Finite Group Databasebeta

← #482130

Subgroups — #482130

Order 512.

1083 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35010C211yesmaximal, characteristic
2C2×#1800C211yesmaximal, characteristic
3C2×#1459C211yesmaximal, characteristic
4#11906C214yesmaximal, complemented
5#11904C212yesmaximal, complemented
6#11903C212yesmaximal, complemented
7#11722C214yesmaximal, complemented
8C22×C22⋊Q8C411yescharacteristic
9C22×C4×D4C2211yescharacteristic
10C2×#423—21no—
11C22×#100—21no—
12C22×C22⋊C8—21no—
13#2855—21no—
14#2848—21no—
15#2627C2212yes—
16#2626C412yes—
17#2579—21no—
18#2579—21no—
19#2564C412yes—
20#2564C2212yes—
21#2487C411yescharacteristic
22#2487C2211yescharacteristic
23#2417—21no—
24#2417—21no—
25#2409C411yescharacteristic
26#2409C2211yescharacteristic
27#2315—21no—
28#2308—21no—
29C2×#380C411yescharacteristic
30#1800—21no—
31#1800—21no—
32#1800C2212yes—
33#1800C2212yes—
34C2×#343C2212yes—
35#1459—21no—
36#1459—21no—
37#1459C2212yes—
38#1459C2212yes—
39C2×#324C2212yes—
40C2×#323—21no—
41#1354C2218yescomplemented
42#1353C2218yescomplemented
43C23×Q8C2×C411yescharacteristic
44C23×D4D411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4D411yescharacteristic
47C22×M16—22no—
48C23×C8—41no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—41no—
54C2×C22⋊Q8—41no—
55C2×C22⋊Q8—41no—
56C2×C22⋊Q8—41no—
57C2×C22⋊Q8C2×C412yes—
58C2×C22⋊Q8C2×C412yes—
59C2×C4×D4—41no—
60C2×C4×D4—41no—
61C2×C4×D4—41no—
62C2×C4×D4—41no—
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4C2×C411yescharacteristic
78C22×C22⋊C4D411yescharacteristic
79C22×C42D411yescharacteristic
80#423—22no—
81#423—22no—
82#423—41no—
83#423—41no—
84C2×C4⋊C8—41no—
85C2×C4⋊C8—41no—
86C2×#100—22no—
87C2×#100—22no—
88C2×#100—44no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—41no—
93C2×C22⋊C8—41no—
94C2×C22⋊C8—41no—
95#400C2×C412yes—
96#400C2312yes—
97#399C2×C412yes—
98#399C2×C412yes—
99#398—22no—
100#398—22no—
101#396—22no—
102#396—22no—
103#395C2×C412yes—
104#395C2312yes—
105#393C2×C412yes—
106#393C2×C412yes—
107#391—21no—
108#391—21no—
109#389—21no—
110#389—21no—
111#387—22no—
112#387—22no—
113#386—22no—
114#386—21no—
115#386—21no—
116#386—22no—
117#386—21no—
118#386—21no—
119#385—22no—
120#385—22no—
121#385—22no—
122#385—22no—
123#385—21no—
124#385—21no—
125#385—21no—
126#385—21no—
127#381—22no—
128#380—21no—
129#380—21no—
130#380—21no—
131#380—21no—
132#380—22no—
133#380—22no—
134#380C2×C412yes—
135#380C2×C411yescharacteristic
136#380C2×C411yescharacteristic
137#379—22no—
138C2×#95C2×C412yes—
139#343—22no—
140#343C2314yes—
141#324—22no—
142#324C2314yes—
143#323—22no—
144#323—22no—
145C25C22⋊C411yescharacteristic
146C25C22⋊C411yescharacteristic
147C22×Q8—21no—
148C22×Q8—41no—
149C22×Q8—41no—
150C22×Q8—82no—
151C22×Q8C22×C412yes—
152C22×D4—41no—
153C22×D4—41no—
154C22×D4—42no—
155C22×D4C22⋊C411yescharacteristic
156C22×D4C22⋊C411yescharacteristic
157C22×D4SD1614yes—
158C22×D4Q1614yes—
159C22×D4C2×D411yescharacteristic
160C22×D4C2×D411yescharacteristic
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—22no—
172C23×C4—82no—
173C23×C4—82no—
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C2×M16—24no—
188C2×M16—42no—
189C2×M16—44no—
190C2×M16—44no—
191C22×C8—42no—
192C22×C8—42no—
193C22×C8—42no—
194C22×C8—42no—
195C22×C8—84no—
196C22⋊Q8—42no—
197C22⋊Q8—42no—
198C22⋊Q8—41no—
199C22⋊Q8—41no—
200C22⋊Q8—41no—
201C22⋊Q8—41no—
202C22⋊Q8—42no—
203C22⋊Q8—42no—
204C22⋊Q8—42no—
205C22⋊Q8—42no—
206C22⋊Q8—82no—
207C22⋊Q8—82no—
208C4×D4—41no—
209C4×D4—41no—
210C4×D4—41no—
211C4×D4—41no—
212C4×D4—41no—
213C4×D4—41no—
214C4×D4—41no—
215C4×D4—41no—
216C4×D4—41no—
217C4×D4—41no—
218C4×D4—41no—
219C4×D4—41no—
220C4×D4—41no—
221C4×D4—41no—
222C4×D4—41no—
223C4×D4—41no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—82no—
241C2×C4⋊C4C22×C412yes—
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2411yescharacteristic
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4C22⋊C411yescharacteristic
263C2×C22⋊C4C22⋊C411yescharacteristic
264C2×C22⋊C4C22⋊C411yescharacteristic
265C2×C22⋊C4C22⋊C411yescharacteristic
266C2×C22⋊C4C22×C411yescharacteristic
267C2×C22⋊C4C22×C411yescharacteristic
268C2×C22⋊C4C2×D411yescharacteristic
269C2×C22⋊C4C2×D411yescharacteristic
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42C22⋊C411yescharacteristic
275C2×C42C22⋊C411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C4⋊C8—42no—
279C4⋊C8—42no—
280C4⋊C8—42no—
281C4⋊C8—42no—
282#100—44no—
283#100—44no—
284C22⋊C8—24no—
285C22⋊C8—42no—
286C22⋊C8—42no—
287C22⋊C8—42no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95C22×C412yes—
298#95C22×C412yes—
299C24—21no—
300C24—21no—
301C24—21no—
302C24—21no—
303C24—22no—
304C24—22no—
305C24—81no—
306C24—81no—
307C24—81no—
308C24—81no—
309C24#10314yes—
310C24#10314yes—
311C24C2×C22⋊C411yescharacteristic
312C24C2×C22⋊C411yescharacteristic
313C24C2×C22⋊C411yescharacteristic
314C24C2×C22⋊C411yescharacteristic
315C24C2×C22⋊C411yescharacteristic
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—82no—
322C2×Q8—84no—
323C2×Q8—84no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4#10314yes—
337C2×D4#10314yes—
338C2×D4C2×SD1612yes—
339C2×D4C2×SD1612yes—
340C2×D4C2×Q1612yes—
341C2×D4C2×Q1612yes—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—42no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4—82no—
383C22×C4—81no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4—81no—
387C22×C4—82no—
388C22×C4—82no—
389C22×C4—82no—
390C22×C4—82no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4—81no—
394C22×C4—81no—
395C22×C4—81no—
396C22×C4—82no—
397C22×C4—82no—
398C22×C4#9911yescharacteristic
399C22×C4#9911yescharacteristic
400C22×C4#9911yescharacteristic
401C22×C4#9911yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×SD1612yes—
410C22×C4C2×Q1612yes—
411C22×C4C23×C411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414M16—44no—
415M16—44no—
416M16—84no—
417C2×C8—44no—
418C2×C8—44no—
419C2×C8—84no—
420C2×C8—84no—
421C2×C8—84no—
422C2×C8—84no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—82no—
436C4⋊C4—82no—
437C4⋊C4—82no—
438C4⋊C4—82no—
439C4⋊C4—82no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—44no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—44no—
454C22⋊C4—82no—
455C22⋊C4—82no—
456C22⋊C4—82no—
457C22⋊C4—82no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—24no—
473C23—21no—
474C23—21no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—42no—
485C23—42no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—81no—
491C23—81no—
492C23—81no—
493C23—81no—
494C23—81no—
495C23—81no—
496C23—84no—
497C23—84no—
498C23—84no—
499C23—84no—
500C23C2×#9911yescharacteristic
501C23C2×#9911yescharacteristic
502C23C2×#10312yes—
503C23C2×#10312yes—
504C23C2×#10312yes—
505C23C2×#10312yes—
506C23C2×#10312yes—
507C23#42112yes—
508C23C22×C22⋊C411yescharacteristic
509Q8—82no—
510Q8—84no—
511Q8—82no—
512D4—44no—
513D4—44no—
514D4—42no—
515D4—42no—
516D4—42no—
517D4—42no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
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—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—164no—
581C2×C4C2×#9911yescharacteristic
582C2×C4C2×#9911yescharacteristic
583C2×C4C2×#9911yescharacteristic
584C2×C4C2×#9911yescharacteristic
585C2×C4C2×#10312yes—
586C2×C4C22×SD1611yescharacteristic
587C2×C4C22×Q1611yescharacteristic
588C8—84no—
589C8—84no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—48no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—81no—
618C22—81no—
619C22—84no—
620C22—84no—
621C22—84no—
622C22#152012yes—
623C22#152012yes—
624C22C22×#9911yescharacteristic
625C22C22×#10311yescharacteristic
626C22C2×#42111yescharacteristic
627C4—21no—
628C4—21no—
629C4—21no—
630C4—21no—
631C4—21no—
632C4—22no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—41no—
637C4—41no—
638C4—41no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4#212112yes—
644C2—21no—
645C2—21no—
646C2—22no—
647C2—22no—
648C2—42no—
649C2—42no—
650C2—42no—
651C2—42no—
652C2C2×#152011yescharacteristic
653C2C2×#212111yescharacteristic
654C2#3336911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11906C2semidirect
5. #11904C2semidirect
6. #11903C2semidirect
7. #11722C2semidirect
41. #1354C22semidirect
42. #1353C22semidirect

Maximal subgroups

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

subgroupindex
#350102
C2×#18002
C2×#14592
#119062
#119042
#119032
#117222