Finite Group Databasebeta

← #204887

Subgroups — #204887

Order 512.

1232 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35293C211yesmaximal, characteristic
2#21978C211yesmaximal, characteristic, complemented
3#20431C211yesmaximal, characteristic, complemented
4C2×#1560C211yesmaximal, characteristic, complemented
5C2×#1557C211yesmaximal, characteristic, complemented
6C2×#1551C211yesmaximal, characteristic, complemented
7C2×#1547C211yesmaximal, characteristic, complemented
8#9710C211yesmaximal, characteristic, complemented
9#9707C211yesmaximal, characteristic, complemented
10#9706C211yesmaximal, characteristic, complemented
11#9701C211yesmaximal, characteristic, complemented
12#9699C211yesmaximal, characteristic, complemented
13#9698C211yesmaximal, characteristic, complemented
14#9695C211yesmaximal, characteristic, complemented
15#9693C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#532C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C2×#405—21no—
23#2867C2211yescharacteristic
24#2862C2211yescharacteristic
25#2858C2211yescharacteristic
26#2858C2211yescharacteristic
27#2619—21no—
28#2619—21no—
29#2544—22no—
30#2544C414yes—
31#2431—22no—
32#2431C414yes—
33#2328—21no—
34#2328—21no—
35C4×#117C2211yescharacteristic
36#1940—22no—
37#1851—22no—
38#1842C2211yescharacteristic
39#1842C2211yescharacteristic
40#1831C2211yescharacteristic
41#1831C2211yescharacteristic
42#1768—21no—
43#1768—21no—
44#1767C2211yescharacteristic
45#1767C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48#1729—21no—
49#1560—24no—
50#1557—24no—
51#1551—24no—
52#1547—24no—
53C2×#333C2211yescharacteristic
54C2×#333C2211yescharacteristic
55C2×#333C2211yescharacteristic
56C2×#333C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×#332C2211yescharacteristic
61C2×#331C2211yescharacteristic
62C2×#331C2211yescharacteristic
63C2×#331C2211yescharacteristic
64C2×#331C2211yescharacteristic
65C2×#331C2211yescharacteristic
66C2×#331C2211yescharacteristic
67C2×#331C2211yescharacteristic
68C2×#331C2211yescharacteristic
69C22×#41—21no—
70C22×#41C2×C412yes—
71C22×M16—21no—
72C22×M16—21no—
73C23×C8—21no—
74#532—22no—
75#532—22no—
76#532—22no—
77#532—22no—
78#532—22no—
79#532—22no—
80#532—22no—
81#532—22no—
82C2×C4⋊Q8C2311yescharacteristic
83C2×C4⋊Q8C2311yescharacteristic
84C2×#127—21no—
85C2×#127—21no—
86C2×#127—22no—
87C2×#127C2×C414yes—
88C2×#127C2311yescharacteristic
89C2×#127C2311yescharacteristic
90C2×#124—21no—
91C2×#124—22no—
92C2×#124C2×C412yes—
93C2×#124C2×C414yes—
94C2×#121—21no—
95C2×#121—21no—
96C2×#121—21no—
97C2×#121C2×C412yes—
98C2×C4×D4—21no—
99C2×C4×D4—21no—
100C2×C4×D4—21no—
101C2×C4×D4C2×C412yes—
102C2×#117D411yescharacteristic
103C2×#117D411yescharacteristic
104C22×C42C2311yescharacteristic
105#432—22no—
106#432—22no—
107#423—22no—
108#423—22no—
109#423—22no—
110#423D414yescomplemented
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—22no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—22no—
116C2×C4⋊C8—22no—
117C2×C4⋊C8D414yescomplemented
118C2×C4⋊C8C2311yescharacteristic
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C22⋊C8—21no—
127#405—44no—
128C2×#97—21no—
129C2×#97—21no—
130C2×#97—21no—
131C2×C4×C8—21no—
132C2×C4×C8—21no—
133#394—24no—
134#394—24no—
135#392—24no—
136#392—24no—
137#391—22no—
138#391C2×C414yes—
139#390—21no—
140#390—21no—
141#390—21no—
142#390—21no—
143#390—22no—
144#390C2×C414yes—
145#386—24no—
146#386—24no—
147#386—24no—
148#386—24no—
149#382—21no—
150#382—21no—
151C4×C4⋊C4C2×C412yes—
152C4×C4⋊C4D412yes—
153C4×C22⋊C4C2×C412yes—
154C4×C22⋊C4D412yes—
155#376D411yescharacteristic
156#376D411yescharacteristic
157C43C2311yescharacteristic
158C43C2311yescharacteristic
159#336—22no—
160#336—22no—
161#336—21no—
162#336—21no—
163#336—21no—
164#336—21no—
165#333—22no—
166#333—22no—
167#333—22no—
168#333—22no—
169#332—22no—
170#332—22no—
171#332—22no—
172#332—22no—
173#331—22no—
174#331—22no—
175#331—22no—
176#331—22no—
177#331—22no—
178#331—22no—
179#331—22no—
180#331—22no—
181C2×#41—24no—
182C2×#41—24no—
183C2×#41—42no—
184C2×#41—42no—
185C2×#41—82no—
186C2×#41—82no—
187C2×#41—84no—
188C22×Q8—21no—
189C22×Q8C22×C412yes—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—21no—
195C22×D4—21no—
196C22×D4—21no—
197C22×D4—21no—
198C22×D4C22×C412yes—
199C22×D4C22×C412yes—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4C22×C411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C2×M16—24no—
206C2×M16—42no—
207C2×M16—42no—
208C2×M16—42no—
209C2×M16—82no—
210C22×C8—24no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—42no—
220C22×C8—42no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8—21no—
225C4⋊Q8—21no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229#127—21no—
230#127—21no—
231#127—21no—
232#127—21no—
233#127—21no—
234#127—21no—
235#127—21no—
236#127—21no—
237#127—42no—
238#127—42no—
239#127—42no—
240#127—42no—
241#127—82no—
242#127—82no—
243#127—84no—
244#124—24no—
245#124—24no—
246#124—42no—
247#124—42no—
248#124—82no—
249#124—82no—
250#124—84no—
251#121—24no—
252#121—24no—
253#121—42no—
254#121—42no—
255#121—42no—
256#121—42no—
257#121—42no—
258#121—42no—
259C4×D4—24no—
260C4×D4—24no—
261C4×D4—42no—
262C4×D4—42no—
263C4×D4—42no—
264C4×D4—42no—
265C4×D4—42no—
266C4×D4—42no—
267#117—22no—
268#117—22no—
269#117—41no—
270#117—41no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4C22⋊C412yes—
275C2×C4⋊C4C22⋊C412yes—
276C2×C4⋊C4C22×C412yes—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4C22⋊C412yes—
286C2×C22⋊C4C22⋊C412yes—
287C2×C42C22×C412yes—
288C2×C42C22×C412yes—
289C2×C42C22×C411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D412yes—
294C2×C42C2×D412yes—
295C2×C42C2×D412yes—
296C2×C42C2×D412yes—
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42#4112yes—
301C2×C42#4112yes—
302C2×C42C2411yescharacteristic
303C4⋊C8—24no—
304C4⋊C8—24no—
305C4⋊C8—21no—
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8—21no—
309C4⋊C8—21no—
310C4⋊C8—21no—
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8—21no—
314C4⋊C8—21no—
315C4⋊C8—21no—
316C4⋊C8—21no—
317C4⋊C8—42no—
318C4⋊C8—82no—
319C4⋊C8—82no—
320C4⋊C8C2×D412yes—
321C4⋊C8C2×D412yes—
322C4⋊C8C2×D412yes—
323C4⋊C8C2×D412yes—
324C22⋊C8—42no—
325#97—42no—
326#97—42no—
327#97—42no—
328C4×C8—42no—
329C4×C8—42no—
330#95—24no—
331#95—24no—
332#95C22⋊C412yes—
333#95C22⋊C412yes—
334#95C22⋊C412yes—
335#95C22⋊C412yes—
336C24—22no—
337C24—22no—
338C24C2×C22⋊C411yescharacteristic
339#41—88no—
340#41—84no—
341#41—84no—
342#41—88no—
343#41—88no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—44no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—42no—
350C2×Q8—42no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—44no—
356C2×D4—44no—
357C2×D4—48no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—48no—
381C2×D4—84no—
382C2×D4—84no—
383C2×D4—84no—
384C2×D4—84no—
385C2×D4—84no—
386C2×D4—84no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—42no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—84no—
399C22×C4—84no—
400C22×C4C2×C22⋊C412yes—
401C22×C4C2×C22⋊C412yes—
402C22×C4C2×C22⋊C412yes—
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C412yes—
407C22×C4C4×D412yes—
408C22×C4C4×D412yes—
409C22×C4#12112yes—
410C22×C4#12112yes—
411C22×C4#12412yes—
412C22×C4#12712yes—
413C22×C4C23×C411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416M16—48no—
417M16—84no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—48no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—42no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C22⋊C4—24no—
450C22⋊C4—42no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—84no—
456C22⋊C4—84no—
457C22⋊C4—84no—
458C22⋊C4—84no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—42no—
470C42—42no—
471C42C4×D412yes—
472C42C4×D412yes—
473C42#12112yes—
474C42#12112yes—
475C42#12412yes—
476C42#12712yes—
477C42AGL(1,ℤ/8)12yes—
478C42#13712yes—
479C42C22×D411yescharacteristic
480C42C22×D411yescharacteristic
481C42C2×#4111yescharacteristic
482C42C2×#4111yescharacteristic
483C23—22no—
484C23—21no—
485C23—21no—
486C23—22no—
487C23—44no—
488C23—44no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23—44no—
495C23—42no—
496C23—42no—
497C23#39012yes—
498C23C22×C22⋊C411yescharacteristic
499Q8—82no—
500Q8—82no—
501Q8—82no—
502Q8—81no—
503Q8—81no—
504D4—82no—
505D4—82no—
506D4—82no—
507D4—82no—
508D4—88no—
509D4—88no—
510D4—82no—
511D4—82no—
512D4—81no—
513D4—81no—
514D4—81no—
515D4—81no—
516D4—81no—
517D4—81no—
518D4—81no—
519D4—81no—
520D4—81no—
521D4—81no—
522D4—81no—
523D4—81no—
524D4—168no—
525D4—168no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—88no—
556C2×C4—88no—
557C2×C4#39012yes—
558C2×C4#39012yes—
559C2×C4#39012yes—
560C2×C4#41714yes—
561C2×C4C2×C4×D411yescharacteristic
562C2×C4C2×C4×D411yescharacteristic
563C2×C4C2×#12111yescharacteristic
564C2×C4C2×#12111yescharacteristic
565C2×C4C2×#12411yescharacteristic
566C2×C4C2×#12711yescharacteristic
567C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
568C2×C4C2×#13711yescharacteristic
569C2×C4#57511yescharacteristic
570C2×C4#57511yescharacteristic
571C2×C4#57612yes—
572C2×C4#57712yes—
573C8—42no—
574C8—42no—
575C8—42no—
576C8—42no—
577C8—81no—
578C8—81no—
579C8—81no—
580C8—81no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—42no—
585C22—42no—
586C22—82no—
587C22—82no—
588C22—84no—
589C22—84no—
590C22—84no—
591C22—84no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22C2×#39011yescharacteristic
597C22#288812yes—
598C22C2×#41711yescharacteristic
599C22#289611yescharacteristic
600C22#289712yes—
601C4—22no—
602C4—24no—
603C4—22no—
604C4—22no—
605C4—21no—
606C4—21no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—82no—
611C4—82no—
612C4#303611yescharacteristic
613C4#303611yescharacteristic
614C4#303711yescharacteristic
615C4#303711yescharacteristic
616C4#313612yes—
617C4#315411yescharacteristic
618C4#315511yescharacteristic
619C2—42no—
620C2—162no—
621C2—162no—
622C2#1776212yes—
623C2#1776612yes—
624C2#2120611yescharacteristic
625C2#2121911yescharacteristic
626C2#2125811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#35293C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21978C2semidirect
3. #20431C2semidirect
4. C2×#1560C2semidirect
5. C2×#1557C2semidirect
6. C2×#1551C2semidirect
7. C2×#1547C2semidirect
8. #9710C2semidirect
9. #9707C2semidirect
10. #9706C2semidirect
11. #9701C2semidirect
12. #9699C2semidirect
13. #9698C2semidirect
14. #9695C2semidirect
15. #9693C2semidirect
110. #423D4semidirect
117. C2×C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
#352932
#219782
#204312
C2×#15602
C2×#15572
C2×#15512
C2×#15472
#97102
#97072
#97062
#97012
#96992
#96982
#96952
#96932