Finite Group Databasebeta

← #369386

Subgroups — #369386

Order 512.

1710 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#489C211yesmaximal, characteristic
2C2×#2326C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic, complemented
4C2×#1958C211yesmaximal, characteristic
5#14282C211yesmaximal, characteristic, complemented
6#11190C212yesmaximal, complemented
7#11189C212yesmaximal
8#10843C211yesmaximal, characteristic, complemented
9#10789C212yesmaximal, complemented
10#10788C212yesmaximal
11#10694C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C2×#489—24no—
17C2×#489—24no—
18C2×#489C418yescomplemented
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#102C2212yes—
22C22×#102C2212yes—
23C22×#97C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25#2326—24no—
26#2326—24no—
27C2×#391C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#388—21no—
30C2×#388—21no—
31C2×#385—21no—
32C2×#385—21no—
33C2×#384C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×#376C2211yescharacteristic
36#1990—24no—
37#1958—24no—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1446—21no—
45#1446—21no—
46#1445—21no—
47#1445—21no—
48C4⋊#95C2212yes—
49C4⋊#95C2212yes—
50C23×Q8—21no—
51C23×Q8C2311yescharacteristic
52C23×D4C2311yescharacteristic
53C24×C4—21no—
54C23×C8C2312yes—
55C23×C8C2312yes—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8C2×C414yescomplemented
62C2×#124—22no—
63C2×#124—22no—
64C2×#124—22no—
65C2×#124C2×C414yes—
66C2×#124D414yes—
67C2×#124D414yes—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—24no—
71C2×C4×Q8—42no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—24no—
75C2×C4×D4—42no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4—21no—
89C22×C42—21no—
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93#489—232no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—42no—
99C2×#103C2×C418yescomplemented
100C2×#102—24no—
101C2×#102—24no—
102C2×#102—24no—
103C2×#102—24no—
104C2×#102—42no—
105C2×#102C2×C418yes—
106C2×#97—22no—
107C2×#97—22no—
108C2×#97C2×C414yescomplemented
109C2×C4×C8—22no—
110C2×C4×C8—22no—
111C2×C4×C8—22no—
112#391—22no—
113#391—22no—
114#390—22no—
115#390—22no—
116#388—28no—
117#388—28no—
118#385—28no—
119#385—28no—
120#384—22no—
121#384D414yes—
122#382—22no—
123#382—22no—
124#376—22no—
125#376—22no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95D411yescharacteristic
135C2×#95D411yescharacteristic
136C2×#95D411yescharacteristic
137C2×#95D411yescharacteristic
138#340—24no—
139#340—24no—
140#340—42no—
141#340—42no—
142#336—24no—
143#336—24no—
144C25—21no—
145C22×Q8—24no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—42no—
154C22×Q8—42no—
155C22×Q8—82no—
156C22×Q8—82no—
157C22×Q8C22×C412yes—
158C22×Q8C2×D412yes—
159C22×Q8C2×D412yes—
160C22×D4—21no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4C22×C412yes—
168C22×D4C2×D412yes—
169C22×D4C2×D412yes—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—82no—
179C23×C4—82no—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2411yescharacteristic
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—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8C22×C414yes—
200C22×C8#4114yes—
201C4⋊Q8—28no—
202C4⋊Q8—28no—
203C4⋊Q8—44no—
204C4⋊Q8—44no—
205#124—28no—
206#124—28no—
207#124—28no—
208#124C22⋊C4116yes—
209C4×Q8—48no—
210C4×Q8—48no—
211C4×D4—48no—
212C4×D4—48no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—82no—
239C2×C4⋊C4—82no—
240C2×C4⋊C4C22×C414yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—82no—
255C2×C22⋊C4—82no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—22no—
259C2×C42—24no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—42no—
265C2×C42—42no—
266C2×C42C22×C412yes—
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42#4114yes—
271C2×C42#4112yes—
272C2×C42#4112yes—
273#103—216no—
274#103—48no—
275#102—216no—
276#102—48no—
277#97—28no—
278C4×C8—44no—
279#95—24no—
280#95—24no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—42no—
286#95—42no—
287#95—82no—
288#95—82no—
289#95—82no—
290#95—82no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—41no—
295C24—41no—
296C24—41no—
297C24—41no—
298C24—42no—
299C24—42no—
300C24—42no—
301C24—42no—
302C24C22×D411yescharacteristic
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—24no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—48no—
319C2×Q8—88no—
320C2×Q8—84no—
321C2×Q8—84no—
322C2×Q8—88no—
323C2×Q8C2×C22⋊C414yes—
324C2×D4—22no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4C2×C22⋊C414yes—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—48no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—88no—
395C22×C4—84no—
396C22×C4—84no—
397C22×C4—88no—
398C22×C4C4×D412yes—
399C22×C4C4×D412yes—
400C22×C4#12012yes—
401C22×C4#12012yes—
402C22×C4#12112yes—
403C22×C4#12112yes—
404C22×C4#12412yes—
405C22×C4#12412yes—
406C22×C4C23×C411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4112yes—
412C22×C4C2×#4112yes—
413C22×C4C2×#4111yescharacteristic
414C22×C4H(5,𝔽2)12yes—
415C22×C42−1+412yes—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—48no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—48no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—84no—
445C4⋊C4—84no—
446C22⋊C4—216no—
447C22⋊C4—216no—
448C22⋊C4—48no—
449C22⋊C4—84no—
450C22⋊C4—84no—
451C42—24no—
452C42—24no—
453C42—48no—
454C42—42no—
455C42—42no—
456C42C2×C22⋊C414yes—
457C42C4×D414yes—
458C42#12114yes—
459C42#12314yes—
460C23—24no—
461C23—42no—
462C23—42no—
463C23—44no—
464C23—42no—
465C23—44no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—44no—
471C23—44no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—82no—
476C23—82no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23C2×C4×D411yescharacteristic
484C23C2×#12011yescharacteristic
485C23C2×#12111yescharacteristic
486C23C2×#12411yescharacteristic
487C23#53211yescharacteristic
488C23#53511yescharacteristic
489C23#53611yescharacteristic
490C23#57612yes—
491C23#57712yes—
492C23#57712yes—
493C23#57812yes—
494Q8—28no—
495Q8—44no—
496Q8—44no—
497Q8—44no—
498Q8—816no—
499D4—28no—
500D4—44no—
501D4—44no—
502D4—44no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—44no—
547C2×C4—816no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4#38614yes—
564C2×C4#38614yes—
565C2×C4C22×C22⋊C411yescharacteristic
566C2×C4C2×C4×D411yescharacteristic
567C2×C4C4×#4112yes—
568C2×C4#52012yes—
569C2×C4C2×#12111yescharacteristic
570C2×C4C2×#12311yescharacteristic
571C2×C4#52912yes—
572C2×C4#54212yes—
573C2×C4#54612yes—
574C2×C4#54612yes—
575C2×C4#54612yes—
576C2×C4#54712yes—
577C2×C4#54712yes—
578C2×C4#54812yes—
579C2×C4#55612yes—
580C2×C4#55912yes—
581C8—44no—
582C8—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—82no—
587C22—82no—
588C22—84no—
589C22—82no—
590C22—82no—
591C22—84no—
592C22—84no—
593C22—84no—
594C22C2×#38611yescharacteristic
595C22C4×#12411yescharacteristic
596C22#232611yescharacteristic
597C22#243211yescharacteristic
598C22#243511yescharacteristic
599C22#254411yescharacteristic
600C22#254711yescharacteristic
601C22#294612yes—
602C22#294712yes—
603C22#300811yescharacteristic
604C22#300911yescharacteristic
605C22#301011yescharacteristic
606C22#301111yescharacteristic
607C22#304512yes—
608C22#304612yes—
609C22#313612yes—
610C22#313712yes—
611C22#317312yes—
612C22#317512yes—
613C22#322412yes—
614C22#322512yes—
615C22#324312yes—
616C22#324412yes—
617C4—24no—
618C4—24no—
619C4—42no—
620C4—42no—
621C4—44no—
622C4—82no—
623C4—82no—
624C4—81no—
625C4—81no—
626C4—84no—
627C4—162no—
628C4#233912yes—
629C4#236712yes—
630C4#242712yes—
631C4#244712yes—
632C2—84no—
633C2#1402711yescharacteristic
634C2#2093111yescharacteristic
635C2#2093211yescharacteristic
636C2#2165412yes—
637C2#2223312yes—
638C2#2568011yescharacteristic
639C2#2568111yescharacteristic
640C2#2586511yescharacteristic
641C2#2586611yescharacteristic
642C2#2862612yes—
643C2#2873012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×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
3. C2×#1990C2semidirect
5. #14282C2semidirect
6. #11190C2semidirect
8. #10843C2semidirect
9. #10789C2semidirect
11. #10694C2semidirect
18. C2×#489C4semidirect
61. C2×C4⋊Q8C2×C4semidirect
99. C2×#103C2×C4semidirect
108. C2×#97C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4892
C2×#23262
C2×#19902
C2×#19582
#142822
#111902
#111892
#108432
#107892
#107882
#106942