Finite Group Databasebeta

← #363243

Subgroups — #363243

Order 512.

2942 conjugacy classes of subgroups.

The subgroup lattice has 609 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#555C211yesmaximal, characteristic
2C2×#1993C211yesmaximal, characteristic, complemented
3C2×#1988C211yesmaximal, characteristic, complemented
4C2×#1955C211yesmaximal, characteristic, complemented
5C2×#1804C211yesmaximal, characteristic, complemented
6C2×#1762C211yesmaximal, characteristic, complemented
7#13905C211yesmaximal, characteristic, complemented
8#10969C214yesmaximal, complemented
9#10731C214yesmaximal, complemented
10C2×#555—24no—
11C2×#555—24no—
12C2×#555—24no—
13C2×#555—24no—
14C2×#555—24no—
15C2×#555C418yescomplemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#124—22no—
18C22×#121—21no—
19C22×#121C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×#117—21no—
23C22×#117C2212yes—
24C22×C4⋊C8C2211yescharacteristic
25C22×#102C2212yes—
26C22×#102C2212yes—
27C22×C22⋊C8C2214yes—
28C22×#97C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#380C2214yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32#1993—24no—
33#1988—24no—
34#1955—24no—
35#1804—24no—
36#1762—24no—
37C2×#340C2214yes—
38C2×#340C2214yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41#1445—21no—
42#1438—21no—
43#1437—22no—
44C23×Q8—21no—
45C23×D4—22no—
46C23×D4D411yescharacteristic
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C24×C4C2312yes—
50C23×C8C2312yes—
51C23×C8C2312yes—
52#555—232no—
53#555—232no—
54#555—232no—
55#555—232no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8C2×C414yes—
60C2×C4⋊Q8D414yes—
61C2×C4⋊Q8D414yes—
62C2×#124—216no—
63C2×#124—216no—
64C2×#124—216no—
65C2×#121—28no—
66C2×#121—28no—
67C2×#121—28no—
68C2×#121—24no—
69C2×#121—24no—
70C2×#121—24no—
71C2×#121—24no—
72C2×#121—24no—
73C2×#121C2×C418yes—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4—22no—
78C2×C4×D4—22no—
79C2×C4×D4—22no—
80C2×C4×D4—22no—
81C2×C4×D4C2×C414yes—
82C2×C4×D4C2×C414yes—
83C2×C4×D4D414yes—
84C2×C4×D4D414yes—
85C2×C4×D4D414yes—
86C2×#117—28no—
87C2×#117—28no—
88C2×#117—24no—
89C2×#117—28no—
90C2×#117—24no—
91C2×#117—24no—
92C2×#117—24no—
93C2×#117—24no—
94C2×#117C2×C418yes—
95C22×C4⋊C4—21no—
96C22×C4⋊C4D411yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C22⋊C4—22no—
102C22×C22⋊C4—22no—
103C22×C22⋊C4—21no—
104C22×C22⋊C4—24no—
105C22×C22⋊C4D412yes—
106C22×C42—22no—
107C22×C42—21no—
108C22×C42C2311yescharacteristic
109C2×C4⋊C8—22no—
110C2×C4⋊C8—22no—
111C2×C4⋊C8—22no—
112C2×#102—24no—
113C2×#102—24no—
114C2×#102—24no—
115C2×#102—24no—
116C2×#102—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—
137C25—21no—
138C25C2×D412yes—
139C22×Q8—24no—
140C22×Q8—24no—
141C22×Q8—24no—
142C22×Q8—82no—
143C22×Q8—82no—
144C22×D4—28no—
145C22×D4—28no—
146C22×D4—28no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—44no—
157C22×D4—84no—
158C22×D4—84no—
159C22×D4C22⋊C414yes—
160C22×D4C22×C412yes—
161C22×D4C2×D412yes—
162C22×D4C2×D412yes—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—48no—
179C23×C4C22×C414yes—
180C23×C4C2×D412yes—
181C23×C4C2×D414yes—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2411yescharacteristic
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—24no—
195C22×C8—24no—
196C22×C8#4114yes—
197C4⋊Q8—28no—
198C4⋊Q8—28no—
199C4⋊Q8—28no—
200C4⋊Q8C22⋊C4116yescomplemented
201#124—2128no—
202#121—216no—
203#121—216no—
204#121—264no—
205#121—216no—
206#121—216no—
207C4×D4—28no—
208C4×D4—28no—
209C4×D4—28no—
210C4×D4—28no—
211C4×D4—28no—
212C4×D4—28no—
213C4×D4—28no—
214C4×D4C22⋊C4116yescomplemented
215#117—216no—
216#117—216no—
217#117—264no—
218#117—216no—
219#117—216no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
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—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4C22⋊C414yes—
241C2×C4⋊C4C22×C412yes—
242C2×C4⋊C4C22×C414yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C4⋊C4C2×D414yes—
246C2×C4⋊C4C2×D414yes—
247C2×C4⋊C4#4114yes—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—48no—
260C2×C22⋊C4—48no—
261C2×C22⋊C4—84no—
262C2×C22⋊C4C22⋊C418yes—
263C2×C42—28no—
264C2×C42—28no—
265C2×C42—28no—
266C2×C42—42no—
267C2×C42—82no—
268C2×C42C22×C412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C4⋊C8—44no—
275#102—216no—
276#102—48no—
277C22⋊C8—416no—
278#97—44no—
279#95—48no—
280#95—42no—
281#95—42no—
282#95—42no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—24no—
289C24—24no—
290C24—24no—
291C24—24no—
292C24—24no—
293C24—24no—
294C24—42no—
295C24—42no—
296C24—44no—
297C24—42no—
298C24—42no—
299C24C2×C22⋊C414yes—
300C24C22×D411yescharacteristic
301C2×Q8—216no—
302C2×Q8—88no—
303C2×Q8—84no—
304C2×Q8—84no—
305C2×Q8—88no—
306C2×D4—232no—
307C2×D4—22no—
308C2×D4—24no—
309C2×D4—24no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—24no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—24no—
318C2×D4—28no—
319C2×D4—48no—
320C2×D4—48no—
321C2×D4—48no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—816no—
329C2×D4—88no—
330C2×D4—88no—
331C2×D4—816no—
332C2×D4C2×C22⋊C414yes—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—28no—
349C22×C4—28no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—216no—
355C22×C4—24no—
356C22×C4—416no—
357C22×C4—416no—
358C22×C4—416no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4C2×C22⋊C414yes—
382C22×C4C2×C22⋊C412yes—
383C22×C4C2×C22⋊C412yes—
384C22×C4#12012yes—
385C22×C4#12012yes—
386C22×C4#12012yes—
387C22×C4#12012yes—
388C22×C4C23×C411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C2×#4112yes—
395C22×C4C2×#4112yes—
396C22×C4H(5,𝔽2)12yes—
397C22×C42−1+412yes—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—44no—
409C2×C8—44no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—216no—
417C4⋊C4—28no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—44no—
425C4⋊C4C2×C22⋊C414yes—
426C4⋊C4C2×C22⋊C418yes—
427C22⋊C4—232no—
428C22⋊C4—232no—
429C22⋊C4—216no—
430C22⋊C4—216no—
431C22⋊C4—816no—
432C42—232no—
433C42—84no—
434C42C2×C22⋊C414yes—
435C42C2×C22⋊C414yes—
436C42#12014yes—
437C42#12014yes—
438C23—28no—
439C23—24no—
440C23—24no—
441C23—28no—
442C23—24no—
443C23—24no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—84no—
472C23—84no—
473C23—84no—
474C23—84no—
475C23C22×C22⋊C411yescharacteristic
476C23C2×#12011yescharacteristic
477C23C2×#12011yescharacteristic
478C23#54612yes—
479C23#54712yes—
480C23#57514yes—
481C23#57612yes—
482C23#57712yes—
483Q8—816no—
484D4—24no—
485D4—24no—
486D4—416no—
487D4—44no—
488D4—44no—
489D4—42no—
490D4—42no—
491D4—832no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—28no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—28no—
503C2×C4—432no—
504C2×C4—44no—
505C2×C4—48no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—48no—
514C2×C4—44no—
515C2×C4—48no—
516C2×C4—48no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—48no—
522C2×C4—84no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—84no—
529C2×C4—84no—
530C2×C4#37914yes—
531C2×C4#37914yes—
532C2×C4C22×C22⋊C411yescharacteristic
533C2×C4C22×C22⋊C411yescharacteristic
534C2×C4C4×#4112yes—
535C2×C4#52012yes—
536C2×C4C2×#12011yescharacteristic
537C2×C4C2×#12011yescharacteristic
538C2×C4#52512yes—
539C2×C4#53211yescharacteristic
540C2×C4#53211yescharacteristic
541C2×C4#53511yescharacteristic
542C2×C4#53611yescharacteristic
543C2×C4#53712yes—
544C2×C4#54612yes—
545C2×C4#54712yes—
546C2×C4#54812yes—
547C2×C4#54812yes—
548C2×C4#55012yes—
549C2×C4#55012yes—
550C8—44no—
551C8—82no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—44no—
556C22—44no—
557C22—44no—
558C22—48no—
559C22—48no—
560C22—84no—
561C22—84no—
562C22—84no—
563C22—84no—
564C22—84no—
565C22—84no—
566C22—88no—
567C22—84no—
568C22—84no—
569C22C2×#37911yescharacteristic
570C22#233912yes—
571C22#240712yes—
572C22#243111yescharacteristic
573C22#243211yescharacteristic
574C22#289612yes—
575C22#289712yes—
576C22#300512yes—
577C22#300512yes—
578C22#300811yescharacteristic
579C22#300911yescharacteristic
580C22#301212yes—
581C22#331914yes—
582C22#332114yes—
583C22#334714yes—
584C22#334814yes—
585C4—24no—
586C4—24no—
587C4—44no—
588C4—44no—
589C4—48no—
590C4—84no—
591C4—162no—
592C4#229711yescharacteristic
593C4#231111yescharacteristic
594C4#234112yes—
595C4#240712yes—
596C4#243111yescharacteristic
597C4#243211yescharacteristic
598C2—84no—
599C2—84no—
600C2#1393711yescharacteristic
601C2#2060011yescharacteristic
602C2#2060211yescharacteristic
603C2#2249214yes—
604C2#2412912yes—
605C2#2413312yes—
606C2#2557812yes—
607C2#2571012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#555C2
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
4. C2×#1955C2semidirect
5. C2×#1804C2semidirect
6. C2×#1762C2semidirect
7. #13905C2semidirect
8. #10969C2semidirect
9. #10731C2semidirect
15. C2×#555C4semidirect
200. C4⋊Q8C22⋊C4semidirect
214. C4×D4C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5552
C2×#19932
C2×#19882
C2×#19552
C2×#18042
C2×#17622
#139052
#109692
#107312