Finite Group Databasebeta

← #368529

Subgroups — #368529

Order 512.

1630 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1895C211yesmaximal, characteristic
2C2×#1875C211yesmaximal, characteristic
3C2×#1736C211yesmaximal, characteristic, complemented
4#14137C211yesmaximal, characteristic
5#13801C211yesmaximal, characteristic, complemented
6#11315C212yesmaximal, complemented
7#11189C212yesmaximal, complemented
8#10931C212yesmaximal, complemented
9#10807C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic, complemented
11#10715C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C22×#121C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#102C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C22×#102C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C2×#390C2212yes—
21C2×#386—22no—
22C2×#380C2211yescharacteristic
23C2×C4×C4⋊C4C2212yes—
24C2×C4×C22⋊C4C2211yescharacteristic
25#1895C418yescomplemented
26#1875—24no—
27#1736—24no—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37#1439—21no—
38#1439—21no—
39#1437—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42#1434—21no—
43C4×#95—21no—
44C23×D4—21no—
45C23×D4C2311yescharacteristic
46C24×C4—21no—
47C24×C4C2311yescharacteristic
48C22×M16—22no—
49C22×M16—22no—
50C22×M16—42no—
51C22×M16—42no—
52C22×M16C2×C414yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2312yes—
56C2×#121—22no—
57C2×#121—22no—
58C2×#121—22no—
59C2×#121—22no—
60C2×#121—22no—
61C2×#121C2×C414yes—
62C2×#117—22no—
63C2×#117—24no—
64C2×#117—42no—
65C2×#117D414yes—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—42no—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4D412yes—
82C22×C42—21no—
83C22×C42—22no—
84C22×C42C2312yes—
85C2×#102—22no—
86C2×#102—22no—
87C2×#102—24no—
88C2×#102—42no—
89C2×#102—42no—
90C2×#102—42no—
91C2×#102C2×C418yescomplemented
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95#390—24no—
96#390—42no—
97#386—48no—
98#380—22no—
99#380C2×C414yes—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—42no—
102C4×C22⋊C4—22no—
103C4×C22⋊C4—22no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95D411yescharacteristic
111C2×#95D411yescharacteristic
112#340—22no—
113#340—22no—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—42no—
118#340C2×C418yescomplemented
119#336—42no—
120#336—42no—
121C25—21no—
122C25C2×D411yescharacteristic
123C22×D4—24no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—44no—
134C22×D4—44no—
135C22×D4—42no—
136C22×D4—42no—
137C22×D4C22×C412yes—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—44no—
148C23×C4—42no—
149C23×C4—44no—
150C23×C4—81no—
151C23×C4—81no—
152C23×C4C22×C412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4#4112yes—
158C23×C4#4112yes—
159C23×C4#4112yes—
160C23×C4#4112yes—
161C23×C4C2411yescharacteristic
162C2×M16—28no—
163C2×M16—48no—
164C2×M16—44no—
165C2×M16—44no—
166C2×M16—48no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8C22×C414yes—
184#121—28no—
185#121—28no—
186#121—44no—
187#121—44no—
188#117—28no—
189#117—44no—
190#117—48no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—48no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C22×C414yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4#4114yes—
221C2×C4⋊C4#4114yes—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—82no—
236C2×C22⋊C4—82no—
237C2×C22⋊C4—81no—
238C2×C22⋊C4—81no—
239C2×C42—24no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—44no—
247C2×C42—44no—
248C2×C42—44no—
249C2×C42—82no—
250#102—48no—
251#102—44no—
252#102—44no—
253C22⋊C8—44no—
254#95—24no—
255#95—22no—
256#95—22no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—44no—
261#95—82no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—21no—
267C24—21no—
268C24—21no—
269C24—21no—
270C24—22no—
271C24—41no—
272C24—41no—
273C24—42no—
274C24—42no—
275C24—42no—
276C24—42no—
277C24—41no—
278C24—41no—
279C24—41no—
280C24—41no—
281C24C4×D412yes—
282C24#12112yes—
283C24#12312yes—
284C24C22×D411yescharacteristic
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—416no—
290C2×D4—48no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—42no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—88no—
306C2×D4—84no—
307C2×D4—84no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—48no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
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—42no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—88no—
368C22×C4—88no—
369C22×C4—84no—
370C22×C4—84no—
371C22×C4#11714yes—
372C22×C4C4×D414yes—
373C22×C4C4×D412yes—
374C22×C4#12112yes—
375C22×C4#12312yes—
376C22×C4#12414yes—
377C22×C4#12614yes—
378C22×C4AGL(1,ℤ/8)14yes—
379C22×C4C23×C411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4112yes—
385C22×C4C2×#4112yes—
386C22×C4C2×#4111yescharacteristic
387C22×C4H(5,𝔽2)12yes—
388C22×C4H(5,𝔽2)12yes—
389M16—416no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C4⋊C4—22no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—48no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4#12314yes—
426C22⋊C4—48no—
427C22⋊C4—48no—
428C22⋊C4—48no—
429C22⋊C4—88no—
430C22⋊C4—84no—
431C22⋊C4—88no—
432C42—48no—
433C42—42no—
434C42—42no—
435C42—42no—
436C42—42no—
437C42—88no—
438C23—24no—
439C23—22no—
440C23—22no—
441C23—24no—
442C23—22no—
443C23—22no—
444C23—24no—
445C23—22no—
446C23—22no—
447C23—24no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—44no—
461C23—44no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—82no—
466C23—82no—
467C23—81no—
468C23—81no—
469C23—81no—
470C23—81no—
471C23—82no—
472C23—81no—
473C23—81no—
474C23—81no—
475C23—81no—
476C23—82no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23#38814yes—
484C23C2×C4×D411yescharacteristic
485C23C2×#12111yescharacteristic
486C23C2×#12311yescharacteristic
487C23#54612yes—
488C23#54612yes—
489C23C2×AGL(1,ℤ/8)12yes—
490C23#57512yes—
491C23#57611yescharacteristic
492C23#57711yescharacteristic
493C23#57711yescharacteristic
494C23#57811yescharacteristic
495D4—42no—
496D4—44no—
497D4—42no—
498D4—816no—
499D4—81no—
500D4—81no—
501D4—82no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—816no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4#38814yes—
557C2×C4C2×#11711yescharacteristic
558C2×C4C2×C4×D411yescharacteristic
559C2×C4C4×#4112yes—
560C2×C4#51812yes—
561C2×C4C2×#12311yescharacteristic
562C2×C4#52511yescharacteristic
563C2×C4C2×#12411yescharacteristic
564C2×C4C2×#12611yescharacteristic
565C2×C4#52912yes—
566C2×C4#53411yescharacteristic
567C2×C4#53511yescharacteristic
568C2×C4#53812yes—
569C2×C4#55112yes—
570C2×C4#55112yes—
571C2×C4#55312yes—
572C2×C4#55312yes—
573C2×C4#55312yes—
574C2×C4#55312yes—
575C8—44no—
576C8—81no—
577C8—81no—
578C22—24no—
579C22—24no—
580C22—24no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—82no—
590C22—82no—
591C22—84no—
592C22—84no—
593C22—82no—
594C22—162no—
595C22—162no—
596C22—161no—
597C22—161no—
598C22—161no—
599C22—161no—
600C22C2×#38811yescharacteristic
601C22#236412yes—
602C22#241111yescharacteristic
603C22#241211yescharacteristic
604C22#244412yes—
605C22C4×AGL(1,ℤ/8)12yes—
606C22#294611yescharacteristic
607C22#294711yescharacteristic
608C22#303811yescharacteristic
609C22#304011yescharacteristic
610C22#304511yescharacteristic
611C22#304611yescharacteristic
612C22#309412yes—
613C22#309811yescharacteristic
614C22#309911yescharacteristic
615C22#330712yes—
616C22#330812yes—
617C22#331812yes—
618C22#332112yes—
619C22#332212yes—
620C22#332412yes—
621C22#332712yes—
622C22#333212yes—
623C4—24no—
624C4—44no—
625C4—44no—
626C4—82no—
627C4—84no—
628C4—82no—
629C4—84no—
630C4—82no—
631C4C4×#12311yescharacteristic
632C4#232111yescharacteristic
633C4#235312yes—
634C4#250411yescharacteristic
635C4#251012yes—
636C4#259111yescharacteristic
637C2—44no—
638C2—162no—
639C2#1402011yescharacteristic
640C2#2106011yescharacteristic
641C2#2106211yescharacteristic
642C2#2314012yes—
643C2#2314112yes—
644C2#2449211yescharacteristic
645C2#2449911yescharacteristic
646C2#2454211yescharacteristic
647C2#2454911yescharacteristic
648C2#2621512yes—
649C2#2622912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1736C2semidirect
5. #13801C2semidirect
6. #11315C2semidirect
7. #11189C2semidirect
8. #10931C2semidirect
9. #10807C2semidirect
10. #10730C2semidirect
11. #10715C2semidirect
25. #1895C4semidirect
91. C2×#102C2×C4semidirect
118. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18952
C2×#18752
C2×#17362
#141372
#138012
#113152
#111892
#109312
#108072
#107302
#107152