Finite Group Databasebeta

← #480321

Subgroups — #480321

Order 512.

705 conjugacy classes of subgroups.

The subgroup lattice has 347 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60126C211yesmaximal, characteristic
2#33427C212yesmaximal, complemented
3#11524C214yesmaximal, complemented
4#11523C214yesmaximal, complemented
5#11458C212yesmaximal, complemented
6C2×#1321C212yesmaximal
7#3428C411yescharacteristic
8#2925—24no—
9#2924—22no—
10#2924—22no—
11#2923C412yes—
12#2923C2212yes—
13#2922C412yes—
14#2922C2212yes—
15#2905C2211yescharacteristic
16C2×#424C411yescharacteristic
17#2879C411yescharacteristic
18C2×#408C2211yescharacteristic
19#2873C2211yescharacteristic
20C2×#405C411yescharacteristic
21#2166—24no—
22#2165C2218yescomplemented
23#2153—22no—
24C2×#349C2214yes—
25#2110—22no—
26#1328C2218yescomplemented
27#1321C2214yes—
28#1321C2214yes—
29C22×#41C2×C411yescharacteristic
30C22×M16C2×C411yescharacteristic
31C22×M16C2×C411yescharacteristic
32C22×M16C2311yescharacteristic
33C23×C8C2×C411yescharacteristic
34#520—21no—
35#520—21no—
36#520—22no—
37#519—21no—
38#518—21no—
39#518—22no—
40C4×#41—22no—
41C4×#41C2×C412yes—
42C4×#41C2×C412yes—
43C2×#117—21no—
44C2×#117—21no—
45C2×#117C2×C411yescharacteristic
46C2×#117C2×C411yescharacteristic
47C2×M32—24no—
48#433—24no—
49#433—24no—
50#433—24no—
51#433—24no—
52#432—24no—
53#432—24no—
54#431—24no—
55#431—24no—
56#424—22no—
57#424—22no—
58#424C2×C412yes—
59#424C2×C412yes—
60#424C2×C412yes—
61#424C2312yes—
62#423—22no—
63C2×C4⋊C8—22no—
64#408—22no—
65#408—22no—
66#408C2×C411yescharacteristic
67#408C2×C411yescharacteristic
68#408C2×C412yes—
69#408C2×C412yes—
70#408C2311yescharacteristic
71#408C2311yescharacteristic
72#407—22no—
73C2×C22⋊C8—22no—
74#405—21no—
75#405—22no—
76#405—21no—
77#405D412yes—
78#405D414yes—
79#405D412yes—
80C4×M16C2×C412yes—
81C4×M16C2312yes—
82C2×#97C2×C412yes—
83C2×C4×C8C2×C412yes—
84#364—28no—
85#349—24no—
86#349C2318yes—
87#347—28no—
88C2×#41—21no—
89C2×#41—21no—
90C2×#41—42no—
91C2×#41—42no—
92C2×#41M1611yescharacteristic
93C2×#41M1611yescharacteristic
94C2×#41C22×C411yescharacteristic
95C2×#41C22×C411yescharacteristic
96C22×Q8M1611yescharacteristic
97C22×D4—21no—
98C22×D4M1611yescharacteristic
99C23×C4—21no—
100C23×C4C22×C411yescharacteristic
101C2×M16—21no—
102C2×M16—21no—
103C2×M16—42no—
104C2×M16—42no—
105C2×M16—42no—
106C2×M16C22⋊C412yes—
107C2×M16C22×C412yes—
108C2×M16C22×C411yescharacteristic
109C2×M16C2×D412yes—
110C2×M16C2411yescharacteristic
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—42no—
114C22×C8C22⋊C412yes—
115C22×C8C22×C411yescharacteristic
116C22×C8C22×C411yescharacteristic
117C22×C8C22×C412yes—
118C22×C8C2×D412yes—
119C4×Q8—24no—
120C4×Q8—22no—
121C4×Q8—22no—
122C4×D4—24no—
123C4×D4—24no—
124C4×D4—24no—
125C4×D4—22no—
126C4×D4—22no—
127C4×D4—24no—
128C4×D4—24no—
129#117—22no—
130#117—22no—
131#117—22no—
132#117—22no—
133#117—21no—
134#117—21no—
135#117—21no—
136#117—21no—
137#117—22no—
138#117—22no—
139#117C22⋊C411yescharacteristic
140#117C22⋊C412yes—
141#117C22⋊C411yescharacteristic
142#117M1612yes—
143#117C22×C412yes—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4M1612yes—
152C2×C42—21no—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42C22×C412yes—
156M32—28no—
157M32—44no—
158C2×C16—44no—
159C4⋊C8—24no—
160C4⋊C8—24no—
161C4⋊C8—24no—
162C4⋊C8—24no—
163C22⋊C8—22no—
164C22⋊C8—22no—
165C22⋊C8—24no—
166C22⋊C8—24no—
167C22⋊C8—22no—
168C22⋊C8—22no—
169#97—22no—
170#97—22no—
171#97C22⋊C414yes—
172#97C2×D414yes—
173C4×C8—22no—
174C4×C8—22no—
175C4×C8C22⋊C414yes—
176C4×C8C2×D414yes—
177C24—21no—
178C24C2×M1611yescharacteristic
179#41—42no—
180#41—42no—
181#41—44no—
182#41—42no—
183#41—42no—
184#41—41no—
185#41—41no—
186#41—41no—
187#41—41no—
188C2×Q8—21no—
189C2×Q8—42no—
190C2×Q8C2×M1611yescharacteristic
191C2×Q8C2×M1611yescharacteristic
192C2×D4—21no—
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—42no—
198C2×D4—44no—
199C2×D4C2×M1611yescharacteristic
200C2×D4C2×M1611yescharacteristic
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—21no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4C2×C22⋊C411yescharacteristic
215C22×C4C2×C22⋊C411yescharacteristic
216C22×C4C2×M1611yescharacteristic
217C22×C4C2×M1611yescharacteristic
218C22×C4C2×M1611yescharacteristic
219C22×C4C23×C411yescharacteristic
220M16—24no—
221M16—41no—
222M16—42no—
223M16—42no—
224M16—41no—
225M16—42no—
226M16—42no—
227C2×C8—24no—
228C2×C8—21no—
229C2×C8—22no—
230C2×C8—21no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—42no—
237C2×C8C2×C22⋊C412yes—
238C2×C8C2×C22⋊C414yes—
239C2×C8C22×D412yes—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—22no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C22⋊C4—24no—
248C22⋊C4—24no—
249C22⋊C4—21no—
250C22⋊C4—21no—
251C22⋊C4—22no—
252C22⋊C4—22no—
253C22⋊C4C2×M1614yes—
254C42—24no—
255C42—24no—
256C42—22no—
257C42—22no—
258C42—22no—
259C42C2×C22⋊C412yes—
260C42C2×C22⋊C412yes—
261C16—48no—
262C23—22no—
263C23—21no—
264C23—21no—
265C23—42no—
266C23—81no—
267C23—82no—
268C23#40711yescharacteristic
269C23#40711yescharacteristic
270C23C22×M1611yescharacteristic
271Q8—41no—
272Q8—41no—
273Q8—42no—
274D4—41no—
275D4—41no—
276D4—42no—
277D4—42no—
278D4—42no—
279D4—42no—
280D4—42no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—84no—
303C2×C4—84no—
304C2×C4#40711yescharacteristic
305C2×C4#40711yescharacteristic
306C2×C4#41012yes—
307C2×C4#41312yes—
308C2×C4C22×C22⋊C411yescharacteristic
309C2×C4C22×M1611yescharacteristic
310C8—22no—
311C8—22no—
312C8—24no—
313C8—42no—
314C8—41no—
315C8—41no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—42no—
322C22—42no—
323C22—41no—
324C22—41no—
325C22—82no—
326C22—162no—
327C22#146412yes—
328C22C2×#40711yescharacteristic
329C4—21no—
330C4—21no—
331C4—44no—
332C4—42no—
333C4—42no—
334C4—41no—
335C4—41no—
336C4#146412yes—
337C4C2×#41011yescharacteristic
338C4C2×#41311yescharacteristic
339C2—21no—
340C2—22no—
341C2—41no—
342C2—41no—
343C2—81no—
344C2—81no—
345C2C2×#146411yescharacteristic
subgroupG/H
centerC4C2×#410
commutatorC22×C4C23×C4
Frattini, cosocleC2×M16C24
socleC2C2×#1464
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ #117 ◁ #2873 ◁ G
LECSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33427C2semidirect
3. #11524C2semidirect
4. #11523C2semidirect
5. #11458C2semidirect
22. #2165C22semidirect
26. #1328C22semidirect

Maximal subgroups

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

subgroupindex
#601262
#334272
#115242
#115232
#114582
C2×#13212