Finite Group Databasebeta

← #374587

Subgroups — #374587

Order 512.

1946 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#478C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3C2×#1992C211yesmaximal, characteristic, complemented
4C2×#1762C211yesmaximal, characteristic
5#14160C211yesmaximal, characteristic, complemented
6#11220C212yesmaximal, complemented
7#11219C212yesmaximal, complemented
8#10849C211yesmaximal, characteristic, complemented
9#10835C211yesmaximal, characteristic, complemented
10#10788C212yesmaximal
11#10788C212yesmaximal
12C22×#125C2211yescharacteristic
13C22×C4×D4C2212yes—
14C22×C4×D4C2211yescharacteristic
15C23×C4⋊C4—21no—
16C2×#478—24no—
17C2×#478—24no—
18C2×#478C418yescomplemented
19C22×#107C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#102C2211yescharacteristic
23C22×#102C2211yescharacteristic
24C22×#102C2212yes—
25C22×#97C2211yescharacteristic
26D4×C4⋊C4—24no—
27D4×C4⋊C4—24no—
28C2×#389C2212yes—
29C2×#385—22no—
30C2×#384C2211yescharacteristic
31C2×#380—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1992—24no—
35#1762—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1441—21no—
44#1441—21no—
45#1436—22no—
46C4⋊#95C2212yes—
47C4⋊#95C2212yes—
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C24×C4—22no—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2312yes—
54C2×#125—22no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125—22no—
59C2×#125C2×C414yescomplemented
60C2×C4×D4—24no—
61C2×C4×D4—24no—
62C2×C4×D4—24no—
63C2×C4×D4—24no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—42no—
68C2×C4×D4—42no—
69C2×C4×D4C2×C414yes—
70C2×C4×D4D414yes—
71C2×C4×D4D414yes—
72C22×C4⋊C4—28no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—48no—
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4D411yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C22⋊C4—21no—
87C22×C22⋊C4—22no—
88C22×C42—21no—
89C22×C42C2312yes—
90C22×C42C2311yescharacteristic
91#478—232no—
92C2×#107—22no—
93C2×#107—22no—
94C2×#107C2×C414yescomplemented
95C2×#106—22no—
96C2×#106—22no—
97C2×#106C2×C414yescomplemented
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8C2×C414yescomplemented
101C2×#102—22no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#102—22no—
105C2×#102—24no—
106C2×#102—24no—
107C2×#102C2×C414yes—
108C2×#102C2×C414yes—
109C2×#102Q818yes—
110C2×#97—22no—
111C2×#97—22no—
112C2×#97—22no—
113#389—24no—
114#389—24no—
115#385—216no—
116#384—22no—
117#384D414yes—
118#380—28no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—21no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95D412yes—
130#340—24no—
131#340—24no—
132#340—24no—
133#340—42no—
134#340—42no—
135#336—42no—
136#336—42no—
137C25—21no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4C22×C412yes—
144C22×D4C2×D412yes—
145C22×D4C2×D412yes—
146C22×D4C2×Q814yes—
147C22×D4#4114yes—
148C23×C4—21no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—28no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4—44no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—82no—
164C23×C4—82no—
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2411yescharacteristic
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8C22×C412yes—
185C22×C8C22×C412yes—
186C22×C8C2×Q814yes—
187#125—28no—
188#125—28no—
189#125—44no—
190#125—44no—
191C4×D4—28no—
192C4×D4—28no—
193C4×D4—28no—
194C4×D4—48no—
195C4×D4—48no—
196C4×D4—48no—
197C4×D4—48no—
198C4×D4C22⋊C4116yes—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
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—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—432no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—81no—
240C2×C4⋊C4—81no—
241C2×C4⋊C4C22×C412yes—
242C2×C4⋊C4C22×C412yes—
243C2×C4⋊C4C22×C412yes—
244C2×C4⋊C4C2×D414yes—
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×Q814yes—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—42no—
261C2×C42—81no—
262C2×C42—81no—
263C2×C42C22×C412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42#4112yes—
268C2×C42#4112yes—
269#107—28no—
270#106—28no—
271C4⋊C8—28no—
272#102—28no—
273#102—28no—
274#102—216no—
275#97—44no—
276#95—28no—
277#95—24no—
278#95—44no—
279#95—42no—
280#95—44no—
281#95—82no—
282#95—82no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—24no—
287C24—24no—
288C24—42no—
289C24—42no—
290C24—41no—
291C24—41no—
292C24—41no—
293C24—41no—
294C24C22×D411yescharacteristic
295C2×D4—22no—
296C2×D4—24no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—24no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—24no—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4C2×C22⋊C414yes—
309C2×D4C4×Q818yes—
310C2×D4C22⋊Q818yes—
311C2×D4#12414yes—
312C2×D4C4⋊Q814yes—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—216no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—48no—
341C22×C4—416no—
342C22×C4—48no—
343C22×C4—416no—
344C22×C4—416no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—416no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—88no—
371C22×C4—88no—
372C22×C4—88no—
373C22×C4—88no—
374C22×C4—88no—
375C22×C4C4×D412yes—
376C22×C4C4×D412yes—
377C22×C4#12012yes—
378C22×C4#12012yes—
379C22×C4#12112yes—
380C22×C4#12112yes—
381C22×C4AGL(1,ℤ/8)14yes—
382C22×C4C23×C411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×Q812yes—
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4112yes—
389C22×C4C2×#4111yescharacteristic
390C22×C4H(5,𝔽2)12yes—
391C22×C42−1+412yes—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—42no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C4⋊C4—28no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—44no—
422C4⋊C4—48no—
423C4⋊C4—416no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—48no—
433C4⋊C4—48no—
434C4⋊C4—84no—
435C4⋊C4C2×C22⋊C414yes—
436C22⋊C4—216no—
437C22⋊C4—416no—
438C42—42no—
439C42—42no—
440C42—42no—
441C42—42no—
442C42—84no—
443C42C2×C22⋊C414yes—
444C42C4×D414yes—
445C42#12114yes—
446C42#12314yes—
447C23—28no—
448C23—28no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—44no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23C2×C4×D411yescharacteristic
471C23C2×#12011yescharacteristic
472C23C2×#12111yescharacteristic
473C23#54612yes—
474C23D4×Q812yes—
475C23C2×AGL(1,ℤ/8)12yes—
476C23#57512yes—
477C23#57611yescharacteristic
478C23#57711yescharacteristic
479C23#57711yescharacteristic
480C23#57811yescharacteristic
481D4—24no—
482D4—28no—
483D4—28no—
484D4—24no—
485D4—24no—
486D4#39118yes—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—432no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—832no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4#38614yes—
545C2×C4#38614yes—
546C2×C4C22×C22⋊C411yescharacteristic
547C2×C4C2×C4×D411yescharacteristic
548C2×C4C2×C4×Q812yes—
549C2×C4#52012yes—
550C2×C4C2×#12111yescharacteristic
551C2×C4C2×C22⋊Q812yes—
552C2×C4C2×#12311yescharacteristic
553C2×C4C2×#12411yescharacteristic
554C2×C4C2×C4⋊Q811yescharacteristic
555C2×C4#53511yescharacteristic
556C2×C4#53611yescharacteristic
557C2×C4#53712yes—
558C2×C4#54612yes—
559C2×C4D4×Q812yes—
560C2×C4#55212yes—
561C2×C4#55212yes—
562C2×C4#55412yes—
563C2×C4#55412yes—
564C8—42no—
565C8—42no—
566C8—44no—
567C22—28no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—48no—
572C22—48no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—48no—
577C22—48no—
578C22C2×#38611yescharacteristic
579C22#236012yes—
580C22#241712yes—
581C22#243411yescharacteristic
582C22#243511yescharacteristic
583C22C4×AGL(1,ℤ/8)12yes—
584C22#295111yescharacteristic
585C22#295211yescharacteristic
586C22#300311yescharacteristic
587C22#300511yescharacteristic
588C22#301211yescharacteristic
589C22#301311yescharacteristic
590C22#303612yes—
591C22#304511yescharacteristic
592C22#304611yescharacteristic
593C22#330712yes—
594C22#331012yes—
595C22#331512yes—
596C22#331812yes—
597C22#338412yes—
598C22#338612yes—
599C22#338812yes—
600C22#338912yes—
601C4—24no—
602C4—24no—
603C4—44no—
604C4—44no—
605C4—84no—
606C4—82no—
607C4—82no—
608C4—82no—
609C4—84no—
610C4C2×#39111yescharacteristic
611C4#231111yescharacteristic
612C4#236012yes—
613C4#241712yes—
614C4#250211yescharacteristic
615C4#250311yescharacteristic
616C2—48no—
617C2#1413911yescharacteristic
618C2#2092511yescharacteristic
619C2#2093711yescharacteristic
620C2#2297412yes—
621C2#2297812yes—
622C2#2482312yes—
623C2#2482512yes—
624C2#2584411yescharacteristic
625C2#2584511yescharacteristic
626C2#2589111yescharacteristic
627C2#2589211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1992C2semidirect
5. #14160C2semidirect
6. #11220C2semidirect
7. #11219C2semidirect
8. #10849C2semidirect
9. #10835C2semidirect
18. C2×#478C4semidirect
59. C2×#125C2×C4semidirect
94. C2×#107C2×C4semidirect
97. C2×#106C2×C4semidirect
100. C2×C4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4782
C2×D4×C4⋊C42
C2×#19922
C2×#17622
#141602
#112202
#112192
#108492
#108352
#107882