Finite Group Databasebeta

← #471508

Subgroups — #471508

Order 512.

1304 conjugacy classes of subgroups.

The subgroup lattice has 607 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1956C211yesmaximal, characteristic, complemented
2C2×#1923C211yesmaximal, characteristic
3#15178C211yesmaximal, characteristic
4#15150C211yesmaximal, characteristic
5#11373C211yesmaximal, 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×#397—21no—
23C2×#397—21no—
24C2×#391C2212yes—
25C2×#384C2211yescharacteristic
26C2×#380C2212yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28#1956—24no—
29#1923—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36#1445—21no—
37#1445—21no—
38#1444—21no—
39#1444—21no—
40#1439—21no—
41#1439—21no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
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—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
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#397—44no—
96#397—44no—
97#391—24no—
98#391—42no—
99#384—22no—
100#384—22no—
101#380—42no—
102#380—42no—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95D411yescharacteristic
115C2×#95D411yescharacteristic
116#340—24no—
117#340—42no—
118#340—42no—
119#340—42no—
120#336—22no—
121#336—22no—
122C25C2×D411yescharacteristic
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—22no—
135C22×Q8—22no—
136C22×Q8#4114yes—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2411yescharacteristic
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8C2×D412yes—
176C22×C8C2×D412yes—
177C4⋊Q8—28no—
178C4⋊Q8—48no—
179C4⋊Q8—44no—
180C22⋊Q8—48no—
181C22⋊Q8—48no—
182C22⋊Q8—48no—
183C22⋊Q8—48no—
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—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4#4114yes—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
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⋊C4SD1614yes—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42C2×D412yes—
234C2×C42#4112yes—
235C4⋊C8—44no—
236#103—48no—
237#103—48no—
238#103—48no—
239#103—48no—
240C22⋊C8—44no—
241C22⋊C8—44no—
242C4×C8—44no—
243#95—22no—
244#95—22no—
245#95—44no—
246#95—44no—
247#95—44no—
248#95—44no—
249#95—42no—
250#95—44no—
251#95—42no—
252#95—42no—
253#95—42no—
254C24—21no—
255C24—21no—
256C24—21no—
257C24—22no—
258C24—22no—
259C24—21no—
260C24—21no—
261C24—21no—
262C24—22no—
263C24—22no—
264C24C2×SD1612yes—
265C24C22×D411yescharacteristic
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—24no—
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—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
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—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
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—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4#12112yes—
359C22×C4#12112yes—
360C22×C4C2×SD1612yes—
361C22×C4C2×SD1612yes—
362C22×C4#13512yes—
363C22×C4AGL(1,ℤ/8)12yes—
364C22×C4#13712yes—
365C22×C4C22×D411yescharacteristic
366C22×C4C2×#4112yes—
367C22×C4C2×#4112yes—
368C22×C4C2×#4111yescharacteristic
369C22×C4H(5,𝔽2)12yes—
370C22×C4H(5,𝔽2)11yescharacteristic
371C22×C4H(5,𝔽2)11yescharacteristic
372C22×C4H(5,𝔽2)12yes—
373C22×C4H(5,𝔽2)11yescharacteristic
374C22×C4H(5,𝔽2)11yescharacteristic
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—48no—
398C4⋊C4—48no—
399C4⋊C4—416no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—84no—
411C4⋊C4—84no—
412C22⋊C4—28no—
413C22⋊C4—416no—
414C22⋊C4—44no—
415C22⋊C4—88no—
416C42—22no—
417C42—22no—
418C42—24no—
419C42—41no—
420C42—41no—
421C42—42no—
422C23—22no—
423C23—22no—
424C23—24no—
425C23—24no—
426C23—22no—
427C23—42no—
428C23—42no—
429C23—41no—
430C23—42no—
431C23—41no—
432C23—41no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—41no—
438C23—42no—
439C23—41no—
440C23—42no—
441C23—41no—
442C23—42no—
443C23—41no—
444C23—41no—
445C23C2×#12111yescharacteristic
446C23#53511yescharacteristic
447C23#53511yescharacteristic
448C23#54612yes—
449C23#54612yes—
450C23C22×SD1611yescharacteristic
451C23C2×#13511yescharacteristic
452C23C2×AGL(1,ℤ/8)11yescharacteristic
453C23C2×#13711yescharacteristic
454C23#57812yes—
455C23#57812yes—
456Q8—44no—
457Q8—44no—
458Q8—44no—
459Q8—44no—
460Q8—82no—
461Q8—82no—
462Q8—82no—
463Q8—82no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—41no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—41no—
488C2×C4—44no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—81no—
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—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4#46514yes—
522C2×C4#46814yes—
523C2×C4#52511yescharacteristic
524C2×C4#53211yescharacteristic
525C2×C4#53411yescharacteristic
526C2×C4#53511yescharacteristic
527C2×C4#53511yescharacteristic
528C2×C4#53511yescharacteristic
529C2×C4#53812yes—
530C2×C4#53812yes—
531C2×C4#53812yes—
532C2×C4#54012yes—
533C2×C4#54011yescharacteristic
534C2×C4#54011yescharacteristic
535C2×C4#55112yes—
536C2×C4#55112yes—
537C2×C4#55312yes—
538C2×C4#55912yes—
539C2×C4#56012yes—
540C2×C4#56012yes—
541C8—42no—
542C8—42no—
543C8—81no—
544C8—81no—
545C22—44no—
546C22—44no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—82no—
551C22—82no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22#253411yescharacteristic
557C22#254411yescharacteristic
558C22#261911yescharacteristic
559C22#268512yes—
560C22#270312yes—
561C22C2×#46511yescharacteristic
562C22C2×#46811yescharacteristic
563C22#305911yescharacteristic
564C22#305911yescharacteristic
565C22#320611yescharacteristic
566C22#320711yescharacteristic
567C22#320811yescharacteristic
568C22#321011yescharacteristic
569C22#322312yes—
570C22#322312yes—
571C22#329812yes—
572C22#329812yes—
573C22#330012yes—
574C22#330212yes—
575C22#330812yes—
576C22#331212yes—
577C22#332812yes—
578C22#333012yes—
579C4—24no—
580C4—24no—
581C4—42no—
582C4—84no—
583C4—81no—
584C4—84no—
585C4—81no—
586C4—162no—
587C4—162no—
588C4#260311yescharacteristic
589C4#263111yescharacteristic
590C4#266111yescharacteristic
591C4#275212yes—
592C4#279712yes—
593C4#281511yescharacteristic
594C2—82no—
595C2#1679911yescharacteristic
596C2#2809811yescharacteristic
597C2#2810511yescharacteristic
598C2#2860011yescharacteristic
599C2#2860611yescharacteristic
600C2#2980711yescharacteristic
601C2#2981211yescharacteristic
602C2#3128712yes—
603C2#3128812yes—
604C2#3183612yes—
605C2#3184112yes—
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. #11373C2semidirect
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
#151782
#151502
#113732
#111902
#110772
#110102
#109702
#108202