Finite Group Databasebeta

← #196653

Subgroups — #196653

Order 512.

1568 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2906C211yesmaximal, characteristic, complemented
2#35273C211yesmaximal, characteristic
3#21847C211yesmaximal, characteristic, complemented
4C2×#1547C211yesmaximal, characteristic, complemented
5C2×#1540C211yesmaximal, characteristic, complemented
6#9141C212yesmaximal, complemented
7#9139C212yesmaximal, complemented
8#9133C212yesmaximal, complemented
9#9131C212yesmaximal, complemented
10#8043C211yesmaximal, characteristic, complemented
11C2×#1296C211yesmaximal, characteristic, complemented
12C22×#131—21no—
13C2×#537—21no—
14C2×#532C2211yescharacteristic
15C2×C4×#41C2211yescharacteristic
16C2×#431C2211yescharacteristic
17#2906—24no—
18#2906—24no—
19C2×#424—21no—
20C2×#424—22no—
21C2×#423C2212yes—
22C2×#423C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×C4⋊C8C2212yes—
25C22×C4⋊C8C2211yescharacteristic
26#2857C2212yes—
27#2856C2212yes—
28#2620—22no—
29#2538—21no—
30#2538C412yes—
31#2537—21no—
32#2537C412yes—
33#2416—22no—
34#2416C414yes—
35#2311—21no—
36C2×#384C2211yescharacteristic
37#1984—21no—
38#1954—21no—
39#1869—22no—
40#1797C2212yes—
41#1797C2212yes—
42#1547—24no—
43#1540—24no—
44C2×#332C2211yescharacteristic
45C2×#332C2211yescharacteristic
46C2×#332C2212yes—
47C2×#331C2211yescharacteristic
48C2×#331C2211yescharacteristic
49C2×#331C2212yes—
50#1296C2212yes—
51#1296C2212yes—
52#1296C2212yes—
53#1296C2212yes—
54C22×#41—21no—
55C22×#41D411yescharacteristic
56C2×#131—28no—
57C2×#131—44no—
58C2×#131—44no—
59C22×M16—22no—
60C22×M16—21no—
61C23×C8—22no—
62C23×C8—21no—
63C23×C8—21no—
64#537—44no—
65#537—44no—
66#532—22no—
67#532—22no—
68#532—22no—
69#532—22no—
70C2×C4⋊Q8C2311yescharacteristic
71C2×#127C2311yescharacteristic
72C2×#124—21no—
73C2×C22⋊Q8—22no—
74C2×C22⋊Q8—22no—
75C2×#121—24no—
76C2×#121—22no—
77C2×#121—22no—
78C2×#121—21no—
79C2×#121—22no—
80C2×#121C2×C412yes—
81C2×#121C2×C414yes—
82C4×#41—22no—
83C4×#41—22no—
84C4×#41—22no—
85C4×#41—22no—
86C2×C4×Q8C2×C411yescharacteristic
87C2×C4×Q8D411yescharacteristic
88C2×C4×D4—21no—
89C2×C4×D4C2×C412yes—
90C2×C4×D4C2×C412yes—
91C2×C4×D4C2×C411yescharacteristic
92C2×C4×D4D412yes—
93C2×C4×D4D411yescharacteristic
94C2×#117D411yescharacteristic
95C2×#117D412yes—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4C2×C412yes—
99C22×C42C2312yes—
100C22×C42C2311yescharacteristic
101#431—22no—
102#431—22no—
103#424—28no—
104#424—216no—
105#423—22no—
106#423—24no—
107#423—24no—
108#423D414yescomplemented
109C2×C4⋊C8—22no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—22no—
112C2×C4⋊C8—22no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—24no—
115C2×C4⋊C8—24no—
116C2×C4⋊C8—22no—
117C2×C4⋊C8—22no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8D414yescomplemented
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2312yes—
125C2×C4⋊C8C2311yescharacteristic
126C2×C22⋊C8—21no—
127C2×C22⋊C8—22no—
128C2×C22⋊C8—22no—
129C2×C4×C8—21no—
130#396—24no—
131#396—24no—
132#394—24no—
133#394—24no—
134#391—22no—
135#390—22no—
136#390—22no—
137#390C2×C414yes—
138#389—22no—
139#389C2×C414yes—
140#386—24no—
141#384—21no—
142#384—21no—
143#384C2×C412yes—
144#384C2312yes—
145#380—22no—
146#380—22no—
147#380—22no—
148#380C2×C414yes—
149#336—22no—
150#336—22no—
151#332—22no—
152#332—22no—
153#332—24no—
154#331—22no—
155#331—22no—
156#331—24no—
157C2×#41—22no—
158C2×#41—22no—
159C2×#41—22no—
160C2×#41—22no—
161C2×#41—42no—
162C2×#41—42no—
163C2×#41—42no—
164C2×#41—42no—
165C2×#41—42no—
166C2×#41—42no—
167C2×#41D814yes—
168C2×#41SD1614yes—
169C22×Q8—21no—
170C22×Q8C22⋊C411yescharacteristic
171C22×D4—22no—
172C22×D4—24no—
173C22×D4—21no—
174C22×D4—21no—
175C22×D4C22⋊C412yes—
176C22×D4C22⋊C411yescharacteristic
177C22×D4C22×C412yes—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4C22×C412yes—
181C23×C4C22×C411yescharacteristic
182C23×C4C2×D412yes—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D412yes—
185C23×C4C2×D411yescharacteristic
186#131—432no—
187C2×M16—28no—
188C2×M16—24no—
189C2×M16—44no—
190C2×M16—44no—
191C2×M16—42no—
192C2×M16—42no—
193C22×C8—28no—
194C22×C8—24no—
195C22×C8—21no—
196C22×C8—22no—
197C22×C8—21no—
198C22×C8—22no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—44no—
202C22×C8—44no—
203C22×C8—42no—
204C22×C8—42no—
205C22×C8—42no—
206C22×C8—81no—
207C22×C8—81no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212#127—21no—
213#127—21no—
214#127—21no—
215#127—21no—
216#124—42no—
217#124—42no—
218C22⋊Q8—48no—
219C22⋊Q8—44no—
220C22⋊Q8—44no—
221#121—24no—
222#121—24no—
223#121—48no—
224#121—416no—
225#121—42no—
226#121—42no—
227#121—44no—
228#121—44no—
229#121—82no—
230#121—82no—
231#121—84no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8—22no—
235C4×Q8—22no—
236C4×D4—24no—
237C4×D4—24no—
238C4×D4—24no—
239C4×D4—24no—
240C4×D4—22no—
241C4×D4—22no—
242C4×D4—24no—
243C4×D4—24no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—42no—
247C4×D4—42no—
248#117—22no—
249#117—22no—
250#117—24no—
251#117—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—81no—
262C2×C4⋊C4—81no—
263C2×C4⋊C4—82no—
264C2×C4⋊C4C22⋊C411yescharacteristic
265C2×C4⋊C4C22⋊C412yes—
266C2×C4⋊C4C22⋊C411yescharacteristic
267C2×C4⋊C4C22⋊C412yes—
268C2×C4⋊C4C22×C412yes—
269C2×C4⋊C4C22×C414yes—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4C22⋊C412yes—
277C2×C22⋊C4C22⋊C412yes—
278C2×C22⋊C4C22⋊C412yes—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42C22×C412yes—
284C2×C42C22×C411yescharacteristic
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42#4112yes—
292C2×C42#4112yes—
293C2×C42C2411yescharacteristic
294C4⋊C8—28no—
295C4⋊C8—28no—
296C4⋊C8—24no—
297C4⋊C8—24no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300C4⋊C8—22no—
301C4⋊C8—22no—
302C4⋊C8—21no—
303C4⋊C8—21no—
304C4⋊C8—22no—
305C4⋊C8—21no—
306C4⋊C8—21no—
307C4⋊C8—42no—
308C4⋊C8C2×D412yes—
309C4⋊C8C2×D414yes—
310C4⋊C8C2×D412yes—
311C22⋊C8—42no—
312C22⋊C8—82no—
313C22⋊C8—82no—
314C4×C8—42no—
315#95—24no—
316#95—24no—
317C24—22no—
318C24C2×C22⋊C412yes—
319C24C2×C22⋊C411yescharacteristic
320#41—28no—
321#41—28no—
322#41—28no—
323#41—28no—
324#41—84no—
325#41—84no—
326#41—84no—
327#41—84no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8#10214yes—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—48no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—84no—
362C2×D4—84no—
363C2×D4#10218yes—
364C2×D4#10214yes—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4C2×C22⋊C412yes—
394C22×C4C2×C22⋊C412yes—
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C412yes—
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C4×D412yes—
400C22×C4C4×D412yes—
401C22×C4#12012yes—
402C22×C4#12112yes—
403C22×C4#12112yes—
404C22×C4#12312yes—
405C22×C4C2×D814yes—
406C22×C4C2×D812yes—
407C22×C4C2×SD1614yes—
408C22×C4C2×SD1612yes—
409C22×C4AGL(1,ℤ/8)12yes—
410C22×C4#13712yes—
411C22×C4C23×C411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414M16—416no—
415M16—48no—
416C2×C8—22no—
417C2×C8—24no—
418C2×C8—22no—
419C2×C8—416no—
420C2×C8—41no—
421C2×C8—42no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—42no—
425C2×C8—41no—
426C2×C8—48no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—42no—
430C2×C8—41no—
431C2×C8—42no—
432C2×C8—41no—
433C2×C8—42no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—84no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—44no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C4⋊C4—48no—
452C4⋊C4—82no—
453C4⋊C4—82no—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—48no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—48no—
461C22⋊C4—84no—
462C22⋊C4—84no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42C4×D412yes—
470C42C4×D412yes—
471C42#12012yes—
472C42#12112yes—
473C42#12112yes—
474C42#12312yes—
475C42C22×D411yescharacteristic
476C42C22×D411yescharacteristic
477C42C2×#4111yescharacteristic
478C42C2×#4111yescharacteristic
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23#38612yes—
497C23C2×#10214yes—
498C23C2×#10212yes—
499C23#41712yes—
500C23C22×C22⋊C411yescharacteristic
501Q8—28no—
502Q8—81no—
503Q8—81no—
504Q8—81no—
505Q8—81no—
506D4—216no—
507D4—28no—
508D4—82no—
509D4—82no—
510D4—88no—
511D4—81no—
512D4—81no—
513D4—81no—
514D4—81no—
515D4—81no—
516D4—81no—
517D4—81no—
518D4—81no—
519D4—168no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
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—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—88no—
557C2×C4—164no—
558C2×C4#38612yes—
559C2×C4#38612yes—
560C2×C4#38612yes—
561C2×C4C2×#10214yes—
562C2×C4C2×#10212yes—
563C2×C4#41712yes—
564C2×C4#44714yes—
565C2×C4#44814yes—
566C2×C4#48014yes—
567C2×C4#48114yes—
568C2×C4C2×C4×D411yescharacteristic
569C2×C4C2×C4×D411yescharacteristic
570C2×C4C2×#12011yescharacteristic
571C2×C4C2×#12111yescharacteristic
572C2×C4C2×#12111yescharacteristic
573C2×C4C2×#12311yescharacteristic
574C2×C4C22×D811yescharacteristic
575C2×C4C22×SD1611yescharacteristic
576C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
577C2×C4C2×#13711yescharacteristic
578C2×C4#57611yescharacteristic
579C2×C4#57711yescharacteristic
580C2×C4#57711yescharacteristic
581C2×C4#57811yescharacteristic
582C8—42no—
583C8—44no—
584C8—42no—
585C8—81no—
586C8—81no—
587C8—82no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—24no—
595C22—24no—
596C22—24no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22#187414yes—
609C22C2×#38611yescharacteristic
610C22#288811yescharacteristic
611C22#288911yescharacteristic
612C22C22×#10211yescharacteristic
613C22C2×#41711yescharacteristic
614C22#289711yescharacteristic
615C22#289811yescharacteristic
616C4—21no—
617C4—21no—
618C4—22no—
619C4—24no—
620C4—22no—
621C4—42no—
622C4—42no—
623C4—84no—
624C4—82no—
625C4#187414yes—
626C4C2×#44711yescharacteristic
627C4C2×#44811yescharacteristic
628C4#306011yescharacteristic
629C4#306111yescharacteristic
630C4#306211yescharacteristic
631C4#306311yescharacteristic
632C4C2×#48011yescharacteristic
633C4C2×#48111yescharacteristic
634C2—24no—
635C2—44no—
636C2—162no—
637C2#1772011yescharacteristic
638C2#1772111yescharacteristic
639C2#1776411yescharacteristic
640C2#1776511yescharacteristic
641C2C2×#187411yescharacteristic
642C2#2097611yescharacteristic
643C2#2099211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#35273C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2906C2semidirect
3. #21847C2semidirect
4. C2×#1547C2semidirect
5. C2×#1540C2semidirect
6. #9141C2semidirect
7. #9139C2semidirect
8. #9133C2semidirect
9. #9131C2semidirect
10. #8043C2semidirect
11. C2×#1296C2semidirect
108. #423D4semidirect
119. C2×C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29062
#352732
#218472
C2×#15472
C2×#15402
#91412
#91392
#91332
#91312
#80432
C2×#12962