Finite Group Databasebeta

← #7624263

Subgroups — #7624263

Order 512.

1380 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62245C211yesmaximal, characteristic
2C2×#3336C211yesmaximal, characteristic, complemented
3#33910C218yesmaximal, complemented
4#33877C218yesmaximal, complemented
5#33800C212yesmaximal, complemented
6#33759C212yesmaximal, complemented
7#33753C212yesmaximal, complemented
8#33672C212yesmaximal, complemented
9#33648C212yesmaximal, complemented
10C2×#2232C212yesmaximal, complemented
11#33487C211yesmaximal, characteristic, complemented
12#3496—21no—
13C2×#552—21no—
14#3336—24no—
15#3336C2218yescomplemented
16#3334C2214yes—
17#3294—24no—
18D4×D8C2214yes—
19#3245—21no—
20#3186—21no—
21#3186—21no—
22#3169C2212yes—
23C2×#499C2211yescharacteristic, complemented
24C2×#485—21no—
25C2×#483—21no—
26C2×#479—21no—
27C2×#467C2211yescharacteristic, complemented
28C2×#467C2211yescharacteristic, complemented
29C2×#466C2212yes—
30C2×#466C2212yes—
31C2×C4×Q16C2211yescharacteristic, complemented
32C2×C8×D4C2211yescharacteristic
33C2×#427C2211yescharacteristic, complemented
34C22×#107C2211yescharacteristic, complemented
35#2255C2214yescomplemented
36#2244C2214yescomplemented
37#2236C2214yescomplemented
38#2235C2214yescomplemented
39#2234C2218yescomplemented
40#2233C2214yescomplemented
41#2232C2214yescomplemented
42#2232C2218yescomplemented
43#2232C2214yescomplemented
44#2228C2218yescomplemented
45#2227C2214yescomplemented
46#2220C2218yescomplemented
47#2217C2218yescomplemented
48#2207C2214yescomplemented
49#2187C2214yescomplemented
50#2187C2214yescomplemented
51#2185C2214yescomplemented
52#2175C2218yescomplemented
53#2168C2218yescomplemented
54#2163C2212yescomplemented
55#2159C2214yescomplemented
56C2×#366C2214yescomplemented
57#2140C2214yescomplemented
58C2×#357C2214yescomplemented
59#2112C2212yescomplemented
60C2×C22⋊C16C2212yescomplemented
61C2×H(5,𝔽2)—41no—
62C22×Q16C2311yescharacteristic
63C22×D8C2312yes—
64C22×D8C2312yes—
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67#552—28no—
68#552—44no—
69#547—24no—
70#546—44no—
71D42—24no—
72#537—41no—
73#534—41no—
74#534—41no—
75#532—22no—
76C2×#126—41no—
77C2×#125—21no—
78C2×#125—41no—
79C2×#123—41no—
80C2×C22⋊Q8—21no—
81C2×C22⋊Q8—21no—
82C2×#121—22no—
83C2×#121—22no—
84C2×C4×Q8—21no—
85C2×C4×D4D411yescharacteristic
86C2×#117—21no—
87C2×#117—41no—
88C22×C4⋊C4—21no—
89C2×SD32C2318yes—
90C2×M32C2314yes—
91C22×C16C2314yes—
92#500C2312yes—
93#499—22no—
94#499C2314yes—
95#493C2312yes—
96#486—22no—
97#485—28no—
98#483—28no—
99#480—24no—
100#480—24no—
101#479—28no—
102#474—24no—
103#467—22no—
104#467—22no—
105#467C2314yes—
106#467C2314yes—
107#466—22no—
108#466—22no—
109#466—22no—
110#466—22no—
111#466—22no—
112#466C2314yes—
113#466C2314yes—
114#466C2314yes—
115#459—24no—
116#447—24no—
117#447—24no—
118C4×Q16—22no—
119C4×Q16C2314yes—
120C4×D8—22no—
121C4×D8C2314yes—
122C8×D4D414yes—
123C8×D4C2314yes—
124#427—22no—
125#427C2314yes—
126C2×#107—21no—
127C2×#107—22no—
128C2×#107C2312yes—
129C2×#107C2312yes—
130C2×#107C2314yes—
131C2×#107C2311yescharacteristic
132C2×#107C2311yescharacteristic
133C2×#107C2311yescharacteristic
134C2×#106—21no—
135C2×#106—21no—
136C2×C4⋊C8D411yescharacteristic
137C2×#103—21no—
138C2×#103—21no—
139C2×#103—21no—
140C2×#102—22no—
141C2×#102—22no—
142C2×C22⋊C8D411yescharacteristic
143C2×C22⋊C8D411yescharacteristic
144C2×C4×C8C2311yescharacteristic
145#367C2318yes—
146#366C2318yes—
147#366C2318yes—
148C4⋊C16C2314yes—
149#358C2318yes—
150#357C2318yes—
151#357C2318yes—
152#357C2318yes—
153C22⋊C16C2314yes—
154C22⋊C16C2314yes—
155C22⋊C16C2314yes—
156#346C2314yes—
157H(5,𝔽2)—88no—
158C2×#41—44no—
159C2×#41—42no—
160C22×Q8—21no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—42no—
164C22×D4—42no—
165C22×D4—44no—
166C22×D4D811yescharacteristic
167C23×C4—41no—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C2×Q16—21no—
171C2×Q16—42no—
172C2×Q16C2412yes—
173C2×D8—22no—
174C2×D8—22no—
175C2×D8—22no—
176C2×D8—44no—
177C2×D8—44no—
178C2×D8C2414yes—
179C22×C8—42no—
180C22×C8—42no—
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D412yes—
184C22×C8C2×D412yes—
185C22×C8C2×D411yescharacteristic
186C22×C8C2411yescharacteristic
187C22×C8C2412yes—
188C22×C8C2412yes—
189C4⋊Q8—22no—
190#127—22no—
191#126—48no—
192#125—24no—
193#125—48no—
194#125—42no—
195#124—42no—
196#123—48no—
197#123—44no—
198#123—44no—
199C22⋊Q8—24no—
200C22⋊Q8—24no—
201C22⋊Q8—42no—
202C22⋊Q8—42no—
203C22⋊Q8—44no—
204#121—24no—
205#121—24no—
206#121—24no—
207#121—42no—
208#121—42no—
209#121—42no—
210#121—42no—
211#121—42no—
212#121—44no—
213#120—44no—
214#120—44no—
215C4×Q8—24no—
216C4×Q8—42no—
217C4×D4—24no—
218C4×D4—42no—
219C4×D4D814yes—
220C4×D4C2×D414yes—
221#117—24no—
222#117—42no—
223#117—48no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4D811yescharacteristic
238C2×C22⋊C4—41no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—41no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4D811yescharacteristic
244C2×C22⋊C4D811yescharacteristic
245C2×C42—41no—
246C2×C42—41no—
247C2×C42C2×D411yescharacteristic
248SD32—48no—
249M32—44no—
250C2×C16—44no—
251C2×C16C2418yes—
252#107—21no—
253#107—21no—
254#107—24no—
255#107—21no—
256#107—22no—
257#107—22no—
258#107—22no—
259#107C2412yes—
260#107C2412yes—
261#107C2412yes—
262#107C2414yes—
263#106—24no—
264#106—24no—
265C4⋊C8C2×D414yes—
266#103—24no—
267#103—24no—
268#103—24no—
269#102—24no—
270#102—24no—
271#102—24no—
272#102—24no—
273C22⋊C8C2×D414yes—
274C22⋊C8C2×D414yes—
275C4×C8C2×D412yes—
276C4×C8C2412yes—
277C24—42no—
278C24—42no—
279C24C2×D811yescharacteristic
280C24C2×D811yescharacteristic
281#41—88no—
282#41—816no—
283C2×Q8—22no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—41no—
287C2×Q8—42no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—82no—
298C2×D4—82no—
299C2×D4—82no—
300C2×D4—82no—
301C2×D4—816no—
302C2×D4—88no—
303C2×D4—88no—
304C2×D4C2×D814yes—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—84no—
318C22×C4C2×D811yescharacteristic
319C22×C4C2×D811yescharacteristic
320C22×C4C2×D811yescharacteristic
321C22×C4C2×D812yes—
322C22×C4C2×D812yes—
323C22×C4C2×D811yescharacteristic
324C22×C4C22×D412yes—
325C22×C4C22×D412yes—
326C22×C4C22×D411yescharacteristic
327Q16—24no—
328Q16—42no—
329D8—28no—
330D8—82no—
331D8—82no—
332D8—82no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—22no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8C22×D411yescharacteristic
342C2×C8C22×D412yes—
343C2×C8C22×D412yes—
344C2×C8C22×D412yes—
345C2×C8C2511yescharacteristic
346C4⋊C4—24no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—44no—
353C4⋊C4—41no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—42no—
362C4⋊C4C2×D814yes—
363C22⋊C4—44no—
364C22⋊C4—44no—
365C22⋊C4—44no—
366C22⋊C4—44no—
367C22⋊C4—44no—
368C22⋊C4—44no—
369C22⋊C4—44no—
370C22⋊C4C2×D814yes—
371C22⋊C4C2×D814yes—
372C42—44no—
373C42—44no—
374C42C2×D812yes—
375C42C22×D412yes—
376C16—28no—
377C23—22no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—44no—
383C23—84no—
384C23—84no—
385C23—162no—
386C23—162no—
387C23—162no—
388C23—162no—
389C23#50914yes—
390C23C22×D812yes—
391C23C22×D812yes—
392C23C22×D811yescharacteristic
393Q8—42no—
394Q8—42no—
395Q8—82no—
396Q8—84no—
397D4—28no—
398D4—44no—
399D4—44no—
400D4—84no—
401D4—88no—
402D4—88no—
403D4—88no—
404D4—162no—
405D4—162no—
406D4—162no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—88no—
425C2×C4—82no—
426C2×C4—84no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—88no—
431C2×C4—84no—
432C2×C4C22×D812yes—
433C2×C4C22×D812yes—
434C2×C4C22×D811yescharacteristic
435C2×C4C22×D812yes—
436C2×C4C22×D812yes—
437C2×C4C23×D411yescharacteristic
438C8—22no—
439C8—21no—
440C8—22no—
441C8—22no—
442C8#58512yes—
443C22—21no—
444C22—22no—
445C22—21no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—44no—
451C22—82no—
452C22—84no—
453C22—84no—
454C22—84no—
455C22—324no—
456C22—324no—
457C22C2×#50912yes—
458C22C2×#50914yes—
459C22C23×D811yescharacteristic
460C4—22no—
461C4—22no—
462C4—21no—
463C4—22no—
464C4—22no—
465C4—82no—
466C4—82no—
467C4—84no—
468C4—82no—
469C4—82no—
470C4#348511yescharacteristic
471C4#358611yescharacteristic
472C2—22no—
473C2—24no—
474C2—42no—
475C2—164no—
476C2#6169411yescharacteristic
477C2C22×#50911yescharacteristic
478C2#6263711yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
prime radical#62245C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3336C2semidirect
3. #33910C2semidirect
4. #33877C2semidirect
5. #33800C2semidirect
6. #33759C2semidirect
7. #33753C2semidirect
8. #33672C2semidirect
9. #33648C2semidirect
10. C2×#2232C2semidirect
11. #33487C2semidirect
15. #3336C22semidirect
23. C2×#499C22semidirect
27. C2×#467C22semidirect
28. C2×#467C22semidirect
31. C2×C4×Q16C22semidirect
33. C2×#427C22semidirect
34. C22×#107C22semidirect
35. #2255C22semidirect
36. #2244C22semidirect
37. #2236C22semidirect
38. #2235C22semidirect
39. #2234C22semidirect
40. #2233C22semidirect
41. #2232C22semidirect
42. #2232C22semidirect
43. #2232C22semidirect
44. #2228C22semidirect
45. #2227C22semidirect
46. #2220C22semidirect
47. #2217C22semidirect
48. #2207C22semidirect
49. #2187C22semidirect
50. #2187C22semidirect
51. #2185C22semidirect
52. #2175C22semidirect
53. #2168C22semidirect
54. #2163C22semidirect
55. #2159C22semidirect
56. C2×#366C22semidirect
57. #2140C22semidirect
58. C2×#357C22semidirect
59. #2112C22semidirect
60. C2×C22⋊C16C22semidirect

Maximal subgroups

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

subgroupindex
#622452
C2×#33362
#339102
#338772
#338002
#337592
#337532
#336722
#336482
C2×#22322
#334872