Finite Group Databasebeta

← #416953

Subgroups — #416953

Order 512.

1514 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2C2×#1985C211yesmaximal, characteristic, complemented
3C2×#1836C211yesmaximal, characteristic, complemented
4C2×#1804C211yesmaximal, characteristic
5C2×#1762C211yesmaximal, characteristic
6#13925C211yesmaximal, characteristic, complemented
7#11293C212yesmaximal, complemented
8#11292C212yesmaximal, complemented
9#11189C212yesmaximal
10#11189C212yesmaximal
11#10849C211yesmaximal, characteristic, complemented
12C22×C22⋊Q8—21no—
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#102C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C22×#102C2212yes—
19C22×#97C2211yescharacteristic
20#2538—24no—
21#2538—24no—
22C2×#396—22no—
23C2×#390C2212yes—
24C2×#389C2212yes—
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#380—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29#1985—24no—
30#1836—24no—
31#1804—24no—
32#1762—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1446—21no—
43#1445—21no—
44#1436—22no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8D411yescharacteristic
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2312yes—
53C2×C22⋊Q8—28no—
54C2×C22⋊Q8—48no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4D414yes—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D412yes—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C42—21no—
81C22×C42C2312yes—
82C22×C42C2311yescharacteristic
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×#102—24no—
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—22no—
90C2×#102—22no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#97—22no—
94C2×#97—22no—
95C2×#97D414yes—
96#396—216no—
97#390—24no—
98#390—24no—
99#389—24no—
100#389—24no—
101#384—22no—
102#384D414yes—
103#382—22no—
104#382—22no—
105#380—28no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95—22no—
116#340—24no—
117#340—22no—
118#340—22no—
119#340—24no—
120#340—24no—
121#340—24no—
122#340—24no—
123#336—24no—
124#336—24no—
125C25—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—24no—
130C22×Q8—42no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4C2×D412yes—
140C22×D4#4114yes—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—24no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—44no—
148C23×C4—44no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2411yescharacteristic
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8C2×D414yes—
173C22⋊Q8—432no—
174C4×Q8—44no—
175C4×Q8—44no—
176C4×Q8—44no—
177C4×Q8—44no—
178C4×D4—28no—
179C4×D4—28no—
180C4×D4—44no—
181C4×D4—44no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—81no—
218C2×C4⋊C4—81no—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D414yes—
221C2×C4⋊C4#4114yes—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—82no—
230C2×C22⋊C4—82no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—42no—
238C2×C42—81no—
239C2×C42—81no—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42#4112yes—
244C2×C42#4112yes—
245C2×C42#4112yes—
246C4⋊C8—44no—
247#102—216no—
248#102—44no—
249#102—44no—
250#97—28no—
251#95—28no—
252#95—44no—
253#95—42no—
254#95—42no—
255#95—42no—
256#95—44no—
257#95—82no—
258#95—82no—
259C24—22no—
260C24—24no—
261C24—41no—
262C24—41no—
263C24—42no—
264C24—42no—
265C24—42no—
266C24—41no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24C22×D411yescharacteristic
271C2×Q8—28no—
272C2×Q8—48no—
273C2×Q8—48no—
274C2×Q8—44no—
275C2×Q8—44no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4#12414yes—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—48no—
317C22×C4—48no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—48no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—88no—
346C22×C4—88no—
347C22×C4#12012yes—
348C22×C4#12012yes—
349C22×C4#12112yes—
350C22×C4#12112yes—
351C22×C4#12312yes—
352C22×C4#12312yes—
353C22×C4AGL(1,ℤ/8)14yes—
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4112yes—
361C22×C4C2×#4112yes—
362C22×C42−1+412yes—
363C22×C42−1+412yes—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—41no—
380C2×C8—41no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—48no—
388C4⋊C4—416no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—48no—
396C4⋊C4—44no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—84no—
404C22⋊C4—416no—
405C22⋊C4—48no—
406C22⋊C4—88no—
407C42—42no—
408C42—42no—
409C42—42no—
410C42—42no—
411C42—84no—
412C42#12114yes—
413C42#12114yes—
414C42#12314yes—
415C42#12414yes—
416C23—24no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—42no—
422C23—44no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—44no—
429C23—82no—
430C23—81no—
431C23—81no—
432C23—82no—
433C23—82no—
434C23—82no—
435C23—82no—
436C23—82no—
437C23—81no—
438C23—81no—
439C23C2×#12011yescharacteristic
440C23C2×#12111yescharacteristic
441C23C2×#12311yescharacteristic
442C23#54812yes—
443C23#54812yes—
444C23C2×AGL(1,ℤ/8)12yes—
445C23#57512yes—
446C23#57611yescharacteristic
447C23#57711yescharacteristic
448C23#57711yescharacteristic
449C23#57811yescharacteristic
450Q8—416no—
451D4—24no—
452D4—24no—
453D4—42no—
454D4—44no—
455D4—44no—
456D4—42no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—816no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4#39414yes—
512C2×C4#39414yes—
513C2×C4C2×#12111yescharacteristic
514C2×C4C2×#12111yescharacteristic
515C2×C4C2×#12311yescharacteristic
516C2×C4#52512yes—
517C2×C4C2×#12411yescharacteristic
518C2×C4C2×#12411yescharacteristic
519C2×C4#52912yes—
520C2×C4#53211yescharacteristic
521C2×C4#53612yes—
522C2×C4#53611yescharacteristic
523C2×C4#53611yescharacteristic
524C2×C4#54112yes—
525C2×C4#54712yes—
526C2×C4#54812yes—
527C2×C4#55212yes—
528C2×C4#55212yes—
529C2×C4#55212yes—
530C2×C4#55612yes—
531C8—44no—
532C8—81no—
533C8—81no—
534C22—44no—
535C22—44no—
536C22—44no—
537C22—84no—
538C22—84no—
539C22—82no—
540C22—82no—
541C22—84no—
542C22—84no—
543C22—82no—
544C22—82no—
545C22C2×#39411yescharacteristic
546C22#242112yes—
547C22#243211yescharacteristic
548C22#243611yescharacteristic
549C22#253612yes—
550C22#300311yescharacteristic
551C22#300511yescharacteristic
552C22#301211yescharacteristic
553C22#301311yescharacteristic
554C22#305512yes—
555C22#306211yescharacteristic
556C22#306311yescharacteristic
557C22#309412yes—
558C22#309811yescharacteristic
559C22#309911yescharacteristic
560C22#336612yes—
561C22#336712yes—
562C22#336812yes—
563C22#337012yes—
564C22#337312yes—
565C22#337612yes—
566C22#337612yes—
567C22#337712yes—
568C4—24no—
569C4—24no—
570C4—44no—
571C4—84no—
572C4—84no—
573C4—82no—
574C4—82no—
575C4—161no—
576C4—161no—
577C4#241512yes—
578C4#248912yes—
579C4#249911yescharacteristic
580C4#250311yescharacteristic
581C4#262111yescharacteristic
582C4#285011yescharacteristic
583C2—84no—
584C2#1523611yescharacteristic
585C2#2348011yescharacteristic
586C2#2348511yescharacteristic
587C2#2500612yes—
588C2#2500712yes—
589C2#2565511yescharacteristic
590C2#2565611yescharacteristic
591C2#2593911yescharacteristic
592C2#2594011yescharacteristic
593C2#2819712yes—
594C2#2819912yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1985C2semidirect
3. C2×#1836C2semidirect
6. #13925C2semidirect
7. #11293C2semidirect
8. #11292C2semidirect
11. #10849C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#19852
C2×#18362
C2×#18042
C2×#17622
#139252
#112932
#112922
#111892
#108492