Finite Group Databasebeta

← #7623930

Subgroups — #7623930

Order 512.

1234 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#61372C211yesmaximal, characteristic, complemented
2#61370C211yesmaximal, characteristic
3#33955C214yesmaximal, complemented
4#33953C212yesmaximal, complemented
5#33952C212yesmaximal, complemented
6#33948C212yesmaximal, complemented
7#33948C214yesmaximal, complemented
8Q8×SD32C212yesmaximal, complemented
9#33730C212yesmaximal, complemented
10#33726C211yesmaximal, characteristic, complemented
11#33632C212yesmaximal, complemented
12#33631C212yesmaximal
13#33628C211yesmaximal, characteristic, complemented
14#33627C211yesmaximal, characteristic
15#33474C211yesmaximal, characteristic, complemented
16#33457C212yesmaximal
17C2×C4×SD32C211yesmaximal, characteristic
18#3502—21no—
19#3501—21no—
20C22×SD32C2211yescharacteristic
21#3390—22no—
22#3389—22no—
23#3387—22no—
24#3384—22no—
25#3383C2212yescomplemented
26#3383C2214yescomplemented
27Q8×Q16C2212yescomplemented
28#3380C2212yes—
29#3380C2214yes—
30Q8×D8C2212yes—
31#3164C2212yes—
32#3164C2212yes—
33#3160C2211yescharacteristic
34#3160C2211yescharacteristic
35#3082—22no—
36#3081—22no—
37#3078—21no—
38#3077—21no—
39#2960C2211yescharacteristic
40#2950C2212yes—
41#2948C2212yes—
42C2×C4×Q16C2211yescharacteristic
43C2×C4×D8C2211yescharacteristic
44#2255C2214yescomplemented
45#2255C2214yescomplemented
46#2254C2211yescharacteristic, complemented
47#2254C2214yescomplemented
48#2254C2211yescharacteristic, complemented
49#2253C2212yescomplemented
50#2253C2214yescomplemented
51#2253C2212yescomplemented
52#2252C2211yescharacteristic, complemented
53#2252C2211yescharacteristic, complemented
54#2229C2214yescomplemented
55#2229C2214yescomplemented
56#2229C2214yescomplemented
57#2228C2214yes—
58#2228C2214yes—
59#2228C2214yes—
60#2227C2212yescomplemented
61#2227C2214yescomplemented
62#2226C2212yes—
63#2226C2214yes—
64#2225C2212yescomplemented
65#2225C2214yescomplemented
66#2224C2212yes—
67#2224C2214yes—
68#2183C2212yescomplemented
69#2183C2212yescomplemented
70Q8×C16C2212yescomplemented
71Q8×C16C2212yescomplemented
72#2175C2214yes—
73#2175C2214yes—
74C4×SD32C2212yes—
75C4×SD32C2214yes—
76C4×SD32C2212yes—
77C4×SD32C2214yes—
78C4×SD32C2212yes—
79C4×SD32C2212yes—
80#2159C2212yes—
81#2159C2212yes—
82#2158C2212yes—
83#2158C2212yes—
84C2×#367C2211yescharacteristic
85#2151C2212yes—
86#2150C2211yescharacteristic
87#2141C2212yes—
88#2140C2212yes—
89C2×#358C2211yescharacteristic
90C2×#357C2211yescharacteristic
91#2108C2212yes—
92C2×C4×C16C2211yescharacteristic
93C22×Q16C2311yescharacteristic
94C22×D8C2311yescharacteristic
95Q82—22no—
96Q82—41no—
97#557—22no—
98#557—24no—
99#557—41no—
100#557—42no—
101#554—22no—
102#554—24no—
103#554—41no—
104#554—42no—
105D4×Q8—22no—
106D4×Q8—41no—
107#544—22no—
108#544—22no—
109#544—42no—
110#543—42no—
111#533—21no—
112#533—21no—
113#533—42no—
114#533D411yescharacteristic
115C2×C22⋊Q8—42no—
116#519—22no—
117#518—22no—
118C2×C4×Q8—21no—
119C2×C4×D4—21no—
120C2×SD32Q818yescomplemented
121C2×SD32C2312yes—
122C2×SD32C2312yes—
123C2×M32C2312yes—
124C22×C16C2311yescharacteristic
125C22×C16C2311yescharacteristic
126#501—24no—
127#501—24no—
128#500C2311yescharacteristic
129#500C2314yes—
130#500C2311yescharacteristic
131#500C2311yescharacteristic
132#500C2314yes—
133#500C2311yescharacteristic
134#499—21no—
135#499—21no—
136#499C2312yes—
137#499C2314yes—
138#499C2312yes—
139#499C2314yes—
140#498—21no—
141#498—21no—
142#479—24no—
143#479—24no—
144#479—24no—
145#478—24no—
146#478—24no—
147#478—24no—
148#477—22no—
149#477—24no—
150#476—22no—
151#476—24no—
152#475—22no—
153#475—24no—
154#474—22no—
155#474—24no—
156#446D412yes—
157#446D412yes—
158C8×Q8C2312yes—
159C8×Q8C2312yes—
160#442—24no—
161#441—24no—
162C4×Q16—21no—
163C4×Q16—21no—
164C4×Q16C2312yes—
165C4×Q16C2314yes—
166C4×Q16C2312yes—
167C4×Q16C2314yes—
168C4×D8—21no—
169C4×D8—21no—
170C4×D8C2312yes—
171C4×D8C2314yes—
172C4×D8C2312yes—
173C4×D8C2314yes—
174#428—22no—
175#428—22no—
176#427—21no—
177#427—21no—
178#427C2312yes—
179#427C2312yes—
180C2×#107C2311yescharacteristic
181C2×#107C2311yescharacteristic
182#424D412yes—
183#423D411yescharacteristic
184#419—22no—
185#418—22no—
186C2×#103—21no—
187C2×#102—21no—
188#405C2312yes—
189C2×C4×C8C2311yescharacteristic
190#367C2312yes—
191#367C2312yes—
192#367C2312yes—
193#367C2312yes—
194#367C2312yes—
195#367C2312yes—
196#367C2314yes—
197#367C2314yes—
198#366C2312yes—
199#366C2312yes—
200#366C2312yes—
201#366C2312yes—
202#366C2314yes—
203#366C2314yes—
204C4⋊C16C2312yes—
205C4⋊C16C2312yes—
206C4⋊C16C2314yes—
207C4⋊C16C2314yes—
208#358C2312yes—
209#358C2312yes—
210#358C2314yes—
211#358C2314yes—
212#357C2312yes—
213#357C2312yes—
214#357C2314yes—
215#357C2314yes—
216#346C2314yes—
217C4×C16C2311yescharacteristic
218C4×C16C2311yescharacteristic
219C4×C16C2311yescharacteristic
220C4×C16C2311yescharacteristic
221C4×C16C2314yes—
222C22×Q8—21no—
223C22×Q8—42no—
224C22×D4—21no—
225C23×C4—41no—
226C2×Q16—21no—
227C2×Q16—22no—
228C2×Q16C2×Q814yescomplemented
229C2×Q16C2412yes—
230C2×D8—21no—
231C2×D8—22no—
232C2×D8C2×Q814yes—
233C2×D8C2412yes—
234C2×M16C2×D412yes—
235C22×C8C2×D411yescharacteristic
236C22×C8C2411yescharacteristic
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—24no—
242C4⋊Q8—24no—
243C4⋊Q8—42no—
244C4⋊Q8—44no—
245C4⋊Q8—42no—
246C4⋊Q8—42no—
247C4⋊Q8—42no—
248C4⋊Q8—44no—
249C4⋊Q8D811yescharacteristic
250C4⋊Q8D811yescharacteristic
251#125—22no—
252#125—22no—
253#125—24no—
254#125—24no—
255#125—44no—
256#125—42no—
257#125—42no—
258#125—44no—
259#125—44no—
260#125D812yes—
261C22⋊Q8—42no—
262C22⋊Q8—44no—
263C22⋊Q8—41no—
264C22⋊Q8—41no—
265C22⋊Q8—41no—
266C22⋊Q8—41no—
267C22⋊Q8—42no—
268C22⋊Q8—44no—
269C22⋊Q8—44no—
270C22⋊Q8—44no—
271C22⋊Q8—44no—
272C22⋊Q8—44no—
273C22⋊Q8—44no—
274C22⋊Q8D812yes—
275C22⋊Q8D812yes—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—42no—
281C4×Q8—42no—
282C4×Q8—42no—
283C4×Q8—44no—
284C4×Q8—42no—
285C4×Q8—41no—
286C4×Q8—41no—
287C4×Q8—41no—
288C4×Q8—41no—
289C4×Q8—41no—
290C4×Q8—41no—
291C4×Q8C2×D412yes—
292C4×Q8C2×D412yes—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—42no—
298C4×D4—42no—
299C4×D4—41no—
300C4×D4—41no—
301#117—22no—
302#117—22no—
303#117—44no—
304#117C2×D412yes—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—41no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—42no—
310C2×C22⋊C4—41no—
311C2×C22⋊C4—44no—
312C2×C42—41no—
313C2×C42C2×D411yescharacteristic
314SD32C2×Q818yescomplemented
315SD32C2×Q818yescomplemented
316M32—24no—
317C2×C16—22no—
318C2×C16C2×Q814yescomplemented
319C2×C16C2411yescharacteristic
320C2×C16C2411yescharacteristic
321C2×C16C2414yes—
322C2×C16C2411yescharacteristic
323C2×C16C2411yescharacteristic
324#107—21no—
325#107—21no—
326#107—21no—
327#107—21no—
328#107—21no—
329#107—21no—
330#107—24no—
331#107—24no—
332#107C2412yes—
333#107C2412yes—
334#107C2412yes—
335#107C2412yes—
336#107C2412yes—
337#107C2412yes—
338#106—22no—
339#106—22no—
340#106—22no—
341#106—22no—
342#106—24no—
343#106—24no—
344C4⋊C8C2×D412yes—
345C4⋊C8C2×D412yes—
346C4⋊C8C2×D414yes—
347C4⋊C8C2×D414yes—
348#103—22no—
349#103—22no—
350#103—24no—
351#103—24no—
352#102—22no—
353#102—22no—
354#102—24no—
355#102—24no—
356#97C2×D414yes—
357C4×C8C2×D411yescharacteristic
358C4×C8C2×D411yescharacteristic
359C4×C8C2411yescharacteristic
360C4×C8C2411yescharacteristic
361C4×C8C2414yes—
362C24—41no—
363C2×Q8—24no—
364C2×Q8—22no—
365C2×Q8—41no—
366C2×Q8—41no—
367C2×Q8—41no—
368C2×Q8—41no—
369C2×Q8—41no—
370C2×Q8—44no—
371C2×Q8—44no—
372C2×Q8—42no—
373C2×Q8—164no—
374C2×Q8C2×D812yes—
375C2×Q8C2×D812yes—
376C2×D4—24no—
377C2×D4—22no—
378C2×D4—41no—
379C2×D4—42no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—42no—
387C22×C4—162no—
388C22×C4C2×D811yescharacteristic
389C22×C4C2×D811yescharacteristic
390C22×C4C22×D411yescharacteristic
391Q16—24no—
392Q16—22no—
393Q16C22×Q814yes—
394D8—24no—
395D8—22no—
396D8C22×Q814yes—
397M16—24no—
398C2×C8—22no—
399C2×C8C22×D411yescharacteristic
400C2×C8C22×D411yescharacteristic
401C2×C8C22×D414yes—
402C2×C8C22×D411yescharacteristic
403C2×C8C22×Q812yes—
404C2×C82−1+412yes—
405C2×C8C2511yescharacteristic
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—42no—
415C4⋊C4—41no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—42no—
424C4⋊C4—41no—
425C4⋊C4—42no—
426C4⋊C4—41no—
427C4⋊C4—44no—
428C4⋊C4—42no—
429C4⋊C4—44no—
430C4⋊C4—41no—
431C4⋊C4—44no—
432C4⋊C4C2×D814yes—
433C4⋊C4C2×D812yes—
434C4⋊C4C2×D812yes—
435C4⋊C4C2×D812yes—
436C4⋊C4C2×D812yes—
437C4⋊C4C2×D814yes—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4C2×D814yes—
445C42—42no—
446C42—42no—
447C42—42no—
448C42—42no—
449C42—44no—
450C42—44no—
451C42C2×D811yescharacteristic
452C42C2×D811yescharacteristic
453C42C22×D411yescharacteristic
454C42C22×D411yescharacteristic
455C42C22×D414yes—
456C16—21no—
457C16—21no—
458C16—24no—
459C16C22×Q812yes—
460C16C22×Q812yes—
461C23—42no—
462C23—44no—
463C23—161no—
464C23—161no—
465C23C22×D811yescharacteristic
466Q8—24no—
467Q8—24no—
468Q8—42no—
469Q8—44no—
470Q8—42no—
471Q8—82no—
472Q8—82no—
473Q8—161no—
474Q8—161no—
475Q8—164no—
476Q8—164no—
477D4—24no—
478D4—44no—
479D4—42no—
480C2×C4—22no—
481C2×C4—41no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—84no—
491C2×C4—164no—
492C2×C4—162no—
493C2×C4—164no—
494C2×C4—161no—
495C2×C4—161no—
496C2×C4D4×Q812yes—
497C2×C4C22×D811yescharacteristic
498C2×C4C22×D814yes—
499C2×C4C22×D814yes—
500C2×C4C22×D811yescharacteristic
501C2×C4C22×D811yescharacteristic
502C2×C4C22×D811yescharacteristic
503C2×C4C22×D811yescharacteristic
504C2×C4#57812yes—
505C2×C4C23×D411yescharacteristic
506C8—21no—
507C8—21no—
508C8—24no—
509C8D4×Q812yes—
510C8C23×Q811yescharacteristic
511C8C2×2−1+411yescharacteristic
512C22—21no—
513C22—21no—
514C22—44no—
515C22—82no—
516C22—82no—
517C22—164no—
518C22Q8×D812yes—
519C22C23×D811yescharacteristic
520C4—21no—
521C4—24no—
522C4—21no—
523C4—21no—
524C4—24no—
525C4—44no—
526C4—82no—
527C4—82no—
528C4—82no—
529C4—82no—
530C4—82no—
531C4—82no—
532C4Q8×D812yes—
533C4#341612yes—
534C4C2×D4×Q811yescharacteristic
535C4C2×#57811yescharacteristic
536C2—22no—
537C2—84no—
538C2C2×Q8×D811yescharacteristic
539C2C2×#341611yescharacteristic
540C2#6263511yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radicalC22×D8C23
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ #533 ◁ #2960 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #61372C2semidirect
3. #33955C2semidirect
4. #33953C2semidirect
5. #33952C2semidirect
6. #33948C2semidirect
7. #33948C2semidirect
8. Q8×SD32C2semidirect
9. #33730C2semidirect
10. #33726C2semidirect
11. #33632C2semidirect
13. #33628C2semidirect
15. #33474C2semidirect
25. #3383C22semidirect
26. #3383C22semidirect
27. Q8×Q16C22semidirect
44. #2255C22semidirect
45. #2255C22semidirect
46. #2254C22semidirect
47. #2254C22semidirect
48. #2254C22semidirect
49. #2253C22semidirect
50. #2253C22semidirect
51. #2253C22semidirect
52. #2252C22semidirect
53. #2252C22semidirect
54. #2229C22semidirect
55. #2229C22semidirect
56. #2229C22semidirect
60. #2227C22semidirect
61. #2227C22semidirect
64. #2225C22semidirect
65. #2225C22semidirect
68. #2183C22semidirect
69. #2183C22semidirect
70. Q8×C16C22semidirect
71. Q8×C16C22semidirect
120. C2×SD32Q8semidirect
228. C2×Q16C2×Q8semidirect
314. SD32C2×Q8semidirect
315. SD32C2×Q8semidirect
318. C2×C16C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#613722
#613702
#339552
#339532
#339522
#339482
Q8×SD322
#337302
#337262
#336322
#336312
#336282
#336272
#334742
#334572
C2×C4×SD322