Finite Group Databasebeta

← #480046

Subgroups — #480046

Order 512.

1066 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34184C211yesmaximal, characteristic
2#17351C212yesmaximal
3#11449C212yesmaximal, complemented
4#11448C214yesmaximal, complemented
5#11447C212yesmaximal, complemented
6C2×#1324C214yesmaximal, complemented
7C2×#520C411yescharacteristic
8C2×#520C2211yescharacteristic
9C2×C4×#41C411yescharacteristic
10#2972—22no—
11#2351—22no—
12#2349—24no—
13#2349—22no—
14#2317C412yes—
15#2317C411yescharacteristic
16#2317C2211yescharacteristic
17#2317C2212yes—
18#2285C412yes—
19#2285C2211yescharacteristic
20C2×#382C411yescharacteristic
21C2×#382C2212yes—
22#1461—22no—
23#1459—24no—
24#1458C2214yes—
25C2×#324C2218yes—
26#1325C2214yes—
27#1324C2214yes—
28#1324C2218yes—
29C22×#41C2×C411yescharacteristic
30#520—44no—
31#520—44no—
32C4×#41—24no—
33C4×#41C818yescomplemented
34C2×C4×Q8—21no—
35C2×C4×Q8—21no—
36C2×C4×Q8D412yes—
37C2×C4×D4—22no—
38C2×C4×D4—21no—
39C2×C4×D4—22no—
40C2×C4×D4—21no—
41C2×C4×D4D414yes—
42C2×C4×D4D412yes—
43C2×#117C2×C411yescharacteristic
44C2×#117C2×C412yes—
45C2×#117C2×C412yes—
46C2×#117C2×C412yes—
47C2×#117C2×C411yescharacteristic
48C2×#117C2311yescharacteristic
49C22×C4⋊C4C2×C412yes—
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C42C2×C411yescharacteristic
54C22×C42C2×C412yes—
55C8×D4—48no—
56#431—24no—
57C2×C22⋊C8—28no—
58#389—22no—
59#389—24no—
60#389—24no—
61#389—22no—
62#385—22no—
63#385—24no—
64#385—24no—
65#385—22no—
66#384C2×C411yescharacteristic
67#384C2×C412yes—
68#384C2×C412yes—
69#384C2311yescharacteristic
70#382—22no—
71#382—24no—
72#382C2×C412yes—
73#382C2×C412yes—
74#382C2×C412yes—
75#382C2×C411yescharacteristic
76#382C2312yes—
77#382C2311yescharacteristic
78#380—24no—
79#380—22no—
80C4×C4⋊C4—24no—
81C4×C4⋊C4—22no—
82C2×#95—22no—
83C2×#95—24no—
84#324—28no—
85#324C2318yes—
86#323—24no—
87C2×#41—24no—
88C2×#41—24no—
89C2×#41C2×C814yes—
90C2×#41M1614yes—
91C22×Q8C22⋊C411yescharacteristic
92C22×D4C22⋊C412yes—
93C22×D4C22⋊C411yescharacteristic
94C23×C4—21no—
95C23×C4—22no—
96C23×C4—21no—
97C23×C4—22no—
98C23×C4C22⋊C412yes—
99C23×C4C22×C411yescharacteristic
100C23×C4C22×C412yes—
101C23×C4C2×D414yes—
102C22×C8—48no—
103C4×Q8—28no—
104C4×Q8—44no—
105C4×Q8—44no—
106C4×D4—216no—
107C4×D4—28no—
108C4×D4—28no—
109C4×D4—48no—
110C4×D4—48no—
111C4×D4—44no—
112#117—24no—
113#117—22no—
114#117—42no—
115#117—44no—
116#117—44no—
117#117—42no—
118#117C2×C818yescomplemented
119#117C2×C814yescomplemented
120C2×C4⋊C4—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—44no—
130C2×C4⋊C4—44no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4C22⋊C412yes—
133C2×C4⋊C4C22×C411yescharacteristic
134C2×C4⋊C4C22×C412yes—
135C2×C4⋊C4C22×C411yescharacteristic
136C2×C4⋊C4C22×C412yes—
137C2×C4⋊C4C22×C411yescharacteristic
138C2×C4⋊C4C2×D414yes—
139C2×C4⋊C4C2411yescharacteristic
140C2×C22⋊C4—21no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4C22⋊C414yes—
145C2×C22⋊C4C2×D412yes—
146C2×C42—22no—
147C2×C42—21no—
148C2×C42—22no—
149C2×C42—24no—
150C2×C42—22no—
151C2×C42—21no—
152C2×C42C22⋊C414yes—
153C2×C42C2×C814yescomplemented
154C2×C42C2×C818yescomplemented
155C2×C42C22×C411yescharacteristic
156C2×C42C22×C412yes—
157C2×C42C22×C411yescharacteristic
158C2×C42C2×D412yes—
159C4⋊C8—44no—
160C22⋊C8—28no—
161C22⋊C8—48no—
162C4×C8—44no—
163#95—24no—
164#95—24no—
165#95—24no—
166#95—22no—
167#95—22no—
168#95—24no—
169C24C2×C22⋊C411yescharacteristic
170C24C2×C22⋊C412yes—
171#41—216no—
172#41—216no—
173C2×Q8—22no—
174C2×Q8—22no—
175C2×Q8C22⋊C814yes—
176C2×D4—24no—
177C2×D4—24no—
178C2×D4C22⋊C818yes—
179C2×D4C22⋊C814yes—
180C2×D4#9914yes—
181C2×D4#10114yes—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—24no—
190C22×C4—28no—
191C22×C4—28no—
192C22×C4—22no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—44no—
198C22×C4—82no—
199C22×C4—82no—
200C22×C4—81no—
201C22×C4—81no—
202C22×C4—84no—
203C22×C4—84no—
204C22×C4—82no—
205C22×C4—82no—
206C22×C4C2×C22⋊C412yes—
207C22×C4C2×C22⋊C414yes—
208C22×C4C2×C22⋊C412yes—
209C22×C4C2×C22⋊C411yescharacteristic
210C22×C4C22×C814yes—
211C22×C4C22×C812yes—
212C22×C4C2×M1614yes—
213C22×C4C2×M1612yes—
214C22×C4C23×C411yescharacteristic
215C22×C4C22×D412yes—
216C2×C8—48no—
217C2×C8—168no—
218C4⋊C4—28no—
219C4⋊C4—24no—
220C4⋊C4—24no—
221C4⋊C4—42no—
222C4⋊C4—41no—
223C4⋊C4—44no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—42no—
227C4⋊C4—41no—
228C4⋊C4—44no—
229C4⋊C4—42no—
230C4⋊C4—84no—
231C4⋊C4—82no—
232C4⋊C4—84no—
233C4⋊C4—82no—
234C22⋊C4—24no—
235C22⋊C4—28no—
236C22⋊C4—28no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—44no—
240C42—21no—
241C42—21no—
242C42—22no—
243C42—28no—
244C42—24no—
245C42—24no—
246C42—44no—
247C42—44no—
248C42—42no—
249C42C22×C812yes—
250C42C22×C812yes—
251C42C22×C814yes—
252C23—21no—
253C23—21no—
254C23—42no—
255C23—42no—
256C23—41no—
257C23—41no—
258C23C2×C22⋊C812yes—
259C23C2×C22⋊C814yes—
260C23C2×#9914yes—
261C23C2×#10114yes—
262C23C22×C22⋊C411yescharacteristic
263Q8—28no—
264D4—216no—
265D4#324116yes—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—28no—
271C2×C4—28no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—44no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—44no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—88no—
296C2×C4—88no—
297C2×C4—84no—
298C2×C4—162no—
299C2×C4—164no—
300C2×C4—162no—
301C2×C4C2×C22⋊C814yes—
302C2×C4C2×C22⋊C812yes—
303C2×C4C2×#9912yes—
304C2×C4C2×#10112yes—
305C2×C4C23×C811yescharacteristic
306C2×C4C22×M1611yescharacteristic
307C8—88no—
308C22—21no—
309C22—24no—
310C22—21no—
311C22—24no—
312C22—22no—
313C22—44no—
314C22—44no—
315C22—42no—
316C22—42no—
317C22—41no—
318C22—41no—
319C22C2×#32418yes—
320C22C22×C22⋊C811yescharacteristic
321C22C22×#9911yescharacteristic
322C22C22×#10111yescharacteristic
323C4—21no—
324C4—21no—
325C4—24no—
326C4—44no—
327C4—42no—
328C4—42no—
329C4—42no—
330C4—44no—
331C4—42no—
332C4—81no—
333C4—81no—
334C4—82no—
335C4C2×#32414yes—
336C2—21no—
337C2—21no—
338C2—28no—
339C2—42no—
340C2C22×#32411yescharacteristic
341C2#3336111yescharacteristic
342C2#3337011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC2×C4C23×C8
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#41C2×C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11449C2semidirect
4. #11448C2semidirect
5. #11447C2semidirect
6. C2×#1324C2semidirect
33. C4×#41C8semidirect
118. #117C2×C8semidirect
119. #117C2×C8semidirect
153. C2×C42C2×C8semidirect
154. C2×C42C2×C8semidirect

Maximal subgroups

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

subgroupindex
#341842
#173512
#114492
#114482
#114472
C2×#13242