Finite Group Databasebeta

← #472041

Subgroups — #472041

Order 512.

1310 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2096C211yesmaximal, characteristic, complemented
2C2×#2093C211yesmaximal, characteristic
3C2×#2085C212yesmaximal, complemented
4C2×#2042C212yesmaximal
5#14712C211yesmaximal, characteristic, complemented
6#14708C211yesmaximal, characteristic
7#11124C212yesmaximal, complemented
8#11121C212yesmaximal
9#11050C211yesmaximal, characteristic, complemented
10#11049C211yesmaximal, characteristic
11#10795C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C2×#391C2211yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#389—22no—
24C2×#389C2212yes—
25C2×#384C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×C43C2211yescharacteristic
29#2096—24no—
30#2093—24no—
31#2085—28no—
32#2042—28no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38C2×#336C2212yes—
39#1437—22no—
40#1437—22no—
41C4⋊#95—22no—
42C4⋊#95C2212yes—
43C23×Q8C2311yescharacteristic
44C23×D4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×#124—22no—
48C2×#124—22no—
49C2×#124—24no—
50C2×#124—42no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C42D412yes—
63C22×C42D412yes—
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C2×#107—24no—
68C2×#107—24no—
69C2×#107—42no—
70C2×#106—24no—
71C2×#106—24no—
72C2×#106—42no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—24no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#102—24no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—42no—
84C2×#102—42no—
85#391—22no—
86#391—22no—
87#391—42no—
88#390—22no—
89#390—22no—
90#390—42no—
91#389—24no—
92#389—24no—
93#389—48no—
94#384—22no—
95#384—42no—
96#384D414yes—
97#382—24no—
98C2×#95—22no—
99C2×#95—24no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C43—42no—
105#340—24no—
106#340—24no—
107#336—22no—
108#336—22no—
109#336—42no—
110#336D418yes—
111C25—21no—
112C22×Q8—21no—
113C22×Q8—22no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8#4112yes—
122C22×D4—22no—
123C22×D4—21no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4#4112yes—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2411yescharacteristic
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8C2×D414yes—
155C22×C8C2×D414yes—
156#124—48no—
157#124—48no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
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—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—48no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4C2×D414yes—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—24no—
204C2×C42—24no—
205C2×C42—24no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—42no—
211C2×C42—42no—
212C2×C42—42no—
213C2×C42C2×D412yes—
214C2×C42C2×D414yes—
215C2×C42#4112yes—
216#107—48no—
217#106—48no—
218#103—48no—
219#103—48no—
220#102—48no—
221#102—48no—
222#95—48no—
223#95—44no—
224#95—44no—
225#95—42no—
226#95—42no—
227#95—44no—
228C24—22no—
229C24—41no—
230C24—41no—
231C24—41no—
232C24—41no—
233C24—41no—
234C24—41no—
235C24—42no—
236C24—42no—
237C24—42no—
238C24—42no—
239C24C22×D411yescharacteristic
240C2×Q8—24no—
241C2×Q8—24no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—41no—
268C2×D4—42no—
269C2×D4—41no—
270C2×D4—42no—
271C2×D4—42no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4#12114yes—
318C22×C4#12712yes—
319C22×C4#12712yes—
320C22×C4#13514yes—
321C22×C4AGL(1,ℤ/8)14yes—
322C22×C4#13714yes—
323C22×C4C22×D412yes—
324C22×C4C22×D411yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4H(5,𝔽2)12yes—
329C22×C4H(5,𝔽2)12yes—
330C22×C42−1+412yes—
331C22×C42−1+411yescharacteristic
332C22×C42−1+411yescharacteristic
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—416no—
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—88no—
361C4⋊C4—88no—
362C22⋊C4—48no—
363C22⋊C4—48no—
364C22⋊C4—88no—
365C22⋊C4—88no—
366C42—24no—
367C42—24no—
368C42—24no—
369C42—24no—
370C42—48no—
371C42—48no—
372C42—42no—
373C42—42no—
374C23—44no—
375C23—44no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—82no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23—82no—
391C23—82no—
392C23—82no—
393C23—82no—
394C23—82no—
395C23—82no—
396C23—81no—
397C23C2×#12111yescharacteristic
398C23#52511yescharacteristic
399C23C2×#12711yescharacteristic
400C23#53411yescharacteristic
401C23#53511yescharacteristic
402C23#53611yescharacteristic
403C23#53611yescharacteristic
404C23C2×#13512yes—
405C23C2×AGL(1,ℤ/8)12yes—
406C23C2×#13712yes—
407C23#57512yes—
408Q8—44no—
409Q8—44no—
410Q8—82no—
411Q8—82no—
412D4—44no—
413D4—44no—
414D4—82no—
415D4—82no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—48no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—81no—
449C2×C4—84no—
450C2×C4—81no—
451C2×C4—84no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—84no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4#46318yes—
467C2×C4#53612yes—
468C2×C4#53612yes—
469C2×C4#53712yes—
470C2×C4#53912yes—
471C2×C4#53912yes—
472C2×C4#53912yes—
473C2×C4D4212yes—
474C2×C4#54612yes—
475C2×C4#54612yes—
476C2×C4#54811yescharacteristic
477C2×C4#54811yescharacteristic
478C2×C4#55012yes—
479C2×C4#55211yescharacteristic
480C2×C4#55211yescharacteristic
481C2×C4#56212yes—
482C2×C4#56212yes—
483C8—44no—
484C8—44no—
485C22—84no—
486C22—84no—
487C22—82no—
488C22—82no—
489C22—82no—
490C22—162no—
491C22—161no—
492C22—161no—
493C22—162no—
494C22—161no—
495C22—161no—
496C22#253611yescharacteristic
497C22#253911yescharacteristic
498C22#254111yescharacteristic
499C22#261811yescharacteristic
500C22#263311yescharacteristic
501C22#265811yescharacteristic
502C22#281611yescharacteristic
503C22C2×#46312yes—
504C22#303712yes—
505C22#310712yes—
506C22#311512yes—
507C22#314711yescharacteristic
508C22#315011yescharacteristic
509C22#315111yescharacteristic
510C22#315211yescharacteristic
511C22#319012yes—
512C22#319412yes—
513C22#320712yes—
514C22#321512yes—
515C22#323212yes—
516C22#323312yes—
517C22#323812yes—
518C22#323912yes—
519C4—28no—
520C4—44no—
521C4—42no—
522C4—42no—
523C4—84no—
524C4—82no—
525C4—161no—
526C4—161no—
527C4—162no—
528C4#268112yes—
529C4#268112yes—
530C4#269412yes—
531C4#269412yes—
532C2—162no—
533C2#1681111yescharacteristic
534C2#2820312yes—
535C2#2830811yescharacteristic
536C2#2831611yescharacteristic
537C2#2844311yescharacteristic
538C2#2844711yescharacteristic
539C2#2976012yes—
540C2#3008312yes—
541C2#3028211yescharacteristic
542C2#3028511yescharacteristic
543C2#3277512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2096C2semidirect
3. C2×#2085C2semidirect
5. #14712C2semidirect
7. #11124C2semidirect
9. #11050C2semidirect
11. #10795C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20962
C2×#20932
C2×#20852
C2×#20422
#147122
#147082
#111242
#111212
#110502
#110492
#107952