Finite Group Databasebeta

← C3×#63867 #11723192

Subgroups — C3×#63867 #11723192

Order 864.

322 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 96 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
1C3×#3750C211yesmaximal, characteristic
2C32×Dic3⋊C4C211yesmaximal, characteristic
3C32×C4×Dic3C211yesmaximal, characteristic
4C3×#3724C211yesmaximal, characteristic
5C3×Dic3⋊Dic3C211yesmaximal, characteristic
6C3×Dic3⋊Dic3C211yesmaximal, characteristic
7C3×Dic3⋊Dic3C211yesmaximal, characteristic
8C3×#889—31nomaximal
9C3×#870—31nomaximal
10#63867C311yesmaximal, characteristic, direct product factor
11C2×C33×C4C2211yescharacteristic
12C2×C3×C3⋊Dic3C2211yescharacteristic
13C2×C3×C3⋊Dic3C2211yescharacteristic
14C2×C32×Dic3C2211yescharacteristic
15C2×C32×Dic3C2211yescharacteristic
16C2×C32×Dic3C2211yescharacteristic
17C2×C32×Dic3C2211yescharacteristic
18C32×C4⋊C4—31no—
19C32×C42—31no—
20#3750C2×C311yescharacteristic
21C3×C4⋊Dic3—31no—
22C3×C4⋊Dic3—31no—
23C3×C4⋊Dic3—31no—
24C3×Dic3⋊C4—21no—
25C3×Dic3⋊C4—31no—
26C3×Dic3⋊C4—31no—
27C3×Dic3⋊C4—31no—
28C3×Dic3⋊C4—31no—
29C3×Dic3⋊C4—31no—
30C3×Dic3⋊C4—31no—
31C3×Dic3⋊C4—31no—
32C3×Dic3⋊C4—61no—
33C3×Dic3⋊C4D311yescharacteristic
34C3×Dic3⋊C4C2×C311yescharacteristic
35C3×C4×Dic3—21no—
36C3×C4×Dic3D311yescharacteristic
37C3×C4×Dic3C2×C311yescharacteristic
38#3724C2×C311yescharacteristic
39Dic3⋊Dic3C2×C311yescharacteristic
40Dic3⋊Dic3C2×C311yescharacteristic
41Dic3⋊Dic3C2×C311yescharacteristic
42C22×C33C2311yescharacteristic
43C33×C4—21no—
44C3×C3⋊Dic3—21no—
45C3×C3⋊Dic3—21no—
46C32×Dic3—21no—
47C32×Dic3—21no—
48C32×Dic3—21no—
49C32×Dic3Q812yes—
50C3×#125—91no—
51#889—31no—
52#870—31no—
53C2×C32×C4—21no—
54C2×C32×C4—21no—
55C2×C32×C4—21no—
56C2×C32×C4—31no—
57C2×C32×C4—31no—
58C2×C32×C4—31no—
59C2×C32×C4—31no—
60C2×C32×C4—41no—
61C2×C32×C4C2×D311yescharacteristic
62C2×C32×C4C2×D311yescharacteristic
63C2×C32×C4C22×C311yescharacteristic
64C2×C3⋊Dic3C22×C311yescharacteristic
65C2×C3⋊Dic3C22×C311yescharacteristic
66C2×C3×Dic3—21no—
67C2×C3×Dic3—21no—
68C2×C3×Dic3—21no—
69C2×C3×Dic3—21no—
70C2×C3×Dic3—31no—
71C2×C3×Dic3—31no—
72C2×C3×Dic3—31no—
73C2×C3×Dic3—31no—
74C2×C3×Dic3—61no—
75C2×C3×Dic3—61no—
76C2×C3×Dic3C2×D311yescharacteristic
77C2×C3×Dic3C2×D311yescharacteristic
78C2×C3×Dic3C2×D311yescharacteristic
79C2×C3×Dic3C2×D311yescharacteristic
80C2×C3×Dic3C22×C311yescharacteristic
81C2×C3×Dic3C22×C311yescharacteristic
82C2×C3×Dic3C22×C311yescharacteristic
83C2×C3×Dic3C22×C311yescharacteristic
84C2×C33C2×Q811yescharacteristic
85C2×C33#4111yescharacteristic
86C2×C33#4111yescharacteristic
87C3×C4⋊C4—31no—
88C3×C4⋊C4—31no—
89C3×C4⋊C4—61no—
90C3×C4⋊C4—91no—
91C3×C4⋊C4—91no—
92C3×C4⋊C4—91no—
93C3×C4⋊C4—91no—
94C3×C4⋊C4—91no—
95C3×C42—31no—
96C3×C42—31no—
97C3×C42—61no—
98C4⋊Dic3—31no—
99C4⋊Dic3—31no—
100C4⋊Dic3—31no—
101Dic3⋊C4—31no—
102Dic3⋊C4—31no—
103Dic3⋊C4—31no—
104Dic3⋊C4—31no—
105Dic3⋊C4—31no—
106Dic3⋊C4—31no—
107Dic3⋊C4—31no—
108Dic3⋊C4—61no—
109Dic3⋊C4C3×D311yescharacteristic
110C4×Dic3C3×D311yescharacteristic
111C22×C32—21no—
112C22×C32—21no—
113C22×C32—21no—
114C22×C32—41no—
115C22×C32C22×D311yescharacteristic
116C22×C32C22×D311yescharacteristic
117C22×C32C23×C311yescharacteristic
118C32×C4—21no—
119C32×C4—22no—
120C32×C4—21no—
121C32×C4—21no—
122C32×C4—32no—
123C32×C4—41no—
124C32×C4—41no—
125C32×C4—42no—
126C32×C4—61no—
127C32×C4—61no—
128C32×C4—61no—
129C3⋊Dic3—21no—
130C3⋊Dic3—21no—
131C3×Dic3—22no—
132C3×Dic3—22no—
133C3×Dic3—21no—
134C3×Dic3—21no—
135C3×Dic3—21no—
136C3×Dic3—21no—
137C3×Dic3—21no—
138C3×Dic3—21no—
139C3×Dic3—22no—
140C3×Dic3—41no—
141C3×Dic3—62no—
142C3×Dic3—61no—
143C3×Dic3—61no—
144C3×Dic3—61no—
145C3×Dic3—61no—
146C3×Dic3—121no—
147C3×Dic3Dic612yes—
148C3×Dic3C3×Q812yes—
149#125—91noSylow
150C33#12511yescharacteristic, complemented, Sylow
151C2×C3×C4—21no—
152C2×C3×C4—21no—
153C2×C3×C4—21no—
154C2×C3×C4—31no—
155C2×C3×C4—31no—
156C2×C3×C4—31no—
157C2×C3×C4—31no—
158C2×C3×C4—31no—
159C2×C3×C4—31no—
160C2×C3×C4—31no—
161C2×C3×C4—31no—
162C2×C3×C4—41no—
163C2×C3×C4—61no—
164C2×C3×C4—61no—
165C2×C3×C4—61no—
166C2×C3×C4—61no—
167C2×C3×C4—91no—
168C2×C3×C4—91no—
169C2×C3×C4D3211yescharacteristic
170C2×C3×C4C2×C3×D311yescharacteristic
171C2×C3×C4C2×C3×D311yescharacteristic
172C2×Dic3—31no—
173C2×Dic3—31no—
174C2×Dic3—31no—
175C2×Dic3—31no—
176C2×Dic3—61no—
177C2×Dic3—61no—
178C2×Dic3C2×C3×D311yescharacteristic
179C2×Dic3C2×C3×D311yescharacteristic
180C2×Dic3C2×C3×D311yescharacteristic
181C2×Dic3C2×C3×D311yescharacteristic
182C2×C32—21no—
183C2×C32—21no—
184C2×C32—21no—
185C2×C32—21no—
186C2×C32—21no—
187C2×C32—21no—
188C2×C32—21no—
189C2×C32—21no—
190C2×C32—21no—
191C2×C32—41no—
192C2×C32—41no—
193C2×C32—41no—
194C2×C32C2×Dic611yescharacteristic
195C2×C32#23511yescharacteristic
196C2×C32#23511yescharacteristic
197C2×C32#23511yescharacteristic
198C2×C32#23711yescharacteristic
199C2×C32D3×Q811yescharacteristic
200C2×C32C2×C3×Q811yescharacteristic
201C2×C32C3×#4111yescharacteristic
202C2×C32C3×#4111yescharacteristic
203C4⋊C4—31no—
204C4⋊C4—91no—
205C4⋊C4—91no—
206C4⋊C4—91no—
207C4⋊C4—91no—
208C4⋊C4—91no—
209C42—31no—
210C22×C3—21no—
211C22×C3—21no—
212C22×C3—21no—
213C22×C3—41no—
214C22×C3C2×D3211yescharacteristic
215C22×C3C22×C3×D311yescharacteristic
216C22×C3C22×C3×D311yescharacteristic
217C3×C4—22no—
218C3×C4—21no—
219C3×C4—21no—
220C3×C4—21no—
221C3×C4—32no—
222C3×C4—32no—
223C3×C4—41no—
224C3×C4—41no—
225C3×C4—42no—
226C3×C4—62no—
227C3×C4—62no—
228C3×C4—62no—
229C3×C4—61no—
230C3×C4—61no—
231C3×C4—61no—
232C3×C4—61no—
233C3×C4—61no—
234C3×C4—61no—
235C3×C4—121no—
236C3×C4—181no—
237C3×C4—181no—
238Dic3—21no—
239Dic3—21no—
240Dic3—21no—
241Dic3—62no—
242Dic3—61no—
243Dic3—61no—
244Dic3—61no—
245Dic3—61no—
246Dic3—121no—
247Dic3C3×Dic612yes—
248C32—21no—
249C32—21no—
250C32—21no—
251C32—41no—
252C32#87011yescharacteristic, complemented
253C32#88911yescharacteristic, complemented
254C32C3×#12511yescharacteristic, complemented
255C2×C4—31no—
256C2×C4—31no—
257C2×C4—31no—
258C2×C4—31no—
259C2×C4—91no—
260C2×C4—91no—
261C2×C4C3×D3211yescharacteristic
262C2×C3—21no—
263C2×C3—21no—
264C2×C3—21no—
265C2×C3—21no—
266C2×C3—21no—
267C2×C3—21no—
268C2×C3—21no—
269C2×C3—21no—
270C2×C3—21no—
271C2×C3—41no—
272C2×C3—41no—
273C2×C3—41no—
274C2×C3D3×Dic611yescharacteristic
275C2×C3#379811yescharacteristic
276C2×C3#380411yescharacteristic
277C2×C3C2×C3×Dic611yescharacteristic
278C2×C3C3×#23511yescharacteristic
279C2×C3C3×#23511yescharacteristic
280C2×C3C3×#23511yescharacteristic
281C2×C3C3×#23711yescharacteristic
282C2×C3C3×D3×Q811yescharacteristic
283C22C2×C3×D3211yescharacteristic
284C4—21no—
285C4—32no—
286C4—61no—
287C4—61no—
288C4—61no—
289C4—181no—
290C4—181no—
291C3—21no—
292C3—21no—
293C3—21no—
294C3—41no—
295C3#6386711yescharacteristic, direct product factor
296C3C3×#87011yescharacteristic, complemented
297C3C3×#88911yescharacteristic, complemented
298C2C3×D3×Dic611yescharacteristic
299C2C3×#379811yescharacteristic
300C2C3×#380411yescharacteristic
subgroupG/H
centerC22×C3C2×D32
commutator, cosocleC22×C32C23×C3
FrattiniC22C2×C3×D32
FittingC2×C33×C4C22
socle, prime radicalC22×C33C23
hypercenterC2×C3×C4D32
hypocenterC32C3×#125
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
10. #63867C3direct (N × H)
150. C33#125semidirect
252. C32#870semidirect
253. C32#889semidirect
254. C32C3×#125semidirect
295. C3#63867direct (N × H)
296. C3C3×#870semidirect
297. C3C3×#889semidirect

Maximal subgroups

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

subgroupindex
C3×#37502
C32×Dic3⋊C42
C32×C4×Dic32
C3×#37242
C3×Dic3⋊Dic32
C3×#8893
C3×#8703
#638673