Finite Group Databasebeta

← #214644

Subgroups — #214644

Order 512.

896 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37390C211yesmaximal, characteristic
2#21756C211yesmaximal, characteristic
3#21751C211yesmaximal, characteristic
4#9462C214yesmaximal, complemented
5#9443C214yesmaximal
6#8741C211yesmaximal, characteristic, complemented
7#8740C211yesmaximal, characteristic, complemented
8#8521C211yesmaximal, characteristic, complemented
9#8521C211yesmaximal, characteristic, complemented
10C2×#541C2211yescharacteristic
11C2×#536C2211yescharacteristic
12#2879C2212yes—
13#2868C411yescharacteristic, complemented
14#2861C411yescharacteristic, complemented
15#2720—24no—
16#2718—24no—
17#2658C411yescharacteristic, complemented
18#2642C411yescharacteristic, complemented
19#2546C412yescomplemented
20#2545C412yescomplemented
21#2434C2212yes—
22#2432C2212yes—
23#2310C2212yes—
24#2298C2211yescharacteristic
25#1901—24no—
26#1899—24no—
27#1822C2211yescharacteristic
28#1822C2211yescharacteristic
29#1820C2211yescharacteristic
30#1820C2211yescharacteristic
31#1784C2212yes—
32#1782C2212yes—
33#1511C2214yes—
34#1511C2214yes—
35#1284C2214yes—
36#1284C2214yes—
37C22×#41C2311yescharacteristic
38C23×Q8C2311yescharacteristic
39C22×M16—22no—
40#541—44no—
41#536—44no—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C412yes—
44C2×C4⋊Q8C2×C412yes—
45C2×C4⋊Q8C2×C412yes—
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×#126—22no—
49C2×#125—22no—
50C2×#125C2×C411yescharacteristic
51C2×#124C2×C412yes—
52C2×#124C2×C411yescharacteristic
53C2×#124C2×C411yescharacteristic
54C2×#123D414yes—
55C2×C22⋊Q8D414yes—
56C2×C22⋊Q8C2312yes—
57C2×C4×Q8C2×C412yes—
58C2×#117—22no—
59C2×#117C2311yescharacteristic
60C2×#101—24no—
61C2×#99—24no—
62#408—44no—
63#407—22no—
64#407C2312yes—
65#407C2312yes—
66#400—24no—
67#399—24no—
68#398—24no—
69#396—24no—
70#395—24no—
71#394—24no—
72#391—44no—
73#391C2×C412yes—
74#391C2×C412yes—
75#391C2×C412yes—
76#390C2×C412yes—
77#389—24no—
78#388—24no—
79#388—24no—
80#388—22no—
81#387—22no—
82#385—24no—
83#385—24no—
84#384—24no—
85#381—22no—
86#381C2312yes—
87#381C2312yes—
88C4×C4⋊C4C2×C412yes—
89C4×C22⋊C4C2312yes—
90#376C2×C411yescharacteristic
91C43C2×C411yescharacteristic
92#343—22no—
93#343—22no—
94#342—22no—
95#342—22no—
96#328—44no—
97#328—44no—
98C2×#41—24no—
99C2×#41—82no—
100C22×Q8—24no—
101C22×Q8—82no—
102C22×Q8C22×C412yes—
103C22×Q8C22×C411yescharacteristic
104C22×Q8C22×C411yescharacteristic
105C22×Q8C22×C411yescharacteristic
106C22×Q8C2×D412yes—
107C22×D4C22×C411yescharacteristic
108C22×D4C2×D412yes—
109C23×C4C2×D412yes—
110C23×C4C2×D412yes—
111C23×C4C2411yescharacteristic
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—42no—
115C2×M16—84no—
116C22×C8—42no—
117C4⋊Q8—42no—
118C4⋊Q8—42no—
119C4⋊Q8—42no—
120C4⋊Q8—82no—
121C4⋊Q8—82no—
122C4⋊Q8—82no—
123#126—48no—
124#125—42no—
125#125—48no—
126#124—42no—
127#124—42no—
128#124—82no—
129#123—44no—
130#123—44no—
131C22⋊Q8—24no—
132C22⋊Q8—24no—
133C22⋊Q8—44no—
134C22⋊Q8—44no—
135C4×Q8—42no—
136C4×Q8—42no—
137#117—24no—
138#117—84no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4C22⋊C414yes—
149C2×C4⋊C4C22⋊C414yes—
150C2×C4⋊C4C22×C412yes—
151C2×C4⋊C4C2×D412yes—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4C22⋊C414yes—
159C2×C22⋊C4C22⋊C414yes—
160C2×C22⋊C4C2×D412yes—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—42no—
164C2×C42C22×C411yescharacteristic
165C2×C42C22×C411yescharacteristic
166C2×C42C22×C412yes—
167C2×C42C22×C412yes—
168#101—84no—
169#99—84no—
170C22⋊C8—24no—
171C22⋊C8—24no—
172C22⋊C8—44no—
173#95—22no—
174#95—22no—
175#95—24no—
176#95—24no—
177#95—24no—
178#95—22no—
179#95—24no—
180#95—24no—
181#95—44no—
182C24C2×C22⋊C412yes—
183C24C22×D411yescharacteristic
184#41—84no—
185#41—84no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8—22no—
189C2×Q8—24no—
190C2×Q8—22no—
191C2×Q8—22no—
192C2×Q8—44no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—42no—
197C2×Q8—42no—
198C2×Q8—84no—
199C2×Q8—84no—
200C2×D4—24no—
201C2×D4—22no—
202C2×D4—44no—
203C2×D4—42no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4C2×C22⋊C412yes—
218C22×C4C2×C22⋊C412yes—
219C22×C4C2×C22⋊C412yes—
220C22×C4C2×C22⋊C412yes—
221C22×C4C2×C22⋊C412yes—
222C22×C4C23×C411yescharacteristic
223C22×C4C22×D411yescharacteristic
224C22×C4H(5,𝔽2)14yes—
225M16—44no—
226M16—44no—
227M16—84no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—84no—
231C4⋊C4—24no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—44no—
238C4⋊C4—44no—
239C4⋊C4—44no—
240C4⋊C4—44no—
241C4⋊C4—84no—
242C4⋊C4—84no—
243C4⋊C4—84no—
244C4⋊C4—84no—
245C22⋊C4—24no—
246C22⋊C4—44no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C22⋊C4—48no—
250C22⋊C4—44no—
251C22⋊C4—44no—
252C22⋊C4—44no—
253C22⋊C4—84no—
254C42—22no—
255C42—22no—
256C42—42no—
257C42—42no—
258C42—44no—
259C42—44no—
260C42—84no—
261C23—22no—
262C23—22no—
263C23—24no—
264C23—42no—
265C23—44no—
266C23C22×C22⋊C411yescharacteristic
267C23#53412yes—
268Q8—84no—
269Q8—82no—
270Q8—82no—
271Q8—82no—
272Q8—82no—
273Q8—82no—
274D4—82no—
275D4—84no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—44no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—84no—
309C2×C4—84no—
310C2×C4#51812yes—
311C2×C4#53412yes—
312C8—82no—
313C8—82no—
314C22—22no—
315C22—24no—
316C22—22no—
317C22—44no—
318C22—42no—
319C22—82no—
320C22—84no—
321C22#230811yescharacteristic
322C22#288312yes—
323C4—22no—
324C4—22no—
325C4—44no—
326C4—42no—
327C4—42no—
328C4—42no—
329C4—42no—
330C4—42no—
331C4—82no—
332C4—82no—
333C4—82no—
334C4—162no—
335C4—162no—
336C2—22no—
337C2—42no—
338C2—82no—
339C2#2175011yescharacteristic
340C2#2178312yes—
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9462C2semidirect
6. #8741C2semidirect
7. #8740C2semidirect
8. #8521C2semidirect
9. #8521C2semidirect
13. #2868C4semidirect
14. #2861C4semidirect
17. #2658C4semidirect
18. #2642C4semidirect
19. #2546C4semidirect
20. #2545C4semidirect

Maximal subgroups

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

subgroupindex
#373902
#217562
#217512
#94622
#94432
#87412
#87402
#85212