Finite Group Databasebeta

← #368986

Subgroups — #368986

Order 512.

1652 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2502C211yesmaximal, characteristic
2C2×#2345C211yesmaximal, characteristic
3C2×#1996C211yesmaximal, characteristic
4C2×#1996C211yesmaximal, characteristic
5#13924C211yesmaximal, characteristic
6#11203C214yesmaximal
7#10912C214yesmaximal
8#10846C211yesmaximal, characteristic
9C4×#342C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12#2502—22no—
13#2502—22no—
14#2502—21no—
15#2502—22no—
16#2502—21no—
17C2×#398C414yescomplemented
18#2345—24no—
19#2345—24no—
20C2×#391C412yescomplemented
21C2×#389—21no—
22C2×#387—22no—
23C2×#385C2214yes—
24C2×#384C412yescomplemented
25C2×#380C2212yes—
26C2×#380C2211yescharacteristic
27C2×#380C2211yescharacteristic
28C2×C4×C4⋊C4—21no—
29C2×C4×C22⋊C4C2212yes—
30C2×C4×C22⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33#1996—24no—
34#1996—24no—
35C2×#342C2212yes—
36C2×#342C2214yes—
37C2×#342C2212yes—
38C2×#342C2214yes—
39C2×#342C2212yes—
40C2×#342C2212yes—
41#1445—22no—
42C4⋊#95—22no—
43C22⋊#95C2214yes—
44C23×Q8C2×C411yescharacteristic
45C24×C4C2312yes—
46C24×C4C2311yescharacteristic
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2×C412yes—
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4C2×C412yes—
66C22×C4⋊C4C2×C418yes—
67C22×C4⋊C4D412yes—
68C22×C4⋊C4D412yes—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D412yes—
71C22×C4⋊C4D411yescharacteristic
72C22×C22⋊C4C2312yes—
73C22×C22⋊C4C2312yes—
74C22×C22⋊C4C2312yes—
75C22×C22⋊C4C2311yescharacteristic
76C22×C22⋊C4C2312yes—
77C22×C22⋊C4C2312yes—
78C22×C22⋊C4C2311yescharacteristic
79C22×C42—21no—
80C22×C42—22no—
81C22×C42—21no—
82C22×C42C2×C412yes—
83#398—28no—
84#398—24no—
85#398—24no—
86#391—24no—
87#391—22no—
88#391—22no—
89#389—28no—
90#387—216no—
91#385—24no—
92#385—28no—
93#385—24no—
94#384—24no—
95#384—22no—
96#384—22no—
97#380—22no—
98#380—24no—
99#380—24no—
100#380—22no—
101#380—22no—
102#380D414yes—
103C4×C4⋊C4—28no—
104C4×C22⋊C4—22no—
105C4×C22⋊C4—21no—
106C4×C22⋊C4—21no—
107C4×C22⋊C4—21no—
108C4×C22⋊C4—21no—
109C4×C22⋊C4—24no—
110C4×C22⋊C4—24no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4D414yes—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—28no—
121C2×#95C2×C412yes—
122C2×#95C2×C418yes—
123#342—28no—
124#342—24no—
125#342—24no—
126#342—24no—
127#342—28no—
128#342Q818yescomplemented
129C25C2411yescharacteristic
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—44no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—44no—
145C23×C4—84no—
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C412yes—
148C23×C4C22×C412yes—
149C23×C4C22×C414yes—
150C23×C4C22×C414yes—
151C23×C4C22×C411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×D414yes—
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4#4112yes—
162C23×C4#4112yes—
163C22⋊Q8—42no—
164C22⋊Q8—42no—
165C22⋊Q8—42no—
166C22⋊Q8—42no—
167C22⋊Q8—42no—
168C22⋊Q8—42no—
169C22⋊Q8—42no—
170C22⋊Q8—42no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—84no—
201C2×C4⋊C4—88no—
202C2×C4⋊C4C22⋊C414yes—
203C2×C4⋊C4C22⋊C412yes—
204C2×C4⋊C4C22⋊C412yes—
205C2×C4⋊C4C22⋊C414yes—
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22⋊C412yes—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4C2×D414yes—
236C2×C22⋊C4C2×D412yes—
237C2×C22⋊C4C2×D412yes—
238C2×C22⋊C4C2×Q814yes—
239C2×C22⋊C4C2×Q814yes—
240C2×C22⋊C4C2×Q814yes—
241C2×C22⋊C4#4114yes—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—24no—
245C2×C42—28no—
246C2×C42—44no—
247C2×C42—44no—
248C2×C42—44no—
249C2×C42—42no—
250C2×C42—84no—
251#95—28no—
252#95—28no—
253#95—22no—
254#95—22no—
255#95—28no—
256#95—28no—
257#95—24no—
258#95—24no—
259#95—42no—
260#95—42no—
261#95—88no—
262C24—42no—
263C24—42no—
264C24C23×C411yescharacteristic
265C24C22×D411yescharacteristic
266C24C22×D411yescharacteristic
267C24C22×D411yescharacteristic
268C24C22×D411yescharacteristic
269C24C22×Q812yes—
270C24C2×#4111yescharacteristic
271C24C2×#4111yescharacteristic
272C24C2×#4112yes—
273C242−1+412yes—
274C242−1+412yes—
275C2×Q8—44no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—44no—
279C2×Q8—84no—
280C2×Q8—84no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—48no—
315C22×C4—48no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—42no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—44no—
336C22×C4—84no—
337C22×C4—84no—
338C22×C4—84no—
339C22×C4—84no—
340C22×C4C2×C22⋊C414yes—
341C22×C4C2×C22⋊C414yes—
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C4×D412yes—
347C22×C4C4×D412yes—
348C22×C4C4×D412yes—
349C22×C4C4×D412yes—
350C22×C4C4×Q818yes—
351C22×C4#12012yes—
352C22×C4#12012yes—
353C22×C4#12112yes—
354C22×C4#12112yes—
355C22×C4#12112yes—
356C22×C4#12112yes—
357C22×C4#12312yes—
358C22×C4#12312yes—
359C4⋊C4—24no—
360C4⋊C4—28no—
361C4⋊C4—24no—
362C4⋊C4—28no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—48no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—88no—
371C4⋊C4—88no—
372C4⋊C4—84no—
373C4⋊C4—88no—
374C22⋊C4—22no—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—24no—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—22no—
381C22⋊C4—24no—
382C22⋊C4—22no—
383C22⋊C4—28no—
384C22⋊C4—24no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4—44no—
397C22⋊C4C4⋊Q814yes—
398C42—44no—
399C42—48no—
400C42—44no—
401C42—84no—
402C42—88no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23C22×C22⋊C411yescharacteristic
422C23C2×C4×D411yescharacteristic
423C23C2×C4×D411yescharacteristic
424C23C2×C4×Q812yes—
425C23#51912yes—
426C23C2×#12011yescharacteristic
427C23C2×#12111yescharacteristic
428C23C2×#12111yescharacteristic
429C23C2×#12311yescharacteristic
430C23C2×C4⋊Q811yescharacteristic
431C23#53211yescharacteristic
432C23#53312yes—
433C23#53611yescharacteristic
434C23#53611yescharacteristic
435C23#54112yes—
436C23#54712yes—
437C23#54812yes—
438C23#54812yes—
439C23D4×Q812yes—
440C23D4×Q812yes—
441C23D4×Q812yes—
442C23#55612yes—
443C23#55712yes—
444Q8—84no—
445Q8—84no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—28no—
456C2×C4—28no—
457C2×C4—28no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—48no—
472C2×C4—48no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—48no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—48no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—84no—
491C2×C4—84no—
492C2×C4—84no—
493C2×C4—84no—
494C2×C4—82no—
495C2×C4—84no—
496C2×C4—84no—
497C2×C4—88no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—88no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4#38612yes—
508C2×C4#38612yes—
509C2×C4#38612yes—
510C2×C4#38612yes—
511C2×C4#41014yes—
512C22—24no—
513C22—24no—
514C22—24no—
515C22—24no—
516C22—24no—
517C22—24no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—41no—
531C22—41no—
532C22—42no—
533C22—41no—
534C22—41no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22C2×#38611yescharacteristic
541C22C4×C4⋊Q811yescharacteristic
542C22#232711yescharacteristic
543C22Q8×C22⋊C412yes—
544C22#236512yes—
545C22#242912yes—
546C22#243211yescharacteristic
547C22#243411yescharacteristic
548C22#244612yes—
549C22#254511yescharacteristic
550C22#254611yescharacteristic
551C22C2×#41012yes—
552C22#288312yes—
553C4—22no—
554C4—22no—
555C4—22no—
556C4—44no—
557C4—48no—
558C4—44no—
559C4—44no—
560C4—82no—
561C4—84no—
562C4—84no—
563C4—82no—
564C4—82no—
565C4—162no—
566C4#189914yes—
567C2—21no—
568C2—21no—
569C2—22no—
570C2—21no—
571C2—21no—
572C2—21no—
573C2—22no—
574C2—22no—
575C2—21no—
576C2#1402411yescharacteristic
577C2C2×#189911yescharacteristic
578C2#2098511yescharacteristic
579C2#2251314yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
17. C2×#398C4semidirect
20. C2×#391C4semidirect
24. C2×#384C4semidirect
128. #342Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25022
C2×#23452
C2×#19962
#139242
#112032
#109122
#108462
C4×#3422