Finite Group Databasebeta

← C3×#63856 #11723181

Subgroups — C3×#63856 #11723181

Order 864.

430 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 112 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×C32×C4×D3C211yesmaximal, characteristic
2C2×C32×Dic6C211yesmaximal, characteristic, complemented
3C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
4C3×#3724C212yesmaximal, complemented
5C3×(C2×D3)⋊Dic3C212yesmaximal
6C3×#946—31nomaximal
7C3×#924—31nomaximal
8#63856C311yesmaximal, characteristic, direct product factor
9C22×C32×D3C2211yescharacteristic
10C2×C33×C4C2211yescharacteristic
11C2×C3×C3⋊Dic3C2212yes—
12C2×C32×Dic3C2212yes—
13C2×C32×Dic3C2211yescharacteristic
14C32×C4×D3—22no—
15C32×Dic6—22no—
16C2×C32×Q8—31no—
17C22×C32×C4—31no—
18C2×C3×C4×D3—21no—
19C2×C3×C4×D3D311yescharacteristic
20C2×C3×C4×D3C2×C311yescharacteristic
21C2×C3×Dic6—21no—
22C2×C3×Dic6D311yescharacteristic, complemented
23C2×C3×Dic6C2×C311yescharacteristic, complemented
24C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
25C3×#217—32no—
26C3×(C2×D3)⋊C4—32no—
27C3×C4⋊Dic3—31no—
28C3×C4⋊Dic3—31no—
29C3×C4⋊Dic3—61no—
30C3×Dic3⋊C4—32no—
31C3×Dic3⋊C4—32no—
32#3724C2×C312yescomplemented
33(C2×D3)⋊Dic3C2×C312yes—
34C22×C33C2311yescharacteristic
35C2×C32×D3—21no—
36C2×C32×D3—21no—
37C2×C32×D3Q812yescomplemented
38C33×C4D412yes—
39C3×C3⋊Dic3—22no—
40C32×Dic3—22no—
41C32×Dic3—21no—
42C3×C22⋊Q8—91no—
43#946—31no—
44#924—31no—
45C23×C32—31no—
46C22×C3×D3—21no—
47C22×C3×D3C2×D311yescharacteristic
48C22×C3×D3C22×C311yescharacteristic
49C32×Q8—62no—
50C2×C32×C4—21no—
51C2×C32×C4—21no—
52C2×C32×C4—21no—
53C2×C32×C4—31no—
54C2×C32×C4—32no—
55C2×C32×C4—41no—
56C2×C32×C4—62no—
57C2×C32×C4C2×D311yescharacteristic
58C2×C32×C4C2×D311yescharacteristic
59C2×C32×C4C22×C311yescharacteristic
60C2×C3⋊Dic3C22×C312yes—
61C2×C3×Dic3—22no—
62C2×C3×Dic3—21no—
63C2×C3×Dic3—32no—
64C2×C3×Dic3—32no—
65C2×C3×Dic3—62no—
66C2×C3×Dic3C2×D311yescharacteristic
67C2×C3×Dic3C2×D312yes—
68C2×C3×Dic3C22×C312yes—
69C2×C3×Dic3C22×C311yescharacteristic
70C3×C4×D3—24no—
71C3×C4×D3—22no—
72C3×C4×D3—22no—
73C3×Dic6—24no—
74C3×Dic6—22no—
75C3×Dic6—22no—
76C2×C33C2×D411yescharacteristic
77C2×C33C2×Q811yescharacteristic
78C2×C33#4111yescharacteristic
79C32×D3—22no—
80C2×C3×Q8—31no—
81C2×C3×Q8—31no—
82C2×C3×Q8—61no—
83C22×C3×C4—31no—
84C22×C3×C4—31no—
85C22×C3×C4—61no—
86C2×C4×D3C3×D311yescharacteristic
87C2×Dic6C3×D311yescharacteristic, complemented
88C3×C4⋊C4—92no—
89C3×C4⋊C4—91no—
90C3×C22⋊C4—92no—
91#217—32no—
92(C2×D3)⋊C4—32no—
93C4⋊Dic3—31no—
94C4⋊Dic3—31no—
95C4⋊Dic3—61no—
96Dic3⋊C4—32no—
97Dic3⋊C4—32no—
98C22×C32—21no—
99C22×C32—21no—
100C22×C32—21no—
101C22×C32—32no—
102C22×C32—41no—
103C22×C32—61no—
104C22×C32—61no—
105C22×C32C22×D311yescharacteristic
106C22×C32C22×D311yescharacteristic
107C22×C32C23×C311yescharacteristic
108C2×C3×D3—22no—
109C2×C3×D3—22no—
110C2×C3×D3—21no—
111C2×C3×D3—21no—
112C2×C3×D3—21no—
113C2×C3×D3—21no—
114C2×C3×D3—22no—
115C2×C3×D3Dic612yescomplemented
116C2×C3×D3C3×Q812yescomplemented
117C32×C4—22no—
118C32×C4—22no—
119C32×C4—22no—
120C32×C4—42no—
121C32×C4—61no—
122C32×C4—62no—
123C32×C4#6712yes—
124C32×C4#6712yes—
125C32×C4C3×D412yes—
126C3⋊Dic3—22no—
127C3×Dic3—24no—
128C3×Dic3—22no—
129C3×Dic3—22no—
130C3×Dic3—21no—
131C3×Dic3—22no—
132C3×Dic3—21no—
133C3×Dic3—62no—
134C3×Dic3—62no—
135C3×Dic3—122no—
136C22⋊Q8—91noSylow
137C33C22⋊Q811yescharacteristic, complemented, Sylow
138C23×C3—31no—
139C23×C3—31no—
140C23×C3—61no—
141C22×D3C2×C3×D311yescharacteristic
142C3×Q8—64no—
143C3×Q8—62no—
144C3×Q8—62no—
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—21no—
148C2×C3×C4—32no—
149C2×C3×C4—31no—
150C2×C3×C4—31no—
151C2×C3×C4—32no—
152C2×C3×C4—41no—
153C2×C3×C4—61no—
154C2×C3×C4—62no—
155C2×C3×C4—64no—
156C2×C3×C4—62no—
157C2×C3×C4—62no—
158C2×C3×C4—92no—
159C2×C3×C4D3211yescharacteristic
160C2×C3×C4C2×C3×D311yescharacteristic
161C2×C3×C4C2×C3×D311yescharacteristic
162C2×Dic3—32no—
163C2×Dic3—32no—
164C2×Dic3—62no—
165C2×Dic3C2×C3×D311yescharacteristic
166C2×Dic3C2×C3×D312yes—
167C4×D3—22no—
168Dic6—22no—
169C2×C32—21no—
170C2×C32—21no—
171C2×C32—21no—
172C2×C32—21no—
173C2×C32—21no—
174C2×C32—21no—
175C2×C32—21no—
176C2×C32—21no—
177C2×C32—21no—
178C2×C32—41no—
179C2×C32—41no—
180C2×C32—41no—
181C2×C32—62no—
182C2×C32C2×Dic611yescharacteristic
183C2×C32#23511yescharacteristic
184C2×C32D3×Q811yescharacteristic
185C2×C32#23911yescharacteristic
186C2×C32C2×#6711yescharacteristic
187C2×C32C2×#6711yescharacteristic
188C2×C32C2×C3×D411yescharacteristic
189C2×C32C2×C3×Q811yescharacteristic
190C2×C32C3×#4111yescharacteristic
191C3×D3—24no—
192C3×D3—22no—
193C3×D3—22no—
194C2×Q8—31no—
195C22×C4—31no—
196C4⋊C4—92no—
197C4⋊C4—91no—
198C22⋊C4—92no—
199C22×C3—21no—
200C22×C3—21no—
201C22×C3—21no—
202C22×C3—32no—
203C22×C3—32no—
204C22×C3—41no—
205C22×C3—62no—
206C22×C3—62no—
207C22×C3—62no—
208C22×C3—61no—
209C22×C3—61no—
210C22×C3—61no—
211C22×C3—61no—
212C22×C3C2×D3211yescharacteristic
213C22×C3C22×C3×D311yescharacteristic
214C22×C3C22×C3×D311yescharacteristic
215C2×D3—21no—
216C2×D3—21no—
217C2×D3C3×Dic612yescomplemented
218C3×C4—22no—
219C3×C4—22no—
220C3×C4—22no—
221C3×C4—42no—
222C3×C4—62no—
223C3×C4—64no—
224C3×C4—61no—
225C3×C4—62no—
226C3×C4—61no—
227C3×C4—62no—
228C3×C4—182no—
229C3×C4#62512yes—
230C3×C4C3×#6712yes—
231C3×C4C3×#6712yes—
232Dic3—22no—
233Dic3—21no—
234Dic3—62no—
235Dic3—62no—
236Dic3—122no—
237C32—21no—
238C32—21no—
239C32—21no—
240C32—41no—
241C32#92411yescharacteristic, complemented
242C32#94611yescharacteristic, complemented
243C32C3×C22⋊Q811yescharacteristic, complemented
244C23—31no—
245Q8—62no—
246C2×C4—32no—
247C2×C4—31no—
248C2×C4—62no—
249C2×C4—92no—
250C2×C4C3×D3211yescharacteristic
251C2×C3—21no—
252C2×C3—21no—
253C2×C3—21no—
254C2×C3—21no—
255C2×C3—21no—
256C2×C3—21no—
257C2×C3—21no—
258C2×C3—21no—
259C2×C3—21no—
260C2×C3—41no—
261C2×C3—41no—
262C2×C3—41no—
263C2×C3—64no—
264C2×C3—62no—
265C2×C3—62no—
266C2×C3D3×Dic611yescharacteristic
267C2×C3#379911yescharacteristic
268C2×C3C2×#62511yescharacteristic
269C2×C3C2×C3×Dic611yescharacteristic
270C2×C3C3×#23511yescharacteristic
271C2×C3C3×D3×Q811yescharacteristic
272C2×C3C3×#23911yescharacteristic
273C2×C3C2×C3×#6711yescharacteristic
274C2×C3C2×C3×#6711yescharacteristic
275D3—22no—
276C22—32no—
277C22—61no—
278C22—61no—
279C22C2×C3×D3211yescharacteristic
280C4—61no—
281C4—62no—
282C4—182no—
283C4C3×#62512yes—
284C3—21no—
285C3—21no—
286C3—21no—
287C3—41no—
288C3#6385611yescharacteristic, direct product factor
289C3C3×#92411yescharacteristic, complemented
290C3C3×#94611yescharacteristic, complemented
291C2—62no—
292C2C3×D3×Dic611yescharacteristic
293C2C3×#379911yescharacteristic
294C2C2×C3×#62511yescharacteristic
subgroupG/H
centerC22×C3C2×D32
commutator, cosocleC22×C32C23×C3
FrattiniC22C2×C3×D32
FittingC2×C33×C4C22
socleC22×C33C23
prime radicalC22×C32×D3C22
hypercenterC2×C3×C4D32
hypocenterC32C3×C22⋊Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C32×Dic6C2semidirect
3. C3×C4⋊(C3⋊Dic3)C2semidirect
4. C3×#3724C2semidirect
8. #63856C3direct (N × H)
22. C2×C3×Dic6D3semidirect
23. C2×C3×Dic6C2×C3semidirect
24. C4⋊(C3⋊Dic3)C2×C3semidirect
32. #3724C2×C3semidirect
37. C2×C32×D3Q8semidirect
87. C2×Dic6C3×D3semidirect
115. C2×C3×D3Dic6semidirect
116. C2×C3×D3C3×Q8semidirect
137. C33C22⋊Q8semidirect
217. C2×D3C3×Dic6semidirect
241. C32#924semidirect
242. C32#946semidirect
243. C32C3×C22⋊Q8semidirect
288. C3#63856direct (N × H)
289. C3C3×#924semidirect
290. C3C3×#946semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×D32
C2×C32×Dic62
C3×C4⋊(C3⋊Dic3)2
C3×#37242
C3×(C2×D3)⋊Dic32
C3×#9463
C3×#9243
#638563