Finite Group Databasebeta

← C2×D3×#214 #10595103

Subgroups — C2×D3×#214 #10595103

Order 576.

630 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 160 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D3×D4C211yesmaximal, characteristic
2C2×D3×Dic6C211yesmaximal, characteristic, complemented
3C2×#3754C211yesmaximal, characteristic, complemented
4C2×C3×#214C211yesmaximal, characteristic, complemented
5D3×#214C218yesmaximal, direct product factor
6C2×#3716C211yesmaximal, characteristic, complemented
7C2×#3715C211yesmaximal, characteristic, complemented
8C2×D3×C3⋊C8C211yesmaximal, characteristic, complemented
9C22×#214—31nomaximal
10C2×D3×SD16—31nomaximal
11C23×C3×D3—21no—
12C2×C32×D4C2211yescharacteristic
13C2×#634C2211yescharacteristic, complemented
14C2×C3×#67—21no—
15C3×D3×D4—22no—
16C3×D3×D4C2214yes—
17C2×C3×D12C2211yescharacteristic
18C2×C3×C4×D3C2211yescharacteristic
19C2×C3×Dic6C2211yescharacteristic, complemented
20C2×#627—21no—
21C2×D3×Dic3—21no—
22D3×Dic6—22no—
23D3×Dic6C2214yescomplemented
24#3754C2214yescomplemented
25C2×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
26C3×#214C2214yescomplemented
27C2×C3×C3⋊C8C2211yescharacteristic, complemented
28#3716C2214yescomplemented
29#3715C2214yescomplemented
30D3×C3⋊C8C2214yescomplemented
31C22×C3×D4—31no—
32C2×D3×Q8—31no—
33C2×D3×D4D311yescharacteristic
34C22×Dic6—31no—
35C2×C3×SD16—31no—
36C2×#215—31no—
37C2×#214—31no—
38C2×#214—31no—
39C2×#214—31no—
40C2×#214—38no—
41C2×#214—31no—
42C2×#214—61no—
43C2×#214D311yescharacteristic, direct product factor
44C22×C3⋊C8—31no—
45D3×SD16—38no—
46C2×#204—31no—
47C2×D3×C8—31no—
48C23×C32—21no—
49C22×C3×D3—21no—
50C22×C3×D3—24no—
51C22×C3×D3—44no—
52C22×C3×D3D411yescharacteristic
53C32×D4—21no—
54C32×D4C2312yes—
55C2×C32×C4C2311yescharacteristic
56C2×C3⋊Dic3—21no—
57#634—21no—
58#634C2312yescomplemented
59C3×#67—24no—
60C2×C3×Dic3—21no—
61C2×C3×Dic3D411yescharacteristic
62C3×D12—21no—
63C3×D12C2312yes—
64C3×C4×D3D412yes—
65C3×C4×D3C2312yes—
66C3×Dic6—21no—
67C3×Dic6C2312yescomplemented
68#627—24no—
69D3×Dic3—24no—
70C3⋊(C3⋊C8)C2312yescomplemented
71C3×C3⋊C8C2312yescomplemented
72C22×SD16—91noSylow
73C24×C3—61no—
74C23×D3—21no—
75C2×C3×Q8—31no—
76C2×C3×D4—21no—
77C2×C3×D4—34no—
78C2×C3×D4—31no—
79C2×C3×D4—62no—
80C2×C3×D4—61no—
81C2×C3×D4C2×D311yescharacteristic
82C2×C3×D4C2×D311yescharacteristic
83C22×C3×C4—31no—
84C2×#67—21no—
85C22×Dic3—61no—
86D3×Q8—34no—
87D3×Q8—62no—
88D3×D4—22no—
89D3×D4C2×D314yes—
90C2×D12C2×D311yescharacteristic
91C2×C4×D3—61no—
92C2×C4×D3C2×D311yescharacteristic
93C2×Dic6—31no—
94C2×Dic6—31no—
95C2×Dic6—34no—
96C2×Dic6—61no—
97C2×Dic6—62no—
98C2×Dic6—61no—
99C2×Dic6—61no—
100C2×Dic6C2×D311yescharacteristic, complemented
101C3×SD16—34no—
102C2×C3×C8—31no—
103#215—34no—
104#214—34no—
105#214—34no—
106#214—34no—
107#214—34no—
108#214—64no—
109#214C2×D314yesdirect product factor
110C2×C3⋊C8—31no—
111C2×C3⋊C8—31no—
112C2×C3⋊C8—34no—
113C2×C3⋊C8—61no—
114C2×C3⋊C8C2×D311yescharacteristic, complemented
115#204—34no—
116D3×C8—34no—
117C22×C32—22no—
118C22×C32—42no—
119C22×C32C2×D411yescharacteristic
120C2×C3×D3—22no—
121C2×C3×D3—42no—
122C2×C3×D3—48no—
123C2×C3×D3SD1614yescomplemented
124C2×C3×D3C2×D412yes—
125C32×C4C2×D411yescharacteristic
126C32×C4C2411yescharacteristic
127C3⋊Dic3—22no—
128C3×Dic3—22no—
129C3×Dic3C2×D412yes—
130C22×Q8—91no—
131C22×D4—31no—
132C2×SD16—31no—
133C2×SD16—91no—
134C2×SD16—98no—
135C2×SD16—91no—
136C2×SD16—91no—
137C22×C8—91no—
138C23×C3—21no—
139C23×C3—21no—
140C23×C3—31no—
141C23×C3—41no—
142C23×C3—64no—
143C23×C3—61no—
144C23×C3—124no—
145C22×D3—21no—
146C22×D3—24no—
147C22×D3—44no—
148C22×D3#6711yescharacteristic
149C3×Q8—32no—
150C3×Q8—61no—
151C3×D4—21no—
152C3×D4—21no—
153C3×D4—22no—
154C3×D4—32no—
155C3×D4—41no—
156C3×D4—64no—
157C3×D4—61no—
158C3×D4C22×D312yes—
159C3×D4C22×D312yes—
160C2×C3×C4—21no—
161C2×C3×C4—31no—
162C2×C3×C4—32no—
163C2×C3×C4—32no—
164C2×C3×C4—61no—
165C2×C3×C4C22×D311yescharacteristic
166C2×C3×C4C22×D311yescharacteristic
167#67—24no—
168C2×Dic3—21no—
169C2×Dic3—64no—
170C2×Dic3—61no—
171C2×Dic3—61no—
172C2×Dic3—121no—
173C2×Dic3#6711yescharacteristic
174D12—21no—
175D12C22×D312yes—
176C4×D3—64no—
177C4×D3#6712yes—
178C4×D3C22×D312yes—
179Dic6—21no—
180Dic6—32no—
181Dic6—32no—
182Dic6—62no—
183Dic6—64no—
184Dic6—61no—
185Dic6—61no—
186Dic6—64no—
187Dic6—121no—
188Dic6C22×D312yescomplemented
189C3×C8—32no—
190C3⋊C8—32no—
191C3⋊C8—32no—
192C3⋊C8—62no—
193C3⋊C8C22×D312yescomplemented
194C2×C32—42no—
195C2×C32C2×SD1612yescomplemented
196C2×C32C22×D411yescharacteristic
197C3×D3—42no—
198C3×D3C2×SD1614yescomplemented
199C24—61no—
200C2×Q8—31no—
201C2×Q8—94no—
202C2×Q8—91no—
203C2×Q8—181no—
204C2×Q8—182no—
205C2×D4—34no—
206C2×D4—31no—
207C2×D4—62no—
208C2×D4—61no—
209C2×D4D3211yescharacteristic
210C22×C4—31no—
211C22×C4—181no—
212SD16—34no—
213SD16—94no—
214SD16—94no—
215SD16—94no—
216C2×C8—31no—
217C2×C8—94no—
218C2×C8—91no—
219C22×C3—22no—
220C22×C3—22no—
221C22×C3—21no—
222C22×C3—34no—
223C22×C3—32no—
224C22×C3—42no—
225C22×C3—42no—
226C22×C3—42no—
227C22×C3—62no—
228C22×C3—82no—
229C22×C3—128no—
230C22×C3—122no—
231C22×C3D3×D411yescharacteristic
232C22×C3C2×#6711yescharacteristic
233C2×D3—22no—
234C2×D3—42no—
235C2×D3—48no—
236C2×D3#21414yesdirect product factor
237C2×D3C2×#6712yes—
238C3×C4—21no—
239C3×C4—21no—
240C3×C4—32no—
241C3×C4—62no—
242C3×C4D3×D411yescharacteristic
243C3×C4C2×#6711yescharacteristic
244C3×C4C23×D311yescharacteristic
245C3×C4C23×D311yescharacteristic
246Dic3—22no—
247Dic3—62no—
248Dic3—62no—
249Dic3—122no—
250Dic3C2×#6712yes—
251C32C22×SD1611yescharacteristic, complemented, Sylow
252C23—21no—
253C23—31no—
254C23—64no—
255C23—61no—
256C23—124no—
257Q8—32no—
258Q8—61no—
259Q8—92no—
260Q8—184no—
261Q8—181no—
262D4—21no—
263D4—32no—
264D4—64no—
265D4—61no—
266D4C2×D3212yes—
267C2×C4—31no—
268C2×C4—32no—
269C2×C4—32no—
270C2×C4—61no—
271C2×C4—184no—
272C2×C4—181no—
273C2×C4C2×D3211yescharacteristic
274C8—32no—
275C8—92no—
276C2×C3—21no—
277C2×C3—22no—
278C2×C3—34no—
279C2×C3—42no—
280C2×C3—42no—
281C2×C3—82no—
282C2×C3—122no—
283C2×C3D3×SD1612yescomplemented
284C2×C3C2×#21412yescomplemented
285C2×C3C2×D3×D411yescharacteristic
286C2×C3C22×#6711yescharacteristic
287D3—42no—
288D3C2×#21414yesdirect product factor
289C22—22no—
290C22—32no—
291C22—34no—
292C22—42no—
293C22—62no—
294C22—128no—
295C22—122no—
296C22D3×#6711yescharacteristic
297C4—32no—
298C4—62no—
299C4—182no—
300C4D3×#6711yescharacteristic
301C4C22×D3211yescharacteristic
302C3—21no—
303C3C2×D3×SD1611yescharacteristic, complemented
304C3C22×#21411yescharacteristic, complemented
305C2—34no—
306C2—42no—
307C2—122no—
308C2D3×#21412yesdirect product factor
309C2C2×D3×#6711yescharacteristic
subgroupG/H
centerC22D3×#67
commutator, cosocleC32×C4C24
FrattiniC4C22×D32
FittingC2×C32×D4C22
socleC22×C32C2×D4
prime radicalC2×C3×D3×D4C2
hypercenterC2×D4D32
hypocenterC32C22×SD16
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×D3×Dic6C2semidirect
3. C2×#3754C2semidirect
4. C2×C3×#214C2semidirect
5. D3×#214C2direct (N × H)
6. C2×#3716C2semidirect
7. C2×#3715C2semidirect
8. C2×D3×C3⋊C8C2semidirect
13. C2×#634C22semidirect
19. C2×C3×Dic6C22semidirect
23. D3×Dic6C22semidirect
24. #3754C22semidirect
25. C2×C3⋊(C3⋊C8)C22semidirect
26. C3×#214C22semidirect
27. C2×C3×C3⋊C8C22semidirect
28. #3716C22semidirect
29. #3715C22semidirect
30. D3×C3⋊C8C22semidirect
43. C2×#214D3direct (N × H)
58. #634C23semidirect
67. C3×Dic6C23semidirect
70. C3⋊(C3⋊C8)C23semidirect
71. C3×C3⋊C8C23semidirect
100. C2×Dic6C2×D3semidirect
109. #214C2×D3direct (N × H)
114. C2×C3⋊C8C2×D3semidirect
123. C2×C3×D3SD16semidirect
188. Dic6C22×D3semidirect
193. C3⋊C8C22×D3semidirect
195. C2×C32C2×SD16semidirect
198. C3×D3C2×SD16semidirect
236. C2×D3#214direct (N × H)
251. C32C22×SD16semidirect
283. C2×C3D3×SD16semidirect
284. C2×C3C2×#214semidirect
288. D3C2×#214direct (N × H)
303. C3C2×D3×SD16semidirect
304. C3C22×#214semidirect
308. C2D3×#214direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D3×D42
C2×D3×Dic62
C2×#37542
C2×C3×#2142
D3×#2142
C2×#37162
C2×#37152
C2×D3×C3⋊C82
C22×#2143
C2×D3×SD163