Finite Group Databasebeta

← #416835

Subgroups — #416835

Order 512.

1174 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1804C211yesmaximal, characteristic
2C2×#1802C211yesmaximal, characteristic
3#14348C211yesmaximal, characteristic, complemented
4#13785C211yesmaximal, characteristic
5#11316C212yesmaximal, complemented
6#11315C212yesmaximal, complemented
7#11189C212yesmaximal
8#11189C212yesmaximal
9#10850C211yesmaximal, characteristic, complemented
10#10849C211yesmaximal, characteristic, complemented
11#10833C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2211yescharacteristic
16C22×#102C2211yescharacteristic
17C22×#102C2212yes—
18C2×#390C2212yes—
19C2×#384C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×#376C2211yescharacteristic
23#1804—24no—
24#1802—24no—
25C2×#340C2211yescharacteristic
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2212yes—
33C2×#336C2212yes—
34#1442—21no—
35#1442—21no—
36#1442—22no—
37#1437—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C22⋊#95—21no—
42C22⋊#95—21no—
43C23×D4C2311yescharacteristic
44C24×C4—21no—
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C2×C4×D4—22no—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4D414yes—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C42—21no—
65C22×C42C2312yes—
66C22×C42C2311yescharacteristic
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×#102—24no—
74C2×#102—22no—
75C2×#102—22no—
76C2×#102—24no—
77C2×#102—24no—
78C2×#102—42no—
79C2×#102—42no—
80#390—24no—
81#390—24no—
82#384—22no—
83#384—22no—
84#382—22no—
85#382—22no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88#376—22no—
89#376—22no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95D411yescharacteristic
100C2×#95D411yescharacteristic
101C2×#95D412yes—
102#340—24no—
103#340—22no—
104#340—22no—
105#340—24no—
106#340—24no—
107#340—42no—
108#340—42no—
109#336—42no—
110#336—42no—
111C25—21no—
112C22×D4—21no—
113C22×D4—21no—
114C22×D4—21no—
115C22×D4—21no—
116C22×D4—21no—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4C2×D412yes—
121C22×D4#4114yes—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—24no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—42no—
129C23×C4—42no—
130C23×C4—82no—
131C23×C4—82no—
132C23×C4C2×D412yes—
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2411yescharacteristic
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C4×D4—28no—
155C4×D4—28no—
156C4×D4—44no—
157C4×D4—44no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
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—22no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—81no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—81no—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4#4114yes—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—82no—
195C2×C22⋊C4—82no—
196C2×C22⋊C4—82no—
197C2×C22⋊C4—82no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—42no—
205C2×C42—81no—
206C2×C42—81no—
207C2×C42C2×D412yes—
208C2×C42#4112yes—
209C2×C42#4112yes—
210C2×C42#4112yes—
211C2×C42#4112yes—
212C2×C42#4112yes—
213C4⋊C8—44no—
214C4⋊C8—44no—
215#102—48no—
216#102—44no—
217#102—44no—
218#95—24no—
219#95—22no—
220#95—22no—
221#95—42no—
222#95—42no—
223#95—42no—
224#95—82no—
225#95—82no—
226#95—82no—
227#95—82no—
228#95—82no—
229#95—82no—
230C24—22no—
231C24—24no—
232C24—41no—
233C24—41no—
234C24—42no—
235C24—41no—
236C24—42no—
237C24—41no—
238C24—42no—
239C24—41no—
240C24—41no—
241C24C22×D411yescharacteristic
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—24no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—24no—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4#12414yes—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—48no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—84no—
311C22×C4—84no—
312C22×C4—88no—
313C22×C4—88no—
314C22×C4#12112yes—
315C22×C4#12112yes—
316C22×C4#12312yes—
317C22×C4#12312yes—
318C22×C4#12412yes—
319C22×C4#12412yes—
320C22×C4C22×D411yescharacteristic
321C22×C4C2×#4111yescharacteristic
322C22×C4C2×#4111yescharacteristic
323C22×C4C2×#4111yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C22×C4C2×#4112yes—
326C22×C4C2×#4111yescharacteristic
327C22×C4C2×#4112yes—
328C22×C4H(5,𝔽2)12yes—
329C22×C42−1+412yes—
330C2×C8—42no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—41no—
338C2×C8—41no—
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—22no—
348C4⋊C4—22no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—84no—
366C4⋊C4—84no—
367C22⋊C4—48no—
368C22⋊C4—88no—
369C22⋊C4—88no—
370C42—42no—
371C42—42no—
372C42—42no—
373C42—42no—
374C42—84no—
375C42#12314yes—
376C42#12314yes—
377C42#12614yes—
378C42#12614yes—
379C23—24no—
380C23—44no—
381C23—44no—
382C23—44no—
383C23—44no—
384C23—44no—
385C23—42no—
386C23—42no—
387C23—44no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—82no—
393C23—81no—
394C23—81no—
395C23—82no—
396C23—81no—
397C23—81no—
398C23—82no—
399C23—82no—
400C23—82no—
401C23—82no—
402C23C2×#12111yescharacteristic
403C23C2×#12311yescharacteristic
404C23#52512yes—
405C23C2×#12411yescharacteristic
406C23#53712yes—
407C23#57612yes—
408C23#57611yescharacteristic
409C23#57712yes—
410C23#57711yescharacteristic
411C23#57711yescharacteristic
412C23#57811yescharacteristic
413D4—24no—
414D4—24no—
415D4—44no—
416D4—44no—
417D4—42no—
418D4—42no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—816no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—81no—
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—81no—
469C2×C4—81no—
470C2×C4—82no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4#39714yes—
474C2×C4#39714yes—
475C2×C4C2×#12311yescharacteristic
476C2×C4C2×#12311yescharacteristic
477C2×C4C2×#12411yescharacteristic
478C2×C4C2×#12611yescharacteristic
479C2×C4C2×#12611yescharacteristic
480C2×C4#52912yes—
481C2×C4#53211yescharacteristic
482C2×C4#53511yescharacteristic
483C2×C4#53611yescharacteristic
484C2×C4#54212yes—
485C2×C4#55212yes—
486C2×C4#55212yes—
487C2×C4#55212yes—
488C2×C4#55312yes—
489C2×C4#55312yes—
490C2×C4#55312yes—
491C2×C4#55512yes—
492C2×C4#55612yes—
493C8—82no—
494C8—81no—
495C8—81no—
496C22—44no—
497C22—44no—
498C22—44no—
499C22—82no—
500C22—82no—
501C22—82no—
502C22—82no—
503C22—84no—
504C22—84no—
505C22—84no—
506C22—84no—
507C22C2×#39711yescharacteristic
508C22#248512yes—
509C22#256212yes—
510C22#262011yescharacteristic
511C22#285011yescharacteristic
512C22#304111yescharacteristic
513C22#304211yescharacteristic
514C22#304511yescharacteristic
515C22#304611yescharacteristic
516C22#309612yes—
517C22#309811yescharacteristic
518C22#309911yescharacteristic
519C22#311812yes—
520C22#311812yes—
521C22#312412yes—
522C22#312512yes—
523C22#313612yes—
524C22#313611yescharacteristic
525C22#313711yescharacteristic
526C22#325712yes—
527C22#325812yes—
528C22#325912yes—
529C22#326012yes—
530C4—24no—
531C4—24no—
532C4—82no—
533C4—84no—
534C4—82no—
535C4—82no—
536C4—161no—
537C4—161no—
538C4—162no—
539C4#249412yes—
540C4#249512yes—
541C4#249911yescharacteristic
542C4#250011yescharacteristic
543C4#250311yescharacteristic
544C4#250411yescharacteristic
545C2—84no—
546C2#1523311yescharacteristic
547C2#2363211yescharacteristic
548C2#2363411yescharacteristic
549C2#2764012yes—
550C2#2764112yes—
551C2#2913612yes—
552C2#2913712yes—
553C2#2983411yescharacteristic
554C2#2983611yescharacteristic
555C2#3303411yescharacteristic
556C2#3303611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #14348C2semidirect
5. #11316C2semidirect
6. #11315C2semidirect
9. #10850C2semidirect
10. #10849C2semidirect
11. #10833C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18042
C2×#18022
#143482
#137852
#113162
#113152
#111892
#108502
#108492
#108332