Finite Group Databasebeta

← #471666

Subgroups — #471666

Order 512.

1140 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1990C211yesmaximal, characteristic, complemented
2C2×#1923C211yesmaximal, characteristic
3#15135C211yesmaximal, characteristic
4#15125C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic, complemented
6#11357C211yesmaximal, characteristic, complemented
7#11292C212yesmaximal, complemented
8#11228C212yesmaximal, complemented
9#11200C212yesmaximal
10#11199C212yesmaximal
11#10820C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17C2×#399—21no—
18C2×#393—21no—
19C2×#389C2212yes—
20C2×#384C2211yescharacteristic
21C2×#380C2212yes—
22C2×#380C2212yes—
23C2×C4×C22⋊C4C2211yescharacteristic
24#1990—24no—
25#1923—24no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36#1446—21no—
37#1445—21no—
38#1444—21no—
39#1442—21no—
40#1439—21no—
41#1439—21no—
42C4⋊#95C2212yes—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C23×Q8—21no—
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—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4D411yescharacteristic
66C22×C22⋊C4D411yescharacteristic
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C2×C4⋊C8—22no—
70C2×C4⋊C8—42no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—42no—
74C2×C22⋊C8—42no—
75C2×C4×C8—42no—
76C2×C4×C8D414yes—
77#399—44no—
78#393—44no—
79#389—24no—
80#389—42no—
81#384—22no—
82#384—22no—
83#380—24no—
84#380—24no—
85#380—42no—
86#380—42no—
87C4×C22⋊C4—22no—
88C4×C22⋊C4—22no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95D411yescharacteristic
99C2×#95D411yescharacteristic
100#340—24no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—42no—
105#340—42no—
106#340—42no—
107#340—42no—
108#336—22no—
109#336—22no—
110C25C2×D411yescharacteristic
111C22×Q8—44no—
112C22×Q8—42no—
113C22×Q8—42no—
114C22×Q8—42no—
115C22×Q8—82no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2411yescharacteristic
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8C2×D412yes—
155C22×C8C2×D412yes—
156C4⋊Q8—28no—
157C4⋊Q8—48no—
158C4⋊Q8—44no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4#4114yes—
192C2×C4⋊C4#4114yes—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—82no—
203C2×C22⋊C4—82no—
204C2×C22⋊C4SD1614yes—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42C2×D412yes—
212C2×C42#4112yes—
213C4⋊C8—44no—
214C22⋊C8—44no—
215C22⋊C8—44no—
216C4×C8—44no—
217#95—22no—
218#95—22no—
219#95—44no—
220#95—44no—
221#95—44no—
222#95—44no—
223#95—42no—
224#95—42no—
225#95—42no—
226#95—42no—
227#95—82no—
228C24—21no—
229C24—21no—
230C24—21no—
231C24—22no—
232C24—22no—
233C24—21no—
234C24—21no—
235C24—21no—
236C24—22no—
237C24—22no—
238C24C2×SD1612yes—
239C24C22×D411yescharacteristic
240C2×Q8—48no—
241C2×Q8—88no—
242C2×Q8—88no—
243C2×Q8—84no—
244C2×Q8—84no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4#12112yes—
310C22×C4#12112yes—
311C22×C4C2×SD1612yes—
312C22×C4C2×SD1612yes—
313C22×C4#13512yes—
314C22×C4AGL(1,ℤ/8)12yes—
315C22×C4#13712yes—
316C22×C4C22×D411yescharacteristic
317C22×C4C2×#4112yes—
318C22×C4C2×#4112yes—
319C22×C4C2×#4111yescharacteristic
320C22×C4H(5,𝔽2)12yes—
321C22×C4H(5,𝔽2)12yes—
322C22×C42−1+411yescharacteristic
323C22×C42−1+411yescharacteristic
324C22×C42−1+411yescharacteristic
325C22×C42−1+411yescharacteristic
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—48no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—84no—
364C4⋊C4—88no—
365C4⋊C4—84no—
366C22⋊C4—28no—
367C22⋊C4—44no—
368C22⋊C4—88no—
369C22⋊C4—88no—
370C42—22no—
371C42—22no—
372C42—24no—
373C42—41no—
374C42—41no—
375C42—42no—
376C23—22no—
377C23—22no—
378C23—24no—
379C23—24no—
380C23—22no—
381C23—42no—
382C23—41no—
383C23—42no—
384C23—42no—
385C23—41no—
386C23—41no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—41no—
391C23—42no—
392C23—41no—
393C23—42no—
394C23—41no—
395C23—42no—
396C23—41no—
397C23—41no—
398C23—42no—
399C23C2×#12111yescharacteristic
400C23#53611yescharacteristic
401C23#53611yescharacteristic
402C23#54612yes—
403C23#54612yes—
404C23C22×SD1611yescharacteristic
405C23C2×#13511yescharacteristic
406C23C2×AGL(1,ℤ/8)11yescharacteristic
407C23C2×#13711yescharacteristic
408C23#57612yes—
409C23#57812yes—
410Q8—816no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—41no—
433C2×C4—44no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4#46514yes—
469C2×C4#46814yes—
470C2×C4#52511yescharacteristic
471C2×C4#53211yescharacteristic
472C2×C4#53411yescharacteristic
473C2×C4#53511yescharacteristic
474C2×C4#53611yescharacteristic
475C2×C4#53611yescharacteristic
476C2×C4#53912yes—
477C2×C4#54111yescharacteristic
478C2×C4#54111yescharacteristic
479C2×C4#54212yes—
480C2×C4#54212yes—
481C2×C4#54212yes—
482C2×C4#55112yes—
483C2×C4#55212yes—
484C2×C4#55612yes—
485C2×C4#55912yes—
486C2×C4#56212yes—
487C2×C4#56212yes—
488C8—42no—
489C8—42no—
490C8—81no—
491C8—81no—
492C22—44no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—44no—
497C22—82no—
498C22—82no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22#253411yescharacteristic
504C22#254411yescharacteristic
505C22#262111yescharacteristic
506C22#268712yes—
507C22#270012yes—
508C22C2×#46511yescharacteristic
509C22C2×#46811yescharacteristic
510C22#305811yescharacteristic
511C22#305911yescharacteristic
512C22#322711yescharacteristic
513C22#322811yescharacteristic
514C22#322911yescharacteristic
515C22#323111yescharacteristic
516C22#324312yes—
517C22#324312yes—
518C22#329512yes—
519C22#329712yes—
520C22#329812yes—
521C22#330212yes—
522C22#330812yes—
523C22#331212yes—
524C22#332512yes—
525C22#333312yes—
526C4—24no—
527C4—24no—
528C4—42no—
529C4—84no—
530C4—81no—
531C4—81no—
532C4—84no—
533C4—162no—
534C4—162no—
535C4#260411yescharacteristic
536C4#263211yescharacteristic
537C4#266311yescharacteristic
538C4#275412yes—
539C4#279812yes—
540C4#281611yescharacteristic
541C2—82no—
542C2#1680311yescharacteristic
543C2#2809711yescharacteristic
544C2#2810511yescharacteristic
545C2#2860011yescharacteristic
546C2#2860511yescharacteristic
547C2#2987711yescharacteristic
548C2#2988011yescharacteristic
549C2#3135412yes—
550C2#3135712yes—
551C2#3174012yes—
552C2#3174512yes—
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×#1990C2semidirect
5. #11384C2semidirect
6. #11357C2semidirect
7. #11292C2semidirect
8. #11228C2semidirect
11. #10820C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19902
C2×#19232
#151352
#151252
#113842
#113572
#112922
#112282
#112002
#111992
#108202