Finite Group Databasebeta

← #368985

Subgroups — #368985

Order 512.

2336 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2434C211yesmaximal, characteristic
2C2×Q8×C22⋊C4C211yesmaximal, characteristic
3C2×#1996C211yesmaximal, characteristic
4C2×#1996C211yesmaximal, characteristic
5#13922C211yesmaximal, characteristic
6#11203C214yesmaximal
7#10912C214yesmaximal
8#10846C211yesmaximal, characteristic
9C4×#342C211yesmaximal, characteristic
10C24×Q8C2211yescharacteristic
11C22×C4⋊Q8C412yescomplemented
12C22×C22⋊Q8C2211yescharacteristic
13C22×C4×Q8—21no—
14#2434—21no—
15#2434—22no—
16#2434—22no—
17#2434—21no—
18#2434—22no—
19C2×#395C414yescomplemented
20Q8×C22⋊C4—24no—
21Q8×C22⋊C4—24no—
22C2×#391—22no—
23C2×#391—21no—
24C2×#391—22no—
25C2×#391C412yescomplemented
26C2×#385C2214yes—
27C2×#380C2212yes—
28C2×#380C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×C4×C22⋊C4C2212yes—
31C2×C4×C22⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34#1996—24no—
35#1996—24no—
36C2×#342C2212yes—
37C2×#342C2214yes—
38C2×#342C2212yes—
39C2×#342C2214yes—
40C2×#342C2212yes—
41C2×#342C2212yes—
42#1446—22no—
43C22⋊#95C2214yes—
44C23×Q8—22no—
45C23×Q8—22no—
46C23×Q8—22no—
47C23×Q8—22no—
48C23×Q8—48no—
49C23×Q8C2×C412yes—
50C23×Q8C2×C411yescharacteristic
51C23×Q8C2×C412yes—
52C23×Q8D411yescharacteristic
53C23×Q8D412yes—
54C23×Q8D412yes—
55C23×Q8D412yes—
56C23×Q8D411yescharacteristic
57C24×C4C2312yes—
58C24×C4C2311yescharacteristic
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—24no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—48no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8—22no—
71C2×C4×Q8—28no—
72C2×C4×Q8—48no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4C2×C412yes—
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C4⋊C4C2×C418yes—
79C22×C22⋊C4C2312yes—
80C22×C22⋊C4C2312yes—
81C22×C22⋊C4C2312yes—
82C22×C22⋊C4C2311yescharacteristic
83C22×C22⋊C4C2312yes—
84C22×C22⋊C4C2312yes—
85C22×C22⋊C4C2311yescharacteristic
86C22×C42—22no—
87C22×C42—21no—
88C22×C42—21no—
89C22×C42C2×C412yes—
90#395—24no—
91#395—28no—
92#395—24no—
93#391—28no—
94#391—216no—
95#391—22no—
96#391—24no—
97#391—22no—
98#391—48no—
99#385—24no—
100#385—28no—
101#385—24no—
102#380—22no—
103#380—24no—
104#380—24no—
105#380—22no—
106#380—22no—
107#380D414yes—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—21no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4—21no—
113C4×C22⋊C4—24no—
114C4×C22⋊C4—24no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4D414yes—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—21no—
122C2×#95—28no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95C2×C412yes—
126C2×#95C2×C418yes—
127#342—28no—
128#342—24no—
129#342—24no—
130#342—24no—
131#342—28no—
132#342Q818yescomplemented
133C25C2411yescharacteristic
134C22×Q8—28no—
135C22×Q8—28no—
136C22×Q8—28no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—24no—
140C22×Q8—24no—
141C22×Q8—24no—
142C22×Q8—24no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—24no—
147C22×Q8—22no—
148C22×Q8—24no—
149C22×Q8—24no—
150C22×Q8—24no—
151C22×Q8—24no—
152C22×Q8—22no—
153C22×Q8—24no—
154C22×Q8—22no—
155C22×Q8—44no—
156C22×Q8—48no—
157C22×Q8—44no—
158C22×Q8—48no—
159C22×Q8—44no—
160C22×Q8—48no—
161C22×Q8—44no—
162C22×Q8—416no—
163C22×Q8—416no—
164C22×Q8—416no—
165C22×Q8—416no—
166C22×Q8C22⋊C412yes—
167C22×Q8C22⋊C412yes—
168C22×Q8C22⋊C414yes—
169C22×Q8C22⋊C414yes—
170C22×Q8C22⋊C412yes—
171C22×Q8C22⋊C412yes—
172C23×C4—21no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—21no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—44no—
182C23×C4—48no—
183C23×C4C22×C414yes—
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C412yes—
186C23×C4C22×C412yes—
187C23×C4C22×C414yes—
188C23×C4C22×C411yescharacteristic
189C23×C4C2×D412yes—
190C23×C4C2×D414yes—
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D412yes—
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D412yes—
197C23×C4C2×D412yes—
198C23×C4#4112yes—
199C23×C4#4112yes—
200C4⋊Q8—416no—
201C4⋊Q8—48no—
202C4⋊Q8—48no—
203C22⋊Q8—42no—
204C22⋊Q8—42no—
205C22⋊Q8—42no—
206C22⋊Q8—42no—
207C22⋊Q8—42no—
208C22⋊Q8—42no—
209C22⋊Q8—42no—
210C22⋊Q8—42no—
211C4×Q8—432no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—48no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—416no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4C2×D414yes—
257C2×C22⋊C4C2×D412yes—
258C2×C22⋊C4C2×D412yes—
259C2×C22⋊C4C2×Q814yes—
260C2×C22⋊C4C2×Q814yes—
261C2×C22⋊C4C2×Q814yes—
262C2×C22⋊C4#4114yes—
263C2×C42—28no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—24no—
267C2×C42—48no—
268C2×C42—44no—
269C2×C42—44no—
270C2×C42—44no—
271C2×C42—42no—
272#95—28no—
273#95—28no—
274#95—22no—
275#95—22no—
276#95—28no—
277#95—28no—
278#95—24no—
279#95—24no—
280#95—416no—
281#95—42no—
282#95—42no—
283C24—42no—
284C24—42no—
285C24C23×C411yescharacteristic
286C24C22×D411yescharacteristic
287C24C22×D411yescharacteristic
288C24C22×D411yescharacteristic
289C24C22×D411yescharacteristic
290C24C22×Q812yes—
291C24C2×#4111yescharacteristic
292C24C2×#4111yescharacteristic
293C24C2×#4112yes—
294C242−1+412yes—
295C242−1+412yes—
296C2×Q8—216no—
297C2×Q8—216no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—28no—
301C2×Q8—28no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—28no—
305C2×Q8—24no—
306C2×Q8—28no—
307C2×Q8—24no—
308C2×Q8—28no—
309C2×Q8—24no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—44no—
313C2×Q8—416no—
314C2×Q8—416no—
315C2×Q8—416no—
316C2×Q8—416no—
317C2×Q8—416no—
318C2×Q8—416no—
319C2×Q8—48no—
320C2×Q8—48no—
321C2×Q8—44no—
322C2×Q8—416no—
323C2×Q8—416no—
324C2×Q8—416no—
325C2×Q8—416no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—44no—
329C2×Q8—84no—
330C2×Q8—84no—
331C2×Q8#9918yes—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—28no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—28no—
349C22×C4—28no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—42no—
364C22×C4—44no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—42no—
380C22×C4—44no—
381C22×C4—42no—
382C22×C4—44no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—42no—
392C22×C4—44no—
393C22×C4C2×C22⋊C414yes—
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C414yes—
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C4×D412yes—
400C22×C4C4×D412yes—
401C22×C4C4×D412yes—
402C22×C4C4×D412yes—
403C22×C4C4×Q818yescomplemented
404C22×C4#12012yes—
405C22×C4#12012yes—
406C22×C4#12112yes—
407C22×C4#12112yes—
408C22×C4#12112yes—
409C22×C4#12112yes—
410C22×C4#12312yes—
411C22×C4#12312yes—
412C4⋊C4—416no—
413C4⋊C4—48no—
414C4⋊C4—416no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—416no—
420C4⋊C4—84no—
421C22⋊C4—22no—
422C22⋊C4—24no—
423C22⋊C4—24no—
424C22⋊C4—24no—
425C22⋊C4—24no—
426C22⋊C4—24no—
427C22⋊C4—22no—
428C22⋊C4—24no—
429C22⋊C4—22no—
430C22⋊C4—28no—
431C22⋊C4—24no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—44no—
444C22⋊C4C4⋊Q814yescomplemented
445C42—416no—
446C42—48no—
447C42—44no—
448C42—44no—
449C42—84no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—44no—
463C23—44no—
464C23—41no—
465C23—41no—
466C23—44no—
467C23—44no—
468C23C22×C22⋊C411yescharacteristic
469C23C2×C4×D411yescharacteristic
470C23C2×C4×D411yescharacteristic
471C23C2×C4×Q812yes—
472C23#51912yes—
473C23C2×#12011yescharacteristic
474C23C2×#12111yescharacteristic
475C23C2×#12111yescharacteristic
476C23C2×#12311yescharacteristic
477C23C2×C4⋊Q811yescharacteristic
478C23#53211yescharacteristic
479C23#53312yes—
480C23#53611yescharacteristic
481C23#53611yescharacteristic
482C23#54112yes—
483C23#54712yes—
484C23#54812yes—
485C23#54812yes—
486C23D4×Q812yes—
487C23D4×Q812yes—
488C23D4×Q812yes—
489C23#55612yes—
490C23#55712yes—
491Q8—216no—
492Q8—216no—
493Q8—416no—
494Q8—416no—
495Q8—48no—
496Q8—48no—
497Q8—84no—
498Q8—84no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—28no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—48no—
524C2×C4—48no—
525C2×C4—48no—
526C2×C4—48no—
527C2×C4—42no—
528C2×C4—48no—
529C2×C4—42no—
530C2×C4—48no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—48no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—48no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—82no—
549C2×C4—84no—
550C2×C4—84no—
551C2×C4—88no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—88no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4#38612yes—
562C2×C4#38612yes—
563C2×C4#38612yes—
564C2×C4#38612yes—
565C2×C4C2×#9918yes—
566C2×C4C2×#9914yes—
567C22—24no—
568C22—24no—
569C22—24no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—41no—
586C22—41no—
587C22—42no—
588C22—41no—
589C22—41no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22C2×#38611yescharacteristic
596C22C4×C4⋊Q811yescharacteristic
597C22#232711yescharacteristic
598C22Q8×C22⋊C412yes—
599C22#236512yes—
600C22#242912yes—
601C22#243211yescharacteristic
602C22#243411yescharacteristic
603C22#244612yes—
604C22#254511yescharacteristic
605C22#254611yescharacteristic
606C22C22×#9912yes—
607C22#288412yes—
608C4—22no—
609C4—22no—
610C4—22no—
611C4—28no—
612C4—44no—
613C4—48no—
614C4—44no—
615C4—82no—
616C4—84no—
617C4—84no—
618C4—82no—
619C4—82no—
620C4—162no—
621C4#189814yes—
622C2—21no—
623C2—21no—
624C2—22no—
625C2—21no—
626C2—21no—
627C2—21no—
628C2—22no—
629C2—22no—
630C2—21no—
631C2#1402411yescharacteristic
632C2C2×#189811yescharacteristic
633C2#2098611yescharacteristic
634C2Q8×#9914yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×Q8 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
11. C22×C4⋊Q8C4semidirect
19. C2×#395C4semidirect
25. C2×#391C4semidirect
132. #342Q8semidirect
403. C22×C4C4×Q8semidirect
444. C22⋊C4C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24342
C2×Q8×C22⋊C42
C2×#19962
#139222
#112032
#109122
#108462
C4×#3422