Finite Group Databasebeta

← #414668

Subgroups — #414668

Order 512.

1674 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#477C211yesmaximal, characteristic
2C22×#441C211yesmaximal, characteristic
3C2×#2568C211yesmaximal, characteristic
4C2×#1993C211yesmaximal, characteristic
5C2×#1805C211yesmaximal, characteristic
6#13942C211yesmaximal, characteristic
7#11325C212yesmaximal
8#11124C212yesmaximal
9#11052C212yesmaximal
10#11024C212yesmaximal
11#10835C211yesmaximal, characteristic
12C23×Q16C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C2×#477—24no—
17C2×#477—24no—
18C2×#477—24no—
19C2×#441—24no—
20C2×#441—24no—
21C2×#441—24no—
22C22×#107C2212yes—
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×#103C2212yes—
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2212yes—
29C22×#97C2211yescharacteristic
30#2568—24no—
31#2568—24no—
32C2×#400—22no—
33C2×#391—22no—
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#382C2212yes—
37C2×#382C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39#1993—24no—
40#1805—24no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46#1441—22no—
47C4⋊#95C2212yes—
48#1434—22no—
49C23×Q8—21no—
50C23×Q8C2311yescharacteristic
51C23×Q8C2311yescharacteristic
52C22×Q16—22no—
53C22×Q16—22no—
54C22×Q16—22no—
55C22×Q16—22no—
56C22×Q16—22no—
57C22×Q16—22no—
58C22×Q16Q814yescomplemented
59C23×C8C2311yescharacteristic
60C23×C8C2312yes—
61C23×C8C2311yescharacteristic
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8—22no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8D414yes—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8—22no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8—22no—
77C2×C4×Q8—22no—
78C2×C4×Q8D414yes—
79C2×C4×Q8D414yes—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4D412yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C42—21no—
89C22×C42—21no—
90C22×C42C2312yes—
91C22×C42C2311yescharacteristic
92#477—232no—
93#441—232no—
94C2×#107—24no—
95C2×#107—24no—
96C2×#107—24no—
97C2×#106—22no—
98C2×#106—22no—
99C2×#106—22no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8D414yes—
103C2×#103—24no—
104C2×#103—22no—
105C2×#103—22no—
106C2×#103—24no—
107C2×#103—22no—
108C2×#103—22no—
109C2×#103—24no—
110C2×#103—22no—
111C2×#103—22no—
112C2×#103—24no—
113C2×#103—24no—
114C2×#103—42no—
115C2×#97—22no—
116C2×#97—22no—
117C2×#97D414yes—
118#400—216no—
119#391—216no—
120#391—24no—
121#391—24no—
122#391—24no—
123#391—42no—
124#382—24no—
125#382—24no—
126#382—22no—
127#382—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C2×#95—24no—
131C2×#95—24no—
132C2×#95—21no—
133C2×#95—24no—
134C2×#95—24no—
135C2×#95—21no—
136C2×#95D412yes—
137#340—24no—
138#340—24no—
139#340—24no—
140#336—24no—
141#336—42no—
142C22×Q8—24no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—48no—
166C22×Q8—42no—
167C22×Q8—42no—
168C22×Q8C2×D412yes—
169C22×Q8C2×D412yes—
170C22×Q8C2×Q812yes—
171C22×Q8C2×Q812yes—
172C22×Q8#4112yes—
173C22×Q8#4112yes—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—22no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—22no—
180C23×C4C2×D414yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2411yescharacteristic
184C2×Q16—28no—
185C2×Q16—28no—
186C2×Q16—28no—
187C2×Q16—28no—
188C2×Q16—28no—
189C2×Q16—28no—
190C2×Q16—28no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—21no—
202C22×C8—22no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—22no—
207C22×C8C2×D412yes—
208C22×C8C2×D414yes—
209C22×C8C2×D412yes—
210C22×C8C2×Q812yes—
211C22×C8#4112yes—
212C4⋊Q8—28no—
213C4⋊Q8—28no—
214C4⋊Q8—44no—
215C4⋊Q8—44no—
216C4×Q8—28no—
217C4×Q8—28no—
218C4×Q8—28no—
219C4×Q8—28no—
220C4×Q8—44no—
221C4×Q8—44no—
222C4×Q8—44no—
223C4×Q8—44no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2×D414yes—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42C2×D412yes—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42#4112yes—
277#107—216no—
278#106—28no—
279C4⋊C8—28no—
280#103—28no—
281#103—28no—
282#103—216no—
283#103—48no—
284#97—28no—
285#95—216no—
286#95—216no—
287#95—24no—
288#95—48no—
289#95—48no—
290#95—42no—
291#95—42no—
292C24C22×D411yescharacteristic
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—416no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8—48no—
349C2×Q8—88no—
350C2×Q8—88no—
351C2×Q8C22⋊Q814yes—
352C2×Q8C22⋊Q814yes—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—44no—
386C22×C4—48no—
387C22×C4—44no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4#12012yes—
394C22×C4#12012yes—
395C22×C4#12112yes—
396C22×C4#12112yes—
397C22×C4#12712yes—
398C22×C4#12712yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×Q811yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4H(5,𝔽2)11yescharacteristic
410C22×C42−1+411yescharacteristic
411C22×C42−1+411yescharacteristic
412C22×C42−1+411yescharacteristic
413Q16—232no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—24no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—42no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8C22⋊Q814yes—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—24no—
441C4⋊C4—48no—
442C4⋊C4—41no—
443C4⋊C4—42no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—42no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C42—42no—
456C42—42no—
457C42—42no—
458C42—42no—
459C42—48no—
460C42—48no—
461C42#12014yes—
462C42#12014yes—
463C42#12114yes—
464C42#12114yes—
465C23C2×#12011yescharacteristic
466C23C2×#12111yescharacteristic
467C23C2×#12711yescharacteristic
468C23#54611yescharacteristic
469C23#54711yescharacteristic
470C23#54811yescharacteristic
471C23D4×Q811yescharacteristic
472C23#57514yes—
473C23#57814yes—
474Q8—24no—
475Q8—24no—
476Q8—24no—
477Q8—24no—
478Q8—42no—
479Q8—42no—
480Q8—42no—
481Q8—42no—
482Q8—42no—
483Q8—42no—
484Q8—42no—
485Q8—42no—
486Q8—42no—
487Q8—42no—
488Q8—42no—
489Q8—42no—
490Q8—816no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—48no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—82no—
525C2×C4—84no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—84no—
529C2×C4—82no—
530C2×C4—84no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—84no—
535C2×C4#39214yes—
536C2×C4#39214yes—
537C2×C4C2×#12011yescharacteristic
538C2×C4C2×#12011yescharacteristic
539C2×C4C2×#12111yescharacteristic
540C2×C4C2×#12111yescharacteristic
541C2×C4C2×C22⋊Q811yescharacteristic
542C2×C4C2×C22⋊Q811yescharacteristic
543C2×C4C2×C22⋊Q811yescharacteristic
544C2×C4#52511yescharacteristic
545C2×C4#52511yescharacteristic
546C2×C4#52511yescharacteristic
547C2×C4#53611yescharacteristic
548C2×C4#53611yescharacteristic
549C2×C4#53611yescharacteristic
550C2×C4#53711yescharacteristic
551C2×C4#53711yescharacteristic
552C2×C4#53711yescharacteristic
553C2×C4#54611yescharacteristic
554C2×C4#54711yescharacteristic
555C2×C4#54811yescharacteristic
556C2×C4#54811yescharacteristic
557C2×C4#54811yescharacteristic
558C2×C4D4×Q811yescharacteristic
559C2×C4D4×Q811yescharacteristic
560C2×C4#55011yescharacteristic
561C2×C4#55211yescharacteristic
562C2×C4#55211yescharacteristic
563C2×C4#55411yescharacteristic
564C2×C4#55611yescharacteristic
565C8—24no—
566C8—42no—
567C8—44no—
568C22C2×#39211yescharacteristic
569C22#240711yescharacteristic
570C22#240911yescharacteristic
571C22#241511yescharacteristic
572C22#241711yescharacteristic
573C22#254011yescharacteristic
574C22#254211yescharacteristic
575C22#300512yes—
576C22#301112yes—
577C22#304012yes—
578C22#304412yes—
579C22#315312yes—
580C22#315612yes—
581C22#331314yes—
582C22#334914yes—
583C22#337714yes—
584C22#339014yes—
585C4—24no—
586C4—24no—
587C4—44no—
588C4—44no—
589C4—84no—
590C4—82no—
591C4—82no—
592C4—84no—
593C4—162no—
594C4#240711yescharacteristic
595C4#240911yescharacteristic
596C4#241511yescharacteristic
597C4#241711yescharacteristic
598C4#253611yescharacteristic
599C4#253811yescharacteristic
600C4#261611yescharacteristic
601C4#284811yescharacteristic
602C2#1517711yescharacteristic
603C2#2342511yescharacteristic
604C2#2343111yescharacteristic
605C2#2413512yes—
606C2#2434412yes—
607C2#2469912yes—
608C2#2483212yes—
609C2#2842412yes—
610C2#2854612yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
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
58. C22×Q16Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4772
C22×#4412
C2×#25682
C2×#19932
C2×#18052
#139422
#113252
#111242
#110522
#110242
#108352