Finite Group Databasebeta

← #374181

Subgroups — #374181

Order 512.

1680 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#485C211yesmaximal, characteristic
2C2×#2346C211yesmaximal, characteristic
3C2×#1892C211yesmaximal, characteristic
4C2×#1735C211yesmaximal, characteristic
5#14138C211yesmaximal, characteristic
6#11220C212yesmaximal, complemented
7#11067C212yesmaximal, complemented
8#10932C212yesmaximal, complemented
9#10907C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C2×#485—24no—
17C2×#485—24no—
18C2×#485C418yescomplemented
19C22×#107C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×#103C2212yes—
22C22×#103C2211yescharacteristic
23C22×#103C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25#2346—24no—
26#2346—24no—
27C2×#391C2212yes—
28C2×#389C2212yes—
29C2×#388—22no—
30C2×#382—21no—
31C2×#380C2211yescharacteristic
32C2×C4×C4⋊C4C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34C2×#376—21no—
35#1892—24no—
36#1735—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1440—22no—
45#1436—21no—
46#1436—21no—
47C4⋊#95C2212yes—
48C23×Q8C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C23×C8—22no—
51C23×C8—24no—
52C23×C8—22no—
53C23×C8—22no—
54C23×C8—42no—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C23×C8C2312yes—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8C2×C414yes—
64C2×#117—22no—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—24no—
70C2×#117—42no—
71C2×#117C2×C414yes—
72C2×#117D414yes—
73C2×#117D414yes—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4D412yes—
86C22×C42—21no—
87C22×C42—22no—
88C22×C42—21no—
89C22×C42C2312yes—
90#485—232no—
91C2×#107—22no—
92C2×#107—22no—
93C2×#107C2×C414yes—
94C2×#106—22no—
95C2×#106—22no—
96C2×#106C2×C414yes—
97C2×#103—22no—
98C2×#103—22no—
99C2×#103—22no—
100C2×#103—22no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—42no—
104C2×#103C2×C414yes—
105C2×#103C2×C414yes—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8C2×C414yes—
109#391—24no—
110#391—42no—
111#389—24no—
112#389—24no—
113#388—216no—
114#382—28no—
115#380—22no—
116#380D414yes—
117C4×C4⋊C4—24no—
118C4×C4⋊C4—24no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121#376—28no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95D411yescharacteristic
130C2×#95D411yescharacteristic
131#340—24no—
132#340—24no—
133#340—24no—
134#340—24no—
135#340—42no—
136#336—24no—
137#336—42no—
138C25C2×D411yescharacteristic
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8C22×C412yes—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4C22×C412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4#4112yes—
167C23×C4#4112yes—
168C23×C4C2411yescharacteristic
169C22×C8—28no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—48no—
185C22×C8—48no—
186C22×C8—44no—
187C22×C8—44no—
188C22×C8C22×C412yes—
189C22×C8C22×C412yes—
190C22×C8#4114yes—
191C22⋊Q8—28no—
192C22⋊Q8—28no—
193C22⋊Q8—44no—
194C22⋊Q8—44no—
195#117—28no—
196#117—28no—
197#117—28no—
198#117—28no—
199#117—48no—
200#117—44no—
201#117C22⋊C4116yes—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—82no—
228C2×C4⋊C4—82no—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C22×C412yes—
232C2×C4⋊C4C2×D414yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4#4114yes—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—82no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—28no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—44no—
262C2×C42—44no—
263C2×C42—42no—
264C2×C42—44no—
265#107—28no—
266#106—28no—
267#103—28no—
268#103—28no—
269#103—48no—
270C22⋊C8—28no—
271#95—28no—
272#95—28no—
273#95—24no—
274#95—22no—
275#95—22no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24C4×D412yes—
287C24#12012yes—
288C24#12112yes—
289C24C22×D411yescharacteristic
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—42no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
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—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4C2×C22⋊C414yes—
372C22×C4C4×D414yes—
373C22×C4C4×D412yes—
374C22×C4#12012yes—
375C22×C4#12114yes—
376C22×C4#12112yes—
377C22×C4#12314yes—
378C22×C4#13514yes—
379C22×C4C23×C411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4112yes—
385C22×C4C2×#4112yes—
386C22×C4C2×#4111yescharacteristic
387C22×C4H(5,𝔽2)12yes—
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—416no—
397C2×C8—42no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—42no—
401C2×C8—41no—
402C2×C8—42no—
403C2×C8—41no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C4⋊C4—22no—
408C4⋊C4—22no—
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—24no—
418C4⋊C4—24no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—48no—
426C4⋊C4—88no—
427C4⋊C4C2×C22⋊C414yes—
428C4⋊C4C2×C22⋊C414yes—
429C22⋊C4—216no—
430C22⋊C4—48no—
431C22⋊C4—48no—
432C22⋊C4—48no—
433C22⋊C4—84no—
434C42—216no—
435C42—416no—
436C42—48no—
437C42—42no—
438C42—42no—
439C42—42no—
440C42—42no—
441C23—24no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—24no—
446C23—22no—
447C23—24no—
448C23—24no—
449C23—22no—
450C23—22no—
451C23—24no—
452C23—24no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23#38614yes—
462C23C2×C4×D411yescharacteristic
463C23C2×#12011yescharacteristic
464C23C2×#12111yescharacteristic
465C23#54612yes—
466C23#54712yes—
467C23C2×#13512yes—
468C23#57512yes—
469C23#57712yes—
470C23#57812yes—
471Q8—42no—
472Q8—44no—
473Q8—42no—
474Q8—81no—
475Q8—81no—
476Q8—82no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—28no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—28no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4#38614yes—
536C2×C4C22×C22⋊C411yescharacteristic
537C2×C4C2×C4×D411yescharacteristic
538C2×C4C4×#4112yes—
539C2×C4#52012yes—
540C2×C4C2×#12111yescharacteristic
541C2×C4C2×#12311yescharacteristic
542C2×C4#52511yescharacteristic
543C2×C4#52511yescharacteristic
544C2×C4#53212yes—
545C2×C4#53511yescharacteristic
546C2×C4#53611yescharacteristic
547C2×C4#53712yes—
548C2×C4#54612yes—
549C2×C4#54712yes—
550C2×C4#55112yes—
551C2×C4#55112yes—
552C2×C4#55212yes—
553C2×C4#55212yes—
554C8—42no—
555C8—44no—
556C8—42no—
557C22—24no—
558C22—24no—
559C22—24no—
560C22—44no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22C2×#38611yescharacteristic
567C22#235912yes—
568C22#241111yescharacteristic
569C22#241311yescharacteristic
570C22#243012yes—
571C22C4×#13511yescharacteristic
572C22#294311yescharacteristic
573C22#294712yes—
574C22#300212yes—
575C22#300911yescharacteristic
576C22#301111yescharacteristic
577C22#303512yes—
578C22#304612yes—
579C22#330012yes—
580C22#330312yes—
581C22#331512yes—
582C22#332112yes—
583C22#333612yes—
584C22#334012yes—
585C22#334812yes—
586C22#334912yes—
587C4—24no—
588C4—44no—
589C4—44no—
590C4—44no—
591C4—82no—
592C4—84no—
593C4—82no—
594C4—82no—
595C4—84no—
596C4#229711yescharacteristic
597C4#231111yescharacteristic
598C4#235912yes—
599C4#241111yescharacteristic
600C4#241311yescharacteristic
601C4#249812yes—
602C2—44no—
603C2#1413511yescharacteristic
604C2#2092411yescharacteristic
605C2#2093111yescharacteristic
606C2#2291712yes—
607C2#2293912yes—
608C2#2448712yes—
609C2#2458612yes—
610C2#2553612yes—
611C2#2554612yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11220C2semidirect
7. #11067C2semidirect
8. #10932C2semidirect
9. #10907C2semidirect
18. C2×#485C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4852
C2×#23462
C2×#18922
C2×#17352
#141382
#112202
#110672
#109322
#109072
#107312
#107142