Finite Group Databasebeta

← #204817

Subgroups — #204817

Order 512.

1204 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35293C211yesmaximal, characteristic
2#21972C211yesmaximal, characteristic, complemented
3#20470C211yesmaximal, characteristic, complemented
4C2×#1553C211yesmaximal, characteristic, complemented
5C2×#1551C211yesmaximal, characteristic, complemented
6C2×#1543C211yesmaximal, characteristic, complemented
7C2×#1540C211yesmaximal, 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×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22#2867C2211yescharacteristic
23#2862C2211yescharacteristic
24#2858C2211yescharacteristic
25#2858C2211yescharacteristic
26#2619—21no—
27#2619—21no—
28#2544—22no—
29#2544C414yes—
30#2431—22no—
31#2431C414yes—
32#2328—21no—
33#2328—21no—
34C4×#117C2211yescharacteristic
35#1939—22no—
36#1850—22no—
37#1842C2211yescharacteristic
38#1842C2211yescharacteristic
39#1831C2211yescharacteristic
40#1831C2211yescharacteristic
41#1767C2211yescharacteristic
42#1767C2211yescharacteristic
43C4×C4⋊C8C2211yescharacteristic
44C4×C4⋊C8C2211yescharacteristic
45#1750—21no—
46#1750—21no—
47#1746—21no—
48#1726—21no—
49#1553—24no—
50#1551—24no—
51#1543—24no—
52C2×#333C2211yescharacteristic
53C2×#333C2211yescharacteristic
54C2×#333C2211yescharacteristic
55C2×#333C2211yescharacteristic
56#1540—24no—
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—
71C23×C8—21no—
72#532—22no—
73#532—22no—
74#532—22no—
75#532—22no—
76#532—22no—
77#532—22no—
78#532—22no—
79#532—22no—
80C2×C4⋊Q8C2311yescharacteristic
81C2×C4⋊Q8C2311yescharacteristic
82C2×#127—21no—
83C2×#127—21no—
84C2×#127—22no—
85C2×#127C2×C414yes—
86C2×#127C2311yescharacteristic
87C2×#127C2311yescharacteristic
88C2×#124—21no—
89C2×#124—22no—
90C2×#124C2×C412yes—
91C2×#124C2×C414yes—
92C2×#121—21no—
93C2×#121—21no—
94C2×#121—21no—
95C2×#121C2×C412yes—
96C2×C4×D4—21no—
97C2×C4×D4—21no—
98C2×C4×D4—21no—
99C2×C4×D4C2×C412yes—
100C2×#117D411yescharacteristic
101C2×#117D411yescharacteristic
102C22×C42C2311yescharacteristic
103#432—22no—
104#432D414yescomplemented
105#431—22no—
106#431—22no—
107#431—22no—
108#431D414yescomplemented
109C2×C4⋊C8—22no—
110C2×C4⋊C8—22no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—22no—
113C2×C4⋊C8—22no—
114C2×C4⋊C8C2311yescharacteristic
115C2×C4⋊C8C2311yescharacteristic
116C2×C4⋊C8C2311yescharacteristic
117C2×C4⋊C8C2311yescharacteristic
118C2×C4⋊C8C2311yescharacteristic
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×#97—21no—
126C2×#97—21no—
127C2×#97—21no—
128C2×C4×C8—21no—
129C2×C4×C8—21no—
130C2×C4×C8—21no—
131C2×C4×C8—21no—
132#394—24no—
133#394—24no—
134#392—24no—
135#392—24no—
136#391—22no—
137#391C2×C414yes—
138#390—21no—
139#390—21no—
140#390—21no—
141#390—21no—
142#390—22no—
143#390C2×C414yes—
144#386—24no—
145#386—24no—
146#386—24no—
147#386—24no—
148#382—21no—
149#382—21no—
150C4×C4⋊C4C2×C412yes—
151C4×C4⋊C4D412yes—
152C4×C22⋊C4C2×C412yes—
153C4×C22⋊C4D412yes—
154#376D411yescharacteristic
155#376D411yescharacteristic
156C43C2311yescharacteristic
157C43C2311yescharacteristic
158#336—22no—
159#336—22no—
160#336—21no—
161#336—21no—
162#336—21no—
163#336—21no—
164#333—22no—
165#333—22no—
166#333—22no—
167#333—22no—
168#332—22no—
169#332—22no—
170#332—22no—
171#332—22no—
172#331—22no—
173#331—22no—
174#331—22no—
175#331—22no—
176#331—22no—
177#331—22no—
178#331—22no—
179#331—22no—
180C2×#41—24no—
181C2×#41—24no—
182C2×#41—42no—
183C2×#41—42no—
184C2×#41—82no—
185C2×#41—82no—
186C2×#41—84no—
187C22×Q8—21no—
188C22×Q8C22×C412yes—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—21no—
194C22×D4—21no—
195C22×D4—21no—
196C22×D4—21no—
197C22×D4C22×C412yes—
198C22×D4C22×C412yes—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4C22×C411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—42no—
213C22×C8—82no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222#127—21no—
223#127—21no—
224#127—21no—
225#127—21no—
226#127—21no—
227#127—21no—
228#127—21no—
229#127—21no—
230#127—42no—
231#127—42no—
232#127—42no—
233#127—42no—
234#127—82no—
235#127—82no—
236#127—84no—
237#124—24no—
238#124—24no—
239#124—42no—
240#124—42no—
241#124—82no—
242#124—82no—
243#124—84no—
244#121—24no—
245#121—24no—
246#121—42no—
247#121—42no—
248#121—42no—
249#121—42no—
250#121—42no—
251#121—42no—
252C4×D4—24no—
253C4×D4—24no—
254C4×D4—42no—
255C4×D4—42no—
256C4×D4—42no—
257C4×D4—42no—
258C4×D4—42no—
259C4×D4—42no—
260#117—22no—
261#117—22no—
262#117—41no—
263#117—41no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4C22⋊C412yes—
268C2×C4⋊C4C22⋊C412yes—
269C2×C4⋊C4C22×C412yes—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4C22⋊C412yes—
279C2×C22⋊C4C22⋊C412yes—
280C2×C42C22×C412yes—
281C2×C42C22×C412yes—
282C2×C42C22×C411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42#4112yes—
294C2×C42#4112yes—
295C2×C42C2411yescharacteristic
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—21no—
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8—42no—
309C4⋊C8—82no—
310C4⋊C8—82no—
311C4⋊C8C2×D412yes—
312C4⋊C8C2×D412yes—
313C4⋊C8C2×D412yes—
314C4⋊C8C2×D412yes—
315C22⋊C8—24no—
316C22⋊C8—24no—
317C22⋊C8—42no—
318#97—24no—
319#97—42no—
320#97—42no—
321C4×C8—24no—
322C4×C8—42no—
323C4×C8—42no—
324C4×C8—42no—
325#95—24no—
326#95—24no—
327#95C22⋊C412yes—
328#95C22⋊C412yes—
329#95C22⋊C412yes—
330#95C22⋊C412yes—
331C24—22no—
332C24—22no—
333C24C2×C22⋊C411yescharacteristic
334#41—88no—
335#41—84no—
336#41—84no—
337#41—88no—
338#41—88no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—44no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—48no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
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—42no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—48no—
376C2×D4—84no—
377C2×D4—84no—
378C2×D4—84no—
379C2×D4—84no—
380C2×D4—84no—
381C2×D4—84no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—84no—
394C22×C4—84no—
395C22×C4C2×C22⋊C412yes—
396C22×C4C2×C22⋊C412yes—
397C22×C4C2×C22⋊C412yes—
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C412yes—
402C22×C4C4×D412yes—
403C22×C4C4×D412yes—
404C22×C4#12112yes—
405C22×C4#12112yes—
406C22×C4#12412yes—
407C22×C4#12712yes—
408C22×C4C23×C411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—84no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—42no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C22⋊C4—24no—
443C22⋊C4—42no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—84no—
449C22⋊C4—84no—
450C22⋊C4—84no—
451C22⋊C4—84no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—42no—
463C42—42no—
464C42C4×D412yes—
465C42C4×D412yes—
466C42#12112yes—
467C42#12112yes—
468C42#12412yes—
469C42#12712yes—
470C42#13512yes—
471C42#13512yes—
472C42C22×D411yescharacteristic
473C42C22×D411yescharacteristic
474C42C2×#4111yescharacteristic
475C42C2×#4111yescharacteristic
476C23—22no—
477C23—21no—
478C23—21no—
479C23—22no—
480C23—44no—
481C23—44no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—44no—
487C23—44no—
488C23—42no—
489C23—42no—
490C23#39012yes—
491C23C22×C22⋊C411yescharacteristic
492Q8—82no—
493Q8—82no—
494Q8—82no—
495Q8—81no—
496Q8—81no—
497D4—82no—
498D4—82no—
499D4—82no—
500D4—82no—
501D4—88no—
502D4—88no—
503D4—82no—
504D4—82no—
505D4—81no—
506D4—81no—
507D4—81no—
508D4—81no—
509D4—81no—
510D4—81no—
511D4—81no—
512D4—81no—
513D4—81no—
514D4—81no—
515D4—81no—
516D4—81no—
517D4—168no—
518D4—168no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
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—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—88no—
549C2×C4—88no—
550C2×C4#39012yes—
551C2×C4#39012yes—
552C2×C4#39012yes—
553C2×C4#41614yes—
554C2×C4C2×C4×D411yescharacteristic
555C2×C4C2×C4×D411yescharacteristic
556C2×C4C2×#12111yescharacteristic
557C2×C4C2×#12111yescharacteristic
558C2×C4C2×#12411yescharacteristic
559C2×C4C2×#12711yescharacteristic
560C2×C4C2×#13511yescharacteristic
561C2×C4C2×#13511yescharacteristic
562C2×C4#57511yescharacteristic
563C2×C4#57511yescharacteristic
564C2×C4#57612yes—
565C2×C4#57712yes—
566C8—42no—
567C8—42no—
568C8—42no—
569C8—42no—
570C8—81no—
571C8—81no—
572C8—81no—
573C8—81no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—42no—
578C22—42no—
579C22—82no—
580C22—82no—
581C22—84no—
582C22—84no—
583C22—84no—
584C22—84no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22C2×#39011yescharacteristic
590C22#288812yes—
591C22C2×#41611yescharacteristic
592C22#289611yescharacteristic
593C22#289712yes—
594C4—22no—
595C4—24no—
596C4—22no—
597C4—22no—
598C4—21no—
599C4—21no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—82no—
604C4—82no—
605C4#303511yescharacteristic
606C4#303511yescharacteristic
607C4#303511yescharacteristic
608C4#303511yescharacteristic
609C4#313612yes—
610C4#315411yescharacteristic
611C4#315511yescharacteristic
612C2—42no—
613C2—162no—
614C2—162no—
615C2#1772412yes—
616C2#1776612yes—
617C2#2120611yescharacteristic
618C2#2121911yescharacteristic
619C2#2125711yescharacteristic
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. #21972C2semidirect
3. #20470C2semidirect
4. C2×#1553C2semidirect
5. C2×#1551C2semidirect
6. C2×#1543C2semidirect
7. C2×#1540C2semidirect
8. #9710C2semidirect
9. #9707C2semidirect
10. #9706C2semidirect
11. #9701C2semidirect
12. #9699C2semidirect
13. #9698C2semidirect
14. #9695C2semidirect
15. #9693C2semidirect
104. #432D4semidirect
108. #431D4semidirect

Maximal subgroups

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

subgroupindex
#352932
#219722
#204702
C2×#15532
C2×#15512
C2×#15432
C2×#15402
#97102
#97072
#97062
#97012
#96992
#96982
#96952
#96932