Finite Group Databasebeta

← #368537

Subgroups — #368537

Order 512.

1530 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1873C211yesmaximal, characteristic
2C2×#1873C211yesmaximal, characteristic
3C2×C4×C22⋊C8C211yesmaximal, characteristic
4#14702C211yesmaximal, characteristic
5#13756C211yesmaximal, characteristic
6#11315C212yesmaximal, complemented
7#11190C212yesmaximal, complemented
8#10926C212yesmaximal, complemented
9#10820C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic
11#10731C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×C4×C8C2212yes—
21C2×#391C2211yescharacteristic
22C2×#391C2211yescharacteristic
23C2×#380C2211yescharacteristic
24#1873—24no—
25#1873C418yescomplemented
26C4×C22⋊C8—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2212yes—
34#1436—22no—
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2212yes—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39C22⋊#95—22no—
40C22⋊#95—22no—
41C4×#95—22no—
42C23×Q8C2311yescharacteristic
43C24×C4D412yes—
44C24×C4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8C2×C414yes—
53C2×#117—22no—
54C2×#117—24no—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117—42no—
58C2×#117—42no—
59C2×#117—42no—
60C2×#117C2×C414yes—
61C22×C4⋊C4—21no—
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×C42—22no—
72C22×C42—21no—
73C22×C42—21no—
74C22×C42—22no—
75C22×C42—24no—
76C22×C42—42no—
77C22×C42D412yes—
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—42no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#103C2×C418yescomplemented
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8C2×C414yes—
91C2×C22⋊C8D414yes—
92C2×C4×C8—24no—
93C2×C4×C8—24no—
94C2×C4×C8—42no—
95#391—24no—
96#391—42no—
97#380—22no—
98#380—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—21no—
108#340—24no—
109#340—24no—
110#340—42no—
111#340—42no—
112#340—42no—
113#340C2×C418yescomplemented
114#336—24no—
115C25C2×D411yescharacteristic
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8C22×C412yes—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—24no—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4C22×C412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4#4112yes—
149C23×C4#4112yes—
150C23×C4#4112yes—
151C23×C4#4112yes—
152C23×C4C2411yescharacteristic
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8C22×C414yes—
166C22×C8C2×D414yes—
167C22⋊Q8—28no—
168C22⋊Q8—28no—
169C22⋊Q8—44no—
170C22⋊Q8—44no—
171#117—48no—
172#117—48no—
173#117—48no—
174#117—48no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4C22×C414yes—
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4#4114yes—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—42no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—24no—
219C2×C42—24no—
220C2×C42—24no—
221C2×C42—24no—
222C2×C42—21no—
223C2×C42—22no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—48no—
227C2×C42—48no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42#4114yes—
234#103—48no—
235#103—48no—
236C22⋊C8—28no—
237C22⋊C8—28no—
238C4×C8—48no—
239#95—44no—
240#95—44no—
241#95—44no—
242#95—42no—
243#95—44no—
244#95—44no—
245#95—44no—
246#95—44no—
247#95—44no—
248C24—21no—
249C24—21no—
250C24—21no—
251C24—21no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24C4×D412yes—
257C24#12112yes—
258C24#12412yes—
259C24C22×D411yescharacteristic
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—216no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—48no—
308C22×C4—48no—
309C22×C4—48no—
310C22×C4—48no—
311C22×C4—48no—
312C22×C4—48no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4#11714yes—
346C22×C4C4×D414yes—
347C22×C4C4×D412yes—
348C22×C4#12112yes—
349C22×C4#12314yes—
350C22×C4#12314yes—
351C22×C4#12412yes—
352C22×C4#12614yes—
353C22×C4#13514yes—
354C22×C4#13514yes—
355C22×C4#13714yes—
356C22×C4C23×C411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4112yes—
362C22×C4C2×#4112yes—
363C22×C4C2×#4111yescharacteristic
364C22×C4H(5,𝔽2)12yes—
365C22×C4H(5,𝔽2)12yes—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—48no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C22⋊C4—48no—
398C22⋊C4—48no—
399C22⋊C4—48no—
400C22⋊C4—84no—
401C42—24no—
402C42—24no—
403C42—48no—
404C42—48no—
405C42—48no—
406C42—42no—
407C42—42no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—24no—
417C23—24no—
418C23—22no—
419C23—24no—
420C23—24no—
421C23—22no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23#38814yes—
431C23C2×C4×D411yescharacteristic
432C23C2×#12111yescharacteristic
433C23C2×#12311yescharacteristic
434C23#52511yescharacteristic
435C23C2×#12411yescharacteristic
436C23#53411yescharacteristic
437C23#53511yescharacteristic
438C23C2×#13512yes—
439C23C2×#13512yes—
440C23C2×#13712yes—
441C23#57512yes—
442Q8—44no—
443Q8—44no—
444Q8—82no—
445Q8—82no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—28no—
459C2×C4—28no—
460C2×C4—28no—
461C2×C4—28no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—432no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—81no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—81no—
506C2×C4#38814yes—
507C2×C4#48518yes—
508C2×C4C2×#11711yescharacteristic
509C2×C4C2×C4×D411yescharacteristic
510C2×C4C4×#4112yes—
511C2×C4#51812yes—
512C2×C4C2×#12311yescharacteristic
513C2×C4C2×#12611yescharacteristic
514C2×C4#52912yes—
515C2×C4#53812yes—
516C2×C4#54612yes—
517C2×C4#54612yes—
518C2×C4#55112yes—
519C2×C4#55112yes—
520C2×C4#55312yes—
521C2×C4#55312yes—
522C2×C4#55312yes—
523C2×C4#55312yes—
524C8—44no—
525C8—44no—
526C22—24no—
527C22—24no—
528C22—24no—
529C22—44no—
530C22—42no—
531C22—42no—
532C22—44no—
533C22—42no—
534C22—42no—
535C22—44no—
536C22—44no—
537C22C2×#38811yescharacteristic
538C22C4×#12311yescharacteristic
539C22#232111yescharacteristic
540C22#241111yescharacteristic
541C22#241211yescharacteristic
542C22#250411yescharacteristic
543C22#259111yescharacteristic
544C22C4×#13512yes—
545C22C4×#13712yes—
546C22#305012yes—
547C22#305612yes—
548C22C2×#48512yes—
549C22#309112yes—
550C22#310212yes—
551C22#310712yes—
552C22#313111yescharacteristic
553C22#313111yescharacteristic
554C22#313911yescharacteristic
555C22#314011yescharacteristic
556C22#319012yes—
557C22#319112yes—
558C22#321012yes—
559C22#321212yes—
560C4—24no—
561C4—28no—
562C4—44no—
563C4—84no—
564C4—82no—
565C4—84no—
566C4—82no—
567C4—161no—
568C4—161no—
569C4#235312yes—
570C4#236412yes—
571C4#244412yes—
572C4#251012yes—
573C2—44no—
574C2#1402011yescharacteristic
575C2#2108311yescharacteristic
576C2#2109411yescharacteristic
577C2C4×#48512yes—
578C2#2204212yes—
579C2#2450512yes—
580C2#2455412yes—
581C2#2797511yescharacteristic
582C2#2797611yescharacteristic
583C2#2933211yescharacteristic
584C2#2933311yescharacteristic
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
6. #11315C2semidirect
7. #11190C2semidirect
8. #10926C2semidirect
9. #10820C2semidirect
25. #1873C4semidirect
85. C2×#103C2×C4semidirect
113. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18732
C2×C4×C22⋊C82
#147022
#137562
#113152
#111902
#109262
#108202
#107312