Finite Group Databasebeta

← #363246

Subgroups — #363246

Order 512.

2878 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic
2C2×#1993C211yesmaximal, characteristic, complemented
3C2×#1988C211yesmaximal, characteristic, complemented
4C2×#1955C211yesmaximal, characteristic
5C2×#1805C211yesmaximal, characteristic, complemented
6C2×#1763C211yesmaximal, characteristic, complemented
7#13905C211yesmaximal, characteristic
8#10970C214yesmaximal, complemented
9#10730C214yesmaximal, complemented
10C2×#554—24no—
11C2×#554—24no—
12C2×#554—24no—
13C2×#554—24no—
14C2×#554—24no—
15C2×#554C418yescomplemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#125—21no—
18C22×C22⋊Q8—22no—
19C22×C22⋊Q8C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4—21no—
22C22×C4×D4C2211yescharacteristic
23C23×C4⋊C4C2212yes—
24C22×C4⋊C8C2211yescharacteristic
25C22×#103C2212yes—
26C22×#103C2212yes—
27C22×C22⋊C8C2214yes—
28C22×#97C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#380C2214yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32#1993—24no—
33#1988—24no—
34#1955—24no—
35#1805—24no—
36#1763—24no—
37C2×#340C2214yes—
38C2×#340C2214yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41#1445—21no—
42#1438—21no—
43#1437—22no—
44C23×Q8—21no—
45C23×Q8C2311yescharacteristic
46C23×D4D411yescharacteristic
47C24×C4—22no—
48C24×C4C2312yes—
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—28no—
62C2×#125—28no—
63C2×#125—28no—
64C2×C22⋊Q8—216no—
65C2×C22⋊Q8—216no—
66C2×C22⋊Q8—24no—
67C2×C22⋊Q8—24no—
68C2×C22⋊Q8—24no—
69C2×C22⋊Q8—24no—
70C2×C22⋊Q8—24no—
71C2×C22⋊Q8—216no—
72C2×C22⋊Q8C2×C418yes—
73C2×C4×Q8—22no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8C2×C414yes—
77C2×C4×Q8D414yes—
78C2×C4×Q8D414yes—
79C2×C4×D4—28no—
80C2×C4×D4—28no—
81C2×C4×D4—28no—
82C2×C4×D4—22no—
83C2×C4×D4—22no—
84C2×C4×D4—22no—
85C2×C4×D4—22no—
86C2×C4×D4C2×C414yes—
87C2×C4×D4D414yes—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—22no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4C2×C418yes—
98C22×C4⋊C4D411yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C22⋊C4—22no—
104C22×C22⋊C4—24no—
105C22×C22⋊C4D412yes—
106C22×C42—21no—
107C22×C42—21no—
108C22×C42C2311yescharacteristic
109C2×C4⋊C8—22no—
110C2×C4⋊C8—22no—
111C2×C4⋊C8—22no—
112C2×#103—24no—
113C2×#103—24no—
114C2×#103—24no—
115C2×#103—24no—
116C2×#103—44no—
117C2×C22⋊C8—28no—
118C2×C22⋊C8—28no—
119C2×C22⋊C8—44no—
120C2×#97—22no—
121C2×#97—22no—
122C2×#97—22no—
123#382—22no—
124#382—22no—
125#380—28no—
126#380—44no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C2×#95—21no—
130C2×#95—24no—
131C2×#95—21no—
132C2×#95—21no—
133#340—24no—
134#340—28no—
135#340—28no—
136#340—24no—
137C25C2×D412yes—
138C22×Q8—24no—
139C22×Q8—24no—
140C22×Q8—24no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—82no—
149C22×Q8—82no—
150C22×Q8C22×C412yes—
151C22×Q8C2×D412yes—
152C22×Q8C2×D412yes—
153C22×D4—28no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4C22⋊C414yes—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—28no—
161C23×C4—28no—
162C23×C4—28no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—84no—
174C23×C4—84no—
175C23×C4C22×C414yes—
176C23×C4C2×D412yes—
177C23×C4C2×D414yes—
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—24no—
191C22×C8—24no—
192C22×C8#4114yes—
193C4⋊Q8—28no—
194C4⋊Q8—28no—
195C4⋊Q8—28no—
196C4⋊Q8C22⋊C4116yescomplemented
197#125—264no—
198C22⋊Q8—216no—
199C22⋊Q8—216no—
200C22⋊Q8—2128no—
201C22⋊Q8—216no—
202C22⋊Q8—216no—
203C4×Q8—28no—
204C4×Q8—28no—
205C4×Q8—28no—
206C4×Q8C22⋊C4116yescomplemented
207C4×D4—264no—
208C4×D4—28no—
209C4×D4—28no—
210C4×D4—28no—
211C4×D4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—216no—
215C2×C4⋊C4—216no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—216no—
235C2×C4⋊C4—216no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4C22⋊C414yes—
246C2×C4⋊C4C22×C412yes—
247C2×C4⋊C4C22×C414yes—
248C2×C4⋊C4C2×D412yes—
249C2×C4⋊C4C2×D412yes—
250C2×C4⋊C4C2×D414yes—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4#4114yes—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—48no—
259C2×C22⋊C4—48no—
260C2×C22⋊C4—84no—
261C2×C22⋊C4C22⋊C418yes—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—42no—
266C2×C42—82no—
267C2×C42C22×C412yes—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D412yes—
273C4⋊C8—44no—
274#103—216no—
275#103—48no—
276C22⋊C8—416no—
277#97—44no—
278#95—48no—
279#95—42no—
280#95—42no—
281#95—42no—
282C24—22no—
283C24—22no—
284C24—24no—
285C24—24no—
286C24—24no—
287C24—24no—
288C24—24no—
289C24C2×C22⋊C414yes—
290C24#13718yes—
291C24C22×D411yescharacteristic
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—216no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—24no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—88no—
311C2×Q8—84no—
312C2×Q8—84no—
313C2×Q8—88no—
314C2×Q8C2×C22⋊C414yes—
315C2×D4—216no—
316C2×D4—216no—
317C2×D4—216no—
318C2×D4—28no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—232no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—28no—
343C22×C4—28no—
344C22×C4—24no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—44no—
349C22×C4—42no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—42no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—816no—
371C22×C4—88no—
372C22×C4—88no—
373C22×C4—816no—
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×C4C23×C411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C2×#4112yes—
388C22×C4C2×#4112yes—
389C22×C4H(5,𝔽2)12yes—
390C22×C42−1+412yes—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—44no—
402C2×C8—44no—
403C4⋊C4—232no—
404C4⋊C4—232no—
405C4⋊C4—22no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—216no—
412C4⋊C4—216no—
413C4⋊C4—28no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—44no—
421C4⋊C4C2×C22⋊C414yes—
422C4⋊C4C2×C22⋊C418yes—
423C22⋊C4—232no—
424C22⋊C4—216no—
425C22⋊C4—816no—
426C42—216no—
427C42—84no—
428C42C2×C22⋊C414yes—
429C42C2×C22⋊C414yes—
430C42#12014yes—
431C42#12014yes—
432C23—28no—
433C23—28no—
434C23—28no—
435C23—24no—
436C23—28no—
437C23—28no—
438C23—28no—
439C23—24no—
440C23—28no—
441C23—28no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23#41918yes—
453C23C22×C22⋊C411yescharacteristic
454C23C2×#12011yescharacteristic
455C23C2×#12011yescharacteristic
456C23#54612yes—
457C23#54712yes—
458C23C2×#13714yes—
459C23#57612yes—
460C23#57712yes—
461Q8—24no—
462Q8—24no—
463Q8—44no—
464Q8—44no—
465Q8—42no—
466Q8—42no—
467Q8—816no—
468D4—232no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—28no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—28no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—48no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—48no—
496C2×C4—44no—
497C2×C4—48no—
498C2×C4—48no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—48no—
504C2×C4—832no—
505C2×C4—84no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—84no—
512C2×C4—84no—
513C2×C4#37914yes—
514C2×C4#37914yes—
515C2×C4C22×C22⋊C411yescharacteristic
516C2×C4C22×C22⋊C411yescharacteristic
517C2×C4C4×#4112yes—
518C2×C4#52012yes—
519C2×C4C2×#12011yescharacteristic
520C2×C4C2×#12011yescharacteristic
521C2×C4#52512yes—
522C2×C4#53211yescharacteristic
523C2×C4#53211yescharacteristic
524C2×C4#53511yescharacteristic
525C2×C4#53611yescharacteristic
526C2×C4#53712yes—
527C2×C4#54612yes—
528C2×C4#54712yes—
529C2×C4#54812yes—
530C2×C4#54812yes—
531C2×C4#55012yes—
532C2×C4#55012yes—
533C8—44no—
534C8—82no—
535C22—28no—
536C22—216no—
537C22—48no—
538C22—48no—
539C22—48no—
540C22—44no—
541C22—44no—
542C22—48no—
543C22—48no—
544C22C2×#37911yescharacteristic
545C22#233912yes—
546C22#240712yes—
547C22#243111yescharacteristic
548C22#243211yescharacteristic
549C22C2×#41912yes—
550C22#289712yes—
551C22#300412yes—
552C22#300412yes—
553C22#300811yescharacteristic
554C22#300911yescharacteristic
555C22#301212yes—
556C22#331414yes—
557C22#331614yes—
558C22#334414yes—
559C22#334614yes—
560C4—24no—
561C4—24no—
562C4—44no—
563C4—44no—
564C4—44no—
565C4—48no—
566C4—84no—
567C4—162no—
568C4#229711yescharacteristic
569C4#231111yescharacteristic
570C4#234112yes—
571C4#240712yes—
572C4#243111yescharacteristic
573C4#243211yescharacteristic
574C2—48no—
575C2#1393711yescharacteristic
576C2#2059911yescharacteristic
577C2#2060211yescharacteristic
578C2#2249714yes—
579C2#2413012yes—
580C2#2413412yes—
581C2#2557912yes—
582C2#2571212yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1993C2semidirect
3. C2×#1988C2semidirect
5. C2×#1805C2semidirect
6. C2×#1763C2semidirect
8. #10970C2semidirect
9. #10730C2semidirect
15. C2×#554C4semidirect
196. C4⋊Q8C22⋊C4semidirect
206. C4×Q8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#19932
C2×#19882
C2×#19552
C2×#18052
C2×#17632
#139052
#109702
#107302