Finite Group Databasebeta

← #405901

Subgroups — #405901

Order 512.

1616 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1895C211yesmaximal, characteristic
2C2×#1875C211yesmaximal, characteristic
3C2×#1851C211yesmaximal, characteristic, complemented
4C2×#1736C211yesmaximal, characteristic, complemented
5#14137C211yesmaximal, characteristic
6#13951C211yesmaximal, characteristic, complemented
7#11316C212yesmaximal, complemented
8#11227C212yesmaximal, complemented
9#11076C212yesmaximal, complemented
10#11066C212yesmaximal, complemented
11#10730C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×#102C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C2×#390C2212yes—
22C2×#389C2212yes—
23C2×#386—22no—
24C2×#380C2211yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26#1895—24no—
27#1875—24no—
28#1851—24no—
29#1736—24no—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#336C2212yes—
38#1439—21no—
39#1439—21no—
40#1439—22no—
41C4⋊#95—21no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44#1434—21no—
45C23×D4—21no—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C22×M16—22no—
50C22×M16—22no—
51C22×M16—42no—
52C22×M16—42no—
53C22×M16D414yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C23×C8C2312yes—
57C2×#121—22no—
58C2×#121—22no—
59C2×#121—22no—
60C2×#121—22no—
61C2×#121—22no—
62C2×#121Q814yes—
63C2×#117—22no—
64C2×#117—24no—
65C2×#117—42no—
66C2×#117D414yes—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—24no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—42no—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4D412yes—
84C22×C42—21no—
85C22×C42C2312yes—
86C2×C4⋊C8—24no—
87C2×C4⋊C8—42no—
88C2×C4⋊C8—42no—
89C2×#102—24no—
90C2×#102—22no—
91C2×#102—22no—
92C2×#102—24no—
93C2×#102—42no—
94C2×#102—42no—
95C2×#102—42no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99#390—24no—
100#390—42no—
101#389—24no—
102#389—42no—
103#386—48no—
104#380—22no—
105#380—22no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95D411yescharacteristic
115C2×#95D411yescharacteristic
116#340—24no—
117#340—22no—
118#340—22no—
119#340—24no—
120#340—24no—
121#340—24no—
122#340—42no—
123#336—24no—
124C25—21no—
125C25C2×D411yescharacteristic
126C22×D4—24no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—21no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—44no—
137C22×D4—44no—
138C22×D4—42no—
139C22×D4—42no—
140C22×D4C2×Q812yes—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—44no—
151C23×C4—42no—
152C23×C4—44no—
153C23×C4—81no—
154C23×C4—81no—
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×Q812yes—
161C23×C4#4112yes—
162C23×C4#4112yes—
163C23×C4#4112yes—
164C23×C4C2411yescharacteristic
165C2×M16—28no—
166C2×M16—48no—
167C2×M16—44no—
168C2×M16—44no—
169C2×M16—48no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8C2×D414yes—
187#121—28no—
188#121—28no—
189#121—44no—
190#121—44no—
191#117—28no—
192#117—44no—
193#117—48no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
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—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—48no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—82no—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×Q812yes—
226C2×C4⋊C4#4114yes—
227C2×C4⋊C4#4114yes—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—82no—
242C2×C22⋊C4—82no—
243C2×C22⋊C4—81no—
244C2×C22⋊C4—81no—
245C2×C42—24no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—44no—
253C4⋊C8—48no—
254#102—44no—
255#102—48no—
256#102—44no—
257C22⋊C8—44no—
258#95—22no—
259#95—22no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—42no—
264#95—44no—
265#95—82no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—21no—
271C24—21no—
272C24—21no—
273C24—21no—
274C24—22no—
275C24—41no—
276C24—41no—
277C24—42no—
278C24—42no—
279C24—42no—
280C24—42no—
281C24—41no—
282C24—41no—
283C24—41no—
284C24—41no—
285C24#12112yes—
286C24#12112yes—
287C24C22⋊Q812yes—
288C24C22×D411yescharacteristic
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—416no—
294C2×D4—48no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—84no—
310C2×D4—84no—
311C2×D4—88no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
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—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—48no—
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—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—88no—
372C22×C4—88no—
373C22×C4—84no—
374C22×C4—84no—
375C22×C4#12114yes—
376C22×C4#12112yes—
377C22×C4#12112yes—
378C22×C4C22⋊Q814yes—
379C22×C4C22⋊Q814yes—
380C22×C4C22⋊Q812yes—
381C22×C4#12614yes—
382C22×C4AGL(1,ℤ/8)14yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×Q811yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4112yes—
390C22×C4C2×#4112yes—
391C22×C4H(5,𝔽2)12yes—
392C22×C4H(5,𝔽2)12yes—
393M16—416no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—41no—
408C2×C8—42no—
409C2×C8—41no—
410C2×C8—42no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—48no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—88no—
430C4⋊C4#12314yes—
431C22⋊C4—48no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—88no—
435C22⋊C4—88no—
436C22⋊C4—84no—
437C42—48no—
438C42—42no—
439C42—42no—
440C42—42no—
441C42—42no—
442C23—24no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—24no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—24no—
453C23—24no—
454C23—22no—
455C23—22no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—44no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—44no—
469C23—82no—
470C23—81no—
471C23—81no—
472C23—81no—
473C23—81no—
474C23—81no—
475C23—81no—
476C23—82no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23—81no—
481C23—82no—
482C23—81no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23#39614yes—
488C23C2×#12111yescharacteristic
489C23C2×#12111yescharacteristic
490C23C2×C22⋊Q811yescharacteristic
491C23#54612yes—
492C23#54612yes—
493C23C2×AGL(1,ℤ/8)12yes—
494C23#57512yes—
495C23#57611yescharacteristic
496C23#57711yescharacteristic
497C23#57711yescharacteristic
498C23#57811yescharacteristic
499D4—44no—
500D4—42no—
501D4—42no—
502D4—816no—
503D4—81no—
504D4—81no—
505D4—82no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—28no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—816no—
544C2×C4—82no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4#39614yes—
561C2×C4C2×#12111yescharacteristic
562C2×C4C2×C22⋊Q811yescharacteristic
563C2×C4C2×C22⋊Q811yescharacteristic
564C2×C4C2×#12311yescharacteristic
565C2×C4#52511yescharacteristic
566C2×C4C2×#12611yescharacteristic
567C2×C4#52912yes—
568C2×C4#53212yes—
569C2×C4#53412yes—
570C2×C4#53411yescharacteristic
571C2×C4#53511yescharacteristic
572C2×C4#53812yes—
573C2×C4#55112yes—
574C2×C4#55112yes—
575C2×C4#55312yes—
576C2×C4#55312yes—
577C2×C4#55412yes—
578C2×C4#55412yes—
579C8—44no—
580C8—81no—
581C8—81no—
582C22—24no—
583C22—24no—
584C22—24no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—84no—
594C22—82no—
595C22—82no—
596C22—84no—
597C22—82no—
598C22—162no—
599C22—162no—
600C22—161no—
601C22—161no—
602C22—161no—
603C22—161no—
604C22C2×#39611yescharacteristic
605C22#241111yescharacteristic
606C22#241211yescharacteristic
607C22#245012yes—
608C22#254312yes—
609C22#303811yescharacteristic
610C22#304011yescharacteristic
611C22#304511yescharacteristic
612C22#304611yescharacteristic
613C22#305512yes—
614C22#306211yescharacteristic
615C22#306311yescharacteristic
616C22#307712yes—
617C22#308311yescharacteristic
618C22#308411yescharacteristic
619C22#330612yes—
620C22#330912yes—
621C22#331812yes—
622C22#332112yes—
623C22#332412yes—
624C22#332612yes—
625C22#332712yes—
626C22#333012yes—
627C4—24no—
628C4—44no—
629C4—44no—
630C4—82no—
631C4—82no—
632C4—84no—
633C4—82no—
634C4—162no—
635C4#248711yescharacteristic
636C4#251012yes—
637C4#256412yes—
638C4#257911yescharacteristic
639C4#261711yescharacteristic
640C4#285411yescharacteristic
641C2—44no—
642C2—162no—
643C2#1492111yescharacteristic
644C2#2361611yescharacteristic
645C2#2361811yescharacteristic
646C2#2449211yescharacteristic
647C2#2449911yescharacteristic
648C2#2454411yescharacteristic
649C2#2454711yescharacteristic
650C2#2642412yes—
651C2#2643812yes—
652C2#2855712yes—
653C2#2856912yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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
3. C2×#1851C2semidirect
4. C2×#1736C2semidirect
6. #13951C2semidirect
7. #11316C2semidirect
8. #11227C2semidirect
9. #11076C2semidirect
10. #11066C2semidirect
11. #10730C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18952
C2×#18752
C2×#18512
C2×#17362
#141372
#139512
#113162
#112272
#110762
#110662
#107302