Finite Group Databasebeta

← #369320

Subgroups — #369320

Order 512.

1530 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2568C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic
3C2×#1985C211yesmaximal, characteristic
4C2×#1981C211yesmaximal, characteristic
5C2×#1960C211yesmaximal, characteristic
6#11052C212yesmaximal
7#11050C212yesmaximal
8#10932C212yesmaximal
9#10932C212yesmaximal
10#10850C211yesmaximal, characteristic
11#10703C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#97C2211yescharacteristic
20#2568—24no—
21#2568C418yescomplemented
22C2×#400—21no—
23C2×#400—21no—
24#2355—24no—
25#2355—24no—
26C2×#391—21no—
27C2×#391—21no—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#387—21no—
31C2×#387—21no—
32C2×#384C2211yescharacteristic
33C2×#382—21no—
34C2×#382C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×C4×C4⋊C4—21no—
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41#1985—24no—
42#1981—24no—
43#1960—24no—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47C2×#340C2212yes—
48C2×#336C2212yes—
49C2×#336C2212yes—
50C23×Q8D411yescharacteristic
51C23×Q8C2312yes—
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8C2×C414yes—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8C2×C414yes—
65C2×C4×Q8D414yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C42—21no—
79C22×C42—21no—
80C22×C42—21no—
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—42no—
97C2×#103—42no—
98C2×#103—42no—
99C2×#103—42no—
100C2×#97—22no—
101C2×#97—22no—
102C2×#97—22no—
103#400—28no—
104#400—28no—
105#391—28no—
106#391—28no—
107#391—24no—
108#391—24no—
109#391C2×C418yes—
110#391C2×C418yes—
111#387—28no—
112#387—28no—
113#384—22no—
114#384—22no—
115#382—28no—
116#382—22no—
117#382—22no—
118#382—22no—
119#382D414yes—
120C4×C4⋊C4—28no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4C2×C414yes—
128C4×C4⋊C4D414yes—
129C2×#95—21no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95—21no—
141#340—24no—
142#340—24no—
143#340—42no—
144#340—42no—
145#336—24no—
146#336—24no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—22no—
160C22×Q8—22no—
161C22×Q8—22no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—41no—
165C22×Q8—41no—
166C22×Q8C22⋊C414yes—
167C22×Q8C22×C414yes—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—21no—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2411yescharacteristic
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8#4114yes—
195C4⋊Q8—28no—
196C4⋊Q8—28no—
197C4⋊Q8—44no—
198C4⋊Q8—44no—
199C4×Q8—28no—
200C4×Q8—28no—
201C4×Q8—28no—
202C4×Q8—28no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
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—24no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—82no—
236C2×C4⋊C4C22⋊C414yes—
237C2×C4⋊C4C22⋊C414yes—
238C2×C4⋊C4C22⋊C414yes—
239C2×C4⋊C4C22×C414yes—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4#4114yes—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42—82no—
262C2×C42—82no—
263C2×C42C22×C412yes—
264C2×C42C22×C412yes—
265C2×C42C22×C412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42#4112yes—
272C2×C42#4112yes—
273C4⋊C8—44no—
274#103—48no—
275#103—48no—
276#103—48no—
277#103—48no—
278#97—44no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—24no—
289#95—24no—
290#95—24no—
291C24C22×D411yescharacteristic
292C2×Q8—28no—
293C2×Q8—28no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—44no—
301C2×Q8—44no—
302C2×Q8—44no—
303C2×Q8—44no—
304C2×Q8—44no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—24no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—41no—
359C22×C4—41no—
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×C4—42no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4C2×C22⋊C412yes—
386C22×C4C2×C22⋊C412yes—
387C22×C4C2×C22⋊C412yes—
388C22×C4C2×C22⋊C412yes—
389C22×C4C2×C22⋊C412yes—
390C22×C4C2×C22⋊C412yes—
391C22×C4#12012yes—
392C22×C4#12012yes—
393C22×C4#12312yes—
394C22×C4#12312yes—
395C22×C4#13712yes—
396C22×C4#13712yes—
397C22×C4C23×C411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4111yescharacteristic
403C22×C4C2×#4112yes—
404C22×C4C2×#4112yes—
405C22×C4H(5,𝔽2)12yes—
406C22×C42−1+412yes—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—28no—
424C4⋊C4—28no—
425C4⋊C4—28no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—48no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—84no—
443C4⋊C4#12414yes—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—48no—
453C42—48no—
454C42—84no—
455C42—84no—
456C42C4×D414yes—
457C42C4×D414yes—
458C42#12114yes—
459C42#12114yes—
460C23C22×C22⋊C411yescharacteristic
461C23C2×#12011yescharacteristic
462C23C2×#12311yescharacteristic
463C23#54812yes—
464C23#55012yes—
465C23C2×#13711yescharacteristic
466C23C2×#13711yescharacteristic
467C23#57511yescharacteristic
468C23#57511yescharacteristic
469C23#57712yes—
470C23#57812yes—
471Q8—416no—
472Q8—44no—
473Q8—44no—
474Q8—44no—
475Q8—44no—
476Q8—82no—
477Q8—82no—
478Q8—82no—
479Q8—82no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—82no—
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—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4#38614yes—
548C2×C4#38614yes—
549C2×C4#41914yes—
550C2×C4C2×C4×D411yescharacteristic
551C2×C4C2×C4×D411yescharacteristic
552C2×C4C4×#4112yes—
553C2×C4#52012yes—
554C2×C4C2×#12111yescharacteristic
555C2×C4C2×#12111yescharacteristic
556C2×C4C2×#12411yescharacteristic
557C2×C4#52912yes—
558C2×C4#53211yescharacteristic
559C2×C4#53511yescharacteristic
560C2×C4#53611yescharacteristic
561C2×C4#54212yes—
562C2×C4#54612yes—
563C2×C4#54812yes—
564C2×C4#54812yes—
565C2×C4#55012yes—
566C2×C4#55612yes—
567C2×C4#55912yes—
568C8—44no—
569C8—82no—
570C22C2×#38611yescharacteristic
571C22#234112yes—
572C22#243111yescharacteristic
573C22#243611yescharacteristic
574C22#244812yes—
575C22C2×#41911yescharacteristic
576C22#289611yescharacteristic
577C22#289812yes—
578C22#300711yescharacteristic
579C22#300711yescharacteristic
580C22#300911yescharacteristic
581C22#301111yescharacteristic
582C22#309511yescharacteristic
583C22#309511yescharacteristic
584C22#309712yes—
585C22#337212yes—
586C22#337412yes—
587C22#337612yes—
588C22#337712yes—
589C22#339812yes—
590C22#340012yes—
591C22#340112yes—
592C22#340212yes—
593C4—24no—
594C4—24no—
595C4—24no—
596C4—44no—
597C4—42no—
598C4—42no—
599C4—42no—
600C4—84no—
601C4—82no—
602C4—82no—
603C4—82no—
604C4—84no—
605C4C4×#12411yescharacteristic
606C4#232611yescharacteristic
607C4#236712yes—
608C4#242612yes—
609C4#254511yescharacteristic
610C4#254811yescharacteristic
611C2#1402611yescharacteristic
612C2#2091111yescharacteristic
613C2#2091311yescharacteristic
614C2#2254712yes—
615C2#2254812yes—
616C2#2564111yescharacteristic
617C2#2564311yescharacteristic
618C2#2594311yescharacteristic
619C2#2594511yescharacteristic
620C2#2633312yes—
621C2#2633512yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×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
21. #2568C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25682
C2×#23552
C2×#19852
C2×#19812
C2×#19602
#110522
#110502
#109322
#108502
#107032