Finite Group Databasebeta

← #6510749

Subgroups — #6510749

Order 512.

1682 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35020C211yesmaximal, characteristic, complemented
2C2×#2500C211yesmaximal, characteristic
3#29110C212yesmaximal, complemented
4#29104C212yesmaximal, complemented
5#27965C212yesmaximal, complemented
6#27964C212yesmaximal, complemented
7#27726C218yesmaximal, complemented
8#27710C218yesmaximal, complemented
9#21764C212yesmaximal
10#20702C212yesmaximal, complemented
11#20524C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C2×#407C2212yes—
15#2851C2212yes—
16#2851C2212yes—
17#2646—22no—
18#2569—24no—
19#2500—24no—
20#2500C2218yes—
21#2442C2214yescomplemented
22#2442C2214yescomplemented
23C2×#394—22no—
24#2315—22no—
25C2×#390C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#388—22no—
28C2×#384C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#380C2212yes—
31C2×C4×C22⋊C4C2211yescharacteristic
32#2101C2218yescomplemented
33#2099C2218yescomplemented
34#2073C2214yescomplemented
35#2073C2214yescomplemented
36#2025C2214yescomplemented
37#2025C2214yescomplemented
38#1992C2214yescomplemented
39#1991C2214yescomplemented
40#1936C2218yescomplemented
41#1934C2218yescomplemented
42#1895C2218yes—
43#1874C2214yes—
44#1874C2214yes—
45#1836C2214yescomplemented
46#1835C2214yescomplemented
47#1825C2218yescomplemented
48#1824C2214yes—
49#1824C2214yes—
50#1823C2214yes—
51#1786C2214yes—
52#1753C2218yescomplemented
53#1749C2212yes—
54C4×C22⋊C8C2212yes—
55#1736C2214yescomplemented
56#1736C2214yescomplemented
57C23×D4—21no—
58C23×D4C2311yescharacteristic
59C24×C4C2311yescharacteristic
60C22×M16C2314yes—
61C2×#125C2312yes—
62C2×#125C2312yes—
63C2×#121—22no—
64C2×#121—28no—
65C2×#121C2314yes—
66C2×#117—22no—
67C2×#117—28no—
68C2×#117C2314yes—
69C22×C4⋊C4—22no—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4D412yes—
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C2×#102C2318yes—
85C2×#102C2318yes—
86#407—28no—
87C2×C22⋊C8C2314yes—
88C2×C22⋊C8C2314yes—
89C2×#97C2314yes—
90C2×C4×C8C2314yes—
91#400—24no—
92#400—24no—
93#398—24no—
94#398—24no—
95#394—28no—
96#394—28no—
97#390—22no—
98#390—22no—
99#390C2314yes—
100#390C2314yes—
101#389C2314yes—
102#389C2314yes—
103#388—28no—
104#388—28no—
105#387—24no—
106#387—24no—
107#385—24no—
108#385—24no—
109#385—24no—
110#385—24no—
111#384—22no—
112#384C2314yes—
113#382—22no—
114#382C2312yes—
115#382C2312yes—
116#382C2314yes—
117#380—22no—
118#380—22no—
119#380—24no—
120#380—24no—
121#380C2314yes—
122#380C2314yes—
123C4×C22⋊C4D414yes—
124C4×C22⋊C4C2314yes—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95D412yes—
128#340C2318yes—
129#340C2318yes—
130#340C2318yes—
131#340C2318yes—
132#340C2318yes—
133#340C2318yes—
134#336C2314yes—
135#336C2314yes—
136C25—21no—
137C25C2×D411yescharacteristic
138C22×D4—24no—
139C22×D4—21no—
140C22×D4—24no—
141C22×D4—42no—
142C22×D4—42no—
143C22×D4—44no—
144C22×D4—84no—
145C22×D4C2412yes—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4#4118yes—
158C23×C4C2411yescharacteristic
159C23×C4C2412yes—
160C2×M16—28no—
161C2×M16—44no—
162C2×M16—44no—
163C22×C8C2418yes—
164#125—44no—
165#125—44no—
166#121—28no—
167#121—28no—
168#121—48no—
169#117—28no—
170#117—28no—
171#117—48no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—84no—
188C2×C4⋊C4C2412yes—
189C2×C4⋊C4C2412yes—
190C2×C4⋊C4C2412yes—
191C2×C4⋊C4C2414yes—
192C2×C4⋊C4C2414yes—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—48no—
201C2×C22⋊C4—48no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—84no—
205C2×C22⋊C4C2×D414yes—
206C2×C22⋊C4C2×D414yes—
207C2×C42—24no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42—44no—
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2412yes—
214C2×C42C2412yes—
215#102—48no—
216#102—48no—
217C22⋊C8—28no—
218C22⋊C8—28no—
219#97—44no—
220C4×C8—44no—
221#95—24no—
222#95—24no—
223#95—28no—
224#95—24no—
225#95—24no—
226#95—24no—
227#95—24no—
228#95C2×D414yes—
229#95C2×D414yes—
230C24—24no—
231C24—22no—
232C24—41no—
233C24—41no—
234C24—42no—
235C24—44no—
236C24—42no—
237C24—42no—
238C24—42no—
239C24—44no—
240C24#12414yes—
241C24C22×D412yes—
242C24C22×D411yescharacteristic
243C2×D4—24no—
244C2×D4—24no—
245C2×D4—24no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—48no—
249C2×D4—42no—
250C2×D4—88no—
251C2×D4—84no—
252C2×D4—84no—
253C2×D4—816no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—416no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4#12414yes—
287C22×C4#12418yes—
288C22×C4C22×D414yes—
289C22×C4C22×D414yes—
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D412yes—
292C22×C4C2×#4114yes—
293C22×C4C2×#4114yes—
294C22×C4C2×#4114yes—
295C22×C4C2511yescharacteristic
296M16—88no—
297C2×C8—28no—
298C2×C8—28no—
299C2×C8—28no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—28no—
304C4⋊C4—28no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—84no—
309C4⋊C4—84no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4—88no—
313C4⋊C4—84no—
314C4⋊C4—84no—
315C22⋊C4—44no—
316C22⋊C4—44no—
317C22⋊C4—416no—
318C22⋊C4—48no—
319C22⋊C4—48no—
320C22⋊C4—48no—
321C22⋊C4—48no—
322C22⋊C4—88no—
323C42—24no—
324C42—24no—
325C42—42no—
326C42—42no—
327C42—48no—
328C23—22no—
329C23—21no—
330C23—21no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—28no—
338C23—44no—
339C23—44no—
340C23—44no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—44no—
345C23—84no—
346C23—84no—
347C23—84no—
348C23—84no—
349C23—88no—
350C23—88no—
351C23—84no—
352C23C2×#12414yes—
353C23C2×#12412yes—
354C23#57514yes—
355C23C23×D411yescharacteristic
356D4—48no—
357D4—84no—
358D4—816no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—28no—
364C2×C4—28no—
365C2×C4—28no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—44no—
373C2×C4—48no—
374C2×C4—48no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—84no—
388C2×C4—84no—
389C2×C4—84no—
390C2×C4—84no—
391C2×C4—164no—
392C2×C4—164no—
393C2×C4C2×#12414yes—
394C2×C4C2×#12412yes—
395C2×C4C22×#4112yes—
396C2×C4#58512yes—
397C2×C4#58512yes—
398C8—48no—
399C22—24no—
400C22—24no—
401C22—22no—
402C22—22no—
403C22—24no—
404C22—24no—
405C22—24no—
406C22—42no—
407C22—42no—
408C22—42no—
409C22—42no—
410C22—88no—
411C22—84no—
412C22—84no—
413C22—82no—
414C22—82no—
415C22—82no—
416C22—168no—
417C22#314214yes—
418C22C22×#12411yescharacteristic
419C22#348511yescharacteristic
420C22#348511yescharacteristic
421C22C2×#57512yes—
422C22#358512yes—
423C4—24no—
424C4—22no—
425C4—44no—
426C4—44no—
427C4—84no—
428C4—84no—
429C4—82no—
430C4—82no—
431C4—82no—
432C4#349114yes—
433C2—24no—
434C2—44no—
435C2—162no—
436C2#3545511yescharacteristic
437C2C2×#314211yescharacteristic
438C2#6061911yescharacteristic
439C2#6171012yes—
440C2#6171812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35020C2semidirect
3. #29110C2semidirect
4. #29104C2semidirect
5. #27965C2semidirect
6. #27964C2semidirect
7. #27726C2semidirect
8. #27710C2semidirect
10. #20702C2semidirect
11. #20524C2semidirect
21. #2442C22semidirect
22. #2442C22semidirect
32. #2101C22semidirect
33. #2099C22semidirect
34. #2073C22semidirect
35. #2073C22semidirect
36. #2025C22semidirect
37. #2025C22semidirect
38. #1992C22semidirect
39. #1991C22semidirect
40. #1936C22semidirect
41. #1934C22semidirect
45. #1836C22semidirect
46. #1835C22semidirect
47. #1825C22semidirect
52. #1753C22semidirect
55. #1736C22semidirect
56. #1736C22semidirect

Maximal subgroups

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

subgroupindex
#350202
C2×#25002
#291102
#291042
#279652
#279642
#277262
#277102
#217642
#207022
#205242