Finite Group Databasebeta

← #11724122

Subgroups — #11724122

Order 864.

296 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 36 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#88955C211yesmaximal, characteristic
2D3×#624C211yesmaximal, characteristic, complemented
3#88939C211yesmaximal, characteristic, complemented
4#88934C211yesmaximal, characteristic, complemented
5#88932C211yesmaximal, characteristic, complemented
6C3×#3773C211yesmaximal, characteristic, complemented
7(C2×D3)⋊#157C211yesmaximal, characteristic, complemented
8#64233—31nomaximal
9C2×#1113C2211yescharacteristic
10C2×D3×C3⋊D3C2211yescharacteristic
11C2×C3⋊#157C2211yescharacteristic
12C2×C3×#157C2211yescharacteristic
13#6294C2211yescharacteristic
14#6292—21no—
15#6289—21no—
16C3×#626—21no—
17C3×#624C2211yescharacteristic
18C3×#624C2211yescharacteristic
19C3×D3×Dic3—21no—
20#3811—31no—
21C2×#624—31no—
22D3×D12—61no—
23C4×D32—61no—
24C22⋊#157—31no—
25#3773—31no—
26#3773D311yescharacteristic, complemented
27D32⋊C4—31no—
28D32⋊C4—31no—
29C2×C3⋊(C3⋊D3)D411yescharacteristic
30C2×C3×C3⋊D3—21no—
31C2×C3×C3⋊D3C2311yescharacteristic
32C2×C32×D3D411yescharacteristic
33#1113—21no—
34#1113—21no—
35D3×C3⋊D3—21no—
36D3×C3⋊D3—21no—
37C3⋊#157—21no—
38C3×#157—21no—
39C3×C3⋊Dic3—21no—
40C32×Dic3—21no—
41C32×Dic3—21no—
42#894—91nomaximal
43C22×C3⋊D3—31no—
44C2×D32—31no—
45C2×D32—61no—
46C2×D32—61no—
47C2×D32—61no—
48C2×#157—31no—
49C2×#157C2×D311yescharacteristic
50C4⋊(C3⋊D3)—61no—
51C4×C3⋊D3—61no—
52C3×#67—61no—
53C2×C3×Dic3—61no—
54C3×D12—61no—
55C3×C4×D3—61no—
56C3×C4×D3—61no—
57C3×C4×D3—61no—
58#626—21no—
59#626—61no—
60#626—61no—
61#626—121no—
62#624—31no—
63#624—61no—
64#624—61no—
65#624—121no—
66#624C2×D311yescharacteristic
67#624C2×D311yescharacteristic
68D3×Dic3—21no—
69D3×Dic3—61no—
70C2×C33C2×D411yescharacteristic
71C3⋊(C3⋊D3)—21no—
72C3×C3⋊D3—41no—
73C3×C3⋊D3#4111yescharacteristic
74C3×C3⋊D3#4111yescharacteristic
75C32×D3—21no—
76D3×D4—181no—
77C2×D12—91no—
78C2×C4×D3—91no—
79C2×C4×D3—181no—
80C3×C4⋊C4—91no—
81(C2×D3)⋊C4—91no—
82(C2×D3)⋊C4—91no—
83(C2×D3)⋊C4—91no—
84Dic3⋊C4—91no—
85C22×C32—31no—
86C2×C3⋊D3—21no—
87C2×C3⋊D3—31no—
88C2×C3⋊D3—61no—
89C2×C3⋊D3—61no—
90C2×C3⋊D3—61no—
91C2×C3⋊D3—61no—
92C2×C3⋊D3—121no—
93C2×C3⋊D3—121no—
94C2×C3⋊D3C22×D311yescharacteristic
95C2×C3×D3—21no—
96C2×C3×D3—21no—
97C2×C3×D3—61no—
98C2×C3×D3—61no—
99C2×C3×D3—61no—
100C2×C3×D3—61no—
101C2×C3×D3—121no—
102D32—62no—
103D32—62no—
104D32—61no—
105D32—61no—
106D32—121no—
107D32—121no—
108D32—121no—
109D32—121no—
110#157—21no—
111#157—61no—
112C32×C4—61no—
113C32×C4—61no—
114C3⋊Dic3—21no—
115C3×Dic3—21no—
116C3×Dic3—21no—
117C3×Dic3—21no—
118C3×Dic3—21no—
119C3×Dic3—41no—
120C3×Dic3—41no—
121C3×Dic3—61no—
122C3×Dic3—61no—
123C3×Dic3—121no—
124#123—271noSylow
125C33#12311yescharacteristic, complemented, Sylow
126C22×D3—91no—
127C22×D3—91no—
128C22×D3—181no—
129C22×D3—181no—
130C22×D3—181no—
131C3×D4—181no—
132C2×C3×C4—91no—
133C2×C3×C4—91no—
134C2×C3×C4—91no—
135C2×C3×C4—181no—
136#67—61no—
137#67—181no—
138C2×Dic3—61no—
139C2×Dic3—91no—
140C2×Dic3—91no—
141D12—61no—
142D12—181no—
143D12—181no—
144D12—361no—
145C4×D3—61no—
146C4×D3—61no—
147C4×D3—61no—
148C4×D3—181no—
149C4×D3—181no—
150C4×D3—181no—
151C4×D3—181no—
152C4×D3—361no—
153C2×C32—21no—
154C2×C32—21no—
155C2×C32—41no—
156C2×C32—41no—
157C2×C32—61no—
158C2×C32D3×D411yescharacteristic
159C3⋊D3—41no—
160C3⋊D3—62no—
161C3⋊D3—61no—
162C3⋊D3—121no—
163C3⋊D3—122no—
164C3⋊D3—241no—
165C3⋊D3#23511yescharacteristic
166C3⋊D3#23911yescharacteristic
167C3×D3—22no—
168C3×D3—41no—
169C3×D3—61no—
170C3×D3—61no—
171C3×D3—61no—
172C3×D3—61no—
173C3×D3—121no—
174C3×D3—121no—
175C3×D3—241no—
176C2×D4—271no—
177C22×C4—271no—
178C4⋊C4—91no—
179C4⋊C4—271no—
180C22⋊C4—271no—
181C22⋊C4—271no—
182C22⋊C4—271no—
183C22×C3—61no—
184C22×C3—61no—
185C22×C3—91no—
186C22×C3—181no—
187C2×D3—61no—
188C2×D3—61no—
189C2×D3—61no—
190C2×D3—61no—
191C2×D3—91no—
192C2×D3—121no—
193C2×D3—181no—
194C2×D3—181no—
195C2×D3—181no—
196C2×D3—181no—
197C2×D3—182no—
198C2×D3—182no—
199C2×D3—181no—
200C2×D3—181no—
201C2×D3—361no—
202C2×D3—361no—
203C2×D3—361no—
204C2×D3—361no—
205C2×D3—361no—
206C2×D3—361no—
207C2×D3GO+(4,𝔽2)11yescharacteristic
208C3×C4—61no—
209C3×C4—61no—
210C3×C4—61no—
211C3×C4—61no—
212C3×C4—121no—
213C3×C4—121no—
214C3×C4—181no—
215C3×C4—181no—
216Dic3—21no—
217Dic3—21no—
218Dic3—61no—
219Dic3—61no—
220Dic3—121no—
221Dic3—181no—
222C32—21no—
223C32—21no—
224C32—41no—
225C32—41no—
226C32#89411yescharacteristic, complemented
227C23—271no—
228C23—271no—
229D4—181no—
230D4—541no—
231C2×C4—91no—
232C2×C4—91no—
233C2×C4—91no—
234C2×C4—181no—
235C2×C4—271no—
236C2×C4—271no—
237C2×C4—541no—
238C2×C3—21no—
239C2×C3—21no—
240C2×C3—41no—
241C2×C3—41no—
242C2×C3—62no—
243C2×C3—91no—
244C2×C3—91no—
245C2×C3—121no—
246C2×C3—361no—
247C2×C3C2×GO+(4,𝔽2)11yescharacteristic
248D3—21no—
249D3—61no—
250D3—61no—
251D3—61no—
252D3—61no—
253D3—121no—
254D3—121no—
255D3—182no—
256D3—181no—
257D3—241no—
258D3—361no—
259D3—362no—
260D3—721no—
261C22—31no—
262C22—91no—
263C22—181no—
264C22—271no—
265C22—541no—
266C22—541no—
267C22—541no—
268C22—541no—
269C4—61no—
270C4—61no—
271C4—181no—
272C4—181no—
273C4—541no—
274C3—21no—
275C3—21no—
276C3—41no—
277C3—41no—
278C3#6423311yescharacteristic, complemented
279C2—61no—
280C2—91no—
281C2—91no—
282C2—361no—
283C2—541no—
284C2D3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D3×GO+(4,𝔽2)
commutator, cosocleC2×C3×C3⋊D3C23
Fitting, socleC2×C33C2×D4
prime radical#88955C2
hypocenterC33#123
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. D3×#624C2semidirect
3. #88939C2semidirect
4. #88934C2semidirect
5. #88932C2semidirect
6. C3×#3773C2semidirect
7. (C2×D3)⋊#157C2semidirect
26. #3773D3semidirect
125. C33#123semidirect
226. C32#894semidirect
278. C3#64233semidirect

Maximal subgroups

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

subgroupindex
#889552
D3×#6242
#889392
#889342
#889322
C3×#37732
(C2×D3)⋊#1572
#642333
#8949