Finite Group Databasebeta

← C2×#63925 #10595093

Subgroups — C2×#63925 #10595093

Order 576.

634 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 144 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×#237C211yesmaximal, characteristic, complemented
2C2×D3×D12C211yesmaximal, characteristic, complemented
3C2×#3753C211yesmaximal, characteristic, complemented
4C2×C3×#213C211yesmaximal, characteristic, complemented
5#63925C218yesmaximal, direct product factor
6C2×#3717C211yesmaximal, characteristic, complemented
7C2×#3715C211yesmaximal, characteristic, complemented
8C2×#3711C211yesmaximal, characteristic, complemented
9C2×#949—31nomaximal
10C2×#911—31nomaximal
11C22×D32—21no—
12C2×C32×D4C2211yescharacteristic, complemented
13C2×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C3×#67—21no—
15C22×C3×Dic3—21no—
16C3×#237—22no—
17C3×#237C2214yescomplemented
18C2×C3×D12C2211yescharacteristic
19C2×C3×C4×D3C2211yescharacteristic
20C2×C3×Dic6C2211yescharacteristic, complemented
21C2×#626—21no—
22D3×D12—22no—
23D3×D12C2214yescomplemented
24#3753C2214yescomplemented
25C2×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
26C3×#213C2214yescomplemented
27C2×C3×C3⋊C8C2211yescharacteristic, complemented
28#3717C2214yescomplemented
29#3715C2214yescomplemented
30#3711C2214yescomplemented
31C2×C3×#41—31no—
32C2×#237D311yescharacteristic, complemented
33C2×D3×D4—31no—
34C22×D12—31no—
35C2×C3×D8—31no—
36#949—38no—
37C2×#215—31no—
38C2×#215—31no—
39C2×#214—31no—
40C2×#213—31no—
41C2×#213—31no—
42C2×#213—61no—
43C2×#213D311yescharacteristic, complemented
44C2×#208—31no—
45#911—38no—
46C2×#204—31no—
47C2×ΓL(1,𝔽25)—31no—
48C23×C32—21no—
49C22×C3⋊D3—21no—
50C22×C3×D3—21no—
51C22×C3×D3D411yescharacteristic
52C2×D32—24no—
53C2×D32—44no—
54C32×D4—21no—
55C32×D4C2312yescomplemented
56C2×C32×C4C2311yescharacteristic
57C4⋊(C3⋊D3)—21no—
58C4⋊(C3⋊D3)C2312yes—
59C3×#67—24no—
60C2×C3×Dic3—21no—
61C2×C3×Dic3—24no—
62C2×C3×Dic3D411yescharacteristic
63C3×D12—21no—
64C3×D12C2312yes—
65C3×C4×D3D412yes—
66C3×C4×D3C2312yes—
67C3×Dic6—21no—
68C3×Dic6C2312yescomplemented
69#626—24no—
70C3⋊(C3⋊C8)C2312yescomplemented
71C3×C3⋊C8C2312yescomplemented
72C2×AGL(1,ℤ/8)—91noSylow
73C23×D3—61no—
74C23×D3—61no—
75C3×#41—34no—
76C3×#41—62no—
77C2×C3×Q8—31no—
78C2×C3×D4—21no—
79C2×C3×D4—31no—
80C2×C3×D4—61no—
81C2×C3×D4C2×D311yescharacteristic, complemented
82C2×C3×D4C2×D311yescharacteristic, complemented
83C22×C3×C4—31no—
84C22×C3×C4—61no—
85C2×#67—21no—
86C2×#67—61no—
87C22×Dic3—21no—
88#237—22no—
89#237C2×D314yescomplemented
90D3×D4—34no—
91D3×D4—62no—
92C2×D12—31no—
93C2×D12—31no—
94C2×D12—34no—
95C2×D12—61no—
96C2×D12—62no—
97C2×D12—61no—
98C2×D12C2×D311yescharacteristic
99C2×C4×D3C2×D311yescharacteristic
100C2×Dic6C2×D311yescharacteristic, complemented
101C3×D8—34no—
102C2×C3×C8—31no—
103#215—34no—
104#215—34no—
105#214—34no—
106#213—34no—
107#213—34no—
108#213—64no—
109#213C2×D314yescomplemented
110#208—34no—
111C2×C3⋊C8—31no—
112C2×C3⋊C8—31no—
113C2×C3⋊C8—61no—
114C2×C3⋊C8C2×D311yescharacteristic, complemented
115#204—34no—
116ΓL(1,𝔽25)—34no—
117C22×C32—22no—
118C22×C32—42no—
119C22×C32C2×D411yescharacteristic
120C2×C3⋊D3—22no—
121C2×C3⋊D3—42no—
122C2×C3×D3—22no—
123C2×C3×D3—22no—
124C2×C3×D3—42no—
125C2×C3×D3C2×D412yes—
126D32—48no—
127C32×C4C2×D411yescharacteristic
128C32×C4C2411yescharacteristic
129C3×Dic3—22no—
130C3×Dic3C2×D412yes—
131C2×#41—31no—
132C22×D4—91no—
133AGL(1,ℤ/8)—98no—
134C2×SD16—91no—
135C2×SD16—91no—
136C2×D8—31no—
137C2×D8—91no—
138C2×M16—91no—
139C23×C3—21no—
140C23×C3—21no—
141C23×C3—31no—
142C23×C3—41no—
143C23×C3—61no—
144C22×D3—21no—
145C22×D3—64no—
146C22×D3—61no—
147C22×D3—61no—
148C22×D3—64no—
149C22×D3—121no—
150C22×D3—124no—
151C22×D3—124no—
152C22×D3#6711yescharacteristic
153C3×Q8—32no—
154C3×Q8—61no—
155C3×D4—21no—
156C3×D4—21no—
157C3×D4—22no—
158C3×D4—32no—
159C3×D4—41no—
160C3×D4—64no—
161C3×D4—61no—
162C3×D4C22×D312yescomplemented
163C3×D4C22×D312yescomplemented
164C2×C3×C4—21no—
165C2×C3×C4—31no—
166C2×C3×C4—32no—
167C2×C3×C4—32no—
168C2×C3×C4—64no—
169C2×C3×C4—61no—
170C2×C3×C4C22×D311yescharacteristic
171C2×C3×C4C22×D311yescharacteristic
172#67—24no—
173#67—64no—
174C2×Dic3—21no—
175C2×Dic3—24no—
176C2×Dic3#6711yescharacteristic
177D12—21no—
178D12—32no—
179D12—32no—
180D12—62no—
181D12—64no—
182D12—61no—
183D12—61no—
184D12—121no—
185D12C22×D312yes—
186C4×D3#6712yes—
187C4×D3C22×D312yes—
188Dic6—21no—
189Dic6C22×D312yescomplemented
190C3×C8—32no—
191C3⋊C8—32no—
192C3⋊C8—32no—
193C3⋊C8—62no—
194C3⋊C8C22×D312yescomplemented
195C2×C32—42no—
196C2×C32AGL(1,ℤ/8)12yescomplemented
197C2×C32C22×D411yescharacteristic
198C3⋊D3—42no—
199C3×D3—22no—
200C3×D3—42no—
201C24—181no—
202#41—34no—
203#41—62no—
204C2×Q8—31no—
205C2×D4—31no—
206C2×D4—61no—
207C2×D4—94no—
208C2×D4—91no—
209C2×D4—182no—
210C2×D4—181no—
211C2×D4D3211yescharacteristic, complemented
212C22×C4—31no—
213C22×C4—61no—
214SD16—94no—
215SD16—94no—
216D8—34no—
217D8—94no—
218M16—94no—
219C2×C8—31no—
220C2×C8—91no—
221C22×C3—22no—
222C22×C3—22no—
223C22×C3—21no—
224C22×C3—32no—
225C22×C3—42no—
226C22×C3—42no—
227C22×C3—42no—
228C22×C3—62no—
229C22×C3—62no—
230C22×C3—82no—
231C22×C3—122no—
232C22×C3D3×D411yescharacteristic
233C22×C3C2×#6711yescharacteristic
234C2×D3—22no—
235C2×D3—22no—
236C2×D3—42no—
237C2×D3—62no—
238C2×D3—62no—
239C2×D3—122no—
240C2×D3—128no—
241C2×D3—128no—
242C2×D3—122no—
243C2×D3—122no—
244C2×D3—242no—
245C2×D3C2×#6712yes—
246C3×C4—21no—
247C3×C4—21no—
248C3×C4—32no—
249C3×C4—62no—
250C3×C4D3×D411yescharacteristic
251C3×C4C2×#6711yescharacteristic
252C3×C4C23×D311yescharacteristic
253C3×C4C23×D311yescharacteristic
254Dic3—22no—
255Dic3C2×#6712yes—
256C32C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
257C23—21no—
258C23—31no—
259C23—61no—
260C23—184no—
261C23—181no—
262C23—364no—
263Q8—32no—
264Q8—61no—
265D4—21no—
266D4—32no—
267D4—64no—
268D4—61no—
269D4—92no—
270D4—184no—
271D4—181no—
272D4C2×D3212yescomplemented
273C2×C4—31no—
274C2×C4—32no—
275C2×C4—32no—
276C2×C4—64no—
277C2×C4—61no—
278C2×C4C2×D3211yescharacteristic
279C8—32no—
280C8—92no—
281C2×C3—21no—
282C2×C3—22no—
283C2×C3—42no—
284C2×C3—42no—
285C2×C3—62no—
286C2×C3—82no—
287C2×C3—122no—
288C2×C3#91112yescomplemented
289C2×C3#94912yescomplemented
290C2×C3C2×D3×D411yescharacteristic
291C2×C3C22×#6711yescharacteristic
292D3—22no—
293D3—42no—
294D3—122no—
295D3—122no—
296D3—242no—
297C22—22no—
298C22—32no—
299C22—42no—
300C22—62no—
301C22—62no—
302C22—122no—
303C22—182no—
304C22—368no—
305C22—362no—
306C22D3×#6711yescharacteristic
307C4—32no—
308C4—62no—
309C4D3×#6711yescharacteristic
310C4C22×D3211yescharacteristic
311C3—21no—
312C3C2×#91111yescharacteristic, complemented
313C3C2×#94911yescharacteristic, complemented
314C2—42no—
315C2—62no—
316C2—122no—
317C2—362no—
318C2#6392512yesdirect product factor
319C2C2×D3×#6711yescharacteristic
subgroupG/H
centerC22D3×#67
commutator, cosocleC32×C4C24
FrattiniC4C22×D32
FittingC2×C32×D4C22
socleC22×C32C2×D4
hypercenterC2×D4D32
hypocenterC32C2×AGL(1,ℤ/8)
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
1. C2×C3×#237C2semidirect
2. C2×D3×D12C2semidirect
3. C2×#3753C2semidirect
4. C2×C3×#213C2semidirect
5. #63925C2direct (N × H)
6. C2×#3717C2semidirect
7. C2×#3715C2semidirect
8. C2×#3711C2semidirect
12. C2×C32×D4C22semidirect
17. C3×#237C22semidirect
20. C2×C3×Dic6C22semidirect
23. D3×D12C22semidirect
24. #3753C22semidirect
25. C2×C3⋊(C3⋊C8)C22semidirect
26. C3×#213C22semidirect
27. C2×C3×C3⋊C8C22semidirect
28. #3717C22semidirect
29. #3715C22semidirect
30. #3711C22semidirect
32. C2×#237D3semidirect
43. C2×#213D3semidirect
55. C32×D4C23semidirect
68. C3×Dic6C23semidirect
70. C3⋊(C3⋊C8)C23semidirect
71. C3×C3⋊C8C23semidirect
81. C2×C3×D4C2×D3semidirect
82. C2×C3×D4C2×D3semidirect
89. #237C2×D3semidirect
100. C2×Dic6C2×D3semidirect
109. #213C2×D3semidirect
114. C2×C3⋊C8C2×D3semidirect
162. C3×D4C22×D3semidirect
163. C3×D4C22×D3semidirect
189. Dic6C22×D3semidirect
194. C3⋊C8C22×D3semidirect
196. C2×C32AGL(1,ℤ/8)semidirect
211. C2×D4D32semidirect
256. C32C2×AGL(1,ℤ/8)semidirect
272. D4C2×D32semidirect
288. C2×C3#911semidirect
289. C2×C3#949semidirect
312. C3C2×#911semidirect
313. C3C2×#949semidirect
318. C2#63925direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#2372
C2×D3×D122
C2×#37532
C2×C3×#2132
#639252
C2×#37172
C2×#37152
C2×#37112
C2×#9493
C2×#9113