Finite Group Databasebeta

← #13138813

Subgroups — #13138813

Order 1728.

379 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 379 conjugacy classes of subgroups — too many to draw. Only its 91 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#11721116C212yesmaximal
2#11720550C211yesmaximal, characteristic
3#63644⋊C3C211yesmaximal, characteristic
4C8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
5#11720307C212yesmaximal
6#10590900—31nomaximal
7#10590889—31nomaximal
8C2×#6196—22no—
9C2×C8×H(3,𝔽3)C2211yescharacteristic
10C4⋊#1084C2212yes—
11C4⋊((C3⋊Dic3)⋊C3)C2212yes—
12C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
13C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
14#88448—22no—
15#63919—32no—
16#63918—32no—
17#63644D311yescharacteristic
18C8×C3⋊Dic3D311yescharacteristic
19C3×#817—31no—
20C3×#817—31no—
21C3×#817—31no—
22C3×#817—61no—
23C3×C8×Dic3—31no—
24#63574—32no—
25#63573—32no—
26C2×C4×H(3,𝔽3)C2311yescharacteristic
27C2×#1084—22no—
28C2×(C3⋊Dic3)⋊C3—22no—
29C2×(C3⋊Dic3)⋊C3D412yes—
30#6196—44no—
31C8×H(3,𝔽3)Q812yescomplemented
32(C3⋊(C3⋊C8))⋊C3Q812yescomplemented
33#4855—91no—
34#4855—91no—
35#4680—91no—
36C2×#627—62no—
37C2×#627—62no—
38C2×C32×C8—21no—
39C2×C32×C8C2×D311yescharacteristic
40C2×C32×C8C2×D311yescharacteristic
41C4⋊(C3⋊Dic3)C2×D312yes—
42C4×C3⋊Dic3C2×D311yescharacteristic
43C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
44C3×C4⋊Dic3—32no—
45C3×C4⋊Dic3—32no—
46C3×C4⋊Dic3—32no—
47C3×C4⋊Dic3—62no—
48C3×C4×Dic3—31no—
49C2×C3×C3⋊C8—31no—
50#3724—62no—
51Dic3⋊Dic3—62no—
52C22×H(3,𝔽3)C2×D411yescharacteristic
53C4×H(3,𝔽3)C2×Q811yescharacteristic
54C4×H(3,𝔽3)C2×Q811yescharacteristic
55#1084—42no—
56(C3⋊Dic3)⋊C3—42no—
57(C3⋊Dic3)⋊C3SD1614yes—
58#888—92no—
59#888—92no—
60#869—92no—
61C3×#106—91no—
62C3×#106—91no—
63C3×C4×C8—91no—
64#817—31no—
65#817—31no—
66#817—31no—
67#817—61no—
68#817—91no—
69#817—92no—
70C8×Dic3—31no—
71C8×Dic3—91no—
72#808—92no—
73#808—92no—
74C2×C32×C4—21no—
75C2×C32×C4C22×D311yescharacteristic
76C2×C32×C4C22×D311yescharacteristic
77C2×C3⋊Dic3—22no—
78C2×C3⋊Dic3D1212yes—
79C2×C3×Dic3—32no—
80C2×C3×Dic3—62no—
81C2×C3×Dic3—62no—
82C2×C3×Dic3—62no—
83C2×C3×Dic3—122no—
84#627—124no—
85#627—124no—
86C32×C8—22no—
87C32×C8Dic612yescomplemented
88C32×C8Dic612yescomplemented
89C3⋊(C3⋊C8)Dic612yescomplemented
90C3×C3⋊C8—32no—
91#498—271noSylow
92C2×H(3,𝔽3)C4⋊Q811yescharacteristic
93C2×H(3,𝔽3)C2×SD1612yes—
94C2×Dic6—182no—
95C2×Dic6—182no—
96C2×Dic6—182no—
97C2×C3×C8—31no—
98C2×C3×C8—31no—
99C2×C3×C8—61no—
100C2×C3×C8—91no—
101C2×C3×C8D3211yescharacteristic
102C3×C4⋊C4—92no—
103C3×C4⋊C4—92no—
104C3×C42—91no—
105C4⋊Dic3—32no—
106C4⋊Dic3—32no—
107C4⋊Dic3—32no—
108C4⋊Dic3—62no—
109C4⋊Dic3—92no—
110C4⋊Dic3—182no—
111Dic3⋊C4—182no—
112Dic3⋊C4—182no—
113C4×Dic3—31no—
114C4×Dic3—91no—
115C2×C3⋊C8—31no—
116C2×C3⋊C8—91no—
117C22×C32—21no—
118C22×C32C2×D1211yescharacteristic
119C22×C32D3×D411yescharacteristic
120C32×C4—21no—
121C32×C4—21no—
122C32×C4C2×Dic611yescharacteristic
123C32×C4C2×Dic611yescharacteristic
124C32×C4C2×Dic611yescharacteristic
125C32×C4D3×Q811yescharacteristic
126C3⋊Dic3—42no—
127C3⋊Dic3#20414yes—
128C3×Dic3—34no—
129C3×Dic3—122no—
130C3×Dic3—122no—
131C3×Dic3—122no—
132C3×Dic3—242no—
133C4⋊Q8—272no—
134#106—91no—
135#106—91no—
136#106—272no—
137C4×C8—91no—
138H(3,𝔽3)#49811yescharacteristic, complemented, Sylow
139C2×C3×C4—31no—
140C2×C3×C4—31no—
141C2×C3×C4—61no—
142C2×C3×C4—92no—
143C2×C3×C4—182no—
144C2×C3×C4—182no—
145C2×C3×C4C2×D3211yescharacteristic
146C2×Dic3—32no—
147C2×Dic3—62no—
148C2×Dic3—62no—
149C2×Dic3—62no—
150C2×Dic3—92no—
151C2×Dic3—122no—
152C2×Dic3—182no—
153Dic6—364no—
154Dic6—364no—
155Dic6—364no—
156C3×C8—32no—
157C3×C8—32no—
158C3×C8—62no—
159C3×C8—92no—
160C3×C8#62712yes—
161C3⋊C8—32no—
162C3⋊C8—92no—
163C2×C32—22no—
164C2×C32—21no—
165C2×C32#86911yescharacteristic
166C2×C32#88811yescharacteristic
167C2×C32C2×#20412yes—
168C2×C32D3×SD1612yes—
169C2×Q8—542no—
170C2×C8—91no—
171C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
172C4⋊C4—92no—
173C4⋊C4—92no—
174C4⋊C4—542no—
175C42—91no—
176C22×C3—31no—
177C22×C3—31no—
178C22×C3—61no—
179C22×C3D3×D1211yescharacteristic
180C3×C4—31no—
181C3×C4—31no—
182C3×C4—31no—
183C3×C4—31no—
184C3×C4—61no—
185C3×C4—61no—
186C3×C4—94no—
187C3×C4—362no—
188C3×C4—362no—
189C3×C4D3×Dic611yescharacteristic
190C3×C4C2×#62711yescharacteristic
191Dic3—34no—
192Dic3—94no—
193Dic3—122no—
194Dic3—122no—
195Dic3—122no—
196Dic3—242no—
197Dic3—362no—
198C32—21no—
199C32#468011yescharacteristic, complemented
200C32#485511yescharacteristic, complemented
201Q8—1084no—
202C2×C4—92no—
203C2×C4—182no—
204C2×C4—182no—
205C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
206C8—92no—
207C8#619612yescomplemented
208C2×C3—32no—
209C2×C3—32no—
210C2×C3—31no—
211C2×C3—31no—
212C2×C3—62no—
213C2×C3—61no—
214C2×C3D3×#20412yes—
215C2×C3#6391811yescharacteristic
216C22#8865411yescharacteristic
217C4—94no—
218C4—362no—
219C4—362no—
220C4((C3⋊D3)⋊C3)⋊Q811yescharacteristic
221C4C2×#619611yescharacteristic
222C3—31no—
223C3—31no—
224C3—61no—
225C3#1059088911yescharacteristic
226C2#1172084612yes—
227C2#1172111611yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#498
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
31. C8×H(3,𝔽3)Q8semidirect
32. (C3⋊(C3⋊C8))⋊C3Q8semidirect
87. C32×C8Dic6semidirect
88. C32×C8Dic6semidirect
89. C3⋊(C3⋊C8)Dic6semidirect
138. H(3,𝔽3)#498semidirect
199. C32#4680semidirect
200. C32#4855semidirect
207. C8#6196semidirect

Maximal subgroups

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

subgroupindex
#117211162
#117205502
#63644⋊C32
C8×(C3⋊Dic3)⋊C32
#117203072
#105909003
#105908893