Finite Group Databasebeta

← #10589331⋊C3 #13135311

Subgroups — #10589331⋊C3 #13135311

Order 1728.

250 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 60 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#64079⋊C3C211yesmaximal, characteristic
2H(3,𝔽3)×#97C211yesmaximal, characteristic
3#63630⋊C3C211yesmaximal, characteristic
4#10589331C311yesmaximal, characteristic, complemented
5C3×#4465—31nomaximal
6C2×C8×H(3,𝔽3)—21no—
7C42×H(3,𝔽3)C2211yescharacteristic
8C4⋊((C3⋊Dic3)⋊C3)C411yescharacteristic
9C4⋊((C3⋊Dic3)⋊C3)C411yescharacteristic
10#3750⋊C3—21no—
11#3750⋊C3—21no—
12C2×(C3⋊(C3⋊C8))⋊C3—21no—
13#64079C2×C311yescharacteristic
14C3×#870—31no—
15C32×#97—21no—
16C32×#97D311yescharacteristic
17C32×#97C2×C311yescharacteristic
18#63630C2×C311yescharacteristic
19C3×#803—31no—
20C2×C4×H(3,𝔽3)C2×C411yescharacteristic
21C2×C4×H(3,𝔽3)D411yescharacteristic
22C2×C4×H(3,𝔽3)D411yescharacteristic
23C2×(C3⋊Dic3)⋊C3—21no—
24C2×(C3⋊Dic3)⋊C3—21no—
25C8×H(3,𝔽3)—22no—
26(C3⋊(C3⋊C8))⋊C3—22no—
27C3×#330—91no—
28#4465—31no—
29#4465—91no—
30C2×C32×C8—22no—
31C2×C32×C8—21no—
32C2×C32×C8—21no—
33C32×C42—21no—
34C32×C42C2×D311yescharacteristic
35C32×C42C22×C311yescharacteristic
36C4⋊(C3⋊Dic3)C3×C411yescharacteristic
37C4⋊(C3⋊Dic3)C3×C411yescharacteristic
38#3750—21no—
39#3750—21no—
40C2×C3⋊(C3⋊C8)—21no—
41C3×C4⋊Dic3—31no—
42C3×C4⋊Dic3—31no—
43C3×Dic3⋊C4—61no—
44C3×Dic3⋊C4—61no—
45C2×C3×C3⋊C8—61no—
46C22×H(3,𝔽3)C22⋊C411yescharacteristic
47C4×H(3,𝔽3)—21no—
48C4×H(3,𝔽3)—21no—
49C4×H(3,𝔽3)—21no—
50(C3⋊Dic3)⋊C3—41no—
51(C3⋊Dic3)⋊C3—41no—
52C3×#125—91no—
53#870—31no—
54#870—91no—
55C3×#97—31no—
56C3×#97—31no—
57C3×#97—61no—
58C3×#97—91no—
59C3×#97C3×D311yescharacteristic
60#803—31no—
61#803—91no—
62C2×C32×C4—21no—
63C2×C32×C4—21no—
64C2×C32×C4—21no—
65C2×C32×C4C4×D311yescharacteristic
66C2×C32×C4D1211yescharacteristic
67C2×C32×C4#6711yescharacteristic
68C2×C32×C4C2×C3×C411yescharacteristic
69C2×C32×C4C3×D411yescharacteristic
70C2×C32×C4C3×D411yescharacteristic
71C2×C3⋊Dic3—21no—
72C2×C3⋊Dic3—21no—
73C2×C3×Dic3—61no—
74C2×C3×Dic3—61no—
75C32×C8—24no—
76C32×C8—22no—
77C32×C8—22no—
78C3⋊(C3⋊C8)—22no—
79C3×C3⋊C8—62no—
80#330—91noSylow
81C2×H(3,𝔽3)#10111yescharacteristic
82C2×H(3,𝔽3)#10411yescharacteristic
83C2×H(3,𝔽3)#10411yescharacteristic
84C2×C3×C8—21no—
85C2×C3×C8—61no—
86C2×C3×C8—61no—
87C2×C3×C8—62no—
88C2×C3×C8—181no—
89C3×C4⋊C4—91no—
90C3×C4⋊C4—91no—
91C3×C4⋊C4—181no—
92C3×C4⋊C4—181no—
93C3×C42—31no—
94C3×C42—31no—
95C3×C42—61no—
96C3×C42C2×C3×D311yescharacteristic
97C4⋊Dic3—31no—
98C4⋊Dic3—31no—
99C4⋊Dic3—91no—
100C4⋊Dic3—91no—
101Dic3⋊C4—61no—
102Dic3⋊C4—61no—
103Dic3⋊C4—181no—
104Dic3⋊C4—181no—
105C2×C3⋊C8—61no—
106C2×C3⋊C8—181no—
107C22×C32—21no—
108C22×C32(C2×D3)⋊C411yescharacteristic
109C22×C32C3×C22⋊C411yescharacteristic
110C32×C4—22no—
111C32×C4—21no—
112C32×C4—21no—
113C32×C4—21no—
114C32×C4—21no—
115C32×C4—21no—
116C32×C4—21no—
117C32×C4—41no—
118C32×C4—41no—
119C3⋊Dic3—41no—
120C3⋊Dic3—41no—
121C3×Dic3—121no—
122C3×Dic3—121no—
123#125—91no—
124#97—91no—
125#97(C3⋊D3)⋊C311yescharacteristic
126H(3,𝔽3)#33011yescharacteristic, complemented, Sylow
127C2×C3×C4—31no—
128C2×C3×C4—31no—
129C2×C3×C4—31no—
130C2×C3×C4—31no—
131C2×C3×C4—31no—
132C2×C3×C4—31no—
133C2×C3×C4—61no—
134C2×C3×C4—61no—
135C2×C3×C4—61no—
136C2×C3×C4—181no—
137C2×C3×C4—181no—
138C2×C3×C4C3×C4×D311yescharacteristic
139C2×C3×C4C3×D1211yescharacteristic
140C2×C3×C4C3×#6711yescharacteristic
141C2×Dic3—61no—
142C2×Dic3—61no—
143C2×Dic3—181no—
144C2×Dic3—181no—
145C3×C8—22no—
146C3×C8—62no—
147C3×C8—62no—
148C3×C8—64no—
149C3×C8—182no—
150C3⋊C8—62no—
151C3⋊C8—182no—
152C2×C32—21no—
153C2×C32—21no—
154C2×C32—21no—
155C2×C32#80511yescharacteristic
156C2×C32#82411yescharacteristic
157C2×C32#82511yescharacteristic
158C2×C32C3×#10111yescharacteristic
159C2×C32C3×#10411yescharacteristic
160C2×C32C3×#10411yescharacteristic
161C2×C8—21no—
162C2×C8—181no—
163C4⋊C4—91no—
164C4⋊C4—91no—
165C4⋊C4—181no—
166C4⋊C4—181no—
167C42C2×(C3⋊D3)⋊C311yescharacteristic
168C22×C3—31no—
169C22×C3—31no—
170C22×C3—61no—
171C22×C3C3×(C2×D3)⋊C411yescharacteristic
172C3×C4—21no—
173C3×C4—21no—
174C3×C4—21no—
175C3×C4—61no—
176C3×C4—61no—
177C3×C4—61no—
178C3×C4—61no—
179C3×C4—61no—
180C3×C4—61no—
181C3×C4—62no—
182C3×C4—121no—
183C3×C4—121no—
184C3×C4—361no—
185C3×C4—361no—
186Dic3—121no—
187Dic3—121no—
188Dic3—361no—
189Dic3—361no—
190C32—21no—
191C32#446511yescharacteristic, complemented
192C32C3×#33011yescharacteristic, complemented
193C2×C4—181no—
194C2×C4—181no—
195C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
196C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
197C2×C4#638⋊C311yescharacteristic
198C8—22no—
199C8—182no—
200C2×C3—31no—
201C2×C3—31no—
202C2×C3—31no—
203C2×C3—31no—
204C2×C3—31no—
205C2×C3—31no—
206C2×C3—61no—
207C2×C3—61no—
208C2×C3—61no—
209C2×C3C3×#80511yescharacteristic
210C2×C3C3×#82411yescharacteristic
211C2×C3C3×#82511yescharacteristic
212C22#3752⋊C311yescharacteristic
213C4—21no—
214C4—21no—
215C4—21no—
216C4—361no—
217C4—361no—
218C3—31no—
219C3—31no—
220C3—61no—
221C3C3×#446511yescharacteristic
222C2#63632⋊C311yescharacteristic
223C2#63651⋊C311yescharacteristic
224C2#63652⋊C311yescharacteristic
subgroupG/H
centerC22#3752⋊C3
commutatorC2×C32×C4C2×C3×C4
FrattiniC3×C42C2×C3×D3
FittingH(3,𝔽3)×#97C2
socleC22×C3C3×(C2×D3)⋊C4
prime radicalC22×H(3,𝔽3)C22⋊C4
cosocleC32×C42C22×C3
hypercenter#97(C3⋊D3)⋊C3
hypocenterC32C3×#330
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #97

Complements

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

NH ≅ G/Ntype
4. #10589331C3semidirect
126. H(3,𝔽3)#330semidirect
191. C32#4465semidirect
192. C32C3×#330semidirect

Maximal subgroups

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

subgroupindex
#64079⋊C32
H(3,𝔽3)×#972
#63630⋊C32
#105893313
C3×#44653