Finite Group Databasebeta

← #215346

Subgroups — #215346

Order 512.

1402 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34936C211yesmaximal, characteristic
2#21430C211yesmaximal, characteristic, complemented
3#20812C211yesmaximal, characteristic, complemented
4#9678C212yesmaximal, complemented
5#9674C212yesmaximal, complemented
6#9644C212yesmaximal, complemented
7#9638C212yesmaximal, complemented
8#8313C212yesmaximal, complemented
9#8310C212yesmaximal, complemented
10C2×#532C2211yescharacteristic
11C2×#525—21no—
12C2×#525—21no—
13C22×C4×D4C2211yescharacteristic
14C2×#433—21no—
15#2542C412yes—
16#2541C412yes—
17#2540—22no—
18#2539—22no—
19#2501C412yes—
20#2433C412yes—
21#2397—21no—
22#2397—21no—
23C4×#127C2212yes—
24C4×#127C2212yes—
25C4×#117C2211yescharacteristic
26#1923—22no—
27#1916—22no—
28#1842C2212yescomplemented
29#1832—21no—
30#1798—21no—
31#1797C2212yes—
32#1768—21no—
33#1767C2211yescharacteristic, complemented
34C4×C4⋊C8C2211yescharacteristic, complemented
35C4×C4⋊C8C2212yescomplemented
36C4×C22⋊C8C2212yes—
37#1744C2211yescharacteristic, complemented
38C42×C8C2211yescharacteristic, complemented
39C2×D4⋊C8C2212yes—
40C2×D4⋊C8C2212yes—
41C2×D4⋊C8C2212yes—
42C2×D4⋊C8C2212yes—
43C2×D4⋊C8C2212yescomplemented
44C2×D4⋊C8C2212yescomplemented
45C2×D4⋊C8C2212yescomplemented
46C2×D4⋊C8C2212yescomplemented
47C22×#41—21no—
48C23×D4C2×C411yescharacteristic
49C24×C4—21no—
50#532—42no—
51#532—42no—
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#127C2×C412yes—
54C2×#127C2×C412yes—
55C2×#127C2×C411yescharacteristic
56C2×#124—21no—
57#525—48no—
58#525—44no—
59#525—44no—
60C2×#123—22no—
61C2×#123—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×#121—21no—
65C2×#121—22no—
66C2×#121—22no—
67C2×#121C2×C412yes—
68C2×#120—22no—
69C2×#120—22no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4—42no—
78C2×C4×D4—42no—
79C2×C4×D4C2×C412yes—
80C2×C4×D4C2312yes—
81C2×C4×D4C2312yes—
82C2×#117D411yescharacteristic
83C2×#117D411yescharacteristic
84C22×C4⋊C4C2×C411yescharacteristic
85C22×C22⋊C4—21no—
86C22×C42C2311yescharacteristic
87#433—44no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8C2312yes—
92C2×C4⋊C8C2312yes—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×#97—21no—
96C2×C4×C8—21no—
97C2×C4×C8—22no—
98C2×C4×C8C2312yes—
99C2×C4×C8C2312yes—
100#399—22no—
101#399—22no—
102#399—22no—
103#399—22no—
104#392—22no—
105#392—22no—
106#392—22no—
107#392—22no—
108#390—22no—
109#390—22no—
110#390—22no—
111#390—22no—
112#390C2×C412yes—
113#390C2×C412yes—
114#390C2×C412yes—
115#390C2×C412yes—
116#384C2×C412yes—
117#384C2×C412yes—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4D412yes—
120C4×C22⋊C4C2×C412yes—
121C4×C22⋊C4D412yes—
122#376D411yescharacteristic
123#376D411yescharacteristic
124C43C2311yescharacteristic
125C43C2311yescharacteristic
126#336—22no—
127#336—21no—
128#336—21no—
129#336—22no—
130#336—22no—
131#336—22no—
132D4⋊C8—42no—
133D4⋊C8—42no—
134D4⋊C8—42no—
135D4⋊C8—42no—
136D4⋊C8—42no—
137D4⋊C8—42no—
138D4⋊C8—42no—
139D4⋊C8—42no—
140C25—21no—
141C2×#41—44no—
142C2×#41—44no—
143C2×#41—42no—
144C2×#41—42no—
145C22×Q8—21no—
146C22×D4—24no—
147C22×D4—22no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—21no—
155C22×D4—41no—
156C22×D4—41no—
157C22×D4—42no—
158C22×D4C22×C412yes—
159C22×D4C22×C412yes—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4C22×C411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C4⋊Q8—41no—
176C4⋊Q8—41no—
177#127—24no—
178#127—24no—
179#127—42no—
180#127—42no—
181#127—41no—
182#127—41no—
183#124—44no—
184#123—48no—
185#123—48no—
186C22⋊Q8—48no—
187C22⋊Q8—48no—
188#121—42no—
189#121—42no—
190#121—48no—
191#121—44no—
192#121—48no—
193#120—48no—
194#120—48no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—22no—
199C4×D4—22no—
200C4×D4—22no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—42no—
204C4×D4—42no—
205C4×D4—44no—
206C4×D4—44no—
207C4×D4—44no—
208C4×D4—44no—
209C4×D4—44no—
210C4×D4—44no—
211C4×D4—44no—
212C4×D4—44no—
213C4×D4—44no—
214C4×D4—44no—
215C4×D4—44no—
216#117—24no—
217#117D814yes—
218#117SD1614yes—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4C22⋊C412yes—
228C2×C4⋊C4C22⋊C412yes—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C22×C412yes—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4C22⋊C412yes—
245C2×C22⋊C4C22⋊C412yes—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C412yes—
250C2×C42C22×C412yes—
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2411yescharacteristic
260C4⋊C8—22no—
261C4⋊C8—22no—
262C4⋊C8—22no—
263C4⋊C8—22no—
264C4⋊C8—44no—
265C4⋊C8—44no—
266C4⋊C8—44no—
267C22⋊C8—44no—
268C22⋊C8—44no—
269#97—42no—
270C4×C8—22no—
271C4×C8—22no—
272C4×C8—22no—
273C4×C8—22no—
274C4×C8—44no—
275C4×C8—42no—
276#95C22⋊C412yes—
277#95C22⋊C412yes—
278#95C22⋊C412yes—
279#95C22⋊C412yes—
280C24—22no—
281C24—22no—
282C24—44no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24—82no—
287C24C2×C22⋊C411yescharacteristic
288#41—88no—
289#41—88no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—44no—
305C2×D4—42no—
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—44no—
315C2×D4—48no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4—42no—
323C2×D4—44no—
324C2×D4—82no—
325C2×D4—82no—
326C2×D4—82no—
327C2×D4—82no—
328C2×D4—82no—
329C2×D4#13114yes—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—42no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—88no—
359C22×C4—88no—
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C412yes—
363C22×C4C2×C22⋊C412yes—
364C22×C4C2×C22⋊C412yes—
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C412yes—
367C22×C4C2×D812yes—
368C22×C4C2×SD1612yes—
369C22×C4C23×C411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C2×C8—24no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C4⋊C4—24no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4#10218yes—
396C22⋊C4—24no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—44no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4#10218yes—
412C42—21no—
413C42—21no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—42no—
422C42—42no—
423C42C2×D812yes—
424C42C2×D812yes—
425C42C2×SD1612yes—
426C42C2×SD1612yes—
427C42AGL(1,ℤ/8)14yes—
428C42H(5,𝔽2)12yes—
429C42H(5,𝔽2)12yes—
430C23—21no—
431C23—22no—
432C23—21no—
433C23—22no—
434C23—24no—
435C23—48no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—88no—
450C23—84no—
451C23—88no—
452C23—84no—
453C23C2×#10212yes—
454C23C22×C22⋊C411yescharacteristic
455Q8—84no—
456D4—24no—
457D4—24no—
458D4—48no—
459D4—42no—
460D4—42no—
461D4—42no—
462D4—42no—
463D4—42no—
464D4—42no—
465D4—88no—
466D4—88no—
467D4—88no—
468D4—84no—
469D4—88no—
470D4—84no—
471D4—88no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—82no—
500C2×C4—88no—
501C2×C4—84no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4C2×#10214yes—
511C2×C4C2×#10214yes—
512C2×C4C2×#10212yes—
513C2×C4#41814yes—
514C2×C4#51811yescharacteristic
515C2×C4#51811yescharacteristic
516C2×C4#53512yes—
517C2×C4#53512yes—
518C2×C4C2×#13112yes—
519C2×C4#56812yes—
520C2×C4C22×D811yescharacteristic
521C2×C4C22×SD1611yescharacteristic
522C2×C4C2×AGL(1,ℤ/8)12yes—
523C8—44no—
524C8—82no—
525C8—82no—
526C8—82no—
527C22—22no—
528C22—24no—
529C22—41no—
530C22—41no—
531C22—42no—
532C22—44no—
533C22—44no—
534C22—44no—
535C22—816no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22#230911yescharacteristic
548C22#288012yes—
549C22C22×#10211yescharacteristic
550C22C2×#41811yescharacteristic
551C22#290112yes—
552C4—22no—
553C4—22no—
554C4—24no—
555C4—24no—
556C4—24no—
557C4—41no—
558C4—41no—
559C4—42no—
560C4—82no—
561C4—82no—
562C4#297612yes—
563C4#297711yescharacteristic
564C4#297811yescharacteristic
565C4#320512yes—
566C4#320812yes—
567C2—42no—
568C2—84no—
569C2—162no—
570C2#1750712yes—
571C2#1751112yes—
572C2#2179211yescharacteristic
573C2#2180311yescharacteristic
574C2#2181711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radical#34936C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21430C2semidirect
3. #20812C2semidirect
4. #9678C2semidirect
5. #9674C2semidirect
6. #9644C2semidirect
7. #9638C2semidirect
8. #8313C2semidirect
9. #8310C2semidirect
28. #1842C22semidirect
33. #1767C22semidirect
34. C4×C4⋊C8C22semidirect
35. C4×C4⋊C8C22semidirect
37. #1744C22semidirect
38. C42×C8C22semidirect
43. C2×D4⋊C8C22semidirect
44. C2×D4⋊C8C22semidirect
45. C2×D4⋊C8C22semidirect
46. C2×D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#349362
#214302
#208122
#96782
#96742
#96442
#96382
#83132
#83102