Finite Group Databasebeta

← #220186

Subgroups — #220186

Order 512.

1146 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2946C211yesmaximal, characteristic
2#17854C212yesmaximal, complemented
3#17351C211yesmaximal, characteristic, complemented
4#17350C211yesmaximal, characteristic, complemented
5#9667C212yesmaximal, complemented
6#9089C212yesmaximal, complemented
7#9087C212yesmaximal, complemented
8#8602C211yesmaximal, characteristic, complemented
9#8324C211yesmaximal, characteristic, complemented
10#8322C211yesmaximal, characteristic, complemented
11#8156C211yesmaximal, characteristic, complemented
12C22×AGL(1,ℤ/8)C411yescharacteristic
13C2×#520C2211yescharacteristic
14C2×#518C2211yescharacteristic
15#2972C2212yes—
16#2972C2212yescomplemented
17#2946—24no—
18#2946—24no—
19C2×#442C411yescharacteristic
20C2×#442C2211yescharacteristic
21C2×#440C411yescharacteristic
22C2×#440C2211yescharacteristic
23C2×C4×SD16C411yescharacteristic
24C2×C4×SD16C2211yescharacteristic
25C2×C4×D8C411yescharacteristic
26C2×C4×D8C2211yescharacteristic
27C2×#428C411yescharacteristic
28C2×#416C411yescharacteristic
29C22×#102C411yescharacteristic
30C2×#405C2211yescharacteristic
31#1580—24no—
32C8×D8—24no—
33#1564C2212yescomplemented
34#1479C2212yescomplemented
35#1478C2212yes—
36#1478C2212yes—
37#1478C2212yescomplemented
38#1461C2212yescomplemented
39#1460C2212yes—
40#1459—22no—
41#1459—22no—
42#1458C2211yescharacteristic
43#1458C2211yescharacteristic
44#1458C2211yescharacteristic
45#1458C2211yescharacteristic
46C2×#324—22no—
47C2×#323—22no—
48#1450C2212yescomplemented
49#1281C2212yescomplemented
50#1275C2212yescomplemented
51C22×#41C2×C411yescharacteristic
52C23×D4C2×C411yescharacteristic
53C2×AGL(1,ℤ/8)—24no—
54C2×AGL(1,ℤ/8)—24no—
55C2×AGL(1,ℤ/8)C818yescomplemented
56C22×SD16C2×C411yescharacteristic
57C22×SD16C2×C411yescharacteristic
58C22×D8C2×C411yescharacteristic
59C22×D8C2×C411yescharacteristic
60C22×M16C2×C411yescharacteristic
61C23×C8C2×C411yescharacteristic
62#520—24no—
63#520—42no—
64#518—24no—
65#518—42no—
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8C2311yescharacteristic
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C2×C4×D4C2311yescharacteristic
78C2×#117—21no—
79C2×#117C2×C411yescharacteristic
80C2×#117C2311yescharacteristic
81C22×C4⋊C4—21no—
82C22×C4⋊C4C2×C411yescharacteristic
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85#442—24no—
86#442—24no—
87#440—24no—
88#440—24no—
89C4×SD16—22no—
90C4×SD16—22no—
91C4×SD16—22no—
92C4×SD16—22no—
93C4×D8—22no—
94C4×D8—22no—
95C4×D8D414yes—
96C4×D8D414yes—
97C8×D4—24no—
98C8×D4—24no—
99C8×D4—42no—
100C8×D4—42no—
101#431C2312yes—
102#431C2312yes—
103#431C2312yes—
104#431C2312yes—
105#428—24no—
106C2×#107C2×C411yescharacteristic
107C2×#107C2×C411yescharacteristic
108C2×#106C2×C411yescharacteristic
109C2×#106C2×C411yescharacteristic
110#416—22no—
111#416—22no—
112C2×#103C2×C411yescharacteristic
113C2×#103C2×C411yescharacteristic
114C2×#102—22no—
115C2×#102—22no—
116C2×#102C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120C2×#102C2×C411yescharacteristic
121C2×#102C2×C411yescharacteristic
122C2×C22⋊C8—24no—
123C2×C22⋊C8—24no—
124#405—22no—
125#405—22no—
126C2×#97C2×C411yescharacteristic
127C2×#97C2311yescharacteristic
128C2×C4×C8C2×C411yescharacteristic
129C2×C4×C8C2311yescharacteristic
130#335—22no—
131#334—22no—
132Q8⋊C8—42no—
133D4⋊C8—24no—
134D4⋊C8—24no—
135D4⋊C8—42no—
136#324—21no—
137#324—21no—
138#324—21no—
139#324—24no—
140#324D412yes—
141#323—21no—
142#323—21no—
143#323—21no—
144#323—24no—
145#323D412yes—
146#322—22no—
147C82—22no—
148C25—21no—
149C2×#41—22no—
150C2×#41—22no—
151C2×#41—22no—
152C2×#41—41no—
153C2×#41—41no—
154C2×#41C2×C814yescomplemented
155C22×Q8C22×C411yescharacteristic
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—22no—
161C22×D4—41no—
162C22×D4—41no—
163C22×D4C2×C814yescomplemented
164C22×D4C22×C411yescharacteristic
165C22×D4C22×C411yescharacteristic
166C22×D4C22×C411yescharacteristic
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4C22×C411yescharacteristic
176AGL(1,ℤ/8)—232no—
177C2×SD16—22no—
178C2×SD16—22no—
179C2×SD16—22no—
180C2×SD16—22no—
181C2×SD16C2×C814yescomplemented
182C2×SD16C2×C814yescomplemented
183C2×D8—22no—
184C2×D8—22no—
185C2×D8C22⋊C414yes—
186C2×D8C2×C814yes—
187C2×D8C2×C814yescomplemented
188C2×D8M1614yes—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16C2×C814yescomplemented
192C22×C8—22no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—41no—
196C22×C8—44no—
197C22×C8—44no—
198C22×C8C22×C411yescharacteristic
199C22×C8C22×C411yescharacteristic
200C22×C8C22×C411yescharacteristic
201C22×C8C22×C411yescharacteristic
202C4×Q8—22no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—41no—
206C4×D4—22no—
207C4×D4—22no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4—22no—
212C4×D4—22no—
213C4×D4—41no—
214C4×D4—41no—
215C4×D4—41no—
216C4×D4—44no—
217C4×D4—44no—
218C4×D4—44no—
219C4×D4—44no—
220C4×D4C2×D414yes—
221C4×D4C2×D414yes—
222#117—22no—
223#117—22no—
224#117—22no—
225#117—44no—
226#117—41no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4C22×C411yescharacteristic
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4#4111yescharacteristic
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4C2×D411yescharacteristic
249C2×C22⋊C4#4111yescharacteristic
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42C22×C411yescharacteristic
254C2×C42C2411yescharacteristic
255#107—22no—
256#107C22⋊C414yes—
257#106—22no—
258#106—22no—
259C4⋊C8—22no—
260C4⋊C8—22no—
261C4⋊C8—22no—
262C4⋊C8—22no—
263#103—21no—
264#103—21no—
265#103—22no—
266#102—22no—
267#102—22no—
268#102—21no—
269#102—21no—
270#102—22no—
271#102C22⋊C412yes—
272#102C22⋊C412yes—
273#102C22⋊C412yes—
274#102C22⋊C412yes—
275C22⋊C8—22no—
276C22⋊C8—22no—
277C22⋊C8—22no—
278C22⋊C8—44no—
279C22⋊C8—44no—
280C22⋊C8C2×D414yes—
281#97—22no—
282#97—22no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—22no—
286C4×C8—22no—
287C4×C8—22no—
288C4×C8—22no—
289C4×C8C2×D412yes—
290C4×C8C2×D412yes—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—24no—
295C24—42no—
296C24—42no—
297C24—161no—
298C24—161no—
299C24C4×D411yescharacteristic
300#41—28no—
301#41—44no—
302#41—44no—
303#41—44no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8C22×C812yes—
310C2×D4—28no—
311C2×D4—24no—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—24no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4C2×C22⋊C412yes—
332C2×D4C2×C22⋊C412yes—
333C2×D4C22×C812yes—
334C2×D4C22×C812yes—
335C2×D4C22×C812yes—
336C2×D4C2×M1612yes—
337C2×D4C2×M1612yes—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—161no—
358C22×C4—161no—
359C22×C4—161no—
360C22×C4—161no—
361C22×C4—161no—
362C22×C4—161no—
363C22×C4—161no—
364C22×C4C4×D411yescharacteristic
365C22×C4C4×D411yescharacteristic
366C22×C4C4×D411yescharacteristic
367C22×C4C22×C812yes—
368C22×C4#13112yes—
369C22×C4C23×C411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C2×#4111yescharacteristic
372SD16—28no—
373SD16—28no—
374D8—28no—
375D8C22⋊C8116yes—
376M16—28no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—24no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—164no—
391C2×C8—164no—
392C2×C8C2×C22⋊C411yescharacteristic
393C2×C8C2×C22⋊C411yescharacteristic
394C2×C8C2×C22⋊C412yes—
395C2×C8C22×C812yes—
396C2×C8C22×C812yes—
397C2×C8C2×M1612yes—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4C2×C22⋊C412yes—
410C4⋊C4C2×C22⋊C412yes—
411C22⋊C4—22no—
412C22⋊C4—24no—
413C22⋊C4—24no—
414C22⋊C4—22no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—42no—
419C22⋊C4—42no—
420C42—21no—
421C42—21no—
422C42—42no—
423C42—42no—
424C42—42no—
425C42C22×D412yes—
426C42H(5,𝔽2)12yes—
427C23—21no—
428C23—22no—
429C23—22no—
430C23—21no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—48no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—81no—
442C23—81no—
443C23—81no—
444C23—81no—
445C23—81no—
446C23—81no—
447C23—162no—
448C23—162no—
449C23—164no—
450C23—164no—
451C23C8×D412yes—
452C23C2×C4×D411yescharacteristic
453Q8—22no—
454Q8—22no—
455Q8—41no—
456Q8—41no—
457D4—22no—
458D4—22no—
459D4—22no—
460D4—22no—
461D4—22no—
462D4—22no—
463D4—22no—
464D4—22no—
465D4—48no—
466D4—48no—
467D4—41no—
468D4—41no—
469D4C2×C22⋊C814yes—
470D4C2×C22⋊C814yes—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—48no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—84no—
492C2×C4—84no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—162no—
498C2×C4—162no—
499C2×C4—162no—
500C2×C4—162no—
501C2×C4—162no—
502C2×C4C8×D412yes—
503C2×C4C22×C22⋊C411yescharacteristic
504C2×C4#51811yescharacteristic
505C2×C4D4212yes—
506C2×C4#54612yes—
507C2×C4C23×C811yescharacteristic
508C2×C4C22×M1611yescharacteristic
509C2×C4C2×#13111yescharacteristic
510C2×C4#56811yescharacteristic
511C8—22no—
512C8—21no—
513C8—21no—
514C8—44no—
515C8—41no—
516C8—82no—
517C8—82no—
518C8—82no—
519C8C2×C22⋊C814yes—
520C22—21no—
521C22—21no—
522C22—22no—
523C22—24no—
524C22—24no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—41no—
530C22—42no—
531C22—41no—
532C22—42no—
533C22—42no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22—81no—
541C22—81no—
542C22—168no—
543C22D4×C22⋊C411yescharacteristic
544C22C2×C8×D411yescharacteristic
545C22D4×M1611yescharacteristic
546C4—22no—
547C4—22no—
548C4—22no—
549C4—42no—
550C4—41no—
551C4—42no—
552C4—42no—
553C4—41no—
554C4—81no—
555C4—81no—
556C4C22×C22⋊C811yescharacteristic
557C4#288011yescharacteristic
558C4#297211yescharacteristic
559C4#297411yescharacteristic
560C2—21no—
561C2—21no—
562C2—42no—
563C2—44no—
564C2—44no—
565C2—82no—
566C2D4×C22⋊C811yescharacteristic
567C2#2245511yescharacteristic
568C2#2245911yescharacteristic
subgroupG/H
center, socleC22D4×C22⋊C4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
prime radicalC22×AGL(1,ℤ/8)C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17854C2semidirect
3. #17351C2semidirect
4. #17350C2semidirect
5. #9667C2semidirect
6. #9089C2semidirect
7. #9087C2semidirect
8. #8602C2semidirect
9. #8324C2semidirect
10. #8322C2semidirect
11. #8156C2semidirect
16. #2972C22semidirect
33. #1564C22semidirect
34. #1479C22semidirect
37. #1478C22semidirect
38. #1461C22semidirect
48. #1450C22semidirect
49. #1281C22semidirect
50. #1275C22semidirect
55. C2×AGL(1,ℤ/8)C8semidirect
154. C2×#41C2×C8semidirect
163. C22×D4C2×C8semidirect
181. C2×SD16C2×C8semidirect
182. C2×SD16C2×C8semidirect
187. C2×D8C2×C8semidirect
191. C2×M16C2×C8semidirect

Maximal subgroups

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

subgroupindex
C2×#29462
#178542
#173512
#173502
#96672
#90892
#90872
#86022
#83242
#83222
#81562