Finite Group Databasebeta

← #412455

Subgroups — #412455

Order 512.

2126 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#459C211yesmaximal, characteristic
2C2×#2054C214yesmaximal, complemented
3C2×#1982C211yesmaximal, characteristic, complemented
4C2×#1842C211yesmaximal, characteristic, complemented
5C2×#1802C211yesmaximal, characteristic, complemented
6#14713C211yesmaximal, characteristic, complemented
7C4×#390C211yesmaximal, characteristic, complemented
8#11076C214yesmaximal, complemented
9#10849C211yesmaximal, characteristic, complemented
10C23×D8C2212yes—
11C22×#127C2211yescharacteristic
12C22×C4×D4C2211yescharacteristic
13C2×#459—24no—
14C2×#459—24no—
15C2×#459—24no—
16C22×C4⋊C8C2211yescharacteristic
17C22×C4⋊C8C2214yes—
18C22×C4⋊C8C2211yescharacteristic
19C22×#102C2212yes—
20C22×#102C2212yes—
21C22×#102C2214yes—
22C2×#390—21no—
23C2×#390—22no—
24C2×#390—21no—
25C2×#390C2214yes—
26C2×#390C2212yes—
27C2×#384C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4—21no—
30C2×C4×C22⋊C4—21no—
31C2×C43C2211yescharacteristic
32#2054—216no—
33#1982—24no—
34#1842—24no—
35#1802—24no—
36C2×#340C2212yes—
37C2×#340C2214yes—
38#1437—22no—
39C4⋊#95C2212yes—
40C23×D4—21no—
41C23×D4—21no—
42C23×D4C2312yes—
43C23×D4C2311yescharacteristic
44C24×C4—21no—
45C22×D8—24no—
46C22×D8—24no—
47C22×D8—28no—
48C22×D8—24no—
49C22×D8—44no—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×#127—22no—
53C2×#127—22no—
54C2×#127—22no—
55C2×#127—22no—
56C2×#127—22no—
57C2×#127D414yes—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4D414yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—24no—
73C22×C22⋊C4—24no—
74C22×C42—21no—
75C22×C42D411yescharacteristic
76C22×C42D412yes—
77C22×C42D411yescharacteristic
78C22×C42C2312yes—
79C22×C42C2311yescharacteristic
80#459—232no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—28no—
84C2×C4⋊C8—28no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—44no—
89C2×C4⋊C8D414yes—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—28no—
94C2×#102—24no—
95C2×#102—28no—
96C2×#102—28no—
97C2×#102—44no—
98#390—28no—
99#390—24no—
100#390—28no—
101#390—44no—
102#390—48no—
103#390—44no—
104#390D418yescomplemented
105#384—22no—
106#384—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C22⋊C4—44no—
110C4×C22⋊C4—44no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C43—22no—
115C43—22no—
116#340—28no—
117#340—24no—
118C25—21no—
119C25—22no—
120C22×D4—24no—
121C22×D4—24no—
122C22×D4—21no—
123C22×D4—21no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—24no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—24no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—24no—
137C22×D4—22no—
138C22×D4—24no—
139C22×D4—44no—
140C22×D4—44no—
141C22×D4—42no—
142C22×D4—42no—
143C22×D4—42no—
144C22×D4—42no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4C2×D414yes—
150C22×D4C2×D412yes—
151C22×D4C2×D414yes—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—24no—
155C23×C4—22no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
167C2×D8—216no—
168C2×D8—416no—
169C2×D8—416no—
170C2×D8—48no—
171C2×D8—48no—
172C22×C8—24no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—24no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—24no—
179C22×C8—22no—
180C22×C8—24no—
181C22×C8C2×D414yes—
182#127—28no—
183#127—28no—
184#127—44no—
185#127—44no—
186C4×D4—28no—
187C4×D4—28no—
188C4×D4—44no—
189C4×D4—44no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D414yes—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—48no—
221C2×C22⋊C4—48no—
222C2×C22⋊C4—48no—
223C2×C22⋊C4—48no—
224C2×C22⋊C4—48no—
225C2×C22⋊C4—48no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—24no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—22no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42D814yescomplemented
242C2×C42C2×D414yes—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42#4114yes—
247C2×C42#4112yes—
248C2×C42#4112yes—
249C2×C42#4112yes—
250C4⋊C8—28no—
251C4⋊C8—416no—
252C4⋊C8—44no—
253#102—216no—
254#102—416no—
255#102—48no—
256#95—44no—
257#95—42no—
258#95—44no—
259C24—24no—
260C24—22no—
261C24—24no—
262C24—41no—
263C24—41no—
264C24—42no—
265C24—41no—
266C24—41no—
267C24—41no—
268C24—42no—
269C24—41no—
270C24—42no—
271C24—44no—
272C24—42no—
273C24—42no—
274C24—44no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24—44no—
279C24—42no—
280C24—44no—
281C24C22×D411yescharacteristic
282C2×D4—22no—
283C2×D4—24no—
284C2×D4—22no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—28no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—28no—
296C2×D4—416no—
297C2×D4—41no—
298C2×D4—42no—
299C2×D4—41no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—44no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—42no—
310C2×D4—44no—
311C2×D4—44no—
312C2×D4—42no—
313C2×D4—44no—
314C2×D4—42no—
315C2×D4—44no—
316C2×D4—42no—
317C2×D4—44no—
318C2×D4—48no—
319C2×D4—48no—
320C2×D4—48no—
321C2×D4—48no—
322C2×D4—84no—
323C2×D4—84no—
324C2×D4—88no—
325C2×D4—84no—
326C2×D4—84no—
327C2×D4—84no—
328C2×D4—84no—
329C2×D4#12714yes—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—48no—
359C22×C4—44no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—44no—
363C22×C4—42no—
364C22×C4—84no—
365C22×C4—84no—
366C22×C4—84no—
367C22×C4—84no—
368C22×C4#12114yes—
369C22×C4#12112yes—
370C22×C4#12112yes—
371C22×C4#12112yes—
372C22×C4#12114yes—
373C22×C4#12112yes—
374C22×C4#12112yes—
375C22×C4#12112yes—
376C22×C4C2×D814yes—
377C22×C4C2×D812yes—
378C22×C4#13512yes—
379C22×C4AGL(1,ℤ/8)14yes—
380C22×C4AGL(1,ℤ/8)12yes—
381C22×C4AGL(1,ℤ/8)12yes—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D412yes—
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4112yes—
390C22×C4H(5,𝔽2)12yes—
391C22×C42−1+412yes—
392D8—432no—
393C2×C8—24no—
394C2×C8—28no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—44no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—44no—
402C2×C8—44no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—44no—
413C4⋊C4—42no—
414C4⋊C4—84no—
415C4⋊C4—84no—
416C22⋊C4—416no—
417C22⋊C4—48no—
418C22⋊C4—416no—
419C22⋊C4—816no—
420C22⋊C4—816no—
421C42—24no—
422C42—24no—
423C42—24no—
424C42—24no—
425C42—28no—
426C42—48no—
427C42—44no—
428C42—84no—
429C42#12014yes—
430C42#12314yes—
431C42#12314yes—
432C42#12414yes—
433C23—24no—
434C23—42no—
435C23—42no—
436C23—44no—
437C23—44no—
438C23—42no—
439C23—42no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—42no—
444C23—42no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—48no—
450C23—48no—
451C23—82no—
452C23—82no—
453C23—82no—
454C23—82no—
455C23—81no—
456C23—81no—
457C23—82no—
458C23—81no—
459C23—81no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23—84no—
464C23—84no—
465C23—84no—
466C23—84no—
467C23—84no—
468C23—84no—
469C23—82no—
470C23—82no—
471C23—84no—
472C23—84no—
473C23—84no—
474C23—82no—
475C23—82no—
476C23—82no—
477C23—82no—
478C23—84no—
479C23C2×#12111yescharacteristic
480C23C2×#12112yes—
481C23C2×#12111yescharacteristic
482C23C2×#12111yescharacteristic
483C23#53712yes—
484C23C22×D811yescharacteristic
485C23C2×#13511yescharacteristic
486C23C2×AGL(1,ℤ/8)12yes—
487C23C2×AGL(1,ℤ/8)11yescharacteristic
488C23C2×AGL(1,ℤ/8)11yescharacteristic
489C23#57512yes—
490D4—24no—
491D4—24no—
492D4—44no—
493D4—48no—
494D4—42no—
495D4—44no—
496D4—44no—
497D4—44no—
498D4—42no—
499D4—816no—
500D4—816no—
501D4—84no—
502D4—82no—
503D4—82no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—82no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—816no—
536C2×C4—82no—
537C2×C4—84no—
538C2×C4—82no—
539C2×C4—84no—
540C2×C4—82no—
541C2×C4#39414yes—
542C2×C4#39414yes—
543C2×C4#45914yes—
544C2×C4#45914yes—
545C2×C4#45918yescomplemented
546C2×C4#46314yes—
547C2×C4#46314yes—
548C2×C4#46614yes—
549C2×C4#46914yes—
550C2×C4C2×#12011yescharacteristic
551C2×C4C2×#12311yescharacteristic
552C2×C4C2×#12311yescharacteristic
553C2×C4C2×#12411yescharacteristic
554C2×C4C2×#12711yescharacteristic
555C2×C4#53212yes—
556C2×C4#53211yescharacteristic
557C2×C4#53511yescharacteristic
558C2×C4#53611yescharacteristic
559C2×C4#54212yes—
560C2×C4D4212yes—
561C2×C4D4212yes—
562C2×C4#54712yes—
563C2×C4#54712yes—
564C2×C4#54712yes—
565C2×C4#54812yes—
566C2×C4#55312yes—
567C2×C4#55512yes—
568C8—44no—
569C8—82no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—84no—
574C22—84no—
575C22—82no—
576C22—82no—
577C22—84no—
578C22—84no—
579C22—82no—
580C22—82no—
581C22—84no—
582C22—88no—
583C22—88no—
584C22—84no—
585C22—84no—
586C22—162no—
587C22—162no—
588C22—162no—
589C22—162no—
590C22—164no—
591C22—164no—
592C22C2×#39411yescharacteristic
593C22#241612yes—
594C22#253712yes—
595C22#262011yescharacteristic
596C22#285611yescharacteristic
597C22C2×#45911yescharacteristic
598C22C2×#45912yes—
599C22C2×#45911yescharacteristic
600C22C2×#46311yescharacteristic
601C22C2×#46311yescharacteristic
602C22#303512yes—
603C22#303612yes—
604C22#303612yes—
605C22#303812yes—
606C22#303811yescharacteristic
607C22#304011yescharacteristic
608C22C2×#46611yescharacteristic
609C22C2×#46911yescharacteristic
610C22#305512yes—
611C22#324512yes—
612C22#324712yes—
613C22#325112yes—
614C22#325312yes—
615C4—28no—
616C4—24no—
617C4—44no—
618C4—84no—
619C4—82no—
620C4—162no—
621C4#205418yescomplemented
622C4#242212yes—
623C4#243211yescharacteristic
624C4#243311yescharacteristic
625C4#249112yes—
626C4#250011yescharacteristic
627C4#250111yescharacteristic
628C2—84no—
629C2—164no—
630C2#1513011yescharacteristic
631C2C2×#205411yescharacteristic
632C2#2354611yescharacteristic
633C2#2475012yes—
634C2#2475812yes—
635C2#2825112yes—
636C2#2825312yes—
637C2#2982411yescharacteristic
638C2#2982511yescharacteristic
639C2#3316011yescharacteristic
640C2#3316411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#459C2
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2054C2semidirect
3. C2×#1982C2semidirect
4. C2×#1842C2semidirect
5. C2×#1802C2semidirect
6. #14713C2semidirect
7. C4×#390C2semidirect
8. #11076C2semidirect
9. #10849C2semidirect
104. #390D4semidirect
241. C2×C42D8semidirect
545. C2×C4#459semidirect
621. C4#2054semidirect

Maximal subgroups

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

subgroupindex
C22×#4592
C2×#20542
C2×#19822
C2×#18422
C2×#18022
#147132
C4×#3902
#110762
#108492