Finite Group Databasebeta

← #397170

Subgroups — #397170

Order 512.

1672 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#467C211yesmaximal, characteristic
2C2×#2056C211yesmaximal, characteristic, complemented
3C2×#1873C212yesmaximal
4C2×#1850C211yesmaximal, characteristic
5C2×#1735C211yesmaximal, characteristic
6#14138C212yesmaximal
7#11256C211yesmaximal, characteristic, complemented
8#11232C211yesmaximal, characteristic, complemented
9#11190C212yesmaximal, complemented
10#11077C212yesmaximal, complemented
11#11024C211yesmaximal, characteristic, complemented
12C23×Q16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C22⋊Q8C2212yes—
15C22×#117C2212yes—
16C2×#467—24no—
17C2×#467—28no—
18C22×#107C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28#2056—24no—
29#1873—28no—
30#1850—24no—
31#1735—24no—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39#1440—22no—
40#1440—22no—
41#1436—22no—
42#1436—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C23×Q8C2312yes—
46C24×C4C2311yescharacteristic
47C22×Q16—24no—
48C22×Q16—22no—
49C22×Q16—22no—
50C22×Q16—22no—
51C22×Q16—42no—
52C23×C8—24no—
53C23×C8—22no—
54C23×C8—22no—
55C23×C8—42no—
56C23×C8D414yes—
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×C22⋊Q8—24no—
62C2×C22⋊Q8—24no—
63C2×C22⋊Q8—24no—
64C2×C22⋊Q8—24no—
65C2×C22⋊Q8—24no—
66C2×C22⋊Q8D418yes—
67C2×#117—24no—
68C2×#117—24no—
69C2×#117—24no—
70C2×#117—24no—
71C2×#117—42no—
72C2×#117—42no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4D411yescharacteristic
80C22×C22⋊C4D411yescharacteristic
81C22×C42—22no—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84#467—232no—
85C2×#107—22no—
86C2×#107—24no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—42no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—42no—
100C2×#103—42no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—24no—
103#391—24no—
104#391—24no—
105#391—42no—
106#391—42no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95D411yescharacteristic
118C2×#95D411yescharacteristic
119#340—22no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340D414yes—
124#336—22no—
125#336—22no—
126C25C2×D411yescharacteristic
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—22no—
139C22×Q8—22no—
140C22×Q8C2×D414yes—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D414yes—
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4#4112yes—
159C23×C4#4114yes—
160C23×C4C2411yescharacteristic
161C2×Q16—28no—
162C2×Q16—48no—
163C2×Q16—48no—
164C2×Q16—44no—
165C2×Q16—44no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—28no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—28no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—22no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—44no—
185C22×C8—48no—
186C22×C8—48no—
187C22×C8C2×D412yes—
188C22×C8C2×D412yes—
189C22×C8C2×D412yes—
190C22×C8#4112yes—
191C22⋊Q8—216no—
192C22⋊Q8—216no—
193C22⋊Q8—48no—
194C22⋊Q8—48no—
195#117—216no—
196#117—48no—
197#117—48no—
198#117—48no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
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—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4C2×D414yes—
218C2×C4⋊C4C2×D414yes—
219C2×C4⋊C4#4114yes—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—22no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—22no—
240C2×C42—21no—
241C2×C42—44no—
242C2×C42—44no—
243C2×C42—44no—
244#107—28no—
245C4⋊C8—44no—
246C4⋊C8—44no—
247#103—216no—
248#103—48no—
249#103—48no—
250C22⋊C8—44no—
251#95—22no—
252#95—22no—
253#95—44no—
254#95—44no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—44no—
261C24—21no—
262C24—22no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—22no—
267C24—21no—
268C24—21no—
269C24—22no—
270C24#12112yes—
271C24#12114yes—
272C24C22×D411yescharacteristic
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—24no—
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—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—21no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
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—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4#12014yes—
368C22×C4#12112yes—
369C22×C4#12114yes—
370C22×C4#12318yes—
371C22×C4#12414yes—
372C22×C4#13512yes—
373C22×C4#13512yes—
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D412yes—
376C22×C4C22×D411yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4112yes—
380C22×C4C2×#4112yes—
381C22×C4H(5,𝔽2)12yes—
382C22×C4H(5,𝔽2)11yescharacteristic
383C22×C42−1+411yescharacteristic
384Q16—416no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—416no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8#12114yes—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—416no—
419C22⋊C4—28no—
420C22⋊C4—44no—
421C22⋊C4—416no—
422C22⋊C4—416no—
423C42—416no—
424C42—41no—
425C42—41no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—42no—
431C23—22no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—41no—
444C23—41no—
445C23—42no—
446C23—42no—
447C23—41no—
448C23—41no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23#39414yes—
454C23C2×#12111yescharacteristic
455C23C2×#12112yes—
456C23D4211yescharacteristic
457C23#54612yes—
458C23#54711yescharacteristic
459C23C2×#13511yescharacteristic
460C23C2×#13511yescharacteristic
461C23#57512yes—
462C23#57811yescharacteristic
463C23#57811yescharacteristic
464C23#57812yes—
465Q8—44no—
466Q8—44no—
467Q8—44no—
468Q8—44no—
469Q8—82no—
470Q8—82no—
471Q8—82no—
472Q8—82no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—41no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—44no—
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—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4#39414yes—
534C2×C4C2×#12011yescharacteristic
535C2×C4C2×#12111yescharacteristic
536C2×C4C2×#12312yes—
537C2×C4#52512yes—
538C2×C4C2×#12411yescharacteristic
539C2×C4#52912yes—
540C2×C4#53211yescharacteristic
541C2×C4#53411yescharacteristic
542C2×C4#53711yescharacteristic
543C2×C4#53812yes—
544C2×C4#53912yes—
545C2×C4D4212yes—
546C2×C4#54612yes—
547C2×C4#54612yes—
548C2×C4#54712yes—
549C2×C4#54811yescharacteristic
550C2×C4#55312yes—
551C2×C4#55311yescharacteristic
552C8—24no—
553C8—42no—
554C8—81no—
555C8—81no—
556C22—24no—
557C22—24no—
558C22—24no—
559C22—44no—
560C22—44no—
561C22—44no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—44no—
567C22C2×#39411yescharacteristic
568C22#241611yescharacteristic
569C22#242412yes—
570C22#242512yes—
571C22#254311yescharacteristic
572C22#303512yes—
573C22#303512yes—
574C22#304412yes—
575C22#304412yes—
576C22#305011yescharacteristic
577C22#305411yescharacteristic
578C22#305911yescharacteristic
579C22#305911yescharacteristic
580C22#326912yes—
581C22#328311yescharacteristic
582C22#328511yescharacteristic
583C22#330012yes—
584C22#330212yes—
585C22#331312yes—
586C22#331512yes—
587C22#333611yescharacteristic
588C22#334211yescharacteristic
589C22#334912yes—
590C4—24no—
591C4—42no—
592C4—42no—
593C4—84no—
594C4—81no—
595C4—84no—
596C4—81no—
597C4—84no—
598C4—84no—
599C4#241011yescharacteristic
600C4#243011yescharacteristic
601C4#244312yes—
602C4#244412yes—
603C4#263911yescharacteristic
604C4#285011yescharacteristic
605C2—44no—
606C2#1474711yescharacteristic
607C2#2353911yescharacteristic
608C2#2354211yescharacteristic
609C2#2477911yescharacteristic
610C2#2478511yescharacteristic
611C2#2507912yes—
612C2#2508912yes—
613C2#2520112yes—
614C2#2521512yes—
615C2#2855411yescharacteristic
616C2#2856411yescharacteristic
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
2. C2×#2056C2semidirect
7. #11256C2semidirect
8. #11232C2semidirect
9. #11190C2semidirect
10. #11077C2semidirect
11. #11024C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4672
C2×#20562
C2×#18732
C2×#18502
C2×#17352
#141382
#112562
#112322
#111902
#110772
#110242