Finite Group Databasebeta

← #6633278

Subgroups — #6633278

Order 512.

2026 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37298C211yesmaximal, characteristic, complemented
2#35554C211yesmaximal, characteristic, complemented
3#32911C212yesmaximal, complemented
4#31641C218yesmaximal, complemented
5#30733C218yesmaximal, complemented
6#29721C212yesmaximal, complemented
7#29331C212yesmaximal, complemented
8#25721C212yesmaximal, complemented
9#23939C212yesmaximal, complemented
10#23939C212yesmaximal, complemented
11#21913C211yesmaximal, characteristic, complemented
12C2×#540C2212yes—
13C22×#127C2211yescharacteristic
14C2×#518C2211yescharacteristic
15C2×#493C2214yes—
16C2×#459C2214yescomplemented
17C2×#447C2214yes—
18C2×#447C2214yes—
19#2856C2212yes—
20#2682—24no—
21#2675—24no—
22#2619C2212yes—
23#2566C2214yescomplemented
24#2541C2214yescomplemented
25#2445—24no—
26#2445C2218yes—
27#2405—24no—
28#2309C2211yescharacteristic
29#2309C2211yescharacteristic
30#2309C2211yescharacteristic
31#2296C2211yescharacteristic
32#2296C2211yescharacteristic
33#2100C2214yescomplemented
34#2073C2214yescomplemented
35#2054C2218yescomplemented
36#2041C2214yescomplemented
37#2025C2214yescomplemented
38#2025C2214yescomplemented
39#2011C2218yes—
40#2011C2214yescomplemented
41#2011C2214yescomplemented
42#2011C2214yescomplemented
43#1957C2214yes—
44#1934C2218yescomplemented
45#1916C2218yescomplemented
46#1879C2218yescomplemented
47#1879C2218yescomplemented
48#1879C2218yescomplemented
49#1875C2214yes—
50#1875C2214yes—
51#1818C2212yes—
52#1804C2214yescomplemented
53#1798C2212yes—
54C4×C22⋊C8C2212yes—
55C23×D4—22no—
56C23×D4C2311yescharacteristic
57C23×D4C2311yescharacteristic
58C22×D8C2318yes—
59C22×D8C2318yes—
60#540—48no—
61C2×#127—22no—
62C2×#127—44no—
63C2×#127C2314yes—
64C2×#127C2312yes—
65C2×#127C2312yes—
66C2×#127C2312yes—
67C2×#125C2312yes—
68C2×#123—24no—
69C2×#123—28no—
70C2×#121—24no—
71C2×#121—28no—
72C2×#121—24no—
73C2×#121C2314yes—
74#518—44no—
75C2×C4×D4—22no—
76C2×C4×D4C2314yes—
77C2×C4×D4C2314yes—
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C2×#117C2311yescharacteristic
82C22×C22⋊C4—22no—
83C22×C42C2311yescharacteristic
84#493—48no—
85#459—48no—
86#447—48no—
87#447—48no—
88C2×C4⋊C8C2314yes—
89C2×#102C2318yes—
90C2×#102C2318yes—
91C2×#102C2318yes—
92C2×#102C2318yes—
93C2×C22⋊C8C2314yes—
94C2×C4×C8C2314yes—
95#396—28no—
96#394—24no—
97#394—28no—
98#394—28no—
99#392—24no—
100#392—28no—
101#390—22no—
102#390—22no—
103#390—22no—
104#390—24no—
105#390C2314yes—
106#390C2314yes—
107#390C2314yes—
108#390C2314yes—
109#390C2314yes—
110#389D414yes—
111#389C2314yes—
112#386—28no—
113#385—22no—
114#385—28no—
115#385D414yes—
116#384C2312yes—
117#382C2312yes—
118#379—22no—
119#379—22no—
120#379—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4C2314yes—
123#340C2318yes—
124#340C2318yes—
125#336C2314yes—
126#336C2314yes—
127#336C2314yes—
128#336C2314yes—
129C25—21no—
130C25—21no—
131C22×D4—24no—
132C22×D4—24no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—28no—
136C22×D4—24no—
137C22×D4—24no—
138C22×D4—24no—
139C22×D4—24no—
140C22×D4—24no—
141C22×D4—42no—
142C22×D4—42no—
143C22×D4—42no—
144C22×D4—48no—
145C22×D4—88no—
146C22×D4C2414yes—
147C22×D4C2412yes—
148C22×D4C2412yes—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4C2×D412yes—
152C23×C4C2411yescharacteristic
153C2×D8—216no—
154C2×D8—48no—
155C2×D8—48no—
156C2×D8—48no—
157C2×D8—48no—
158C2×D8—48no—
159C22×C8C2418yes—
160#127—28no—
161#127—44no—
162#127—44no—
163#127—44no—
164#127—44no—
165#127—48no—
166#125—44no—
167#123—416no—
168#123—816no—
169#121—28no—
170#121—28no—
171#121—416no—
172#121—88no—
173#121—816no—
174C4×D4—28no—
175C4×D4—44no—
176C4×D4—44no—
177C4×D4—44no—
178C4×D4—48no—
179#117—24no—
180#117—42no—
181#117—42no—
182#117—42no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2×D418yes—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2412yes—
190C2×C4⋊C4C2412yes—
191C2×C4⋊C4C2414yes—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—48no—
203C2×C22⋊C4C2×D412yes—
204C2×C22⋊C4C2×D412yes—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—42no—
208C2×C42C2×D412yes—
209C2×C42C2412yes—
210C2×C42C2412yes—
211C2×C42C2412yes—
212C4⋊C8—44no—
213#102—48no—
214#102—48no—
215#102—48no—
216#102—48no—
217C22⋊C8—28no—
218C4×C8—44no—
219#95—24no—
220#95—24no—
221#95C2×D418yes—
222C24—24no—
223C24—22no—
224C24—22no—
225C24—24no—
226C24—42no—
227C24—42no—
228C24—42no—
229C24—42no—
230C24—42no—
231C24—84no—
232C24—84no—
233C24C22×D411yescharacteristic
234C2×D4—24no—
235C2×D4—24no—
236C2×D4—22no—
237C2×D4—24no—
238C2×D4—28no—
239C2×D4—24no—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—24no—
243C2×D4—28no—
244C2×D4—24no—
245C2×D4—416no—
246C2×D4—42no—
247C2×D4—416no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—48no—
251C2×D4—48no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—416no—
255C2×D4—48no—
256C2×D4—88no—
257C2×D4—88no—
258C2×D4—84no—
259C2×D4—88no—
260C2×D4—88no—
261C2×D4—84no—
262C2×D4—84no—
263C2×D4—832no—
264C2×D4—88no—
265C2×D4H(5,𝔽2)14yes—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—28no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—42no—
273C22×C4—48no—
274C22×C4—48no—
275C22×C4—88no—
276C22×C4—88no—
277C22×C4C22×D414yes—
278C22×C4C22×D414yes—
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D414yes—
281C22×C4H(5,𝔽2)14yes—
282C22×C4C2511yescharacteristic
283D8—816no—
284D8—816no—
285C2×C8—28no—
286C2×C8—28no—
287C2×C8—28no—
288C4⋊C4—24no—
289C4⋊C4—216no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—42no—
295C4⋊C4—48no—
296C4⋊C4—48no—
297C4⋊C4—84no—
298C22⋊C4—24no—
299C22⋊C4—42no—
300C22⋊C4—48no—
301C22⋊C4—48no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—48no—
305C22⋊C4—416no—
306C22⋊C4—88no—
307C22⋊C4—88no—
308C22⋊C4—88no—
309C22⋊C4—88no—
310C42—24no—
311C42—24no—
312C42—24no—
313C42—42no—
314C42—42no—
315C42—42no—
316C23—21no—
317C23—21no—
318C23—24no—
319C23—24no—
320C23—48no—
321C23—42no—
322C23—42no—
323C23—48no—
324C23—44no—
325C23—42no—
326C23—42no—
327C23—44no—
328C23—42no—
329C23—44no—
330C23—48no—
331C23—48no—
332C23—48no—
333C23—84no—
334C23—88no—
335C23—88no—
336C23—164no—
337C23—164no—
338C23—164no—
339C23—164no—
340C23#53512yes—
341C23C23×D411yescharacteristic
342D4—28no—
343D4—48no—
344D4—44no—
345D4—48no—
346D4—48no—
347D4—48no—
348D4—832no—
349D4—84no—
350D4—168no—
351D4—168no—
352D4—168no—
353D4—168no—
354C2×C4—24no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—24no—
358C2×C4—28no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—28no—
362C2×C4—28no—
363C2×C4—48no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—48no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—168no—
371C2×C4—168no—
372C2×C4#53512yes—
373C2×C4#57618yes—
374C2×C4C2×H(5,𝔽2)12yes—
375C2×C4C2×H(5,𝔽2)12yes—
376C2×C4#58512yes—
377C8—48no—
378C22—22no—
379C22—24no—
380C22—42no—
381C22—42no—
382C22—44no—
383C22—48no—
384C22—84no—
385C22—84no—
386C22—82no—
387C22—84no—
388C22—84no—
389C22—84no—
390C22—84no—
391C22—84no—
392C22—88no—
393C22—88no—
394C22—84no—
395C22—168no—
396C22—168no—
397C22C2×#53511yescharacteristic
398C22#348411yescharacteristic
399C22#348711yescharacteristic
400C22C2×#57614yes—
401C4—22no—
402C4—28no—
403C4—44no—
404C4—44no—
405C4—84no—
406C4—82no—
407C4—82no—
408C4#353414yes—
409C2—42no—
410C2—84no—
411C2—164no—
412C2—162no—
413C2#3671011yescharacteristic
414C2#6081712yes—
415C2#6163012yes—
416C2#6193212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37298C2semidirect
2. #35554C2semidirect
3. #32911C2semidirect
4. #31641C2semidirect
5. #30733C2semidirect
6. #29721C2semidirect
7. #29331C2semidirect
8. #25721C2semidirect
9. #23939C2semidirect
10. #23939C2semidirect
11. #21913C2semidirect
16. C2×#459C22semidirect
23. #2566C22semidirect
24. #2541C22semidirect
33. #2100C22semidirect
34. #2073C22semidirect
35. #2054C22semidirect
36. #2041C22semidirect
37. #2025C22semidirect
38. #2025C22semidirect
40. #2011C22semidirect
41. #2011C22semidirect
42. #2011C22semidirect
44. #1934C22semidirect
45. #1916C22semidirect
46. #1879C22semidirect
47. #1879C22semidirect
48. #1879C22semidirect
52. #1804C22semidirect

Maximal subgroups

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

subgroupindex
#372982
#355542
#329112
#316412
#307332
#297212
#293312
#257212
#239392
#219132