Finite Group Databasebeta

← #415930

Subgroups — #415930

Order 512.

1814 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#491C211yesmaximal, characteristic, complemented
2C22×#478C211yesmaximal, characteristic
3C2×#2547C211yesmaximal, characteristic
4C2×#1981C211yesmaximal, characteristic, complemented
5C2×#1762C211yesmaximal, characteristic
6#14160C211yesmaximal, characteristic, complemented
7#11124C212yesmaximal, complemented
8#11122C212yesmaximal, complemented
9#11009C212yesmaximal
10#11009C212yesmaximal
11#10835C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×#124C2212yes—
15C22×C4×D4C2211yescharacteristic
16C2×#491—24no—
17C2×#491—24no—
18C2×#491—24no—
19C2×#478—24no—
20C2×#478—24no—
21C2×#478—24no—
22C22×#107C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×#103C2212yes—
26C22×#103C2212yes—
27C22×#103C2212yes—
28C22×#103C2212yes—
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2212yes—
32C22×#97C2211yescharacteristic
33#2547—24no—
34#2547—24no—
35C2×#397—22no—
36C2×#391C2212yes—
37C2×#380—22no—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40#1981—24no—
41#1762—24no—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1441—22no—
46#1436—22no—
47C4⋊#95C2212yes—
48C4⋊#95C2212yes—
49C23×Q8C2312yes—
50C23×D4C2311yescharacteristic
51C24×C4—21no—
52C23×C8C2312yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8D414yes—
61C2×#125—22no—
62C2×#125—22no—
63C2×#125—22no—
64C2×#125—22no—
65C2×#125—22no—
66C2×#125D414yes—
67C2×#124—24no—
68C2×#124—24no—
69C2×#124—24no—
70C2×#124—24no—
71C2×#124—42no—
72C2×#124—42no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C2×C4×D4D414yes—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4D412yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—24no—
90C22×C42—21no—
91C22×C42C2312yes—
92C22×C42C2311yescharacteristic
93#491—232no—
94#478—232no—
95C2×#107—22no—
96C2×#107—22no—
97C2×#107—22no—
98C2×#106—22no—
99C2×#106—22no—
100C2×#106—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8D414yes—
104C2×#103—24no—
105C2×#103—24no—
106C2×#103—24no—
107C2×#103—24no—
108C2×#103—24no—
109C2×#103—24no—
110C2×#103—24no—
111C2×#103—24no—
112C2×#103—24no—
113C2×#103—24no—
114C2×#103—42no—
115C2×#103—42no—
116C2×#102—22no—
117C2×#102—22no—
118C2×#102—24no—
119C2×#102—22no—
120C2×#102—22no—
121C2×#102—24no—
122C2×#102—22no—
123C2×#102—22no—
124C2×#102—24no—
125C2×#97—22no—
126C2×#97—22no—
127C2×#97D414yes—
128#397—216no—
129#391—24no—
130#391—24no—
131#380—216no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C2×#95—24no—
137C2×#95—24no—
138C2×#95—21no—
139C2×#95—24no—
140C2×#95D412yes—
141#340—24no—
142#336—42no—
143#336—42no—
144C25—21no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8C2×D414yes—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4C2×D412yes—
168C22×D4#4114yes—
169C23×C4—22no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—48no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4C2×D414yes—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2411yescharacteristic
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8C2×D414yes—
197C22×C8C2×D412yes—
198C22×C8C2×D412yes—
199C4⋊Q8—28no—
200C4⋊Q8—28no—
201C4⋊Q8—44no—
202C4⋊Q8—44no—
203#125—28no—
204#125—28no—
205#125—44no—
206#125—44no—
207#124—48no—
208#124—48no—
209#124—48no—
210#124—48no—
211C4×D4—28no—
212C4×D4—28no—
213C4×D4—44no—
214C4×D4—44no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4C2×D412yes—
250C2×C4⋊C4C2×D412yes—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4#4114yes—
253C2×C22⋊C4—216no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—48no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—42no—
266C2×C42—82no—
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42#4112yes—
273#107—28no—
274#106—28no—
275C4⋊C8—28no—
276#103—216no—
277#103—216no—
278#103—48no—
279#103—48no—
280#102—216no—
281#102—28no—
282#102—28no—
283#97—28no—
284#95—216no—
285#95—24no—
286#95—48no—
287#95—42no—
288#95—84no—
289C24—22no—
290C24—24no—
291C24—41no—
292C24—41no—
293C24—42no—
294C24—44no—
295C24—42no—
296C24—42no—
297C24C22×D411yescharacteristic
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—24no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4#12414yes—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—416no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—42no—
378C22×C4—44no—
379C22×C4—42no—
380C22×C4—48no—
381C22×C4—88no—
382C22×C4—88no—
383C22×C4#12012yes—
384C22×C4#12012yes—
385C22×C4#12112yes—
386C22×C4#12112yes—
387C22×C4#12712yes—
388C22×C4#12712yes—
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4112yes—
396C22×C4C2×#4112yes—
397C22×C4H(5,𝔽2)12yes—
398C22×C42−1+412yes—
399C2×C8—24no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—48no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C22⋊C4—432no—
438C22⋊C4—48no—
439C42—42no—
440C42—42no—
441C42—42no—
442C42—42no—
443C42—84no—
444C42#12014yes—
445C42#12014yes—
446C42#12114yes—
447C42#12114yes—
448C23—24no—
449C23—42no—
450C23—44no—
451C23—42no—
452C23—44no—
453C23—44no—
454C23—48no—
455C23—44no—
456C23—48no—
457C23—84no—
458C23—84no—
459C23—82no—
460C23—81no—
461C23—81no—
462C23—82no—
463C23—82no—
464C23C2×#12011yescharacteristic
465C23C2×#12111yescharacteristic
466C23C2×#12711yescharacteristic
467C23#54612yes—
468C23#54812yes—
469C23#57514yes—
470C23#57712yes—
471C23#57812yes—
472Q8—44no—
473Q8—44no—
474Q8—44no—
475Q8—44no—
476Q8—82no—
477Q8—82no—
478Q8—82no—
479Q8—82no—
480D4—24no—
481D4—24no—
482D4—42no—
483D4—44no—
484D4—44no—
485D4—42no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—48no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—816no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—84no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4#39214yes—
527C2×C4#39214yes—
528C2×C4C2×#12011yescharacteristic
529C2×C4C2×#12011yescharacteristic
530C2×C4C2×#12111yescharacteristic
531C2×C4C2×#12111yescharacteristic
532C2×C4#52512yes—
533C2×C4C2×#12411yescharacteristic
534C2×C4#52912yes—
535C2×C4#53211yescharacteristic
536C2×C4#53511yescharacteristic
537C2×C4#53611yescharacteristic
538C2×C4#53712yes—
539C2×C4#54212yes—
540C2×C4#54612yes—
541C2×C4#54612yes—
542C2×C4#54712yes—
543C2×C4#54812yes—
544C2×C4#54812yes—
545C2×C4#55012yes—
546C8—44no—
547C8—42no—
548C8—42no—
549C22—48no—
550C22—44no—
551C22—84no—
552C22—84no—
553C22—84no—
554C22—82no—
555C22—82no—
556C22—88no—
557C22C2×#39211yescharacteristic
558C22#242612yes—
559C22#243211yescharacteristic
560C22#243511yescharacteristic
561C22#254012yes—
562C22#300512yes—
563C22#301312yes—
564C22#304012yes—
565C22#304612yes—
566C22#315312yes—
567C22#315411yescharacteristic
568C22#315611yescharacteristic
569C22#330714yes—
570C22#331514yes—
571C22#337614yes—
572C22#337714yes—
573C4—24no—
574C4—24no—
575C4—84no—
576C4—84no—
577C4—84no—
578C4—82no—
579C4—82no—
580C4—82no—
581C4#240712yes—
582C4#242712yes—
583C4#243111yescharacteristic
584C4#243611yescharacteristic
585C4#262011yescharacteristic
586C4#285011yescharacteristic
587C2—84no—
588C2#1521511yescharacteristic
589C2#2342711yescharacteristic
590C2#2343111yescharacteristic
591C2#2528914yes—
592C2#2565612yes—
593C2#2589212yes—
594C2#2842712yes—
595C2#2842912yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#491C2semidirect
4. C2×#1981C2semidirect
6. #14160C2semidirect
7. #11124C2semidirect
8. #11122C2semidirect
11. #10835C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4912
C22×#4782
C2×#25472
C2×#19812
C2×#17622
#141602
#111242
#111222
#110092
#108352