Finite Group Databasebeta

← #471505

Subgroups — #471505

Order 512.

1284 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1956C211yesmaximal, characteristic, complemented
2C2×#1923C211yesmaximal, characteristic
3#15195C211yesmaximal, characteristic
4#15178C211yesmaximal, characteristic
5#11371C211yesmaximal, characteristic, complemented
6#11190C212yesmaximal, complemented
7#11077C212yesmaximal, complemented
8#11010C211yesmaximal, characteristic, complemented
9#10970C212yesmaximal
10#10970C212yesmaximal
11#10820C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22C2×#391C2212yes—
23C2×#384C2211yescharacteristic
24C2×#380C2212yes—
25C2×C4×C22⋊C4C2211yescharacteristic
26#1956—24no—
27#1923—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34#1444—21no—
35#1444—21no—
36#1442—21no—
37#1442—21no—
38#1439—21no—
39#1439—21no—
40#1439—21no—
41#1439—21no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×Q8C2312yes—
46C24×C4C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—42no—
55C2×C22⋊Q8—24no—
56C2×C22⋊Q8—24no—
57C2×C22⋊Q8—42no—
58C2×C22⋊Q8—42no—
59C2×C22⋊Q8—42no—
60C2×C22⋊Q8—42no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4D411yescharacteristic
72C22×C22⋊C4D411yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C2×C4⋊C8—22no—
76C2×C4⋊C8—42no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—42no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#103—42no—
87C2×#103—42no—
88C2×#103—42no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—42no—
92C2×C22⋊C8—42no—
93C2×C4×C8—42no—
94C2×C4×C8D414yes—
95#391—24no—
96#391—42no—
97#384—22no—
98#384—22no—
99#380—42no—
100#380—42no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—22no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95D411yescharacteristic
113C2×#95D411yescharacteristic
114#340—24no—
115#340—42no—
116#340—42no—
117#340—42no—
118#336—22no—
119#336—22no—
120C25C2×D411yescharacteristic
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8#4114yes—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2411yescharacteristic
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8C2×D412yes—
174C22×C8C2×D412yes—
175C4⋊Q8—28no—
176C4⋊Q8—48no—
177C4⋊Q8—44no—
178C22⋊Q8—48no—
179C22⋊Q8—48no—
180C22⋊Q8—48no—
181C22⋊Q8—48no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
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—22no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4#4114yes—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—82no—
221C2×C22⋊C4—82no—
222C2×C22⋊C4Q1614yes—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42C2×D412yes—
230C2×C42#4112yes—
231C4⋊C8—44no—
232#103—48no—
233#103—48no—
234#103—48no—
235#103—48no—
236C22⋊C8—44no—
237C22⋊C8—44no—
238C4×C8—44no—
239#95—22no—
240#95—22no—
241#95—44no—
242#95—44no—
243#95—44no—
244#95—42no—
245#95—42no—
246#95—42no—
247#95—42no—
248#95—82no—
249#95—82no—
250C24—21no—
251C24—21no—
252C24—21no—
253C24—22no—
254C24—22no—
255C24—21no—
256C24—21no—
257C24—21no—
258C24—22no—
259C24—22no—
260C24C2×Q1612yes—
261C24C22×D411yescharacteristic
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
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#12112yes—
355C22×C4#12112yes—
356C22×C4C2×Q1612yes—
357C22×C4C2×Q1612yes—
358C22×C4#13512yes—
359C22×C4#13712yes—
360C22×C4#13712yes—
361C22×C4C22×D411yescharacteristic
362C22×C4C2×#4112yes—
363C22×C4C2×#4112yes—
364C22×C4C2×#4111yescharacteristic
365C22×C4H(5,𝔽2)12yes—
366C22×C4H(5,𝔽2)11yescharacteristic
367C22×C4H(5,𝔽2)11yescharacteristic
368C22×C4H(5,𝔽2)12yes—
369C22×C4H(5,𝔽2)11yescharacteristic
370C22×C4H(5,𝔽2)11yescharacteristic
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—416no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—84no—
407C4⋊C4—84no—
408C22⋊C4—28no—
409C22⋊C4—416no—
410C22⋊C4—44no—
411C22⋊C4—88no—
412C42—22no—
413C42—22no—
414C42—24no—
415C42—41no—
416C42—41no—
417C42—42no—
418C23—22no—
419C23—22no—
420C23—24no—
421C23—24no—
422C23—22no—
423C23—42no—
424C23—42no—
425C23—41no—
426C23—42no—
427C23—41no—
428C23—41no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—41no—
434C23—42no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—42no—
440C23—42no—
441C23C2×#12111yescharacteristic
442C23#53511yescharacteristic
443C23#53511yescharacteristic
444C23#54612yes—
445C23#54612yes—
446C23C22×Q1611yescharacteristic
447C23C2×#13511yescharacteristic
448C23C2×#13711yescharacteristic
449C23C2×#13711yescharacteristic
450C23#57712yes—
451C23#57712yes—
452Q8—44no—
453Q8—44no—
454Q8—44no—
455Q8—44no—
456Q8—82no—
457Q8—82no—
458Q8—82no—
459Q8—82no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—41no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—41no—
484C2×C4—44no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—81no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4#46714yes—
518C2×C4#47014yes—
519C2×C4#52511yescharacteristic
520C2×C4#53211yescharacteristic
521C2×C4#53411yescharacteristic
522C2×C4#53511yescharacteristic
523C2×C4#53511yescharacteristic
524C2×C4#53511yescharacteristic
525C2×C4#53812yes—
526C2×C4#53812yes—
527C2×C4#53812yes—
528C2×C4#54012yes—
529C2×C4#54011yescharacteristic
530C2×C4#54011yescharacteristic
531C2×C4#55112yes—
532C2×C4#55112yes—
533C2×C4#55312yes—
534C2×C4#55912yes—
535C2×C4#56012yes—
536C2×C4#56012yes—
537C8—42no—
538C8—42no—
539C8—81no—
540C8—81no—
541C22—44no—
542C22—44no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—82no—
547C22—82no—
548C22—81no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22#253411yescharacteristic
553C22#254411yescharacteristic
554C22#261911yescharacteristic
555C22#268512yes—
556C22#270312yes—
557C22C2×#46711yescharacteristic
558C22C2×#47011yescharacteristic
559C22#305711yescharacteristic
560C22#305711yescharacteristic
561C22#320911yescharacteristic
562C22#320911yescharacteristic
563C22#321011yescharacteristic
564C22#321011yescharacteristic
565C22#322112yes—
566C22#322212yes—
567C22#329912yes—
568C22#329912yes—
569C22#330112yes—
570C22#330312yes—
571C22#331612yes—
572C22#332012yes—
573C22#332912yes—
574C22#333112yes—
575C4—24no—
576C4—24no—
577C4—42no—
578C4—84no—
579C4—81no—
580C4—84no—
581C4—81no—
582C4—162no—
583C4—162no—
584C4#260311yescharacteristic
585C4#263111yescharacteristic
586C4#266111yescharacteristic
587C4#275212yes—
588C4#279712yes—
589C4#281511yescharacteristic
590C2—82no—
591C2#1679911yescharacteristic
592C2#2809511yescharacteristic
593C2#2810811yescharacteristic
594C2#2859811yescharacteristic
595C2#2860811yescharacteristic
596C2#2980811yescharacteristic
597C2#2981111yescharacteristic
598C2#3129112yes—
599C2#3129212yes—
600C2#3183712yes—
601C2#3184012yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×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
1. C2×#1956C2semidirect
5. #11371C2semidirect
6. #11190C2semidirect
7. #11077C2semidirect
8. #11010C2semidirect
11. #10820C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19562
C2×#19232
#151952
#151782
#113712
#111902
#110772
#110102
#109702
#108202