Finite Group Databasebeta

← #374314

Subgroups — #374314

Order 512.

1794 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#433C211yesmaximal, characteristic, complemented
2C2×#2620C211yesmaximal, characteristic
3C2×#1983C212yesmaximal, complemented
4#13777C211yesmaximal, characteristic, complemented
5#10970C212yesmaximal, complemented
6#10969C212yesmaximal
7#10928C211yesmaximal, characteristic, complemented
8#10927C211yesmaximal, characteristic, complemented
9#10926C211yesmaximal, characteristic, complemented
10#10925C211yesmaximal, characteristic, complemented
11#10849C212yesmaximal, complemented
12C22×#124C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×#121C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C2×#433—28no—
17C2×#433—24no—
18C22×C4⋊C8C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#102C2212yes—
22C22×#102C2212yes—
23C22×C22⋊C8C2212yes—
24C22×#97C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26#2620—24no—
27#2620C418yescomplemented
28C2×#396—22no—
29C2×#394—22no—
30C2×#391C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#384C2211yescharacteristic
33C2×#380C2212yes—
34C2×C4×C4⋊C4C2212yes—
35#1983—28no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40#1437—22no—
41#1436—22no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C23×Q8C2311yescharacteristic
45C23×D4—21no—
46C23×D4C2311yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×#124—22no—
51C2×#124—24no—
52C2×#124—24no—
53C2×#124C2×C414yes—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—42no—
56C2×C22⋊Q8—42no—
57C2×C22⋊Q8C2×C414yes—
58C2×#121—22no—
59C2×#121—42no—
60C2×#121—42no—
61C2×#121C2×C414yes—
62C2×#117—24no—
63C2×#117—24no—
64C2×#117C2×C414yes—
65C2×#117D414yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4D412yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4D412yes—
79C22×C42—22no—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82#433—232no—
83C2×C4⋊C8—24no—
84C2×C4⋊C8—24no—
85C2×C4⋊C8D418yescomplemented
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—42no—
91C2×#103—42no—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—42no—
97C2×#102—42no—
98C2×C22⋊C8—24no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—24no—
101C2×#97—24no—
102C2×#97—22no—
103C2×C4×C8—24no—
104C2×C4×C8—22no—
105#396—216no—
106#394—216no—
107#391—22no—
108#391C2×C414yes—
109#390—22no—
110#390C2×C414yes—
111#384—22no—
112#384C2×C414yes—
113#380—24no—
114#380—24no—
115C4×C4⋊C4—24no—
116C4×C4⋊C4—24no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122#340—42no—
123#340—42no—
124#340—42no—
125#340—42no—
126C25—21no—
127C25C2×D411yescharacteristic
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8C22×C412yes—
136C22×D4—24no—
137C22×D4—22no—
138C22×D4—21no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—42no—
145C22×D4—42no—
146C22×D4—82no—
147C22×D4—82no—
148C22×D4C22×C412yes—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4C22×C412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2411yescharacteristic
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8C2×D414yes—
179C22×C8C2×D414yes—
180#124—216no—
181#124—44no—
182#124—44no—
183C22⋊Q8—48no—
184C22⋊Q8—48no—
185#121—48no—
186#121—48no—
187#117—216no—
188#117—28no—
189#117—28no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
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—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4C22⋊C418yes—
215C2×C4⋊C4C22×C412yes—
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4#4114yes—
218C2×C4⋊C4#4114yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—82no—
233C2×C22⋊C4—82no—
234C2×C22⋊C4—82no—
235C2×C22⋊C4—82no—
236C2×C22⋊C4C22⋊C418yes—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—44no—
240C2×C42—82no—
241C2×C42—82no—
242C2×C42C22×C412yes—
243C2×C42C22×C412yes—
244C2×C42C2×D412yes—
245C2×C42C2×D414yes—
246C2×C42C2×D412yes—
247C2×C42#4114yes—
248C4⋊C8—216no—
249#103—48no—
250#103—48no—
251#102—48no—
252#102—48no—
253C22⋊C8—216no—
254#97—28no—
255C4×C8—28no—
256#95—28no—
257#95—28no—
258#95—44no—
259#95—82no—
260#95—82no—
261C24—22no—
262C24—22no—
263C24—21no—
264C24—21no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—42no—
271C24—42no—
272C24—41no—
273C24—41no—
274C24—42no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24C2×C22⋊C412yes—
279C24C22×D411yescharacteristic
280C2×Q8—24no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—24no—
284C2×Q8—41no—
285C2×Q8—41no—
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×D4—24no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—24no—
300C2×D4—48no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—88no—
314C2×D4—84no—
315C2×D4—84no—
316C2×D4—88no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
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—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4C2×C22⋊C414yes—
377C22×C4C2×C22⋊C412yes—
378C22×C4C2×C22⋊C414yes—
379C22×C4#12114yes—
380C22×C4#12114yes—
381C22×C4C23×C411yescharacteristic
382C22×C4C22×D412yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C2×#4112yes—
385C22×C4C2×#4112yes—
386C22×C4C2×#4112yes—
387C22×C4H(5,𝔽2)12yes—
388C22×C4H(5,𝔽2)12yes—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C4⋊C4—216no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—48no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—88no—
418C22⋊C4—216no—
419C22⋊C4—48no—
420C22⋊C4—48no—
421C22⋊C4—416no—
422C22⋊C4—88no—
423C22⋊C4—88no—
424C42—24no—
425C42—22no—
426C42—22no—
427C42—88no—
428C42C4×D418yes—
429C42#12014yes—
430C42#12314yes—
431C23—22no—
432C23—24no—
433C23—24no—
434C23—22no—
435C23—22no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—44no—
454C23—44no—
455C23—42no—
456C23—42no—
457C23—82no—
458C23—82no—
459C23—82no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23—81no—
464C23—81no—
465C23—81no—
466C23—81no—
467C23—82no—
468C23—82no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23C22×C22⋊C411yescharacteristic
474C23C2×#12112yes—
475C23#52512yes—
476C23#53511yescharacteristic
477C23#53511yescharacteristic
478C23#57612yes—
479C23#57712yes—
480C23#57712yes—
481C23#57812yes—
482Q8—44no—
483Q8—44no—
484Q8—82no—
485Q8—82no—
486D4—44no—
487D4—44no—
488D4—816no—
489D4—82no—
490D4—82no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—81no—
529C2×C4—82no—
530C2×C4—81no—
531C2×C4—82no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4#38618yes—
543C2×C4C2×C4×D412yes—
544C2×C4C4×#4111yescharacteristic
545C2×C4C4×#4111yescharacteristic
546C2×C4#51812yes—
547C2×C4C2×#12011yescharacteristic
548C2×C4C2×#12311yescharacteristic
549C2×C4#53212yes—
550C2×C4#53512yes—
551C2×C4#54612yes—
552C2×C4#54612yes—
553C2×C4#55012yes—
554C2×C4#55012yes—
555C2×C4#55312yes—
556C2×C4#55312yes—
557C2×C4#55512yes—
558C2×C4#55512yes—
559C8—44no—
560C8—44no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—44no—
565C22—44no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—84no—
574C22—84no—
575C22—162no—
576C22—162no—
577C22—162no—
578C22—162no—
579C22C2×#38611yescharacteristic
580C22#229711yescharacteristic
581C22#230911yescharacteristic
582C22#241412yes—
583C22#241412yes—
584C22#289712yes—
585C22#289812yes—
586C22#306012yes—
587C22#306112yes—
588C22#306212yes—
589C22#306312yes—
590C22#311812yes—
591C22#312112yes—
592C22#312412yes—
593C22#312512yes—
594C22#322011yescharacteristic
595C22#322111yescharacteristic
596C22#322211yescharacteristic
597C22#322311yescharacteristic
598C22#322412yes—
599C22#322512yes—
600C4—28no—
601C4—42no—
602C4—44no—
603C4—82no—
604C4—82no—
605C4—82no—
606C4—84no—
607C4—84no—
608C4#236312yes—
609C4#236312yes—
610C4#243112yes—
611C4#250012yes—
612C2—82no—
613C2—162no—
614C2#1413611yescharacteristic
615C2#2097612yes—
616C2#2139712yes—
617C2#2181311yescharacteristic
618C2#2181411yescharacteristic
619C2#2466312yes—
620C2#2466512yes—
621C2#2466712yes—
622C2#2466912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#433C2semidirect
3. C2×#1983C2semidirect
4. #13777C2semidirect
5. #10970C2semidirect
7. #10928C2semidirect
8. #10927C2semidirect
9. #10926C2semidirect
10. #10925C2semidirect
11. #10849C2semidirect
27. #2620C4semidirect
85. C2×C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C22×#4332
C2×#26202
C2×#19832
#137772
#109702
#109692
#109282
#109272
#109262
#109252
#108492