Finite Group Databasebeta

← #13135086

Subgroups — #13135086

Order 1728.

224 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 59 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#64110⋊C3C211yesmaximal, characteristic
2C2×#88386C211yesmaximal, characteristic, complemented
3#63639⋊C3C211yesmaximal, characteristic, complemented
4#10589113—31nomaximal
5#10589112—31nomaximal
6#3828⋊C3C411yescharacteristic
7C2×C8×H(3,𝔽3)C2211yescharacteristic
8#3748⋊C3C411yescharacteristic
9#3743⋊C3—21no—
10C8×(C3⋊D3)⋊C3—21no—
11#88386—21no—
12(C3⋊(C3⋊C16))⋊C3—21no—
13#64110D311yescharacteristic
14C3×#901—31no—
15#63639D311yescharacteristic, complemented
16C3×#812—31no—
17C2×C3×C3⋊C16—31no—
18C2×C3×C3⋊C16—31no—
19C2×C3×C3⋊C16—61no—
20C2×C4×H(3,𝔽3)C2×C411yescharacteristic
21#638⋊C3—21no—
22C4⋊((C3⋊D3)⋊C3)C811yescharacteristic
23C4×(C3⋊D3)⋊C3—21no—
24#634⋊C3C811yescharacteristic
25C8×H(3,𝔽3)D411yescharacteristic
26C8×H(3,𝔽3)D411yescharacteristic
27(C3⋊(C3⋊C8))⋊C3—21no—
28#4561—91no—
29#4515—91no—
30#4508—91no—
31#3828Dic311yescharacteristic
32C3×#235—31no—
33C2×C32×C8—21no—
34C2×C32×C8C2×D311yescharacteristic
35C2×C32×C8C2×D311yescharacteristic
36#3748Dic311yescharacteristic
37#3743—21no—
38C8×C3⋊D3—21no—
39C3×#208—31no—
40C3×ΓL(1,𝔽25)—61no—
41C3×D3×C8—61no—
42C3⋊(C3⋊C16)—21no—
43C3×C3⋊C16—62no—
44C3×C3⋊C16—61no—
45C3×C3⋊C16—61no—
46C3×C3⋊C16—61no—
47C22×H(3,𝔽3)M1611yescharacteristic
48C2×(C3⋊D3)⋊C3—21no—
49C4×H(3,𝔽3)C22⋊C411yescharacteristic
50C4×H(3,𝔽3)C2×C811yescharacteristic
51(C3⋊Dic3)⋊C3—21no—
52C3×#131—91no—
53#901—31no—
54#901—91no—
55C3×M32—91no—
56C2×C3×C16—91no—
57#812—31no—
58#812—91no—
59C2×C3⋊C16—31no—
60C2×C3⋊C16—31no—
61C2×C3⋊C16—61no—
62C2×C32×C4—21no—
63C2×C32×C4C4×D311yescharacteristic
64C2×C32×C4C2×Dic311yescharacteristic
65#638—21no—
66C4⋊(C3⋊D3)C3⋊C811yescharacteristic
67C4×C3⋊D3—21no—
68#634C3⋊C811yescharacteristic
69C3×#67—61no—
70C3×D12—31no—
71C3×C4×D3—61no—
72C3×Dic6—31no—
73C32×C8—21no—
74C32×C8—21no—
75C32×C8D1211yescharacteristic
76C32×C8#6711yescharacteristic
77C32×C8#6711yescharacteristic
78C32×C8#6711yescharacteristic
79C3⋊(C3⋊C8)—21no—
80C3×C3⋊C8—61no—
81#350—271noSylow
82C2×H(3,𝔽3)—21no—
83C2×H(3,𝔽3)C22⋊C811yescharacteristic
84(C3⋊D3)⋊C3—41no—
85C3×#41—91no—
86#235—31no—
87#235—91no—
88C3×M16—91no—
89C3×M16—181no—
90C2×C3×C8—31no—
91C2×C3×C8—31no—
92C2×C3×C8—61no—
93C2×C3×C8—181no—
94C2×C3×C8D3211yescharacteristic
95#208—31no—
96#208—91no—
97ΓL(1,𝔽25)—61no—
98ΓL(1,𝔽25)—181no—
99D3×C8—61no—
100D3×C8—181no—
101C3×C16—181no—
102C3×C16—181no—
103C3⋊C16—61no—
104C3⋊C16—61no—
105C3⋊C16—62no—
106C3⋊C16—61no—
107C3⋊C16—181no—
108C22×C32—21no—
109C22×C32ΓL(1,𝔽25)11yescharacteristic
110C22×C32#20811yescharacteristic
111C2×C3⋊D3—21no—
112C2×C3×D3—61no—
113C32×C4—21no—
114C32×C4—21no—
115C32×C4D3×C811yescharacteristic
116C32×C4C2×C3⋊C811yescharacteristic
117C32×C4(C2×D3)⋊C411yescharacteristic
118C32×C4#21711yescharacteristic
119C3⋊Dic3—21no—
120C3×Dic3—61no—
121#131—91no—
122M32—91no—
123C2×C16—91no—
124H(3,𝔽3)#35011yescharacteristic, complemented, Sylow
125C3×Q8—91no—
126C3×D4—91no—
127C3×D4—181no—
128C2×C3×C4—31no—
129C2×C3×C4—31no—
130C2×C3×C4—61no—
131C2×C3×C4—181no—
132C2×C3×C4D3×Dic311yescharacteristic
133#67—61no—
134#67—181no—
135D12—31no—
136D12—91no—
137C4×D3—61no—
138C4×D3—181no—
139Dic6—31no—
140Dic6—91no—
141C3×C8—31no—
142C3×C8—31no—
143C3×C8—31no—
144C3×C8—31no—
145C3×C8—61no—
146C3×C8—61no—
147C3×C8—181no—
148C3×C8#62511yescharacteristic
149C3×C8#62611yescharacteristic
150C3⋊C8—61no—
151C3⋊C8—181no—
152C2×C32—22no—
153C2×C32—21no—
154C2×C32—21no—
155C2×C32—21no—
156C2×C32(C2×D3)⋊C811yescharacteristic
157C2×C32#83011yescharacteristic
158C3⋊D3—41no—
159C3×D3—121no—
160#41—91no—
161M16—91no—
162M16—181no—
163C2×C8—181no—
164C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
165C16—181no—
166C16—181no—
167C22×C3—31no—
168C22×C3—31no—
169C22×C3—61no—
170C22×C3—181no—
171C22×C3#371111yescharacteristic
172C2×D3—61no—
173C2×D3—181no—
174C3×C4—31no—
175C3×C4—31no—
176C3×C4—31no—
177C3×C4—31no—
178C3×C4—61no—
179C3×C4—61no—
180C3×C4—181no—
181C3×C4D3×C3⋊C811yescharacteristic
182C3×C4(C2×D3)⋊Dic311yescharacteristic
183Dic3—61no—
184Dic3—181no—
185C32—21no—
186C32#451511yescharacteristic, complemented
187C32#456111yescharacteristic, complemented
188Q8—91no—
189D4—91no—
190D4—181no—
191C2×C4—181no—
192C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
193C8—181no—
194C8#619811yescharacteristic
195C8#620011yescharacteristic
196C2×C3—21no—
197C2×C3—31no—
198C2×C3—31no—
199C2×C3—61no—
200C2×C3—61no—
201C2×C3—62no—
202C2×C3—61no—
203C2×C3—361no—
204C2×C3(C2×D3)⋊(C3⋊C8)11yescharacteristic
205D3—121no—
206D3—361no—
207C22—181no—
208C22#8843011yescharacteristic
209C4—181no—
210C4((C3⋊D3)⋊C3)⋊C811yescharacteristic
211C4#8845011yescharacteristic
212C3—31no—
213C3—31no—
214C3—61no—
215C3#1058911111yescharacteristic
216C2—21no—
217C2—361no—
218C2#1172031711yescharacteristic
subgroupG/H
centerC8#6198
commutatorC4×H(3,𝔽3)C2×C8
FrattiniC2×C3×C8D32
Fitting, cosocleC2×C8×H(3,𝔽3)C22
socleC2×C3(C2×D3)⋊(C3⋊C8)
prime radical#3828⋊C3C4
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#350
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C8 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×#88386C2semidirect
3. #63639⋊C3C2semidirect
15. #63639D3semidirect
124. H(3,𝔽3)#350semidirect
186. C32#4515semidirect
187. C32#4561semidirect

Maximal subgroups

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

subgroupindex
#64110⋊C32
C2×#883862
#63639⋊C32
#105891133
#105891123