Finite Group Databasebeta

← #181548

Subgroups — #181548

Order 512.

1884 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2975C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic
3C2×#2349C211yesmaximal, characteristic
4#8026C212yesmaximal, complemented
5#8020C214yesmaximal, complemented
6#8016C214yesmaximal, complemented
7#8014C212yesmaximal, complemented
8C22×C4×D4—21no—
9C22×C4×D4C2211yescharacteristic
10#2975—24no—
11#2975C418yescomplemented
12C2×#436—24no—
13C2×#431C2212yes—
14C2×#407C2214yes—
15#2357—24no—
16#2354C418yescomplemented
17#2349—24no—
18#2349C418yescomplemented
19#2312—22no—
20#2309—21no—
21C2×#389—21no—
22C2×#385—22no—
23#2285C2212yes—
24C2×#382C2211yescharacteristic
25#2280C2214yes—
26C2×#380C2212yes—
27#2274—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29C22×#95C2212yes—
30#1294C2214yes—
31#1294C2214yes—
32#1294C2218yes—
33C23×D4D411yescharacteristic
34C24×C4—22no—
35C24×C4C2312yes—
36C22×M16—24no—
37C2×C4×D4—28no—
38C2×C4×D4—48no—
39C2×C4×D4C2×C414yes—
40C2×C4×D4C2×C414yes—
41C2×C4×D4C2×C414yes—
42C2×C4×D4D414yes—
43C2×C4×D4D414yes—
44C2×C4×D4Q814yes—
45C2×#117—22no—
46C2×#117C2×C418yescomplemented
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4D411yescharacteristic
51C22×C22⋊C4—22no—
52C22×C22⋊C4D412yes—
53C22×C42—21no—
54C22×C42C2311yescharacteristic
55#436—232no—
56#431—24no—
57#431C2×C418yescomplemented
58C2×C4⋊C8—22no—
59#407—28no—
60#407C2×C4116yescomplemented
61C2×C22⋊C8—24no—
62C2×C22⋊C8C2314yes—
63C2×C4×C8—22no—
64#390—28no—
65#389—28no—
66#388—216no—
67#385—216no—
68#384C2×C414yescomplemented
69#382—22no—
70#382—22no—
71#382C2×C414yes—
72#381—24no—
73#380—22no—
74#380—24no—
75#380C2×C418yes—
76#380C2×C418yescomplemented
77#380C2314yes—
78#379—24no—
79C4×C4⋊C4—28no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4C2×C414yes—
83C4×C4⋊C4C2×C414yescomplemented
84C4×C22⋊C4—24no—
85C4×C22⋊C4—24no—
86C2×#95—24no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95C2×C418yes—
90C2×#95C2311yescharacteristic
91C2×#95C2312yes—
92C2×#95C2311yescharacteristic
93C25C2×D412yes—
94C22×D4—24no—
95C22×D4—24no—
96C22×D4C22⋊C414yes—
97C22×D4C22⋊C414yes—
98C22×D4C4⋊C414yes—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—22no—
102C23×C4—28no—
103C23×C4—48no—
104C23×C4—48no—
105C23×C4—84no—
106C23×C4—84no—
107C23×C4C22×C414yes—
108C23×C4C22×C414yes—
109C23×C4C22×C414yes—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D412yes—
112C23×C4C2×D414yes—
113C23×C4C2×D414yes—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×Q814yes—
116C23×C4C2411yescharacteristic
117C2×M16—216no—
118C2×M16—88no—
119C22×C8—24no—
120C4×D4—432no—
121C4×D4C42116yes—
122C4×D4C22⋊C4116yes—
123C4×D4C4⋊C4116yes—
124C4×D4C4⋊C4116yes—
125#117—44no—
126#117—44no—
127#117—88no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—28no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—44no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4—84no—
136C2×C4⋊C4C22⋊C414yes—
137C2×C4⋊C4C22⋊C414yes—
138C2×C4⋊C4C4⋊C414yes—
139C2×C4⋊C4C22×C418yes—
140C2×C22⋊C4—28no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—28no—
143C2×C22⋊C4—28no—
144C2×C22⋊C4—48no—
145C2×C22⋊C4C22⋊C418yes—
146C2×C22⋊C4C22⋊C418yes—
147C2×C22⋊C4C4⋊C418yes—
148C2×C42—24no—
149C2×C42—24no—
150C2×C42—28no—
151C2×C42—44no—
152C2×C42C22×C412yes—
153C2×C42C22×C412yes—
154C2×C42C22×C412yes—
155C2×C42C2×D412yes—
156C2×C42C2×D412yes—
157C2×C42C2×Q812yes—
158C4⋊C8—28no—
159C22⋊C8—216no—
160C22⋊C8—24no—
161C22⋊C8C22×C418yes—
162C4×C8—28no—
163#95—28no—
164#95—24no—
165#95—28no—
166#95—24no—
167#95—21no—
168#95—22no—
169#95—21no—
170#95—24no—
171#95C22×C412yes—
172#95C22×C414yes—
173#95C22×C412yes—
174C24—24no—
175C24—24no—
176C24—44no—
177C24—44no—
178C24C2×C22⋊C414yes—
179C24C2×C22⋊C414yes—
180C24C2×C4⋊C414yes—
181C24C22×D411yescharacteristic
182C2×D4—28no—
183C2×D4—28no—
184C2×D4—28no—
185C2×D4—28no—
186C2×D4—216no—
187C2×D4#95116yes—
188C22×C4—24no—
189C22×C4—22no—
190C22×C4—24no—
191C22×C4—22no—
192C22×C4—28no—
193C22×C4—42no—
194C22×C4—44no—
195C22×C4—48no—
196C22×C4—48no—
197C22×C4—42no—
198C22×C4—416no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—44no—
204C22×C4—416no—
205C22×C4—44no—
206C22×C4—42no—
207C22×C4—44no—
208C22×C4—84no—
209C22×C4—84no—
210C22×C4—84no—
211C22×C4—84no—
212C22×C4—88no—
213C22×C4—832no—
214C22×C4C2×C4218yes—
215C22×C4C2×C22⋊C414yes—
216C22×C4C2×C22⋊C412yes—
217C22×C4C2×C22⋊C418yes—
218C22×C4C2×C22⋊C412yes—
219C22×C4C2×C22⋊C414yes—
220C22×C4C2×C22⋊C412yes—
221C22×C4C2×C22⋊C412yes—
222C22×C4C2×C4⋊C418yes—
223C22×C4C2×C4⋊C418yes—
224C22×C4C2×C4⋊C412yes—
225C22×C4C2×C4⋊C414yes—
226C22×C4C2×C4⋊C412yes—
227C22×C4AGL(1,ℤ/8)14yes—
228C22×C4#13714yes—
229C22×C4C23×C411yescharacteristic
230C22×C4C23×C411yescharacteristic
231C22×C4C23×C411yescharacteristic
232C22×C4C22×D411yescharacteristic
233C22×C4C22×D411yescharacteristic
234C22×C4C22×Q811yescharacteristic
235M16—816no—
236C2×C8—28no—
237C2×C8—48no—
238C4⋊C4—48no—
239C4⋊C4—48no—
240C4⋊C4—48no—
241C4⋊C4—48no—
242C4⋊C4—84no—
243C4⋊C4—88no—
244C4⋊C4#95116yes—
245C22⋊C4—416no—
246C22⋊C4—44no—
247C22⋊C4—44no—
248C22⋊C4—88no—
249C22⋊C4—88no—
250C22⋊C4#95132yes—
251C42—48no—
252C42—48no—
253C42—88no—
254C42C2×C4214yes—
255C42C2×C22⋊C414yes—
256C42C2×C4⋊C414yes—
257C42C2×C4⋊C414yes—
258C23—22no—
259C23—22no—
260C23—28no—
261C23—24no—
262C23—24no—
263C23—28no—
264C23—28no—
265C23—28no—
266C23—44no—
267C23—44no—
268C23—44no—
269C23—44no—
270C23—416no—
271C23—416no—
272C23C2×#9518yes—
273C23C22×C22⋊C411yescharacteristic
274C23C22×C22⋊C411yescharacteristic
275C23C22×C4⋊C411yescharacteristic
276C23C2×AGL(1,ℤ/8)12yes—
277C23C2×#13712yes—
278D4—232no—
279C2×C4—28no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—48no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—48no—
292C2×C4—48no—
293C2×C4—48no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—48no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—48no—
304C2×C4—42no—
305C2×C4—44no—
306C2×C4—48no—
307C2×C4—48no—
308C2×C4—832no—
309C2×C4—84no—
310C2×C4—88no—
311C2×C4C2×#9514yes—
312C2×C4C2×#9518yes—
313C2×C4C2×#9514yes—
314C2×C4#41714yes—
315C2×C4#41714yes—
316C2×C4#41814yes—
317C2×C4#41914yes—
318C2×C4#42818yes—
319C2×C4C22×C4211yescharacteristic
320C2×C4C22×C22⋊C411yescharacteristic
321C2×C4C22×C4⋊C411yescharacteristic
322C2×C4C22×C4⋊C411yescharacteristic
323C8—48no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—24no—
329C22—24no—
330C22—28no—
331C22—28no—
332C22—28no—
333C22—48no—
334C22—48no—
335C22—416no—
336C22—416no—
337C22C22×#9511yescharacteristic
338C22#288312yes—
339C22C2×#41811yescharacteristic
340C22C2×#41911yescharacteristic
341C22C2×#41712yes—
342C22#290112yes—
343C22C2×#42812yes—
344C4—24no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—28no—
350C4—44no—
351C4—82no—
352C4—84no—
353C4—88no—
354C4#173618yes—
355C2—21no—
356C2—21no—
357C2—22no—
358C2—48no—
359C2#1763014yes—
360C2#2034311yescharacteristic
361C2C2×#173611yescharacteristic
362C2#2035411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8026C2semidirect
5. #8020C2semidirect
6. #8016C2semidirect
7. #8014C2semidirect
11. #2975C4semidirect
16. #2354C4semidirect
18. #2349C4semidirect
46. C2×#117C2×C4semidirect
57. #431C2×C4semidirect
60. #407C2×C4semidirect
68. #384C2×C4semidirect
76. #380C2×C4semidirect
83. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29752
#341862
C2×#23492
#80262
#80202
#80162
#80142