Finite Group Databasebeta

← #13135102

Subgroups — #13135102

Order 1728.

263 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 263 conjugacy classes of subgroups — too many to draw. Only its 63 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
1C2×#88530C211yesmaximal, characteristic
2#63645⋊C3C211yesmaximal, characteristic
3C2×(C3⋊(C3⋊C16))⋊C3C211yesmaximal, characteristic
4#10589118—31nomaximal
5#10589117—31nomaximal
6C2×#6229—21no—
7C2×C8×H(3,𝔽3)C2211yescharacteristic
8#88530—21no—
9#88530C412yescomplemented
10C4⋊((C3⋊Dic3)⋊C3)—21no—
11(C3⋊(C3⋊C16))⋊C3—21no—
12C2×C3×Dic12—31no—
13C2×C3×Dic12—31no—
14C2×C3×Dic12—61no—
15#63645D311yescharacteristic
16C2×C3⋊(C3⋊C16)D311yescharacteristic
17C3×#818—31no—
18C2×C3×C3⋊C16—31no—
19C2×C4×H(3,𝔽3)D411yescharacteristic
20C2×#1084—41no—
21#6229—22no—
22#6229—41no—
23C2×(C3⋊Dic3)⋊C3—41no—
24C8×H(3,𝔽3)C2×C411yescharacteristic
25C8×H(3,𝔽3)D411yescharacteristic
26#4564—91no—
27#4503—91no—
28#4492—91no—
29C2×C3×Dic6—62no—
30C2×C3×Dic6—61no—
31C2×C3×Dic6—61no—
32C2×C32×C8—21no—
33C2×C32×C8C2×D311yescharacteristic
34C2×C32×C8C2×D311yescharacteristic
35C4⋊(C3⋊Dic3)—21no—
36C3×C4⋊Dic3—61no—
37C3×Dic12—32no—
38C3×Dic12—32no—
39C3×Dic12—62no—
40C3×Dic12—62no—
41C3×Dic12—61no—
42C3×Dic12—61no—
43C3⋊(C3⋊C16)—21no—
44C3×C3⋊C16—61no—
45C22×H(3,𝔽3)D811yescharacteristic
46C4×H(3,𝔽3)C22⋊C411yescharacteristic
47C4×H(3,𝔽3)SD1611yescharacteristic
48#1084—42no—
49(C3⋊Dic3)⋊C3—81no—
50C2×C3×Q16—91no—
51C2×Dic12—31no—
52C2×Dic12—31no—
53C2×Dic12—61no—
54C2×C3×C16—91no—
55C3×#107—91no—
56#818—31no—
57#818—91no—
58C2×C3⋊C16—31no—
59C2×C3⋊C16—91no—
60C2×C32×C4—21no—
61C2×C32×C4D1211yescharacteristic
62C2×C32×C4#6711yescharacteristic
63C2×C3⋊Dic3—41no—
64C2×C3×Dic3—122no—
65C2×C3×Dic3—121no—
66C2×C3×Dic3—121no—
67C2×C3×Dic3—121no—
68C3×Dic6—64no—
69C3×Dic6—62no—
70C3×Dic6—62no—
71C3×Dic6—122no—
72C3×Dic6—121no—
73C3×Dic6—121no—
74C32×C8—21no—
75C32×C8—21no—
76C32×C8C4×D311yescharacteristic
77C32×C8C4×D311yescharacteristic
78C32×C8D1211yescharacteristic
79C32×C8#6711yescharacteristic
80#358—271noSylow
81C2×H(3,𝔽3)#10211yescharacteristic
82C2×H(3,𝔽3)SD3211yescharacteristic
83C2×H(3,𝔽3)Q3211yescharacteristic
84C2×C3×Q8—181no—
85C2×Dic6—61no—
86C2×Dic6—61no—
87C2×Dic6—62no—
88C3×Q16—92no—
89C3×Q16—181no—
90C2×C3×C8—31no—
91C2×C3×C8—31no—
92C2×C3×C8—61no—
93C2×C3×C8D3211yescharacteristic
94C3×C4⋊C4—181no—
95C4⋊Dic3—61no—
96C4⋊Dic3—181no—
97Dic12—32no—
98Dic12—32no—
99Dic12—62no—
100Dic12—61no—
101Dic12—61no—
102Dic12—62no—
103C3×C16—181no—
104C3⋊C16—61no—
105C3⋊C16—181no—
106C22×C32—21no—
107C22×C32D2411yescharacteristic
108C22×C32#21311yescharacteristic
109C32×C4—21no—
110C32×C4—21no—
111C32×C4#20411yescharacteristic
112C32×C4(C2×D3)⋊C411yescharacteristic
113C32×C4(C2×D3)⋊C411yescharacteristic
114C32×C4#21511yescharacteristic
115C3⋊Dic3—81no—
116C3×Dic3—124no—
117C3×Dic3—122no—
118C3×Dic3—122no—
119C3×Dic3—241no—
120C2×Q16—91no—
121C2×C16—91no—
122#107—91no—
123H(3,𝔽3)#35811yescharacteristic, complemented, Sylow
124C3×Q8—182no—
125C3×Q8—361no—
126C2×C3×C4—31no—
127C2×C3×C4—31no—
128C2×C3×C4—61no—
129C2×C3×C4—361no—
130C2×C3×C4—361no—
131C2×C3×C4#62611yescharacteristic
132C2×Dic3—121no—
133C2×Dic3—121no—
134C2×Dic3—122no—
135C2×Dic3—121no—
136C2×Dic3—361no—
137Dic6—62no—
138Dic6—62no—
139Dic6—64no—
140Dic6—121no—
141Dic6—121no—
142Dic6—122no—
143C3×C8—31no—
144C3×C8—31no—
145C3×C8—31no—
146C3×C8—31no—
147C3×C8—61no—
148C3×C8—61no—
149C3×C8#62411yescharacteristic
150C3×C8#62611yescharacteristic
151C2×C32—21no—
152C2×C32—21no—
153C2×C32—21no—
154C2×C32#80011yescharacteristic
155C2×C32Dic2411yescharacteristic
156C2×C32#80911yescharacteristic
157C2×C32#82111yescharacteristic
158C2×C32#82711yescharacteristic
159C2×C32#82911yescharacteristic
160C2×Q8—181no—
161Q16—92no—
162Q16—181no—
163C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
164C4⋊C4—181no—
165C16—181no—
166C22×C3—31no—
167C22×C3—31no—
168C22×C3—61no—
169C22×C3#371411yescharacteristic
170C3×C4—31no—
171C3×C4—31no—
172C3×C4—31no—
173C3×C4—31no—
174C3×C4—61no—
175C3×C4—61no—
176C3×C4—362no—
177C3×C4—721no—
178C3×C4#371711yescharacteristic
179C3×C4#372211yescharacteristic
180Dic3—122no—
181Dic3—122no—
182Dic3—124no—
183Dic3—241no—
184Dic3—721no—
185C32—21no—
186C32#449211yescharacteristic, complemented
187C32#450311yescharacteristic, complemented
188Q8—182no—
189Q8—361no—
190C2×C4—361no—
191C2×C4—361no—
192C2×C4#620011yescharacteristic
193C8#619911yescharacteristic
194C8#620011yescharacteristic
195C2×C3—31no—
196C2×C3—31no—
197C2×C3—31no—
198C2×C3—31no—
199C2×C3—31no—
200C2×C3—31no—
201C2×C3—61no—
202C2×C3—61no—
203C2×C3—61no—
204C2×C3#6354811yescharacteristic
205C2×C3#6355111yescharacteristic
206C2×C3#6356411yescharacteristic
207C22#8843611yescharacteristic
208C4—362no—
209C4—721no—
210C4#8843711yescharacteristic
211C4#8845111yescharacteristic
212C3—31no—
213C3—31no—
214C3—61no—
215C3#1058911911yescharacteristic
216C2#1172025511yescharacteristic
217C2#1172025611yescharacteristic
218C2#1172032411yescharacteristic
subgroupG/H
centerC22#88436
commutatorC8×H(3,𝔽3)C2×C4
FrattiniC2×C3×C8D32
Fitting, cosocleC2×C8×H(3,𝔽3)C22
socleC22×C3#3714
prime radicalC22×H(3,𝔽3)D8
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#358
LCSG ▷ C8×H(3,𝔽3) ▷ 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
9. #88530C4semidirect
123. H(3,𝔽3)#358semidirect
186. C32#4492semidirect
187. C32#4503semidirect

Maximal subgroups

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

subgroupindex
C2×#885302
#63645⋊C32
C2×(C3⋊(C3⋊C16))⋊C32
#105891183
#105891173