Finite Group Databasebeta

← #359003

Subgroups — #359003

Order 512.

2236 conjugacy classes of subgroups.

The subgroup lattice has 607 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#441C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊Q8C212yesmaximal
3C2×#2001C211yesmaximal, characteristic
4C2×#1763C212yesmaximal, complemented
5C2×#1749C211yesmaximal, characteristic
6#11327C211yesmaximal, characteristic, complemented
7#11095C211yesmaximal, characteristic
8#11067C212yesmaximal, complemented
9#11024C211yesmaximal, characteristic, complemented
10#10970C212yesmaximal
11#10703C211yesmaximal, characteristic, complemented
12C23×Q16C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C22×C4×Q8C2212yes—
15C23×C42C2211yescharacteristic
16C2×#441—24no—
17C2×#441—28no—
18C22×#106C2211yescharacteristic
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24C22×#97C2211yescharacteristic
25C22×#97C2211yescharacteristic
26C4×C22⋊Q8—28no—
27C4×C22⋊Q8—28no—
28C2×#391C2212yes—
29C2×#389C2212yes—
30C2×#387—22no—
31C2×#385—22no—
32C2×#380C2212yes—
33C2×C4×C4⋊C4—22no—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C22⋊C4—22no—
36#2001C418yescomplemented
37#1763—28no—
38#1749—24no—
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2211yescharacteristic
44C2×#336C2211yescharacteristic
45C2×#336C2211yescharacteristic
46C23×Q8C2312yes—
47C24×C4D412yes—
48C24×C4C2311yescharacteristic
49C22×Q16—24no—
50C22×Q16—24no—
51C22×Q16—22no—
52C22×Q16—22no—
53C22×Q16C2×C414yescomplemented
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C2×C22⋊Q8—24no—
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—24no—
62C2×C22⋊Q8—24no—
63C2×C22⋊Q8C2×C418yes—
64C2×C4×Q8—24no—
65C2×C4×Q8—24no—
66C2×C4×Q8—24no—
67C2×C4×Q8—24no—
68C2×C4×Q8—24no—
69C2×C4×Q8D418yes—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C42—22no—
79C22×C42—22no—
80C22×C42—22no—
81C22×C42—24no—
82C22×C42—24no—
83C22×C42—22no—
84C22×C42D412yes—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87#441—232no—
88C2×#106—22no—
89C2×#106—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103C2×C418yescomplemented
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8C2×C414yes—
105C2×#97—24no—
106C2×#97—22no—
107C2×#97—24no—
108C2×#97D414yes—
109#391—24no—
110#391—24no—
111#389—24no—
112#389—24no—
113#387—216no—
114#385—216no—
115#380—24no—
116#380—24no—
117C4×C4⋊C4—216no—
118C4×C4⋊C4—24no—
119C4×C4⋊C4D418yes—
120C4×C22⋊C4—216no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126#340—22no—
127#340—24no—
128#340—24no—
129#340—24no—
130#340C2×C414yescomplemented
131#336—22no—
132#336—22no—
133C25C2×D411yescharacteristic
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8C22×C414yes—
146C22×Q8C2×D414yes—
147C22×Q8#4114yes—
148C23×C4—24no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4C22×C412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4#4112yes—
172C23×C4C2411yescharacteristic
173C2×Q16—28no—
174C2×Q16—216no—
175C2×Q16—28no—
176C2×Q16—216no—
177C2×Q16—28no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8C22×C412yes—
193C22×C8C22×C412yes—
194C22×C8C2×D412yes—
195C22×C8C2×D412yes—
196C22×C8#4112yes—
197C22×C8#4112yes—
198C22⋊Q8—216no—
199C22⋊Q8—216no—
200C22⋊Q8—216no—
201C22⋊Q8—216no—
202C4×Q8—216no—
203C4×Q8—216no—
204C4×Q8—48no—
205C4×Q8—48no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—22no—
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—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4C22×C414yes—
229C2×C4⋊C4C2×D414yes—
230C2×C4⋊C4C2×D414yes—
231C2×C4⋊C4C2×D414yes—
232C2×C4⋊C4#4114yes—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C42—28no—
240C2×C42—28no—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—21no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—21no—
248C2×C42—44no—
249C2×C42—44no—
250C2×C42—44no—
251C2×C42—44no—
252C2×C42—44no—
253C2×C42—48no—
254C2×C42—44no—
255C2×C42C2×D414yes—
256C2×C42C2×D412yes—
257C2×C42C2×D414yes—
258C2×C42#4112yes—
259#106—28no—
260#103—216no—
261#103—216no—
262#103—216no—
263C22⋊C8—28no—
264C22⋊C8—28no—
265#97—28no—
266#97—44no—
267#95—28no—
268#95—28no—
269#95—28no—
270#95—28no—
271#95—28no—
272C24—22no—
273C24—21no—
274C24—22no—
275C24—21no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24C4×D412yes—
280C24#13714yes—
281C24C22×D411yescharacteristic
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—24no—
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8C4×D418yes—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—216no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—48no—
342C22×C4—48no—
343C22×C4—48no—
344C22×C4—48no—
345C22×C4—48no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—48no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4C4×D412yes—
378C22×C4#12014yes—
379C22×C4#12014yes—
380C22×C4#12114yes—
381C22×C4#13714yes—
382C22×C4C23×C411yescharacteristic
383C22×C4C22×D412yes—
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D412yes—
386C22×C4C2×#4112yes—
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4112yes—
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392Q16—232no—
393C2×C8—24no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—24no—
397C2×C8—22no—
398C2×C8—24no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8—22no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8C4×D414yes—
413C4⋊C4—232no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—416no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—416no—
427C4⋊C4C4×D418yes—
428C22⋊C4—232no—
429C22⋊C4—416no—
430C42—416no—
431C42—41no—
432C42—42no—
433C42—41no—
434C42—48no—
435C42—416no—
436C42#12014yes—
437C42#12118yes—
438C42#12714yes—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—22no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—22no—
449C23—42no—
450C23—42no—
451C23—41no—
452C23—42no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23#44114yescomplemented
460C23C2×C4×D411yescharacteristic
461C23C2×#12012yes—
462C23C2×#12111yescharacteristic
463C23#52512yes—
464C23#53211yescharacteristic
465C23C2×#13712yes—
466C23C2×#13712yes—
467C23#57512yes—
468C23#57512yes—
469Q8—28no—
470Q8—28no—
471Q8—44no—
472Q8—44no—
473Q8—44no—
474Q8—44no—
475Q8—44no—
476Q8—44no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—432no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—41no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4#39218yes—
531C2×C4C2×C4×D411yescharacteristic
532C2×C4C2×C4×D412yes—
533C2×C4C2×C4×D412yes—
534C2×C4C4×#4111yescharacteristic
535C2×C4C2×#12011yescharacteristic
536C2×C4C2×#12112yes—
537C2×C4#52512yes—
538C2×C4#52511yescharacteristic
539C2×C4#52512yes—
540C2×C4#52512yes—
541C2×C4#52511yescharacteristic
542C2×C4#52512yes—
543C2×C4#52911yescharacteristic
544C2×C4#52912yes—
545C2×C4C2×#12711yescharacteristic
546C2×C4#53211yescharacteristic
547C2×C4#53212yes—
548C2×C4#53212yes—
549C8—24no—
550C8—42no—
551C8—42no—
552C8—42no—
553C22—24no—
554C22—28no—
555C22—44no—
556C22—44no—
557C22—42no—
558C22—42no—
559C22—44no—
560C22—44no—
561C22—44no—
562C22C4×#12012yes—
563C22C4×#12111yescharacteristic
564C22C2×#39211yescharacteristic
565C22#238812yes—
566C22#239211yescharacteristic
567C22C2×#44111yescharacteristic
568C22#294311yescharacteristic
569C22C4×#13712yes—
570C22#300412yes—
571C22#300512yes—
572C22#300712yes—
573C22#300712yes—
574C22#305612yes—
575C22#305612yes—
576C22#311112yes—
577C22#311312yes—
578C22#311612yes—
579C22#311712yes—
580C22#318112yes—
581C22#318311yescharacteristic
582C22#318511yescharacteristic
583C4—28no—
584C4—42no—
585C4—48no—
586C4—44no—
587C4—48no—
588C4—84no—
589C4—84no—
590C4C4×#12011yescharacteristic
591C4C4×#12112yes—
592C4C4×#12711yescharacteristic
593C4#238811yescharacteristic
594C4#239212yes—
595C4#239711yescharacteristic
596C2—44no—
597C2C4×#39211yescharacteristic
598C2#2145512yes—
599C2#2145712yes—
600C2#2152711yescharacteristic
601C2#2153911yescharacteristic
602C2#2339212yes—
603C2#2357712yes—
604C2#2357812yes—
605C2#2370912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#441C2semidirect
4. C2×#1763C2semidirect
6. #11327C2semidirect
8. #11067C2semidirect
9. #11024C2semidirect
11. #10703C2semidirect
36. #2001C4semidirect
53. C22×Q16C2×C4semidirect
98. C2×#103C2×C4semidirect
130. #340C2×C4semidirect
459. C23#441semidirect

Maximal subgroups

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

subgroupindex
C22×#4412
C2×C4×C22⋊Q82
C2×#20012
C2×#17632
C2×#17492
#113272
#110952
#110672
#110242
#109702
#107032