Finite Group Databasebeta

← #385947

Subgroups — #385947

Order 512.

1828 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#466C211yesmaximal, characteristic
2C2×#1921C212yesmaximal, complemented
3#13998C212yesmaximal, complemented
4#11066C212yesmaximal, complemented
5#11022C212yesmaximal, complemented
6#10969C212yesmaximal, complemented
7#10931C212yesmaximal, complemented
8#10721C211yesmaximal, characteristic, complemented
9#10714C211yesmaximal, characteristic, complemented
10C23×D8C2211yescharacteristic
11C24×C8C2211yescharacteristic
12C22×#121C2212yes—
13C2×#466—28no—
14C2×#466C418yescomplemented
15C22×#107C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C2×#390—22no—
21C2×#390C2212yes—
22C2×#390C2212yes—
23C2×#389C2212yes—
24C2×#380C2212yes—
25C2×C4×C4⋊C4C2212yes—
26C2×C4×C22⋊C4C2211yescharacteristic
27#1921—28no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2211yescharacteristic
33C2×#336C2212yes—
34C2×#336C2212yes—
35#1439—22no—
36#1438—22no—
37#1437—22no—
38C23×D4—22no—
39C23×D4C2312yes—
40C24×C4C2311yescharacteristic
41C22×D8—24no—
42C22×D8—41no—
43C22×D8—42no—
44C22×D8—41no—
45C22×D8C2×C414yes—
46C23×C8—24no—
47C23×C8—42no—
48C23×C8—41no—
49C23×C8—41no—
50C23×C8C2×C414yes—
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×#121—24no—
55C2×#121—24no—
56C2×#121—24no—
57C2×#121—42no—
58C2×#121—42no—
59C2×#121C2×C418yes—
60C22×C4⋊C4—22no—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4D412yes—
71C22×C42—22no—
72C22×C42C2312yes—
73#466—232no—
74C2×#107C2×C414yes—
75C2×#107D418yes—
76C2×#102—24no—
77C2×#102—24no—
78C2×#102—24no—
79C2×#102—42no—
80C2×#102—42no—
81C2×#102—42no—
82C2×#102—42no—
83C2×#102—42no—
84C2×#102C2×C418yes—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—41no—
87C2×C22⋊C8—41no—
88#390—24no—
89#390—42no—
90#390—42no—
91#390—42no—
92#390—48no—
93#389—24no—
94#389—42no—
95#380—24no—
96#380—42no—
97C4×C4⋊C4—24no—
98C4×C4⋊C4—42no—
99C4×C22⋊C4—41no—
100C4×C22⋊C4—41no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95D411yescharacteristic
105C2×#95D411yescharacteristic
106#340—24no—
107#340—24no—
108#340—24no—
109#340—41no—
110#340—41no—
111#336—24no—
112#336—42no—
113C25—22no—
114C25C2×D411yescharacteristic
115C22×D4—28no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—44no—
130C22×D4—44no—
131C22×D4—44no—
132C22×D4—44no—
133C22×D4—82no—
134C22×D4—82no—
135C22×D4C22×C414yes—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4C22×C412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D412yes—
151C23×C4#4112yes—
152C23×C4C2411yescharacteristic
153C2×D8—28no—
154C2×D8—44no—
155C2×D8—44no—
156C2×D8—48no—
157C2×D8—48no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—28no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—44no—
174C22×C8—44no—
175C22×C8—48no—
176C22×C8—48no—
177C22×C8C22×C412yes—
178C22×C8C22×C412yes—
179C22×C8C2×D414yes—
180#121—216no—
181#121—48no—
182#121—48no—
183#121—48no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
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—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4C22×C414yes—
202C2×C4⋊C4C2×D414yes—
203C2×C4⋊C4C2×D414yes—
204C2×C4⋊C4#4114yes—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—82no—
222C2×C22⋊C4—82no—
223C2×C22⋊C4—82no—
224C2×C22⋊C4—82no—
225C2×C22⋊C4—82no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—44no—
233C2×C42—44no—
234C2×C42—82no—
235#107C22⋊C4116yescomplemented
236#102—216no—
237#102—48no—
238#102—48no—
239C22⋊C8—44no—
240#95—41no—
241#95—41no—
242#95—44no—
243#95—44no—
244#95—44no—
245C24—21no—
246C24—21no—
247C24—22no—
248C24—22no—
249C24—22no—
250C24—21no—
251C24—21no—
252C24—21no—
253C24—21no—
254C24—24no—
255C24—42no—
256C24—42no—
257C24—42no—
258C24—42no—
259C24—42no—
260C24—42no—
261C24—42no—
262C24—42no—
263C24—42no—
264C24—42no—
265C24—42no—
266C24—42no—
267C24—42no—
268C24—42no—
269C24C4×D412yes—
270C24AGL(1,ℤ/8)14yes—
271C24C22×D411yescharacteristic
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—24no—
278C2×D4—416no—
279C2×D4—416no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—88no—
303C2×D4—88no—
304C2×D4—88no—
305C2×D4—88no—
306C2×D4—88no—
307C22×C4—24no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
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—24no—
329C22×C4—24no—
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—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—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4C4×D412yes—
364C22×C4#13514yes—
365C22×C4C23×C411yescharacteristic
366C22×C4C22×D412yes—
367C22×C4C2×#4112yes—
368C22×C4C2×#4111yescharacteristic
369C22×C4H(5,𝔽2)11yescharacteristic
370C22×C4H(5,𝔽2)11yescharacteristic
371C22×C4H(5,𝔽2)12yes—
372C22×C4H(5,𝔽2)12yes—
373C22×C4H(5,𝔽2)11yescharacteristic
374C22×C42−1+411yescharacteristic
375D8—416no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—416no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8C2×C22⋊C414yes—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—88no—
406C4⋊C4C2×C22⋊C418yes—
407C22⋊C4—416no—
408C22⋊C4—48no—
409C22⋊C4—88no—
410C22⋊C4—88no—
411C22⋊C4—88no—
412C22⋊C4—88no—
413C22⋊C4—88no—
414C42—42no—
415C42—42no—
416C42—42no—
417C42—42no—
418C42—88no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—24no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—24no—
427C23—24no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—42no—
432C23—42no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—82no—
449C23—82no—
450C23—82no—
451C23—82no—
452C23—82no—
453C23—82no—
454C23—82no—
455C23—82no—
456C23—82no—
457C23—82no—
458C23—82no—
459C23—82no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23—82no—
465C23—82no—
466C23—82no—
467C23—82no—
468C23—82no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23#44214yes—
477C23C2×C4×D411yescharacteristic
478C23#53411yescharacteristic
479C23#53711yescharacteristic
480C23D4212yes—
481C23#54612yes—
482C23C2×#13512yes—
483C23C2×AGL(1,ℤ/8)12yes—
484C23#57611yescharacteristic
485C23#57611yescharacteristic
486C23#57611yescharacteristic
487C23#57811yescharacteristic
488D4—44no—
489D4—44no—
490D4—44no—
491D4—44no—
492D4—832no—
493D4—82no—
494D4—82no—
495D4—82no—
496D4—82no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—28no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4C22×C22⋊C411yescharacteristic
544C2×C4C4×#4111yescharacteristic
545C2×C4#51811yescharacteristic
546C2×C4#51811yescharacteristic
547C2×C4#51811yescharacteristic
548C2×C4#52011yescharacteristic
549C2×C4#53512yes—
550C2×C4#53512yes—
551C2×C4#53812yes—
552C2×C4#53811yescharacteristic
553C2×C4#53911yescharacteristic
554C2×C4#54012yes—
555C2×C4#54612yes—
556C2×C4#54812yes—
557C2×C4#55012yes—
558C2×C4#55312yes—
559C2×C4#56012yes—
560C2×C4#56212yes—
561C8—24no—
562C8—42no—
563C8—82no—
564C22—24no—
565C22—24no—
566C22—24no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—84no—
576C22—84no—
577C22—84no—
578C22—84no—
579C22—84no—
580C22—84no—
581C22—84no—
582C22—162no—
583C22—162no—
584C22—162no—
585C22—162no—
586C22—162no—
587C22—162no—
588C22—162no—
589C22—162no—
590C22#232511yescharacteristic
591C22D4×C22⋊C411yescharacteristic
592C22#236411yescharacteristic
593C22#267312yes—
594C22#267612yes—
595C22C4×#13511yescharacteristic
596C22C2×#44211yescharacteristic
597C22#294811yescharacteristic
598C22#295111yescharacteristic
599C22#318812yes—
600C22#319911yescharacteristic
601C22#320311yescharacteristic
602C22#324712yes—
603C22#325911yescharacteristic
604C22#326211yescharacteristic
605C22#326412yes—
606C22#326412yes—
607C22#328512yes—
608C22#328712yes—
609C22#329712yes—
610C22#330512yes—
611C22#330812yes—
612C22#331812yes—
613C4—42no—
614C4—48no—
615C4—44no—
616C4—84no—
617C4—82no—
618C4—162no—
619C4#230911yescharacteristic
620C4#233311yescharacteristic
621C4#234111yescharacteristic
622C4#237711yescharacteristic
623C4#271112yes—
624C4#271412yes—
625C2—44no—
626C2—164no—
627C2#1441811yescharacteristic
628C2#2218411yescharacteristic
629C2#2219611yescharacteristic
630C2#2243611yescharacteristic
631C2#2245111yescharacteristic
632C2#2313011yescharacteristic
633C2#2313211yescharacteristic
634C2#3071412yes—
635C2#3071512yes—
636C2#3091912yes—
637C2#3092112yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#466C2
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
2. C2×#1921C2semidirect
3. #13998C2semidirect
4. #11066C2semidirect
5. #11022C2semidirect
6. #10969C2semidirect
7. #10931C2semidirect
8. #10721C2semidirect
9. #10714C2semidirect
14. C2×#466C4semidirect
235. #107C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4662
C2×#19212
#139982
#110662
#110222
#109692
#109312
#107212
#107142