Finite Group Databasebeta

← #194278

Subgroups — #194278

Order 512.

1176 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3249C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic
3#21860C211yesmaximal, characteristic
4#21311C211yesmaximal, characteristic
5C2×#1481C211yesmaximal, characteristic
6#8987C212yesmaximal, complemented
7#8986C212yesmaximal, complemented
8#8975C212yesmaximal, complemented
9#8974C212yesmaximal, complemented
10C2×#1296C211yesmaximal, characteristic
11#7891C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#533C2211yescharacteristic
14#3249—24no—
15#3249C418yescomplemented
16C2×#477—21no—
17C2×#477C2212yes—
18C2×#467—22no—
19C2×#462—21no—
20C2×#462C2212yes—
21C2×#431C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24#2360—22no—
25#2360—22no—
26#2355—21no—
27#2355C2212yes—
28Q8×C4⋊C4—21no—
29Q8×C4⋊C4C2212yes—
30#2330—22no—
31#2317—21no—
32C4×C4⋊Q8—21no—
33C4×C4⋊Q8—21no—
34C2×#384C2211yescharacteristic
35#2274—21no—
36C4×#117C2211yescharacteristic
37#1990C2212yes—
38#1989—21no—
39#1989—21no—
40#1988C2211yescharacteristic
41#1988C2211yescharacteristic
42#1987C2211yescharacteristic
43#1987C2211yescharacteristic
44#1956C2212yes—
45#1872—22no—
46#1803—22no—
47#1735—21no—
48#1735—21no—
49#1481—24no—
50C2×Q8⋊C8C2212yes—
51C2×Q8⋊C8C2212yes—
52#1296C2212yes—
53#1296C2212yes—
54#1279C2212yes—
55#1279C2212yes—
56C22×Q16—22no—
57C23×C8—21no—
58C23×C8—21no—
59#533—22no—
60#533—22no—
61#533—22no—
62#533—22no—
63#533—22no—
64#533C2×C414yes—
65#533C2×C414yes—
66#533D414yes—
67C2×C4⋊Q8D411yescharacteristic
68C2×C4⋊Q8D411yescharacteristic
69C2×C4⋊Q8C2311yescharacteristic
70C2×C4⋊Q8C2311yescharacteristic
71C2×#125—21no—
72C2×#125—21no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8D412yes—
77C2×C4×Q8—22no—
78C2×C4×Q8—22no—
79C2×C4×Q8—21no—
80C2×C4×Q8—21no—
81C2×C4×Q8—21no—
82C2×C4×Q8C2312yes—
83C2×#117—21no—
84C2×#117—21no—
85C2×#117D411yescharacteristic
86C2×#117D411yescharacteristic
87C22×C4⋊C4—21no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4D412yes—
90C22×C42C2311yescharacteristic
91#477—28no—
92#477—24no—
93#477C2×C418yescomplemented
94#467—216no—
95#462—28no—
96#462—24no—
97#462C2×C418yescomplemented
98#431—22no—
99#431—22no—
100C2×#107—22no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8C2×C414yes—
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2312yes—
111C2×#103—22no—
112C2×#103—22no—
113C2×#103—24no—
114C2×C22⋊C8—21no—
115C2×C4×C8C2312yes—
116#391—22no—
117#391—22no—
118#389—22no—
119#389—22no—
120#389—22no—
121#389—24no—
122#389—22no—
123#389—24no—
124#387—24no—
125#387—24no—
126#385—24no—
127#384—22no—
128#384—21no—
129#384—21no—
130#384D412yes—
131#384C2312yes—
132#382—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—22no—
138C4×C4⋊C4—22no—
139C4×C4⋊C4D411yescharacteristic
140C4×C4⋊C4D411yescharacteristic
141C4×C4⋊C4C2311yescharacteristic
142C4×C4⋊C4C2311yescharacteristic
143C4×C22⋊C4—24no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—21no—
146#376—21no—
147C43—21no—
148#340—24no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#340—21no—
154#340—21no—
155#340—21no—
156#340—21no—
157Q8⋊C8—24no—
158Q8⋊C8—24no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8C2×D412yes—
163C23×C4—22no—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C2×Q16—28no—
168C2×Q16—82no—
169C2×Q16—82no—
170C22×C8—22no—
171C22×C8—24no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8—42no—
176C22×C8—42no—
177C22×C8—42no—
178C22×C8—81no—
179C22×C8—81no—
180C4⋊Q8—22no—
181C4⋊Q8—22no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8C22⋊C414yes—
189C4⋊Q8C22⋊C414yes—
190C4⋊Q8C22×C412yes—
191C4⋊Q8C22×C412yes—
192#125—24no—
193#125—24no—
194#125—24no—
195#125—42no—
196C22⋊Q8—24no—
197C22⋊Q8—24no—
198C22⋊Q8—24no—
199C22⋊Q8—24no—
200C22⋊Q8—24no—
201C22⋊Q8—42no—
202C22⋊Q8—42no—
203C22⋊Q8C22⋊C418yes—
204C4×Q8—24no—
205C4×Q8—24no—
206C4×Q8—22no—
207C4×Q8—22no—
208C4×Q8—24no—
209C4×Q8—24no—
210C4×Q8—42no—
211C4×Q8—42no—
212C4×Q8—82no—
213C4×Q8—82no—
214C4×Q8—84no—
215C4×Q8C22×C414yes—
216C4×Q8C2×D414yes—
217#117—24no—
218#117—24no—
219#117—22no—
220#117—24no—
221#117—22no—
222#117—22no—
223#117—42no—
224#117Q1614yescomplemented
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—82no—
246C2×C4⋊C4—81no—
247C2×C4⋊C4—81no—
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C4⋊C4C2×D411yescharacteristic
250C2×C4⋊C4C2×D412yes—
251C2×C4⋊C4C2×D411yescharacteristic
252C2×C4⋊C4C2×D411yescharacteristic
253C2×C4⋊C4C2×D414yes—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—41no—
258C2×C42—21no—
259C2×C42—22no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—41no—
268C2×C42C22×C412yes—
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D412yes—
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42#4112yes—
274C2×C42#4112yes—
275C2×C42C2411yescharacteristic
276#107—28no—
277C4⋊C8—24no—
278C4⋊C8—22no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—42no—
284C4⋊C8—42no—
285C4⋊C8C22×C414yes—
286#103—28no—
287#103—28no—
288#103—84no—
289C22⋊C8—42no—
290C4×C8—22no—
291C4×C8—22no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300C24#12011yescharacteristic
301C2×Q8—22no—
302C2×Q8—24no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8C2×C22⋊C414yes—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—24no—
330C22×C4—21no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—44no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4C2×C22⋊C412yes—
339C22×C4C4×D412yes—
340C22×C4C4×D412yes—
341C22×C4#12011yescharacteristic
342C22×C4#12011yescharacteristic
343C22×C4#12011yescharacteristic
344C22×C4#12112yes—
345C22×C4#12112yes—
346C22×C4#12312yes—
347C22×C4C2×Q1612yes—
348C22×C4#13512yes—
349C22×C4#13712yes—
350C22×C4#13712yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354Q16—88no—
355C2×C8—24no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—84no—
368C2×C8—84no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—28no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—82no—
400C4⋊C4—84no—
401C4⋊C4—84no—
402C4⋊C4C2×C22⋊C414yes—
403C4⋊C4C2×Q1612yes—
404C4⋊C4C2×Q1612yes—
405C22⋊C4—28no—
406C22⋊C4—24no—
407C22⋊C4—44no—
408C22⋊C4—42no—
409C42—24no—
410C42—24no—
411C42—24no—
412C42—42no—
413C42—42no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—44no—
419C42—42no—
420C42—84no—
421C42—84no—
422C42C2×C22⋊C412yes—
423C42C4×D412yes—
424C42C4×D412yes—
425C42#12112yes—
426C42#12112yes—
427C42#12312yes—
428C42C23×C411yescharacteristic
429C42C22×D411yescharacteristic
430C42C2×#4111yescharacteristic
431C42C2×#4111yescharacteristic
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23#38612yes—
437C23#45112yes—
438C23#45212yes—
439C23C2×#12011yescharacteristic
440Q8—24no—
441Q8—24no—
442Q8—48no—
443Q8—44no—
444Q8—42no—
445Q8—42no—
446Q8—42no—
447Q8—42no—
448Q8—82no—
449Q8—81no—
450Q8—81no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—44no—
478C2×C4—41no—
479C2×C4—44no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—44no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—164no—
491C2×C4#38612yes—
492C2×C4#38612yes—
493C2×C4#38612yes—
494C2×C4C4×Q1614yes—
495C2×C4#44114yes—
496C2×C4#45112yes—
497C2×C4#45212yes—
498C2×C4#46214yes—
499C2×C4#46414yes—
500C2×C4C22×C22⋊C411yescharacteristic
501C2×C4C2×C4×D411yescharacteristic
502C2×C4C2×C4×D411yescharacteristic
503C2×C4C2×#12111yescharacteristic
504C2×C4C2×#12111yescharacteristic
505C2×C4C2×#12311yescharacteristic
506C2×C4C22×Q1611yescharacteristic
507C2×C4C2×#13511yescharacteristic
508C2×C4C2×#13711yescharacteristic
509C2×C4C2×#13711yescharacteristic
510C2×C4#57611yescharacteristic
511C2×C4#57711yescharacteristic
512C2×C4#57711yescharacteristic
513C2×C4#57811yescharacteristic
514C8—44no—
515C8—82no—
516C8—81no—
517C8—81no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22#187814yes—
525C22C2×#38611yescharacteristic
526C22C2×#45111yescharacteristic
527C22C2×#45211yescharacteristic
528C22#301211yescharacteristic
529C22#301311yescharacteristic
530C22#301911yescharacteristic
531C22#302011yescharacteristic
532C4—22no—
533C4—21no—
534C4—21no—
535C4—22no—
536C4—44no—
537C4—42no—
538C4—42no—
539C4—44no—
540C4—82no—
541C4—81no—
542C4—84no—
543C4—81no—
544C4—82no—
545C4#187814yes—
546C4#289711yescharacteristic
547C4#289811yescharacteristic
548C4C2×C4×Q1611yescharacteristic
549C4C2×#44111yescharacteristic
550C4C2×#46211yescharacteristic
551C4C2×#46411yescharacteristic
552C4#309811yescharacteristic
553C4#309911yescharacteristic
554C2—24no—
555C2#1887311yescharacteristic
556C2#1887511yescharacteristic
557C2#1888511yescharacteristic
558C2#1888711yescharacteristic
559C2#2091511yescharacteristic
560C2C2×#187811yescharacteristic
561C2#2094311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8987C2semidirect
7. #8986C2semidirect
8. #8975C2semidirect
9. #8974C2semidirect
15. #3249C4semidirect
93. #477C2×C4semidirect
97. #462C2×C4semidirect
224. #117Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32492
#342142
#218602
#213112
C2×#14812
#89872
#89862
#89752
#89742
C2×#12962
#78912