Finite Group Databasebeta

← #11724346

Subgroups — #11724346

Order 864.

296 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 39 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#89049C211yesmaximal, characteristic
2C2×#6294C211yesmaximal, characteristic, complemented
3C3×C22⋊#157C211yesmaximal, characteristic, complemented
4#88940C212yesmaximal, complemented
5#88939C212yesmaximal, complemented
6#64233—31nomaximal
7C22×C3×C3⋊D3C2211yescharacteristic
8C2×#1113C2211yescharacteristic
9C2×C3×#157C2212yes—
10C3×#638—21no—
11C2×C3×C3⋊Dic3—21no—
12#6295—21no—
13#6294—21no—
14#6294C2212yes—
15#6294C2211yescharacteristic
16#3811—31no—
17D3×#67—61no—
18C2×#624—31no—
19C2×D3×Dic3—61no—
20C22⋊#157D311yescharacteristic, complemented
21#3773—32no—
22D32⋊C4—32no—
23C22×C33D411yescharacteristic
24C2×C3×C3⋊D3—22no—
25C2×C3×C3⋊D3—21no—
26C2×C3×C3⋊D3D411yescharacteristic
27C2×C3×C3⋊D3C2311yescharacteristic
28#1113—22no—
29C3×#157—22no—
30C3×C3⋊Dic3—22no—
31C3×C3⋊Dic3—21no—
32#886—91nomaximal
33C22×C3⋊D3C2×D311yescharacteristic
34C22×C3×D3—61no—
35C22×C3×D3—61no—
36C2×D32—31no—
37C2×D32—61no—
38C2×#157C2×D312yes—
39#638—21no—
40C2×C3⋊Dic3—21no—
41C3×#67—61no—
42C3×#67—61no—
43C3×#67—121no—
44C2×C3×Dic3—61no—
45C2×C3×Dic3—61no—
46C2×C3×Dic3—121no—
47#626—61no—
48#626—61no—
49#625—61no—
50#624—32no—
51#624—31no—
52#624—61no—
53D3×Dic3—62no—
54D3×Dic3—61no—
55D3×Dic3—62no—
56C2×C33—21no—
57C2×C33C2×D411yescharacteristic
58C3×C3⋊D3—21no—
59C3×C3⋊D3—41no—
60C3×C3⋊D3#4112yes—
61C2×#67—91no—
62C22×Dic3—91no—
63D3×D4—181no—
64C2×C4×D3—181no—
65C3×C22⋊C4—91no—
66(C2×D3)⋊C4—92no—
67C4⋊Dic3—92no—
68C22×C32—21no—
69C22×C32—21no—
70C22×C32—41no—
71C22×C32—41no—
72C22×C32D1211yescharacteristic
73C2×C3⋊D3—22no—
74C2×C3⋊D3—21no—
75C2×C3⋊D3D1211yescharacteristic
76C2×C3⋊D3C22×D311yescharacteristic
77C2×C3×D3—61no—
78C2×C3×D3—61no—
79C2×C3×D3—61no—
80C2×C3×D3—61no—
81C2×C3×D3—64no—
82C2×C3×D3—61no—
83C2×C3×D3—61no—
84C2×C3×D3—122no—
85C2×C3×D3—121no—
86D32—62no—
87D32—122no—
88#157—22no—
89C3⋊Dic3—22no—
90C3⋊Dic3—21no—
91C3×Dic3—62no—
92C3×Dic3—61no—
93C3×Dic3—62no—
94C3×Dic3—61no—
95C3×Dic3—122no—
96C3×Dic3—121no—
97#123—271noSylow
98C33#12311yescharacteristic, complemented, Sylow
99C23×C3—91no—
100C22×D3—61no—
101C22×D3—61no—
102C22×D3—91no—
103C22×D3—181no—
104C3×D4—181no—
105C2×C3×C4—92no—
106C2×C3×C4—181no—
107#67—61no—
108#67—61no—
109#67—121no—
110#67—181no—
111#67—181no—
112C2×Dic3—61no—
113C2×Dic3—61no—
114C2×Dic3—91no—
115C2×Dic3—92no—
116C2×Dic3—121no—
117C2×Dic3—181no—
118D12—181no—
119C4×D3—182no—
120C4×D3—181no—
121C4×D3—182no—
122C2×C32—22no—
123C2×C32—21no—
124C2×C32—21no—
125C2×C32—21no—
126C2×C32—41no—
127C2×C32—41no—
128C2×C32—41no—
129C2×C32—42no—
130C2×C32—81no—
131C2×C32C2×D1211yescharacteristic
132C3⋊D3—21no—
133C3⋊D3—41no—
134C3⋊D3#23712yes—
135C3×D3—62no—
136C3×D3—62no—
137C3×D3—62no—
138C3×D3—121no—
139C3×D3—121no—
140C3×D3—121no—
141C3×D3—241no—
142C2×D4—271no—
143C22×C4—271no—
144C4⋊C4—272no—
145C22⋊C4—91no—
146C22⋊C4—272no—
147C22×C3—21no—
148C22×C3—21no—
149C22×C3—41no—
150C22×C3—41no—
151C22×C3—91no—
152C22×C3—91no—
153C22×C3—182no—
154C22×C3—181no—
155C22×C3GO+(4,𝔽2)11yescharacteristic
156C2×D3—61no—
157C2×D3—61no—
158C2×D3—61no—
159C2×D3—61no—
160C2×D3—64no—
161C2×D3—61no—
162C2×D3—61no—
163C2×D3—122no—
164C2×D3—121no—
165C2×D3—182no—
166C2×D3—362no—
167C3×C4—182no—
168C3×C4—181no—
169C3×C4—182no—
170Dic3—62no—
171Dic3—61no—
172Dic3—62no—
173Dic3—61no—
174Dic3—122no—
175Dic3—121no—
176C32—21no—
177C32—21no—
178C32—41no—
179C32—41no—
180C32#88611yescharacteristic, complemented
181C23—91no—
182C23—271no—
183D4—181no—
184D4—541no—
185C2×C4—92no—
186C2×C4—181no—
187C2×C4—272no—
188C2×C4—271no—
189C2×C4—541no—
190C2×C3—22no—
191C2×C3—21no—
192C2×C3—21no—
193C2×C3—21no—
194C2×C3—41no—
195C2×C3—42no—
196C2×C3—41no—
197C2×C3—41no—
198C2×C3—81no—
199C2×C3—92no—
200C2×C3—181no—
201C2×C3—361no—
202C2×C3C2×GO+(4,𝔽2)11yescharacteristic
203D3—62no—
204D3—62no—
205D3—62no—
206D3—121no—
207D3—121no—
208D3—121no—
209D3—241no—
210C22—91no—
211C22—91no—
212C22—182no—
213C22—181no—
214C22—542no—
215C22#157⋊D311yescharacteristic
216C4—182no—
217C4—182no—
218C4—181no—
219C3—21no—
220C3—21no—
221C3—41no—
222C3—41no—
223C3#6423311yescharacteristic, complemented
224C2—21no—
225C2—92no—
226C2—181no—
227C2—361no—
228C2C2×#157⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊D3C23
FittingC22×C33D4
socleC2×C33C2×D4
prime radical#89049C2
hypercenterC22#157⋊D3
hypocenterC33#123
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×#6294C2semidirect
3. C3×C22⋊#157C2semidirect
4. #88940C2semidirect
5. #88939C2semidirect
20. C22⋊#157D3semidirect
98. C33#123semidirect
180. C32#886semidirect
223. C3#64233semidirect

Maximal subgroups

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

subgroupindex
#890492
C2×#62942
C3×C22⋊#1572
#889402
#889392
#642333
#8869