Finite Group Databasebeta

← #416041

Subgroups — #416041

Order 512.

1410 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2640C211yesmaximal, characteristic
2C2×#1988C211yesmaximal, characteristic, complemented
3C2×#1987C211yesmaximal, characteristic, complemented
4C2×#1983C211yesmaximal, characteristic
5C2×#1983C211yesmaximal, characteristic
6#13783C211yesmaximal, characteristic, complemented
7#11190C212yesmaximal, complemented
8#11189C212yesmaximal
9#11067C212yesmaximal, complemented
10#11066C212yesmaximal
11#10833C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2211yescharacteristic
19C22×#102C2211yescharacteristic
20C22×#102C2212yes—
21C22×#102C2211yescharacteristic
22#2640—24no—
23#2640—24no—
24C2×#397—22no—
25C2×#393—21no—
26C2×#393—21no—
27C2×#391C2212yes—
28C2×#390C2212yes—
29C2×#389C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×#376C2211yescharacteristic
33#1988—24no—
34#1987—24no—
35#1983—24no—
36#1983—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1436—22no—
44C4⋊#95—21no—
45C4⋊#95C2212yes—
46#1434—21no—
47C23×Q8—21no—
48C23×Q8C2311yescharacteristic
49C23×D4C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8D414yes—
59C2×#124—22no—
60C2×#124—22no—
61C2×#124—22no—
62C2×#124—22no—
63C2×#124—22no—
64C2×#124D414yes—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C42—21no—
76C22×C42—21no—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×#103—24no—
86C2×#103—22no—
87C2×#103—22no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—42no—
91C2×#103—42no—
92C2×#102—24no—
93C2×#102—22no—
94C2×#102—22no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—42no—
98C2×#102—42no—
99#397—216no—
100#393—28no—
101#393—28no—
102#391—24no—
103#391—24no—
104#390—24no—
105#390—24no—
106#389—24no—
107#389—24no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4D414yes—
112#376—22no—
113#376—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95D411yescharacteristic
122C2×#95D411yescharacteristic
123C2×#95D412yes—
124#340—24no—
125#340—24no—
126#340—24no—
127#340—24no—
128#336—42no—
129#336—42no—
130C25—21no—
131C22×Q8—24no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8—82no—
144C22×Q8—82no—
145C22×Q8C2×D412yes—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4C2×D412yes—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2411yescharacteristic
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C4⋊Q8—28no—
184C4⋊Q8—28no—
185C4⋊Q8—44no—
186C4⋊Q8—44no—
187#124—28no—
188#124—28no—
189#124—44no—
190#124—44no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4C2×D414yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4#4114yes—
216C2×C4⋊C4#4114yes—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—82no—
226C2×C22⋊C4—82no—
227C2×C22⋊C4—82no—
228C2×C22⋊C4—82no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—42no—
237C2×C42—82no—
238C2×C42—81no—
239C2×C42—81no—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42#4112yes—
244C2×C42#4112yes—
245C2×C42#4112yes—
246C4⋊C8—44no—
247C4⋊C8—44no—
248#103—48no—
249#103—44no—
250#103—44no—
251#102—48no—
252#102—44no—
253#102—44no—
254#95—28no—
255#95—28no—
256#95—28no—
257#95—22no—
258#95—22no—
259#95—24no—
260#95—24no—
261#95—42no—
262#95—82no—
263#95—82no—
264C24—22no—
265C24—41no—
266C24—42no—
267C24—41no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24—41no—
275C24C22×D411yescharacteristic
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—48no—
285C2×Q8—41no—
286C2×Q8—42no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—88no—
296C2×Q8—88no—
297C2×Q8—84no—
298C2×Q8—84no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—24no—
302C2×D4—24no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—42no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
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—22no—
342C22×C4—22no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
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—41no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4#12012yes—
367C22×C4#12012yes—
368C22×C4#12112yes—
369C22×C4#12112yes—
370C22×C4#12412yes—
371C22×C4#12412yes—
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4112yes—
379C22×C4C2×#4112yes—
380C22×C4H(5,𝔽2)12yes—
381C22×C42−1+412yes—
382C2×C8—42no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—42no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C4⋊C4—24no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—48no—
413C4⋊C4—88no—
414C4⋊C4#12414yes—
415C22⋊C4—48no—
416C22⋊C4—48no—
417C22⋊C4—88no—
418C22⋊C4—88no—
419C42—42no—
420C42—42no—
421C42—42no—
422C42—42no—
423C42—84no—
424C42—84no—
425C42#12114yes—
426C42#12114yes—
427C42#12314yes—
428C42#12314yes—
429C23—42no—
430C23—44no—
431C23—42no—
432C23—42no—
433C23—44no—
434C23—82no—
435C23—81no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23—82no—
440C23—81no—
441C23—82no—
442C23—81no—
443C23—81no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23—81no—
449C23—81no—
450C23—81no—
451C23—81no—
452C23C2×#12011yescharacteristic
453C23C2×#12111yescharacteristic
454C23#52512yes—
455C23C2×#12411yescharacteristic
456C23#53712yes—
457C23#57612yes—
458C23#57711yescharacteristic
459C23#57712yes—
460C23#57711yescharacteristic
461C23#57811yescharacteristic
462C23#57811yescharacteristic
463Q8—44no—
464Q8—42no—
465Q8—42no—
466Q8—816no—
467Q8—82no—
468Q8—81no—
469Q8—81no—
470D4—44no—
471D4—42no—
472D4—42no—
473D4—82no—
474D4—81no—
475D4—81no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—82no—
517C2×C4—81no—
518C2×C4—81no—
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#39414yes—
530C2×C4#39414yes—
531C2×C4C2×#12111yescharacteristic
532C2×C4C2×#12111yescharacteristic
533C2×C4C2×#12311yescharacteristic
534C2×C4C2×#12311yescharacteristic
535C2×C4C2×#12411yescharacteristic
536C2×C4#52912yes—
537C2×C4#53211yescharacteristic
538C2×C4#53511yescharacteristic
539C2×C4#53611yescharacteristic
540C2×C4#54212yes—
541C2×C4#54612yes—
542C2×C4#54712yes—
543C2×C4#54812yes—
544C2×C4#55012yes—
545C2×C4#55612yes—
546C2×C4#55612yes—
547C2×C4#55912yes—
548C2×C4#55912yes—
549C8—82no—
550C8—81no—
551C8—81no—
552C22—84no—
553C22—84no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—161no—
558C22—161no—
559C22—161no—
560C22—161no—
561C22—162no—
562C22—162no—
563C22C2×#39411yescharacteristic
564C22#240712yes—
565C22#259212yes—
566C22#262011yescharacteristic
567C22#285011yescharacteristic
568C22#301011yescharacteristic
569C22#301111yescharacteristic
570C22#301212yes—
571C22#304111yescharacteristic
572C22#304311yescharacteristic
573C22#304611yescharacteristic
574C22#304611yescharacteristic
575C22#311812yes—
576C22#312112yes—
577C22#312412yes—
578C22#312512yes—
579C22#313612yes—
580C22#313711yescharacteristic
581C22#313711yescharacteristic
582C22#325712yes—
583C22#325812yes—
584C22#326212yes—
585C22#326312yes—
586C4—24no—
587C4—24no—
588C4—44no—
589C4—82no—
590C4—84no—
591C4—82no—
592C4—82no—
593C4—82no—
594C4—84no—
595C4#244712yes—
596C4#244812yes—
597C4#254411yescharacteristic
598C4#254511yescharacteristic
599C4#254711yescharacteristic
600C4#254811yescharacteristic
601C2—162no—
602C2#1521711yescharacteristic
603C2#2351011yescharacteristic
604C2#2351111yescharacteristic
605C2#2416612yes—
606C2#2416812yes—
607C2#2936212yes—
608C2#2936312yes—
609C2#2983911yescharacteristic
610C2#2984111yescharacteristic
611C2#3303611yescharacteristic
612C2#3303711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1988C2semidirect
3. C2×#1987C2semidirect
6. #13783C2semidirect
7. #11190C2semidirect
9. #11067C2semidirect
11. #10833C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26402
C2×#19882
C2×#19872
C2×#19832
#137832
#111902
#111892
#110672
#110662
#108332