Finite Group Databasebeta

← #188288

Subgroups — #188288

Order 512.

1080 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2314C211yesmaximal, characteristic
2#21951C212yesmaximal, complemented
3C2×#1458C212yesmaximal
4#8562C214yesmaximal, complemented
5#8561C214yesmaximal
6#7908C212yesmaximal, complemented
7C22×#117C2211yescharacteristic
8C2×#431C2214yes—
9#2314—28no—
10C2×#388C412yes—
11C2×#388C2212yes—
12C2×#387C412yes—
13C2×#387C2212yes—
14C2×#384C411yescharacteristic
15C2×#382C411yescharacteristic
16C2×#380C411yescharacteristic
17C2×#380C411yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20#1939—28no—
21#1797C2218yes—
22#1458—28no—
23C2×#324C2214yes—
24C2×#323C2214yes—
25#1280C2218yes—
26C24×C4C2×C411yescharacteristic
27C2×#117—24no—
28C2×#117—24no—
29C2×#117D414yes—
30C2×#117Q814yes—
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4C2×C412yes—
37C22×C4⋊C4C2×C412yes—
38C22×C4⋊C4C2311yescharacteristic
39C22×C22⋊C4C2×C411yescharacteristic
40C22×C22⋊C4C2×C412yes—
41C22×C22⋊C4C2×C412yes—
42C22×C22⋊C4C2311yescharacteristic
43C22×C42C2×C411yescharacteristic
44C22×C42C2×C412yes—
45C22×C42C2312yes—
46C22×C42C2311yescharacteristic
47#431—28no—
48#431—28no—
49C2×C4⋊C8—24no—
50C2×C4⋊C8—28no—
51C2×C22⋊C8C2318yes—
52C2×C4×C8—24no—
53#388—28no—
54#388—28no—
55#387—28no—
56#387—28no—
57#384—24no—
58#382—24no—
59#380—24no—
60#380—24no—
61C4×C4⋊C4—24no—
62C4×C4⋊C4D418yes—
63C2×#95C2×C412yes—
64C2×#95C2×C412yes—
65C2×#95C2×C412yes—
66C2×#95C2×C412yes—
67C2×#95C2×C412yes—
68C2×#95C2312yes—
69#336—28no—
70#324—28no—
71#323—28no—
72C25C22×C411yescharacteristic
73C23×C4—22no—
74C23×C4—22no—
75C23×C4—24no—
76C23×C4—24no—
77C23×C4C4⋊C414yes—
78C23×C4C22×C411yescharacteristic
79C23×C4C22×C412yes—
80C23×C4C22×C412yes—
81C23×C4C22×C412yes—
82C23×C4C22×C412yes—
83C23×C4C22×C411yescharacteristic
84C23×C4C22×C411yescharacteristic
85C23×C4C22×C411yescharacteristic
86C23×C4C22×C411yescharacteristic
87C23×C4C2411yescharacteristic
88C22×C8—28no—
89#117—216no—
90#117—216no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—82no—
104C2×C4⋊C4—82no—
105C2×C4⋊C4C22⋊C414yes—
106C2×C4⋊C4C22⋊C414yes—
107C2×C4⋊C4C4⋊C414yes—
108C2×C4⋊C4C2×D412yes—
109C2×C4⋊C4C2×Q812yes—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—24no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—82no—
117C2×C22⋊C4—82no—
118C2×C22⋊C4C4⋊C414yes—
119C2×C22⋊C4C2×D412yes—
120C2×C22⋊C4C2×Q812yes—
121C2×C42—22no—
122C2×C42—22no—
123C2×C42—24no—
124C2×C42—82no—
125C2×C42—82no—
126C2×C42C4⋊C414yes—
127C2×C42C2×D418yes—
128C2×C42C2×D412yes—
129C2×C42C2×D414yes—
130C2×C42C2×Q812yes—
131C2×C42#4114yes—
132C4⋊C8—48no—
133C4⋊C8—88no—
134C22⋊C8—28no—
135C22⋊C8—28no—
136C4×C8—48no—
137#95—24no—
138#95—24no—
139#95—24no—
140#95—24no—
141#95—24no—
142#95C22⋊C418yes—
143C24—21no—
144C24—21no—
145C24—22no—
146C24—22no—
147C24—41no—
148C24—41no—
149C24—41no—
150C24—41no—
151C24C2×C4⋊C412yes—
152C24C23×C411yescharacteristic
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—24no—
162C22×C4—24no—
163C22×C4—28no—
164C22×C4—216no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—44no—
178C22×C4—44no—
179C22×C4—48no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4C2×C22⋊C414yes—
189C22×C4C2×C22⋊C414yes—
190C22×C4C2×C22⋊C412yes—
191C22×C4C2×C22⋊C412yes—
192C22×C4C2×C4⋊C412yes—
193C22×C4C2×C4⋊C412yes—
194C22×C4C2×C4⋊C412yes—
195C22×C4C2×C4⋊C412yes—
196C22×C4C2×C4⋊C412yes—
197C22×C4#11712yes—
198C22×C4#11712yes—
199C22×C4#13114yes—
200C22×C4#13114yes—
201C22×C4C22×D412yes—
202C22×C4C22×D411yescharacteristic
203C22×C4C22×Q811yescharacteristic
204C22×C4C2×#4112yes—
205C2×C8—48no—
206C2×C8—48no—
207C2×C8—48no—
208C4⋊C4—24no—
209C4⋊C4—24no—
210C4⋊C4—28no—
211C4⋊C4—28no—
212C4⋊C4—28no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C4⋊C4—84no—
216C4⋊C4—84no—
217C22⋊C4—24no—
218C22⋊C4—24no—
219C22⋊C4—28no—
220C22⋊C4—84no—
221C22⋊C4—84no—
222C42—28no—
223C42—48no—
224C42—84no—
225C42#12118yes—
226C42C22⋊Q818yes—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—22no—
232C23—24no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—42no—
244C23—82no—
245C23—82no—
246C23—82no—
247C23—82no—
248C23C22×C22⋊C411yescharacteristic
249C23C22×C4⋊C411yescharacteristic
250C23C2×#11711yescharacteristic
251C23C2×#13112yes—
252C23C2×#13112yes—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—416no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—84no—
291C2×C4—84no—
292C2×C4—84no—
293C2×C4—84no—
294C2×C4#38014yes—
295C2×C4#38014yes—
296C2×C4#40814yes—
297C2×C4#40814yes—
298C2×C4#42418yes—
299C2×C4#43318yes—
300C2×C4C2×#12112yes—
301C2×C4C2×C22⋊Q812yes—
302C8—88no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—84no—
315C22—84no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22C2×#38011yescharacteristic
321C22C2×#40811yescharacteristic
322C22C2×#40811yescharacteristic
323C22#288311yescharacteristic
324C22#288411yescharacteristic
325C22#288811yescharacteristic
326C22#288911yescharacteristic
327C22C2×#42412yes—
328C22C2×#43312yes—
329C4—22no—
330C4—22no—
331C4—24no—
332C4—42no—
333C4—42no—
334C4—42no—
335C4—44no—
336C4—44no—
337C4—82no—
338C4—82no—
339C4#179818yes—
340C2—22no—
341C2—22no—
342C2—82no—
343C2#1738811yescharacteristic
344C2#1738911yescharacteristic
345C2#1739011yescharacteristic
346C2#1739111yescharacteristic
347C2C2×#179811yescharacteristic
348C2#2063411yescharacteristic
349C2#2064011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21951C2semidirect
4. #8562C2semidirect
6. #7908C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23142
#219512
C2×#14582
#85622
#85612
#79082