Finite Group Databasebeta

← #399117

Subgroups — #399117

Order 512.

1612 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#427C211yesmaximal, characteristic, complemented
2C2×#2076C211yesmaximal, characteristic, complemented
3C2×#2075C211yesmaximal, characteristic, complemented
4C2×#1869C211yesmaximal, characteristic
5#13871C212yesmaximal
6#11379C211yesmaximal, characteristic, complemented
7#11377C211yesmaximal, characteristic, complemented
8#11257C212yesmaximal, complemented
9#11095C212yesmaximal, complemented
10#10731C212yesmaximal, complemented
11#10721C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×#117C2212yes—
15C2×#427—24no—
16C2×#427—24no—
17C2×#427—24no—
18C22×#107C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#106C2212yes—
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×#380C2212yes—
27#2076—24no—
28#2075—24no—
29#1869—24no—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39#1440—22no—
40#1438—22no—
41#1437—22no—
42#1436—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C23×D4D411yescharacteristic
46C24×C4C2311yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8—24no—
49C23×C8—22no—
50C23×C8—22no—
51C23×C8—24no—
52C23×C8D414yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—24no—
60C2×C4×D4—24no—
61C2×#117—24no—
62C2×#117—24no—
63C2×#117—24no—
64C2×#117—24no—
65C2×#117—42no—
66C2×#117—42no—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4D411yescharacteristic
77C22×C22⋊C4D411yescharacteristic
78C22×C42—22no—
79C22×C42C2311yescharacteristic
80#427—232no—
81C2×#107—22no—
82C2×#107—22no—
83C2×#107—22no—
84C2×#107—22no—
85C2×#107D414yescomplemented
86C2×#107D414yescomplemented
87C2×#106—24no—
88C2×#106—24no—
89C2×#106—24no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—24no—
97#380—24no—
98#380—24no—
99#380—24no—
100#380—24no—
101#380—42no—
102#380—42no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110#340—22no—
111#340—22no—
112#340—24no—
113#340—24no—
114#340—24no—
115#340—42no—
116#340—42no—
117#336—22no—
118#336—22no—
119C25C2×D411yescharacteristic
120C25C2×D411yescharacteristic
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—24no—
124C22×D4—22no—
125C22×D4—42no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D412yes—
151C23×C4#4112yes—
152C23×C4#4114yes—
153C23×C4#4114yes—
154C23×C4C2411yescharacteristic
155C22×C8—216no—
156C22×C8—216no—
157C22×C8—28no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—28no—
164C22×C8—28no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8C2×D412yes—
176C22×C8C2×D412yes—
177C22×C8C2×D412yes—
178C22×C8C2×D412yes—
179C22×C8#4114yes—
180C4×D4—44no—
181C4×D4—44no—
182C4×D4—48no—
183#117—216no—
184#117—48no—
185#117—48no—
186#117—48no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
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—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4C2×D414yes—
212C2×C4⋊C4C2×D414yes—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—82no—
226C2×C22⋊C4—82no—
227C2×C22⋊C4—82no—
228C2×C22⋊C4—82no—
229C2×C42—21no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—21no—
233C2×C42—44no—
234C2×C42—44no—
235C2×C42—44no—
236#107—28no—
237#107—28no—
238#106—216no—
239C22⋊C8—44no—
240C22⋊C8—44no—
241C22⋊C8—44no—
242#95—44no—
243#95—44no—
244#95—44no—
245#95—44no—
246#95—44no—
247#95—82no—
248#95—82no—
249C24—21no—
250C24—21no—
251C24—22no—
252C24—21no—
253C24—21no—
254C24—22no—
255C24—21no—
256C24—22no—
257C24—21no—
258C24—21no—
259C24—22no—
260C24—21no—
261C24—21no—
262C24—22no—
263C24—21no—
264C24—22no—
265C24—21no—
266C24—22no—
267C24—21no—
268C24#12112yes—
269C24#12314yes—
270C24#13512yes—
271C24C22×D411yescharacteristic
272C2×D4—28no—
273C2×D4—48no—
274C2×D4—48no—
275C2×D4—44no—
276C2×D4—44no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
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—42no—
355C22×C4#12112yes—
356C22×C4#12314yes—
357C22×C4#12314yes—
358C22×C4#12414yes—
359C22×C4#12618yes—
360C22×C4#13512yes—
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C2×#4112yes—
365C22×C4C2×#4112yes—
366C22×C4C2×#4112yes—
367C22×C4C2×#4111yescharacteristic
368C22×C4H(5,𝔽2)11yescharacteristic
369C22×C4H(5,𝔽2)11yescharacteristic
370C22×C42−1+412yes—
371C2×C8—232no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8#12014yes—
391C2×C8#12314yes—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—44no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C22⋊C4—416no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—88no—
413C22⋊C4—88no—
414C22⋊C4—88no—
415C42—416no—
416C42—41no—
417C42—41no—
418C42—42no—
419C23—24no—
420C23—24no—
421C23—24no—
422C23—24no—
423C23—22no—
424C23—24no—
425C23—24no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—24no—
432C23—24no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—41no—
453C23—41no—
454C23—42no—
455C23—42no—
456C23—41no—
457C23—42no—
458C23—42no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23#39714yes—
465C23C2×#12111yescharacteristic
466C23C2×#12312yes—
467C23D4211yescharacteristic
468C23D4211yescharacteristic
469C23#54712yes—
470C23C2×#13511yescharacteristic
471C23C2×#13511yescharacteristic
472C23#57512yes—
473C23#57611yescharacteristic
474C23#57712yes—
475C23#57811yescharacteristic
476D4—416no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—28no—
485C2×C4—28no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—81no—
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—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4#39714yes—
537C2×C4C2×#12011yescharacteristic
538C2×C4C2×#12311yescharacteristic
539C2×C4C2×#12311yescharacteristic
540C2×C4#52512yes—
541C2×C4#52512yes—
542C2×C4C2×#12411yescharacteristic
543C2×C4C2×#12612yes—
544C2×C4#53511yescharacteristic
545C2×C4#53611yescharacteristic
546C2×C4#53912yes—
547C2×C4#53912yes—
548C2×C4#54611yescharacteristic
549C2×C4#54611yescharacteristic
550C2×C4#54812yes—
551C2×C4#54812yes—
552C2×C4#55212yes—
553C2×C4#55312yes—
554C2×C4#55312yes—
555C8—24no—
556C8—24no—
557C8—81no—
558C8—81no—
559C22—24no—
560C22—24no—
561C22—24no—
562C22—42no—
563C22—42no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—44no—
572C22—44no—
573C22—42no—
574C22—42no—
575C22—81no—
576C22—82no—
577C22—82no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22C2×#39711yescharacteristic
582C22#240811yescharacteristic
583C22#244312yes—
584C22#244312yes—
585C22#248611yescharacteristic
586C22#305011yescharacteristic
587C22#305411yescharacteristic
588C22#306011yescharacteristic
589C22#306111yescharacteristic
590C22#309112yes—
591C22#309112yes—
592C22#309612yes—
593C22#309712yes—
594C22#326712yes—
595C22#327112yes—
596C22#328011yescharacteristic
597C22#328311yescharacteristic
598C22#328511yescharacteristic
599C22#328711yescharacteristic
600C22#333712yes—
601C22#334012yes—
602C22#334712yes—
603C22#334912yes—
604C4—24no—
605C4—42no—
606C4—81no—
607C4—81no—
608C4—84no—
609C4—84no—
610C4—162no—
611C4—162no—
612C4#243611yescharacteristic
613C4#248112yes—
614C4#248112yes—
615C4#250411yescharacteristic
616C4#262311yescharacteristic
617C4#284911yescharacteristic
618C2—44no—
619C2—82no—
620C2#1477611yescharacteristic
621C2#2363811yescharacteristic
622C2#2364111yescharacteristic
623C2#2426611yescharacteristic
624C2#2427111yescharacteristic
625C2#2612912yes—
626C2#2613112yes—
627C2#2613712yes—
628C2#2613912yes—
629C2#2765911yescharacteristic
630C2#2766411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×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
1. C22×#427C2semidirect
2. C2×#2076C2semidirect
3. C2×#2075C2semidirect
6. #11379C2semidirect
7. #11377C2semidirect
8. #11257C2semidirect
9. #11095C2semidirect
10. #10731C2semidirect
11. #10721C2semidirect
85. C2×#107D4semidirect
86. C2×#107D4semidirect

Maximal subgroups

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

subgroupindex
C22×#4272
C2×#20762
C2×#20752
C2×#18692
#138712
#113792
#113772
#112572
#110952
#107312
#107212