Finite Group Databasebeta

← #363873

Subgroups — #363873

Order 512.

3206 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#529C211yesmaximal, characteristic
2C2×#1874C212yesmaximal
3C2×#1872C212yesmaximal, complemented
4C2×#1817C211yesmaximal, characteristic, complemented
5C2×C4⋊#95C211yesmaximal, characteristic, complemented
6#11385C212yesmaximal, complemented
7#11009C212yesmaximal, complemented
8#10789C212yesmaximal, complemented
9#10788C212yesmaximal, complemented
10C2×#529—24no—
11C2×#529—28no—
12C2×#529—28no—
13C2×#529C418yescomplemented
14C22×#126—21no—
15C22×#125C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C22×#123—21no—
18C22×C22⋊Q8C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4—21no—
22C22×C4×D4C2211yescharacteristic
23C22×#117—21no—
24C23×C4⋊C4C2212yes—
25C23×C42C2211yescharacteristic
26C22×#103C2214yes—
27C22×#102C2214yes—
28C22×C22⋊C8C2212yes—
29C22×#95—21no—
30C22×#95—21no—
31#1874—28no—
32#1872—28no—
33#1817—24no—
34C2×#340C2214yes—
35C2×#340C2214yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C4⋊#95—21no—
39C4⋊#95—21no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C23×Q8C2311yescharacteristic
43C23×D4—21no—
44C23×D4C2311yescharacteristic
45C24×C4—21no—
46C24×C4—22no—
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C24×C4C2311yescharacteristic
50C23×C8C2314yes—
51#529—264no—
52#529—232no—
53#529—232no—
54C2×#126—216no—
55C2×#126—28no—
56C2×#125—24no—
57C2×#125—24no—
58C2×#125—22no—
59C2×#125C2×C414yescomplemented
60C2×#124—24no—
61C2×#124—24no—
62C2×#124—22no—
63C2×#124C2×C414yescomplemented
64C2×#123—216no—
65C2×#123—28no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—24no—
68C2×C22⋊Q8C2×C414yes—
69C2×C22⋊Q8D418yes—
70C2×#121—22no—
71C2×#121—24no—
72C2×#121C2×C414yes—
73C2×#121D418yes—
74C2×C4×Q8—24no—
75C2×C4×Q8—24no—
76C2×C4×Q8—22no—
77C2×C4×Q8C2×C414yescomplemented
78C2×C4×D4—216no—
79C2×C4×D4—24no—
80C2×C4×D4—28no—
81C2×C4×D4—24no—
82C2×C4×D4—22no—
83C2×C4×D4C2×C414yescomplemented
84C2×#117—216no—
85C2×#117—28no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—22no—
91C22×C4⋊C4—28no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—44no—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C22⋊C4—21no—
99C22×C22⋊C4—21no—
100C22×C22⋊C4—21no—
101C22×C22⋊C4—21no—
102C22×C22⋊C4—21no—
103C22×C42—24no—
104C22×C42—24no—
105C22×C42—21no—
106C22×C42C2×C414yes—
107C22×C42D411yescharacteristic
108C22×C42D411yescharacteristic
109C22×C42D414yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C2×#103—28no—
113C2×#103—28no—
114C2×#103—44no—
115C2×#102—28no—
116C2×#102—28no—
117C2×#102—44no—
118C2×C22⋊C8—24no—
119C2×C22⋊C8—24no—
120C2×C22⋊C8—24no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—44no—
126C2×#95—44no—
127#340—28no—
128#340—28no—
129#336—24no—
130#336—24no—
131C25—21no—
132C25C2×D411yescharacteristic
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—24no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8C22×C412yes—
139C22×Q8C2×D414yes—
140C22×D4—24no—
141C22×D4—28no—
142C22×D4—22no—
143C22×D4—21no—
144C22×D4—24no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—44no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4C22×C412yes—
151C22×D4C2×D414yes—
152C23×C4—24no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—28no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—24no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—28no—
168C23×C4—24no—
169C23×C4—42no—
170C23×C4—44no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4C22⋊C414yes—
180C23×C4C22⋊C414yes—
181C23×C4C22×C412yes—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D414yes—
187C23×C4C2×D414yes—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D412yes—
190C23×C4C2411yescharacteristic
191C22×C8—24no—
192C22×C8—24no—
193C22×C8—24no—
194C22×C8—24no—
195C22×C8—24no—
196C22×C8—24no—
197C22×C8—24no—
198#126—264no—
199#125—216no—
200#125—28no—
201#125—28no—
202#124—216no—
203#124—28no—
204#124—28no—
205#123—264no—
206C22⋊Q8—28no—
207C22⋊Q8—216no—
208C22⋊Q8C22⋊C4116yes—
209#121—28no—
210#121—216no—
211#121C22⋊C4116yes—
212C4×Q8—216no—
213C4×Q8—28no—
214C4×Q8—28no—
215C4×D4—264no—
216C4×D4—216no—
217C4×D4—28no—
218C4×D4—28no—
219#117—264no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—416no—
239C2×C4⋊C4—416no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4C22×C412yes—
243C2×C4⋊C4C22×C412yes—
244C2×C4⋊C4C2×D414yes—
245C2×C4⋊C4C2×D414yes—
246C2×C4⋊C4#4118yes—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C42—28no—
258C2×C42—24no—
259C2×C42—28no—
260C2×C42—28no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—216no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42C22⋊C414yes—
269C2×C42C22⋊C414yes—
270C2×C42C22×C412yes—
271C2×C42C22×C412yes—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274#103—416no—
275#102—416no—
276C22⋊C8—48no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281C24—22no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—22no—
287C24—22no—
288C24—24no—
289C24—42no—
290C24—41no—
291C24—41no—
292C24—42no—
293C24—41no—
294C24—41no—
295C24—41no—
296C24—41no—
297C24—42no—
298C24C2×C22⋊C412yes—
299C24#12014yes—
300C24#13514yes—
301C24AGL(1,ℤ/8)12yes—
302C24#13712yes—
303C24C22×D411yescharacteristic
304C2×Q8—22no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—28no—
309C2×Q8—28no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—44no—
313C2×Q8—44no—
314C2×Q8C2×C22⋊C414yes—
315C2×D4—216no—
316C2×D4—22no—
317C2×D4—24no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—28no—
321C2×D4—28no—
322C2×D4—48no—
323C2×D4—48no—
324C2×D4—416no—
325C2×D4—416no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4C2×C22⋊C414yes—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—28no—
344C22×C4—216no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—44no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—416no—
374C22×C4—416no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—48no—
385C22×C4—48no—
386C22×C4—48no—
387C22×C4—416no—
388C22×C4—416no—
389C22×C4—416no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—88no—
397C22×C4—88no—
398C22×C4—88no—
399C22×C4—88no—
400C22×C4C2×C22⋊C412yes—
401C22×C4C2×C22⋊C412yes—
402C22×C4C2×C22⋊C412yes—
403C22×C4C2×C22⋊C414yes—
404C22×C4C2×C22⋊C414yes—
405C22×C4C2×C22⋊C412yes—
406C22×C4C2×C22⋊C412yes—
407C22×C4#12018yes—
408C22×C4#12014yes—
409C22×C4AGL(1,ℤ/8)12yes—
410C22×C4#13712yes—
411C22×C4C23×C411yescharacteristic
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D411yescharacteristic
415C22×C4C2×#4114yes—
416C22×C4H(5,𝔽2)12yes—
417C22×C42−1+412yes—
418C2×C8—44no—
419C2×C8—44no—
420C2×C8—44no—
421C2×C8—44no—
422C2×C8—44no—
423C2×C8—44no—
424C2×C8—44no—
425C4⋊C4—216no—
426C4⋊C4—216no—
427C4⋊C4—216no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—216no—
433C4⋊C4—28no—
434C4⋊C4—28no—
435C4⋊C4—416no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4C2×C22⋊C414yes—
439C4⋊C4C2×C22⋊C414yes—
440C22⋊C4—216no—
441C22⋊C4—216no—
442C22⋊C4—216no—
443C22⋊C4—216no—
444C22⋊C4—216no—
445C42—216no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—24no—
451C42—24no—
452C42—28no—
453C42—28no—
454C23—22no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—22no—
460C23—22no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—22no—
465C23—22no—
466C23—24no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—24no—
471C23—22no—
472C23—24no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—42no—
485C23—82no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23—82no—
490C23—81no—
491C23—81no—
492C23—82no—
493C23—81no—
494C23—81no—
495C23#37914yes—
496C23#41614yes—
497C23#41714yes—
498C23#44918yes—
499C23#45218yes—
500C23C22×C22⋊C411yescharacteristic
501C23C2×#12012yes—
502C23#52512yes—
503C23#53511yescharacteristic
504C23#53611yescharacteristic
505C23C2×#13512yes—
506C23C2×AGL(1,ℤ/8)11yescharacteristic
507C23C2×AGL(1,ℤ/8)11yescharacteristic
508C23C2×#13711yescharacteristic
509C23C2×#13711yescharacteristic
510C23#57512yes—
511Q8—28no—
512Q8—44no—
513Q8—48no—
514D4—28no—
515D4—432no—
516D4—44no—
517D4—48no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—216no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—416no—
542C2×C4—432no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—416no—
562C2×C4—48no—
563C2×C4—48no—
564C2×C4—48no—
565C2×C4—832no—
566C2×C4—84no—
567C2×C4—84no—
568C2×C4—84no—
569C2×C4—84no—
570C2×C4#37914yes—
571C2×C4#41714yes—
572C2×C4C22×C22⋊C411yescharacteristic
573C2×C4C22×C22⋊C411yescharacteristic
574C2×C4C4×#4112yes—
575C2×C4#52012yes—
576C2×C4C2×#12012yes—
577C2×C4#53212yes—
578C2×C4#53712yes—
579C2×C4#54614yes—
580C2×C4#54614yes—
581C2×C4#54714yes—
582C2×C4#54714yes—
583C8—84no—
584C22—28no—
585C22—28no—
586C22—24no—
587C22—24no—
588C22—24no—
589C22—28no—
590C22—28no—
591C22—24no—
592C22—24no—
593C22—24no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—84no—
598C22—84no—
599C22—82no—
600C22—82no—
601C22—84no—
602C22—84no—
603C22—82no—
604C22—82no—
605C22#178818yes—
606C22C2×#37911yescharacteristic
607C22#229711yescharacteristic
608C22#231111yescharacteristic
609C22#240412yes—
610C22#240612yes—
611C22C2×#41611yescharacteristic
612C22C2×#41711yescharacteristic
613C22C2×#41711yescharacteristic
614C22#289611yescharacteristic
615C22C2×#44912yes—
616C22C2×#45212yes—
617C22#300612yes—
618C22#300712yes—
619C22#310512yes—
620C22#310812yes—
621C22#311412yes—
622C22#311712yes—
623C22#321112yes—
624C22#321711yescharacteristic
625C22#321811yescharacteristic
626C22#323212yes—
627C22#323811yescharacteristic
628C22#323911yescharacteristic
629C4—24no—
630C4—24no—
631C4—42no—
632C4—42no—
633C4—44no—
634C4—42no—
635C4—44no—
636C4—44no—
637C4—88no—
638C4#233912yes—
639C4#233912yes—
640C4#243014yes—
641C2—28no—
642C2—84no—
643C2#1395611yescharacteristic
644C2C2×#178811yescharacteristic
645C2#2060111yescharacteristic
646C2#2138811yescharacteristic
647C2#2139111yescharacteristic
648C2#2186111yescharacteristic
649C2#2186711yescharacteristic
650C2#2393612yes—
651C2#2393712yes—
652C2#2406612yes—
653C2#2406712yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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
3. C2×#1872C2semidirect
4. C2×#1817C2semidirect
5. C2×C4⋊#95C2semidirect
6. #11385C2semidirect
7. #11009C2semidirect
8. #10789C2semidirect
9. #10788C2semidirect
13. C2×#529C4semidirect
59. C2×#125C2×C4semidirect
63. C2×#124C2×C4semidirect
77. C2×C4×Q8C2×C4semidirect
83. C2×C4×D4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5292
C2×#18742
C2×#18722
C2×#18172
C2×C4⋊#952
#113852
#110092
#107892
#107882