Finite Group Databasebeta

← #77456⋊C3 #11921191

Subgroups — #77456⋊C3 #11921191

Order 1152.

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

normal subgroups only

This group has 318 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
1C3×#396⋊C3C211yesmaximal, characteristic, complemented
2C3⋊(#399⋊C3)C211yesmaximal, characteristic, complemented
3C3⋊(#396⋊C3)C211yesmaximal, characteristic, complemented
4#77456C311yesmaximal, characteristic, complemented
5#2842⋊C3—31nomaximal
6C22×D3×A4—41nomaximal
7C3×#95⋊C3C2211yescharacteristic, complemented
8C3×#396C2×C311yescharacteristic, complemented
9#4994—31no—
10#4993—31no—
11#4955—31no—
12#4948—31no—
13#4677C2×C311yescharacteristic, complemented
14#4676C2×C311yescharacteristic, complemented
15#399⋊C3—31no—
16#396⋊C3—21no—
17#396⋊C3—31no—
18#396⋊C3D311yescharacteristic, complemented
19C22×C3×A4—41no—
20C2×D3×A4—44no—
21C2×D3×A4—41no—
22C2×D3×A4—41no—
23#2842—31noSylow
24C24×D3A411yescharacteristic
25C23×A4—121no—
26C2×C3×C4⋊C4—31no—
27C2×C3×C22⋊C4—31no—
28C2×#217—31no—
29C2×(C2×D3)⋊C4—31no—
30C2×(C2×D3)⋊C4—31no—
31C2×(C2×D3)⋊C4—31no—
32C2×(C2×D3)⋊C4—31no—
33C2×C4⋊Dic3—31no—
34C2×Dic3⋊C4—31no—
35C2×Dic3⋊C4—31no—
36C3×#95C22×C311yescharacteristic, complemented
37C3⋊#95—31no—
38#95⋊C3—21no—
39#95⋊C3C2×D311yescharacteristic, complemented
40C22×C3×D3—161no—
41C2×C3×A4—42no—
42C2×C3×A4—41no—
43D3×A4—42no—
44D3×A4—42no—
45#399—31no—
46#399—91no—
47#396—31no—
48#396—91no—
49#396—91no—
50#396C3×D311yescharacteristic, complemented
51#379—91no—
52C24×C3C2×A411yescharacteristic
53C23×D3—121no—
54C23×D3—121no—
55C23×D3SL(2,𝔽3)14yes—
56C23×D3C2×A411yescharacteristic
57C23×D3C2×A411yescharacteristic
58C22×A4—41no—
59C22×A4—81no—
60C22×A4—121no—
61C22×A4—121no—
62C22×A4—124no—
63C22×C3×C4—31no—
64C22×C3×C4—31no—
65C22×Dic3—31no—
66C22×Dic3—31no—
67C3×C4⋊C4—121no—
68C3×C22⋊C4—121no—
69#217—121no—
70(C2×D3)⋊C4—121no—
71(C2×D3)⋊C4—121no—
72(C2×D3)⋊C4—121no—
73(C2×D3)⋊C4—121no—
74C4⋊Dic3—121no—
75Dic3⋊C4—121no—
76Dic3⋊C4—121no—
77C22×C32—161no—
78C2×C3×D3—164no—
79C2×C3×D3—161no—
80C2×C3×D3—161no—
81C3×A4—41no—
82C25—31no—
83C2×C4⋊C4—31no—
84C2×C4⋊C4—91no—
85C2×C4⋊C4—91no—
86C2×C4⋊C4—91no—
87C2×C22⋊C4—31no—
88C2×C22⋊C4—91no—
89C2×C22⋊C4—91no—
90C2×C22⋊C4—91no—
91C2×C22⋊C4—91no—
92C2×C22⋊C4—91no—
93#95—91no—
94#95C2×C3×D311yescharacteristic, complemented
95C23×C3—61no—
96C23×C3—61no—
97C23×C3—481no—
98C23×C3C2×SL(2,𝔽3)12yes—
99C23×C3C22×A411yescharacteristic
100C22×D3—41no—
101C22×D3—61no—
102C22×D3—61no—
103C22×D3—61no—
104C22×D3—61no—
105C22×D3—121no—
106C22×D3—124no—
107C22×D3—124no—
108C22×D3C2×SL(2,𝔽3)12yes—
109C22×D3C2×SL(2,𝔽3)12yes—
110C2×A4—41no—
111C2×A4—42no—
112C2×A4—81no—
113C2×A4—82no—
114C2×A4—122no—
115C2×A4—122no—
116C2×C3×C4—61no—
117C2×C3×C4—61no—
118C2×C3×C4—61no—
119C2×C3×C4—61no—
120C2×C3×C4—61no—
121C2×C3×C4—61no—
122C2×Dic3—61no—
123C2×Dic3—61no—
124C2×Dic3—61no—
125C2×Dic3—61no—
126C2×Dic3—61no—
127C2×Dic3—61no—
128C2×C32—162no—
129C2×C32—161no—
130C3×D3—162no—
131C3×D3—162no—
132C24—31no—
133C24—31no—
134C24—34no—
135C24—361no—
136C24—361no—
137C24D3×A411yescharacteristic
138C22×C4—31no—
139C22×C4—31no—
140C22×C4—91no—
141C22×C4—91no—
142C4⋊C4—121no—
143C4⋊C4—361no—
144C4⋊C4—361no—
145C4⋊C4—361no—
146C22⋊C4—121no—
147C22⋊C4—361no—
148C22⋊C4—361no—
149C22⋊C4—361no—
150C22⋊C4—361no—
151C22⋊C4—361no—
152C22×C3—31no—
153C22×C3—31no—
154C22×C3—41no—
155C22×C3—62no—
156C22×C3—121no—
157C22×C3—161no—
158C22×C3—321no—
159C22×C3—484no—
160C22×C3—481no—
161C22×C3—481no—
162C22×C3C22×SL(2,𝔽3)11yescharacteristic
163C2×D3—44no—
164C2×D3—41no—
165C2×D3—41no—
166C2×D3—62no—
167C2×D3—62no—
168C2×D3—121no—
169C2×D3—121no—
170C2×D3—124no—
171A4—41no—
172A4—81no—
173C3×C4—121no—
174C3×C4—121no—
175Dic3—121no—
176Dic3—121no—
177C32—161noSylow
178C23—32no—
179C23—32no—
180C23—61no—
181C23—61no—
182C23—121no—
183C23—181no—
184C23—181no—
185C23—181no—
186C23—181no—
187C23—361no—
188C23—364no—
189C23—364no—
190C23D3×SL(2,𝔽3)12yes—
191C23C2×D3×A411yescharacteristic
192C2×C4—61no—
193C2×C4—61no—
194C2×C4—61no—
195C2×C4—61no—
196C2×C4—61no—
197C2×C4—61no—
198C2×C4—181no—
199C2×C4—181no—
200C2×C4—181no—
201C2×C4—181no—
202C2×C4—181no—
203C2×C4—181no—
204C2×C3—31no—
205C2×C3—31no—
206C2×C3—42no—
207C2×C3—162no—
208C2×C3—161no—
209C2×C3—322no—
210C2×C3—321no—
211C2×C3—482no—
212C2×C3—482no—
213C2×C3#560⋊C311yescharacteristic
214D3—42no—
215D3—42no—
216C22—31no—
217C22—31no—
218C22—41no—
219C22—62no—
220C22—121no—
221C22—124no—
222C22—121no—
223C22—121no—
224C22—182no—
225C22—182no—
226C22—361no—
227C22—361no—
228C22—364no—
229C22C2×D3×SL(2,𝔽3)11yescharacteristic
230C4—121no—
231C4—121no—
232C4—361no—
233C4—361no—
234C3—161no—
235C3—321no—
236C3#2842⋊C311yescharacteristic, complemented
237C2—31no—
238C2—31no—
239C2—42no—
240C2—122no—
241C2—122no—
242C2(#126⋊C3)⋊D311yescharacteristic
subgroupG/H
center, hypercenterC2(#126⋊C3)⋊D3
commutator, cosocle, hypocenterC3×#95C22×C3
FrattiniC23C2×D3×A4
FittingC3×#396C2×C3
socleC23×C3C22×A4
LCSG ▷ C3×#95
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C3×#396⋊C3C2semidirect
2. C3⋊(#399⋊C3)C2semidirect
3. C3⋊(#396⋊C3)C2semidirect
4. #77456C3semidirect
7. C3×#95⋊C3C22semidirect
8. C3×#396C2×C3semidirect
13. #4677C2×C3semidirect
14. #4676C2×C3semidirect
18. #396⋊C3D3semidirect
36. C3×#95C22×C3semidirect
39. #95⋊C3C2×D3semidirect
50. #396C3×D3semidirect
94. #95C2×C3×D3semidirect
236. C3#2842⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#396⋊C32
C3⋊(#399⋊C3)2
C3⋊(#396⋊C3)2
#774563
#2842⋊C33
C22×D3×A44