4⋊Q8C2×C412yes—
81C2×#125C2×C412yes—
82C2×C22⋊Q8—22no—
83C2×C4×Q8—22no—
84C2×#117C2311yescharacteristic
85C2×#117C2311yescharacteristic
86C2×#117C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#423—41no—
91#423—41no—
92C2×C4⋊C8—22no—
93C2×#104—22no—
94C2×#104—22no—
95C2×#104—22no—
96C2×#104—22no—
97C2×#104—41no—
98C2×#104—41no—
99#416—41no—
100#416—41no—
101#416—41no—
102#416—41no—
103C2×#103C2×C412yes—
104C2×#103C2×C412yes—
105C2×#102C2×C412yes—
106C2×#102C2×C412yes—
107#408—22no—
108#408Q814yescomplemented
109C2×C22⋊C8—22no—
110#405—22no—
111#405—22no—
112C2×#97—21no—
113C2×#97—22no—
114C2×#97—22no—
115C2×#97—41no—
116C2×#97—41no—
117C2×#97—41no—
118C2×#97D412yes—
119C2×#97D412yes—
120C2×C4×C8—21no—
121C2×C4×C8—22no—
122C2×C4×C8—22no—
123C2×C4×C8—41no—
124#391C2×C411yescharacteristic
125#391C2×C411yescharacteristic
126#390C2×C411yescharacteristic
127#390C2×C411yescharacteristic
128#386—22no—
129C4×C22⋊C4—22no—
130C2×#95—22no—
131#341—22no—
132#341—22no—
133#341—41no—
134#341—41no—
135#340C2×C412yes—
136#340C2×C412yes—
137#340C2×C412yes—
138#340C2×C412yes—
139#339—22no—
140#339—22no—
141#339—41no—
142#339—41no—
143#339—41no—
144#339—41no—
145#336—22no—
146#336—22no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—22no—
152#336—22no—
153#336—22no—
154#336—22no—
155#336C2×C412yes—
156#336C2×C412yes—
157C2×#41—21no—
158C2×#41—21no—
159C2×#41—21no—
160C2×#41—21no—
161C2×#41—21no—
162C2×#41—21no—
163C2×#41—21no—
164C2×#41—21no—
165C2×#41—21no—
166C2×#41—21no—
167C2×#41C2×Q812yes—
168C2×#41#4112yes—
169C22×Q8—22no—
170C22×Q8C22×C411yescharacteristic
171C22×D4—21no—
172C22×D4C22×C411yescharacteristic
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4#4112yes—
177C23×C4C2411yescharacteristic
178#131—44no—
179#131—44no—
180#131—44no—
181#131—44no—
182C2×M16—24no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—42no—
206C2×M16—42no—
207C2×M16C2×Q812yes—
208C2×M16#4112yes—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—24no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—42no—
235C22×C8—42no—
236C22×C8C22×C412yes—
237C22×C8C22×C412yes—
238C22×C8C2×D412yes—
239C22×C8C2×D411yescharacteristic
240C22×C8C2×D411yescharacteristic
241C22×C8C2×Q812yes—
242C22×C8#4112yes—
243C22×C8#4112yes—
244C22×C8#4112yes—
245C4⋊Q8—42no—
246C4⋊Q8—42no—
247#125—42no—
248#125—42no—
249C22⋊Q8—44no—
250C22⋊Q8—44no—
251C4×Q8—44no—
252C4×Q8—44no—
253#117—21no—
254#117—21no—
255#117—21no—
256#117—21no—
257#117—21no—
258#117—21no—
259#117—21no—
260#117—21no—
261#117—21no—
262#117—21no—
263#117—21no—
264#117—21no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4C22×C412yes—
268C2×C4⋊C4C22×C412yes—
269C2×C4⋊C4C22×C411yescharacteristic
270C2×C4⋊C4C22×C411yescharacteristic
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42C22×C411yescharacteristic
294C2×C42C22×C411yescharacteristic
295C2×C42C2×D412yes—
296C2×C42#4112yes—
297C2×C42#4112yes—
298C4⋊C8—44no—
299#104—44no—
300#104—44no—
301#103—42no—
302#103—42no—
303#102—42no—
304#102—42no—
305C22⋊C8—28no—
306#97—42no—
307#97—42no—
308#97—44no—
309#97—42no—
310C4×C8—44no—
311C4×C8—42no—
312#95—22no—
313#95—22no—
314#95—44no—
315C24—21no—
316C24C2×#4111yescharacteristic
317#41—22no—
318#41—22no—
319#41—22no—
320#41—22no—
321#41—22no—
322#41—22no—
323#41—22no—
324#41—22no—
325#41—41no—
326#41—41no—
327#41—41no—
328#41—41no—
329#41—41no—
330#41—41no—
331#41—41no—
332#41—41no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—44no—
339C2×Q8—82no—
340C2×Q8—82no—
341C2×Q8C4×Q812yes—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4C4×Q812yes—
354C22×C4—24no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4C4×D412yes—
380C22×C4C4×D412yes—
381C22×C4C23×C411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×Q811yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4H(5,𝔽2)11yescharacteristic
392C22×C4H(5,𝔽2)11yescharacteristic
393C22×C4H(5,𝔽2)11yescharacteristic
394C22×C4H(5,𝔽2)11yescharacteristic
395M16—22no—
396M16—22no—
397M16—22no—
398M16—22no—
399M16—48no—
400M16—41no—
401M16—41no—
402M16—41no—
403M16—41no—
404M16—41no—
405M16—41no—
406M16—41no—
407M16—41no—
408M16—41no—
409M16—41no—
410M16—41no—
411M16—41no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—22no—
415C2×C8—21no—
416C2×C8—22no—
417C2×C8—21no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—48no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8C22⋊Q814yes—
454C2×C8#12412yes—
455C2×C8#12412yes—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—48no—
465C4⋊C4—44no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C22⋊C4—82no—
473C22⋊C4—82no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—41no—
485C42—41no—
486C42—41no—
487C42—41no—
488C42—41no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42—41no—
493C42—41no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—41no—
501C23C2×C4×D411yescharacteristic
502C23#52911yescharacteristic
503C23#52911yescharacteristic
504C23#52911yescharacteristic
505C23#53311yescharacteristic
506C23#53811yescharacteristic
507C23#53811yescharacteristic
508Q8—22no—
509Q8—22no—
510Q8—41no—
511Q8—41no—
512Q8—88no—
513D4—22no—
514D4—22no—
515D4—42no—
516D4—42no—
517D4—42no—
518D4—42no—
519D4—41no—
520D4—41no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
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—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4C2×C4×Q811yescharacteristic
557C2×C4C4×#4111yescharacteristic
558C2×C4C4×#4111yescharacteristic
559C2×C4C4×#4111yescharacteristic
560C2×C4#51811yescharacteristic
561C2×C4#51811yescharacteristic
562C2×C4C2×C22⋊Q811yescharacteristic
563C2×C4#52511yescharacteristic
564C2×C4C2×#12411yescharacteristic
565C2×C4#53211yescharacteristic
566C2×C4#53411yescharacteristic
567C2×C4#53511yescharacteristic
568C2×C4#55111yescharacteristic
569C2×C4#55111yescharacteristic
570C2×C4#55111yescharacteristic
571C2×C4#55111yescharacteristic
572C2×C4#55111yescharacteristic
573C2×C4#55111yescharacteristic
574C2×C4#55111yescharacteristic
575C2×C4#55111yescharacteristic
576C2×C4#55411yescharacteristic
577C2×C4#55411yescharacteristic
578C2×C4#55411yescharacteristic
579C2×C4#55411yescharacteristic
580C2×C4#55511yescharacteristic
581C2×C4#55511yescharacteristic
582C2×C4#55511yescharacteristic
583C2×C4#55511yescharacteristic
584C8—22no—
585C8—22no—
586C8—24no—
587C8—42no—
588C8—42no—
589C8—42no—
590C8—42no—
591C8—81no—
592C8—81no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—44no—
602C22—42no—
603C22—82no—
604C22—81no—
605C22—81no—
606C22C4×C22⋊Q811yescharacteristic
607C22C4×#12411yescharacteristic
608C22#232111yescharacteristic
609C22#257211yescharacteristic
610C22#257211yescharacteristic
611C22#257411yescharacteristic
612C22#257411yescharacteristic
613C22#295612yes—
614C22#295712yes—
615C4—44no—
616C4—42no—
617C4—42no—
618C4—42no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4—81no—
623C4—81no—
624C4#236811yescharacteristic
625C4#236811yescharacteristic
626C4#237211yescharacteristic
627C4#237211yescharacteristic
628C4#256111yescharacteristic
629C4#256311yescharacteristic
630C4#256411yescharacteristic
631C4#261011yescharacteristic
632C2—21no—
633C2—21no—
634C2—21no—
635C2—21no—
636C2—82no—
637C2#1424311yescharacteristic
638C2#2162912yes—
639C2#2165911yescharacteristic
640C2#2166211yescharacteristic
641C2#2206212yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×#97 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21301C2semidirect
4. #11386C2semidirect
5. #11269C2semidirect
9. #10819C2semidirect
10. #10818C2semidirect
11. #10809C2semidirect
12. #10809C2semidirect
15. #10696C2semidirect
29. #2026C4semidirect
30. #2025C4semidirect
31. #2024C4semidirect
32. #2023C4semidirect
108. #408Q8semidirect

Maximal subgroups

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

subgroupindex
#234512
#213012
#203842
#113862
#112692
#110532
#109302
#109292
#108192
#108182
#108092
#107912
#106962