Finite Group Databasebeta

← #11723016

Subgroups — #11723016

Order 864.

322 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 72 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
1#88815C211yesmaximal, characteristic
2#88806C211yesmaximal, characteristic
3#88806C211yesmaximal, characteristic
4Dic3×C3⋊Dic3C211yesmaximal, characteristic
5C3×#3724C211yesmaximal, characteristic
6C3×Dic3⋊Dic3C211yesmaximal, characteristic
7C3×Dic3⋊Dic3C211yesmaximal, characteristic
8#63870—31nomaximal
9#63847—31nomaximal
10#63847—31nomaximal
11C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
12C2×C3×C3⋊Dic3C2211yescharacteristic
13C2×C3×C3⋊Dic3C2211yescharacteristic
14C2×C3×C3⋊Dic3C2211yescharacteristic
15C2×C32×Dic3C2211yescharacteristic
16C2×C32×Dic3C2211yescharacteristic
17C2×C32×Dic3C2211yescharacteristic
18#3750—31no—
19#3750—31no—
20C4×C3⋊Dic3—31no—
21C3×C4⋊Dic3—31no—
22C3×C4⋊Dic3—31no—
23C3×Dic3⋊C4—31no—
24C3×Dic3⋊C4—31no—
25C3×Dic3⋊C4—31no—
26C3×Dic3⋊C4—31no—
27#3724—31no—
28#3724—31no—
29#3724—31no—
30#3724—31no—
31#3724—31no—
32#3724—61no—
33#3724—61no—
34#3724D311yescharacteristic
35Dic3⋊Dic3—31no—
36Dic3⋊Dic3—31no—
37Dic3⋊Dic3D311yescharacteristic
38Dic3⋊Dic3D311yescharacteristic
39Dic32—31no—
40Dic32—31no—
41Dic32—61no—
42C22×C33C2311yescharacteristic
43C3⋊(C3⋊Dic3)Q812yes—
44C3×C3⋊Dic3—21no—
45C3×C3⋊Dic3—21no—
46C3×C3⋊Dic3—21no—
47C32×Dic3—21no—
48C32×Dic3—21no—
49C32×Dic3—21no—
50#889—91no—
51#889—91no—
52#889—91no—
53C2×C32×C4—31no—
54C2×C32×C4—31no—
55C2×C32×C4—31no—
56C2×C3⋊Dic3—31no—
57C2×C3⋊Dic3—31no—
58C2×C3⋊Dic3—31no—
59C2×C3⋊Dic3—61no—
60C2×C3⋊Dic3—61no—
61C2×C3⋊Dic3—61no—
62C2×C3⋊Dic3—121no—
63C2×C3⋊Dic3C2×D311yescharacteristic
64C2×C3⋊Dic3C2×D311yescharacteristic
65C2×C3⋊Dic3C2×D311yescharacteristic
66C2×C3×Dic3—21no—
67C2×C3×Dic3—21no—
68C2×C3×Dic3—21no—
69C2×C3×Dic3—31no—
70C2×C3×Dic3—31no—
71C2×C3×Dic3—31no—
72C2×C3×Dic3—31no—
73C2×C3×Dic3—31no—
74C2×C3×Dic3—31no—
75C2×C3×Dic3—61no—
76C2×C3×Dic3—61no—
77C2×C3×Dic3—61no—
78C2×C3×Dic3C2×D311yescharacteristic
79C2×C3×Dic3C2×D311yescharacteristic
80C2×C3×Dic3C2×D311yescharacteristic
81C2×C3×Dic3C2×D311yescharacteristic
82C2×C3×Dic3C2×D311yescharacteristic
83C2×C3×Dic3C2×D311yescharacteristic
84C2×C33C2×Q811yescharacteristic
85C2×C33#4111yescharacteristic
86C2×C33#4111yescharacteristic
87C3×C4⋊C4—91no—
88C3×C4⋊C4—91no—
89C3×C4⋊C4—91no—
90C4⋊Dic3—31no—
91C4⋊Dic3—31no—
92C4⋊Dic3—91no—
93Dic3⋊C4—31no—
94Dic3⋊C4—31no—
95Dic3⋊C4—31no—
96Dic3⋊C4—31no—
97Dic3⋊C4—91no—
98Dic3⋊C4—91no—
99Dic3⋊C4—91no—
100Dic3⋊C4—91no—
101Dic3⋊C4—91no—
102Dic3⋊C4—91no—
103Dic3⋊C4—91no—
104Dic3⋊C4—91no—
105Dic3⋊C4—181no—
106Dic3⋊C4—181no—
107C4×Dic3—91no—
108C4×Dic3—91no—
109C4×Dic3—91no—
110C4×Dic3—181no—
111C22×C32—21no—
112C22×C32—21no—
113C22×C32—21no—
114C22×C32—41no—
115C22×C32C22×D311yescharacteristic
116C22×C32C22×D311yescharacteristic
117C22×C32C22×D311yescharacteristic
118C32×C4—61no—
119C32×C4—61no—
120C32×C4—61no—
121C3⋊Dic3—21no—
122C3⋊Dic3—21no—
123C3⋊Dic3—21no—
124C3⋊Dic3—32no—
125C3⋊Dic3—32no—
126C3⋊Dic3—32no—
127C3⋊Dic3—62no—
128C3⋊Dic3—62no—
129C3⋊Dic3—62no—
130C3⋊Dic3—122no—
131C3×Dic3—22no—
132C3×Dic3—21no—
133C3×Dic3—21no—
134C3×Dic3—21no—
135C3×Dic3—21no—
136C3×Dic3—21no—
137C3×Dic3—21no—
138C3×Dic3—41no—
139C3×Dic3—41no—
140C3×Dic3—62no—
141C3×Dic3—61no—
142C3×Dic3—61no—
143C3×Dic3—61no—
144C3×Dic3—61no—
145C3×Dic3—61no—
146C3×Dic3—61no—
147C3×Dic3—121no—
148C3×Dic3—121no—
149#125—271noSylow
150C33#12511yescharacteristic, complemented, Sylow
151C2×C3×C4—31no—
152C2×C3×C4—31no—
153C2×C3×C4—31no—
154C2×C3×C4—31no—
155C2×C3×C4—31no—
156C2×C3×C4—31no—
157C2×C3×C4—61no—
158C2×C3×C4—61no—
159C2×C3×C4—61no—
160C2×C3×C4—91no—
161C2×C3×C4—91no—
162C2×C3×C4—91no—
163C2×Dic3—31no—
164C2×Dic3—31no—
165C2×Dic3—31no—
166C2×Dic3—31no—
167C2×Dic3—31no—
168C2×Dic3—31no—
169C2×Dic3—61no—
170C2×Dic3—61no—
171C2×Dic3—61no—
172C2×Dic3—91no—
173C2×Dic3—91no—
174C2×Dic3—91no—
175C2×Dic3—181no—
176C2×Dic3—181no—
177C2×Dic3—181no—
178C2×Dic3—361no—
179C2×Dic3D3211yescharacteristic
180C2×Dic3D3211yescharacteristic
181C2×Dic3D3211yescharacteristic
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×C32#23511yescharacteristic
195C2×C32#23511yescharacteristic
196C2×C32#23511yescharacteristic
197C2×C32#23711yescharacteristic
198C2×C32#23711yescharacteristic
199C2×C32#23711yescharacteristic
200C2×C32D3×Q811yescharacteristic
201C2×C32D3×Q811yescharacteristic
202C2×C32D3×Q811yescharacteristic
203C4⋊C4—91no—
204C4⋊C4—91no—
205C4⋊C4—91no—
206C4⋊C4—271no—
207C4⋊C4—271no—
208C4⋊C4—271no—
209C42—271no—
210C22×C3—21no—
211C22×C3—21no—
212C22×C3—21no—
213C22×C3—41no—
214C22×C3C2×D3211yescharacteristic
215C22×C3C2×D3211yescharacteristic
216C22×C3C2×D3211yescharacteristic
217C3×C4—62no—
218C3×C4—61no—
219C3×C4—61no—
220C3×C4—61no—
221C3×C4—61no—
222C3×C4—61no—
223C3×C4—61no—
224C3×C4—121no—
225C3×C4—121no—
226C3×C4—181no—
227C3×C4—181no—
228C3×C4—181no—
229Dic3—21no—
230Dic3—21no—
231Dic3—21no—
232Dic3—62no—
233Dic3—61no—
234Dic3—61no—
235Dic3—61no—
236Dic3—61no—
237Dic3—61no—
238Dic3—61no—
239Dic3—92no—
240Dic3—92no—
241Dic3—92no—
242Dic3—121no—
243Dic3—121no—
244Dic3—182no—
245Dic3—182no—
246Dic3—182no—
247Dic3—362no—
248C32—21no—
249C32—21no—
250C32—21no—
251C32—41no—
252C32#88911yescharacteristic, complemented
253C32#88911yescharacteristic, complemented
254C32#88911yescharacteristic, complemented
255C2×C4—31no—
256C2×C4—31no—
257C2×C4—31no—
258C2×C4—91no—
259C2×C4—91no—
260C2×C4—91no—
261C2×C4—271no—
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×C3#379511yescharacteristic
275C2×C3#379511yescharacteristic
276C2×C3#379711yescharacteristic
277C2×C3#379711yescharacteristic
278C2×C3#379711yescharacteristic
279C2×C3#379811yescharacteristic
280C2×C3#380411yescharacteristic
281C2×C3#380411yescharacteristic
282C2×C3#380511yescharacteristic
283C22D3311yescharacteristic
284C4—61no—
285C4—61no—
286C4—61no—
287C4—181no—
288C4—181no—
289C4—181no—
290C4—272no—
291C3—21no—
292C3—21no—
293C3—21no—
294C3—41no—
295C3#6384711yescharacteristic, complemented
296C3#6384711yescharacteristic, complemented
297C3#6387011yescharacteristic, complemented
298C2#8895811yescharacteristic
299C2#8896011yescharacteristic
300C2#8896111yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D33
commutator, Fitting, socle, prime radical, cosocleC22×C33C23
hypocenterC33#125
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
150. C33#125semidirect
252. C32#889semidirect
253. C32#889semidirect
254. C32#889semidirect
295. C3#63847semidirect
296. C3#63847semidirect
297. C3#63870semidirect

Maximal subgroups

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

subgroupindex
#888152
#888062
Dic3×C3⋊Dic32
C3×#37242
C3×Dic3⋊Dic32
#638703
#638473