Finite Group Databasebeta

← #13135214

Subgroups — #13135214

Order 1728.

248 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 45 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#64153⋊C3C211yesmaximal, characteristic
2#11720584C211yesmaximal, characteristic
3#63663⋊C3C211yesmaximal, characteristic
4#10589185—31nomaximal
5#10589184—31nomaximal
6C2×Q8×H(3,𝔽3)C2211yescharacteristic
7C2×#634⋊C3C411yescharacteristic
8#88538—21no—
9#3750⋊C3—21no—
10#3750⋊C3—21no—
11C4×(C3⋊Dic3)⋊C3C411yescharacteristic
12#3748⋊C3—21no—
13#64153D311yescharacteristic
14C3×#944—31no—
15#63663D311yescharacteristic
16C3×#836—31no—
17C3×#836—31no—
18C3×#836—31no—
19C3×#836—61no—
20Q8×H(3,𝔽3)—41no—
21C2×C4×H(3,𝔽3)C2×C411yescharacteristic
22C2×C4×H(3,𝔽3)D411yescharacteristic
23C2×C4×H(3,𝔽3)D411yescharacteristic
24C2×(C3⋊Dic3)⋊C3—21no—
25C2×(C3⋊Dic3)⋊C3—21no—
26#634⋊C3—41no—
27#6180—41no—
28(C3⋊(C3⋊C8))⋊C3—41no—
29#4548—91no—
30#4478—91no—
31#4478—91no—
32C2×C32×Q8—21no—
33C2×C32×Q8C2×D311yescharacteristic
34C2×C32×Q8C2×D311yescharacteristic
35C2×#634Dic311yescharacteristic
36C2×C3×Dic6—31no—
37#3750—21no—
38#3750—21no—
39C4×C3⋊Dic3Dic311yescharacteristic
40#3748—21no—
41C3×Dic3⋊C4—61no—
42C3×Dic3⋊C4—61no—
43C3×C4×Dic3—31no—
44C3×#208—62no—
45C3×#208—61no—
46C3×#208—61no—
47C3×#208—61no—
48C22×H(3,𝔽3)C22⋊C411yescharacteristic
49C4×H(3,𝔽3)—21no—
50C4×H(3,𝔽3)—21no—
51C4×H(3,𝔽3)—21no—
52(C3⋊Dic3)⋊C3—22no—
53(C3⋊Dic3)⋊C3—41no—
54C3×C4⋊Q8—91no—
55#944—31no—
56#944—91no—
57C3×#101—91no—
58C3×#101—91no—
59#836—31no—
60#836—31no—
61#836—31no—
62#836—61no—
63#836—91no—
64C32×Q8—41no—
65C32×Q8—41no—
66C32×Q8—42no—
67C2×C32×C4—21no—
68C2×C32×C4—21no—
69C2×C32×C4—21no—
70C2×C32×C4C4×D311yescharacteristic
71C2×C32×C4D1211yescharacteristic
72C2×C32×C4C2×Dic311yescharacteristic
73C2×C32×C4#6711yescharacteristic
74C2×C32×C4#6711yescharacteristic
75C2×C32×C4#6711yescharacteristic
76C2×C3⋊Dic3—21no—
77C2×C3⋊Dic3—21no—
78#634—41no—
79C2×C3×Dic3—61no—
80C2×C3×Dic3—61no—
81C3×Dic6—121no—
82C3⋊(C3⋊C8)—41no—
83C3×C3⋊C8—121no—
84C3×C3⋊C8—122no—
85C3×C3⋊C8—121no—
86C3×C3⋊C8—121no—
87#356—271noSylow
88C2×H(3,𝔽3)—21no—
89C2×H(3,𝔽3)#9911yescharacteristic
90C2×C3×Q8—31no—
91C2×C3×Q8—31no—
92C2×C3×Q8—61no—
93C2×C3×Q8—91no—
94C2×C3×Q8D3211yescharacteristic
95C2×Dic6—31no—
96C2×Dic6—91no—
97C3×M16—181no—
98C3×M16—181no—
99C3×C4⋊C4—181no—
100C3×C4⋊C4—181no—
101C3×C42—91no—
102Dic3⋊C4—61no—
103Dic3⋊C4—61no—
104Dic3⋊C4—181no—
105Dic3⋊C4—181no—
106C4×Dic3—31no—
107C4×Dic3—91no—
108#208—61no—
109#208—61no—
110#208—62no—
111#208—61no—
112#208—181no—
113C22×C32—21no—
114C22×C32(C2×D3)⋊C411yescharacteristic
115C22×C32#21711yescharacteristic
116C32×C4—22no—
117C32×C4—21no—
118C32×C4—21no—
119C32×C4—21no—
120C32×C4—21no—
121C32×C4—21no—
122C32×C4—21no—
123C32×C4—41no—
124C32×C4—41no—
125C3⋊Dic3—22no—
126C3⋊Dic3—41no—
127C3×Dic3—62no—
128C3×Dic3—121no—
129C4⋊Q8—91no—
130#101—91no—
131#101—91no—
132H(3,𝔽3)#35611yescharacteristic, complemented, Sylow
133C3×Q8—41no—
134C3×Q8—121no—
135C3×Q8—121no—
136C3×Q8—122no—
137C3×Q8—361no—
138C2×C3×C4—31no—
139C2×C3×C4—31no—
140C2×C3×C4—31no—
141C2×C3×C4—31no—
142C2×C3×C4—31no—
143C2×C3×C4—31no—
144C2×C3×C4—61no—
145C2×C3×C4—61no—
146C2×C3×C4—61no—
147C2×C3×C4—181no—
148C2×C3×C4—181no—
149C2×C3×C4D3×Dic311yescharacteristic
150C2×C3×C4#62511yescharacteristic
151C2×C3×C4#62611yescharacteristic
152C2×Dic3—61no—
153C2×Dic3—61no—
154C2×Dic3—181no—
155C2×Dic3—181no—
156Dic6—121no—
157Dic6—361no—
158C3×C8—361no—
159C3×C8—361no—
160C3⋊C8—121no—
161C3⋊C8—121no—
162C3⋊C8—121no—
163C3⋊C8—122no—
164C3⋊C8—361no—
165C2×C32—21no—
166C2×C32—21no—
167C2×C32—21no—
168C2×C32—41no—
169C2×C32#80611yescharacteristic
170C2×C32#83411yescharacteristic
171C2×Q8—91no—
172C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
173M16—181no—
174M16—181no—
175C4⋊C4—181no—
176C4⋊C4—181no—
177C42—91no—
178C22×C3—31no—
179C22×C3—31no—
180C22×C3—61no—
181C22×C3(C2×D3)⋊Dic311yescharacteristic
182C3×C4—21no—
183C3×C4—21no—
184C3×C4—21no—
185C3×C4—61no—
186C3×C4—61no—
187C3×C4—61no—
188C3×C4—61no—
189C3×C4—61no—
190C3×C4—61no—
191C3×C4—62no—
192C3×C4—121no—
193C3×C4—121no—
194C3×C4—182no—
195C3×C4—361no—
196Dic3—62no—
197Dic3—121no—
198Dic3—182no—
199Dic3—361no—
200C32—21no—
201C32#447811yescharacteristic, complemented
202C32#454811yescharacteristic, complemented
203Q8—41no—
204Q8—361no—
205C2×C4—181no—
206C2×C4—181no—
207C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
208C2×C4#619811yescharacteristic
209C2×C4#620011yescharacteristic
210C8—361no—
211C8—361no—
212C2×C3—21no—
213C2×C3—31no—
214C2×C3—31no—
215C2×C3—61no—
216C2×C3—61no—
217C2×C3—61no—
218C2×C3—121no—
219C2×C3#6358011yescharacteristic
220C22#8845011yescharacteristic
221C4—21no—
222C4—21no—
223C4—21no—
224C4—182no—
225C4—361no—
226C3—31no—
227C3—31no—
228C3—61no—
229C3#1058918411yescharacteristic
230C2—21no—
231C2#1172033311yescharacteristic
subgroupG/H
centerC2#11720333
commutatorC2×C4×H(3,𝔽3)C2×C4
FrattiniC2×C3×Q8D32
Fitting, cosocleC2×Q8×H(3,𝔽3)C22
socleC2×C3#63580
prime radicalC22×H(3,𝔽3)C22⋊C4
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#356
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
132. H(3,𝔽3)#356semidirect
201. C32#4478semidirect
202. C32#4548semidirect

Maximal subgroups

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

subgroupindex
#64153⋊C32
#117205842
#63663⋊C32
#105891853
#105891843