Finite Group Databasebeta

← #13049448

Subgroups — #13049448

Order 1296.

299 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 26 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#10620156C211yesmaximal, characteristic
2C3×#6296C211yesmaximal, characteristic, complemented
3C3×C3⋊#622C211yesmaximal, characteristic, complemented
4#88942—31nomaximal
5#88936—31nomaximal
6C2×C3×C3⋊(C3⋊D3)—21no—
7C32×C3⋊Dic3—21no—
8C32×C3⋊Dic3C2211yescharacteristic
9#6296D311yescharacteristic, complemented
10#6295—31no—
11#6295—31no—
12#6295—61no—
13#6292—61no—
14C3×#627—31no—
15C3×#627—61no—
16C3⋊#622D311yescharacteristic, complemented
17C3×#622—31no—
18C2×C34D411yescharacteristic
19C3×C3⋊(C3⋊D3)—41no—
20#3775—91no—
21#3717—91nomaximal
22C2×C3⋊(C3⋊D3)—21no—
23C2×C3×C3⋊D3—61no—
24C2×C3×C3⋊D3—62no—
25C2×C3×C3⋊D3—61no—
26C2×C3×C3⋊D3—61no—
27C2×C3×C3⋊D3—122no—
28C2×C3×C3⋊D3—121no—
29C2×C3×C3⋊D3—121no—
30C3×C3⋊Dic3—21no—
31C3×C3⋊Dic3—21no—
32C3×C3⋊Dic3—21no—
33C3×C3⋊Dic3—41no—
34C3×C3⋊Dic3C2×D311yescharacteristic
35C3×C3⋊Dic3C2×D311yescharacteristic
36C32×Dic3—61no—
37C32×Dic3—61no—
38C32×Dic3—61no—
39C32×Dic3—61no—
40C32×Dic3—121no—
41C34SD1611yescharacteristic, complemented, Sylow
42C4⋊(C3⋊D3)—91no—
43C3×Dic6—91no—
44C3×Dic6—181no—
45#627—31no—
46#627—61no—
47#626—181no—
48#626—181no—
49#626—182no—
50#625—91no—
51#622—31no—
52C3×C3⋊C8—91no—
53C2×C33—21no—
54C2×C33—21no—
55C2×C33—21no—
56C2×C33—41no—
57C2×C33—41no—
58C2×C33—42no—
59C2×C33—41no—
60C2×C33—41no—
61C2×C33—81no—
62C2×C33D1211yescharacteristic
63C2×C33#6711yescharacteristic
64C3⋊(C3⋊D3)—41no—
65C3×C3⋊D3—121no—
66C3×C3⋊D3—122no—
67C3×C3⋊D3—124no—
68C3×C3⋊D3—122no—
69C3×C3⋊D3—122no—
70C3×C3⋊D3—121no—
71C3×C3⋊D3—121no—
72#215—271no—
73#204—271no—
74C2×C3⋊D3—61no—
75C2×C3⋊D3—62no—
76C2×C3⋊D3—61no—
77C2×C3⋊D3—61no—
78C2×C3⋊D3—122no—
79C2×C3⋊D3—121no—
80C2×C3⋊D3—121no—
81C2×C3×D3—182no—
82C2×C3×D3—181no—
83C2×C3×D3—181no—
84C2×C3×D3—181no—
85C2×C3×D3—361no—
86C2×C3×D3—361no—
87C2×C3×D3—362no—
88C32×C4—91no—
89C32×C4—181no—
90C3⋊Dic3—21no—
91C3⋊Dic3D3211yescharacteristic
92C3×Dic3—61no—
93C3×Dic3—61no—
94C3×Dic3—61no—
95C3×Dic3—61no—
96C3×Dic3—62no—
97C3×Dic3—61no—
98C3×Dic3—61no—
99C3×Dic3—61no—
100C3×Dic3—61no—
101C3×Dic3—121no—
102C3×Dic3—121no—
103C3×Dic3—121no—
104C3×Dic3—121no—
105C3×Dic3—121no—
106C3×Dic3—122no—
107C33—21no—
108C33—21no—
109C33—21no—
110C33—42no—
111C33—41no—
112C33—41no—
113C33—41no—
114C33—41no—
115C33—81no—
116C33#20411yescharacteristic, complemented
117C33#21511yescharacteristic, complemented
118C3×Q8—271no—
119#67—541no—
120D12—271no—
121D12—271no—
122D12—541no—
123Dic6—91no—
124Dic6—181no—
125C3×C8—271no—
126C3⋊C8—91no—
127C2×C32—22no—
128C2×C32—21no—
129C2×C32—21no—
130C2×C32—21no—
131C2×C32—21no—
132C2×C32—41no—
133C2×C32—42no—
134C2×C32—41no—
135C2×C32—41no—
136C2×C32—41no—
137C2×C32—41no—
138C2×C32—42no—
139C2×C32—41no—
140C2×C32—41no—
141C2×C32—42no—
142C2×C32—41no—
143C2×C32—41no—
144C2×C32—81no—
145C2×C32—81no—
146C2×C32—82no—
147C2×C32—82no—
148C2×C32—81no—
149C2×C32#62611yescharacteristic
150C2×C32GO+(4,𝔽2)11yescharacteristic
151C3⋊D3—122no—
152C3⋊D3—121no—
153C3⋊D3—121no—
154C3⋊D3—124no—
155C3⋊D3—122no—
156C3⋊D3—122no—
157C3⋊D3—121no—
158C3×D3—362no—
159C3×D3—361no—
160C3×D3—361no—
161C3×D3—362no—
162C3×D3—362no—
163C3×D3—364no—
164C3×D3—361no—
165SD16—811noSylow
166C22×C3—541no—
167C2×D3—182no—
168C2×D3—181no—
169C2×D3—181no—
170C2×D3—181no—
171C2×D3—362no—
172C2×D3—361no—
173C2×D3—361no—
174C3×C4—91no—
175C3×C4—91no—
176C3×C4—181no—
177C3×C4—181no—
178C3×C4—181no—
179C3×C4—361no—
180Dic3—61no—
181Dic3—61no—
182Dic3—61no—
183Dic3—61no—
184Dic3—121no—
185C32—22no—
186C32—21no—
187C32—21no—
188C32—21no—
189C32—21no—
190C32—41no—
191C32—41no—
192C32—41no—
193C32—41no—
194C32—41no—
195C32—42no—
196C32—42no—
197C32—42no—
198C32—41no—
199C32—41no—
200C32—41no—
201C32—41no—
202C32—81no—
203C32—81no—
204C32—82no—
205C32—82no—
206C32—81no—
207C32#371711yescharacteristic, complemented
208C32#377511yescharacteristic, complemented
209Q8—271no—
210D4—811no—
211C8—271no—
212C2×C3—21no—
213C2×C3—21no—
214C2×C3—21no—
215C2×C3—41no—
216C2×C3—41no—
217C2×C3—42no—
218C2×C3—41no—
219C2×C3—41no—
220C2×C3—81no—
221C2×C3—1081no—
222C2×C3D32⋊D311yescharacteristic
223C2×C3#157⋊D311yescharacteristic
224D3—362no—
225D3—361no—
226D3—364no—
227D3—362no—
228D3—362no—
229D3—361no—
230D3—361no—
231C22—541no—
232C4—91no—
233C4—181no—
234C3—21no—
235C3—21no—
236C3—21no—
237C3—42no—
238C3—41no—
239C3—41no—
240C3—41no—
241C3—41no—
242C3—81no—
243C3#8893611yescharacteristic, complemented
244C3#8894211yescharacteristic, complemented
245C2—1081no—
246C2#1062020411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2#10620204
commutator, cosocleC32×C3⋊Dic3C22
Fitting, socleC2×C34D4
prime radical#10620156C2
hypocenterC34SD16
LCSG ▷ C32×C3⋊Dic3 ▷ C2×C34 ▷ C34
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. C3×#6296C2semidirect
3. C3×C3⋊#622C2semidirect
9. #6296D3semidirect
16. C3⋊#622D3semidirect
41. C34SD16semidirect
116. C33#204semidirect
117. C33#215semidirect
207. C32#3717semidirect
208. C32#3775semidirect
243. C3#88936semidirect
244. C3#88942semidirect

Maximal subgroups

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

subgroupindex
#106201562
C3×#62962
C3×C3⋊#6222
#889423
#889363
#37179