Finite Group Databasebeta

← #6510704

Subgroups — #6510704

Order 512.

2086 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic, complemented
2C2×#2501C211yesmaximal, characteristic
3#29103C214yesmaximal, complemented
4#27960C214yesmaximal, complemented
5#27695C2116yesmaximal, complemented
6#21758C212yesmaximal
7C2×#1823C212yesmaximal, complemented
8#20523C211yesmaximal, characteristic, complemented
9C2×#543—21no—
10C22×#123—21no—
11C22×#123—21no—
12C22×#121C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15#2857C2214yes—
16#2659—22no—
17#2549—24no—
18#2501—24no—
19#2501C2218yes—
20#2440C2218yescomplemented
21C2×#399—22no—
22#2315—21no—
23C2×#390C2212yes—
24C2×#384C2212yes—
25C2×#380C2212yes—
26C2×C4×C22⋊C4C2211yescharacteristic
27#2099C22116yescomplemented
28#2071C2218yescomplemented
29#2023C2218yescomplemented
30#1990C2218yescomplemented
31#1934C22116yescomplemented
32#1893C2218yes—
33#1874C2218yes—
34#1834C2218yescomplemented
35#1823C2214yes—
36#1823C2218yes—
37#1823C2218yescomplemented
38#1786C2214yes—
39C8×C22⋊C4C2218yescomplemented
40C4×C22⋊C8C2214yes—
41C2×#340C2218yescomplemented
42C23×D4—21no—
43C23×D4C2311yescharacteristic
44C24×C4—21no—
45C24×C4C2311yescharacteristic
46C23×C8C2314yes—
47#543—48no—
48C2×C4⋊Q8C2314yes—
49C2×#123—28no—
50C2×#123—28no—
51C2×#123—48no—
52C2×#123—48no—
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8—28no—
55C2×#121—22no—
56C2×#121—28no—
57C2×#121C2314yes—
58C22×C4⋊C4—28no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2314yes—
66C22×C4⋊C4C2314yes—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4D411yescharacteristic
72C22×C22⋊C4D411yescharacteristic
73C22×C42C2312yes—
74C2×#102C23116yes—
75C2×C22⋊C8—28no—
76C2×C22⋊C8C2314yes—
77C2×C22⋊C8C2314yes—
78C2×C4×C8C2318yes—
79#399—216no—
80#398—28no—
81#395—28no—
82#391—24no—
83#390—24no—
84#390C2318yes—
85#389—24no—
86#389C2318yes—
87#385—28no—
88#384—24no—
89#384C2314yes—
90#384C2318yes—
91#380—24no—
92#380—22no—
93#380—22no—
94#380C2314yes—
95#380C2314yes—
96C4×C22⋊C4D414yes—
97C4×C22⋊C4C2314yes—
98C2×#95—22no—
99C2×#95D412yes—
100#340C23116yes—
101#340C23116yes—
102#340C23116yes—
103#336C2318yes—
104C25—21no—
105C25C2×D411yescharacteristic
106C22×Q8—24no—
107C22×D4—24no—
108C22×D4—21no—
109C22×D4—24no—
110C22×D4—44no—
111C22×D4—44no—
112C22×D4—44no—
113C22×D4—44no—
114C22×D4C2412yes—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—24no—
118C23×C4—24no—
119C23×C4—44no—
120C23×C4—44no—
121C23×C4—44no—
122C23×C4—44no—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4C2×D412yes—
127C23×C4#4118yes—
128C23×C4C2412yes—
129C23×C4C2411yescharacteristic
130C22×C8—28no—
131C22×C8—48no—
132C22×C8C2418yes—
133C4⋊Q8—48no—
134#123—432no—
135#123—432no—
136C22⋊Q8—416no—
137C22⋊Q8—432no—
138#121—28no—
139#121—28no—
140#121—48no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—48no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—48no—
158C2×C4⋊C4—48no—
159C2×C4⋊C4—48no—
160C2×C4⋊C4C2414yes—
161C2×C4⋊C4C2412yes—
162C2×C4⋊C4C2418yes—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—48no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—48no—
171C2×C22⋊C4—48no—
172C2×C22⋊C4D818yes—
173C2×C22⋊C4SD1618yes—
174C2×C22⋊C4C2×D414yes—
175C2×C22⋊C4C2×D414yes—
176C2×C42—44no—
177C2×C42C2×D414yes—
178C2×C42C2414yes—
179#102—416no—
180C22⋊C8—28no—
181C22⋊C8—28no—
182C4×C8—48no—
183#95—24no—
184#95—24no—
185#95—28no—
186#95C2×D418yes—
187C24—22no—
188C24—42no—
189C24—42no—
190C24—44no—
191C24—44no—
192C24—42no—
193C24—42no—
194C24—42no—
195C24#12414yes—
196C24C2×D814yes—
197C24C2×SD1614yes—
198C24C22×D411yescharacteristic
199C24C22×D412yes—
200C2×Q8—48no—
201C2×Q8—84no—
202C2×Q8—84no—
203C2×D4—24no—
204C2×D4—28no—
205C2×D4—44no—
206C2×D4—48no—
207C2×D4—48no—
208C2×D4—48no—
209C2×D4—42no—
210C2×D4—88no—
211C2×D4—816no—
212C22×C4—24no—
213C22×C4—28no—
214C22×C4—28no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—416no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—44no—
226C22×C4—48no—
227C22×C4—44no—
228C22×C4—42no—
229C22×C4—48no—
230C22×C4—44no—
231C22×C4—42no—
232C22×C4—48no—
233C22×C4—48no—
234C22×C4—84no—
235C22×C4—84no—
236C22×C4—816no—
237C22×C4#12414yes—
238C22×C4#12418yes—
239C22×C4C2×D818yes—
240C22×C4C2×SD1618yes—
241C22×C4C22×D411yescharacteristic
242C22×C4C22×D412yes—
243C22×C4C22×D414yes—
244C22×C4C22×D414yes—
245C22×C4C2×#4114yes—
246C22×C4C2×#4114yes—
247C22×C4C2×#4114yes—
248C22×C4C2511yescharacteristic
249C2×C8—28no—
250C2×C8—216no—
251C2×C8—88no—
252C4⋊C4—28no—
253C4⋊C4—24no—
254C4⋊C4—216no—
255C4⋊C4—48no—
256C4⋊C4—44no—
257C4⋊C4—42no—
258C4⋊C4—416no—
259C4⋊C4—48no—
260C4⋊C4—416no—
261C4⋊C4—416no—
262C4⋊C4—84no—
263C4⋊C4—84no—
264C4⋊C4—84no—
265C22⋊C4—48no—
266C22⋊C4—48no—
267C22⋊C4—48no—
268C22⋊C4—416no—
269C22⋊C4—416no—
270C22⋊C4C2×D8116yes—
271C22⋊C4C2×SD16116yes—
272C42—28no—
273C42—44no—
274C23—22no—
275C23—24no—
276C23—22no—
277C23—24no—
278C23—44no—
279C23—44no—
280C23—48no—
281C23—48no—
282C23—44no—
283C23—42no—
284C23—44no—
285C23—42no—
286C23—42no—
287C23—84no—
288C23—84no—
289C23—88no—
290C23—88no—
291C23—84no—
292C23#486116yes—
293C23C2×#12412yes—
294C23C2×#12414yes—
295C23C22×D812yes—
296C23C22×D814yes—
297C23C22×SD1614yes—
298C23C22×SD1612yes—
299C23C23×D411yescharacteristic
300Q8—168no—
301D4—48no—
302D4—84no—
303D4—816no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—28no—
311C2×C4—28no—
312C2×C4—28no—
313C2×C4—44no—
314C2×C4—48no—
315C2×C4—48no—
316C2×C4—48no—
317C2×C4—48no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—48no—
321C2×C4—48no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—88no—
327C2×C4—816no—
328C2×C4—84no—
329C2×C4—168no—
330C2×C4C2×#12414yes—
331C2×C4C2×#12412yes—
332C2×C4C22×D818yes—
333C2×C4C22×SD1618yes—
334C2×C4C22×#4112yes—
335C2×C4#58514yes—
336C8—48no—
337C22—22no—
338C22—24no—
339C22—22no—
340C22—24no—
341C22—24no—
342C22—28no—
343C22—24no—
344C22—28no—
345C22—416no—
346C22—82no—
347C22—82no—
348C22—82no—
349C22—84no—
350C22—84no—
351C22—168no—
352C22C2×#48614yes—
353C22C2×#48618yes—
354C22C22×#12411yescharacteristic
355C22#348512yes—
356C22C23×D811yescharacteristic
357C22C23×SD1611yescharacteristic
358C22#358612yes—
359C4—24no—
360C4—22no—
361C4—28no—
362C4—28no—
363C4—84no—
364C4—82no—
365C4—88no—
366C4#349114yes—
367C2—24no—
368C2—28no—
369C2—162no—
370C2#3545511yescharacteristic
371C2C22×#48611yescharacteristic
372C2#6062011yescharacteristic
373C2#6169412yes—
374C2#6169512yes—
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. #35002C2semidirect
3. #29103C2semidirect
4. #27960C2semidirect
5. #27695C2semidirect
7. C2×#1823C2semidirect
8. #20523C2semidirect
20. #2440C22semidirect
27. #2099C22semidirect
28. #2071C22semidirect
29. #2023C22semidirect
30. #1990C22semidirect
31. #1934C22semidirect
34. #1834C22semidirect
37. #1823C22semidirect
39. C8×C22⋊C4C22semidirect
41. C2×#340C22semidirect

Maximal subgroups

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

subgroupindex
#350022
C2×#25012
#291032
#279602
#276952
#217582
C2×#18232
#205232