Finite Group Databasebeta

← #368542

Subgroups — #368542

Order 512.

1656 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1875C211yesmaximal, characteristic
2C2×#1875C211yesmaximal, characteristic
3C2×#1749C211yesmaximal, characteristic, complemented
4#14698C211yesmaximal, characteristic
5#13756C211yesmaximal, characteristic, complemented
6#11315C212yesmaximal, complemented
7#11189C212yesmaximal, complemented
8#10927C212yesmaximal, complemented
9#10821C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic, complemented
11#10731C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×#97C2212yes—
21C2×#390C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#386—22no—
24C2×#380C2211yescharacteristic
25#1875—24no—
26#1875C418yescomplemented
27#1749—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35#1437—22no—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40C22⋊#95—22no—
41C4×#95—22no—
42C23×D4—21no—
43C23×D4C2311yescharacteristic
44C24×C4D412yes—
45C24×C4C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×#121—22no—
49C2×#121—22no—
50C2×#121—22no—
51C2×#121—22no—
52C2×#121—22no—
53C2×#121C2×C414yes—
54C2×#117—22no—
55C2×#117—24no—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—42no—
59C2×#117—42no—
60C2×#117—42no—
61C2×#117C2×C414yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C42—22no—
74C22×C42—21no—
75C22×C42—21no—
76C22×C42—22no—
77C22×C42—24no—
78C22×C42—42no—
79C22×C42D412yes—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C2×#102—24no—
83C2×#102—24no—
84C2×#102—42no—
85C2×#102—42no—
86C2×#102—42no—
87C2×#102C2×C418yescomplemented
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8C2×C414yes—
93C2×C22⋊C8D414yes—
94C2×#97—24no—
95C2×#97—24no—
96C2×#97—42no—
97#390—24no—
98#390—42no—
99#386—48no—
100#380—22no—
101#380—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—21no—
109#340—24no—
110#340—24no—
111#340—42no—
112#340—42no—
113#340—42no—
114#340C2×C418yescomplemented
115#336—24no—
116C25—21no—
117C25C2×D411yescharacteristic
118C22×D4—24no—
119C22×D4—21no—
120C22×D4—21no—
121C22×D4—21no—
122C22×D4—21no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—44no—
129C22×D4—44no—
130C22×D4—42no—
131C22×D4—42no—
132C22×D4C22×C412yes—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4C22×C412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D412yes—
154C23×C4#4112yes—
155C23×C4#4112yes—
156C23×C4#4112yes—
157C23×C4#4112yes—
158C23×C4C2411yescharacteristic
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8C22×C414yes—
172C22×C8C2×D414yes—
173#121—28no—
174#121—28no—
175#121—44no—
176#121—44no—
177#117—48no—
178#117—48no—
179#117—48no—
180#117—48no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
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—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4C22×C414yes—
205C2×C4⋊C4C22×C412yes—
206C2×C4⋊C4#4114yes—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—21no—
234C2×C42—22no—
235C2×C42—48no—
236C2×C42—48no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42#4114yes—
243#102—48no—
244#102—48no—
245C22⋊C8—28no—
246C22⋊C8—28no—
247#97—48no—
248#95—44no—
249#95—44no—
250#95—44no—
251#95—44no—
252#95—42no—
253#95—44no—
254#95—44no—
255C24—21no—
256C24—21no—
257C24—21no—
258C24—21no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—41no—
265C24—41no—
266C24—41no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24—42no—
274C24C4×D412yes—
275C24#12112yes—
276C24#12412yes—
277C24C22×D411yescharacteristic
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—416no—
283C2×D4—48no—
284C2×D4—41no—
285C2×D4—41no—
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—84no—
297C2×D4—84no—
298C2×D4—88no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
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—24no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—48no—
331C22×C4—48no—
332C22×C4—48no—
333C22×C4—48no—
334C22×C4—48no—
335C22×C4—48no—
336C22×C4—48no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
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—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4#11714yes—
364C22×C4C4×D414yes—
365C22×C4C4×D412yes—
366C22×C4#12112yes—
367C22×C4#12314yes—
368C22×C4#12314yes—
369C22×C4#12412yes—
370C22×C4#12614yes—
371C22×C4AGL(1,ℤ/8)14yes—
372C22×C4C23×C411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4112yes—
378C22×C4C2×#4112yes—
379C22×C4C2×#4111yescharacteristic
380C22×C4H(5,𝔽2)12yes—
381C22×C4H(5,𝔽2)12yes—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C22⋊C4—48no—
413C22⋊C4—48no—
414C22⋊C4—48no—
415C22⋊C4—88no—
416C22⋊C4—88no—
417C22⋊C4—84no—
418C42—24no—
419C42—24no—
420C42—48no—
421C42—48no—
422C42—48no—
423C42—42no—
424C42—42no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—24no—
434C23—24no—
435C23—22no—
436C23—24no—
437C23—24no—
438C23—22no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—44no—
449C23—44no—
450C23—42no—
451C23—42no—
452C23—82no—
453C23—82no—
454C23—81no—
455C23—82no—
456C23—82no—
457C23—81no—
458C23—81no—
459C23—81no—
460C23—81no—
461C23—81no—
462C23—81no—
463C23—82no—
464C23—82no—
465C23—82no—
466C23—82no—
467C23—81no—
468C23—82no—
469C23—82no—
470C23#38814yes—
471C23C2×C4×D411yescharacteristic
472C23C2×#12111yescharacteristic
473C23C2×#12311yescharacteristic
474C23#52511yescharacteristic
475C23C2×#12411yescharacteristic
476C23#53411yescharacteristic
477C23#53511yescharacteristic
478C23C2×AGL(1,ℤ/8)12yes—
479C23#57512yes—
480C23#57512yes—
481C23#57512yes—
482D4—44no—
483D4—44no—
484D4—816no—
485D4—82no—
486D4—82no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—28no—
494C2×C4—28no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—432no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—81no—
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—81no—
533C2×C4—82no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—81no—
541C2×C4#38814yes—
542C2×C4C2×#11711yescharacteristic
543C2×C4C2×C4×D411yescharacteristic
544C2×C4C4×#4112yes—
545C2×C4#51812yes—
546C2×C4C2×#12311yescharacteristic
547C2×C4C2×#12611yescharacteristic
548C2×C4#52912yes—
549C2×C4#53812yes—
550C2×C4#54612yes—
551C2×C4#54612yes—
552C2×C4#55112yes—
553C2×C4#55112yes—
554C2×C4#55312yes—
555C2×C4#55312yes—
556C2×C4#55312yes—
557C2×C4#55312yes—
558C8—44no—
559C8—44no—
560C22—24no—
561C22—24no—
562C22—24no—
563C22—44no—
564C22—42no—
565C22—42no—
566C22—44no—
567C22—42no—
568C22—42no—
569C22—44no—
570C22—44no—
571C22—84no—
572C22—84no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—161no—
577C22—161no—
578C22—161no—
579C22—161no—
580C22—162no—
581C22—162no—
582C22C2×#38811yescharacteristic
583C22C4×#12311yescharacteristic
584C22#232111yescharacteristic
585C22#241111yescharacteristic
586C22#241211yescharacteristic
587C22#250411yescharacteristic
588C22#259111yescharacteristic
589C22#294312yes—
590C22C4×AGL(1,ℤ/8)12yes—
591C22#305412yes—
592C22#305512yes—
593C22#309312yes—
594C22#309412yes—
595C22#311112yes—
596C22#311512yes—
597C22#313511yescharacteristic
598C22#313511yescharacteristic
599C22#313811yescharacteristic
600C22#314011yescharacteristic
601C22#319412yes—
602C22#319712yes—
603C22#321412yes—
604C22#321912yes—
605C4—24no—
606C4—44no—
607C4—44no—
608C4—84no—
609C4—82no—
610C4—84no—
611C4—161no—
612C4—161no—
613C4#235312yes—
614C4#236412yes—
615C4#244412yes—
616C4#251012yes—
617C2—44no—
618C2—162no—
619C2#1402011yescharacteristic
620C2#2108611yescharacteristic
621C2#2109511yescharacteristic
622C2#2158212yes—
623C2#2205112yes—
624C2#2451012yes—
625C2#2456012yes—
626C2#2797911yescharacteristic
627C2#2798111yescharacteristic
628C2#2933711yescharacteristic
629C2#2933911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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
3. C2×#1749C2semidirect
5. #13756C2semidirect
6. #11315C2semidirect
7. #11189C2semidirect
8. #10927C2semidirect
9. #10821C2semidirect
10. #10731C2semidirect
11. #10731C2semidirect
26. #1875C4semidirect
87. C2×#102C2×C4semidirect
114. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18752
C2×#17492
#146982
#137562
#113152
#111892
#109272
#108212
#107312