Finite Group Databasebeta

← #188410

Subgroups — #188410

Order 512.

1152 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34127C211yesmaximal, characteristic
2#21974C211yesmaximal, characteristic, complemented
3C4×#424C211yesmaximal, characteristic, complemented
4C2×#1489C211yesmaximal, characteristic
5C2×#1476C211yesmaximal, characteristic
6#8570C212yesmaximal, complemented
7#8568C212yesmaximal, complemented
8#8565C212yesmaximal, complemented
9#8563C212yesmaximal, complemented
10#7897C211yesmaximal, characteristic, complemented
11#7896C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#424—21no—
15C2×#423C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2355C411yescharacteristic
19#2355C411yescharacteristic
20#2352—21no—
21#2352—21no—
22D4×C4⋊C4C411yescharacteristic
23D4×C4⋊C4C411yescharacteristic
24#2347C412yes—
25#2343—22no—
26#2339C412yes—
27#2295C2212yes—
28#2290—22no—
29#2285C2211yescharacteristic
30#2283—21no—
31C42×D4C2212yes—
32C4×#117C2211yescharacteristic
33C2×C4×C4⋊C4—21no—
34#1939—21no—
35#1939—21no—
36#1851—22no—
37#1842C2212yes—
38#1831C2212yes—
39#1768—22no—
40C4×C4⋊C8C2212yes—
41C4×C4⋊C8C2212yes—
42#1729—21no—
43#1489—24no—
44#1476—24no—
45C2×Q8⋊C8C2211yescharacteristic
46C2×Q8⋊C8C2211yescharacteristic
47C2×Q8⋊C8C2212yes—
48C2×D4⋊C8C2211yescharacteristic
49C2×D4⋊C8C2211yescharacteristic
50C2×D4⋊C8C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54#1279C2212yes—
55C22×#41C2×C411yescharacteristic
56C22×M16—21no—
57C22×M16—21no—
58C23×C8—21no—
59C4×#41—22no—
60C4×#41—22no—
61C4×#41D414yes—
62C4×#41Q814yes—
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8C2311yescharacteristic
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4C2×C412yes—
68C2×C4×D4C2×C412yes—
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C2×#117—21no—
72C2×#117—21no—
73C2×#117C2×C411yescharacteristic
74C2×#117D411yescharacteristic
75C2×#117D411yescharacteristic
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C42—21no—
79C22×C42—22no—
80C22×C42C2311yescharacteristic
81#431—22no—
82#431—22no—
83#424—28no—
84#423—22no—
85#423—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2312yes—
93C2×C4⋊C8C2311yescharacteristic
94C2×C22⋊C8—21no—
95C4×M16—22no—
96C4×M16—22no—
97C2×#97—21no—
98C2×C4×C8—22no—
99C2×C4×C8—22no—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2312yes—
105#391C2×C412yes—
106#391C2×C412yes—
107#389C2×C412yes—
108#389C2×C412yes—
109#388—22no—
110#388—22no—
111#387—22no—
112#387—22no—
113#387—22no—
114#387—22no—
115#386—22no—
116#386—22no—
117#385—22no—
118#385—22no—
119#384C2×C411yescharacteristic
120#384C2×C411yescharacteristic
121#383—22no—
122#382—21no—
123#382—21no—
124#382—21no—
125#381—22no—
126#380—21no—
127#380—21no—
128#380—21no—
129#380C2×C412yes—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4C2×C412yes—
136C4×C4⋊C4C2×C412yes—
137C4×C4⋊C4C2×C411yescharacteristic
138C4×C4⋊C4C2×C411yescharacteristic
139C4×C4⋊C4D411yescharacteristic
140C4×C4⋊C4D411yescharacteristic
141C4×C4⋊C4C2312yes—
142C4×C22⋊C4—22no—
143C4×C22⋊C4—22no—
144C4×C22⋊C4—22no—
145C4×C22⋊C4C2×C412yes—
146C4×C22⋊C4D412yes—
147#376—22no—
148#376D412yes—
149C2×#95—22no—
150C43C2312yes—
151#336—22no—
152#336—22no—
153#336—22no—
154#336—22no—
155Q8⋊C8—22no—
156Q8⋊C8—22no—
157Q8⋊C8—24no—
158D4⋊C8—22no—
159D4⋊C8—22no—
160D4⋊C8—24no—
161C2×#41—22no—
162C2×#41—22no—
163C2×#41—22no—
164C2×#41—41no—
165C2×#41—41no—
166C2×#41C4⋊C414yes—
167C22×Q8C22×C411yescharacteristic
168C22×D4—21no—
169C22×D4C22×C411yescharacteristic
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4C22×C411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C2×M16—24no—
179C2×M16—42no—
180C2×M16—42no—
181C2×M16—42no—
182C2×M16—81no—
183C2×M16—81no—
184C22×C8—24no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—42no—
192C22×C8—42no—
193C4×Q8—22no—
194C4×Q8—21no—
195C4×Q8—21no—
196C4×Q8C4⋊C414yes—
197C4×Q8C2×D412yes—
198C4×Q8C2×Q812yes—
199C4×D4—24no—
200C4×D4—24no—
201C4×D4—24no—
202C4×D4—24no—
203C4×D4—24no—
204C4×D4—24no—
205C4×D4—22no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4C4⋊C414yes—
211C4×D4C2×D412yes—
212C4×D4C2×Q812yes—
213#117—22no—
214#117—24no—
215#117—24no—
216#117—24no—
217#117—42no—
218#117—81no—
219#117—81no—
220#117C4⋊C414yes—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4C22⋊C411yescharacteristic
232C2×C4⋊C4C22⋊C411yescharacteristic
233C2×C4⋊C4C22⋊C411yescharacteristic
234C2×C4⋊C4C22⋊C411yescharacteristic
235C2×C4⋊C4C22×C412yes—
236C2×C4⋊C4C22×C412yes—
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4C22⋊C412yes—
246C2×C22⋊C4C22⋊C412yes—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—82no—
256C2×C42—82no—
257C2×C42C22×C412yes—
258C2×C42C22×C412yes—
259C2×C42C22×C411yescharacteristic
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×Q812yes—
268C2×C42#4112yes—
269C2×C42#4112yes—
270C2×C42C2411yescharacteristic
271C4⋊C8—28no—
272C4⋊C8—28no—
273C4⋊C8—22no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—22no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—42no—
280C4⋊C8—82no—
281C4⋊C8—82no—
282C22⋊C8—42no—
283#97—42no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—22no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—22no—
290C4×C8—42no—
291C4×C8—42no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95C22⋊C412yes—
303#95C22⋊C412yes—
304#95C22⋊C412yes—
305#95C22⋊C412yes—
306C24—21no—
307C24C2×C22⋊C411yescharacteristic
308#41—28no—
309#41—44no—
310#41—44no—
311#41—44no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8C2×C4⋊C412yes—
318C2×D4—24no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4C2×C4⋊C412yes—
325C2×D4#13112yes—
326C2×D4#13112yes—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—82no—
359C22×C4—82no—
360C22×C4C2×C22⋊C412yes—
361C22×C4C2×C22⋊C412yes—
362C22×C4C2×C22⋊C412yes—
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C412yes—
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C4⋊C412yes—
368C22×C4#11712yes—
369C22×C4#12112yes—
370C22×C4#12112yes—
371C22×C4C22⋊Q812yes—
372C22×C4C22⋊Q812yes—
373C22×C4C23×C411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376M16—48no—
377M16—84no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—24no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—48no—
384C2×C8—42no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—42no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—42no—
391C2×C8—41no—
392C2×C8—42no—
393C2×C8—41no—
394C2×C8—42no—
395C2×C8—41no—
396C2×C8—41no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—81no—
414C4⋊C4—81no—
415C4⋊C4#10414yes—
416C4⋊C4C2×C4⋊C412yes—
417C4⋊C4C2×C4⋊C412yes—
418C22⋊C4—24no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—24no—
422C22⋊C4—24no—
423C22⋊C4—42no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—82no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—24no—
438C42—24no—
439C42—24no—
440C42—42no—
441C42—42no—
442C42—42no—
443C42—82no—
444C42—82no—
445C42C2×C4⋊C412yes—
446C42#11712yes—
447C42#12112yes—
448C42#12112yes—
449C42C22⋊Q812yes—
450C42C22⋊Q812yes—
451C42C22×D411yescharacteristic
452C42C22×Q811yescharacteristic
453C42C2×#4111yescharacteristic
454C42C2×#4111yescharacteristic
455C23—22no—
456C23—22no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—22no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23#38012yes—
469C23C22×C22⋊C411yescharacteristic
470Q8—22no—
471Q8—22no—
472Q8—41no—
473Q8—41no—
474D4—22no—
475D4—22no—
476D4—22no—
477D4—48no—
478D4#42414yescomplemented
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
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—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—48no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—48no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—88no—
528C2×C4—88no—
529C2×C4#38012yes—
530C2×C4#38012yes—
531C2×C4#38012yes—
532C2×C4C2×#10414yes—
533C2×C4C2×#10412yes—
534C2×C4#42112yes—
535C2×C4C22×C4⋊C411yescharacteristic
536C2×C4C2×#11711yescharacteristic
537C2×C4C2×#12111yescharacteristic
538C2×C4C2×#12111yescharacteristic
539C2×C4C2×C22⋊Q811yescharacteristic
540C2×C4C2×C22⋊Q811yescharacteristic
541C2×C4C2×#13111yescharacteristic
542C2×C4C2×#13111yescharacteristic
543C2×C4#56811yescharacteristic
544C2×C4#56811yescharacteristic
545C2×C4#57611yescharacteristic
546C2×C4#57711yescharacteristic
547C2×C4#57711yescharacteristic
548C2×C4#57811yescharacteristic
549C8—42no—
550C8—42no—
551C8—44no—
552C8—81no—
553C8—82no—
554C8—81no—
555C22—22no—
556C22—24no—
557C22—22no—
558C22—22no—
559C22—42no—
560C22—41no—
561C22—41no—
562C22—42no—
563C22—42no—
564C22—41no—
565C22—41no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22C2×#38011yescharacteristic
570C22#288011yescharacteristic
571C22#288011yescharacteristic
572C22#289711yescharacteristic
573C22#289811yescharacteristic
574C22C22×#10411yescharacteristic
575C22C2×#42111yescharacteristic
576C4—24no—
577C4—22no—
578C4—22no—
579C4—21no—
580C4—21no—
581C4—42no—
582C4—42no—
583C4—42no—
584C4—42no—
585C4—42no—
586C4—42no—
587C4—82no—
588C4—82no—
589C4#180614yes—
590C4C2×#42411yescharacteristic
591C4#290511yescharacteristic
592C4#292411yescharacteristic
593C4#292411yescharacteristic
594C4#304511yescharacteristic
595C4#304611yescharacteristic
596C4#308311yescharacteristic
597C4#308411yescharacteristic
598C2—42no—
599C2—44no—
600C2#1751711yescharacteristic
601C2#1751811yescharacteristic
602C2#1752211yescharacteristic
603C2#1752311yescharacteristic
604C2#2062811yescharacteristic
605C2#2065211yescharacteristic
606C2C2×#180611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21974C2semidirect
3. C4×#424C2semidirect
6. #8570C2semidirect
7. #8568C2semidirect
8. #8565C2semidirect
9. #8563C2semidirect
10. #7897C2semidirect
11. #7896C2semidirect
478. D4#424semidirect

Maximal subgroups

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

subgroupindex
#341272
#219742
C4×#4242
C2×#14892
C2×#14762
#85702
#85682
#85652
#85632
#78972
#78962