Finite Group Databasebeta

← #405049

Subgroups — #405049

Order 512.

1222 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1949C212yesmaximal, complemented
2C2×#1948C211yesmaximal, characteristic, complemented
3C2×#1926C212yesmaximal
4C2×#1925C211yesmaximal, characteristic
5#15244C211yesmaximal, characteristic
6#13810C211yesmaximal, characteristic, complemented
7#11358C212yesmaximal, complemented
8#11357C211yesmaximal, characteristic, complemented
9#11322C211yesmaximal, characteristic, complemented
10#11076C212yesmaximal
11#11076C211yesmaximal, characteristic
12C22×#107C2211yescharacteristic
13C22×#106C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×#102C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C22×#102C2212yes—
19C2×#390C2211yescharacteristic
20C2×#390C2212yes—
21C2×#384C2211yescharacteristic
22C2×#382C2212yes—
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2212yes—
25C2×#376—21no—
26#1949—28no—
27#1948—24no—
28#1926—28no—
29#1925—24no—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2212yes—
37#1439—21no—
38#1439—21no—
39#1439—22no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2212yes—
42#1434—21no—
43#1434—22no—
44C4×#95C2211yescharacteristic
45C23×D4C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
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—22no—
65C2×#107—24no—
66C2×#106—24no—
67C2×#106—24no—
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—22no—
75C2×#102—24no—
76C2×#102—22no—
77C2×#102—24no—
78C2×#102—22no—
79C2×#102—24no—
80C2×#102—22no—
81C2×#102—24no—
82#390—22no—
83#390—24no—
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⋊C4D414yes—
93C4×C4⋊C4D418yes—
94#376—44no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95D411yescharacteristic
104C2×#95D411yescharacteristic
105C2×#95D412yes—
106#340—22no—
107#340—24no—
108#340—24no—
109#340—24no—
110#340—24no—
111#336—22no—
112#336—24no—
113C25—21no—
114C22×D4—21no—
115C22×D4—22no—
116C22×D4—21no—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—21no—
120C22×D4—22no—
121C22×D4—21no—
122C22×D4—21no—
123C22×D4#4112yes—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2411yescharacteristic
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8#4114yes—
150C22×C8#4112yes—
151C2×C4⋊C4—21no—
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—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4—81no—
177C2×C4⋊C4—81no—
178C2×C4⋊C4C2×D412yes—
179C2×C4⋊C4C2×D414yes—
180C2×C4⋊C4C2×D412yes—
181C2×C4⋊C4C2×D414yes—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—82no—
190C2×C22⋊C4—82no—
191C2×C22⋊C4—82no—
192C2×C22⋊C4—82no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—28no—
200C2×C42—24no—
201C2×C42—44no—
202C2×C42—44no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42C2×D414yes—
206C2×C42C2×D412yes—
207C2×C42#4114yes—
208C2×C42#4112yes—
209#107—44no—
210#106—48no—
211C4⋊C8—28no—
212C4⋊C8—216no—
213#102—28no—
214#102—216no—
215#102—44no—
216#95—28no—
217#95—24no—
218#95—22no—
219#95—22no—
220#95—42no—
221#95—44no—
222#95—44no—
223#95—44no—
224#95—42no—
225#95—82no—
226#95—81no—
227C24—22no—
228C24—41no—
229C24—42no—
230C24—42no—
231C24—41no—
232C24—41no—
233C24—41no—
234C24—42no—
235C24—41no—
236C24—41no—
237C24—42no—
238C24C22×D411yescharacteristic
239C2×D4—22no—
240C2×D4—24no—
241C2×D4—22no—
242C2×D4—24no—
243C2×D4—41no—
244C2×D4—42no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—42no—
248C2×D4—41no—
249C2×D4—42no—
250C2×D4—41no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—41no—
256C2×D4—42no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
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—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4#12114yes—
307C22×C4#12112yes—
308C22×C4#12114yes—
309C22×C4#12112yes—
310C22×C4#13512yes—
311C22×C4AGL(1,ℤ/8)12yes—
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D412yes—
314C22×C4C2×#4111yescharacteristic
315C22×C4C2×#4112yes—
316C22×C4C2×#4111yescharacteristic
317C22×C4C2×#4111yescharacteristic
318C22×C4C2×#4112yes—
319C22×C42−1+412yes—
320C22×C42−1+411yescharacteristic
321C22×C42−1+411yescharacteristic
322C2×C8—24no—
323C2×C8—22no—
324C2×C8—24no—
325C2×C8—24no—
326C2×C8—24no—
327C2×C8—42no—
328C2×C8—41no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—42no—
337C2×C8—41no—
338C2×C8—42no—
339C4⋊C4—22no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—24no—
345C4⋊C4—41no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—41no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—41no—
356C4⋊C4—84no—
357C4⋊C4—84no—
358C4⋊C4#12714yes—
359C22⋊C4—84no—
360C22⋊C4—88no—
361C22⋊C4—88no—
362C22⋊C4—84no—
363C42—22no—
364C42—24no—
365C42—22no—
366C42—24no—
367C42—24no—
368C42—24no—
369C42—24no—
370C42—416no—
371C42—48no—
372C23—42no—
373C23—42no—
374C23—44no—
375C23—42no—
376C23—44no—
377C23—82no—
378C23—81no—
379C23—82no—
380C23—82no—
381C23—82no—
382C23—82no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—82no—
387C23—81no—
388C23—82no—
389C23—82no—
390C23—81no—
391C23—82no—
392C23—81no—
393C23—82no—
394C23—81no—
395C23C2×#12111yescharacteristic
396C23C2×#12112yes—
397C23#54712yes—
398C23#54711yescharacteristic
399C23#54811yescharacteristic
400C23C2×#13511yescharacteristic
401C23C2×AGL(1,ℤ/8)11yescharacteristic
402C23#57512yes—
403C23#57611yescharacteristic
404C23#57712yes—
405C23#57711yescharacteristic
406D4—42no—
407D4—44no—
408D4—42no—
409D4—81no—
410D4—82no—
411D4—81no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—81no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—81no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—81no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4#46314yes—
467C2×C4#52511yescharacteristic
468C2×C4#52512yes—
469C2×C4#52511yescharacteristic
470C2×C4#52512yes—
471C2×C4C2×#12711yescharacteristic
472C2×C4#53212yes—
473C2×C4#53211yescharacteristic
474C2×C4#53611yescharacteristic
475C2×C4#53612yes—
476C2×C4#53611yescharacteristic
477C2×C4#53612yes—
478C2×C4#55212yes—
479C2×C4#55212yes—
480C2×C4#55212yes—
481C2×C4#55211yescharacteristic
482C2×C4#55212yes—
483C2×C4#55611yescharacteristic
484C2×C4#55612yes—
485C8—44no—
486C8—42no—
487C8—81no—
488C22—82no—
489C22—82no—
490C22—84no—
491C22—84no—
492C22—82no—
493C22—161no—
494C22—162no—
495C22—162no—
496C22—161no—
497C22—161no—
498C22—161no—
499C22#241311yescharacteristic
500C22#241312yes—
501C22#241511yescharacteristic
502C22#241512yes—
503C22#285611yescharacteristic
504C22C2×#46311yescharacteristic
505C22#303512yes—
506C22#303612yes—
507C22#304011yescharacteristic
508C22#304311yescharacteristic
509C22#304312yes—
510C22#304512yes—
511C22#304511yescharacteristic
512C22#333712yes—
513C22#334111yescharacteristic
514C22#334512yes—
515C22#334712yes—
516C22#334812yes—
517C22#335112yes—
518C22#335411yescharacteristic
519C22#336011yescharacteristic
520C22#336611yescharacteristic
521C22#337612yes—
522C4—24no—
523C4—44no—
524C4—44no—
525C4—42no—
526C4—44no—
527C4—82no—
528C4—84no—
529C4—161no—
530C4—162no—
531C4#239711yescharacteristic
532C4#260411yescharacteristic
533C4#260412yes—
534C4#261111yescharacteristic
535C4#261112yes—
536C4#265811yescharacteristic
537C2—162no—
538C2#1491011yescharacteristic
539C2#2458111yescharacteristic
540C2#2458712yes—
541C2#2458912yes—
542C2#2459911yescharacteristic
543C2#2467711yescharacteristic
544C2#2468512yes—
545C2#2468812yes—
546C2#2469511yescharacteristic
547C2#3316411yescharacteristic
548C2#3316711yescharacteristic
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×#1948C2semidirect
6. #13810C2semidirect
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×#19482
C2×#19262
C2×#19252
#152442
#138102
#113582
#113572
#113222
#110762