Finite Group Databasebeta

← #414363

Subgroups — #414363

Order 512.

1318 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2101C212yesmaximal, complemented
2C2×#2042C212yesmaximal
3#14712C211yesmaximal, characteristic, complemented
4#14708C211yesmaximal, characteristic
5#11050C211yesmaximal, characteristic, complemented
6#11050C211yesmaximal, characteristic, complemented
7#11049C211yesmaximal, characteristic
8#11049C211yesmaximal, characteristic
9#10926C212yesmaximal, complemented
10#10925C212yesmaximal
11C4×#336C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×C4×C8C2212yes—
18C2×#391C2211yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#389—22no—
23C2×#389C2212yes—
24C2×#384C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×C43C2211yescharacteristic
28#2101—28no—
29#2042—28no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2212yes—
37#1437—22no—
38#1437—22no—
39C4⋊#95—22no—
40C4⋊#95C2212yes—
41C23×Q8C2311yescharacteristic
42C23×D4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×#124—22no—
46C2×#124—22no—
47C2×#124—24no—
48C2×#124—42no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C22⋊C4—21no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—22no—
60C22×C42D412yes—
61C22×C42D412yes—
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×#103—24no—
66C2×#103—24no—
67C2×#103—24no—
68C2×#103—24no—
69C2×#103—42no—
70C2×#103—42no—
71C2×#102—24no—
72C2×#102—24no—
73C2×#102—24no—
74C2×#102—24no—
75C2×#102—42no—
76C2×#102—42no—
77C2×C4×C8—24no—
78C2×C4×C8—42no—
79C2×C4×C8—42no—
80#391—22no—
81#391—22no—
82#391—24no—
83#390—22no—
84#390—22no—
85#390—24no—
86#389—24no—
87#389—24no—
88#389—48no—
89#384—24no—
90#384—42no—
91#382—24no—
92C2×#95—22no—
93C2×#95—24no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C43—42no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#336—22no—
104#336—22no—
105#336D418yes—
106C25—21no—
107C22×Q8—21no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—21no—
111C22×Q8—21no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×Q8#4112yes—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—21no—
120C22×D4—21no—
121C22×D4—22no—
122C22×D4—21no—
123C22×D4—22no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4#4112yes—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C23×C4C2411yescharacteristic
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8C2×D414yes—
150C22×C8C2×D414yes—
151#124—48no—
152#124—48no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—48no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4#4114yes—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—24no—
202C2×C42—24no—
203C2×C42—24no—
204C2×C42—24no—
205C2×C42—42no—
206C2×C42—42no—
207C2×C42C2×D414yes—
208C2×C42#4112yes—
209C2×C42#4112yes—
210C2×C42#4114yes—
211#103—48no—
212#103—48no—
213#102—48no—
214#102—48no—
215C4×C8—48no—
216#95—44no—
217#95—44no—
218#95—42no—
219#95—42no—
220#95—44no—
221#95—48no—
222C24—22no—
223C24—41no—
224C24—41no—
225C24—41no—
226C24—41no—
227C24—41no—
228C24—41no—
229C24—42no—
230C24—42no—
231C24—42no—
232C24—42no—
233C24C22×D411yescharacteristic
234C2×Q8—24no—
235C2×Q8—24no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—41no—
241C2×Q8—41no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—41no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—41no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4#12414yes—
310C22×C4#12414yes—
311C22×C4#12712yes—
312C22×C4#12712yes—
313C22×C4#13514yes—
314C22×C4#13514yes—
315C22×C4AGL(1,ℤ/8)14yes—
316C22×C4#13714yes—
317C22×C4C22×D412yes—
318C22×C4C2×#4112yes—
319C22×C4C2×#4111yescharacteristic
320C22×C4C2×#4111yescharacteristic
321C22×C4C2×#4112yes—
322C22×C4C2×#4111yescharacteristic
323C22×C4C2×#4111yescharacteristic
324C22×C4H(5,𝔽2)11yescharacteristic
325C22×C4H(5,𝔽2)12yes—
326C22×C42−1+411yescharacteristic
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—24no—
330C2×C8—24no—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—416no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—88no—
355C4⋊C4—88no—
356C22⋊C4—48no—
357C22⋊C4—48no—
358C22⋊C4—88no—
359C22⋊C4—88no—
360C42—24no—
361C42—24no—
362C42—24no—
363C42—24no—
364C42—24no—
365C42—24no—
366C42—48no—
367C42—48no—
368C23—42no—
369C23—44no—
370C23—44no—
371C23—42no—
372C23—42no—
373C23—81no—
374C23—82no—
375C23—82no—
376C23—81no—
377C23—82no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23—81no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—82no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23—81no—
391C23#52511yescharacteristic
392C23#52511yescharacteristic
393C23C2×#12411yescharacteristic
394C23C2×#12411yescharacteristic
395C23C2×#12711yescharacteristic
396C23#53411yescharacteristic
397C23#53711yescharacteristic
398C23C2×#13512yes—
399C23C2×#13512yes—
400C23C2×AGL(1,ℤ/8)12yes—
401C23C2×#13712yes—
402Q8—44no—
403Q8—44no—
404Q8—82no—
405Q8—82no—
406D4—44no—
407D4—44no—
408D4—82no—
409D4—82no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
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—22no—
427C2×C4—22no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—48no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—82no—
440C2×C4—81no—
441C2×C4—82no—
442C2×C4—84no—
443C2×C4—81no—
444C2×C4—84no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—82no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—84no—
458C2×C4—82no—
459C2×C4#48818yes—
460C2×C4#48918yes—
461C2×C4#52912yes—
462C2×C4#52912yes—
463C2×C4#53212yes—
464C2×C4#53212yes—
465C2×C4#53812yes—
466C2×C4#53912yes—
467C2×C4#54612yes—
468C2×C4#54612yes—
469C2×C4#54612yes—
470C2×C4#54812yes—
471C2×C4#55211yescharacteristic
472C2×C4#55211yescharacteristic
473C2×C4#55312yes—
474C2×C4#55312yes—
475C2×C4#55311yescharacteristic
476C2×C4#55311yescharacteristic
477C8—44no—
478C8—44no—
479C22—84no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22—84no—
484C22—162no—
485C22—161no—
486C22—161no—
487C22—161no—
488C22—161no—
489C22—162no—
490C22#239811yescharacteristic
491C22#253911yescharacteristic
492C22#254011yescharacteristic
493C22#256211yescharacteristic
494C22#259111yescharacteristic
495C22#263911yescharacteristic
496C22#285811yescharacteristic
497C22#310512yes—
498C22#310612yes—
499C22#310712yes—
500C22#310812yes—
501C22C2×#48812yes—
502C22#313112yes—
503C22C2×#48912yes—
504C22#314112yes—
505C22#314711yescharacteristic
506C22#314711yescharacteristic
507C22#315111yescharacteristic
508C22#315211yescharacteristic
509C22#318812yes—
510C22#319012yes—
511C22#324712yes—
512C22#324812yes—
513C4—28no—
514C4—28no—
515C4—42no—
516C4—42no—
517C4—82no—
518C4—84no—
519C4—162no—
520C4—161no—
521C4—161no—
522C4#255812yes—
523C4#255812yes—
524C4#255912yes—
525C4#255912yes—
526C2—162no—
527C2#1517011yescharacteristic
528C2#2381712yes—
529C2#2830711yescharacteristic
530C2#2830811yescharacteristic
531C2#2837511yescharacteristic
532C2#2837711yescharacteristic
533C2#2912212yes—
534C2#2933012yes—
535C2#3018812yes—
536C2#3322611yescharacteristic
537C2#3323011yescharacteristic
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×#2101C2semidirect
3. #14712C2semidirect
5. #11050C2semidirect
6. #11050C2semidirect
9. #10926C2semidirect
11. C4×#336C2semidirect

Maximal subgroups

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

subgroupindex
C2×#21012
C2×#20422
#147122
#147082
#110502
#110492
#109262
#109252
C4×#3362