Finite Group Databasebeta

← #409026

Subgroups — #409026

Order 512.

1420 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2086C211yesmaximal, characteristic, complemented
2C2×#2075C211yesmaximal, characteristic
3C2×#2056C211yesmaximal, characteristic, complemented
4C2×#2001C211yesmaximal, characteristic
5C2×#1969C211yesmaximal, characteristic, complemented
6C2×#1923C211yesmaximal, characteristic
7#13997C212yesmaximal
8#11067C212yesmaximal, complemented
9#10970C212yesmaximal
10#10932C212yesmaximal, complemented
11C4×#340C211yesmaximal, characteristic, complemented
12C23×Q16C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C22×#107C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22C2×#391C2212yes—
23C2×#391C2212yes—
24C2×#389C2212yes—
25C2×#380C2212yes—
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28#2086—24no—
29#2075—24no—
30#2056—24no—
31#2001—24no—
32#1969—24no—
33#1923—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41#1445—22no—
42#1439—22no—
43#1437—22no—
44C4⋊#95—22no—
45C23×Q8C2312yes—
46C24×C4C2311yescharacteristic
47C22×Q16—24no—
48C22×Q16—22no—
49C22×Q16—22no—
50C22×Q16—22no—
51C22×Q16—42no—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×C22⋊Q8—24no—
57C2×C22⋊Q8—24no—
58C2×C22⋊Q8—24no—
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—42no—
61C2×C22⋊Q8—42no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4D411yescharacteristic
70C22×C22⋊C4D411yescharacteristic
71C22×C42—22no—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×#107—22no—
75C2×#107—22no—
76C2×#107—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—42no—
87C2×#103—42no—
88C2×#103—42no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—42no—
94C2×C4×C8—42no—
95C2×C4×C8D414yes—
96#391—24no—
97#391—24no—
98#391—24no—
99#391—42no—
100#389—24no—
101#389—42no—
102#380—24no—
103#380—42no—
104C4×C4⋊C4—24no—
105C4×C4⋊C4—42no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95D411yescharacteristic
116C2×#95D411yescharacteristic
117#340—24no—
118#340—24no—
119#340—24no—
120#340D414yes—
121#340D414yes—
122#336—22no—
123#336—22no—
124C25C2×D411yescharacteristic
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8#4114yes—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2411yescharacteristic
159C2×Q16—28no—
160C2×Q16—48no—
161C2×Q16—48no—
162C2×Q16—44no—
163C2×Q16—44no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8C2×D412yes—
181C22×C8C2×D412yes—
182C22×C8C2×D412yes—
183C22×C8C2×D412yes—
184C22⋊Q8—48no—
185C22⋊Q8—48no—
186C22⋊Q8—48no—
187C22⋊Q8—48no—
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—22no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4C2×D414yes—
209C2×C4⋊C4C2×D414yes—
210C2×C4⋊C4#4114yes—
211C2×C4⋊C4#4114yes—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—82no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—44no—
232C2×C42—44no—
233C2×C42—82no—
234C2×C42C2×D412yes—
235C2×C42#4112yes—
236#107—44no—
237C4⋊C8—44no—
238#103—48no—
239#103—48no—
240#103—48no—
241C22⋊C8—44no—
242C22⋊C8—44no—
243C4×C8—44no—
244#95—22no—
245#95—22no—
246#95—44no—
247#95—44no—
248#95—44no—
249#95—44no—
250#95—44no—
251#95—44no—
252#95—44no—
253C24—21no—
254C24—22no—
255C24—21no—
256C24—21no—
257C24—22no—
258C24—21no—
259C24—22no—
260C24—21no—
261C24—21no—
262C24—22no—
263C24#13512yes—
264C24C22×D411yescharacteristic
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—24no—
298C22×C4—21no—
299C22×C4—24no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—24no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
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—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4#12112yes—
363C22×C4#12112yes—
364C22×C4#13512yes—
365C22×C4#13714yes—
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C2×#4112yes—
370C22×C4C2×#4112yes—
371C22×C4C2×#4112yes—
372C22×C4C2×#4111yescharacteristic
373C22×C4H(5,𝔽2)12yes—
374C22×C4H(5,𝔽2)11yescharacteristic
375C22×C4H(5,𝔽2)11yescharacteristic
376Q16—416no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—41no—
387C2×C8—42no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—42no—
393C2×C8—41no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8#12714yes—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—416no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—88no—
410C4⋊C4#12418yes—
411C22⋊C4—28no—
412C22⋊C4—416no—
413C22⋊C4—44no—
414C22⋊C4—88no—
415C42—22no—
416C42—22no—
417C42—24no—
418C42—41no—
419C42—41no—
420C42—41no—
421C42—41no—
422C42—88no—
423C23—24no—
424C23—22no—
425C23—22no—
426C23—24no—
427C23—22no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—41no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—41no—
437C23—41no—
438C23—42no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—42no—
443C23—42no—
444C23—41no—
445C23—41no—
446C23C2×#12111yescharacteristic
447C23#52512yes—
448C23D4211yescharacteristic
449C23D4211yescharacteristic
450C23#54612yes—
451C23C2×#13511yescharacteristic
452C23C2×#13511yescharacteristic
453C23C2×#13712yes—
454C23#57611yescharacteristic
455C23#57812yes—
456C23#57811yescharacteristic
457Q8—44no—
458Q8—44no—
459Q8—44no—
460Q8—44no—
461Q8—82no—
462Q8—82no—
463Q8—82no—
464Q8—82no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—28no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—41no—
486C2×C4—44no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—82no—
516C2×C4—81no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4#47014yes—
526C2×C4#52511yescharacteristic
527C2×C4C2×#12412yes—
528C2×C4#52912yes—
529C2×C4C2×#12711yescharacteristic
530C2×C4#53211yescharacteristic
531C2×C4#53211yescharacteristic
532C2×C4#53212yes—
533C2×C4#53411yescharacteristic
534C2×C4#53511yescharacteristic
535C2×C4#53812yes—
536C2×C4#54012yes—
537C2×C4#54612yes—
538C2×C4#54612yes—
539C2×C4#55011yescharacteristic
540C2×C4#55011yescharacteristic
541C2×C4#55112yes—
542C2×C4#55312yes—
543C2×C4#55312yes—
544C8—24no—
545C8—42no—
546C8—81no—
547C8—81no—
548C22—44no—
549C22—42no—
550C22—44no—
551C22—42no—
552C22—42no—
553C22—82no—
554C22—81no—
555C22—82no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22#239211yescharacteristic
560C22#246412yes—
561C22#247212yes—
562C22#254111yescharacteristic
563C22#254311yescharacteristic
564C22#305011yescharacteristic
565C22C2×#47011yescharacteristic
566C22#305911yescharacteristic
567C22#306111yescharacteristic
568C22#310712yes—
569C22#310812yes—
570C22#312412yes—
571C22#312712yes—
572C22#327612yes—
573C22#327612yes—
574C22#328011yescharacteristic
575C22#328011yescharacteristic
576C22#328511yescharacteristic
577C22#328511yescharacteristic
578C22#330012yes—
579C22#330212yes—
580C22#331212yes—
581C22#331412yes—
582C4—24no—
583C4—42no—
584C4—48no—
585C4—42no—
586C4—84no—
587C4—81no—
588C4—81no—
589C4—84no—
590C4—162no—
591C4#249012yes—
592C4#253911yescharacteristic
593C4#255412yes—
594C4#260311yescharacteristic
595C4#266111yescharacteristic
596C4#285811yescharacteristic
597C2—82no—
598C2#1500711yescharacteristic
599C2#2369911yescharacteristic
600C2#2370511yescharacteristic
601C2#2693112yes—
602C2#2693712yes—
603C2#2722312yes—
604C2#2723712yes—
605C2#2848411yescharacteristic
606C2#2848911yescharacteristic
607C2#2855411yescharacteristic
608C2#2856311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2086C2semidirect
3. C2×#2056C2semidirect
5. C2×#1969C2semidirect
8. #11067C2semidirect
10. #10932C2semidirect
11. C4×#340C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20862
C2×#20752
C2×#20562
C2×#20012
C2×#19692
C2×#19232
#139972
#110672
#109702
#109322
C4×#3402