Finite Group Databasebeta

← #6510743

Subgroups — #6510743

Order 512.

1636 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35033C211yesmaximal, characteristic
2C2×#2502C211yesmaximal, characteristic
3#29112C212yesmaximal, complemented
4#29105C212yesmaximal, complemented
5#27965C212yesmaximal, complemented
6#27961C212yesmaximal, complemented
7#27721C218yesmaximal, complemented
8#27707C218yesmaximal, complemented
9#21768C212yesmaximal
10#20700C212yesmaximal
11#20524C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C2×#407C2212yes—
15#2857C2212yes—
16#2851C2212yes—
17#2626—22no—
18#2608—22no—
19#2607—22no—
20#2502—22no—
21#2502—22no—
22#2502C2218yes—
23#2441C2214yes—
24#2439C2214yes—
25C2×#398—22no—
26#2329—22no—
27C2×#391C2211yescharacteristic
28C2×#391C2211yescharacteristic
29C2×#385—22no—
30C2×#384C2211yescharacteristic
31C2×#384C2211yescharacteristic
32#2280C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34#2100C2218yescomplemented
35#2099C2218yescomplemented
36#2074C2214yescomplemented
37#2072C2214yescomplemented
38#2026C2214yescomplemented
39#2024C2214yescomplemented
40#1992C2214yescomplemented
41#1990C2214yescomplemented
42#1935C2218yescomplemented
43#1933C2218yescomplemented
44#1894C2218yes—
45#1873C2214yes—
46#1873C2214yes—
47#1836C2214yescomplemented
48#1834C2214yescomplemented
49#1824C2214yes—
50#1824C2214yes—
51#1824C2214yes—
52#1823C2218yes—
53#1787C2214yes—
54#1753C2218yes—
55#1749C2212yes—
56C4×C22⋊C8C2212yes—
57#1736C2214yes—
58#1736C2214yes—
59C23×Q8C2311yescharacteristic
60C24×C4—21no—
61C24×C4C2311yescharacteristic
62C22×M16C2314yes—
63C2×C4⋊Q8C2312yes—
64C2×#125C2312yes—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—28no—
68C2×C22⋊Q8C2314yes—
69C2×#117C2312yes—
70C2×#117C2312yes—
71C22×C4⋊C4—28no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—21no—
82C22×C22⋊C4D412yes—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×#103C2318yes—
86C2×#103C2318yes—
87#407—28no—
88C2×C22⋊C8C2314yes—
89C2×C22⋊C8C2314yes—
90C2×#97C2314yes—
91C2×C4×C8C2314yes—
92#399—24no—
93#398—28no—
94#398—28no—
95#396—24no—
96#396—24no—
97#393—24no—
98#391—22no—
99#391—22no—
100#391—21no—
101#391—21no—
102#391—24no—
103#391C2314yes—
104#391C2314yes—
105#389—22no—
106#389—24no—
107#389C2314yes—
108#389C2314yes—
109#387—24no—
110#387—24no—
111#387—24no—
112#385—24no—
113#385—28no—
114#385—28no—
115#384—22no—
116#384—21no—
117#384—21no—
118#384C2312yes—
119#384C2314yes—
120#384C2314yes—
121#382C2312yes—
122#381—22no—
123#380—22no—
124#380C2314yes—
125#380C2314yes—
126#379—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4D412yes—
130C4×C22⋊C4D412yes—
131C4×C22⋊C4C2314yes—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95D412yes—
135#340C2318yes—
136#340C2318yes—
137#340C2318yes—
138#340C2318yes—
139#340C2318yes—
140#340C2318yes—
141#336C2314yes—
142#336C2314yes—
143C25C2×D411yescharacteristic
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—24no—
147C22×Q8—42no—
148C22×Q8—42no—
149C22×Q8C2412yes—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—24no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—44no—
158C23×C4—84no—
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4#4118yes—
164C23×C4C2411yescharacteristic
165C23×C4C2412yes—
166C2×M16—28no—
167C2×M16—44no—
168C2×M16—44no—
169C22×C8C2418yes—
170C4⋊Q8—44no—
171#125—44no—
172C22⋊Q8—28no—
173C22⋊Q8—28no—
174C22⋊Q8—48no—
175C22⋊Q8—48no—
176#117—44no—
177#117—44no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—48no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—84no—
201C2×C4⋊C4—84no—
202C2×C4⋊C4C2412yes—
203C2×C4⋊C4C2412yes—
204C2×C4⋊C4C2412yes—
205C2×C4⋊C4C2414yes—
206C2×C4⋊C4C2414yes—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4C2×D414yes—
215C2×C22⋊C4C2×D414yes—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42C2×D412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2412yes—
224C2×C42C2412yes—
225#103—48no—
226#103—48no—
227C22⋊C8—28no—
228C22⋊C8—28no—
229#97—44no—
230C4×C8—44no—
231#95—28no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95C2×D414yes—
240#95C2×D414yes—
241C24—24no—
242C24—41no—
243C24—41no—
244C24—42no—
245C24#12414yes—
246C24C22×D412yes—
247C24C22×D411yescharacteristic
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—44no—
255C2×Q8—82no—
256C2×Q8—82no—
257C2×Q8—88no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—416no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—48no—
288C22×C4—88no—
289C22×C4—816no—
290C22×C4#12414yes—
291C22×C4#12418yes—
292C22×C4#13718yes—
293C22×C4C22×D414yes—
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D412yes—
296C22×C4C22×D412yes—
297C22×C4C22×D412yes—
298C22×C4C2×#4114yes—
299C22×C4C2×#4114yes—
300C22×C4C2×#4114yes—
301C22×C4C2511yescharacteristic
302M16—88no—
303C2×C8—28no—
304C2×C8—28no—
305C2×C8—28no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—28no—
310C4⋊C4—28no—
311C4⋊C4—48no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—48no—
316C4⋊C4—44no—
317C4⋊C4—48no—
318C4⋊C4—82no—
319C4⋊C4—84no—
320C4⋊C4—84no—
321C4⋊C4—84no—
322C4⋊C4—88no—
323C4⋊C4—84no—
324C4⋊C4—82no—
325C22⋊C4—28no—
326C22⋊C4—28no—
327C22⋊C4—48no—
328C22⋊C4—48no—
329C22⋊C4—84no—
330C22⋊C4—82no—
331C22⋊C4—82no—
332C42—24no—
333C42—24no—
334C42—41no—
335C42—41no—
336C42—42no—
337C42—82no—
338C42—82no—
339C23—22no—
340C23—21no—
341C23—21no—
342C23—24no—
343C23—28no—
344C23—22no—
345C23—22no—
346C23—24no—
347C23—44no—
348C23—44no—
349C23—48no—
350C23—48no—
351C23C2×#12414yes—
352C23C2×#12412yes—
353C23C2×#13714yes—
354C23C23×D411yescharacteristic
355Q8—48no—
356Q8—84no—
357Q8—164no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—28no—
367C2×C4—28no—
368C2×C4—28no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—44no—
373C2×C4—48no—
374C2×C4—48no—
375C2×C4—48no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—84no—
392C2×C4—84no—
393C2×C4—84no—
394C2×C4—84no—
395C2×C4—816no—
396C2×C4C2×#12414yes—
397C2×C4C2×#12412yes—
398C2×C4C2×#13714yes—
399C2×C4C2×#13714yes—
400C2×C4C22×#4112yes—
401C2×C4#58512yes—
402C2×C4#58512yes—
403C8—48no—
404C22—24no—
405C22—24no—
406C22—22no—
407C22—22no—
408C22—24no—
409C22—24no—
410C22—24no—
411C22—416no—
412C22—44no—
413C22—44no—
414C22#313914yes—
415C22C22×#12411yescharacteristic
416C22#348511yescharacteristic
417C22#348511yescharacteristic
418C22C22×#13712yes—
419C22#358512yes—
420C4—24no—
421C4—24no—
422C4—24no—
423C4—22no—
424C4—44no—
425C4—84no—
426C4—84no—
427C4—82no—
428C4—82no—
429C4—82no—
430C4—82no—
431C4#349114yes—
432C2—24no—
433C2—44no—
434C2#3545511yescharacteristic
435C2C2×#313911yescharacteristic
436C2#6061911yescharacteristic
437C2#6170912yes—
438C2#6171912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
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
3. #29112C2semidirect
4. #29105C2semidirect
5. #27965C2semidirect
6. #27961C2semidirect
7. #27721C2semidirect
8. #27707C2semidirect
34. #2100C22semidirect
35. #2099C22semidirect
36. #2074C22semidirect
37. #2072C22semidirect
38. #2026C22semidirect
39. #2024C22semidirect
40. #1992C22semidirect
41. #1990C22semidirect
42. #1935C22semidirect
43. #1933C22semidirect
47. #1836C22semidirect
48. #1834C22semidirect

Maximal subgroups

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

subgroupindex
#350332
C2×#25022
#291122
#291052
#279652
#279612
#277212
#277072
#217682
#207002
#205242