Finite Group Databasebeta

← #6519411

Subgroups — #6519411

Order 512.

2152 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2540C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic, complemented
3#26245C218yesmaximal, complemented
4#26243C218yesmaximal, complemented
5#24010C214yesmaximal, complemented
6#24009C214yesmaximal, complemented
7#21808C212yesmaximal, complemented
8#21790C211yesmaximal, characteristic, complemented
9#21365C212yesmaximal, complemented
10C22×#124C2211yescharacteristic
11C22×C22⋊Q8C2212yes—
12C22×#121C2212yes—
13C22×C4×D4C2211yescharacteristic
14C2×#485C2218yescomplemented
15C2×#482C2218yescomplemented
16C2×#432C2212yes—
17#2540—24no—
18#2540C2218yes—
19C2×#399—21no—
20C2×#392—21no—
21#2357—24no—
22#2354C2214yescomplemented
23#2354C2214yescomplemented
24#2312—22no—
25#2309—21no—
26C2×#390—22no—
27#2285C2212yes—
28#2280C2214yes—
29#2274—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31#2036C2218yescomplemented
32#2035C2218yescomplemented
33#2034C2218yescomplemented
34#2033C2218yescomplemented
35#1983C2218yescomplemented
36#1959C2214yescomplemented
37#1959C2214yescomplemented
38#1955C2218yescomplemented
39#1948C2214yescomplemented
40#1947C2214yescomplemented
41#1875C2218yescomplemented
42#1873C2218yescomplemented
43#1853C2214yes—
44#1852C2214yes—
45#1787C2214yes—
46#1786C2214yes—
47#1779C2214yes—
48#1776C2218yescomplemented
49C23×Q8C2311yescharacteristic
50C23×D4—22no—
51C23×D4D411yescharacteristic
52C23×D4C2311yescharacteristic
53C24×C4C2312yes—
54C2×#124—22no—
55C2×#124—28no—
56C2×#124C2314yes—
57C2×C22⋊Q8—24no—
58C2×C22⋊Q8—48no—
59C2×C22⋊Q8C2318yes—
60C2×#121—24no—
61C2×#121—48no—
62C2×#121C2318yes—
63C2×C4×D4—28no—
64C2×C4×D4D414yes—
65C2×C4×D4C2314yes—
66C2×#117—22no—
67C2×#117C2314yes—
68C2×#117C2314yes—
69C22×C4⋊C4—22no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4D412yes—
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79#485—416no—
80#482—416no—
81#432—28no—
82C2×#107C2318yes—
83C2×#106C2318yes—
84C2×C4⋊C8C2314yes—
85C2×#103C2318yes—
86C2×#103C2318yes—
87C2×#102C2318yes—
88C2×#102C2318yes—
89C2×C22⋊C8C2314yes—
90C2×C22⋊C8C2318yes—
91C2×C22⋊C8C2318yes—
92C2×#97C2314yes—
93#399—28no—
94#392—28no—
95#390—24no—
96#390—24no—
97#390—28no—
98#390—28no—
99#388—24no—
100#388—24no—
101#388—24no—
102#388—24no—
103#384C2312yes—
104#384C2312yes—
105#382—22no—
106#381—24no—
107#380—22no—
108#380C2314yes—
109#380C2314yes—
110#380C2314yes—
111#379—24no—
112C4×C4⋊C4—24no—
113C4×C4⋊C4—24no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4C2314yes—
117C4×C4⋊C4C2314yes—
118C4×C22⋊C4—24no—
119C4×C22⋊C4—24no—
120C2×#95—21no—
121#340C2318yes—
122#340C2318yes—
123C25—21no—
124C25C2×D412yes—
125C22×Q8—21no—
126C22×Q8—24no—
127C22×Q8—44no—
128C22×Q8C2412yes—
129C22×D4—28no—
130C22×D4—24no—
131C22×D4—21no—
132C22×D4—24no—
133C22×D4—44no—
134C22×D4—44no—
135C22×D4—48no—
136C22×D4—48no—
137C22×D4—84no—
138C22×D4C2×D414yes—
139C22×D4C2412yes—
140C23×C4—22no—
141C23×C4—28no—
142C23×C4—21no—
143C23×C4—44no—
144C23×C4—44no—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D414yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2414yes—
150C23×C4C2411yescharacteristic
151C22×C8C2418yes—
152#124—28no—
153#124—28no—
154#124—48no—
155C22⋊Q8—216no—
156C22⋊Q8—48no—
157C22⋊Q8—416no—
158#121—216no—
159#121—48no—
160#121—416no—
161C4×D4—216no—
162C4×D4—28no—
163C4×D4—28no—
164#117—48no—
165#117—48no—
166#117—48no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—48no—
176C2×C4⋊C4C2×D414yes—
177C2×C4⋊C4C2414yes—
178C2×C4⋊C4C2414yes—
179C2×C4⋊C4C2414yes—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—28no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—48no—
191C2×C22⋊C4—48no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4C2×D414yes—
195C2×C22⋊C4C2×D414yes—
196C2×C42—24no—
197C2×C42—24no—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42—82no—
201C2×C42C2×D412yes—
202C2×C42#4118yes—
203C2×C42C2412yes—
204#107—216no—
205#106—216no—
206C4⋊C8—28no—
207#103—48no—
208#103—48no—
209#102—48no—
210#102—48no—
211C22⋊C8—28no—
212C22⋊C8—48no—
213C22⋊C8—48no—
214#97—28no—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—24no—
220C24—24no—
221C24—24no—
222C24—24no—
223C24—22no—
224C24—42no—
225C24—42no—
226C24—42no—
227C24—42no—
228C24—44no—
229C24—42no—
230C24—42no—
231C24—42no—
232C24—44no—
233C24C22×D414yes—
234C24C22×D411yescharacteristic
235C2×Q8—24no—
236C2×Q8—28no—
237C2×Q8—44no—
238C2×Q8—42no—
239C2×Q8—88no—
240C2×D4—28no—
241C2×D4—24no—
242C2×D4—28no—
243C2×D4—44no—
244C2×D4—44no—
245C2×D4—416no—
246C2×D4—416no—
247C2×D4—44no—
248C2×D4—416no—
249C2×D4—42no—
250C2×D4—88no—
251C2×D4—88no—
252C2×D4—832no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—28no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—28no—
259C22×C4—28no—
260C22×C4—28no—
261C22×C4—28no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—42no—
275C22×C4—816no—
276C22×C4#12314yes—
277C22×C4#12314yes—
278C22×C4C22×D414yes—
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D412yes—
281C22×C4C22×D412yes—
282C22×C4C2×#4114yes—
283C22×C4C2511yescharacteristic
284C2×C8—28no—
285C2×C8—28no—
286C2×C8—28no—
287C4⋊C4—28no—
288C4⋊C4—28no—
289C4⋊C4—24no—
290C4⋊C4—28no—
291C4⋊C4—28no—
292C4⋊C4—416no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C4⋊C4H(5,𝔽2)18yes—
298C22⋊C4—28no—
299C22⋊C4—28no—
300C22⋊C4—416no—
301C22⋊C4—416no—
302C22⋊C4—48no—
303C22⋊C4—48no—
304C22⋊C4—48no—
305C22⋊C4—84no—
306C22⋊C4—84no—
307C22⋊C4—84no—
308C22⋊C4—84no—
309C42—48no—
310C42—84no—
311C42—84no—
312C42—84no—
313C42#12318yes—
314C42C2×#4114yes—
315C42C2×#4114yes—
316C23—21no—
317C23—21no—
318C23—22no—
319C23—24no—
320C23—24no—
321C23—24no—
322C23—24no—
323C23—24no—
324C23—24no—
325C23—24no—
326C23—48no—
327C23—48no—
328C23—48no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—44no—
333C23—44no—
334C23—84no—
335C23—84no—
336C23—84no—
337C23—84no—
338C23—88no—
339C23—84no—
340C23—84no—
341C23—88no—
342C23—84no—
343C23C2×#12312yes—
344C23#57514yes—
345C23C23×D411yescharacteristic
346Q8—48no—
347Q8—84no—
348D4—416no—
349D4—48no—
350D4—84no—
351D4—832no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—28no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—28no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—28no—
363C2×C4—44no—
364C2×C4—416no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—416no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—88no—
377C2×C4—84no—
378C2×C4—84no—
379C2×C4C2×#12312yes—
380C2×C4C2×#12314yes—
381C2×C4C2×#12314yes—
382C2×C4C22×#4112yes—
383C2×C4C2×H(5,𝔽2)14yes—
384C8—48no—
385C22—22no—
386C22—24no—
387C22—24no—
388C22—22no—
389C22—44no—
390C22—44no—
391C22—44no—
392C22—44no—
393C22—44no—
394C22—44no—
395C22—44no—
396C22—816no—
397C22—82no—
398C22—82no—
399C22—84no—
400C22—84no—
401C22—82no—
402C22—168no—
403C22#309314yes—
404C22C22×#12311yescharacteristic
405C22#348412yes—
406C22C2×#57512yes—
407C22#358511yescharacteristic
408C22#358611yescharacteristic
409C4—24no—
410C4—24no—
411C4—22no—
412C4—48no—
413C4—44no—
414C4—84no—
415C4—84no—
416C4—82no—
417C4#350814yes—
418C2—24no—
419C2—84no—
420C2—162no—
421C2#3555411yescharacteristic
422C2C2×#309311yescharacteristic
423C2#6052011yescharacteristic
424C2#6157612yes—
425C2#6157812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2540C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34186C2semidirect
3. #26245C2semidirect
4. #26243C2semidirect
5. #24010C2semidirect
6. #24009C2semidirect
7. #21808C2semidirect
8. #21790C2semidirect
9. #21365C2semidirect
14. C2×#485C22semidirect
15. C2×#482C22semidirect
22. #2354C22semidirect
23. #2354C22semidirect
31. #2036C22semidirect
32. #2035C22semidirect
33. #2034C22semidirect
34. #2033C22semidirect
35. #1983C22semidirect
36. #1959C22semidirect
37. #1959C22semidirect
38. #1955C22semidirect
39. #1948C22semidirect
40. #1947C22semidirect
41. #1875C22semidirect
42. #1873C22semidirect
48. #1776C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25402
#341862
#262452
#262432
#240102
#240092
#218082
#217902
#213652