Finite Group Databasebeta

← #362762

Subgroups — #362762

Order 512.

2802 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×#554C211yesmaximal, characteristic
2C2×#1990C211yesmaximal, characteristic, complemented
3C2×#1987C211yesmaximal, characteristic, complemented
4C2×#1984C211yesmaximal, characteristic, complemented
5C2×#1802C211yesmaximal, characteristic
6C2×C4×#102C211yesmaximal, characteristic
7#13926C211yesmaximal, characteristic, complemented
8#11385C214yesmaximal, complemented
9#10788C214yesmaximal
10C2×#554—24no—
11C2×#554—24no—
12C2×#554—24no—
13C2×#554—24no—
14C2×#554—24no—
15C2×#554C418yescomplemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#125C2212yes—
18C22×C22⋊Q8—21no—
19C22×C22⋊Q8—22no—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2212yes—
22C22×C4×D4C2211yescharacteristic
23C23×C4⋊C4—21no—
24C22×C4⋊C8C2211yescharacteristic
25C22×#102C2212yes—
26C22×#102C2212yes—
27C22×C4×C8C2211yescharacteristic
28C2×#391—21no—
29C2×#384C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1990—24no—
32#1987—24no—
33#1984—24no—
34#1802—24no—
35C4×#102—24no—
36C2×#340C2214yes—
37C2×#340C2214yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2214yes—
41#1441—22no—
42C4⋊#95—21no—
43C4⋊#95C2214yes—
44C23×Q8—21no—
45C23×Q8D411yescharacteristic
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C24×C4—22no—
49C23×C8C2312yes—
50C23×C8C2312yes—
51#554—232no—
52#554—232no—
53#554—232no—
54#554—232no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8C2×C414yes—
59C2×C4⋊Q8D414yes—
60C2×C4⋊Q8D414yes—
61C2×#125—24no—
62C2×#125—24no—
63C2×#125—24no—
64C2×#125—24no—
65C2×#125—24no—
66C2×#125C2×C418yes—
67C2×C22⋊Q8—28no—
68C2×C22⋊Q8—28no—
69C2×C22⋊Q8—216no—
70C2×C22⋊Q8—216no—
71C2×C22⋊Q8—28no—
72C2×C22⋊Q8—216no—
73C2×C4×Q8—22no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8—22no—
77C2×C4×Q8C2×C414yes—
78C2×C4×Q8D414yes—
79C2×C4×D4—24no—
80C2×C4×D4—24no—
81C2×C4×D4—24no—
82C2×C4×D4—24no—
83C2×C4×D4—24no—
84C2×C4×D4—22no—
85C2×C4×D4—22no—
86C2×C4×D4—22no—
87C2×C4×D4C2×C418yes—
88C2×C4×D4C2×C414yes—
89C2×C4×D4D414yes—
90C2×C4×D4D414yes—
91C22×C4⋊C4—28no—
92C22×C4⋊C4—28no—
93C22×C4⋊C4—22no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—22no—
96C22×C4⋊C4—28no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4D411yescharacteristic
100C22×C4⋊C4D412yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C22⋊C4—22no—
106C22×C22⋊C4—21no—
107C22×C42—21no—
108C22×C42C2312yes—
109C22×C42C2311yescharacteristic
110C2×C4⋊C8—22no—
111C2×C4⋊C8—22no—
112C2×C4⋊C8—22no—
113C2×#102—24no—
114C2×#102—24no—
115C2×#102—24no—
116C2×#102—28no—
117C2×#102Q818yescomplemented
118C2×C4×C8—22no—
119C2×C4×C8—22no—
120C2×C4×C8—22no—
121#391—44no—
122#384—22no—
123#384—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C2×#95—24no—
127C2×#95—21no—
128C2×#95—24no—
129#340—24no—
130#340—28no—
131#340—28no—
132#340—24no—
133#336—28no—
134C25—21no—
135C22×Q8—24no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—24no—
139C22×Q8—24no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—82no—
145C22×Q8—82no—
146C22×Q8C22⋊C414yes—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4C22×C412yes—
153C22×D4C2×D412yes—
154C22×D4C2×D412yes—
155C22×D4C2×Q814yes—
156C22×D4#4114yes—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—28no—
160C23×C4—28no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—28no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—84no—
172C23×C4—84no—
173C23×C4C2×D412yes—
174C23×C4C2×D412yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2411yescharacteristic
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—24no—
188C22×C8—24no—
189C22×C8C2×Q814yes—
190C4⋊Q8—28no—
191C4⋊Q8—28no—
192C4⋊Q8—28no—
193C4⋊Q8C22⋊C4116yes—
194#125—216no—
195#125—216no—
196#125—216no—
197#125—216no—
198C22⋊Q8—264no—
199C22⋊Q8—2128no—
200C4×Q8—28no—
201C4×Q8—28no—
202C4×Q8—28no—
203C4×Q8—28no—
204C4×D4—216no—
205C4×D4—216no—
206C4×D4—216no—
207C4×D4—216no—
208C4×D4—28no—
209C4×D4—28no—
210C4×D4—28no—
211C4×D4C22⋊C4116yes—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—264no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—82no—
246C2×C4⋊C4C22⋊C414yes—
247C2×C4⋊C4C22⋊C418yes—
248C2×C4⋊C4C22×C412yes—
249C2×C4⋊C4C22×C414yes—
250C2×C4⋊C4C2×D414yes—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2×Q814yes—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—42no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42C22×C414yes—
269C2×C42C22×C412yes—
270C2×C42C2×D414yes—
271C2×C42C2×D412yes—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C4⋊C8—44no—
277#102—216no—
278#102—48no—
279C4×C8—44no—
280#95—48no—
281#95—42no—
282#95—84no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—24no—
287C24—24no—
288C24—41no—
289C24—41no—
290C24—41no—
291C24—41no—
292C24—42no—
293C24—42no—
294C24C22×D411yescharacteristic
295C2×Q8—28no—
296C2×Q8—216no—
297C2×Q8—28no—
298C2×Q8—28no—
299C2×Q8—44no—
300C2×Q8—44no—
301C2×Q8—44no—
302C2×Q8—44no—
303C2×Q8—88no—
304C2×Q8—84no—
305C2×Q8—84no—
306C2×Q8—88no—
307C2×D4—22no—
308C2×D4—24no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—24no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4C2×C22⋊C414yes—
321C2×D4C4×Q818yes—
322C2×D4C22⋊Q818yes—
323C2×D4#12414yes—
324C2×D4C4⋊Q814yes—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—232no—
344C22×C4—216no—
345C22×C4—24no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—48no—
353C22×C4—48no—
354C22×C4—48no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—44no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—816no—
370C22×C4—88no—
371C22×C4—88no—
372C22×C4—816no—
373C22×C4C2×C22⋊C414yes—
374C22×C4C2×C22⋊C414yes—
375C22×C4C2×C22⋊C412yes—
376C22×C4C2×C22⋊C412yes—
377C22×C4#12012yes—
378C22×C4#12012yes—
379C22×C4#12012yes—
380C22×C4#12012yes—
381C22×C4#13512yes—
382C22×C4#13512yes—
383C22×C4AGL(1,ℤ/8)12yes—
384C22×C4#13712yes—
385C22×C4C23×C411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×Q812yes—
392C22×C4C2×#4112yes—
393C22×C42−1+412yes—
394C22×C42−1+412yes—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—44no—
406C2×C8—44no—
407C4⋊C4—232no—
408C4⋊C4—232no—
409C4⋊C4—22no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—28no—
416C4⋊C4—216no—
417C4⋊C4—216no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—44no—
425C4⋊C4—84no—
426C4⋊C4—84no—
427C4⋊C4C2×C22⋊C418yes—
428C4⋊C4C2×C22⋊C414yes—
429C22⋊C4—232no—
430C22⋊C4—216no—
431C42—24no—
432C42—24no—
433C42—24no—
434C42—24no—
435C42—84no—
436C42C2×C22⋊C414yes—
437C42C2×C22⋊C414yes—
438C42#12014yes—
439C42#12014yes—
440C23—28no—
441C23—28no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—42no—
454C23—42no—
455C23—44no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—44no—
463C23C22×C22⋊C411yescharacteristic
464C23C2×#12011yescharacteristic
465C23C2×#12011yescharacteristic
466C23#54812yes—
467C23D4×Q812yes—
468C23C2×#13511yescharacteristic
469C23C2×#13511yescharacteristic
470C23C2×AGL(1,ℤ/8)11yescharacteristic
471C23C2×#13711yescharacteristic
472C23#57612yes—
473C23#57712yes—
474Q8—416no—
475Q8—816no—
476D4—24no—
477D4—28no—
478D4—28no—
479D4—24no—
480D4—24no—
481D4#39118yescomplemented
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—28no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—48no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—48no—
510C2×C4—432no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—832no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—84no—
520C2×C4—84no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4#37914yes—
525C2×C4#37914yes—
526C2×C4#41614yes—
527C2×C4#41714yes—
528C2×C4#44914yes—
529C2×C4#44914yes—
530C2×C4#45214yes—
531C2×C4#45214yes—
532C2×C4C22×C22⋊C411yescharacteristic
533C2×C4C22×C22⋊C411yescharacteristic
534C2×C4C2×C4×Q812yes—
535C2×C4#51912yes—
536C2×C4C2×#12011yescharacteristic
537C2×C4C2×#12011yescharacteristic
538C2×C4C2×C22⋊Q812yes—
539C2×C4C2×#12411yescharacteristic
540C2×C4C2×C4⋊Q811yescharacteristic
541C2×C4#53612yes—
542C2×C4#53611yescharacteristic
543C2×C4#53611yescharacteristic
544C2×C4#54812yes—
545C2×C4#54812yes—
546C2×C4#54812yes—
547C2×C4D4×Q812yes—
548C2×C4D4×Q812yes—
549C2×C4D4×Q812yes—
550C8—44no—
551C8—82no—
552C22—28no—
553C22—48no—
554C22—48no—
555C22—44no—
556C22—44no—
557C22—44no—
558C22—44no—
559C22—44no—
560C22—44no—
561C22—48no—
562C22—48no—
563C22C2×#37911yescharacteristic
564C22Q8×C22⋊C412yes—
565C22#240912yes—
566C22#243411yescharacteristic
567C22#243611yescharacteristic
568C22C2×#41611yescharacteristic
569C22C2×#41711yescharacteristic
570C22#289712yes—
571C22C2×#44911yescharacteristic
572C22C2×#44911yescharacteristic
573C22C2×#45211yescharacteristic
574C22C2×#45211yescharacteristic
575C22#300811yescharacteristic
576C22#300911yescharacteristic
577C22#301212yes—
578C22#336012yes—
579C22#336112yes—
580C22#336612yes—
581C22#337212yes—
582C22#338012yes—
583C22#338112yes—
584C22#338412yes—
585C22#338512yes—
586C4—24no—
587C4—44no—
588C4—44no—
589C4—44no—
590C4—48no—
591C4—84no—
592C4—162no—
593C4#178818yes—
594C4C2×#39111yescharacteristic
595C4#231011yescharacteristic
596C4Q8×C22⋊C412yes—
597C4#240912yes—
598C4#243411yescharacteristic
599C4#243611yescharacteristic
600C2—48no—
601C2#1392211yescharacteristic
602C2C2×#178811yescharacteristic
603C2#2060211yescharacteristic
604C2#2251912yes—
605C2#2252212yes—
606C2#2432712yes—
607C2#2433112yes—
608C2#2583511yescharacteristic
609C2#2583911yescharacteristic
610C2#2593011yescharacteristic
611C2#2593411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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
2. C2×#1990C2semidirect
3. C2×#1987C2semidirect
4. C2×#1984C2semidirect
7. #13926C2semidirect
8. #11385C2semidirect
15. C2×#554C4semidirect
117. C2×#102Q8semidirect
481. D4#391semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#19902
C2×#19872
C2×#19842
C2×#18022
C2×C4×#1022
#139262
#113852
#107882