Finite Group Databasebeta

← #181540

Subgroups — #181540

Order 512.

1700 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2972C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic
3C2×#2357C211yesmaximal, characteristic
4#8026C212yesmaximal, complemented
5#8024C214yesmaximal, complemented
6#8014C212yesmaximal, complemented
7#8012C214yesmaximal, complemented
8C22×C4×D4C2211yescharacteristic
9#2972—24no—
10#2972C418yescomplemented
11C2×C8×D4—24no—
12C2×#431C2212yes—
13C22×C22⋊C8C2214yes—
14#2357—24no—
15#2357—24no—
16#2357C418yescomplemented
17#2354C418yescomplemented
18#2312—22no—
19#2309—21no—
20C2×#390—21no—
21C2×#388—22no—
22#2285C2212yes—
23C2×#382C2211yescharacteristic
24#2280C2214yes—
25C2×#381C2212yes—
26#2274—21no—
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4C2212yes—
30#1294C2214yes—
31#1294C2218yes—
32#1294C2214yes—
33C23×D4D411yescharacteristic
34C24×C4C2312yes—
35C23×C8—24no—
36C2×C4×D4C2×C414yes—
37C2×C4×D4C2×C414yes—
38C2×C4×D4C2×C414yes—
39C2×C4×D4D414yes—
40C2×C4×D4D414yes—
41C2×C4×D4Q814yes—
42C2×#117—22no—
43C2×#117C2×C418yes—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4D411yescharacteristic
47C22×C22⋊C4—22no—
48C22×C22⋊C4—22no—
49C22×C22⋊C4D412yes—
50C22×C42—21no—
51C22×C42—22no—
52C22×C42C2311yescharacteristic
53C8×D4—232no—
54#431—24no—
55#431C2×C418yes—
56C2×C4⋊C8—22no—
57C2×C22⋊C8—24no—
58C2×C22⋊C8—28no—
59C2×C22⋊C8C2×C4116yes—
60C2×C22⋊C8C2314yes—
61C2×C4×C8—22no—
62#390—28no—
63#390—28no—
64#388—216no—
65#388—216no—
66#384C2×C414yes—
67#382—22no—
68#382—22no—
69#382C2×C414yes—
70#381—24no—
71#381—24no—
72#381C2×C418yes—
73#380—22no—
74#380C2×C418yes—
75#380C2314yes—
76#379—24no—
77C4×C4⋊C4—28no—
78C4×C4⋊C4—28no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4C2×C414yes—
82C4×C4⋊C4C2×C414yes—
83C4×C22⋊C4—24no—
84C4×C22⋊C4—24no—
85C4×C22⋊C4—24no—
86C4×C22⋊C4C2×C418yes—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95C2311yescharacteristic
90C2×#95C2312yes—
91C2×#95C2311yescharacteristic
92C25C2×D412yes—
93C22×D4—24no—
94C22×D4—24no—
95C22×D4C22⋊C414yes—
96C22×D4C22⋊C414yes—
97C22×D4C4⋊C414yes—
98C23×C4—21no—
99C23×C4—22no—
100C23×C4—21no—
101C23×C4—84no—
102C23×C4C22×C414yes—
103C23×C4C22×C414yes—
104C23×C4C22×C414yes—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D412yes—
107C23×C4C2×D414yes—
108C23×C4C2×D414yes—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×Q814yes—
111C23×C4C2411yescharacteristic
112C22×C8—24no—
113C22×C8—216no—
114C22×C8—88no—
115C4×D4C42116yes—
116C4×D4C22⋊C4116yes—
117C4×D4C4⋊C4116yes—
118C4×D4C4⋊C4116yes—
119#117—44no—
120#117—44no—
121#117—88no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—44no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4C22⋊C414yes—
129C2×C4⋊C4C22⋊C414yes—
130C2×C4⋊C4C4⋊C414yes—
131C2×C4⋊C4C22×C418yes—
132C2×C22⋊C4—28no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—28no—
135C2×C22⋊C4—28no—
136C2×C22⋊C4—28no—
137C2×C22⋊C4—48no—
138C2×C22⋊C4—84no—
139C2×C22⋊C4C22⋊C418yes—
140C2×C22⋊C4C22⋊C418yes—
141C2×C22⋊C4C4⋊C418yes—
142C2×C42—24no—
143C2×C42—24no—
144C2×C42—28no—
145C2×C42—28no—
146C2×C42—44no—
147C2×C42—84no—
148C2×C42C22×C412yes—
149C2×C42C22×C412yes—
150C2×C42C22×C412yes—
151C2×C42C2×D412yes—
152C2×C42C2×D412yes—
153C2×C42C2×Q812yes—
154C4⋊C8—28no—
155C22⋊C8—216no—
156C22⋊C8—24no—
157C22⋊C8C22×C418yes—
158C4×C8—28no—
159#95—28no—
160#95—24no—
161#95—28no—
162#95—24no—
163#95—21no—
164#95—22no—
165#95—21no—
166#95—24no—
167#95C22×C412yes—
168#95C22×C414yes—
169#95C22×C412yes—
170C24—24no—
171C24—24no—
172C24—44no—
173C24—44no—
174C24C2×C22⋊C414yes—
175C24C2×C22⋊C414yes—
176C24C2×C4⋊C414yes—
177C24C22×D411yescharacteristic
178C2×D4—28no—
179C2×D4—28no—
180C2×D4—28no—
181C2×D4—28no—
182C2×D4—216no—
183C2×D4#95116yes—
184C22×C4—24no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—28no—
189C22×C4—44no—
190C22×C4—42no—
191C22×C4—48no—
192C22×C4—48no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—44no—
197C22×C4—44no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—44no—
202C22×C4—84no—
203C22×C4—84no—
204C22×C4—84no—
205C22×C4—84no—
206C22×C4C2×C4218yes—
207C22×C4C2×C22⋊C414yes—
208C22×C4C2×C22⋊C412yes—
209C22×C4C2×C22⋊C418yes—
210C22×C4C2×C22⋊C414yes—
211C22×C4C2×C22⋊C412yes—
212C22×C4C2×C22⋊C412yes—
213C22×C4C2×C22⋊C412yes—
214C22×C4C2×C4⋊C418yes—
215C22×C4C2×C4⋊C418yes—
216C22×C4C2×C4⋊C412yes—
217C22×C4C2×C4⋊C414yes—
218C22×C4C2×C4⋊C412yes—
219C22×C4#13514yes—
220C22×C4#13514yes—
221C22×C4C23×C411yescharacteristic
222C22×C4C23×C411yescharacteristic
223C22×C4C23×C411yescharacteristic
224C22×C4C22×D411yescharacteristic
225C22×C4C22×D411yescharacteristic
226C22×C4C22×Q811yescharacteristic
227C2×C8—28no—
228C2×C8—48no—
229C2×C8—816no—
230C4⋊C4—48no—
231C4⋊C4—48no—
232C4⋊C4—48no—
233C4⋊C4—48no—
234C4⋊C4—84no—
235C4⋊C4#95116yes—
236C22⋊C4—416no—
237C22⋊C4—44no—
238C22⋊C4—44no—
239C22⋊C4—88no—
240C22⋊C4—88no—
241C22⋊C4—88no—
242C22⋊C4#95132yes—
243C42—48no—
244C42—48no—
245C42—88no—
246C42—88no—
247C42C2×C4214yes—
248C42C2×C22⋊C414yes—
249C42C2×C4⋊C414yes—
250C42C2×C4⋊C414yes—
251C23—22no—
252C23—22no—
253C23—28no—
254C23—28no—
255C23—28no—
256C23—24no—
257C23—24no—
258C23—28no—
259C23—44no—
260C23—44no—
261C23—44no—
262C23—44no—
263C23—88no—
264C23—88no—
265C23C2×#9518yes—
266C23C22×C22⋊C411yescharacteristic
267C23C22×C22⋊C411yescharacteristic
268C23C22×C4⋊C411yescharacteristic
269C23C2×#13512yes—
270C23C2×#13512yes—
271D4—232no—
272C2×C4—28no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—44no—
278C2×C4—42no—
279C2×C4—48no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—48no—
285C2×C4—48no—
286C2×C4—48no—
287C2×C4—44no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—48no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—48no—
297C2×C4—42no—
298C2×C4—48no—
299C2×C4—44no—
300C2×C4—48no—
301C2×C4—84no—
302C2×C4—88no—
303C2×C4C2×#9514yes—
304C2×C4C2×#9518yes—
305C2×C4C2×#9514yes—
306C2×C4#41614yes—
307C2×C4#41614yes—
308C2×C4#41614yes—
309C2×C4#41614yes—
310C2×C4#42718yes—
311C2×C4C22×C4211yescharacteristic
312C2×C4C22×C22⋊C411yescharacteristic
313C2×C4C22×C4⋊C411yescharacteristic
314C2×C4C22×C4⋊C411yescharacteristic
315C8—48no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—24no—
321C22—24no—
322C22—28no—
323C22—28no—
324C22—28no—
325C22—416no—
326C22—84no—
327C22—84no—
328C22—88no—
329C22C22×#9511yescharacteristic
330C22#288312yes—
331C22C2×#41611yescharacteristic
332C22C2×#41611yescharacteristic
333C22C2×#41612yes—
334C22#290112yes—
335C22C2×#42712yes—
336C4—24no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—28no—
342C4—44no—
343C4—82no—
344C4—84no—
345C4—88no—
346C4#173518yes—
347C2—21no—
348C2—21no—
349C2—22no—
350C2—48no—
351C2#1761714yes—
352C2#2034311yescharacteristic
353C2C2×#173511yescharacteristic
354C2#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. #8024C2semidirect
6. #8014C2semidirect
7. #8012C2semidirect
10. #2972C4semidirect
16. #2357C4semidirect
17. #2354C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29722
#341862
C2×#23572
#80262
#80242
#80142
#80122