Finite Group Databasebeta

← #13173433

Subgroups — #13173433

Order 1728.

680 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 89 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×C3×Q8×C3⋊D3C211yesmaximal, characteristic
2#11724511C212yesmaximal
3#11723105C214yesmaximal, complemented
4#10593836—31nomaximal
5C2×C33×Q8C411yescharacteristic
6C3×Q8×C3⋊D3—22no—
7C3×Q8×C3⋊D3—22no—
8C2×C3×C4×C3⋊D3C412yes—
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C3×#634C411yescharacteristic
11C2×C3×#634C2212yes—
12#88993C2214yes—
13#88989—22no—
14#88988—22no—
15C2×Q8×C3⋊D3D311yescharacteristic
16C2×C3×D3×Q8—62no—
17#64283—32no—
18#63780—34no—
19C22×C3×C3⋊D3C2×C411yescharacteristic
20C33×Q8—22no—
21C2×C33×C4C2×C412yes—
22C2×C33×C4C2×C411yescharacteristic
23C2×C3×C3⋊Dic3C2×C412yes—
24C2×C3×C3⋊Dic3C2311yescharacteristic
25C3×C4×C3⋊D3—22no—
26C3×C4×C3⋊D3—22no—
27C3×C4×C3⋊D3D412yes—
28C3×C4×C3⋊D3D412yes—
29C3×#634—22no—
30C3×#634—24no—
31C3⋊#622—24no—
32C2×#836—91nomaximal
33C2×C32×Q8—22no—
34C2×C32×Q8—42no—
35C2×C32×Q8Dic311yescharacteristic
36Q8×C3⋊D3—22no—
37Q8×C3⋊D3—22no—
38C2×C4×C3⋊D3Dic312yes—
39C2×C4×C3⋊D3C2×D311yescharacteristic
40C2×#634Dic311yescharacteristic
41C2×#634C2×D312yes—
42C3×D3×Q8—68no—
43C3×D3×Q8—68no—
44C2×C3×C4×D3—64no—
45C2×C3×C4×D3—62no—
46C2×C3×Dic6—62no—
47C2×C3×Dic6—64no—
48#3792—34no—
49#3788—62no—
50#3787—62no—
51C22×C33C22×C411yescharacteristic
52C2×C3×C3⋊D3—22no—
53C2×C3×C3⋊D3C22⋊C411yescharacteristic
54C2×C3×C3⋊D3C22⋊C411yescharacteristic
55C33×C4—22no—
56C33×C4C22⋊C412yes—
57C3×C3⋊Dic3—22no—
58C3×C3⋊Dic3C2×D412yes—
59C22×C3×Q8—91no—
60C2×D3×Q8—62no—
61C2×#208—92no—
62#836—94no—
63C22×C3⋊D3C2×Dic311yescharacteristic
64C22×C3×D3—62no—
65C32×Q8—22no—
66C32×Q8—28no—
67C32×Q8—48no—
68C2×C32×C4—24no—
69C2×C32×C4—22no—
70C2×C32×C4—44no—
71C2×C32×C4—42no—
72C2×C32×C4C2×Dic312yes—
73C2×C32×C4C2×Dic311yescharacteristic
74C2×C3⋊Dic3C2×Dic312yes—
75C2×C3⋊Dic3C22×D311yescharacteristic
76C4×C3⋊D3—22no—
77C4×C3⋊D3—22no—
78C4×C3⋊D3#6712yes—
79C4×C3⋊D3#6712yes—
80#634—22no—
81#634—24no—
82C2×C3×Dic3—64no—
83C2×C3×Dic3—62no—
84C3×C4×D3—64no—
85C3×C4×D3—64no—
86C3×C4×D3—68no—
87C3×C4×D3—68no—
88C3×Dic6—616no—
89C3×Dic6—68no—
90#622—64no—
91C2×#101—271noSylow
92C2×C33—21no—
93C2×C33C2×C22⋊C411yescharacteristic
94C3×C3⋊D3—21no—
95C3×C3⋊D3#10112yescomplemented
96C2×C3×Q8—22no—
97C2×C3×Q8—42no—
98C2×C3×Q8—92no—
99C2×C3×Q8—91no—
100C2×C3×Q8—182no—
101C2×C3×Q8—182no—
102C2×C3×Q8#15711yescharacteristic
103C22×C3×C4—91no—
104C22×C3×C4—92no—
105D3×Q8—68no—
106D3×Q8—68no—
107C2×C4×D3—64no—
108C2×C4×D3—62no—
109C2×Dic6—62no—
110C2×Dic6—64no—
111#208—94no—
112#208—182no—
113C2×C3⋊C8—182no—
114C22×C32—22no—
115C22×C32—42no—
116C22×C32C22×Dic311yescharacteristic
117C2×C3⋊D3—22no—
118C2×C3⋊D3#21711yescharacteristic
119C2×C3⋊D3#21711yescharacteristic
120C2×C3×D3—62no—
121C2×C3×D3—62no—
122C2×C3×D3—68no—
123C32×C4—22no—
124C32×C4—28no—
125C32×C4—24no—
126C32×C4—48no—
127C32×C4—44no—
128C32×C4#21712yes—
129C3⋊Dic3—22no—
130C3⋊Dic3C2×#6712yes—
131C3×Dic3—64no—
132C3×Dic3—68no—
133C22×Q8—91no—
134C2×M16—272no—
135#101—274no—
136C33C2×#10111yescharacteristic, complemented, Sylow
137C23×C3—91no—
138C22×D3—62no—
139C3×Q8—28no—
140C3×Q8—22no—
141C3×Q8—48no—
142C3×Q8—184no—
143C3×Q8—182no—
144C2×C3×C4—24no—
145C2×C3×C4—22no—
146C2×C3×C4—44no—
147C2×C3×C4—42no—
148C2×C3×C4—91no—
149C2×C3×C4—92no—
150C2×C3×C4—92no—
151C2×C3×C4—92no—
152C2×C3×C4—182no—
153C2×C3×C4—182no—
154C2×C3×C4C2×#15712yes—
155C2×C3×C4C2×#15711yescharacteristic
156C2×Dic3—64no—
157C2×Dic3—62no—
158C4×D3—64no—
159C4×D3—64no—
160C4×D3—68no—
161C4×D3—68no—
162Dic6—616no—
163Dic6—68no—
164C3⋊C8—184no—
165C2×C32—21no—
166C2×C32—24no—
167C2×C32—22no—
168C2×C32—44no—
169C2×C32—42no—
170C2×C32C2×#21711yescharacteristic
171C3⋊D3—21no—
172C3⋊D3#83612yescomplemented
173C3×D3—64no—
174C3×D3—64no—
175C2×Q8—92no—
176C2×Q8—91no—
177C2×Q8—182no—
178C2×Q8—182no—
179C2×Q8C3⋊#15711yescharacteristic
180C22×C4—91no—
181C22×C4—92no—
182M16—274no—
183M16—542no—
184C2×C8—542no—
185C22×C3—22no—
186C22×C3—42no—
187C22×C3—91no—
188C22×C3—91no—
189C22×C3—182no—
190C22×C3C22×#15711yescharacteristic
191C2×D3—62no—
192C2×D3—62no—
193C2×D3—68no—
194C3×C4—28no—
195C3×C4—22no—
196C3×C4—24no—
197C3×C4—48no—
198C3×C4—44no—
199C3×C4—92no—
200C3×C4—182no—
201C3×C4C22⋊#15712yes—
202Dic3—64no—
203Dic3—68no—
204C32—22no—
205C32—42no—
206C32C2×#83611yescharacteristic, complemented
207C23—91no—
208Q8—22no—
209Q8—184no—
210Q8—182no—
211C2×C4—91no—
212C2×C4—92no—
213C2×C4—92no—
214C2×C4—92no—
215C2×C4—182no—
216C2×C4—182no—
217C2×C4C2×C3⋊#15712yes—
218C2×C4C2×C3⋊#15711yescharacteristic
219C8—544no—
220C2×C3—24no—
221C2×C3—21no—
222C2×C3—22no—
223C2×C3—44no—
224C2×C3—42no—
225C2×C3—92no—
226C2×C3—181no—
227C2×C3C2×C22⋊#15711yescharacteristic
228D3—64no—
229D3—64no—
230C22—91no—
231C22—91no—
232C22—182no—
233C22C22×C3⋊#15711yescharacteristic
234C4—22no—
235C4—92no—
236C4—182no—
237C4#8899412yes—
238C3—22no—
239C3—42no—
240C3#1059383611yescharacteristic, complemented
241C2—21no—
242C2—92no—
243C2—181no—
244C2C2×#8899411yescharacteristic
subgroupG/H
centerC2C2×#88994
commutatorC22×C33C22×C4
FrattiniC22C22×C3⋊#157
FittingC2×C33×Q8C4
socleC2×C33C2×C22⋊C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×Q8C3⋊#157
hypocenterC33C2×#101
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
3. #11723105C2semidirect
95. C3×C3⋊D3#101semidirect
136. C33C2×#101semidirect
172. C3⋊D3#836semidirect
206. C32C2×#836semidirect
240. C3#10593836semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
#117245112
#117231052
#105938363
C2×#8369