Finite Group Databasebeta

← #468059

Subgroups — #468059

Order 512.

1458 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic, complemented
2C2×#2023C211yesmaximal, characteristic
3C2×#1880C211yesmaximal, characteristic
4#15148C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic, complemented
6#11346C211yesmaximal, characteristic, complemented
7#11207C212yesmaximal, complemented
8#11067C212yesmaximal, complemented
9#11049C212yesmaximal
10#11023C212yesmaximal
11#10789C211yesmaximal, characteristic, complemented
12C23×SD16C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2211yescharacteristic
17C22×#102C2212yes—
18C22×#102C2211yescharacteristic
19#2567—24no—
20#2567—24no—
21C2×#399—21no—
22C2×#395—22no—
23C2×#391C2212yes—
24C2×#390C2211yescharacteristic
25C2×#390C2212yes—
26C2×#389—21no—
27C2×#389—21no—
28C2×#389C2212yes—
29C2×#384C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31#2023—24no—
32#1880—24no—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1439—22no—
43#1439—21no—
44C4⋊#95C2211yescharacteristic
45C23×Q8—21no—
46C23×Q8C2311yescharacteristic
47C23×D4C2311yescharacteristic
48C22×SD16—22no—
49C22×SD16—22no—
50C22×SD16—42no—
51C22×SD16—42no—
52C22×SD16Q814yescomplemented
53C23×C8C2312yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—42no—
58C2×C4⋊Q8—42no—
59C2×C4⋊Q8D414yes—
60C2×C4×Q8—22no—
61C2×C4×Q8—42no—
62C2×C4×Q8—42no—
63C2×C4×Q8D414yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—21no—
79C22×C42—21no—
80C22×C42C2312yes—
81C22×C42C2311yescharacteristic
82C2×#103—22no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—42no—
86C2×#103—42no—
87C2×#102—22no—
88C2×#102—24no—
89C2×#102—42no—
90C2×#102—42no—
91C2×#102—42no—
92#399—44no—
93#395—216no—
94#391—24no—
95#391—24no—
96#390—24no—
97#390—22no—
98#390—22no—
99#390—42no—
100#389—28no—
101#389—28no—
102#389—24no—
103#389—42no—
104#384—24no—
105#384—24no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95D412yes—
114#340—22no—
115#340—24no—
116#340—42no—
117#340—42no—
118#340—42no—
119#336—22no—
120#336—22no—
121#336—42no—
122#336—42no—
123C25—21no—
124C22×Q8—24no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—44no—
133C22×Q8—81no—
134C22×Q8—82no—
135C22×Q8—81no—
136C22×Q8C2×D412yes—
137C22×Q8C2×Q812yes—
138C22×Q8#4112yes—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4C2×Q812yes—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2411yescharacteristic
159C2×SD16—28no—
160C2×SD16—48no—
161C2×SD16—44no—
162C2×SD16—44no—
163C2×SD16—48no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—22no—
180C22×C8C2×Q812yes—
181C4⋊Q8—48no—
182C4⋊Q8—44no—
183C4⋊Q8—44no—
184C4×Q8—44no—
185C4×Q8—44no—
186C4×Q8—48no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4SD1614yescomplemented
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D414yes—
224C2×C4⋊C4#4114yes—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—82no—
234C2×C22⋊C4—82no—
235C2×C22⋊C4—82no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—24no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—44no—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248#103—48no—
249#103—44no—
250#102—44no—
251#102—48no—
252#95—28no—
253#95—28no—
254#95—24no—
255#95—44no—
256#95—42no—
257#95—81no—
258C24—22no—
259C24—41no—
260C24—42no—
261C24—42no—
262C24—41no—
263C24—41no—
264C24—41no—
265C24—42no—
266C24—42no—
267C24—41no—
268C24—41no—
269C24C22×D411yescharacteristic
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—48no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—84no—
289C2×Q8—84no—
290C2×Q8—88no—
291C2×Q8—88no—
292C2×Q8C22⋊Q814yes—
293C2×D4—22no—
294C2×D4—24no—
295C2×D4—22no—
296C2×D4—24no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—42no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—41no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4#12012yes—
359C22×C4#12012yes—
360C22×C4C2×SD1614yes—
361C22×C4C2×SD1612yes—
362C22×C4#13714yes—
363C22×C4#13712yes—
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×Q811yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4112yes—
369C22×C4H(5,𝔽2)12yes—
370C22×C4H(5,𝔽2)12yes—
371C22×C4H(5,𝔽2)12yes—
372C22×C42−1+411yescharacteristic
373C22×C42−1+411yescharacteristic
374SD16—416no—
375C2×C8—22no—
376C2×C8—24no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—42no—
380C2×C8—41no—
381C2×C8—42no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C4⋊C4—28no—
393C4⋊C4—28no—
394C4⋊C4—22no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—48no—
401C4⋊C4—48no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—42no—
410C4⋊C4—88no—
411C4⋊C4—88no—
412C4⋊C4—88no—
413C22⋊C4—88no—
414C22⋊C4—88no—
415C22⋊C4—84no—
416C22⋊C4—84no—
417C42—22no—
418C42—22no—
419C42—24no—
420C42—48no—
421C42—42no—
422C42—42no—
423C42—42no—
424C42—42no—
425C23—42no—
426C23—42no—
427C23—44no—
428C23—42no—
429C23—44no—
430C23—82no—
431C23—81no—
432C23—81no—
433C23—82no—
434C23—81no—
435C23—81no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23—82no—
440C23—82no—
441C23—81no—
442C23—81no—
443C23—81no—
444C23—81no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23C2×#12011yescharacteristic
449C23#53512yes—
450C23#54612yes—
451C23#54811yescharacteristic
452C23D4×Q811yescharacteristic
453C23C22×SD1611yescharacteristic
454C23C2×#13711yescharacteristic
455C23C2×#13712yes—
456C23#57612yes—
457C23#57611yescharacteristic
458C23#57711yescharacteristic
459Q8—24no—
460Q8—24no—
461Q8—42no—
462Q8—42no—
463Q8—44no—
464Q8—44no—
465Q8—816no—
466D4—42no—
467D4—42no—
468D4—44no—
469D4—81no—
470D4—81no—
471D4—82no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—82no—
524C2×C4#44814yes—
525C2×C4C2×C22⋊Q811yescharacteristic
526C2×C4#53211yescharacteristic
527C2×C4#53411yescharacteristic
528C2×C4#53411yescharacteristic
529C2×C4#53412yes—
530C2×C4#53511yescharacteristic
531C2×C4#53611yescharacteristic
532C2×C4#53812yes—
533C2×C4#53912yes—
534C2×C4#54212yes—
535C2×C4#54312yes—
536C2×C4#54811yescharacteristic
537C2×C4D4×Q811yescharacteristic
538C2×C4#55012yes—
539C2×C4#55112yes—
540C2×C4#55412yes—
541C2×C4#56212yes—
542C2×C4#56212yes—
543C8—42no—
544C8—81no—
545C8—82no—
546C22—82no—
547C22—82no—
548C22—84no—
549C22—84no—
550C22—82no—
551C22—162no—
552C22—162no—
553C22—161no—
554C22—161no—
555C22—161no—
556C22—161no—
557C22#240911yescharacteristic
558C22#243111yescharacteristic
559C22#261311yescharacteristic
560C22#269212yes—
561C22#269612yes—
562C22C2×#44811yescharacteristic
563C22#300411yescharacteristic
564C22#300811yescharacteristic
565C22#301211yescharacteristic
566C22#320612yes—
567C22#321812yes—
568C22#322012yes—
569C22#322412yes—
570C22#329512yes—
571C22#331412yes—
572C22#331612yes—
573C22#333312yes—
574C22#336211yescharacteristic
575C22#336911yescharacteristic
576C22#337512yes—
577C22Q8×SD1611yescharacteristic
578C22#338511yescharacteristic
579C22#338512yes—
580C4—24no—
581C4—42no—
582C4—44no—
583C4—44no—
584C4—44no—
585C4—84no—
586C4—82no—
587C4—84no—
588C4—161no—
589C4#256411yescharacteristic
590C4#262611yescharacteristic
591C4#266311yescharacteristic
592C4#273012yes—
593C4#273212yes—
594C4#281611yescharacteristic
595C2—162no—
596C2#1660011yescharacteristic
597C2#2432911yescharacteristic
598C2#2434111yescharacteristic
599C2#2556911yescharacteristic
600C2#2557911yescharacteristic
601C2#2962411yescharacteristic
602C2#2963011yescharacteristic
603C2#3152212yes—
604C2#3153412yes—
605C2#3163812yes—
606C2#3165212yes—
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. C2×#2567C2semidirect
5. #11384C2semidirect
6. #11346C2semidirect
7. #11207C2semidirect
8. #11067C2semidirect
11. #10789C2semidirect
52. C22×SD16Q8semidirect
221. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#20232
C2×#18802
#151482
#113842
#113462
#112072
#110672
#110492
#110232
#107892