Finite Group Databasebeta

← #164672

Subgroups — #164672

Order 512.

1305 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36258C211yesmaximal, characteristic
2#22530C211yesmaximal, characteristic, complemented
3#20630C211yesmaximal, characteristic, complemented
4#7714C212yesmaximal, complemented
5#7684C211yesmaximal, characteristic, complemented
6#7664C211yesmaximal, characteristic, complemented
7#7658C212yesmaximal, complemented
8#7640C212yesmaximal, complemented
9#7616C211yesmaximal, characteristic, complemented
10#7572C212yesmaximal, complemented
11#7565C211yesmaximal, characteristic, complemented
12C2×#546—21no—
13C2×D42C411yescharacteristic
14C2×C4×#41C2211yescharacteristic
15C2×C8×Q8—21no—
16C2×#424—21no—
17C2×#408—21no—
18#2856C2211yescharacteristic
19#2854C2212yes—
20#2548C411yescharacteristic
21#2538C411yescharacteristic
22#2426—21no—
23#2424—22no—
24#2416C412yes—
25#2414—21no—
26#2411—22no—
27#2408C412yes—
28D4×C4⋊C4C411yescharacteristic
29D4×C22⋊C4—21no—
30C4×#121C2211yescharacteristic
31C2×#390C2212yes—
32#1954—21no—
33#1954—22no—
34#1798—22no—
35#1797C2212yescomplemented
36#1797C2211yescharacteristic, complemented
37#1797C2212yescomplemented
38#1797C2212yescomplemented
39C4×C22⋊C8C2211yescharacteristic, complemented
40C4×C22⋊C8C2212yescomplemented
41#1728C2211yescharacteristic, complemented
42C2×#336C2211yescharacteristic, complemented
43#1271C2211yescharacteristic, complemented
44#1271C2212yescomplemented
45#1271C2211yescharacteristic, complemented
46#1270C2211yescharacteristic
47#1270C2211yescharacteristic
48#1270C2212yes—
49#1270C2212yescomplemented
50#1270C2212yescomplemented
51#1270C2212yescomplemented
52#1270C2212yescomplemented
53C22×#41D411yescharacteristic
54C23×D4—21no—
55C23×D4C2×C412yes—
56C22×M16—21no—
57C23×C8—21no—
58#546—44no—
59#546—44no—
60D42—44no—
61C2×#127C2×C411yescharacteristic
62C2×#124—21no—
63C2×#123—22no—
64C2×C22⋊Q8—21no—
65C2×#121—22no—
66C2×#121—21no—
67C2×#121—21no—
68C2×#121—21no—
69C2×#121C2×C411yescharacteristic
70C2×#121C2×C412yes—
71C2×#121C2×C412yes—
72C2×#120—22no—
73C2×#120—22no—
74C4×#41—44no—
75C2×C4×Q8C2×C411yescharacteristic
76C2×C4×Q8D411yescharacteristic
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×C4×D4C2×C411yescharacteristic
80C2×C4×D4C2×C411yescharacteristic
81C2×C4×D4C2×C412yes—
82C2×C4×D4D411yescharacteristic
83C2×C4×D4D412yes—
84C2×#117D411yescharacteristic
85C2×#117D412yes—
86C22×C4⋊C4C2×C412yes—
87C22×C22⋊C4—22no—
88C22×C22⋊C4—22no—
89C22×C42C2312yes—
90C22×C42C2311yescharacteristic
91C8×Q8—44no—
92#424—44no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—22no—
97#408—44no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8—22no—
106#396—22no—
107#396—22no—
108#394—22no—
109#394—22no—
110#394—22no—
111#392—22no—
112#390—21no—
113#390—21no—
114#390—21no—
115#390—22no—
116#390—22no—
117#390C2×C412yes—
118#390C2×C412yes—
119#390C2×C412yes—
120#390C2312yes—
121#390C2311yescharacteristic
122#389—21no—
123#389C2×C412yes—
124#389C2311yescharacteristic
125#388—22no—
126#388—22no—
127#388—21no—
128#386—22no—
129#386—21no—
130#386—22no—
131#385—22no—
132#384C2×C411yescharacteristic
133#380—21no—
134#380C2×C412yes—
135#380C2×C411yescharacteristic
136#379—22no—
137C4×C4⋊C4C2×C411yescharacteristic
138C4×C22⋊C4—21no—
139#336—21no—
140#336—21no—
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143#336C2312yes—
144#336C2311yescharacteristic
145#336C2312yes—
146#336C2311yescharacteristic
147C25—22no—
148C2×#41—42no—
149C2×#41—42no—
150C2×#41D814yes—
151C2×#41SD1614yes—
152C22×Q8C22⋊C411yescharacteristic
153C22×D4—24no—
154C22×D4—22no—
155C22×D4—21no—
156C22×D4—22no—
157C22×D4—21no—
158C22×D4—22no—
159C22×D4—21no—
160C22×D4—42no—
161C22×D4—44no—
162C22×D4—44no—
163C22×D4—44no—
164C22×D4—44no—
165C22×D4—82no—
166C22×D4C22⋊C411yescharacteristic
167C22×D4C22⋊C412yes—
168C22×D4C22×C412yes—
169C22×D4C22×C411yescharacteristic
170C23×C4—22no—
171C23×C4—21no—
172C23×C4C22×C412yes—
173C23×C4C22×C411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D412yes—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D412yes—
178C2×M16—42no—
179C2×M16—42no—
180C2×M16—42no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8—21no—
186C22×C8—22no—
187C22×C8—42no—
188C22×C8—42no—
189C22×C8—42no—
190#127—42no—
191#124—44no—
192#123—44no—
193#123—44no—
194C22⋊Q8—44no—
195#121—22no—
196#121—41no—
197#121—48no—
198#121—42no—
199#121—42no—
200#121—44no—
201#121—44no—
202#121—81no—
203#121—81no—
204#121—82no—
205#120—48no—
206#120—48no—
207C4×Q8—24no—
208C4×Q8—42no—
209C4×D4—24no—
210C4×D4—44no—
211C4×D4—42no—
212C4×D4—42no—
213C4×D4—44no—
214C4×D4—44no—
215C4×D4—42no—
216C4×D4—42no—
217#117—24no—
218#117—44no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4C22⋊C412yes—
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22⋊C412yes—
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C412yes—
232C2×C4⋊C4C22×C411yescharacteristic
233C2×C22⋊C4—22no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
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×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—82no—
252C2×C22⋊C4—82no—
253C2×C22⋊C4C22⋊C412yes—
254C2×C22⋊C4C22⋊C412yes—
255C2×C22⋊C4C22⋊C412yes—
256C2×C42—21no—
257C2×C42—44no—
258C2×C42—42no—
259C2×C42C22×C411yescharacteristic
260C2×C42C22×C412yes—
261C2×C42C2×D412yes—
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D412yes—
264C2×C42C2×D411yescharacteristic
265C2×C42C2411yescharacteristic
266C4⋊C8—44no—
267C4⋊C8—44no—
268C4⋊C8—44no—
269C4⋊C8—42no—
270C22⋊C8—44no—
271C22⋊C8—42no—
272C22⋊C8—44no—
273C22⋊C8—44no—
274C22⋊C8—82no—
275C22⋊C8—81no—
276C4×C8—44no—
277C4×C8—42no—
278#95—21no—
279#95—21no—
280#95—22no—
281#95—22no—
282#95—21no—
283C24—21no—
284C24—22no—
285C24—44no—
286C24—44no—
287C24—44no—
288C24—44no—
289C24—82no—
290C24—82no—
291C24—82no—
292C24C2×C22⋊C412yes—
293C24C2×C22⋊C411yescharacteristic
294#41—48no—
295#41—48no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8#10214yes—
299C2×D4—22no—
300C2×D4—21no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—44no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—44no—
314C2×D4—44no—
315C2×D4—42no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—82no—
323C2×D4—81no—
324C2×D4—81no—
325C2×D4—88no—
326C2×D4—88no—
327C2×D4—81no—
328C2×D4—82no—
329C2×D4—82no—
330C2×D4—816no—
331C2×D4#10214yes—
332C2×D4#10218yes—
333C2×D4#13112yes—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—44no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—82no—
363C22×C4—82no—
364C22×C4C2×C22⋊C412yes—
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C412yes—
368C22×C4C2×C22⋊C412yes—
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×D812yes—
371C22×C4C2×D814yes—
372C22×C4C2×SD1612yes—
373C22×C4C2×SD1614yes—
374C22×C4C23×C411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377M16—84no—
378C2×C8—22no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—42no—
382C2×C8—41no—
383C2×C8—42no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—42no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—42no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—42no—
394C2×C8—41no—
395C2×C8—42no—
396C2×C8—84no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—42no—
402C4⋊C4—41no—
403C4⋊C4—42no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—42no—
408C4⋊C4—44no—
409C4⋊C4—82no—
410C4⋊C4—81no—
411C4⋊C4#9914yes—
412C22⋊C4—24no—
413C22⋊C4—24no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—44no—
418C22⋊C4—42no—
419C22⋊C4—41no—
420C22⋊C4—42no—
421C22⋊C4—41no—
422C22⋊C4—42no—
423C22⋊C4—41no—
424C22⋊C4—42no—
425C22⋊C4—41no—
426C22⋊C4—42no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—82no—
432C22⋊C4—82no—
433C22⋊C4—82no—
434C22⋊C4—82no—
435C42—24no—
436C42—22no—
437C42—24no—
438C42—22no—
439C42—41no—
440C42—42no—
441C42—41no—
442C42—42no—
443C23—21no—
444C23—22no—
445C23—21no—
446C23—22no—
447C23—22no—
448C23—21no—
449C23—22no—
450C23—21no—
451C23—21no—
452C23—22no—
453C23—21no—
454C23—22no—
455C23—22no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—88no—
470C23—88no—
471C23—88no—
472C23—88no—
473C23—164no—
474C23—164no—
475C23—164no—
476C23C2×#10214yes—
477C23C2×#10212yes—
478C23C22×C22⋊C411yescharacteristic
479Q8—28no—
480D4—28no—
481D4—48no—
482D4—42no—
483D4—42no—
484D4—88no—
485D4—82no—
486D4—82no—
487D4—84no—
488D4—88no—
489D4—84no—
490D4—162no—
491D4—164no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—82no—
532C2×C4—81no—
533C2×C4—82no—
534C2×C4—81no—
535C2×C4—84no—
536C2×C4—164no—
537C2×C4C2×#9914yes—
538C2×C4C2×#9912yes—
539C2×C4C2×#10214yes—
540C2×C4C2×#10212yes—
541C2×C4#42112yes—
542C2×C4C2×#13111yescharacteristic
543C2×C4#56811yescharacteristic
544C2×C4C22×D811yescharacteristic
545C2×C4C22×SD1611yescharacteristic
546C2×C4#57611yescharacteristic
547C2×C4#57711yescharacteristic
548C8—42no—
549C8—81no—
550C8—82no—
551C8—81no—
552C8—81no—
553C8—82no—
554C22—24no—
555C22—22no—
556C22—22no—
557C22—24no—
558C22—22no—
559C22—24no—
560C22—42no—
561C22—41no—
562C22—41no—
563C22—42no—
564C22—41no—
565C22—42no—
566C22—41no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—1616no—
583C22#288011yescharacteristic
584C22C22×#9911yescharacteristic
585C22#288811yescharacteristic
586C22C22×#10211yescharacteristic
587C22#289711yescharacteristic
588C22C2×#42111yescharacteristic
589C22#290111yescharacteristic
590C4—21no—
591C4—21no—
592C4—24no—
593C4—21no—
594C4—21no—
595C4—24no—
596C4—21no—
597C4—21no—
598C4—41no—
599C4—42no—
600C4—41no—
601C4—81no—
602C4—82no—
603C4—81no—
604C4—81no—
605C4#151914yes—
606C2—42no—
607C2—44no—
608C2—82no—
609C2—162no—
610C2#1741111yescharacteristic
611C2#1750711yescharacteristic
612C2#1752211yescharacteristic
613C2C2×#151911yescharacteristic
614C2#1761111yescharacteristic
615C2#1768111yescharacteristic
616C2#1772011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36258C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22530C2semidirect
3. #20630C2semidirect
4. #7714C2semidirect
5. #7684C2semidirect
6. #7664C2semidirect
7. #7658C2semidirect
8. #7640C2semidirect
9. #7616C2semidirect
10. #7572C2semidirect
11. #7565C2semidirect
35. #1797C22semidirect
36. #1797C22semidirect
37. #1797C22semidirect
38. #1797C22semidirect
39. C4×C22⋊C8C22semidirect
40. C4×C22⋊C8C22semidirect
41. #1728C22semidirect
42. C2×#336C22semidirect
43. #1271C22semidirect
44. #1271C22semidirect
45. #1271C22semidirect
49. #1270C22semidirect
50. #1270C22semidirect
51. #1270C22semidirect
52. #1270C22semidirect

Maximal subgroups

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

subgroupindex
#362582
#225302
#206302
#77142
#76842
#76642
#76582
#76402
#76162
#75722
#75652