Finite Group Databasebeta

← #210504

Subgroups — #210504

Order 512.

777 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22264C211yesmaximal, characteristic, complemented
2C4×#459C211yesmaximal, characteristic
3#20568C211yesmaximal, characteristic, complemented
4C2×#1594C211yesmaximal, characteristic, complemented
5C2×#1589C211yesmaximal, characteristic, complemented
6#9674C211yesmaximal, characteristic, complemented
7#9603C211yesmaximal, characteristic, complemented
8#9546C211yesmaximal, characteristic, complemented
9#9545C211yesmaximal, characteristic, complemented
10#8951C211yesmaximal, characteristic, complemented
11#8951C211yesmaximal, characteristic, complemented
12#8565C211yesmaximal, characteristic, complemented
13#8264C211yesmaximal, characteristic, complemented
14#8208C211yesmaximal, characteristic, complemented
15#8206C211yesmaximal, characteristic, complemented
16C2×#459C411yescharacteristic
17C2×C4×D8C2211yescharacteristic
18C2×C4×D8C2211yescharacteristic
19C2×#436C2211yescharacteristic
20C2×#436C2211yescharacteristic
21C2×#436C2211yescharacteristic
22C2×#436C2211yescharacteristic
23C4×#127C2211yescharacteristic
24C42×D4C2211yescharacteristic
25#1982C411yescharacteristic
26#1954C2211yescharacteristic
27#1954C2211yescharacteristic
28#1947C411yescharacteristic
29#1925C411yescharacteristic
30#1925C411yescharacteristic
31#1879C411yescharacteristic
32#1879C411yescharacteristic
33#1870—21no—
34#1870—21no—
35#1870—21no—
36#1870—21no—
37#1831C2211yescharacteristic
38#1802C411yescharacteristic
39#1770C2211yescharacteristic
40#1770C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42C4×C4⋊C8C2211yescharacteristic
43C4×C4⋊C8C2211yescharacteristic
44C4×#102C2211yescharacteristic
45C4×#102C2211yescharacteristic
46#1753—21no—
47#1753—21no—
48#1749—21no—
49#1744C2211yescharacteristic
50#1728—21no—
51#1594—24no—
52#1589—24no—
53C2×#335C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×C82C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×D8C2×C411yescharacteristic
70C22×D8C2×C411yescharacteristic
71C22×M16—21no—
72C22×M16—21no—
73C22×M16—21no—
74C22×M16—21no—
75C2×#127C2×C411yescharacteristic
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4C2×C411yescharacteristic
79C2×C4×D4C2×C411yescharacteristic
80C2×C4×D4C2311yescharacteristic
81C2×C4×D4C2311yescharacteristic
82C2×C4×D4C2311yescharacteristic
83C22×C42—21no—
84#459—24no—
85C4×D8—22no—
86C4×D8—22no—
87C4×D8—22no—
88C4×D8—22no—
89#436—22no—
90#436—22no—
91#436—22no—
92#436—22no—
93#436—22no—
94#436—22no—
95#436—22no—
96#436D414yescomplemented
97C2×#107C2×C411yescharacteristic
98C2×#107C2×C411yescharacteristic
99C2×C4⋊C8—21no—
100C2×C4⋊C8C2×C411yescharacteristic
101C2×C4⋊C8C2×C411yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×#102C2×C411yescharacteristic
108C2×#102C2×C411yescharacteristic
109C2×#102C2×C411yescharacteristic
110C2×#102C2×C411yescharacteristic
111C2×#102C2×C411yescharacteristic
112C2×#102C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×#97—21no—
123C2×#97—21no—
124C2×C4×C8—21no—
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131#390—21no—
132#390—21no—
133#390C2×C411yescharacteristic
134#390C2×C411yescharacteristic
135#384C2×C411yescharacteristic
136C4×C4⋊C4C2×C411yescharacteristic
137C4×C4⋊C4C2×C411yescharacteristic
138C4×C4⋊C4C2311yescharacteristic
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142C43C2311yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336C2×C411yescharacteristic
154#336C2×C411yescharacteristic
155#336D411yescharacteristic
156#336D411yescharacteristic
157#336D411yescharacteristic
158#336D411yescharacteristic
159#335—22no—
160#334—22no—
161D4⋊C8—22no—
162D4⋊C8—22no—
163D4⋊C8—22no—
164D4⋊C8—22no—
165D4⋊C8—22no—
166D4⋊C8—22no—
167#322—22no—
168C82—22no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4C22×C411yescharacteristic
172C22×D4C22×C411yescharacteristic
173C22×D4C22×C411yescharacteristic
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C2×D8—22no—
180C2×D8—22no—
181C2×D8—22no—
182C2×D8—22no—
183C2×D8—41no—
184C2×D8—41no—
185C2×M16—24no—
186C2×M16—24no—
187C2×M16—42no—
188C2×M16—42no—
189C2×M16—81no—
190C2×M16—81no—
191C2×M16—81no—
192C2×M16—81no—
193C2×M16—81no—
194C2×M16—81no—
195C2×M16—81no—
196C2×M16—81no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8C22×C411yescharacteristic
204C22×C8C22×C411yescharacteristic
205C22×C8C22×C411yescharacteristic
206C22×C8C22×C411yescharacteristic
207C22×C8C2×D411yescharacteristic
208C22×C8C2×D411yescharacteristic
209C22×C8C2×D411yescharacteristic
210C22×C8C2×D411yescharacteristic
211#127—22no—
212#127—22no—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4—21no—
216C4×D4—21no—
217C4×D4—21no—
218C4×D4—21no—
219C4×D4—21no—
220C4×D4—21no—
221C4×D4—21no—
222C4×D4—21no—
223C4×D4—21no—
224C4×D4—21no—
225C4×D4—41no—
226C4×D4—41no—
227C4×D4—81no—
228C4×D4—81no—
229C4×D4—81no—
230C4×D4—81no—
231C4×D4C2×D412yes—
232C4×D4#4112yes—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—81no—
253C2×C42—81no—
254C2×C42C22×C411yescharacteristic
255C2×C42C22×C411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42#4111yescharacteristic
259C2×C42#4111yescharacteristic
260C2×C42C2411yescharacteristic
261#107—22no—
262#107—22no—
263C4⋊C8—22no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—42no—
271C4⋊C8M1614yescomplemented
272C4⋊C8C2×D412yes—
273C4⋊C8#4112yes—
274#102—22no—
275#102—22no—
276#102—22no—
277#102—22no—
278#102—22no—
279#102—22no—
280#102—41no—
281#102—41no—
282C22⋊C8—24no—
283C22⋊C8—24no—
284C22⋊C8—42no—
285C22⋊C8—42no—
286C22⋊C8—81no—
287C22⋊C8—81no—
288C22⋊C8—81no—
289C22⋊C8—81no—
290#97—81no—
291#97—81no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—21no—
296C4×C8—21no—
297C4×C8—21no—
298C4×C8—21no—
299C4×C8—21no—
300C4×C8—21no—
301C4×C8—21no—
302C4×C8—81no—
303C4×C8C2×D412yes—
304C4×C8#4112yes—
305#95—21no—
306#95—21no—
307#95—21no—
308C24—21no—
309C24—21no—
310C24—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—81no—
331C2×D4—81no—
332C2×D4—81no—
333C2×D4—81no—
334C2×D4C4×D412yes—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—81no—
361C22×C4—81no—
362C22×C4—81no—
363C22×C4—81no—
364C22×C4C4×D411yescharacteristic
365C22×C4C4×D411yescharacteristic
366C22×C4C4×D411yescharacteristic
367C22×C4C4×D411yescharacteristic
368C22×C4C23×C411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C2×#4111yescharacteristic
371D8—44no—
372D8—44no—
373M16—84no—
374M16—84no—
375M16—84no—
376M16—84no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—22no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—22no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—22no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8#12112yes—
416C2×C8#12112yes—
417C2×C8C2×M1612yes—
418C2×C8C2×M1612yes—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—81no—
429C4⋊C4—81no—
430C4⋊C4C4×D412yes—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—42no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—42no—
439C22⋊C4—81no—
440C22⋊C4—81no—
441C22⋊C4—81no—
442C22⋊C4—81no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—81no—
454C42—81no—
455C42C2×M1612yes—
456C42#13112yes—
457C42C22×D411yescharacteristic
458C42C2×#4111yescharacteristic
459C42C2×#4111yescharacteristic
460C42C2×#4111yescharacteristic
461C23—22no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23C2×C4×D411yescharacteristic
483D4—22no—
484D4—22no—
485D4—41no—
486D4—41no—
487D4—41no—
488D4—41no—
489D4—41no—
490D4—41no—
491D4—41no—
492D4—41no—
493D4—41no—
494D4—41no—
495D4—84no—
496D4—84no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—84no—
529C2×C4—84no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4#43612yes—
539C2×C4#43612yes—
540C2×C4C2×C4×D411yescharacteristic
541C2×C4C4×#4111yescharacteristic
542C2×C4C2×#12111yescharacteristic
543C2×C4#52511yescharacteristic
544C2×C4#52911yescharacteristic
545C2×C4#53211yescharacteristic
546C2×C4C22×M1611yescharacteristic
547C2×C4C2×#13111yescharacteristic
548C2×C4#56811yescharacteristic
549C2×C4#56811yescharacteristic
550C2×C4#57611yescharacteristic
551C2×C4#57611yescharacteristic
552C2×C4#57611yescharacteristic
553C2×C4#57711yescharacteristic
554C8—22no—
555C8—22no—
556C8—22no—
557C8—22no—
558C8—42no—
559C8—41no—
560C8—41no—
561C8—42no—
562C8—41no—
563C8—41no—
564C8—42no—
565C8—81no—
566C8—81no—
567C8—81no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22C4×#12111yescharacteristic
590C22C2×#43611yescharacteristic
591C22#293011yescharacteristic
592C22#294611yescharacteristic
593C22#294811yescharacteristic
594C22#295611yescharacteristic
595C22#295711yescharacteristic
596C4—21no—
597C4—21no—
598C4—22no—
599C4—22no—
600C4—42no—
601C4—41no—
602C4—41no—
603C4—81no—
604C4—81no—
605C4—81no—
606C4—81no—
607C4D4×M1611yescharacteristic
608C4#293511yescharacteristic
609C4#296911yescharacteristic
610C4#297111yescharacteristic
611C4#304111yescharacteristic
612C4#304511yescharacteristic
613C4#311911yescharacteristic
614C4#312211yescharacteristic
615C2—82no—
616C2—161no—
617C2—161no—
618C2#1789311yescharacteristic
619C2#1789511yescharacteristic
620C2#1794811yescharacteristic
621C2#1795011yescharacteristic
622C2#2148811yescharacteristic
623C2#2151311yescharacteristic
624C2#2151911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#459C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ #1744 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22264C2semidirect
3. #20568C2semidirect
4. C2×#1594C2semidirect
5. C2×#1589C2semidirect
6. #9674C2semidirect
7. #9603C2semidirect
8. #9546C2semidirect
9. #9545C2semidirect
10. #8951C2semidirect
11. #8951C2semidirect
12. #8565C2semidirect
13. #8264C2semidirect
14. #8208C2semidirect
15. #8206C2semidirect
96. #436D4semidirect
271. C4⋊C8M16semidirect

Maximal subgroups

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

subgroupindex
#222642
C4×#4592
#205682
C2×#15942
C2×#15892
#96742
#96032
#95462
#95452
#89512
#85652
#82642
#82082
#82062