Finite Group Databasebeta

← #464318

Subgroups — #464318

Order 512.

1478 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2862C211yesmaximal, characteristic
2C2×#2056C212yesmaximal
3C2×#2024C211yesmaximal, characteristic
4C2×#2016C211yesmaximal, characteristic
5#14703C211yesmaximal, characteristic
6#11384C211yesmaximal, characteristic
7#11077C212yesmaximal
8#11067C212yesmaximal
9#11024C212yesmaximal
10#11010C211yesmaximal, characteristic
11#10832C211yesmaximal, characteristic
12C23×Q16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20#2862—28no—
21C2×#391—21no—
22C2×#391—21no—
23C2×#391C2212yes—
24C2×#391C2211yescharacteristic
25C2×#391C2212yes—
26C2×#391C2211yescharacteristic
27C2×#391C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#382—21no—
30C2×#382—21no—
31C2×C43C2211yescharacteristic
32#2056—28no—
33#2024—24no—
34#2016—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1436—21no—
42#1436—21no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45#1434—21no—
46#1434—21no—
47C23×Q8—21no—
48C23×Q8C2312yes—
49C23×Q8C2311yescharacteristic
50C22×Q16—24no—
51C22×Q16—24no—
52C22×Q16—24no—
53C22×Q16—42no—
54C22×Q16—42no—
55C22×Q16—42no—
56C22×Q16—42no—
57C23×C8C2312yes—
58C23×C8C2312yes—
59C2×C4⋊Q8—24no—
60C2×C4⋊Q8—24no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8—42no—
66C2×C4⋊Q8—42no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C2×C4⋊C8—24no—
83C2×C4⋊C8—24no—
84C2×C4⋊C8—42no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—42no—
91C2×#103—42no—
92C2×#103—42no—
93C2×#103—42no—
94C2×#103—42no—
95C2×#103—42no—
96C2×#103—42no—
97#391—28no—
98#391—28no—
99#391—24no—
100#391—24no—
101#391—24no—
102#391—22no—
103#391—24no—
104#391—42no—
105#391D418yes—
106#391D414yes—
107#389—42no—
108#382—28no—
109#382—28no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C43—24no—
126#340—24no—
127#340—24no—
128#340—42no—
129#336—22no—
130#336—22no—
131#336—42no—
132C22×Q8—24no—
133C22×Q8—22no—
134C22×Q8—21no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—21no—
152C22×Q8—22no—
153C22×Q8—21no—
154C22×Q8—22no—
155C22×Q8—42no—
156C22×Q8—42no—
157C22×Q8—42no—
158C22×Q8—42no—
159C22×Q8—42no—
160C22×Q8—42no—
161C22×Q8C2×D414yes—
162C22×Q8C2×D412yes—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2411yescharacteristic
176C2×Q16—48no—
177C2×Q16—48no—
178C2×Q16—48no—
179C2×Q16—48no—
180C2×Q16—48no—
181C2×Q16—48no—
182C2×Q16—48no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8#4114yes—
197C4⋊Q8—48no—
198C4⋊Q8—48no—
199C4⋊Q8—48no—
200C4⋊Q8—48no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4C2×D414yes—
230C2×C4⋊C4C2×D412yes—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—21no—
247C2×C42—22no—
248C2×C42—21no—
249C2×C42—41no—
250C2×C42—41no—
251C2×C42Q1614yescomplemented
252C2×C42C2×D414yes—
253C2×C42C2×D412yes—
254C2×C42#4112yes—
255C4⋊C8—48no—
256#103—48no—
257#103—48no—
258#103—48no—
259#103—48no—
260#95—28no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—24no—
265#95—24no—
266#95—42no—
267#95—42no—
268#95—42no—
269#95—42no—
270#95—42no—
271#95—42no—
272#95—42no—
273#95—42no—
274#95—42no—
275C24C22×D411yescharacteristic
276C2×Q8—24no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—48no—
287C2×Q8—48no—
288C2×Q8—48no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—84no—
324C2×Q8—84no—
325C2×Q8—84no—
326C2×Q8—84no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4#12012yes—
400C22×C4#12012yes—
401C22×C4#12112yes—
402C22×C4#12112yes—
403C22×C4C2×Q1612yes—
404C22×C4C2×Q1612yes—
405C22×C4C2×Q1612yes—
406C22×C4#13512yes—
407C22×C4#13712yes—
408C22×C4#13712yes—
409C22×C4#13712yes—
410C22×C4#13712yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D412yes—
413C22×C4C2×#4112yes—
414C22×C4C2×#4111yescharacteristic
415C22×C4H(5,𝔽2)11yescharacteristic
416C22×C4H(5,𝔽2)12yes—
417C22×C4H(5,𝔽2)11yescharacteristic
418C22×C4H(5,𝔽2)12yes—
419C22×C42−1+411yescharacteristic
420C22×C42−1+411yescharacteristic
421Q16—432no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—84no—
450C42—28no—
451C42—24no—
452C42—24no—
453C42—24no—
454C42—24no—
455C42—28no—
456C42—44no—
457C42—44no—
458C42—42no—
459C42—42no—
460C23C2×#12011yescharacteristic
461C23C2×#12111yescharacteristic
462C23#53211yescharacteristic
463C23#53511yescharacteristic
464C23#53511yescharacteristic
465C23#53711yescharacteristic
466C23#53711yescharacteristic
467C23C22×Q1611yescharacteristic
468C23C2×#13511yescharacteristic
469C23C2×#13711yescharacteristic
470C23C2×#13711yescharacteristic
471C23C2×#13711yescharacteristic
472C23C2×#13711yescharacteristic
473C23#57511yescharacteristic
474C23#57511yescharacteristic
475Q8—44no—
476Q8—44no—
477Q8—44no—
478Q8—44no—
479Q8—44no—
480Q8—44no—
481Q8—816no—
482Q8—82no—
483Q8—82no—
484Q8—82no—
485Q8—82no—
486Q8—82no—
487Q8—82no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—81no—
540C2×C4—82no—
541C2×C4—81no—
542C2×C4—82no—
543C2×C4—81no—
544C2×C4—82no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—82no—
549C2×C4—81no—
550C2×C4—82no—
551C2×C4—81no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4#45114yes—
570C2×C4#46214yes—
571C2×C4#46414yes—
572C2×C4#53511yescharacteristic
573C2×C4#53512yes—
574C2×C4#53611yescharacteristic
575C2×C4#53712yes—
576C2×C4#53812yes—
577C2×C4#54212yes—
578C2×C4#54211yescharacteristic
579C2×C4#54211yescharacteristic
580C2×C4D4212yes—
581C2×C4#54612yes—
582C2×C4#54612yes—
583C2×C4#54612yes—
584C2×C4#54712yes—
585C2×C4#55912yes—
586C2×C4H(3,𝔽4)12yes—
587C2×C4#56212yes—
588C8—44no—
589C8—82no—
590C22#240811yescharacteristic
591C22#243011yescharacteristic
592C22#254411yescharacteristic
593C22#261511yescharacteristic
594C22#262011yescharacteristic
595C22#266211yescharacteristic
596C22#281911yescharacteristic
597C22C2×#45111yescharacteristic
598C22#300211yescharacteristic
599C22#300411yescharacteristic
600C22#300711yescharacteristic
601C22C2×#46211yescharacteristic
602C22C2×#46411yescharacteristic
603C22#303911yescharacteristic
604C22#304011yescharacteristic
605C22#317011yescharacteristic
606C22#317211yescharacteristic
607C22#318111yescharacteristic
608C22#318511yescharacteristic
609C22#320911yescharacteristic
610C22#320911yescharacteristic
611C22#321011yescharacteristic
612C22#321111yescharacteristic
613C22#321511yescharacteristic
614C22#321511yescharacteristic
615C22#321611yescharacteristic
616C22#321811yescharacteristic
617C22#324811yescharacteristic
618C22#324811yescharacteristic
619C22#324911yescharacteristic
620C22#324911yescharacteristic
621C22#325211yescharacteristic
622C22#325211yescharacteristic
623C22#325511yescharacteristic
624C22#325511yescharacteristic
625C4—24no—
626C4—24no—
627C4—24no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—84no—
632C4—82no—
633C4—84no—
634C4—82no—
635C4—161no—
636C4—161no—
637C4#267012yes—
638C4#267212yes—
639C4#271012yes—
640C4#271212yes—
641C2#1650411yescharacteristic
642C2#2421211yescharacteristic
643C2#2421711yescharacteristic
644C2#2549811yescharacteristic
645C2#2550411yescharacteristic
646C2#2861611yescharacteristic
647C2#2862011yescharacteristic
648C2#2966411yescharacteristic
649C2#2966811yescharacteristic
650C2#2983011yescharacteristic
651C2#2983111yescharacteristic
652C2#3033811yescharacteristic
653C2#3034411yescharacteristic
654C2#3280811yescharacteristic
655C2#3281111yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
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
251. C2×C42Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#28622
C2×#20562
C2×#20242
C2×#20162
#147032
#113842
#110772
#110672
#110242
#110102
#108322