Finite Group Databasebeta

← #13165798

Subgroups — #13165798

Order 1728.

378 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 84 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×Dic3⋊#157C212yesmaximal
2C2×Dic3×#157C211yesmaximal, characteristic
3C3⋊#63747C212yesmaximal
4C3⋊#63747C211yesmaximal, characteristic
5C3×#63747C211yesmaximal, characteristic
6#10593545—31nomaximal
7C22×C3⋊#157C2212yes—
8C22×C3⋊#157C2211yescharacteristic
9C22×C3×#157C2212yes—
10C22×C3×#157C2211yescharacteristic
11C2×Dic3×C3⋊D3C2211yescharacteristic
12Dic3⋊#157—24no—
13Dic3×#157C414yescomplemented
14C2×C4⋊#157—32no—
15C2×C4×#157—31no—
16#63747—32no—
17#63747—31no—
18#63747D311yescharacteristic
19C22×C3×C3⋊D3C2311yescharacteristic
20C2×C3⋊#157—22no—
21C2×C3⋊#157—22no—
22C2×C3⋊#157—22no—
23C2×C3⋊#157—22no—
24C2×C3⋊#157C2×C412yes—
25C2×C3×#157—22no—
26C2×C3×#157—22no—
27C2×C3×#157—22no—
28C2×C3×#157—22no—
29C2×C3×#157C2×C412yes—
30C2×C3⋊(C3⋊Dic3)Q811yescharacteristic
31C2×C32×Dic3Q811yescharacteristic
32Dic3×C3⋊D3C2×C412yes—
33Dic3×C3⋊D3D412yes—
34#4650—91nomaximal
35C22×#157—31no—
36C22×#157—32no—
37C22×#157C2×D312yes—
38C22×#157C2×D311yescharacteristic
39C2×C4×C3⋊D3—31no—
40C2×D3×Dic3—121no—
41C4⋊#157—64no—
42C4×#157—34no—
43C22×C33C2×Q811yescharacteristic
44C2×C3×C3⋊D3C22×C411yescharacteristic
45C2×C3×C3⋊D3C2×D411yescharacteristic
46C2×C3×C3⋊D3#4112yes—
47C2×C3×C3⋊D3#4112yes—
48C3⋊#157—22no—
49C3⋊#157—42no—
50C3×#157—22no—
51C3×#157—42no—
52C3⋊(C3⋊Dic3)C4⋊C412yes—
53C32×Dic3C4⋊C412yes—
54C2×Dic3⋊C4—92no—
55C2×C4×Dic3—91no—
56C3×#95—91no—
57C3⋊#95—92no—
58C3⋊#95—91no—
59C22×C3⋊D3C22×D311yescharacteristic
60C22×C3×D3—121no—
61C2×#157—22no—
62C2×#157—22no—
63C2×#157—22no—
64C2×#157—22no—
65C2×#157—32no—
66C2×#157—62no—
67C2×#157—62no—
68C2×#157—62no—
69C2×#157—62no—
70C2×#157C4×D312yes—
71C2×C32×C4—31no—
72C2×C3⋊Dic3—31no—
73C2×C3⋊Dic3—121no—
74C2×C3⋊Dic3—241no—
75C4×C3⋊D3—32no—
76C4×C3⋊D3—32no—
77C2×C3×Dic3—41no—
78D3×Dic3—122no—
79D3×Dic3—122no—
80#382—271noSylow
81C2×C33C2×C4⋊C411yescharacteristic
82C2×C33#12512yes—
83C3×C3⋊D3#11711yescharacteristic
84C3×C3⋊D3#11711yescharacteristic
85C3×C3⋊D3#12412yes—
86C22×C3×C4—91no—
87C22×C3×C4—92no—
88C22×Dic3—91no—
89C22×Dic3—91no—
90C22×Dic3—92no—
91C2×C4×D3—361no—
92Dic3⋊C4—184no—
93C4×Dic3—94no—
94C22×C32—41no—
95C22×C32—81no—
96C22×C32D3×Q811yescharacteristic
97C2×C3⋊D3C2×C4×D311yescharacteristic
98C2×C3⋊D3#23512yes—
99C2×C3⋊D3D3×D411yescharacteristic
100C2×C3⋊D3#23712yes—
101C2×C3×D3—121no—
102C2×C3×D3—121no—
103C2×C3×D3—122no—
104C2×C3×D3—122no—
105#157—22no—
106#157—42no—
107#157—62no—
108#157—122no—
109C32×C4—32no—
110C3⋊Dic3—32no—
111C3⋊Dic3—122no—
112C3⋊Dic3—242no—
113C3×Dic3—42no—
114C2×C4⋊C4—272no—
115C2×C42—271no—
116#95—91no—
117#95—271no—
118#95—272no—
119C33#38211yescharacteristic, complemented, Sylow
120C23×C3—91no—
121C22×D3—121no—
122C2×C3×C4—92no—
123C2×C3×C4—121no—
124C2×C3×C4—182no—
125C2×C3×C4—182no—
126C2×C3×C4—182no—
127C2×C3×C4—182no—
128C2×Dic3—91no—
129C2×Dic3—92no—
130C2×Dic3—92no—
131C2×Dic3—92no—
132C2×Dic3—182no—
133C2×Dic3—182no—
134C2×Dic3—182no—
135C2×Dic3—182no—
136C2×Dic3—361no—
137C2×Dic3—721no—
138C2×Dic3PSU(3,𝔽2)11yescharacteristic
139C4×D3—362no—
140C4×D3—362no—
141C2×C32—42no—
142C2×C32—41no—
143C2×C32—81no—
144C2×C32—82no—
145C2×C32#88912yes—
146C2×C32D3×C4⋊C411yescharacteristic
147C3⋊D3#87211yescharacteristic
148C3⋊D3#87711yescharacteristic
149C3⋊D3#88512yes—
150C3×D3—121no—
151C3×D3—121no—
152C3×D3—122no—
153C22×C4—91no—
154C22×C4—92no—
155C22×C4—271no—
156C22×C4—271no—
157C22×C4—272no—
158C4⋊C4—544no—
159C42—274no—
160C22×C3—41no—
161C22×C3—81no—
162C22×C3—92no—
163C22×C3—92no—
164C22×C3—91no—
165C22×C3—91no—
166C22×C3C2×PSU(3,𝔽2)11yescharacteristic
167C2×D3—121no—
168C2×D3—121no—
169C2×D3—122no—
170C2×D3—122no—
171C3×C4—122no—
172C3×C4—182no—
173C3×C4—362no—
174Dic3—92no—
175Dic3—182no—
176Dic3—362no—
177Dic3—362no—
178Dic3—722no—
179Dic3#377712yes—
180C32—41no—
181C32—81no—
182C32#465011yescharacteristic, complemented
183C23—91no—
184C2×C4—31no—
185C2×C4—92no—
186C2×C4—182no—
187C2×C4—182no—
188C2×C4—182no—
189C2×C4—182no—
190C2×C4—272no—
191C2×C4—272no—
192C2×C4—271no—
193C2×C4—272no—
194C2×C4—542no—
195C2×C4—542no—
196C2×C4—542no—
197C2×C4—542no—
198C2×C3—41no—
199C2×C3—42no—
200C2×C3—81no—
201C2×C3—82no—
202C2×C3—91no—
203C2×C3—92no—
204C2×C3—91no—
205C2×C3#6424312yes—
206C2×C3C2×#377711yescharacteristic
207D3—121no—
208D3—121no—
209D3—122no—
210C22—91no—
211C22—91no—
212C22—92no—
213C22—92no—
214C22D3×PSU(3,𝔽2)11yescharacteristic
215C4—32no—
216C4—182no—
217C4—272no—
218C4—362no—
219C4—542no—
220C4—1082no—
221C3—41no—
222C3—81no—
223C3#1059354511yescharacteristic, complemented
224C2—91no—
225C2—92no—
226C2—91no—
227C2D3×#377711yescharacteristic
228C2#1172415212yes—
subgroupG/H
center, Frattini, hypercenterC22D3×PSU(3,𝔽2)
commutatorC2×C3×C3⋊D3C22×C4
Fitting, socleC22×C33C2×Q8
prime radical, cosocleC22×C3×C3⋊D3C23
hypocenterC33#382
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
13. Dic3×#157C4semidirect
119. C33#382semidirect
182. C32#4650semidirect
223. C3#10593545semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3⋊#1572
C2×Dic3×#1572
C3⋊#637472
C3×#637472
#105935453
#46509