Finite Group Databasebeta

← #464468

Subgroups — #464468

Order 512.

1464 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic, complemented
2C2×#2071C211yesmaximal, characteristic
3C2×#2023C212yesmaximal
4C2×#1925C211yesmaximal, characteristic
5#15147C211yesmaximal, characteristic
6#11384C212yesmaximal, complemented
7#11373C211yesmaximal, characteristic, complemented
8#11346C211yesmaximal, characteristic, complemented
9#11219C212yesmaximal, complemented
10#11066C212yesmaximal
11#10806C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8—21no—
13C22×C4⋊Q8C2212yes—
14C22×#107C2211yescharacteristic
15C22×#102C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2211yescharacteristic
18#2551—24no—
19#2551—24no—
20C2×#397—21no—
21C2×#395—22no—
22C2×#390C2211yescharacteristic
23C2×#390C2212yes—
24C2×#389—21no—
25C2×#389C2212yes—
26C2×#384C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#2071—24no—
30#2023—28no—
31#1925—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1440—22no—
43#1439—21no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2212yes—
46C23×Q8—22no—
47C23×D4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×C4⋊Q8—28no—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—42no—
57C2×C4⋊Q8—42no—
58C2×C4⋊Q8—44no—
59C2×C4⋊Q8—44no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—21no—
75C22×C42—21no—
76C22×C42C2312yes—
77C22×C42C2311yescharacteristic
78C2×#107—22no—
79C2×#107—24no—
80C2×#102—22no—
81C2×#102—22no—
82C2×#102—24no—
83C2×#102—24no—
84C2×#102—22no—
85C2×#102—24no—
86C2×#102—42no—
87C2×#102D414yes—
88#397—44no—
89#395—216no—
90#390—24no—
91#390—22no—
92#390—22no—
93#390—42no—
94#389—28no—
95#389—24no—
96#389—24no—
97#384—22no—
98#384D414yes—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4D414yes—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95D412yes—
110#340—24no—
111#340—24no—
112#340—22no—
113#340—24no—
114#340—24no—
115#340—42no—
116#336—24no—
117#336—22no—
118#336—22no—
119#336—42no—
120C25—21no—
121C22×Q8—28no—
122C22×Q8—44no—
123C22×Q8—44no—
124C22×Q8—44no—
125C22×Q8—44no—
126C22×Q8—82no—
127C22×Q8—82no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—22no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4C2×D412yes—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2411yescharacteristic
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8C2×D412yes—
165C4⋊Q8—48no—
166C4⋊Q8—48no—
167C4⋊Q8—48no—
168C4⋊Q8—432no—
169C4⋊Q8—48no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
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—24no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—81no—
205C2×C4⋊C4—81no—
206C2×C4⋊C4D814yes—
207C2×C4⋊C4C2×D414yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4#4112yes—
212C2×C4⋊C4#4114yes—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—82no—
222C2×C22⋊C4—82no—
223C2×C22⋊C4—82no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—24no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237#107—44no—
238#102—28no—
239#102—48no—
240#102—44no—
241#95—28no—
242#95—24no—
243#95—24no—
244#95—44no—
245#95—42no—
246#95—81no—
247#95—82no—
248C24—22no—
249C24—41no—
250C24—42no—
251C24—41no—
252C24—42no—
253C24—41no—
254C24—41no—
255C24—42no—
256C24—41no—
257C24—42no—
258C24—41no—
259C24C22×D411yescharacteristic
260C2×Q8—416no—
261C2×Q8—416no—
262C2×Q8—88no—
263C2×Q8—88no—
264C2×Q8—88no—
265C2×Q8—88no—
266C2×Q8—88no—
267C2×D4—22no—
268C2×D4—24no—
269C2×D4—22no—
270C2×D4—24no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—41no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4#12012yes—
335C22×C4#12012yes—
336C22×C4C2×D814yes—
337C22×C4C2×D812yes—
338C22×C4AGL(1,ℤ/8)12yes—
339C22×C4#13714yes—
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4112yes—
345C22×C4H(5,𝔽2)11yescharacteristic
346C22×C4H(5,𝔽2)11yescharacteristic
347C22×C42−1+412yes—
348C22×C42−1+412yes—
349C22×C42−1+412yes—
350C2×C8—22no—
351C2×C8—24no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—42no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C4⋊C4—28no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—28no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—416no—
377C4⋊C4—416no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—48no—
381C4⋊C4—41no—
382C4⋊C4—42no—
383C4⋊C4—41no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—84no—
389C4⋊C4#12114yes—
390C4⋊C4#12414yes—
391C22⋊C4—88no—
392C22⋊C4—88no—
393C22⋊C4—84no—
394C22⋊C4—84no—
395C42—22no—
396C42—22no—
397C42—24no—
398C42—42no—
399C42—42no—
400C42—42no—
401C42—42no—
402C42—48no—
403C23—42no—
404C23—44no—
405C23—44no—
406C23—42no—
407C23—42no—
408C23—82no—
409C23—82no—
410C23—82no—
411C23—82no—
412C23—81no—
413C23—81no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23—82no—
418C23—82no—
419C23—81no—
420C23—81no—
421C23—81no—
422C23—81no—
423C23—81no—
424C23—82no—
425C23—81no—
426C23C2×#12011yescharacteristic
427C23#53612yes—
428C23D4211yescharacteristic
429C23#54611yescharacteristic
430C23#54812yes—
431C23C22×D811yescharacteristic
432C23C2×AGL(1,ℤ/8)11yescharacteristic
433C23C2×#13712yes—
434C23#57611yescharacteristic
435C23#57712yes—
436C23#57711yescharacteristic
437Q8—832no—
438D4—42no—
439D4—44no—
440D4—42no—
441D4—81no—
442D4—81no—
443D4—82no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4#44714yes—
499C2×C4C2×#12111yescharacteristic
500C2×C4C2×#12411yescharacteristic
501C2×C4#53511yescharacteristic
502C2×C4#53611yescharacteristic
503C2×C4#53611yescharacteristic
504C2×C4#53611yescharacteristic
505C2×C4#53612yes—
506C2×C4#53612yes—
507C2×C4#54112yes—
508C2×C4#54212yes—
509C2×C4#54212yes—
510C2×C4D4211yescharacteristic
511C2×C4#54611yescharacteristic
512C2×C4#54712yes—
513C2×C4#54812yes—
514C2×C4#55612yes—
515C2×C4#56312yes—
516C2×C4#56312yes—
517C8—42no—
518C8—81no—
519C8—82no—
520C22—84no—
521C22—84no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—161no—
526C22—161no—
527C22—161no—
528C22—161no—
529C22—162no—
530C22—162no—
531C22#240811yescharacteristic
532C22#243611yescharacteristic
533C22#261411yescharacteristic
534C22#267412yes—
535C22#271612yes—
536C22C2×#44711yescharacteristic
537C22#300411yescharacteristic
538C22#301011yescharacteristic
539C22#301211yescharacteristic
540C22#322612yes—
541C22#323812yes—
542C22#324112yes—
543C22#324412yes—
544C22#326612yes—
545C22#327211yescharacteristic
546C22D4×D811yescharacteristic
547C22#329411yescharacteristic
548C22#332012yes—
549C22#332511yescharacteristic
550C22#335812yes—
551C22#336912yes—
552C22#337312yes—
553C22#337512yes—
554C4—24no—
555C4—44no—
556C4—44no—
557C4—44no—
558C4—44no—
559C4—42no—
560C4—84no—
561C4—161no—
562C4—162no—
563C4#254711yescharacteristic
564C4#262111yescharacteristic
565C4#264011yescharacteristic
566C4#267412yes—
567C4#271612yes—
568C4#282111yescharacteristic
569C2—162no—
570C2#1650611yescharacteristic
571C2#2423611yescharacteristic
572C2#2425111yescharacteristic
573C2#2593311yescharacteristic
574C2#2594211yescharacteristic
575C2#2963811yescharacteristic
576C2#2964711yescharacteristic
577C2#3080512yes—
578C2#3082912yes—
579C2#3222612yes—
580C2#3224612yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#2551C2semidirect
6. #11384C2semidirect
7. #11373C2semidirect
8. #11346C2semidirect
9. #11219C2semidirect
11. #10806C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
C2×#20712
C2×#20232
C2×#19252
#151472
#113842
#113732
#113462
#112192
#110662
#108062