Finite Group Databasebeta

← #13049447

Subgroups — #13049447

Order 1296.

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

normal subgroups only

This group has 279 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
1C3⋊#6296C211yesmaximal, characteristic, complemented
2C3×#6295C211yesmaximal, characteristic
3C3×C3⋊#622C211yesmaximal, characteristic, complemented
4#88942—31nomaximal
5#88937—31nomaximal
6C2×C32×C3⋊D3—21no—
7C3×C3⋊(C3⋊Dic3)—21no—
8C32×C3⋊Dic3C2211yescharacteristic
9#6296—31no—
10#6296—31no—
11#6296—61no—
12#6295D311yescharacteristic
13#6293—61no—
14C3×#626—61no—
15C3×#625—31no—
16C3⋊#622D311yescharacteristic, complemented
17C3×#622—31no—
18C2×C34D411yescharacteristic
19C32×C3⋊D3—41no—
20#3775—91no—
21#3716—91nomaximal
22C2×C3×C3⋊D3—21no—
23C2×C3×C3⋊D3—21no—
24C2×C3×C3⋊D3—41no—
25C2×C32×D3—61no—
26C2×C32×D3—61no—
27C2×C32×D3—121no—
28C3⋊(C3⋊Dic3)—21no—
29C3×C3⋊Dic3—21no—
30C3×C3⋊Dic3—61no—
31C3×C3⋊Dic3—62no—
32C3×C3⋊Dic3—61no—
33C3×C3⋊Dic3—61no—
34C3×C3⋊Dic3—122no—
35C3×C3⋊Dic3—121no—
36C3×C3⋊Dic3—121no—
37C3×C3⋊Dic3C2×D311yescharacteristic
38C3×C3⋊Dic3C2×D311yescharacteristic
39C32×Dic3—61no—
40C32×Dic3—61no—
41C34SD1611yescharacteristic, complemented, Sylow
42#634—91no—
43C3×#67—181no—
44C3×D12—91no—
45#627—91no—
46#627—181no—
47#627—181no—
48#627—182no—
49#626—61no—
50#625—31no—
51#622—31no—
52C3×C3⋊C8—91no—
53C2×C33—21no—
54C2×C33—21no—
55C2×C33—21no—
56C2×C33—42no—
57C2×C33—41no—
58C2×C33—41no—
59C2×C33—41no—
60C2×C33—41no—
61C2×C33—81no—
62C2×C33D1211yescharacteristic
63C2×C33#6711yescharacteristic
64C3×C3⋊D3—41no—
65C3×C3⋊D3—42no—
66C3×C3⋊D3—41no—
67C32×D3—121no—
68C32×D3—122no—
69C32×D3—121no—
70#214—271no—
71#204—271no—
72C22×C32—181no—
73C2×C3⋊D3—21no—
74C2×C3×D3—61no—
75C2×C3×D3—61no—
76C2×C3×D3—61no—
77C2×C3×D3—61no—
78C2×C3×D3—121no—
79C2×C3×D3—121no—
80C2×C3×D3—121no—
81C2×C3×D3—121no—
82C2×C3×D3—122no—
83C32×C4—91no—
84C3⋊Dic3—61no—
85C3⋊Dic3—62no—
86C3⋊Dic3—61no—
87C3⋊Dic3—61no—
88C3⋊Dic3—122no—
89C3⋊Dic3—121no—
90C3⋊Dic3—121no—
91C3⋊Dic3D3211yescharacteristic
92C3×Dic3—61no—
93C3×Dic3—61no—
94C3×Dic3—61no—
95C3×Dic3—61no—
96C3×Dic3—62no—
97C3×Dic3—121no—
98C3×Dic3—182no—
99C3×Dic3—181no—
100C3×Dic3—181no—
101C3×Dic3—181no—
102C3×Dic3—361no—
103C3×Dic3—361no—
104C3×Dic3—362no—
105C33—21no—
106C33—21no—
107C33—21no—
108C33—41no—
109C33—41no—
110C33—42no—
111C33—41no—
112C33—41no—
113C33—81no—
114C33#20411yescharacteristic, complemented
115C33#21411yescharacteristic, complemented
116C3×D4—271no—
117#67—181no—
118D12—91no—
119Dic6—271no—
120Dic6—271no—
121Dic6—541no—
122Dic6—541no—
123C3×C8—271no—
124C3⋊C8—91no—
125C2×C32—22no—
126C2×C32—21no—
127C2×C32—21no—
128C2×C32—21no—
129C2×C32—21no—
130C2×C32—41no—
131C2×C32—42no—
132C2×C32—41no—
133C2×C32—41no—
134C2×C32—41no—
135C2×C32—41no—
136C2×C32—42no—
137C2×C32—41no—
138C2×C32—41no—
139C2×C32—42no—
140C2×C32—41no—
141C2×C32—41no—
142C2×C32—81no—
143C2×C32—81no—
144C2×C32—82no—
145C2×C32—82no—
146C2×C32—81no—
147C2×C32—361no—
148C2×C32#62611yescharacteristic
149C2×C32GO+(4,𝔽2)11yescharacteristic
150C3⋊D3—41no—
151C3×D3—121no—
152C3×D3—121no—
153C3×D3—122no—
154C3×D3—122no—
155C3×D3—124no—
156C3×D3—122no—
157C3×D3—122no—
158C3×D3—121no—
159C3×D3—121no—
160SD16—811noSylow
161C22×C3—181no—
162C22×C3—181no—
163C22×C3—361no—
164C2×D3—61no—
165C2×D3—61no—
166C2×D3—121no—
167C3×C4—91no—
168C3×C4—91no—
169C3×C4—181no—
170C3×C4—541no—
171Dic3—61no—
172Dic3—61no—
173Dic3—181no—
174Dic3—182no—
175Dic3—181no—
176Dic3—181no—
177Dic3—362no—
178Dic3—361no—
179Dic3—361no—
180C32—22no—
181C32—21no—
182C32—21no—
183C32—21no—
184C32—21no—
185C32—41no—
186C32—41no—
187C32—41no—
188C32—41no—
189C32—41no—
190C32—42no—
191C32—42no—
192C32—42no—
193C32—41no—
194C32—41no—
195C32—41no—
196C32—41no—
197C32—81no—
198C32—81no—
199C32—82no—
200C32—82no—
201C32—81no—
202C32#371611yescharacteristic, complemented
203C32#377511yescharacteristic, complemented
204Q8—811no—
205D4—271no—
206C8—271no—
207C2×C3—21no—
208C2×C3—21no—
209C2×C3—21no—
210C2×C3—42no—
211C2×C3—41no—
212C2×C3—41no—
213C2×C3—41no—
214C2×C3—41no—
215C2×C3—81no—
216C2×C3—361no—
217C2×C3—362no—
218C2×C3—361no—
219C2×C3D32⋊D311yescharacteristic
220C2×C3#157⋊D311yescharacteristic
221D3—121no—
222D3—122no—
223D3—121no—
224C22—181no—
225C4—91no—
226C4—541no—
227C3—21no—
228C3—21no—
229C3—21no—
230C3—41no—
231C3—41no—
232C3—42no—
233C3—41no—
234C3—41no—
235C3—81no—
236C3#8893711yescharacteristic, complemented
237C3#8894211yescharacteristic, complemented
238C2—361no—
239C2#1062020411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2#10620204
commutator, cosocleC32×C3⋊Dic3C22
Fitting, socleC2×C34D4
prime radicalC3×#6295C2
hypocenterC34SD16
LCSG ▷ C32×C3⋊Dic3 ▷ C2×C34 ▷ C34
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C3⋊#6296C2semidirect
3. C3×C3⋊#622C2semidirect
16. C3⋊#622D3semidirect
41. C34SD16semidirect
114. C33#204semidirect
115. C33#214semidirect
202. C32#3716semidirect
203. C32#3775semidirect
236. C3#88937semidirect
237. C3#88942semidirect

Maximal subgroups

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

subgroupindex
C3⋊#62962
C3×#62952
C3×C3⋊#6222
#889423
#889373
#37169