Finite Group Databasebeta

← #198024

Subgroups — #198024

Order 512.

1185 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3104C211yesmaximal, characteristic
2Q8×#117C211yesmaximal, characteristic
3#21329C211yesmaximal, characteristic
4#17575C211yesmaximal, characteristic
5C2×#1487C211yesmaximal, characteristic
6#9236C214yesmaximal, complemented
7#9232C214yesmaximal, complemented
8#8095C211yesmaximal, characteristic
9#7866C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#3104—22no—
13#3104—21no—
14#3104—21no—
15#3104C414yescomplemented
16#3104C2212yes—
17#3104C2212yes—
18C2×#484—21no—
19C2×#477C412yescomplemented
20C2×#462C412yescomplemented
21C2×#451—21no—
22C2×C4×Q16—22no—
23C2×#431C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×#423C2211yescharacteristic
26#2355—22no—
27#2355—21no—
28Q8×C4⋊C4—21no—
29#2345—22no—
30#2345—21no—
31Q8×C22⋊C4—21no—
32#2295—22no—
33#2295C2212yes—
34#2285—21no—
35#2285C2211yescharacteristic
36C42×Q8C2212yes—
37C4×#117—21no—
38C4×#117C2211yescharacteristic
39#1995—22no—
40#1994C2214yes—
41#1733—21no—
42#1733—21no—
43#1520—22no—
44#1518—22no—
45#1503C2214yes—
46#1487—22no—
47#1487C2212yes—
48#1487C2212yes—
49C2×Q8⋊C8—22no—
50#1303C2214yes—
51#1276C2214yes—
52C23×Q8D411yescharacteristic
53C22×Q16—22no—
54C22×M16—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533—21no—
59#533—21no—
60#533—21no—
61#533C2×C412yes—
62#533C2312yes—
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×C4⋊Q8C2×C411yescharacteristic
65C2×#125—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C4×Q8—22no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×Q8C2×C412yes—
78C2×C4×Q8C2×C412yes—
79C2×C4×Q8D412yes—
80C2×C4×Q8D412yes—
81C2×C4×Q8D411yescharacteristic
82C2×C4×Q8D411yescharacteristic
83C2×C4×Q8C2311yescharacteristic
84C2×C4×Q8C2311yescharacteristic
85C2×#117—21no—
86C2×#117—21no—
87C2×#117D411yescharacteristic
88C2×#117D411yescharacteristic
89C22×C4⋊C4—21no—
90C22×C4⋊C4D411yescharacteristic
91C22×C42—21no—
92C22×C42C2311yescharacteristic
93#484—24no—
94#484—22no—
95#484—22no—
96#477—22no—
97#477—22no—
98#477C2×C414yes—
99#477C2×C414yes—
100#462—22no—
101#462—22no—
102#462C2×C414yes—
103#462C2×C414yes—
104#451—24no—
105#451—22no—
106#451—22no—
107C4×Q16—24no—
108C4×Q16—24no—
109C4×Q16—24no—
110C4×Q16—24no—
111#431—21no—
112#431—21no—
113#431—21no—
114#431—21no—
115#431—21no—
116#431C2×C412yes—
117#431C2312yes—
118#431C2312yes—
119C2×#107—22no—
120#423—21no—
121#423—21no—
122#423—21no—
123#423C2312yes—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8—21no—
127C2×C4⋊C8C2×C412yes—
128C2×#103—22no—
129C2×#103—22no—
130C2×C22⋊C8—21no—
131C2×C22⋊C8—21no—
132C2×C4×C8—21no—
133C2×C4×C8—21no—
134#391—24no—
135#391—22no—
136#389—24no—
137#389—22no—
138#387—24no—
139#387—24no—
140#387—24no—
141#384—22no—
142#384—21no—
143#382—22no—
144#382D412yes—
145#380—22no—
146#380—22no—
147#380—21no—
148#380D412yes—
149C4×C4⋊C4—22no—
150C4×C4⋊C4—22no—
151C4×C4⋊C4—22no—
152C4×C4⋊C4—21no—
153C4×C4⋊C4—21no—
154C4×C4⋊C4—22no—
155C4×C4⋊C4—22no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—24no—
158C4×C4⋊C4C2312yes—
159C4×C22⋊C4—22no—
160C4×C22⋊C4—22no—
161C4×C22⋊C4—21no—
162C4×C22⋊C4—21no—
163#376—22no—
164#376D412yes—
165C43—22no—
166C43C2312yes—
167#339—24no—
168#339—24no—
169#328—28no—
170Q8⋊C8—24no—
171Q8⋊C8—24no—
172C22×Q8—21no—
173C22×Q8—21no—
174C22×Q8—21no—
175C22×Q8—21no—
176C22×Q8—21no—
177C22×Q8—44no—
178C22×Q8C22⋊C412yes—
179C22×Q8C22⋊C412yes—
180C22×Q8C2×D411yescharacteristic
181C22×Q8C2×D411yescharacteristic
182C23×C4—21no—
183C23×C4—21no—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C2×Q16—24no—
188C2×Q16—24no—
189C2×Q16—48no—
190C2×M16—22no—
191C2×M16—41no—
192C2×M16—42no—
193C2×M16—42no—
194C22×C8—22no—
195C22×C8—41no—
196C22×C8—41no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8C22×C412yes—
204C4⋊Q8C22×C412yes—
205#125—22no—
206#125—22no—
207#125—41no—
208#125—41no—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212C22⋊Q8—22no—
213C22⋊Q8—41no—
214C22⋊Q8—41no—
215C22⋊Q8—41no—
216C22⋊Q8—41no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—22no—
221C4×Q8—22no—
222C4×Q8—22no—
223C4×Q8—22no—
224C4×Q8—22no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8—21no—
232C4×Q8—21no—
233C4×Q8—22no—
234C4×Q8—22no—
235C4×Q8—22no—
236C4×Q8—22no—
237C4×Q8—48no—
238C4×Q8—41no—
239C4×Q8—41no—
240C4×Q8—41no—
241C4×Q8—41no—
242C4×Q8C22⋊C412yes—
243C4×Q8C22⋊C412yes—
244C4×Q8C22⋊C412yes—
245C4×Q8C22⋊C412yes—
246C4×Q8C22×C412yes—
247C4×Q8C22×C412yes—
248C4×Q8C2×D412yes—
249C4×Q8C2×D412yes—
250#117—22no—
251#117—22no—
252#117—22no—
253#117—22no—
254#117—22no—
255#117—41no—
256#117—41no—
257#117—42no—
258#117—42no—
259#117Q1614yescomplemented
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—41no—
283C2×C4⋊C4—41no—
284C2×C4⋊C4C22⋊C412yes—
285C2×C4⋊C4C22⋊C412yes—
286C2×C4⋊C4C2×D411yescharacteristic
287C2×C4⋊C4C2×D411yescharacteristic
288C2×C22⋊C4—21no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—41no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—41no—
306C2×C42C22×C411yescharacteristic
307C2×C42C22×C411yescharacteristic
308C2×C42C2×D412yes—
309C2×C42C2×D412yes—
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42#4111yescharacteristic
316C2×C42#4111yescharacteristic
317C2×C42#4111yescharacteristic
318C2×C42#4111yescharacteristic
319C2×C42C2411yescharacteristic
320#107—24no—
321#107—24no—
322C4⋊C8—22no—
323C4⋊C8—22no—
324C4⋊C8—22no—
325C4⋊C8—22no—
326C4⋊C8—41no—
327C4⋊C8—41no—
328C4⋊C8—41no—
329C4⋊C8C22×C414yes—
330#103—24no—
331#103—24no—
332#103—24no—
333#103—24no—
334C22⋊C8—22no—
335C22⋊C8—22no—
336C22⋊C8—22no—
337C22⋊C8—41no—
338C4×C8—22no—
339C4×C8—22no—
340C4×C8—22no—
341C4×C8—41no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95C2×D414yes—
350C24#12011yescharacteristic
351C2×Q8—24no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—24no—
357C2×Q8—24no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—44no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×Q8—41no—
368C2×Q8—41no—
369C2×Q8—82no—
370C2×Q8C2×C22⋊C412yes—
371C2×Q8C2×C22⋊C412yes—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—42no—
384C22×C4—44no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—81no—
388C22×C4—82no—
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C4×D411yescharacteristic
392C22×C4C4×D411yescharacteristic
393C22×C4C4×D411yescharacteristic
394C22×C4C4×D411yescharacteristic
395C22×C4#12011yescharacteristic
396C22×C4#12011yescharacteristic
397C22×C4#12011yescharacteristic
398C22×C4#12111yescharacteristic
399C22×C4#12111yescharacteristic
400C22×C4#12111yescharacteristic
401C22×C4#12111yescharacteristic
402C22×C4#12311yescharacteristic
403C22×C4#12311yescharacteristic
404C22×C4C2×Q1612yes—
405C22×C4#13512yes—
406C22×C4#13712yes—
407C22×C4#13712yes—
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411Q16—48no—
412Q16—48no—
413M16—42no—
414M16—42no—
415M16—82no—
416C2×C8—24no—
417C2×C8—44no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—82no—
421C2×C8—82no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—48no—
440C4⋊C4—44no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C4⋊C4—82no—
452C4⋊C4—84no—
453C4⋊C4C2×C22⋊C412yes—
454C4⋊C4C2×C22⋊C412yes—
455C22⋊C4—24no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—44no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—48no—
472C42—44no—
473C42—42no—
474C42—42no—
475C42—42no—
476C42—42no—
477C42—82no—
478C42—84no—
479C42C2×C22⋊C411yescharacteristic
480C42C2×C22⋊C411yescharacteristic
481C42C4×D411yescharacteristic
482C42C4×D411yescharacteristic
483C42C4×D411yescharacteristic
484C42C4×D411yescharacteristic
485C42#12111yescharacteristic
486C42#12111yescharacteristic
487C42#12111yescharacteristic
488C42#12111yescharacteristic
489C42#12311yescharacteristic
490C42#12311yescharacteristic
491C42C2×Q1614yescomplemented
492C42C23×C411yescharacteristic
493C42C22×D411yescharacteristic
494C42C2×#4111yescharacteristic
495C42C2×#4111yescharacteristic
496C23—41no—
497C23—41no—
498C23#38611yescharacteristic
499C23#38611yescharacteristic
500C23#45112yes—
501C23#45212yes—
502C23C2×#12011yescharacteristic
503Q8—24no—
504Q8—22no—
505Q8—22no—
506Q8—22no—
507Q8—44no—
508Q8—44no—
509Q8—42no—
510Q8—44no—
511Q8—82no—
512Q8—82no—
513Q8—82no—
514Q8#42114yes—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4#38611yescharacteristic
553C2×C4#38611yescharacteristic
554C2×C4#38611yescharacteristic
555C2×C4#38611yescharacteristic
556C2×C4#38611yescharacteristic
557C2×C4#38611yescharacteristic
558C2×C4C4×Q1612yes—
559C2×C4C4×Q1612yes—
560C2×C4#44112yes—
561C2×C4#44112yes—
562C2×C4#45112yes—
563C2×C4#45212yes—
564C2×C4#46212yes—
565C2×C4#46212yes—
566C2×C4#46412yes—
567C2×C4#46412yes—
568C2×C4C22×C22⋊C411yescharacteristic
569C2×C4C2×C4×D411yescharacteristic
570C2×C4C2×C4×D411yescharacteristic
571C2×C4C2×#12111yescharacteristic
572C2×C4C2×#12111yescharacteristic
573C2×C4C2×#12311yescharacteristic
574C2×C4C22×Q1611yescharacteristic
575C2×C4C2×#13511yescharacteristic
576C2×C4C2×#13711yescharacteristic
577C2×C4C2×#13711yescharacteristic
578C8—44no—
579C8—82no—
580C8—82no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—42no—
588C22—42no—
589C22#187812yes—
590C22#187812yes—
591C22C2×#38611yescharacteristic
592C22C2×#45111yescharacteristic
593C22C2×#45211yescharacteristic
594C4—21no—
595C4—21no—
596C4—21no—
597C4—21no—
598C4—24no—
599C4—21no—
600C4—21no—
601C4—21no—
602C4—44no—
603C4—42no—
604C4—42no—
605C4—42no—
606C4—82no—
607C4—84no—
608C4—82no—
609C4—82no—
610C4#187812yes—
611C4#187812yes—
612C4C2×#42112yes—
613C4C2×C4×Q1611yescharacteristic
614C4C2×#44111yescharacteristic
615C4C2×#46211yescharacteristic
616C4C2×#46411yescharacteristic
617C2—21no—
618C2—21no—
619C2—22no—
620C2—22no—
621C2C2×#187811yescharacteristic
622C2#2097911yescharacteristic
623C2#2100011yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9236C2semidirect
7. #9232C2semidirect
15. #3104C4semidirect
19. C2×#477C4semidirect
20. C2×#462C4semidirect
259. #117Q16semidirect
491. C42C2×Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#31042
Q8×#1172
#213292
#175752
C2×#14872
#92362
#92322
#80952
#78662