Finite Group Databasebeta

← #6510709

Subgroups — #6510709

Order 512.

1702 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35033C211yesmaximal, characteristic, complemented
2C2×#2500C211yesmaximal, characteristic
3#29104C212yesmaximal, complemented
4#29103C212yesmaximal, complemented
5#27964C212yesmaximal, complemented
6#27961C212yesmaximal, complemented
7#27702C218yesmaximal, complemented
8#27696C218yesmaximal, complemented
9#21761C212yesmaximal
10#20699C212yesmaximal, complemented
11#20523C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15#2857C2212yes—
16#2851C2212yes—
17#2626—22no—
18#2608—22no—
19#2607—22no—
20#2500—22no—
21#2500—22no—
22#2500C2218yes—
23#2441C2214yescomplemented
24#2439C2214yescomplemented
25C2×#394—22no—
26#2329—22no—
27C2×#390C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#388—22no—
30C2×#384C2211yescharacteristic
31C2×#382C2211yescharacteristic
32#2280C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34#2101C2218yescomplemented
35#2100C2218yescomplemented
36#2073C2214yescomplemented
37#2071C2214yescomplemented
38#2025C2214yescomplemented
39#2023C2214yescomplemented
40#1991C2214yescomplemented
41#1990C2214yescomplemented
42#1934C2218yescomplemented
43#1934C2218yescomplemented
44#1893C2218yes—
45#1875C2214yes—
46#1875C2214yes—
47#1835C2214yescomplemented
48#1834C2214yescomplemented
49#1825C2218yescomplemented
50#1824C2214yes—
51#1824C2214yes—
52#1823C2214yes—
53#1786C2214yes—
54C8×C22⋊C4C2218yescomplemented
55C4×C22⋊C8C2212yes—
56C4×C22⋊C8C2212yes—
57#1735C2214yescomplemented
58#1735C2214yescomplemented
59C23×D4—21no—
60C23×D4C2311yescharacteristic
61C24×C4C2311yescharacteristic
62C23×C8C2314yes—
63C2×C4⋊Q8C2312yes—
64C2×#125C2312yes—
65C2×C22⋊Q8—22no—
66C2×#121—22no—
67C2×#121—28no—
68C2×#121C2314yes—
69C2×#117—22no—
70C2×#117—28no—
71C2×#117C2314yes—
72C2×#117C2312yes—
73C2×#117C2312yes—
74C22×C4⋊C4—22no—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4D412yes—
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×#102C2318yes—
87C2×#102C2318yes—
88C2×C22⋊C8—28no—
89C2×C22⋊C8C2314yes—
90C2×C22⋊C8C2314yes—
91C2×C4×C8C2314yes—
92C2×C4×C8C2314yes—
93#399—24no—
94#396—24no—
95#396—24no—
96#394—28no—
97#394—28no—
98#393—24no—
99#391—22no—
100#391—24no—
101#390—22no—
102#390—21no—
103#390—21no—
104#390C2314yes—
105#390C2314yes—
106#389—22no—
107#389—24no—
108#389C2314yes—
109#389C2314yes—
110#388—28no—
111#388—28no—
112#387—24no—
113#387—24no—
114#387—24no—
115#385—24no—
116#384—21no—
117#384—21no—
118#384C2312yes—
119#384C2314yes—
120#382—22no—
121#382C2312yes—
122#382C2314yes—
123#381—22no—
124#380—22no—
125#380C2314yes—
126#380C2314yes—
127#379—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4D412yes—
131C4×C22⋊C4D412yes—
132C4×C22⋊C4C2314yes—
133C2×#95—22no—
134C2×#95D412yes—
135#340C2318yes—
136#340C2318yes—
137#340C2318yes—
138#340C2318yes—
139#340C2318yes—
140#340C2318yes—
141#336C2314yes—
142#336C2314yes—
143C25—21no—
144C25C2×D411yescharacteristic
145C22×Q8—22no—
146C22×D4—24no—
147C22×D4—21no—
148C22×D4—24no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4—44no—
152C22×D4—84no—
153C22×D4C2412yes—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4#4118yes—
164C23×C4C2411yescharacteristic
165C23×C4C2412yes—
166C22×C8—28no—
167C22×C8—44no—
168C22×C8—44no—
169C22×C8C2418yes—
170C4⋊Q8—44no—
171#125—44no—
172C22⋊Q8—48no—
173#121—28no—
174#121—28no—
175#121—48no—
176#117—28no—
177#117—28no—
178#117—48no—
179#117—44no—
180#117—44no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—84no—
196C2×C4⋊C4C2412yes—
197C2×C4⋊C4C2412yes—
198C2×C4⋊C4C2412yes—
199C2×C4⋊C4C2414yes—
200C2×C4⋊C4C2414yes—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—48no—
214C2×C22⋊C4—84no—
215C2×C22⋊C4C2×D414yes—
216C2×C22⋊C4C2×D414yes—
217C2×C42—24no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—42no—
221C2×C42—42no—
222C2×C42—44no—
223C2×C42C2×D412yes—
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2412yes—
227C2×C42C2412yes—
228#102—48no—
229#102—48no—
230C22⋊C8—28no—
231C22⋊C8—28no—
232C4×C8—44no—
233C4×C8—44no—
234#95—28no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95C2×D414yes—
241#95C2×D414yes—
242C24—24no—
243C24—22no—
244C24—41no—
245C24—41no—
246C24—42no—
247C24—44no—
248C24—42no—
249C24—42no—
250C24—42no—
251C24—44no—
252C24#12414yes—
253C24C22×D412yes—
254C24C22×D411yescharacteristic
255C2×Q8—44no—
256C2×Q8—82no—
257C2×Q8—82no—
258C2×D4—24no—
259C2×D4—24no—
260C2×D4—24no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—48no—
264C2×D4—42no—
265C2×D4—88no—
266C2×D4—88no—
267C2×D4—816no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—416no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4#12414yes—
294C22×C4#12418yes—
295C22×C4#13518yes—
296C22×C4C22×D414yes—
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D412yes—
299C22×C4C22×D412yes—
300C22×C4C22×D412yes—
301C22×C4C2×#4114yes—
302C22×C4C2×#4114yes—
303C22×C4C2×#4114yes—
304C22×C4C2511yescharacteristic
305C2×C8—28no—
306C2×C8—28no—
307C2×C8—28no—
308C2×C8—88no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—28no—
313C4⋊C4—28no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—44no—
318C4⋊C4—48no—
319C4⋊C4—82no—
320C4⋊C4—84no—
321C4⋊C4—84no—
322C4⋊C4—84no—
323C4⋊C4—84no—
324C4⋊C4—82no—
325C22⋊C4—28no—
326C22⋊C4—28no—
327C22⋊C4—48no—
328C22⋊C4—48no—
329C22⋊C4—48no—
330C22⋊C4—48no—
331C22⋊C4—48no—
332C22⋊C4—88no—
333C22⋊C4—84no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C42—24no—
337C42—24no—
338C42—41no—
339C42—41no—
340C42—42no—
341C42—48no—
342C42—82no—
343C42—82no—
344C23—22no—
345C23—21no—
346C23—21no—
347C23—24no—
348C23—22no—
349C23—22no—
350C23—24no—
351C23—28no—
352C23—44no—
353C23—44no—
354C23—44no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—44no—
359C23—84no—
360C23—84no—
361C23—84no—
362C23—84no—
363C23—88no—
364C23—88no—
365C23—84no—
366C23C2×#12414yes—
367C23C2×#12412yes—
368C23C2×#13514yes—
369C23C23×D411yescharacteristic
370Q8—164no—
371D4—48no—
372D4—84no—
373D4—816no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—28no—
383C2×C4—28no—
384C2×C4—28no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—44no—
389C2×C4—48no—
390C2×C4—48no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—84no—
404C2×C4—84no—
405C2×C4—84no—
406C2×C4—84no—
407C2×C4C2×#12414yes—
408C2×C4C2×#12412yes—
409C2×C4C2×#13514yes—
410C2×C4C2×#13514yes—
411C2×C4C22×#4112yes—
412C2×C4#58512yes—
413C2×C4#58512yes—
414C8—48no—
415C22—24no—
416C22—24no—
417C22—22no—
418C22—22no—
419C22—24no—
420C22—24no—
421C22—24no—
422C22—44no—
423C22—44no—
424C22—88no—
425C22—84no—
426C22—84no—
427C22—82no—
428C22—82no—
429C22—82no—
430C22—168no—
431C22#313114yes—
432C22C22×#12411yescharacteristic
433C22#348511yescharacteristic
434C22#348511yescharacteristic
435C22C22×#13512yes—
436C22#358512yes—
437C4—24no—
438C4—24no—
439C4—24no—
440C4—22no—
441C4—44no—
442C4—84no—
443C4—84no—
444C4—82no—
445C4—82no—
446C4—82no—
447C4#349114yes—
448C2—24no—
449C2—44no—
450C2—162no—
451C2#3545511yescharacteristic
452C2C2×#313111yescharacteristic
453C2#6061911yescharacteristic
454C2#6169912yes—
455C2#6170012yes—
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. #35033C2semidirect
3. #29104C2semidirect
4. #29103C2semidirect
5. #27964C2semidirect
6. #27961C2semidirect
7. #27702C2semidirect
8. #27696C2semidirect
10. #20699C2semidirect
11. #20523C2semidirect
23. #2441C22semidirect
24. #2439C22semidirect
34. #2101C22semidirect
35. #2100C22semidirect
36. #2073C22semidirect
37. #2071C22semidirect
38. #2025C22semidirect
39. #2023C22semidirect
40. #1991C22semidirect
41. #1990C22semidirect
42. #1934C22semidirect
43. #1934C22semidirect
47. #1835C22semidirect
48. #1834C22semidirect
49. #1825C22semidirect
54. C8×C22⋊C4C22semidirect
57. #1735C22semidirect
58. #1735C22semidirect

Maximal subgroups

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

subgroupindex
#350332
C2×#25002
#291042
#291032
#279642
#279612
#277022
#276962
#217612
#206992
#205232