Finite Group Databasebeta

← #385840

Subgroups — #385840

Order 512.

2016 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#466C211yesmaximal, characteristic
2C2×#1879C212yesmaximal, complemented
3#13970C212yesmaximal, complemented
4#11377C212yesmaximal, complemented
5#11189C212yesmaximal, complemented
6#10969C212yesmaximal, complemented
7#10788C212yesmaximal, complemented
8#10721C211yesmaximal, characteristic, complemented
9#10714C211yesmaximal, characteristic, complemented
10C23×D8C2211yescharacteristic
11C24×C8C2211yescharacteristic
12C22×#121C2212yes—
13C22×C4×D4C2212yes—
14C2×#466—28no—
15C2×#466C418yescomplemented
16C22×#107C2211yescharacteristic
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C2×#390C2212yes—
22C2×#386—22no—
23C2×#380C2212yes—
24C2×C4×C22⋊C4C2211yescharacteristic
25#1879—28no—
26C2×#340C2212yes—
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#336C2212yes—
32C2×#336C2212yes—
33#1440—22no—
34#1438—22no—
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C22⋊#95—22no—
38C23×D4—22no—
39C23×D4C2312yes—
40C24×C4—22no—
41C24×C4C2311yescharacteristic
42C22×D8—41no—
43C22×D8—42no—
44C22×D8—41no—
45C22×D8C2×C414yes—
46C22×D8D418yes—
47C23×C8—24no—
48C23×C8—42no—
49C23×C8—41no—
50C23×C8—41no—
51C23×C8C2×C414yescomplemented
52C23×C8C2312yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C2×#121—24no—
56C2×#121—24no—
57C2×#121—24no—
58C2×#121—42no—
59C2×#121—42no—
60C2×#121C2×C418yescomplemented
61C2×C4×D4—24no—
62C2×C4×D4—24no—
63C2×C4×D4—42no—
64C2×C4×D4—42no—
65C2×C4×D4—42no—
66C2×C4×D4—42no—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4D412yes—
76C22×C42C2312yes—
77#466—232no—
78C2×#107—24no—
79C2×#107C2×C414yescomplemented
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—42no—
86C2×#102—42no—
87C2×#102—42no—
88C2×#102C2×C418yescomplemented
89C2×C22⋊C8—22no—
90C2×C22⋊C8—41no—
91C2×C22⋊C8—41no—
92#390—24no—
93#390—42no—
94#386—48no—
95#380—24no—
96#380—42no—
97C4×C22⋊C4—41no—
98C4×C22⋊C4—41no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95D411yescharacteristic
105C2×#95D411yescharacteristic
106#340—24no—
107#340—41no—
108#340—42no—
109#340—42no—
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—44no—
128C22×D4—44no—
129C22×D4—44no—
130C22×D4—44no—
131C22×D4—82no—
132C22×D4—82no—
133C22×D4C22×C414yes—
134C22×D4C2×D414yes—
135C22×D4C2×D414yes—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—44no—
148C23×C4—44no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4—44no—
152C23×C4—82no—
153C23×C4—82no—
154C23×C4C22×C412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4#4112yes—
159C23×C4C2411yescharacteristic
160C2×D8—44no—
161C2×D8—44no—
162C2×D8—48no—
163C2×D8—48no—
164C2×D8C22⋊C4116yescomplemented
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—28no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—44no—
181C22×C8—44no—
182C22×C8—48no—
183C22×C8—48no—
184C22×C8C22×C412yes—
185C22×C8C22×C412yes—
186C22×C8C2×D414yes—
187#121—216no—
188#121—48no—
189#121—48no—
190#121—48no—
191C4×D4—48no—
192C4×D4—48no—
193C4×D4—48no—
194C4×D4—48no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
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⋊C4C22×C414yes—
212C2×C4⋊C4#4114yes—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—42no—
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—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—82no—
229C2×C22⋊C4—82no—
230C2×C22⋊C4—82no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237#107—28no—
238#102—216no—
239#102—48no—
240#102—48no—
241C22⋊C8—44no—
242#95—41no—
243#95—41no—
244#95—44no—
245#95—44no—
246#95—44no—
247#95—82no—
248#95—82no—
249C24—21no—
250C24—21no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24—21no—
255C24—21no—
256C24—21no—
257C24—21no—
258C24—24no—
259C24—24no—
260C24—24no—
261C24—42no—
262C24—42no—
263C24—42no—
264C24—42no—
265C24—42no—
266C24—42no—
267C24—42no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24C4×D412yes—
274C24AGL(1,ℤ/8)14yes—
275C24C22×D411yescharacteristic
276C2×D4—24no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—416no—
292C2×D4—416no—
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—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—88no—
306C2×D4—88no—
307C2×D4—88no—
308C2×D4—88no—
309C2×D4—88no—
310C2×D4C2×C22⋊C418yes—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
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—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
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—416no—
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×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—88no—
369C22×C4—88no—
370C22×C4—88no—
371C22×C4—88no—
372C22×C4—88no—
373C22×C4—88no—
374C22×C4C4×D412yes—
375C22×C4#13514yes—
376C22×C4C23×C411yescharacteristic
377C22×C4C22×D412yes—
378C22×C4C2×#4112yes—
379C22×C4C2×#4111yescharacteristic
380C22×C4H(5,𝔽2)11yescharacteristic
381C22×C4H(5,𝔽2)11yescharacteristic
382C22×C4H(5,𝔽2)12yes—
383C22×C4H(5,𝔽2)12yes—
384C22×C4H(5,𝔽2)11yescharacteristic
385C22×C42−1+411yescharacteristic
386D8—416no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—416no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8C2×C22⋊C414yes—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C22⋊C4—48no—
418C22⋊C4—416no—
419C22⋊C4—416no—
420C22⋊C4—88no—
421C22⋊C4—88no—
422C22⋊C4—88no—
423C42—42no—
424C42—42no—
425C42—42no—
426C42—42no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—24no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23—28no—
440C23—42no—
441C23—42no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—82no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23—82no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23#44214yes—
485C23C2×C4×D411yescharacteristic
486C23#53411yescharacteristic
487C23#53711yescharacteristic
488C23D4212yes—
489C23#54612yes—
490C23C2×#13512yes—
491C23C2×AGL(1,ℤ/8)12yes—
492C23#57611yescharacteristic
493C23#57611yescharacteristic
494C23#57611yescharacteristic
495C23#57811yescharacteristic
496D4—28no—
497D4—28no—
498D4—44no—
499D4—44no—
500D4—44no—
501D4—44no—
502D4—44no—
503D4—44no—
504D4—832no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—832no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4C22×C22⋊C411yescharacteristic
553C2×C4C4×#4111yescharacteristic
554C2×C4#51811yescharacteristic
555C2×C4#51811yescharacteristic
556C2×C4#51811yescharacteristic
557C2×C4#52011yescharacteristic
558C2×C4#53412yes—
559C2×C4#53412yes—
560C2×C4#53812yes—
561C2×C4#53811yescharacteristic
562C2×C4#53911yescharacteristic
563C2×C4#54012yes—
564C2×C4#54612yes—
565C2×C4#54612yes—
566C2×C4#54712yes—
567C2×C4#55312yes—
568C2×C4#56012yes—
569C2×C4#56212yes—
570C8—24no—
571C8—42no—
572C8—82no—
573C22—24no—
574C22—24no—
575C22—24no—
576C22—42no—
577C22—42no—
578C22—44no—
579C22—44no—
580C22—42no—
581C22—42no—
582C22—44no—
583C22—44no—
584C22—48no—
585C22—48no—
586C22—48no—
587C22—84no—
588C22—84no—
589C22—84no—
590C22—84no—
591C22—84no—
592C22—84no—
593C22—84no—
594C22—84no—
595C22—84no—
596C22—84no—
597C22—84no—
598C22—84no—
599C22#232511yescharacteristic
600C22D4×C22⋊C411yescharacteristic
601C22#236411yescharacteristic
602C22#266812yes—
603C22#267112yes—
604C22C4×#13511yescharacteristic
605C22C2×#44211yescharacteristic
606C22#294811yescharacteristic
607C22#295111yescharacteristic
608C22#318812yes—
609C22#319911yescharacteristic
610C22#320311yescharacteristic
611C22#324712yes—
612C22#325911yescharacteristic
613C22#326211yescharacteristic
614C22#326412yes—
615C22#326412yes—
616C22#328012yes—
617C22#328312yes—
618C22#329712yes—
619C22#330512yes—
620C22#330812yes—
621C22#331812yes—
622C4—42no—
623C4—44no—
624C4—84no—
625C4—84no—
626C4—82no—
627C4—162no—
628C4#230811yescharacteristic
629C4#233311yescharacteristic
630C4#233911yescharacteristic
631C4#237711yescharacteristic
632C4#268812yes—
633C4#270812yes—
634C2—44no—
635C2—88no—
636C2#1441711yescharacteristic
637C2#2218411yescharacteristic
638C2#2219611yescharacteristic
639C2#2243611yescharacteristic
640C2#2244811yescharacteristic
641C2#2313011yescharacteristic
642C2#2313211yescharacteristic
643C2#3040012yes—
644C2#3040212yes—
645C2#3059312yes—
646C2#3059512yes—
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×#1879C2semidirect
3. #13970C2semidirect
4. #11377C2semidirect
5. #11189C2semidirect
6. #10969C2semidirect
7. #10788C2semidirect
8. #10721C2semidirect
9. #10714C2semidirect
15. C2×#466C4semidirect
51. C23×C8C2×C4semidirect
60. C2×#121C2×C4semidirect
79. C2×#107C2×C4semidirect
88. C2×#102C2×C4semidirect
164. C2×D8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4662
C2×#18792
#139702
#113772
#111892
#109692
#107882
#107212
#107142