Finite Group Databasebeta

← #405050

Subgroups — #405050

Order 512.

1200 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1949C212yesmaximal, complemented
2C2×#1947C211yesmaximal, characteristic, complemented
3C2×#1926C211yesmaximal, characteristic
4C2×#1925C212yesmaximal
5#15244C211yesmaximal, characteristic
6#13812C211yesmaximal, characteristic, complemented
7#11358C212yesmaximal, complemented
8#11357C211yesmaximal, characteristic, complemented
9#11322C211yesmaximal, characteristic, complemented
10#11076C211yesmaximal, characteristic
11#11076C212yesmaximal
12C22×#107C2212yes—
13C22×#106C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×#102C2211yescharacteristic
17C22×#102C2212yes—
18C22×#102C2211yescharacteristic
19C2×#390C2212yes—
20C2×#390C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#382C2212yes—
23C2×C4×C4⋊C4C2212yes—
24C2×C4×C4⋊C4C2211yescharacteristic
25#1949—28no—
26#1947—24no—
27#1926—24no—
28#1925—28no—
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2211yescharacteristic
36#1439—21no—
37#1439—21no—
38#1439—22no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41#1434—21no—
42#1434—22no—
43C4×#95—21no—
44C4×#95C2211yescharacteristic
45C23×D4C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2312yes—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C22⋊C4—21no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—22no—
60C22×C42—22no—
61C22×C42—21no—
62C22×C42C2312yes—
63C22×C42C2311yescharacteristic
64C2×#107—24no—
65C2×#107—24no—
66C2×#107—24no—
67C2×#106—22no—
68C2×#106—24no—
69C2×C4⋊C8—24no—
70C2×C4⋊C8—24no—
71C2×C4⋊C8—24no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—24no—
74C2×#102—24no—
75C2×#102—22no—
76C2×#102—24no—
77C2×#102—22no—
78C2×#102—24no—
79C2×#102—22no—
80C2×#102—24no—
81C2×#102—22no—
82#390—24no—
83#390—22no—
84#390—24no—
85#390—22no—
86#384—22no—
87#384—22no—
88#382—24no—
89#382—24no—
90C4×C4⋊C4—24no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4D418yes—
93C4×C4⋊C4D414yes—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95D411yescharacteristic
103C2×#95D411yescharacteristic
104C2×#95D412yes—
105#340—24no—
106#340—22no—
107#340—24no—
108#340—24no—
109#340—24no—
110#336—24no—
111#336—22no—
112C25—21no—
113C22×D4—22no—
114C22×D4—21no—
115C22×D4—21no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—21no—
119C22×D4—22no—
120C22×D4—21no—
121C22×D4—21no—
122C22×D4#4112yes—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D412yes—
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2411yescharacteristic
133C22×C8—21no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8#4112yes—
149C22×C8#4114yes—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4—81no—
176C2×C4⋊C4—81no—
177C2×C4⋊C4C2×D414yes—
178C2×C4⋊C4C2×D412yes—
179C2×C4⋊C4C2×D414yes—
180C2×C4⋊C4C2×D412yes—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—82no—
189C2×C22⋊C4—82no—
190C2×C22⋊C4—82no—
191C2×C22⋊C4—82no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—28no—
199C2×C42—24no—
200C2×C42—82no—
201C2×C42—82no—
202C2×C42—82no—
203C2×C42C2×D414yes—
204C2×C42C2×D412yes—
205C2×C42#4114yes—
206C2×C42#4112yes—
207#107—48no—
208#106—44no—
209C4⋊C8—28no—
210C4⋊C8—216no—
211#102—216no—
212#102—28no—
213#102—44no—
214#95—28no—
215#95—24no—
216#95—22no—
217#95—22no—
218#95—42no—
219#95—44no—
220#95—44no—
221#95—44no—
222#95—42no—
223#95—82no—
224#95—81no—
225C24—22no—
226C24—41no—
227C24—42no—
228C24—42no—
229C24—41no—
230C24—41no—
231C24—41no—
232C24—42no—
233C24—41no—
234C24—42no—
235C24—41no—
236C24C22×D411yescharacteristic
237C2×D4—22no—
238C2×D4—24no—
239C2×D4—22no—
240C2×D4—24no—
241C2×D4—41no—
242C2×D4—42no—
243C2×D4—41no—
244C2×D4—42no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—42no—
248C2×D4—41no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—41no—
254C2×D4—42no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4#12114yes—
303C22×C4#12112yes—
304C22×C4#12114yes—
305C22×C4#12112yes—
306C22×C4#13512yes—
307C22×C4#13712yes—
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D412yes—
310C22×C4C2×#4111yescharacteristic
311C22×C4C2×#4112yes—
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C22×C4C2×#4112yes—
315C22×C42−1+412yes—
316C22×C42−1+411yescharacteristic
317C22×C42−1+411yescharacteristic
318C2×C8—22no—
319C2×C8—24no—
320C2×C8—24no—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—41no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—41no—
329C2×C8—42no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—42no—
333C2×C8—41no—
334C2×C8—42no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—24no—
338C4⋊C4—22no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—42no—
342C4⋊C4—41no—
343C4⋊C4—42no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—41no—
351C4⋊C4—42no—
352C4⋊C4—84no—
353C4⋊C4—84no—
354C4⋊C4#12714yes—
355C22⋊C4—84no—
356C22⋊C4—88no—
357C22⋊C4—88no—
358C22⋊C4—84no—
359C42—22no—
360C42—24no—
361C42—22no—
362C42—24no—
363C42—24no—
364C42—24no—
365C42—24no—
366C42—84no—
367C42—88no—
368C23—42no—
369C23—44no—
370C23—42no—
371C23—42no—
372C23—44no—
373C23—82no—
374C23—81no—
375C23—82no—
376C23—82no—
377C23—82no—
378C23—82no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23—82no—
383C23—81no—
384C23—82no—
385C23—82no—
386C23—81no—
387C23—82no—
388C23—81no—
389C23—82no—
390C23—81no—
391C23C2×#12111yescharacteristic
392C23C2×#12112yes—
393C23#54712yes—
394C23#54711yescharacteristic
395C23#54811yescharacteristic
396C23C2×#13511yescharacteristic
397C23C2×#13711yescharacteristic
398C23#57512yes—
399C23#57612yes—
400C23#57611yescharacteristic
401C23#57711yescharacteristic
402D4—42no—
403D4—44no—
404D4—42no—
405D4—81no—
406D4—82no—
407D4—81no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—81no—
449C2×C4—82no—
450C2×C4—81no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—81no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4#46414yes—
461C2×C4#52511yescharacteristic
462C2×C4#52512yes—
463C2×C4#52511yescharacteristic
464C2×C4#52512yes—
465C2×C4C2×#12711yescharacteristic
466C2×C4#53212yes—
467C2×C4#53211yescharacteristic
468C2×C4#53611yescharacteristic
469C2×C4#53612yes—
470C2×C4#53611yescharacteristic
471C2×C4#53612yes—
472C2×C4#55212yes—
473C2×C4#55212yes—
474C2×C4#55212yes—
475C2×C4#55211yescharacteristic
476C2×C4#55212yes—
477C2×C4#55611yescharacteristic
478C2×C4#55612yes—
479C8—42no—
480C8—44no—
481C8—81no—
482C22—82no—
483C22—82no—
484C22—84no—
485C22—84no—
486C22—82no—
487C22—161no—
488C22—162no—
489C22—162no—
490C22—161no—
491C22—161no—
492C22—161no—
493C22#241311yescharacteristic
494C22#241312yes—
495C22#241511yescharacteristic
496C22#241512yes—
497C22#285611yescharacteristic
498C22C2×#46411yescharacteristic
499C22#303512yes—
500C22#303712yes—
501C22#304011yescharacteristic
502C22#304111yescharacteristic
503C22#304112yes—
504C22#304512yes—
505C22#304511yescharacteristic
506C22#333912yes—
507C22#334011yescharacteristic
508C22#334612yes—
509C22#334712yes—
510C22#335012yes—
511C22#335512yes—
512C22#335711yescharacteristic
513C22#336111yescharacteristic
514C22#337012yes—
515C22#337211yescharacteristic
516C4—24no—
517C4—44no—
518C4—44no—
519C4—42no—
520C4—44no—
521C4—84no—
522C4—82no—
523C4—161no—
524C4—162no—
525C4#239711yescharacteristic
526C4#260411yescharacteristic
527C4#260412yes—
528C4#261111yescharacteristic
529C4#261112yes—
530C4#265811yescharacteristic
531C2—162no—
532C2#1491011yescharacteristic
533C2#2458311yescharacteristic
534C2#2458512yes—
535C2#2459112yes—
536C2#2459711yescharacteristic
537C2#2467911yescharacteristic
538C2#2468412yes—
539C2#2468912yes—
540C2#2469311yescharacteristic
541C2#3316511yescharacteristic
542C2#3316611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1949C2semidirect
2. C2×#1947C2semidirect
6. #13812C2semidirect
7. #11358C2semidirect
8. #11357C2semidirect
9. #11322C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19492
C2×#19472
C2×#19262
C2×#19252
#152442
#138122
#113582
#113572
#113222
#110762