Finite Group Databasebeta

← #7623432

Subgroups — #7623432

Order 512.

1280 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60714C211yesmaximal, characteristic
2#60667C211yesmaximal, characteristic, complemented
3#33807C218yesmaximal, complemented
4#33794C212yesmaximal, complemented
5#33794C212yesmaximal, complemented
6#33793C212yesmaximal, complemented
7#33792C212yesmaximal, complemented
8#33731C211yesmaximal, characteristic, complemented
9#33726C211yesmaximal, characteristic, complemented
10#33704C211yesmaximal, characteristic, complemented
11C2×#2240C211yesmaximal, characteristic, complemented
12#33650C214yesmaximal, complemented
13#33631C214yesmaximal, complemented
14#33420C211yesmaximal, characteristic, complemented
15#3508—21no—
16#3503—21no—
17#3184—22no—
18#3176—22no—
19#3176C2214yes—
20#3169C2212yes—
21#3169C2212yes—
22#3165—21no—
23#3165—22no—
24#3164—21no—
25#3164—22no—
26#3164C2212yes—
27#3164C2214yescomplemented
28#3160C2211yescharacteristic
29#3160C2211yescharacteristic
30#3160C2212yescomplemented
31C2×#500C2211yescharacteristic
32C2×#500C2212yescomplemented
33C2×#493C2211yescharacteristic
34#3138—21no—
35#3058—21no—
36#3058C2212yes—
37#2948—21no—
38#2948C2212yes—
39#2913—21no—
40#2913C2212yes—
41C4×#131C2211yescharacteristic
42#2255C2214yescomplemented
43#2254C2211yescharacteristic, complemented
44#2254C2212yescomplemented
45#2254C2211yescharacteristic, complemented
46#2253C2214yescomplemented
47#2252C2211yescharacteristic, complemented
48#2252C2212yescomplemented
49#2252C2211yescharacteristic, complemented
50#2244C2214yes—
51#2240C2212yes—
52#2240C2212yes—
53#2240C2212yes—
54#2240C2212yes—
55#2240C2214yes—
56#2234C2218yescomplemented
57#2234C2218yescomplemented
58#2233C2218yescomplemented
59#2232C2218yescomplemented
60#2228C2218yescomplemented
61#2228C2218yescomplemented
62#2226C2218yescomplemented
63#2224C2218yescomplemented
64#2164C2214yescomplemented
65#2162C2212yescomplemented
66#2162C2212yescomplemented
67#2159C2214yes—
68#2158C2214yes—
69#2151C2212yes—
70#2140C2214yes—
71C2×#357C2214yes—
72#2113C2212yes—
73C4×M32C2211yescharacteristic
74C2×C4×C16C2211yescharacteristic
75C22×D8—21no—
76C22×D8C2312yes—
77C2×#131D411yescharacteristic
78C2×#131C2311yescharacteristic
79#559—41no—
80#559—41no—
81Q82—41no—
82#557—41no—
83#554—42no—
84#554—41no—
85#550—41no—
86#550—41no—
87D4×Q8—42no—
88D4×Q8—41no—
89#546—42no—
90#546—42no—
91#544—22no—
92#544—24no—
93#540—24no—
94#535—22no—
95#533—21no—
96#533—21no—
97#533—22no—
98#532—22no—
99#532—22no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—22no—
102C2×#127—21no—
103#520—22no—
104#518—22no—
105C4×#41D411yescharacteristic
106C2×M32C2312yes—
107C22×C16C2312yes—
108#501—24no—
109#501—24no—
110#501—24no—
111#500—21no—
112#500—21no—
113#500—22no—
114#500—22no—
115#500C2312yes—
116#500C2312yes—
117#500C2311yescharacteristic
118#500C2314yes—
119#500C2311yescharacteristic
120#500C2311yescharacteristic
121#500C2311yescharacteristic
122#499—21no—
123#499—21no—
124#499—22no—
125#499C2312yes—
126#499C2312yes—
127#499C2314yes—
128#498—21no—
129#498—21no—
130#498—22no—
131#493—21no—
132#493—21no—
133#493C2312yes—
134#493C2312yes—
135#490—24no—
136#486—22no—
137#486—22no—
138#469—24no—
139#469—24no—
140#466—22no—
141#466—22no—
142#466C2314yes—
143#466C2314yes—
144#442—24no—
145#442—24no—
146C4×D8—22no—
147C4×D8—22no—
148C4×D8C2314yes—
149C4×D8C2314yes—
150#428—22no—
151#428—24no—
152#428—22no—
153#428—24no—
154#427—21no—
155#427—21no—
156#427—22no—
157#427C2312yes—
158#427C2312yes—
159#427C2314yes—
160C2×#107—21no—
161C2×#107—22no—
162C2×#107C2312yes—
163C2×#107C2314yes—
164#418—22no—
165#418—22no—
166#405D412yes—
167#405D411yescharacteristic
168#405C2311yescharacteristic
169#405C2312yes—
170C4×M16D411yescharacteristic
171C4×M16D412yes—
172C2×C4×C8C2311yescharacteristic
173C2×C4×C8C2312yes—
174#367C2314yes—
175#367C2314yes—
176#367C2318yes—
177#366C2314yes—
178#366C2314yes—
179#366C2318yes—
180C4⋊C16C2314yes—
181C4⋊C16C2314yes—
182#357C2318yes—
183#357C2318yes—
184#357C2318yes—
185#357C2318yes—
186C22⋊C16C2314yes—
187C22⋊C16C2314yes—
188#346C2312yes—
189C4×C16C2311yescharacteristic
190C4×C16C2311yescharacteristic
191C4×C16C2312yes—
192C4×C16C2312yes—
193C2×#41—42no—
194C2×#41—42no—
195C2×#41C2×D411yescharacteristic
196C22×Q8—42no—
197C22×Q8—41no—
198C22×D4—22no—
199C22×D4—41no—
200C22×D4—41no—
201C2×D8—22no—
202C2×D8—22no—
203C2×D8—22no—
204C2×D8—28no—
205C2×D8—44no—
206C2×D8C2414yes—
207#131—24no—
208#131—24no—
209C2×M16C2×D411yescharacteristic
210C2×M16C2×D412yes—
211C2×M16C2×D411yescharacteristic
212C2×M16C2×D412yes—
213C22×C8C2×D412yes—
214C22×C8C2×D411yescharacteristic
215C22×C8C2412yes—
216C22×C8C2411yescharacteristic
217C4⋊Q8—22no—
218C4⋊Q8—21no—
219C4⋊Q8—24no—
220C4⋊Q8—22no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8—42no—
225C4⋊Q8—44no—
226C4⋊Q8—42no—
227C4⋊Q8—41no—
228C4⋊Q8—42no—
229C4⋊Q8—41no—
230#127—22no—
231#127—22no—
232#127—42no—
233#127—41no—
234#127—41no—
235#125—22no—
236#125—24no—
237#125—22no—
238#125—42no—
239#125—44no—
240#124—42no—
241#124—42no—
242#124—42no—
243#123—44no—
244#123—44no—
245C22⋊Q8—41no—
246C22⋊Q8—41no—
247C22⋊Q8—42no—
248C22⋊Q8—41no—
249C22⋊Q8—42no—
250C22⋊Q8—41no—
251C22⋊Q8—44no—
252C22⋊Q8—44no—
253C22⋊Q8—44no—
254#121—24no—
255#121—24no—
256#121—42no—
257#121—42no—
258#121—44no—
259#121—44no—
260#120—44no—
261C4×Q8—42no—
262C4×Q8—42no—
263C4×Q8—41no—
264C4×Q8—41no—
265C4×Q8—41no—
266C4×Q8—42no—
267C4×Q8—41no—
268C4×Q8D811yescharacteristic
269C4×Q8D811yescharacteristic
270C4×D4—24no—
271C4×D4—24no—
272C4×D4—44no—
273C4×D4—42no—
274C4×D4—42no—
275C4×D4—42no—
276C4×D4—42no—
277C4×D4—42no—
278C4×D4D812yes—
279C4×D4D812yes—
280C4×D4D811yescharacteristic
281C4×D4D811yescharacteristic
282#117—22no—
283#117—24no—
284#117—22no—
285#117C2×D411yescharacteristic
286#117C2×D412yes—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—41no—
291C2×C22⋊C4—44no—
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D412yes—
294M32—24no—
295C2×C16—24no—
296C2×C16C2412yes—
297C2×C16C2412yes—
298C2×C16C2414yes—
299#107—22no—
300#107—21no—
301#107—24no—
302#107—22no—
303#107—24no—
304#107—22no—
305#107—21no—
306#107—21no—
307#107—21no—
308#107C2414yes—
309#107C2412yes—
310#107C2412yes—
311#107C2412yes—
312#107C2412yes—
313#106—24no—
314#106—22no—
315#106—22no—
316#106—24no—
317#106—22no—
318#106—22no—
319#102—24no—
320#102—24no—
321#102—24no—
322#102—24no—
323#97C2×D412yes—
324#97C2×D412yes—
325#97C2×D412yes—
326C4×C8C2×D412yes—
327C4×C8C2×D411yescharacteristic
328C4×C8C2×D412yes—
329C4×C8C2×D411yescharacteristic
330C4×C8C2411yescharacteristic
331C4×C8C2411yescharacteristic
332C4×C8C2412yes—
333C24—42no—
334#41—22no—
335#41—22no—
336#41—164no—
337#41—164no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—42no—
342C2×Q8—41no—
343C2×Q8—44no—
344C2×Q8—42no—
345C2×Q8—162no—
346C2×Q8—164no—
347C2×Q8C2×D811yescharacteristic
348C2×D4—24no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—44no—
358C2×D4—162no—
359C2×D4—162no—
360C2×D4C2×D812yes—
361C2×D4C2×D811yescharacteristic
362C22×C4—44no—
363C22×C4—42no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4C2×D812yes—
367C22×C4C2×D811yescharacteristic
368C22×C4C2×D812yes—
369C22×C4C2×D811yescharacteristic
370C22×C4C22×D412yes—
371C22×C4C22×D411yescharacteristic
372D8—28no—
373D8—44no—
374D8—44no—
375D8—44no—
376M16—22no—
377M16—22no—
378M16—24no—
379M16—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8#13714yes—
383C2×C8C22×D412yes—
384C2×C8C22×D412yes—
385C2×C8C22×D411yescharacteristic
386C2×C8C22×D411yescharacteristic
387C2×C8C22×D411yescharacteristic
388C2×C82−1+414yes—
389C2×C8C2511yescharacteristic
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—44no—
396C4⋊C4—42no—
397C4⋊C4—41no—
398C4⋊C4—42no—
399C4⋊C4—44no—
400C4⋊C4—42no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—44no—
407C4⋊C4—42no—
408C4⋊C4—44no—
409C4⋊C4—42no—
410C4⋊C4C2×D811yescharacteristic
411C4⋊C4C2×D811yescharacteristic
412C4⋊C4C2×D812yes—
413C4⋊C4C2×D812yes—
414C22⋊C4—42no—
415C22⋊C4—44no—
416C22⋊C4—42no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4C2×D812yes—
422C22⋊C4C2×D812yes—
423C22⋊C4C2×D812yes—
424C42—42no—
425C42—44no—
426C42—42no—
427C42C2×D812yes—
428C42C2×D812yes—
429C42C2×D811yescharacteristic
430C42C2×D811yescharacteristic
431C42C22×D411yescharacteristic
432C42C22×D412yes—
433C42C22×D411yescharacteristic
434C16—22no—
435C16—22no—
436C16—24no—
437C23—44no—
438C23—84no—
439C23—162no—
440C23—162no—
441C23C22×D812yes—
442C23C22×D811yescharacteristic
443Q8—22no—
444Q8—84no—
445Q8—82no—
446Q8—82no—
447Q8—82no—
448Q8—82no—
449Q8—164no—
450Q8—162no—
451D4—24no—
452D4—22no—
453D4—44no—
454D4—44no—
455D4—84no—
456D4—84no—
457D4—84no—
458D4—162no—
459D4—162no—
460D4—164no—
461D4—164no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—164no—
474C2×C4—164no—
475C2×C4—162no—
476C2×C4—164no—
477C2×C4#53612yes—
478C2×C4C22×D812yes—
479C2×C4C22×D812yes—
480C2×C4C22×D812yes—
481C2×C4C22×D811yescharacteristic
482C2×C4C22×D811yescharacteristic
483C2×C4C22×D811yescharacteristic
484C2×C4C22×D811yescharacteristic
485C2×C4C22×D811yescharacteristic
486C2×C4C2×#13712yes—
487C2×C4C23×D411yescharacteristic
488C8—22no—
489C8—22no—
490C8#53612yes—
491C8C2×#13712yes—
492C8C2×#13712yes—
493C8C2×2−1+412yes—
494C22—21no—
495C22—21no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—84no—
500C22—84no—
501C22—84no—
502C22—324no—
503C22#322612yes—
504C22C23×D811yescharacteristic
505C4—22no—
506C4—22no—
507C4—21no—
508C4—22no—
509C4—21no—
510C4—82no—
511C4—84no—
512C4—82no—
513C4—82no—
514C4—82no—
515C4#322612yes—
516C4#341512yes—
517C4#341612yes—
518C4C2×#53611yescharacteristic
519C4C22×#13711yescharacteristic
520C2—22no—
521C2—42no—
522C2—164no—
523C2C2×#322611yescharacteristic
524C2C2×#341511yescharacteristic
525C2C2×#341611yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radical#60714C2
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
2. #60667C2semidirect
3. #33807C2semidirect
4. #33794C2semidirect
5. #33794C2semidirect
6. #33793C2semidirect
7. #33792C2semidirect
8. #33731C2semidirect
9. #33726C2semidirect
10. #33704C2semidirect
11. C2×#2240C2semidirect
12. #33650C2semidirect
13. #33631C2semidirect
14. #33420C2semidirect
27. #3164C22semidirect
30. #3160C22semidirect
32. C2×#500C22semidirect
42. #2255C22semidirect
43. #2254C22semidirect
44. #2254C22semidirect
45. #2254C22semidirect
46. #2253C22semidirect
47. #2252C22semidirect
48. #2252C22semidirect
49. #2252C22semidirect
56. #2234C22semidirect
57. #2234C22semidirect
58. #2233C22semidirect
59. #2232C22semidirect
60. #2228C22semidirect
61. #2228C22semidirect
62. #2226C22semidirect
63. #2224C22semidirect
64. #2164C22semidirect
65. #2162C22semidirect
66. #2162C22semidirect

Maximal subgroups

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

subgroupindex
#607142
#606672
#338072
#337942
#337932
#337922
#337312
#337262
#337042
C2×#22402
#336502
#336312
#334202