Finite Group Databasebeta

← D3×#451 #80181

Subgroups — D3×#451 #80181

Order 384.

740 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 155 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
1C22×D3×Q8C211yesmaximal, characteristic
2C2×D3×Q16C212yesmaximal, complemented
3D3×C22⋊Q8C211yesmaximal, characteristic
4C3×#451C211yesmaximal, characteristic, complemented
5#5228C211yesmaximal, characteristic, complemented
6#5186C212yesmaximal, complemented
7#5050C211yesmaximal, characteristic, complemented
8#4809C212yesmaximal, complemented
9D3×#103C212yesmaximal, complemented
10#4739C211yesmaximal, characteristic, complemented
11D3×C22⋊C8C211yesmaximal, characteristic
12C2×#451—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C2×D3×Q8—28no—
15C2×D3×Q8—21no—
16C2×D3×Q8—21no—
17C2×D3×Q8C2212yes—
18C2×D3×Q8C2211yescharacteristic
19C2×D3×Q8C2212yes—
20C22×C4×D3—21no—
21C22×C4×D3C2211yescharacteristic
22C22×Dic6—21no—
23C22×Dic6C2211yescharacteristic
24C2×C3×Q16C2212yescomplemented
25C3×C22⋊Q8C2211yescharacteristic
26#946C2211yescharacteristic
27C2×#216C2212yescomplemented
28C2×#216C2212yescomplemented
29#924C2211yescharacteristic
30D3×Q16—28no—
31C2×Dic12C2212yescomplemented
32C2×D3×C8C2212yes—
33#897C2211yescharacteristic
34#896—21no—
35D3×C4⋊C4—21no—
36D3×C4⋊C4C2211yescharacteristic
37D3×C22⋊C4—21no—
38#878—21no—
39C3×#103C2212yescomplemented
40C3×C22⋊C8C2211yescharacteristic
41#835C2212yescomplemented
42#830C2211yescharacteristic
43(C2×D3)⋊C8C2212yes—
44#816C2212yescomplemented
45#810C2212yescomplemented
46C23×Q8—31no—
47C22×Q16—32no—
48C2×C22⋊Q8—31no—
49#451—31no—
50#451—32no—
51#451—31no—
52#451—32no—
53#451—31no—
54#451D311yescharacteristic, direct product factor
55C2×#103—32no—
56C2×C22⋊C8—31no—
57C23×D3D411yescharacteristic
58C2×C3×Q8—24no—
59C2×C3×Q8—21no—
60C2×C3×Q8C2311yescharacteristic
61C2×C3×Q8C2312yes—
62C22×C3×C4—21no—
63C22×C3×C4C2311yescharacteristic
64C22×Dic3—21no—
65C22×Dic3D411yescharacteristic
66D3×Q8—28no—
67D3×Q8—28no—
68D3×Q8—24no—
69D3×Q8—24no—
70D3×Q8—24no—
71D3×Q8—24no—
72D3×Q8D418yes—
73C2×C4×D3—28no—
74C2×C4×D3—22no—
75C2×C4×D3—22no—
76C2×C4×D3—24no—
77C2×C4×D3—21no—
78C2×C4×D3—21no—
79C2×C4×D3D411yescharacteristic
80C2×C4×D3D411yescharacteristic
81C2×C4×D3C2311yescharacteristic
82C2×C4×D3C2311yescharacteristic
83C2×Dic6—28no—
84C2×Dic6—21no—
85C2×Dic6—22no—
86C2×Dic6—22no—
87C2×Dic6—24no—
88C2×Dic6—21no—
89C2×Dic6C2311yescharacteristic
90C2×Dic6C2312yes—
91C3×Q16—24no—
92C2×C3×C8C2312yes—
93C3×C4⋊C4—21no—
94C3×C4⋊C4C2311yescharacteristic
95C3×C22⋊C4—21no—
96#217—21no—
97#216—24no—
98#216—24no—
99(C2×D3)⋊C4—21no—
100(C2×D3)⋊C4—21no—
101C4⋊Dic3—21no—
102C4⋊Dic3C2311yescharacteristic
103Dic3⋊C4—21no—
104Dic3⋊C4—21no—
105Dic3⋊C4—21no—
106C2×C3⋊C8C2312yes—
107Dic12—24no—
108D3×C8—24no—
109C22×Q8—32no—
110C22×Q8—31no—
111C22×Q8—32no—
112C22×Q8—31no—
113C22×Q8—61no—
114C22×Q8—61no—
115C22×Q8—61no—
116C22×Q8—68no—
117C22×Q8C2×D311yescharacteristic
118C23×C4—31no—
119C23×C4—61no—
120C2×Q16—32no—
121C2×Q16—32no—
122C2×Q16—32no—
123C2×Q16—68no—
124C2×Q16C2×D312yescomplemented
125C22×C8—32no—
126C22⋊Q8—31no—
127C22⋊Q8—31no—
128C22⋊Q8—31no—
129C22⋊Q8—61no—
130C22⋊Q8—61no—
131C22⋊Q8C2×D311yescharacteristic
132C2×C4⋊C4—31no—
133C2×C4⋊C4—61no—
134C2×C22⋊C4—61no—
135#103—32no—
136#103—32no—
137#103—32no—
138#103C2×D312yescomplemented
139C22⋊C8—31no—
140C22⋊C8—32no—
141C22⋊C8C2×D311yescharacteristic
142C23×C3C2×D411yescharacteristic
143C22×D3—22no—
144C22×D3—22no—
145C22×D3Q1614yescomplemented
146C22×D3C2×D411yescharacteristic
147C22×D3C2×D411yescharacteristic
148C3×Q8—24no—
149C3×Q8—22no—
150C3×Q8—22no—
151C3×Q8C2×D414yes—
152C2×C3×C4—24no—
153C2×C3×C4—22no—
154C2×C3×C4—22no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4C2×D411yescharacteristic
158C2×C3×C4C2411yescharacteristic
159C2×Dic3—24no—
160C2×Dic3—22no—
161C2×Dic3—22no—
162C2×Dic3—21no—
163C2×Dic3—21no—
164C2×Dic3C2×D411yescharacteristic
165C2×Dic3C2×D411yescharacteristic
166C4×D3—28no—
167C4×D3—28no—
168C4×D3—22no—
169C4×D3—22no—
170C4×D3—22no—
171C4×D3—42no—
172C4×D3—42no—
173C4×D3C2×D414yes—
174Dic6—28no—
175Dic6—24no—
176Dic6—22no—
177Dic6—28no—
178Dic6—22no—
179Dic6—42no—
180Dic6C2×D414yes—
181C3×C8—22no—
182C3⋊C8—22no—
183C24—31no—
184C2×Q8—24no—
185C2×Q8—21no—
186C2×Q8—38no—
187C2×Q8—32no—
188C2×Q8—31no—
189C2×Q8—62no—
190C2×Q8—62no—
191C2×Q8—61no—
192C2×Q8—64no—
193C2×Q8—68no—
194C2×Q8—64no—
195C2×Q8—64no—
196C2×Q8—64no—
197C2×Q8—68no—
198C2×Q8—68no—
199C2×Q8—64no—
200C2×Q8—61no—
201C2×Q8C22×D311yescharacteristic
202C2×Q8C22×D312yes—
203C22×C4—21no—
204C22×C4—31no—
205C22×C4—31no—
206C22×C4—31no—
207C22×C4—31no—
208C22×C4—31no—
209C22×C4—64no—
210C22×C4—62no—
211C22×C4—62no—
212C22×C4—68no—
213C22×C4—61no—
214C22×C4—61no—
215C22×C4—61no—
216C22×C4C22×D311yescharacteristic
217Q16—24no—
218Q16—64no—
219Q16—64no—
220Q16—64no—
221C2×C8—32no—
222C2×C8—64no—
223C2×C8C22×D312yes—
224C4⋊C4—21no—
225C4⋊C4—31no—
226C4⋊C4—61no—
227C4⋊C4—61no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4C22×D311yescharacteristic
231C22⋊C4—21no—
232C22⋊C4—61no—
233C22⋊C4—61no—
234C22⋊C4—61no—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3C2×Q1612yescomplemented
238C22×C3C22×D411yescharacteristic
239C2×D3—21no—
240C2×D3—21no—
241C2×D3—28no—
242C2×D3#12012yes—
243C2×D3C2×Q1612yes—
244C2×D3C2×Q1612yescomplemented
245C2×D3#13712yes—
246C3×C4—24no—
247C3×C4—21no—
248C3×C4—41no—
249C3×C4—41no—
250C3×C4C22×D412yes—
251Dic3—21no—
252Dic3—24no—
253Dic3—41no—
254Dic3—41no—
255Dic3#12012yes—
256C23—31no—
257C23—34no—
258C23—31no—
259C23—62no—
260C23—62no—
261C23D3×D411yescharacteristic
262Q8—24no—
263Q8—22no—
264Q8—22no—
265Q8—34no—
266Q8—68no—
267Q8—68no—
268Q8—64no—
269Q8—62no—
270Q8—62no—
271Q8—122no—
272Q8D3×D414yes—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—31no—
279C2×C4—34no—
280C2×C4—31no—
281C2×C4—62no—
282C2×C4—62no—
283C2×C4—68no—
284C2×C4—68no—
285C2×C4—62no—
286C2×C4—64no—
287C2×C4—62no—
288C2×C4—62no—
289C2×C4—61no—
290C2×C4—61no—
291C2×C4—122no—
292C2×C4—122no—
293C2×C4D3×D411yescharacteristic
294C2×C4C23×D311yescharacteristic
295C8—22no—
296C8—62no—
297C2×C3—22no—
298C2×C3C2×#12011yescharacteristic
299C2×C3C22×Q1611yescharacteristic
300C2×C3C2×#13711yescharacteristic
301D3—22no—
302D3#45114yesdirect product factor
303C22—21no—
304C22—21no—
305C22—32no—
306C22—32no—
307C22—32no—
308C22—32no—
309C22—68no—
310C22—61no—
311C22—61no—
312C22D3×Q1612yescomplemented
313C22C2×D3×D411yescharacteristic
314C4—24no—
315C4—21no—
316C4—32no—
317C4—41no—
318C4—41no—
319C4—61no—
320C4—64no—
321C4—121no—
322C4—121no—
323C4C2×D3×D412yes—
324C3C2×#45111yescharacteristic, complemented, Sylow
325C2—22no—
326C2—34no—
327C2—62no—
328C2D3×#12011yescharacteristic
329C2C2×D3×Q1611yescharacteristic
330C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#451C2
socleC22×C3C22×D4
prime radicalC23×D3D4
hypercenter#451D3
hypocenterC3C2×#451
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C2×D3×Q16C2semidirect
4. C3×#451C2semidirect
5. #5228C2semidirect
6. #5186C2semidirect
7. #5050C2semidirect
8. #4809C2semidirect
9. D3×#103C2semidirect
10. #4739C2semidirect
24. C2×C3×Q16C22semidirect
27. C2×#216C22semidirect
28. C2×#216C22semidirect
31. C2×Dic12C22semidirect
39. C3×#103C22semidirect
41. #835C22semidirect
44. #816C22semidirect
45. #810C22semidirect
54. #451D3direct (N × H)
124. C2×Q16C2×D3semidirect
138. #103C2×D3semidirect
145. C22×D3Q16semidirect
237. C22×C3C2×Q16semidirect
244. C2×D3C2×Q16semidirect
302. D3#451direct (N × H)
312. C22D3×Q16semidirect
324. C3C2×#451semidirect

Maximal subgroups

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

subgroupindex
C22×D3×Q82
C2×D3×Q162
D3×C22⋊Q82
C3×#4512
#52282
#51862
#50502
#48092
D3×#1032
#47392
D3×C22⋊C82
C2×#4513