Finite Group Databasebeta

← #414736

Subgroups — #414736

Order 512.

1600 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#479C211yesmaximal, characteristic
2C2×#2582C211yesmaximal, characteristic
3C2×#2584C211yesmaximal, characteristic
4C2×#1992C211yesmaximal, characteristic
5C2×#1985C211yesmaximal, characteristic
6C2×#1763C211yesmaximal, characteristic
7#11328C212yesmaximal
8#11327C212yesmaximal
9#11067C212yesmaximal
10#11067C212yesmaximal
11#10849C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C2×#479—24no—
16C2×#479—24no—
17C2×#479—24no—
18C22×#107C2211yescharacteristic
19C22×#106C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#103C2211yescharacteristic
22C22×#103C2211yescharacteristic
23C22×#103C2212yes—
24C22×#97C2211yescharacteristic
25#2584—24no—
26#2584—24no—
27#2582—24no—
28#2582—24no—
29C2×#398—21no—
30C2×#398—22no—
31C2×#395—21no—
32C2×#391—21no—
33C2×#391C2212yes—
34C2×#389—22no—
35C2×#389—21no—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#384C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41#1992—24no—
42#1985—24no—
43#1763—24no—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47C2×#340C2212yes—
48C2×#340C2212yes—
49C2×#336C2212yes—
50C2×#336C2212yes—
51C23×Q8D411yescharacteristic
52C23×Q8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2312yes—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125—22no—
59C2×#125—22no—
60C2×#125—22no—
61C2×#125D414yes—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8D414yes—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4D412yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C42—21no—
90C22×C42—21no—
91C22×C42C2312yes—
92C22×C42C2311yescharacteristic
93#479—232no—
94C2×#107—22no—
95C2×#107—22no—
96C2×#107—22no—
97C2×#106—22no—
98C2×#106—22no—
99C2×#106—22no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8D414yes—
103C2×#103—22no—
104C2×#103—22no—
105C2×#103—22no—
106C2×#103—22no—
107C2×#103—22no—
108C2×#103—22no—
109C2×#103—24no—
110C2×#103—24no—
111C2×#103Q818yescomplemented
112C2×#97—22no—
113C2×#97—22no—
114C2×#97—22no—
115#398—216no—
116#398—28no—
117#395—28no—
118#391—28no—
119#391—24no—
120#391—24no—
121#389—28no—
122#389—216no—
123#389—24no—
124#389—24no—
125#389—24no—
126#389—24no—
127#389—24no—
128#389—24no—
129#384—22no—
130#384D414yes—
131C4×C4⋊C4—22no—
132C4×C4⋊C4D414yes—
133C2×#95—21no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—22no—
139#340—24no—
140#340—24no—
141#340—24no—
142#340—24no—
143#340—24no—
144#336—24no—
145#336—24no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—24no—
150C22×Q8—21no—
151C22×Q8—22no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—42no—
159C22×Q8C2×D412yes—
160C22×Q8C2×Q814yes—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—21no—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2411yescharacteristic
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8C2×D412yes—
187C22×C8C2×D412yes—
188C22×C8C2×Q814yes—
189#125—28no—
190#125—28no—
191#125—44no—
192#125—44no—
193C4×Q8—28no—
194C4×Q8—28no—
195C4×Q8—44no—
196C4×Q8—44no—
197C4×Q8—44no—
198C4×Q8—44no—
199C4×Q8—44no—
200C4×Q8—44no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—81no—
248C2×C4⋊C4—81no—
249C2×C4⋊C4—82no—
250C2×C4⋊C4—82no—
251C2×C4⋊C4—82no—
252C2×C4⋊C4—82no—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×D414yes—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2×Q814yes—
259C2×C4⋊C4#4114yes—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—81no—
271C2×C42—81no—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42#4112yes—
278#107—28no—
279#106—28no—
280C4⋊C8—28no—
281#103—28no—
282#103—28no—
283#103—216no—
284#97—44no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—24no—
291C24C22×D411yescharacteristic
292C2×Q8—28no—
293C2×Q8—22no—
294C2×Q8—24no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—24no—
298C2×Q8—22no—
299C2×Q8—24no—
300C2×Q8—22no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—48no—
304C2×Q8—48no—
305C2×Q8—44no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—44no—
314C2×Q8C4⋊Q814yes—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
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—22no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4#12012yes—
373C22×C4#12012yes—
374C22×C4#12012yes—
375C22×C4#12012yes—
376C22×C4#12112yes—
377C22×C4#12112yes—
378C22×C4#13714yes—
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×Q812yes—
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4112yes—
387C22×C42−1+412yes—
388C22×C42−1+412yes—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—42no—
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—42no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—48no—
418C4⋊C4—416no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—48no—
428C4⋊C4—44no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—84no—
433C4⋊C4—88no—
434C4⋊C4—88no—
435C4⋊C4C22⋊Q818yes—
436C42—42no—
437C42—42no—
438C42—42no—
439C42—42no—
440C42—48no—
441C42—84no—
442C42#12014yes—
443C42#12114yes—
444C42#12114yes—
445C42#12714yes—
446C23C2×#12011yescharacteristic
447C23C2×#12011yescharacteristic
448C23C2×#12111yescharacteristic
449C23#54812yes—
450C23D4×Q812yes—
451C23C2×#13712yes—
452C23#57512yes—
453C23#57611yescharacteristic
454C23#57711yescharacteristic
455C23#57711yescharacteristic
456C23#57811yescharacteristic
457Q8—24no—
458Q8—24no—
459Q8—416no—
460Q8—42no—
461Q8—44no—
462Q8—42no—
463Q8—44no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—28no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
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—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—82no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4#39214yes—
527C2×C4#39214yes—
528C2×C4C2×#12011yescharacteristic
529C2×C4C2×#12111yescharacteristic
530C2×C4C2×#12111yescharacteristic
531C2×C4C2×C22⋊Q812yes—
532C2×C4C2×#12711yescharacteristic
533C2×C4C2×C4⋊Q811yescharacteristic
534C2×C4#53211yescharacteristic
535C2×C4#53312yes—
536C2×C4#53611yescharacteristic
537C2×C4#53611yescharacteristic
538C2×C4#53612yes—
539C2×C4#54112yes—
540C2×C4#54712yes—
541C2×C4#54712yes—
542C2×C4#54812yes—
543C2×C4D4×Q812yes—
544C2×C4D4×Q812yes—
545C2×C4D4×Q812yes—
546C8—42no—
547C8—42no—
548C8—44no—
549C22C2×#39211yescharacteristic
550C22#240912yes—
551C22#242912yes—
552C22#243211yescharacteristic
553C22#243411yescharacteristic
554C22#300411yescharacteristic
555C22#300511yescharacteristic
556C22#300712yes—
557C22#301211yescharacteristic
558C22#301211yescharacteristic
559C22#301311yescharacteristic
560C22#301311yescharacteristic
561C22#305612yes—
562C22#306011yescharacteristic
563C22#306111yescharacteristic
564C22#337012yes—
565C22#337112yes—
566C22#337212yes—
567C22#337412yes—
568C22#338512yes—
569C22#338712yes—
570C22#338812yes—
571C22#339012yes—
572C4—24no—
573C4—24no—
574C4—48no—
575C4—44no—
576C4—44no—
577C4—84no—
578C4—82no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4#242912yes—
583C4#243211yescharacteristic
584C4#243411yescharacteristic
585C4#254212yes—
586C4#262111yescharacteristic
587C4#285711yescharacteristic
588C2#1517811yescharacteristic
589C2#2339011yescharacteristic
590C2#2339211yescharacteristic
591C2#2434812yes—
592C2#2434912yes—
593C2#2548412yes—
594C2#2548812yes—
595C2#2565711yescharacteristic
596C2#2565811yescharacteristic
597C2#2584611yescharacteristic
598C2#2584711yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
111. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4792
C2×#25822
C2×#25842
C2×#19922
C2×#19852
C2×#17632
#113282
#113272
#110672
#108492