Finite Group Databasebeta

← #202382

Subgroups — #202382

Order 512.

791 conjugacy classes of subgroups.

The subgroup lattice has 609 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21388C211yesmaximal, characteristic
2#21378C211yesmaximal, characteristic
3C4⋊C4×M16C211yesmaximal, characteristic, complemented
4C2×#1574C211yesmaximal, characteristic, complemented
5C2×#1495C211yesmaximal, characteristic
6#9548C211yesmaximal, characteristic, complemented
7#9547C211yesmaximal, characteristic, complemented
8#9546C211yesmaximal, characteristic, complemented
9#9545C211yesmaximal, characteristic, complemented
10#9544C211yesmaximal, characteristic, complemented
11#9543C211yesmaximal, characteristic, complemented
12#9542C211yesmaximal, characteristic, complemented
13#9541C211yesmaximal, characteristic, complemented
14#7892C211yesmaximal, characteristic, complemented
15#7850C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C2×C4×M16—21no—
22C2×C4×M16C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1989C411yescharacteristic
25#1989C411yescharacteristic
26#1984C2211yescharacteristic
27#1984C2211yescharacteristic
28#1983C411yescharacteristic
29#1983C411yescharacteristic
30#1954C2211yescharacteristic
31#1954C2211yescharacteristic
32#1874C411yescharacteristic
33#1874C411yescharacteristic
34#1872C411yescharacteristic
35#1872C411yescharacteristic
36#1870—21no—
37#1870—21no—
38#1771—21no—
39#1771—21no—
40#1770C2211yescharacteristic
41#1770C2211yescharacteristic
42C8×C4⋊C4C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44C4×#103C2211yescharacteristic
45C4×#103C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48#1750—21no—
49#1750—21no—
50#1728—21no—
51#1574—24no—
52C2×#335C2211yescharacteristic
53C2×#335C2211yescharacteristic
54C2×#334C2211yescharacteristic
55C2×#334C2211yescharacteristic
56#1495—24no—
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×M16—21no—
70C22×M16D411yescharacteristic
71C22×M16D411yescharacteristic
72#529—22no—
73#529—22no—
74#529—22no—
75#529—22no—
76C2×#126—21no—
77C2×#125C2×C411yescharacteristic
78C2×#124C2×C411yescharacteristic
79C2×#123—21no—
80C2×C22⋊Q8C2×C411yescharacteristic
81C2×#121C2×C411yescharacteristic
82C2×C4×Q8C2311yescharacteristic
83C2×C4×D4—21no—
84C2×C4×D4C2311yescharacteristic
85C2×#117—21no—
86C22×C4⋊C4C2×C411yescharacteristic
87C22×C4⋊C4C2×C411yescharacteristic
88C22×C42—21no—
89C22×C42C2311yescharacteristic
90#431—22no—
91#431—22no—
92#431—22no—
93#431—22no—
94#431—22no—
95#431—22no—
96#423—22no—
97#423D414yes—
98C2×C4⋊C8—21no—
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×#103C2×C411yescharacteristic
107C2×#103C2×C411yescharacteristic
108C2×#103C2×C411yescharacteristic
109C2×#103C2×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⋊C8C2×C411yescharacteristic
121C2×C22⋊C8C2×C411yescharacteristic
122C4×M16—22no—
123C4×M16—22no—
124C4×M16—44no—
125C2×#97—21no—
126C2×#97—21no—
127C2×#97D411yescharacteristic
128C2×#97D411yescharacteristic
129C2×C4×C8—21no—
130C2×C4×C8—21no—
131C2×C4×C8—21no—
132C2×C4×C8C2311yescharacteristic
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137C2×C4×C8C2311yescharacteristic
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4C2×C411yescharacteristic
141C4×C4⋊C4C2×C411yescharacteristic
142C4×C4⋊C4C2311yescharacteristic
143C4×C4⋊C4C2311yescharacteristic
144C2×#95—21no—
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#340C2×C411yescharacteristic
150#340C2×C411yescharacteristic
151#340C2×C411yescharacteristic
152#340C2×C411yescharacteristic
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336—21no—
159#336—21no—
160#336—21no—
161#335—22no—
162#335—22no—
163#334—22no—
164#334—22no—
165Q8⋊C8—22no—
166Q8⋊C8—22no—
167D4⋊C8—22no—
168D4⋊C8—22no—
169C22×Q8C22×C411yescharacteristic
170C22×D4—21no—
171C22×D4C22×C411yescharacteristic
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4C22×C411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4#4111yescharacteristic
178C2×M16—22no—
179C2×M16—24no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—41no—
184C2×M16—41no—
185C2×M16—42no—
186C2×M16—42no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8C22×C411yescharacteristic
194C22×C8C22×C411yescharacteristic
195C22×C8C22×C411yescharacteristic
196C22×C8C22×C411yescharacteristic
197C22×C8C2×D411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C22×C8C2×D411yescharacteristic
200C22×C8C2×D411yescharacteristic
201#126—42no—
202#126—42no—
203#125—22no—
204#125—22no—
205#124—22no—
206#124—22no—
207#123—42no—
208#123—42no—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211#121—22no—
212#121—22no—
213C4×Q8—21no—
214C4×Q8—21no—
215C4×Q8—21no—
216C4×Q8—21no—
217C4×D4—21no—
218C4×D4—21no—
219C4×D4—21no—
220C4×D4—21no—
221C4×D4—42no—
222C4×D4—42no—
223#117—42no—
224#117—42no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42C22×C411yescharacteristic
254C2×C42C2×D412yes—
255C2×C42C2×D411yescharacteristic
256C2×C42#4112yes—
257C2×C42#4112yes—
258C2×C42#4112yes—
259C2×C42#4111yescharacteristic
260C2×C42C2411yescharacteristic
261C4⋊C8—22no—
262C4⋊C8—22no—
263C4⋊C8—24no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—42no—
271C4⋊C8C2×D412yes—
272C4⋊C8C2×D412yes—
273#103—22no—
274#103—22no—
275#103—41no—
276#103—41no—
277#102—22no—
278#102—22no—
279#102—41no—
280#102—41no—
281C22⋊C8—22no—
282C22⋊C8—22no—
283C22⋊C8—42no—
284C22⋊C8—42no—
285C22⋊C8—42no—
286C22⋊C8—42no—
287C22⋊C8—42no—
288C22⋊C8—42no—
289#97—22no—
290#97—22no—
291#97—42no—
292#97—42no—
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—21no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—42no—
306C4×C8—42no—
307C4×C8—42no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312C24—21no—
313C24C4×D411yescharacteristic
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—42no—
327C2×D4—81no—
328C2×D4—81no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—81no—
357C22×C4—81no—
358C22×C4#11712yes—
359C22×C4C4×D412yes—
360C22×C4C4×D411yescharacteristic
361C22×C4C4×D411yescharacteristic
362C22×C4C4×D411yescharacteristic
363C22×C4#12112yes—
364C22×C4#12312yes—
365C22×C4#12412yes—
366C22×C4#12612yes—
367C22×C4C23×C411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C2×#4111yescharacteristic
370M16—28no—
371M16—44no—
372M16—48no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
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×C8AGL(1,ℤ/8)12yes—
414C2×C8#13712yes—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4#13112yes—
433C4⋊C4#13112yes—
434C22⋊C4—42no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C42—42no—
441C42—42no—
442C42—41no—
443C42—41no—
444C42—41no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42#11712yes—
449C42C4×D412yes—
450C42#12112yes—
451C42#12312yes—
452C42#12412yes—
453C42#12612yes—
454C42C22×D411yescharacteristic
455C42C2×#4111yescharacteristic
456C42C2×#4111yescharacteristic
457C42C2×#4111yescharacteristic
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23#38812yes—
472C23C2×C4×D411yescharacteristic
473Q8—41no—
474Q8—41no—
475Q8—41no—
476Q8—41no—
477D4—41no—
478D4—41no—
479D4—41no—
480D4—41no—
481D4—84no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—84no—
522C2×C4—84no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4#38812yes—
528C2×C4#38812yes—
529C2×C4#38812yes—
530C2×C4C2×#11711yescharacteristic
531C2×C4C2×C4×D411yescharacteristic
532C2×C4C2×#12111yescharacteristic
533C2×C4C2×#12311yescharacteristic
534C2×C4C2×#12411yescharacteristic
535C2×C4C2×#12611yescharacteristic
536C2×C4C2×#13111yescharacteristic
537C2×C4C2×#13111yescharacteristic
538C2×C4#56811yescharacteristic
539C2×C4#56811yescharacteristic
540C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
541C2×C4C2×#13711yescharacteristic
542C2×C4#57511yescharacteristic
543C2×C4#57511yescharacteristic
544C8—22no—
545C8—22no—
546C8—41no—
547C8—41no—
548C8—41no—
549C8—41no—
550C8—41no—
551C8—41no—
552C8—42no—
553C8—42no—
554C8—81no—
555C8—81no—
556C8—81no—
557C8—81no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22C2×#38811yescharacteristic
573C22#293311yescharacteristic
574C22#293311yescharacteristic
575C22C4×AGL(1,ℤ/8)11yescharacteristic
576C22C4×#13711yescharacteristic
577C22#295711yescharacteristic
578C22#295711yescharacteristic
579C4—22no—
580C4—22no—
581C4—21no—
582C4—21no—
583C4—22no—
584C4—42no—
585C4—42no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4#292211yescharacteristic
592C4#292311yescharacteristic
593C4#293511yescharacteristic
594C4#293511yescharacteristic
595C4#303611yescharacteristic
596C4#303711yescharacteristic
597C4#309211yescharacteristic
598C4#309311yescharacteristic
599C2—42no—
600C2—161no—
601C2#1800611yescharacteristic
602C2#1800811yescharacteristic
603C2#1801011yescharacteristic
604C2#1801211yescharacteristic
605C2#2104811yescharacteristic
606C2#2105911yescharacteristic
607C2#2107211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C4⋊C4×M16C2semidirect
4. C2×#1574C2semidirect
6. #9548C2semidirect
7. #9547C2semidirect
8. #9546C2semidirect
9. #9545C2semidirect
10. #9544C2semidirect
11. #9543C2semidirect
12. #9542C2semidirect
13. #9541C2semidirect
14. #7892C2semidirect
15. #7850C2semidirect

Maximal subgroups

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

subgroupindex
#213882
#213782
C4⋊C4×M162
C2×#15742
C2×#14952
#95482
#95472
#95462
#95452
#95442
#95432
#95422
#95412
#78922
#78502