Finite Group Databasebeta

← #7623277

Subgroups — #7623277

Order 512.

1422 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60718C211yesmaximal, characteristic, complemented
2#60714C211yesmaximal, characteristic, complemented
3#33751C212yesmaximal, complemented
4#33748C212yesmaximal, complemented
5#33744C218yesmaximal, complemented
6#33736C212yesmaximal, complemented
7#33733C212yesmaximal, complemented
8#33731C211yesmaximal, characteristic, complemented
9#33717C211yesmaximal, characteristic, complemented
10#33697C211yesmaximal, characteristic, complemented
11C2×#2240C211yesmaximal, characteristic, complemented
12#33612C214yesmaximal, complemented
13#33454C214yesmaximal, complemented
14C4×#504C211yesmaximal, characteristic, complemented
15#3511—21no—
16#3508—21no—
17C2×#509C2214yes—
18C2×#504C2212yes—
19#3184—22no—
20#3182—22no—
21#3176—22no—
22#3176C2214yes—
23#3175—22no—
24#3175C2214yescomplemented
25#3170C2212yescomplemented
26#3169C2212yes—
27#3169C2212yes—
28#3169C2212yescomplemented
29#3160C2211yescharacteristic
30C2×#500C2211yescharacteristic
31#3147C2211yescharacteristic
32C2×#493C2211yescharacteristic
33#3141—21no—
34#3138—21no—
35#3061—21no—
36#3061C2212yes—
37#3058—21no—
38#3058C2212yes—
39#2951—21no—
40#2951C2212yes—
41#2948—21no—
42#2948C2212yes—
43C4×#131C2211yescharacteristic
44#2255C2214yescomplemented
45#2254C2212yescomplemented
46#2252C2212yescomplemented
47#2250C2214yes—
48#2248C2212yes—
49#2246C2212yes—
50#2242C2214yescomplemented
51#2241C2211yescharacteristic, complemented
52#2241C2211yescharacteristic, complemented
53#2240C2212yes—
54#2240C2214yes—
55#2240C2212yes—
56#2240C2211yescharacteristic, complemented
57#2240C2211yescharacteristic, complemented
58#2220C2218yescomplemented
59#2218C2218yescomplemented
60#2217C2218yescomplemented
61#2215C2218yescomplemented
62#2177C2218yescomplemented
63#2175C2218yescomplemented
64C4×SD32C2218yescomplemented
65C4×D16C2218yescomplemented
66#2159C2214yes—
67#2138C2214yes—
68C2×#357C2214yes—
69#2108C2212yes—
70#2108C2214yescomplemented
71C4×M32C2211yescharacteristic
72C4×M32C2212yescomplemented
73C2×C4×C16C2211yescharacteristic
74C2×C4×C16C2212yescomplemented
75C2×#135—21no—
76C2×#135C2312yes—
77C22×D8—21no—
78C22×D8C2312yes—
79C2×#131D411yescharacteristic
80C2×#131C2311yescharacteristic
81#559—41no—
82#559—41no—
83#556—41no—
84#555—41no—
85#554—41no—
86#553—42no—
87#550—41no—
88#550—41no—
89#548—42no—
90#547—41no—
91#546—42no—
92#546—42no—
93#542—24no—
94#540—24no—
95#537—22no—
96#535—22no—
97#533—21no—
98#533—22no—
99#532—22no—
100#532—22no—
101#532—22no—
102#532—21no—
103C2×C4⋊Q8—21no—
104C2×#127—21no—
105#520—22no—
106#518—22no—
107C4×#41D411yescharacteristic
108#509—216no—
109C2×SD32C2318yes—
110C2×D16C2318yes—
111#504—28no—
112C2×M32C2314yes—
113C2×M32C2312yes—
114C22×C16C2312yes—
115C22×C16C2314yes—
116#500—21no—
117#500—21no—
118#500C2312yes—
119#500C2312yes—
120#500C2311yescharacteristic
121#500C2311yescharacteristic
122#499—21no—
123#499C2312yes—
124#498—21no—
125#495—21no—
126#495C2312yes—
127#494C2311yescharacteristic
128#493—21no—
129#493—21no—
130#493C2312yes—
131#493C2312yes—
132#493C2311yescharacteristic
133#492—21no—
134#490—24no—
135#489—24no—
136#487—22no—
137#486—22no—
138#486—22no—
139#486—22no—
140#469—24no—
141#469—24no—
142#468—24no—
143#468—24no—
144#467—22no—
145#467C2314yes—
146#466—22no—
147#466—22no—
148#466—22no—
149#466C2314yes—
150#466C2314yes—
151#466C2314yes—
152#442—24no—
153#442—24no—
154#440—24no—
155#440—24no—
156C4×Q16—22no—
157C4×Q16C2314yes—
158C4×D8—22no—
159C4×D8—22no—
160C4×D8—22no—
161C4×D8C2314yes—
162C4×D8C2314yes—
163C4×D8C2314yes—
164#427—21no—
165#427C2312yes—
166C2×#107—21no—
167C2×#107C2312yes—
168#418—22no—
169#418—22no—
170#417—22no—
171#417—22no—
172#405D412yes—
173#405D411yescharacteristic
174#405C2311yescharacteristic
175#405C2312yes—
176C4×M16D411yescharacteristic
177C4×M16D412yes—
178C2×C4×C8C2311yescharacteristic
179C2×C4×C8C2312yes—
180#367C2318yes—
181#366C2318yes—
182#358C2314yes—
183#358C2314yes—
184#357C2318yes—
185#357C2318yes—
186#357C2314yes—
187#357C2314yes—
188#346C2312yes—
189#346C2314yes—
190C4×C16C2311yescharacteristic
191C4×C16C2311yescharacteristic
192C4×C16C2312yes—
193C4×C16C2312yes—
194C4×C16C2314yes—
195C2×#41—22no—
196C2×#41—42no—
197C2×#41—42no—
198C2×#41—42no—
199C2×#41C2×D411yescharacteristic
200C22×Q8—41no—
201C22×D4—22no—
202C22×D4—41no—
203C22×D4—41no—
204#135—28no—
205#135—44no—
206C2×Q16—21no—
207C2×Q16C2412yes—
208C2×SD16—22no—
209C2×SD16—22no—
210C2×D8—22no—
211C2×D8—22no—
212C2×D8—22no—
213C2×D8—21no—
214C2×D8—28no—
215C2×D8—44no—
216C2×D8C2414yes—
217C2×D8C2412yes—
218#131—24no—
219#131—24no—
220C2×M16C2×D411yescharacteristic
221C2×M16C2×D412yes—
222C2×M16C2×D411yescharacteristic
223C2×M16C2×D412yes—
224C22×C8C2×D412yes—
225C22×C8C2×D411yescharacteristic
226C22×C8C2412yes—
227C22×C8C2411yescharacteristic
228C4⋊Q8—22no—
229C4⋊Q8—22no—
230C4⋊Q8—21no—
231C4⋊Q8—21no—
232C4⋊Q8—21no—
233C4⋊Q8—41no—
234C4⋊Q8—41no—
235#127—22no—
236#127—22no—
237#127—21no—
238#127—42no—
239#127—41no—
240#127—41no—
241#126—44no—
242#125—22no—
243#125—42no—
244#124—22no—
245#124—42no—
246#124—44no—
247#124—42no—
248#124—42no—
249#124—42no—
250#123—44no—
251#123—44no—
252#123—44no—
253C22⋊Q8—24no—
254C22⋊Q8—42no—
255C22⋊Q8—41no—
256C22⋊Q8—41no—
257C22⋊Q8—42no—
258C22⋊Q8—44no—
259#121—24no—
260#121—24no—
261#121—24no—
262#121—42no—
263#121—44no—
264#121—42no—
265#121—42no—
266#121—41no—
267#121—41no—
268#121—42no—
269#121—44no—
270#121—44no—
271#120—44no—
272C4×Q8—24no—
273C4×Q8—42no—
274C4×Q8—41no—
275C4×Q8—41no—
276C4×Q8D811yescharacteristic
277C4×Q8D811yescharacteristic
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—42no—
282C4×D4—42no—
283C4×D4—44no—
284C4×D4—42no—
285C4×D4—42no—
286C4×D4—41no—
287C4×D4—42no—
288C4×D4—41no—
289C4×D4D812yes—
290C4×D4D812yes—
291C4×D4D811yescharacteristic
292C4×D4D811yescharacteristic
293#117—22no—
294#117—42no—
295#117—42no—
296#117C2×D411yescharacteristic
297#117C2×D412yes—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—41no—
300C2×C4⋊C4—42no—
301C2×C22⋊C4—44no—
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D412yes—
304SD32—216no—
305D16—216no—
306M32—28no—
307M32#4118yescomplemented
308C2×C16—24no—
309C2×C16—28no—
310C2×C16C2412yes—
311C2×C16C2412yes—
312C2×C16C2414yes—
313#107—22no—
314#107—22no—
315#107—22no—
316#107—21no—
317#107—21no—
318#107C2414yes—
319#107C2412yes—
320#107C2412yes—
321#106—22no—
322#106—22no—
323#103—24no—
324#103—24no—
325#102—24no—
326#102—24no—
327#102—24no—
328#102—24no—
329#102—24no—
330#102—24no—
331#97C2×D412yes—
332#97C2×D412yes—
333#97C2×D412yes—
334C4×C8C2×D412yes—
335C4×C8C2×D411yescharacteristic
336C4×C8C2×D412yes—
337C4×C8C2×D411yescharacteristic
338C4×C8C2411yescharacteristic
339C4×C8C2412yes—
340C4×C8C2411yescharacteristic
341C24—42no—
342#41—22no—
343#41—22no—
344#41—44no—
345#41—44no—
346#41—164no—
347#41—164no—
348#41—164no—
349C2×Q8—22no—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8—162no—
355C2×Q8C2×D811yescharacteristic
356C2×D4—22no—
357C2×D4—24no—
358C2×D4—42no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—41no—
364C2×D4—42no—
365C2×D4—41no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—44no—
371C2×D4—162no—
372C2×D4—162no—
373C2×D4C2×D812yes—
374C2×D4C2×D811yescharacteristic
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4C2×D811yescharacteristic
381C22×C4C2×D812yes—
382C22×C4C2×D811yescharacteristic
383C22×C4C2×D812yes—
384C22×C4C22×D412yes—
385C22×C4C22×D411yescharacteristic
386Q16—24no—
387Q16—42no—
388SD16—44no—
389SD16—44no—
390D8—24no—
391D8—28no—
392D8—44no—
393D8—44no—
394D8—44no—
395D8—42no—
396M16—22no—
397M16—22no—
398M16—24no—
399M16—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8#13514yes—
403C2×C8C22×D412yes—
404C2×C8C22×D412yes—
405C2×C8C22×D411yescharacteristic
406C2×C8C22×D411yescharacteristic
407C2×C8C22×D411yescharacteristic
408C2×C8C2×#4114yes—
409C2×C8C2511yescharacteristic
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—44no—
416C4⋊C4—42no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—44no—
424C4⋊C4C2×D811yescharacteristic
425C4⋊C4C2×D811yescharacteristic
426C4⋊C4C2×D812yes—
427C4⋊C4C2×D812yes—
428C22⋊C4—42no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4C2×D812yes—
439C22⋊C4C2×D812yes—
440C22⋊C4C2×D812yes—
441C42—42no—
442C42—44no—
443C42C2×D812yes—
444C42C2×D812yes—
445C42C2×D811yescharacteristic
446C42C2×D811yescharacteristic
447C42C22×D411yescharacteristic
448C42C22×D412yes—
449C42C22×D411yescharacteristic
450C16—22no—
451C16—22no—
452C16C2×#4118yes—
453C23—42no—
454C23—44no—
455C23—84no—
456C23—162no—
457C23—162no—
458C23C22×D812yes—
459C23C22×D811yescharacteristic
460Q8—22no—
461Q8—42no—
462Q8—42no—
463Q8—84no—
464Q8—82no—
465Q8—82no—
466Q8—162no—
467D4—24no—
468D4—22no—
469D4—44no—
470D4—42no—
471D4—44no—
472D4—42no—
473D4—84no—
474D4—84no—
475D4—82no—
476D4—82no—
477D4—84no—
478D4—84no—
479D4—162no—
480D4—162no—
481D4—164no—
482D4—164no—
483D4—164no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—84no—
495C2×C4—164no—
496C2×C4—164no—
497C2×C4—164no—
498C2×C4—162no—
499C2×C4#53212yes—
500C2×C4C22×D812yes—
501C2×C4C22×D812yes—
502C2×C4C22×D811yescharacteristic
503C2×C4C22×D812yes—
504C2×C4C22×D811yescharacteristic
505C2×C4C22×D811yescharacteristic
506C2×C4C22×D811yescharacteristic
507C2×C4C22×D811yescharacteristic
508C2×C4C2×#13512yes—
509C2×C4C23×D411yescharacteristic
510C8—22no—
511C8—22no—
512C8#53212yes—
513C8C2×#13512yes—
514C8C2×#13512yes—
515C8C22×#4112yes—
516C22—21no—
517C22—21no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—82no—
522C22—84no—
523C22—84no—
524C22—82no—
525C22—82no—
526C22—84no—
527C22—324no—
528C22#316912yes—
529C22C23×D811yescharacteristic
530C4—22no—
531C4—21no—
532C4—22no—
533C4—21no—
534C4—22no—
535C4—82no—
536C4—84no—
537C4—82no—
538C4—82no—
539C4#316912yes—
540C4#341512yes—
541C4#341612yes—
542C4C2×#53211yescharacteristic
543C4C22×#13511yescharacteristic
544C2—22no—
545C2—42no—
546C2—162no—
547C2—164no—
548C2C2×#316911yescharacteristic
549C2C2×#341511yescharacteristic
550C2C2×#341611yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60718C2semidirect
2. #60714C2semidirect
3. #33751C2semidirect
4. #33748C2semidirect
5. #33744C2semidirect
6. #33736C2semidirect
7. #33733C2semidirect
8. #33731C2semidirect
9. #33717C2semidirect
10. #33697C2semidirect
11. C2×#2240C2semidirect
12. #33612C2semidirect
13. #33454C2semidirect
14. C4×#504C2semidirect
24. #3175C22semidirect
25. #3170C22semidirect
28. #3169C22semidirect
44. #2255C22semidirect
45. #2254C22semidirect
46. #2252C22semidirect
50. #2242C22semidirect
51. #2241C22semidirect
52. #2241C22semidirect
56. #2240C22semidirect
57. #2240C22semidirect
58. #2220C22semidirect
59. #2218C22semidirect
60. #2217C22semidirect
61. #2215C22semidirect
62. #2177C22semidirect
63. #2175C22semidirect
64. C4×SD32C22semidirect
65. C4×D16C22semidirect
70. #2108C22semidirect
72. C4×M32C22semidirect
74. C2×C4×C16C22semidirect
307. M32#41semidirect

Maximal subgroups

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

subgroupindex
#607182
#607142
#337512
#337482
#337442
#337362
#337332
#337312
#337172
#336972
C2×#22402
#336122
#334542
C4×#5042