Finite Group Databasebeta

← #183113

Subgroups — #183113

Order 512.

1186 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34155C211yesmaximal, characteristic
2#20546C211yesmaximal, characteristic, complemented
3#20462C211yesmaximal, characteristic, complemented
4C2×#1496C211yesmaximal, characteristic
5C2×#1480C211yesmaximal, characteristic
6#8183C211yesmaximal, characteristic, complemented
7#8181C211yesmaximal, characteristic, complemented
8#8179C211yesmaximal, characteristic, complemented
9#8177C211yesmaximal, characteristic, complemented
10#8174C211yesmaximal, characteristic, complemented
11#8172C211yesmaximal, characteristic, complemented
12#8170C211yesmaximal, characteristic, complemented
13C4×D4⋊C8C211yesmaximal, characteristic, complemented
14#7902C211yesmaximal, characteristic, complemented
15#7891C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21#2372—22no—
22#2369C412yes—
23#2366C412yes—
24#2363—22no—
25#2363C412yes—
26#2363C412yes—
27#2297—21no—
28#2297—21no—
29#2293—21no—
30#2293C2211yescharacteristic
31#2293C2211yescharacteristic
32C42×D4—21no—
33C42×D4C2211yescharacteristic
34C42×D4C2211yescharacteristic
35C4×#117C2211yescharacteristic
36C4×#117C2211yescharacteristic
37#1771—21no—
38#1771—21no—
39#1768—22no—
40#1767C2211yescharacteristic
41#1767C2211yescharacteristic
42C4×C4⋊C8C2211yescharacteristic
43C4×C4⋊C8C2211yescharacteristic
44#1752—21no—
45C8×C22⋊C4—21no—
46#1750—22no—
47#1744C2211yescharacteristic
48#1744C2211yescharacteristic
49C4×#97C2211yescharacteristic
50C42×C8C2211yescharacteristic
51#1496C418yescomplemented
52#1480C418yescomplemented
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68#1279C2211yescharacteristic
69C22×#41—21no—
70C23×C8—21no—
71#532—22no—
72#532—22no—
73#532C2×C414yes—
74#532C2×C414yes—
75C2×C4⋊Q8C2×C411yescharacteristic
76C2×#127C2×C411yescharacteristic
77C2×#124—21no—
78C2×#121—21no—
79C2×#121C2×C411yescharacteristic
80C2×#121C2×C411yescharacteristic
81C2×C4×D4—22no—
82C2×C4×D4—21no—
83C2×C4×D4C2×C412yes—
84C2×C4×D4C2×C412yes—
85C2×C4×D4C2311yescharacteristic
86C2×C4×D4C2311yescharacteristic
87C2×#117—21no—
88C2×#117—21no—
89C2×#117C2×C412yes—
90C2×#117D412yes—
91C22×C42—21no—
92C22×C42—21no—
93C22×C42C2311yescharacteristic
94#432—22no—
95#432C2×C414yescomplemented
96#431—22no—
97#431—22no—
98#431C2×C414yescomplemented
99#431C2×C414yescomplemented
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8C2×C414yescomplemented
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×#97—21no—
113C2×#97—22no—
114C2×#97—21no—
115C2×#97—21no—
116C2×C4×C8—22no—
117C2×C4×C8—21no—
118C2×C4×C8—21no—
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123#391—22no—
124#391—22no—
125#390—22no—
126#390—22no—
127#390—22no—
128#390C2×C412yes—
129#390C2×C412yes—
130#389—22no—
131#389C2×C412yes—
132#389C2×C412yes—
133#388—22no—
134#388—22no—
135#388—22no—
136#388—22no—
137#386—22no—
138#386—22no—
139#386—22no—
140#386—22no—
141C4×C4⋊C4—22no—
142C4×C4⋊C4C2×C412yes—
143C4×C4⋊C4C2×C412yes—
144C4×C4⋊C4C2×C412yes—
145C4×C4⋊C4C2×C412yes—
146C4×C4⋊C4D411yescharacteristic
147C4×C4⋊C4D411yescharacteristic
148C4×C4⋊C4C2311yescharacteristic
149C4×C4⋊C4C2311yescharacteristic
150C4×C22⋊C4—21no—
151C4×C22⋊C4—21no—
152C4×C22⋊C4—22no—
153C4×C22⋊C4—22no—
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4—22no—
159C4×C22⋊C4—22no—
160C4×C22⋊C4C2×C412yes—
161C4×C22⋊C4D411yescharacteristic
162C4×C22⋊C4D411yescharacteristic
163#376—21no—
164#376D412yes—
165C43—21no—
166C43C2311yescharacteristic
167C43C2311yescharacteristic
168#336—22no—
169#336—22no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174D4⋊C8C2×C414yescomplemented
175D4⋊C8C2×C414yescomplemented
176D4⋊C8C2×C414yescomplemented
177D4⋊C8C2×C414yescomplemented
178D4⋊C8C2×C414yescomplemented
179D4⋊C8C2×C414yescomplemented
180D4⋊C8C2×C414yescomplemented
181D4⋊C8C2×C414yescomplemented
182C2×#41—24no—
183C2×#41—24no—
184C2×#41—81no—
185C2×#41—81no—
186C2×#41—81no—
187C2×#41—81no—
188C22×Q8—21no—
189C22×D4—21no—
190C22×D4—21no—
191C22×D4—21no—
192C22×D4—21no—
193C22×D4C22×C411yescharacteristic
194C22×D4C22×C411yescharacteristic
195C23×C4—21no—
196C23×C4—21no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4C22×C411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—24no—
211C22×C8—81no—
212C22×C8—81no—
213C4⋊Q8—22no—
214C4⋊Q8C4214yes—
215#127—22no—
216#127C4214yes—
217#124—24no—
218#124—24no—
219#121—22no—
220#121—22no—
221#121—24no—
222#121—24no—
223#121C4214yes—
224#121C4214yes—
225C4×D4—24no—
226C4×D4—24no—
227C4×D4—24no—
228C4×D4—24no—
229C4×D4—21no—
230C4×D4—21no—
231C4×D4—21no—
232C4×D4—21no—
233C4×D4—42no—
234C4×D4—42no—
235C4×D4—48no—
236C4×D4C22×C412yes—
237C4×D4C22×C412yes—
238C4×D4C22×C412yes—
239C4×D4C22×C412yes—
240#117—24no—
241#117—44no—
242#117—42no—
243#117—81no—
244#117—81no—
245#117—81no—
246#117—81no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4C22⋊C412yes—
251C2×C4⋊C4C22⋊C412yes—
252C2×C4⋊C4C22×C412yes—
253C2×C4⋊C4C22×C411yescharacteristic
254C2×C4⋊C4C22×C411yescharacteristic
255C2×C4⋊C4C22×C412yes—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4C22⋊C412yes—
268C2×C22⋊C4C22⋊C412yes—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—81no—
276C2×C42—81no—
277C2×C42—81no—
278C2×C42—81no—
279C2×C42C22×C412yes—
280C2×C42C22×C412yes—
281C2×C42C22×C411yescharacteristic
282C2×C42C22×C412yes—
283C2×C42C22×C412yes—
284C2×C42C2×D412yes—
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D412yes—
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42#4112yes—
293C2×C42#4112yes—
294C2×C42C2411yescharacteristic
295C4⋊C8—24no—
296C4⋊C8—24no—
297C4⋊C8—24no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300C4⋊C8—21no—
301C4⋊C8—21no—
302C4⋊C8C22×C412yes—
303C4⋊C8C22×C412yes—
304C4⋊C8C22×C412yes—
305C4⋊C8C22×C412yes—
306C22⋊C8—24no—
307C22⋊C8—24no—
308C22⋊C8—24no—
309#97—24no—
310#97—44no—
311#97—81no—
312#97—81no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—44no—
318C4×C8—81no—
319C4×C8—81no—
320C4×C8C22×C412yes—
321C4×C8C22×C412yes—
322C4×C8C22×C412yes—
323C4×C8C22×C412yes—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—21no—
330#95—21no—
331#95C22⋊C412yes—
332#95C22⋊C412yes—
333#95C22⋊C412yes—
334#95C22⋊C412yes—
335C24—21no—
336C24—21no—
337C24C2×C22⋊C411yescharacteristic
338#41—84no—
339#41—84no—
340#41—84no—
341#41—84no—
342C2×Q8—24no—
343C2×Q8—81no—
344C2×Q8—81no—
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—44no—
356C2×D4—81no—
357C2×D4—81no—
358C2×D4—81no—
359C2×D4—81no—
360C2×D4—81no—
361C2×D4—81no—
362C2×D4C2×C4212yes—
363C2×D4C2×C4212yes—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—44no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4C2×C4212yes—
394C22×C4C2×C22⋊C412yes—
395C22×C4C2×C22⋊C412yes—
396C22×C4C2×C22⋊C412yes—
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C412yes—
401C22×C4#11712yes—
402C22×C4C4×D412yes—
403C22×C4C4×D412yes—
404C22×C4C4×D412yes—
405C22×C4C4×D412yes—
406C22×C4C23×C411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—84no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—42no—
438C4⋊C4—82no—
439C4⋊C4—82no—
440C4⋊C4C2×C4212yes—
441C4⋊C4C2×C4212yes—
442C4⋊C4#13114yes—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—42no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—82no—
457C22⋊C4—82no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—24no—
471C42—42no—
472C42—44no—
473C42—44no—
474C42—41no—
475C42—41no—
476C42—41no—
477C42—41no—
478C42—82no—
479C42—82no—
480C42C2×C4212yes—
481C42#11712yes—
482C42C4×D412yes—
483C42C4×D412yes—
484C42C4×D412yes—
485C42C4×D412yes—
486C42C23×C411yescharacteristic
487C42C23×C411yescharacteristic
488C42C2×#4111yescharacteristic
489C42C2×#4111yescharacteristic
490C23—22no—
491C23—21no—
492C23—21no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—42no—
497C23—42no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—42no—
503C23—42no—
504C23C4×C22⋊C412yes—
505C23C22×C22⋊C411yescharacteristic
506Q8—84no—
507D4—22no—
508D4—22no—
509D4—22no—
510D4—22no—
511D4—48no—
512D4—42no—
513D4—42no—
514D4—84no—
515D4—84no—
516D4—84no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
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—24no—
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—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—88no—
563C2×C4—88no—
564C2×C4C4×C22⋊C412yes—
565C2×C4C4×C22⋊C412yes—
566C2×C4C4×C22⋊C412yes—
567C2×C4#42114yes—
568C2×C4C22×C4211yescharacteristic
569C2×C4C2×#11711yescharacteristic
570C2×C4C2×C4×D411yescharacteristic
571C2×C4C2×C4×D411yescharacteristic
572C2×C4C2×C4×D411yescharacteristic
573C2×C4C2×C4×D411yescharacteristic
574C2×C4C2×#13112yes—
575C2×C4#56812yes—
576C2×C4#57611yescharacteristic
577C2×C4#57611yescharacteristic
578C2×C4#57711yescharacteristic
579C2×C4#57711yescharacteristic
580C8—42no—
581C8—42no—
582C8—42no—
583C8—42no—
584C8—42no—
585C8—42no—
586C8—42no—
587C8—42no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22C2×C4×C22⋊C411yescharacteristic
604C22#288012yes—
605C22#289711yescharacteristic
606C22#289711yescharacteristic
607C22C2×#42112yes—
608C4—24no—
609C4—22no—
610C4—21no—
611C4—21no—
612C4—22no—
613C4—22no—
614C4—42no—
615C4—44no—
616C4—42no—
617C4—42no—
618C4—42no—
619C4—42no—
620C4—82no—
621C4C4×#13111yescharacteristic
622C4#287511yescharacteristic
623C4#292512yes—
624C4#294611yescharacteristic
625C4#294611yescharacteristic
626C4#294811yescharacteristic
627C4#294911yescharacteristic
628C2—42no—
629C2—82no—
630C2—82no—
631C2#1752112yes—
632C2#1752612yes—
633C2#2037811yescharacteristic
634C2C4×#42111yescharacteristic
635C2#2041611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#532C22
LCSG ▷ C2×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. #20546C2semidirect
3. #20462C2semidirect
6. #8183C2semidirect
7. #8181C2semidirect
8. #8179C2semidirect
9. #8177C2semidirect
10. #8174C2semidirect
11. #8172C2semidirect
12. #8170C2semidirect
13. C4×D4⋊C8C2semidirect
14. #7902C2semidirect
15. #7891C2semidirect
51. #1496C4semidirect
52. #1480C4semidirect
95. #432C2×C4semidirect
98. #431C2×C4semidirect
99. #431C2×C4semidirect
104. C2×C4⋊C8C2×C4semidirect
174. D4⋊C8C2×C4semidirect
175. D4⋊C8C2×C4semidirect
176. D4⋊C8C2×C4semidirect
177. D4⋊C8C2×C4semidirect
178. D4⋊C8C2×C4semidirect
179. D4⋊C8C2×C4semidirect
180. D4⋊C8C2×C4semidirect
181. D4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341552
#205462
#204622
C2×#14962
C2×#14802
#81832
#81812
#81792
#81772
#81742
#81722
#81702
C4×D4⋊C82
#79022
#78912