Finite Group Databasebeta

← #415567

Subgroups — #415567

Order 512.

1204 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×#2330C211yesmaximal, characteristic
2C2×#1988C211yesmaximal, characteristic
3C2×#1987C211yesmaximal, characteristic
4#14282C211yesmaximal, characteristic
5#11385C212yesmaximal
6#11293C212yesmaximal
7#11292C212yesmaximal
8#11267C212yesmaximal
9#10850C211yesmaximal, characteristic
10#10850C211yesmaximal, characteristic
11#10833C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16#2330—24no—
17#2330—24no—
18C2×#389—21no—
19C2×#389—21no—
20C2×#389C2211yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#387—21no—
23C2×#387—21no—
24C2×#384C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×#376C2211yescharacteristic
29#1988—24no—
30#1987—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1446—21no—
42#1446—21no—
43#1445—21no—
44#1445—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8—21no—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8Q814yes—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125—22no—
59C2×#125—22no—
60C2×#125—22no—
61C2×#125Q814yescomplemented
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C42—21no—
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86#389—28no—
87#389—28no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389—22no—
92#387—28no—
93#387—28no—
94#384—22no—
95#384—22no—
96#382—22no—
97#382—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4D414yes—
101C4×C4⋊C4D414yes—
102#376—22no—
103#376—22no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95D411yescharacteristic
115C2×#95D411yescharacteristic
116C2×#95D411yescharacteristic
117C2×#95D411yescharacteristic
118#340—24no—
119#340—24no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—24no—
124#340—42no—
125#340—42no—
126#336—24no—
127#336—42no—
128C22×Q8—24no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8—82no—
132C22×Q8—82no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2411yescharacteristic
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C4⋊Q8—28no—
161C4⋊Q8—28no—
162C4⋊Q8—44no—
163C4⋊Q8—44no—
164#125—28no—
165#125—28no—
166#125—28no—
167#125—28no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×Q814yes—
217C2×C4⋊C4C2×Q812yes—
218C2×C4⋊C4C2×Q812yes—
219C2×C4⋊C4#4112yes—
220C2×C4⋊C4#4112yes—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—82no—
234C2×C42—82no—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2×Q812yes—
238C2×C42C2×Q812yes—
239C2×C42#4112yes—
240C2×C42#4112yes—
241C2×C42#4112yes—
242C2×C42#4112yes—
243C4⋊C8—44no—
244C4⋊C8—44no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—42no—
254#95—42no—
255#95—82no—
256#95—82no—
257#95—82no—
258#95—82no—
259C24C22×D411yescharacteristic
260C2×Q8—48no—
261C2×Q8—88no—
262C2×Q8—88no—
263C2×Q8—84no—
264C2×Q8—84no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4#12012yes—
322C22×C4#12012yes—
323C22×C4C22⋊Q812yes—
324C22×C4C22⋊Q812yes—
325C22×C4C22⋊Q812yes—
326C22×C4C22⋊Q812yes—
327C22×C4#12412yes—
328C22×C4#12412yes—
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×Q811yescharacteristic
332C22×C4C22×Q811yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4H(5,𝔽2)11yescharacteristic
340C22×C42−1+411yescharacteristic
341C22×C42−1+411yescharacteristic
342C22×C42−1+411yescharacteristic
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C4⋊C4—216no—
358C4⋊C4—216no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—48no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—84no—
383C4⋊C4—84no—
384C4⋊C4—84no—
385C4⋊C4—84no—
386C4⋊C4C22⋊Q814yes—
387C4⋊C4C22⋊Q814yes—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—41no—
393C42—41no—
394C42—41no—
395C42—41no—
396C42—84no—
397C42—84no—
398C42C22⋊Q814yes—
399C42C22⋊Q814yes—
400C42#12414yes—
401C42#12414yes—
402C23C2×#12011yescharacteristic
403C23C2×C22⋊Q811yescharacteristic
404C23C2×C22⋊Q811yescharacteristic
405C23#52511yescharacteristic
406C23C2×#12411yescharacteristic
407C23#53611yescharacteristic
408C23#53711yescharacteristic
409C23#57612yes—
410C23#57712yes—
411C23#57712yes—
412C23#57812yes—
413Q8—816no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4#39314yes—
477C2×C4#39314yes—
478C2×C4C2×C22⋊Q811yescharacteristic
479C2×C4C2×C22⋊Q811yescharacteristic
480C2×C4C2×C22⋊Q811yescharacteristic
481C2×C4C2×C22⋊Q811yescharacteristic
482C2×C4#52511yescharacteristic
483C2×C4#52511yescharacteristic
484C2×C4C2×#12411yescharacteristic
485C2×C4C2×#12411yescharacteristic
486C2×C4#53611yescharacteristic
487C2×C4#53611yescharacteristic
488C2×C4#53711yescharacteristic
489C2×C4#53711yescharacteristic
490C2×C4#55611yescharacteristic
491C2×C4#55611yescharacteristic
492C2×C4#55611yescharacteristic
493C2×C4#55611yescharacteristic
494C2×C4#55611yescharacteristic
495C2×C4#55611yescharacteristic
496C2×C4#55711yescharacteristic
497C2×C4#55711yescharacteristic
498C2×C4#55711yescharacteristic
499C2×C4#55711yescharacteristic
500C2×C4#55711yescharacteristic
501C2×C4#55711yescharacteristic
502C2×C4Q8211yescharacteristic
503C2×C4#55911yescharacteristic
504C2×C4#55911yescharacteristic
505C2×C4#55911yescharacteristic
506C8—82no—
507C8—82no—
508C22C2×#39311yescharacteristic
509C22#256811yescharacteristic
510C22#258111yescharacteristic
511C22#258211yescharacteristic
512C22#259211yescharacteristic
513C22#261611yescharacteristic
514C22#284811yescharacteristic
515C22#300811yescharacteristic
516C22#300911yescharacteristic
517C22#301011yescharacteristic
518C22#301111yescharacteristic
519C22#308112yes—
520C22#308212yes—
521C22#308312yes—
522C22#308412yes—
523C22#312412yes—
524C22#312512yes—
525C22#313612yes—
526C22#313712yes—
527C22#324112yes—
528C22#324212yes—
529C22#325712yes—
530C22#325812yes—
531C4—24no—
532C4—24no—
533C4—42no—
534C4—44no—
535C4—42no—
536C4—44no—
537C4—81no—
538C4—82no—
539C4—82no—
540C4—81no—
541C4—84no—
542C4—162no—
543C4#256811yescharacteristic
544C4#256811yescharacteristic
545C4#258111yescharacteristic
546C4#258111yescharacteristic
547C4#258211yescharacteristic
548C4#258211yescharacteristic
549C4#259211yescharacteristic
550C4#259211yescharacteristic
551C2#1520111yescharacteristic
552C2#2345611yescharacteristic
553C2#2345711yescharacteristic
554C2#2922812yes—
555C2#2928212yes—
556C2#2929212yes—
557C2#2936312yes—
558C2#2969811yescharacteristic
559C2#2969911yescharacteristic
560C2#3296311yescharacteristic
561C2#3296511yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
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
61. C2×#125Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23302
C2×#19882
C2×#19872
#142822
#113852
#112932
#112922
#112672
#108502
#108332