Finite Group Databasebeta

← #6374828

Subgroups — #6374828

Order 512.

1588 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34012C211yesmaximal, characteristic
2#21928C218yesmaximal, complemented
3#21920C212yesmaximal, complemented
4#21918C212yesmaximal, complemented
5#21910C212yesmaximal
6#21908C212yesmaximal
7#21435C214yesmaximal, complemented
8#21435C214yesmaximal, complemented
9#20811C212yesmaximal, complemented
10#20715C212yesmaximal
11#20467C212yesmaximal, complemented
12C23×C42C2211yescharacteristic
13C2×#432C2214yes—
14#2285C412yes—
15#2285C412yes—
16#2285C2211yescharacteristic
17#2285C2211yescharacteristic
18#2283C414yes—
19#2283C412yes—
20#2283C412yes—
21#2283C2214yes—
22#2280C414yes—
23#2280C2211yescharacteristic
24#2280C2211yescharacteristic
25C4×#117C2214yes—
26C4×#117C2211yescharacteristic
27C4×#117C2211yescharacteristic
28#1924C2218yes—
29#1924C2218yes—
30#1923C2218yes—
31#1923C2218yes—
32#1917C2218yescomplemented
33#1917C2218yescomplemented
34#1916C2218yescomplemented
35#1916C2218yescomplemented
36#1843C2214yescomplemented
37#1832C2214yescomplemented
38#1817C2214yes—
39#1798C2214yes—
40#1798C2214yes—
41#1797C2214yes—
42#1797C2214yes—
43#1779C2214yes—
44#1768C2214yescomplemented
45#1768C2214yescomplemented
46#1767C2212yescomplemented
47#1767C2214yescomplemented
48#1767C2212yescomplemented
49#1750C2214yes—
50#1750C2214yes—
51#1749C2212yes—
52#1749C2214yes—
53#1749C2212yes—
54#1745C2214yescomplemented
55C4×#97C2212yescomplemented
56C4×#97C2212yescomplemented
57C24×C4C2×C412yes—
58C24×C4C2311yescharacteristic
59C2×#117C2×C414yes—
60C2×#117C2×C414yes—
61C2×#117C2311yescharacteristic
62C2×#117C2311yescharacteristic
63C2×#117C2311yescharacteristic
64C2×#117C2311yescharacteristic
65C22×C42—28no—
66C22×C42C2×C412yes—
67C22×C42C2×C412yes—
68C22×C42C2×C412yes—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71C22×C42C2312yes—
72C22×C42C2311yescharacteristic
73#432—216no—
74C2×C4⋊C8C2318yes—
75C2×C4⋊C8C2314yes—
76C2×C4⋊C8C2318yes—
77C2×C4⋊C8C2314yes—
78C2×C22⋊C8C2318yes—
79C2×C22⋊C8C2314yes—
80C2×C22⋊C8C2318yes—
81C2×C22⋊C8C2314yes—
82C2×#97C2314yes—
83C2×#97C2314yes—
84C2×#97C2314yes—
85C2×#97C2314yes—
86C2×C4×C8C2314yes—
87C2×C4×C8C2314yes—
88#384C2×C411yescharacteristic
89#384C2×C412yes—
90#384C2×C411yescharacteristic
91#383C2×C414yes—
92#383C2×C414yes—
93#382C2×C412yes—
94#382C2×C411yescharacteristic
95#382C2×C411yescharacteristic
96#382C2×C414yes—
97#382C2×C414yes—
98#381C2×C414yes—
99#381C2×C414yes—
100#381C2×C412yes—
101#381C2×C412yes—
102#380C2×C414yes—
103#380C2×C414yes—
104#380C2×C414yes—
105#380C2×C414yes—
106#380C2×C412yes—
107#380C2×C412yes—
108#380C2×C412yes—
109#380C2×C412yes—
110#379C2×C412yes—
111#379C2×C412yes—
112C4×C4⋊C4C2×C414yes—
113C4×C4⋊C4C2×C414yes—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2314yes—
119C4×C4⋊C4C2314yes—
120C4×C22⋊C4C2×C414yes—
121C4×C22⋊C4C2×C414yes—
122C4×C22⋊C4C2×C414yes—
123C4×C22⋊C4C2×C414yes—
124C4×C22⋊C4C2314yes—
125C4×C22⋊C4C2314yes—
126#376C2×C414yes—
127#376C2×C414yes—
128#376C2312yes—
129#376C2312yes—
130C43C2312yes—
131C43C2312yes—
132#336C2314yes—
133#336C2318yes—
134#336C2314yes—
135#336C2314yes—
136#336C2318yes—
137#336C2314yes—
138#336C2318yes—
139#336C2318yes—
140C25C22×C411yescharacteristic
141C23×C4—28no—
142C23×C4—24no—
143C23×C4—44no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—44no—
147C23×C4C22×C412yes—
148C23×C4C22×C412yes—
149C23×C4C22×C412yes—
150C23×C4C22×C412yes—
151C23×C4C22×C412yes—
152C23×C4C22×C412yes—
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C412yes—
155C23×C4C2412yes—
156C23×C4C2411yescharacteristic
157C22×C8C2418yes—
158C22×C8C2418yes—
159#117—42no—
160#117—42no—
161#117—42no—
162#117—42no—
163#117—48no—
164#117—48no—
165C2×C4⋊C4C22×C414yes—
166C2×C4⋊C4C22×C414yes—
167C2×C4⋊C4C22×C414yes—
168C2×C4⋊C4C22×C414yes—
169C2×C4⋊C4C22×C411yescharacteristic
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C411yescharacteristic
172C2×C4⋊C4C22×C411yescharacteristic
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C22×C411yescharacteristic
175C2×C22⋊C4C22×C414yes—
176C2×C22⋊C4C22×C414yes—
177C2×C22⋊C4C22×C414yes—
178C2×C22⋊C4C22×C414yes—
179C2×C22⋊C4C22×C412yes—
180C2×C22⋊C4C22×C412yes—
181C2×C22⋊C4C22×C412yes—
182C2×C22⋊C4C22×C412yes—
183C2×C42—28no—
184C2×C42—24no—
185C2×C42—28no—
186C2×C42—28no—
187C2×C42—28no—
188C2×C42—28no—
189C2×C42C22×C412yes—
190C2×C42C22×C412yes—
191C2×C42C22×C412yes—
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C412yes—
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C414yes—
196C2×C42C22×C414yes—
197C2×C42C22×C414yes—
198C2×C42C22×C414yes—
199C2×C42C22×C412yes—
200C2×C42#4118yes—
201C2×C42C2412yes—
202C2×C42C2412yes—
203C2×C42C2412yes—
204C2×C42C2412yes—
205C2×C42C2412yes—
206C2×C42C2411yescharacteristic
207C2×C42C2411yescharacteristic
208C4⋊C8—216no—
209C4⋊C8—44no—
210C4⋊C8—48no—
211C4⋊C8—44no—
212C22⋊C8—216no—
213C22⋊C8—44no—
214C22⋊C8—48no—
215C22⋊C8—44no—
216#97—28no—
217#97—28no—
218#97—44no—
219#97—44no—
220C4×C8—44no—
221C4×C8—44no—
222#95C22×C414yes—
223#95C22×C414yes—
224#95C22×C414yes—
225#95C22×C414yes—
226#95C22×C414yes—
227#95C22×C414yes—
228#95C22×C414yes—
229#95C22×C414yes—
230#95C22×C414yes—
231#95C22×C414yes—
232#95C22×C414yes—
233#95C22×C414yes—
234C24—22no—
235C24—24no—
236C24—41no—
237C24—41no—
238C24—42no—
239C24C23×C411yescharacteristic
240C24C23×C412yes—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—432no—
270C22×C4—416no—
271C22×C4#11718yes—
272C22×C4C23×C412yes—
273C22×C4C23×C412yes—
274C22×C4C23×C412yes—
275C22×C4C23×C412yes—
276C22×C4C23×C414yes—
277C22×C4C23×C414yes—
278C22×C4C23×C412yes—
279C22×C4C23×C412yes—
280C22×C4C23×C412yes—
281C22×C4C23×C412yes—
282C22×C4C23×C411yescharacteristic
283C22×C4C23×C412yes—
284C22×C4C2×#4114yes—
285C22×C4C2511yescharacteristic
286C2×C8—28no—
287C2×C8—28no—
288C2×C8—28no—
289C2×C8—28no—
290C2×C8—28no—
291C2×C8—28no—
292C4⋊C4—28no—
293C4⋊C4—28no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C22⋊C4—28no—
303C22⋊C4—28no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—44no—
309C22⋊C4—44no—
310C42—22no—
311C42—24no—
312C42—24no—
313C42—22no—
314C42—24no—
315C42—24no—
316C42—24no—
317C42—24no—
318C42—24no—
319C42—24no—
320C42—24no—
321C42—44no—
322C42—44no—
323C42—44no—
324C42—44no—
325C42#11718yes—
326C42C2×#4114yes—
327C42C2×#4114yes—
328C23—22no—
329C23—21no—
330C23—21no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—48no—
339C23—44no—
340C23—44no—
341C23—48no—
342C23—44no—
343C23C2×#11712yes—
344C23#56814yes—
345C23C24×C411yescharacteristic
346C2×C4—216no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—28no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—48no—
377C2×C4—48no—
378C2×C4—48no—
379C2×C4—48no—
380C2×C4—48no—
381C2×C4—48no—
382C2×C4—44no—
383C2×C4—48no—
384C2×C4C2×#11712yes—
385C2×C4C2×#11714yes—
386C2×C4C2×#11714yes—
387C2×C4#56814yes—
388C2×C4C22×#4112yes—
389C2×C4#58512yes—
390C2×C4#58512yes—
391C8—48no—
392C8—48no—
393C22—22no—
394C22—24no—
395C22—24no—
396C22—24no—
397C22—24no—
398C22—416no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—41no—
403C22—41no—
404C22—42no—
405C22—42no—
406C22C22×#11711yescharacteristic
407C22#342811yescharacteristic
408C22#342811yescharacteristic
409C22C2×#56812yes—
410C22C2×#56812yes—
411C4—22no—
412C4—24no—
413C4—24no—
414C4—24no—
415C4—42no—
416C4—44no—
417C4—44no—
418C4—42no—
419C4—42no—
420C4—42no—
421C4—42no—
422C4—42no—
423C4#345514yes—
424C2—42no—
425C2—44no—
426C2#3404811yescharacteristic
427C2#6012412yes—
428C2#6028512yes—
429C2#6028611yescharacteristic
430C2#6028711yescharacteristic
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socleC23C24×C4
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21928C2semidirect
3. #21920C2semidirect
4. #21918C2semidirect
7. #21435C2semidirect
8. #21435C2semidirect
9. #20811C2semidirect
11. #20467C2semidirect
32. #1917C22semidirect
33. #1917C22semidirect
34. #1916C22semidirect
35. #1916C22semidirect
36. #1843C22semidirect
37. #1832C22semidirect
44. #1768C22semidirect
45. #1768C22semidirect
46. #1767C22semidirect
47. #1767C22semidirect
48. #1767C22semidirect
54. #1745C22semidirect
55. C4×#97C22semidirect
56. C4×#97C22semidirect

Maximal subgroups

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

subgroupindex
#340122
#219282
#219202
#219182
#219102
#219082
#214352
#208112
#207152
#204672