Finite Group Databasebeta

← #6806144

Subgroups — #6806144

Order 512.

1826 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36602C211yesmaximal, characteristic, complemented
2Q8×#127C211yesmaximal, characteristic, complemented
3#33230C211yesmaximal, characteristic, complemented
4#31636C214yesmaximal, complemented
5#31634C214yesmaximal, complemented
6#30216C212yesmaximal, complemented
7#29226C211yesmaximal, characteristic, complemented
8#29148C212yesmaximal, complemented
9#28618C212yesmaximal, complemented
10#28445C211yesmaximal, characteristic
11#25939C214yesmaximal, complemented
12#25334C214yesmaximal, complemented
13#22272C212yesmaximal, complemented
14#21656C211yesmaximal, characteristic, complemented
15#21346C211yesmaximal, characteristic, complemented
16C2×D4×Q8—22no—
17C2×D4×Q8C2212yes—
18C2×#496C2212yes—
19C2×#490C2211yescharacteristic
20C2×#489C2214yes—
21C2×#463C2214yes—
22#2868C2212yes—
23#2868C2211yescharacteristic
24#2861C2212yes—
25#2858C2211yescharacteristic
26#2597—22no—
27#2593—22no—
28#2592C2212yes—
29#2542—22no—
30#2542—22no—
31#2542—22no—
32#2542C2214yes—
33#2542C2212yes—
34#2458—22no—
35#2446C2212yes—
36#2445C2212yes—
37#2374—22no—
38#2367C2212yes—
39C4×#127C2212yes—
40C4×#127C2211yescharacteristic
41C4×#125C2211yescharacteristic
42C4×#124C2212yes—
43C42×Q8C2211yescharacteristic
44#2099C2214yes—
45#2072C2214yes—
46#2071C2214yes—
47#2055C2214yes—
48#2054C2214yes—
49#2024C2214yes—
50#2024C2214yes—
51#2023C2214yes—
52#2023C2214yes—
53#2013C2212yes—
54#2011C2212yes—
55#1992C2211yescharacteristic
56#1992C2212yes—
57#1987C2214yes—
58#1985C2212yes—
59#1961C2211yescharacteristic
60#1959C2212yes—
61#1959C2212yes—
62#1959C2212yes—
63#1957C2212yes—
64#1956C2212yes—
65#1940C2212yes—
66#1934C2214yes—
67#1916C2214yes—
68#1880C2214yes—
69#1879C2214yes—
70#1879C2214yes—
71#1879C2214yes—
72#1877C2214yes—
73#1877C2214yes—
74#1836C2212yes—
75#1836C2211yescharacteristic
76#1803C2214yes—
77#1802C2214yes—
78#1767C2211yescharacteristic
79#1767C2212yes—
80#1762C2214yes—
81#1745C2212yes—
82C4×#97C2211yescharacteristic
83C23×Q8—22no—
84C22×SD16C2314yes—
85C22×D8C2314yes—
86D4×Q8—48no—
87D4×Q8—416no—
88C2×C4⋊Q8—22no—
89C2×C4⋊Q8—22no—
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2312yes—
93C2×C4⋊Q8C2314yes—
94C2×#127C2311yescharacteristic
95C2×#127C2311yescharacteristic
96C2×#125C2311yescharacteristic
97C2×#124C2312yes—
98C2×#124C2312yes—
99C2×#123—22no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8—24no—
104C2×C22⋊Q8—22no—
105C2×C4×Q8D412yes—
106C2×C4×Q8C2312yes—
107C2×C4×Q8C2311yescharacteristic
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4—22no—
111C2×C4×D4C2312yes—
112C2×C4×D4C2314yes—
113C2×C4×D4C2312yes—
114#496—28no—
115#490—24no—
116#489—48no—
117#463—48no—
118C2×C4⋊C8C2312yes—
119C2×C4⋊C8C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#97C2311yescharacteristic
129C2×#97C2312yes—
130C2×#97C2312yes—
131C2×#97C2311yescharacteristic
132C2×C4×C8C2312yes—
133#400—22no—
134#399—22no—
135#399—24no—
136#399—22no—
137#399—24no—
138#398—22no—
139#397—22no—
140#394—22no—
141#394—22no—
142#393—22no—
143#391C2312yes—
144#391C2312yes—
145#391C2312yes—
146#390—22no—
147#390—22no—
148#390—22no—
149#390—22no—
150#390—22no—
151#390C2314yes—
152#390C2312yes—
153#390C2312yes—
154#390C2312yes—
155#390C2312yes—
156#389C2312yes—
157#388—22no—
158#388—22no—
159#388—22no—
160#388—22no—
161#388—22no—
162#388—22no—
163#388—22no—
164#388—22no—
165#388—22no—
166#388—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#386—22no—
171#385—22no—
172#384—22no—
173#384—22no—
174#384—22no—
175#384C2312yes—
176#384C2311yescharacteristic
177#384C2314yes—
178#384C2312yes—
179#382C2311yescharacteristic
180C4×C4⋊C4—22no—
181C4×C4⋊C4D412yes—
182C4×C4⋊C4D412yes—
183C4×C4⋊C4D412yes—
184C4×C4⋊C4C2312yes—
185C4×C4⋊C4C2311yescharacteristic
186C4×C4⋊C4C2312yes—
187C4×C22⋊C4—22no—
188C4×C22⋊C4—22no—
189C4×C22⋊C4—22no—
190C43C2311yescharacteristic
191C43C2312yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2312yes—
200#336C2314yes—
201#336C2314yes—
202C22×Q8—22no—
203C22×Q8—24no—
204C22×Q8—48no—
205C22×Q8—416no—
206C22×Q8C2×D411yescharacteristic
207C22×Q8C2412yes—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4C2412yes—
211C22×D4C2412yes—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—22no—
215C2×SD16—28no—
216C2×SD16—44no—
217C2×SD16—44no—
218C2×D8—28no—
219C2×D8—44no—
220C2×D8—44no—
221C22×C8C2414yes—
222C22×C8C2412yes—
223C22×C8C2412yes—
224C4⋊Q8—28no—
225C4⋊Q8—44no—
226C4⋊Q8—44no—
227C4⋊Q8—48no—
228C4⋊Q8—48no—
229C4⋊Q8—48no—
230C4⋊Q8—42no—
231#127—22no—
232#127—42no—
233#127#4114yes—
234#125—22no—
235#125—22no—
236#124—24no—
237#124—28no—
238#124—24no—
239#123—48no—
240C22⋊Q8—48no—
241C22⋊Q8—416no—
242C22⋊Q8—416no—
243C22⋊Q8—416no—
244C22⋊Q8—416no—
245C4×Q8—28no—
246C4×Q8—24no—
247C4×Q8—44no—
248C4×D4—28no—
249C4×D4—44no—
250C4×D4—44no—
251C4×D4—48no—
252C4×D4—48no—
253C4×D4—48no—
254C4×D4—44no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D411yescharacteristic
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2414yes—
275C2×C4⋊C4C2412yes—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42C2×D412yes—
295C2×C42C2×D412yes—
296C2×C42C2×D412yes—
297C2×C42C2×D412yes—
298C2×C42C2×D412yes—
299C2×C42C2×D412yes—
300C2×C42C2412yes—
301C2×C42C2412yes—
302C2×C42C2411yescharacteristic
303C2×C42C2411yescharacteristic
304C2×C42C2411yescharacteristic
305C4⋊C8—24no—
306C4⋊C8—44no—
307#103—44no—
308#103—44no—
309#103—44no—
310#102—28no—
311#102—44no—
312#102—44no—
313#102—44no—
314#102—44no—
315#97—22no—
316#97—24no—
317#97—24no—
318#97—22no—
319C4×C8—42no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95C2×D411yescharacteristic
330#95C2×D412yes—
331#95C2×D411yescharacteristic
332#95C2×D412yes—
333C24—22no—
334C24—22no—
335C2×Q8—22no—
336C2×Q8—24no—
337C2×Q8—22no—
338C2×Q8—24no—
339C2×Q8—22no—
340C2×Q8—24no—
341C2×Q8—48no—
342C2×Q8—48no—
343C2×Q8—48no—
344C2×Q8—44no—
345C2×Q8—48no—
346C2×Q8—832no—
347C2×D4—28no—
348C2×D4—22no—
349C2×D4—24no—
350C2×D4—24no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—44no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—82no—
359C2×D4—82no—
360C2×D4C2×#4114yes—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—44no—
370C22×C4—48no—
371C22×C4—44no—
372C22×C4—82no—
373C22×C4—82no—
374C22×C4C22×D412yes—
375C22×C4C22×D412yes—
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D412yes—
379C22×C4C22×D412yes—
380C22×C4C22×D412yes—
381C22×C4C22×D412yes—
382C22×C4C2511yescharacteristic
383SD16—88no—
384D8—88no—
385C2×C8—22no—
386C2×C8—24no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—24no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—24no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—48no—
408C4⋊C4—48no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—48no—
412C4⋊C4—48no—
413C4⋊C4—44no—
414C4⋊C4—82no—
415C4⋊C4—82no—
416C4⋊C4AGL(1,ℤ/8)18yes—
417C22⋊C4—48no—
418C22⋊C4—48no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—44no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—48no—
425C22⋊C4—48no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—82no—
430C22⋊C4—82no—
431C22⋊C4—82no—
432C22⋊C4—82no—
433C42—24no—
434C42—24no—
435C42—24no—
436C42—24no—
437C42—22no—
438C42—24no—
439C42—42no—
440C42—42no—
441C42—42no—
442C42—82no—
443C42—82no—
444C42C2×#4112yes—
445C42H(5,𝔽2)14yes—
446C422−1+412yes—
447C23—24no—
448C23—44no—
449C23—42no—
450C23—42no—
451C23—44no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23C23×D411yescharacteristic
460Q8—28no—
461Q8—44no—
462Q8—44no—
463Q8—88no—
464Q8—816no—
465D4—28no—
466D4—416no—
467D4—44no—
468D4—44no—
469D4—44no—
470D4—164no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—88no—
502C2×C4—816no—
503C2×C4—164no—
504C2×C4#53512yes—
505C2×C4#53512yes—
506C2×C4#54814yes—
507C2×C4C2×AGL(1,ℤ/8)14yes—
508C2×C4C2×AGL(1,ℤ/8)14yes—
509C2×C4C2×AGL(1,ℤ/8)14yes—
510C2×C4C22×#4111yescharacteristic
511C2×C4C2×H(5,𝔽2)12yes—
512C2×C4C2×2−1+411yescharacteristic
513C2×C4#58512yes—
514C8—42no—
515C8—44no—
516C8—42no—
517C22—44no—
518C22—48no—
519C22—84no—
520C22—82no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22C2×#53511yescharacteristic
530C22C2×#54811yescharacteristic
531C22#350011yescharacteristic
532C22C22×AGL(1,ℤ/8)12yes—
533C22#358612yes—
534C4—24no—
535C4—24no—
536C4—22no—
537C4—24no—
538C4—24no—
539C4—22no—
540C4—42no—
541C4—42no—
542C4—82no—
543C4—82no—
544C4—84no—
545C4—82no—
546C4—82no—
547C4#351412yes—
548C4#355412yes—
549C2—84no—
550C2—162no—
551C2#3917111yescharacteristic
552C2#6079811yescharacteristic
553C2#6083111yescharacteristic
554C2#6131712yes—
555C2#6258312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#28445C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36602C2semidirect
2. Q8×#127C2semidirect
3. #33230C2semidirect
4. #31636C2semidirect
5. #31634C2semidirect
6. #30216C2semidirect
7. #29226C2semidirect
8. #29148C2semidirect
9. #28618C2semidirect
11. #25939C2semidirect
12. #25334C2semidirect
13. #22272C2semidirect
14. #21656C2semidirect
15. #21346C2semidirect

Maximal subgroups

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

subgroupindex
#366022
Q8×#1272
#332302
#316362
#316342
#302162
#292262
#291482
#286182
#284452
#259392
#253342
#222722
#216562
#213462