Finite Group Databasebeta

← #392046

Subgroups — #392046

Order 512.

1246 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1817C211yesmaximal, characteristic
2#14686C211yesmaximal, characteristic
3#14682C211yesmaximal, characteristic
4#11325C212yesmaximal, complemented
5#11324C211yesmaximal, characteristic, complemented
6#11323C211yesmaximal, characteristic, complemented
7#11268C212yesmaximal, complemented
8#11267C211yesmaximal, characteristic, complemented
9#11267C211yesmaximal, characteristic, complemented
10#11096C212yesmaximal
11#11095C212yesmaximal
12C23×C42C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×C22⋊C8C2212yes—
16C2×#384C2211yescharacteristic
17C2×#384C2211yescharacteristic
18C2×#382C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×#380—21no—
21C2×#380—21no—
22C2×#380C2211yescharacteristic
23C2×#380C2212yes—
24C2×#380C2211yescharacteristic
25#1817C418yescomplemented
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34C2×#336C2212yes—
35C4⋊#95—21no—
36C4⋊#95—21no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40#1434—21no—
41#1434—21no—
42C22⋊#95—21no—
43C22⋊#95—21no—
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2312yes—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—22no—
62C22×C42—42no—
63C22×C42—42no—
64C22×C42C2×C414yes—
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42Q814yes—
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C2×#107—42no—
71C2×#107—42no—
72C2×#107—42no—
73C2×#106—42no—
74C2×#106—42no—
75C2×#106—42no—
76C2×C22⋊C8—42no—
77C2×C22⋊C8—42no—
78C2×C22⋊C8C2×C418yes—
79#384—42no—
80#384—42no—
81#382—42no—
82#382—42no—
83#380—42no—
84#380—42no—
85#380—42no—
86#380—42no—
87#380—44no—
88#380—44no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102#340—42no—
103#340—42no—
104#340—42no—
105#340—42no—
106#340—42no—
107#340—42no—
108#336C2×C414yes—
109#336C2×C414yes—
110#336C2×C418yes—
111C25C2×D411yescharacteristic
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—41no—
133C23×C4—41no—
134C23×C4—41no—
135C23×C4—41no—
136C23×C4—41no—
137C23×C4—41no—
138C23×C4C4⋊C414yes—
139C23×C4C4⋊C414yes—
140C23×C4C22×C412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×Q812yes—
147C23×C4C2411yescharacteristic
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8C22×C414yes—
161C22×C8C22×C414yes—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—42no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—48no—
216C2×C42—48no—
217C2×C42—41no—
218C2×C42—41no—
219C2×C42—41no—
220C2×C42—41no—
221C2×C42C4⋊C414yes—
222C2×C42C4⋊C414yes—
223C2×C42C22×C412yes—
224C2×C42C22×C412yes—
225C2×C42C2×Q812yes—
226C2×C42C2×Q812yes—
227#107—48no—
228#106—48no—
229C22⋊C8—48no—
230#95—42no—
231#95—42no—
232#95—44no—
233#95—44no—
234#95—44no—
235#95—42no—
236#95—42no—
237#95—42no—
238#95—42no—
239#95—42no—
240#95—42no—
241#95—42no—
242#95—42no—
243C24—21no—
244C24—21no—
245C24—21no—
246C24—21no—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—21no—
251C24—21no—
252C24—21no—
253C24—21no—
254C24—21no—
255C24C2×C4⋊C412yes—
256C24AGL(1,ℤ/8)12yes—
257C24#13712yes—
258C24C22×D411yescharacteristic
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—28no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—28no—
295C22×C4—28no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4C2×C4⋊C412yes—
358C22×C4C2×C4⋊C412yes—
359C22×C4C2×C4⋊C412yes—
360C22×C4C2×C4⋊C412yes—
361C22×C4C2×C4⋊C412yes—
362C22×C4#13512yes—
363C22×C4#13512yes—
364C22×C4AGL(1,ℤ/8)12yes—
365C22×C4#13712yes—
366C22×C4C23×C411yescharacteristic
367C22×C4C22×Q811yescharacteristic
368C22×C4H(5,𝔽2)12yes—
369C22×C4H(5,𝔽2)12yes—
370C22×C4H(5,𝔽2)12yes—
371C22×C4H(5,𝔽2)11yescharacteristic
372C22×C4H(5,𝔽2)11yescharacteristic
373C22×C42−1+412yes—
374C22×C42−1+411yescharacteristic
375C22×C42−1+411yescharacteristic
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—84no—
407C4⋊C4—84no—
408C4⋊C4—84no—
409C4⋊C4—84no—
410C22⋊C4—84no—
411C22⋊C4—84no—
412C22⋊C4—88no—
413C42—24no—
414C42—24no—
415C42—24no—
416C42—24no—
417C42—44no—
418C42—44no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23#42814yes—
454C23C22×C4⋊C411yescharacteristic
455C23#53411yescharacteristic
456C23#53411yescharacteristic
457C23#53511yescharacteristic
458C23#53511yescharacteristic
459C23#53611yescharacteristic
460C23#53611yescharacteristic
461C23C2×#13511yescharacteristic
462C23C2×#13511yescharacteristic
463C23C2×AGL(1,ℤ/8)11yescharacteristic
464C23C2×AGL(1,ℤ/8)11yescharacteristic
465C23C2×#13711yescharacteristic
466C23C2×#13711yescharacteristic
467C23#57511yescharacteristic
468C23#57511yescharacteristic
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—416no—
491C2×C4—416no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—82no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—82no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—82no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—82no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4#42714yes—
551C2×C4#42814yes—
552C2×C4#51812yes—
553C2×C4#52012yes—
554C2×C4#52011yescharacteristic
555C2×C4#52011yescharacteristic
556C2×C4#54312yes—
557C2×C4#54412yes—
558C2×C4#54411yescharacteristic
559C2×C4#54411yescharacteristic
560C2×C4#56012yes—
561C2×C4#56012yes—
562C2×C4#56212yes—
563C2×C4#56212yes—
564C2×C4#56212yes—
565C2×C4#56212yes—
566C2×C4#56312yes—
567C2×C4#56312yes—
568C8—44no—
569C8—44no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22#231511yescharacteristic
586C22#231711yescharacteristic
587C22#231711yescharacteristic
588C22#281511yescharacteristic
589C22#281611yescharacteristic
590C22#281611yescharacteristic
591C22#281711yescharacteristic
592C22C2×#42711yescharacteristic
593C22C2×#42811yescharacteristic
594C22C2×#42811yescharacteristic
595C22#291111yescharacteristic
596C22#318811yescharacteristic
597C22#318811yescharacteristic
598C22#319011yescharacteristic
599C22#319011yescharacteristic
600C22#319411yescharacteristic
601C22#319411yescharacteristic
602C22#319611yescharacteristic
603C22#319611yescharacteristic
604C22#320711yescharacteristic
605C22#321011yescharacteristic
606C22#321211yescharacteristic
607C22#321211yescharacteristic
608C22#321311yescharacteristic
609C22#321511yescharacteristic
610C22#321711yescharacteristic
611C22#321811yescharacteristic
612C22#322811yescharacteristic
613C22#323111yescharacteristic
614C22#323311yescharacteristic
615C22#323311yescharacteristic
616C22#323411yescharacteristic
617C22#323611yescharacteristic
618C22#323811yescharacteristic
619C22#323911yescharacteristic
620C4—24no—
621C4—24no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—161no—
626C4—161no—
627C4—162no—
628C4—162no—
629C4—161no—
630C4—161no—
631C4#237612yes—
632C4#237612yes—
633C4#284112yes—
634C4#284112yes—
635C2—44no—
636C2#1467411yescharacteristic
637C2#2194111yescharacteristic
638C2#2194411yescharacteristic
639C2#2198411yescharacteristic
640C2#2198511yescharacteristic
641C2#2198611yescharacteristic
642C2#2198711yescharacteristic
643C2#3275911yescharacteristic
644C2#3276111yescharacteristic
645C2#3277211yescharacteristic
646C2#3277511yescharacteristic
647C2#3277611yescharacteristic
648C2#3277711yescharacteristic
649C2#3278711yescharacteristic
650C2#3278911yescharacteristic
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
4. #11325C2semidirect
5. #11324C2semidirect
6. #11323C2semidirect
7. #11268C2semidirect
8. #11267C2semidirect
9. #11267C2semidirect
25. #1817C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18172
#146862
#146822
#113252
#113242
#113232
#112682
#112672
#110962
#110952