Finite Group Databasebeta

← #6528495

Subgroups — #6528495

Order 512.

1620 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35104C211yesmaximal, characteristic, complemented
2C2×#2566C211yesmaximal, characteristic
3#29104C212yesmaximal
4#29103C212yesmaximal
5#27964C212yesmaximal, complemented
6#27961C212yesmaximal, complemented
7#27817C218yesmaximal, complemented
8#27731C218yesmaximal, complemented
9#21836C212yesmaximal, complemented
10#20733C212yesmaximal
11#20533C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C2×#431C2212yes—
15#2857C2212yes—
16#2851C2212yes—
17#2627—22no—
18#2606—22no—
19#2605—22no—
20#2570C2214yescomplemented
21#2566—22no—
22#2566—22no—
23#2566C2218yes—
24#2550C2214yescomplemented
25C2×#396—22no—
26#2329—22no—
27C2×#390C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#385—22no—
30C2×#384C2211yescharacteristic
31#2283C2212yes—
32C2×#382C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#2101C2218yescomplemented
35#2100C2218yescomplemented
36#2073C2214yes—
37#2071C2214yes—
38#2025C2214yes—
39#2023C2214yes—
40#1991C2214yescomplemented
41#1991C2214yescomplemented
42#1991C2214yescomplemented
43#1990C2214yescomplemented
44#1989C2218yescomplemented
45#1934C2218yes—
46#1934C2218yes—
47#1835C2214yescomplemented
48#1834C2214yescomplemented
49#1824C2214yes—
50#1824C2214yes—
51#1823C2214yes—
52#1804C2214yes—
53#1804C2214yes—
54#1786C2214yes—
55C8×C4⋊C4C2218yescomplemented
56C4×#102C2218yes—
57C4×C22⋊C8C2212yes—
58C4×C22⋊C8C2212yes—
59C23×D4C2311yescharacteristic
60C24×C4—21no—
61C2×C4⋊Q8C2312yes—
62C2×#125—22no—
63C2×#125—28no—
64C2×#125C2314yes—
65C2×#125C2312yes—
66C2×C22⋊Q8—22no—
67C2×C4×D4—22no—
68C2×C4×D4—28no—
69C2×C4×D4C2314yes—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4D412yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C22⋊C4—21no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#431—28no—
84C2×C4⋊C8C2314yes—
85C2×#102C2318yes—
86C2×#102C2318yes—
87C2×C22⋊C8C2314yes—
88C2×C4×C8C2314yes—
89C2×C4×C8C2314yes—
90C2×C4×C8C2314yes—
91#400—24no—
92#398—24no—
93#398—24no—
94#396—28no—
95#396—28no—
96#395—24no—
97#391—22no—
98#391—24no—
99#390—22no—
100#390—21no—
101#390—21no—
102#390C2314yes—
103#390C2314yes—
104#389—22no—
105#389—24no—
106#389C2314yes—
107#389C2314yes—
108#387—24no—
109#387—24no—
110#387—24no—
111#385—24no—
112#385—28no—
113#385—28no—
114#384—21no—
115#384—21no—
116#384C2314yes—
117#384C2312yes—
118#383—22no—
119#382—22no—
120#382C2314yes—
121#382C2312yes—
122#382C2312yes—
123#382C2312yes—
124#381—22no—
125#380—22no—
126#380C2314yes—
127C4×C4⋊C4D412yes—
128C4×C4⋊C4D412yes—
129C4×C4⋊C4C2314yes—
130C4×C4⋊C4C2314yes—
131C4×C22⋊C4—22no—
132#376—22no—
133C2×#95—22no—
134C2×#95D412yes—
135#340C2318yes—
136#340C2318yes—
137#340C2318yes—
138#340C2318yes—
139#340C2318yes—
140#340C2318yes—
141#336C2314yes—
142#336C2314yes—
143C25—21no—
144C22×Q8—22no—
145C22×D4—24no—
146C22×D4—21no—
147C22×D4—24no—
148C22×D4—42no—
149C22×D4C2412yes—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—24no—
154C23×C4—44no—
155C23×C4—84no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2411yescharacteristic
161C22×C8C2418yes—
162C4⋊Q8—44no—
163#125—28no—
164#125—28no—
165#125—44no—
166#125—48no—
167C22⋊Q8—48no—
168C4×D4—28no—
169C4×D4—28no—
170C4×D4—48no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—84no—
191C2×C4⋊C4C2×D414yes—
192C2×C4⋊C4C2×D414yes—
193C2×C4⋊C4C2412yes—
194C2×C4⋊C4C2412yes—
195C2×C4⋊C4C2412yes—
196C2×C4⋊C4C2414yes—
197C2×C4⋊C4C2414yes—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—48no—
205C2×C22⋊C4—84no—
206C2×C42—24no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42C2×D412yes—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D412yes—
213C2×C42#4118yes—
214C2×C42C2412yes—
215C2×C42C2412yes—
216C2×C42C2412yes—
217C4⋊C8—28no—
218#102—216no—
219#102—48no—
220C22⋊C8—28no—
221C4×C8—28no—
222C4×C8—44no—
223C4×C8—44no—
224#95—28no—
225#95—24no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95C2×D414yes—
235#95C2×D414yes—
236C24—24no—
237C24—22no—
238C24—44no—
239C24—42no—
240C24—42no—
241C24—44no—
242C24C22×D411yescharacteristic
243C2×Q8—44no—
244C2×Q8—82no—
245C2×Q8—82no—
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—22no—
249C2×D4—24no—
250C2×D4—416no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D42−1+414yes—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—48no—
275C22×C4—88no—
276C22×C4—816no—
277C22×C4#12414yes—
278C22×C4#12414yes—
279C22×C4#13518yes—
280C22×C4C22×D414yes—
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D412yes—
283C22×C4C22×D412yes—
284C22×C4C22×D412yes—
285C22×C4C22×D412yes—
286C22×C4C2×#4114yes—
287C22×C4C2511yescharacteristic
288C2×C8—28no—
289C2×C8—28no—
290C2×C8—28no—
291C4⋊C4—28no—
292C4⋊C4—28no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—28no—
297C4⋊C4—28no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—48no—
302C4⋊C4—48no—
303C4⋊C4—44no—
304C4⋊C4—48no—
305C4⋊C4—82no—
306C4⋊C4—84no—
307C4⋊C4—84no—
308C4⋊C4—84no—
309C4⋊C4—84no—
310C4⋊C4—82no—
311C22⋊C4—48no—
312C22⋊C4—48no—
313C22⋊C4—48no—
314C22⋊C4—48no—
315C22⋊C4—88no—
316C42—24no—
317C42—24no—
318C42—41no—
319C42—41no—
320C42—42no—
321C42—84no—
322C42#12418yes—
323C42C2×#4114yes—
324C42C2×#4114yes—
325C23—22no—
326C23—21no—
327C23—21no—
328C23—24no—
329C23—48no—
330C23—416no—
331C23—44no—
332C23—42no—
333C23—42no—
334C23—44no—
335C23—84no—
336C23—88no—
337C23—84no—
338C23C2×#12412yes—
339C23C2×#13514yes—
340C23C23×D411yescharacteristic
341Q8—164no—
342D4—28no—
343D4—48no—
344D4—44no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—28no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—24no—
354C2×C4—28no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—44no—
359C2×C4—48no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—816no—
373C2×C4—84no—
374C2×C4—84no—
375C2×C4—84no—
376C2×C4—84no—
377C2×C4C2×#12412yes—
378C2×C4C2×#12414yes—
379C2×C4C2×#12414yes—
380C2×C4C2×#13514yes—
381C2×C4C2×#13514yes—
382C2×C4C22×#4112yes—
383C2×C4C2×2−1+412yes—
384C2×C4#58512yes—
385C8—48no—
386C22—24no—
387C22—22no—
388C22—24no—
389C22—22no—
390C22—44no—
391C22—48no—
392C22—816no—
393C22—84no—
394C22—82no—
395C22—82no—
396C22#313114yes—
397C22C22×#12411yescharacteristic
398C22#348511yescharacteristic
399C22#348611yescharacteristic
400C22C22×#13512yes—
401C22#358512yes—
402C4—24no—
403C4—24no—
404C4—22no—
405C4—24no—
406C4—24no—
407C4—44no—
408C4—84no—
409C4—84no—
410C4—82no—
411C4—82no—
412C4—82no—
413C4#349514yes—
414C2—24no—
415C2—84no—
416C2#3564411yescharacteristic
417C2C2×#313111yescharacteristic
418C2#6061911yescharacteristic
419C2#6169912yes—
420C2#6181912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35104C2semidirect
5. #27964C2semidirect
6. #27961C2semidirect
7. #27817C2semidirect
8. #27731C2semidirect
9. #21836C2semidirect
11. #20533C2semidirect
20. #2570C22semidirect
24. #2550C22semidirect
34. #2101C22semidirect
35. #2100C22semidirect
40. #1991C22semidirect
41. #1991C22semidirect
42. #1991C22semidirect
43. #1990C22semidirect
44. #1989C22semidirect
47. #1835C22semidirect
48. #1834C22semidirect
55. C8×C4⋊C4C22semidirect

Maximal subgroups

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

subgroupindex
#351042
C2×#25662
#291042
#291032
#279642
#279612
#278172
#277312
#218362
#207332
#205332