Finite Group Databasebeta

← #13145744

Subgroups — #13145744

Order 1728.

422 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C3×#64153C211yesmaximal, characteristic
2C3×#63663C212yesmaximal
3#10589185—31nomaximal
4#10589184—32nomaximal
5C2×C33×Q8C2211yescharacteristic
6C2×C3×#634C411yescharacteristic
7C3×#3750—22no—
8C3×C4×C3⋊Dic3C411yescharacteristic
9C3×#3748—22no—
10#64153D311yescharacteristic
11C3×#944—32no—
12C3×#944—61no—
13#63663D312yes—
14C3×#836—32no—
15C3×#836—32no—
16C3×#836—62no—
17C33×Q8—41no—
18C2×C33×C4C2×C411yescharacteristic
19C2×C33×C4D412yes—
20C2×C3×C3⋊Dic3—21no—
21C2×C3×C3⋊Dic3—21no—
22C3×#634—41no—
23C3×C3⋊(C3⋊C8)—42no—
24#4548—91no—
25#4478—92no—
26C2×C32×Q8—21no—
27C2×C32×Q8—22no—
28C2×C32×Q8—41no—
29C2×C32×Q8C2×D312yes—
30C2×C32×Q8C2×D311yescharacteristic
31C2×#634Dic311yescharacteristic
32C2×C3×Dic6—32no—
33C2×C3×Dic6—61no—
34#3750—22no—
35C4×C3⋊Dic3Dic311yescharacteristic
36#3748—22no—
37C3×Dic3⋊C4—64no—
38C3×Dic3⋊C4—62no—
39C3×Dic3⋊C4—62no—
40C3×C4×Dic3—32no—
41C3×C4×Dic3—61no—
42C3×#208—64no—
43C3×#208—62no—
44C3×#208—62no—
45C22×C33C22⋊C411yescharacteristic
46C33×C4—22no—
47C33×C4—21no—
48C3×C3⋊Dic3—22no—
49C3×C3⋊Dic3—41no—
50C3×C4⋊Q8—91no—
51#944—32no—
52#944—61no—
53C3×#101—92no—
54#836—32no—
55#836—32no—
56#836—62no—
57C32×Q8—42no—
58C32×Q8—41no—
59C32×Q8—44no—
60C32×Q8—42no—
61C32×Q8—44no—
62C2×C32×C4—22no—
63C2×C32×C4—21no—
64C2×C32×C4—22no—
65C2×C32×C4—22no—
66C2×C32×C4—22no—
67C2×C32×C4—41no—
68C2×C32×C4—42no—
69C2×C32×C4C4×D312yes—
70C2×C32×C4D1212yes—
71C2×C32×C4C2×Dic311yescharacteristic
72C2×C32×C4#6712yes—
73C2×C32×C4#6712yes—
74C2×C3⋊Dic3—21no—
75C2×C3⋊Dic3—21no—
76#634—41no—
77C2×C3×Dic3—62no—
78C2×C3×Dic3—62no—
79C2×C3×Dic3—62no—
80C2×C3×Dic3—62no—
81C3×Dic6—122no—
82C3×Dic6—122no—
83C3⋊(C3⋊C8)—42no—
84C3×C3⋊C8—124no—
85C3×C3⋊C8—122no—
86C3×C3⋊C8—122no—
87#356—271noSylow
88C2×C33—21no—
89C2×C33#9911yescharacteristic
90C2×C3×Q8—22no—
91C2×C3×Q8—21no—
92C2×C3×Q8—41no—
93C2×C3×Q8—91no—
94C2×C3×Q8D3212yes—
95C2×C3×Q8D3211yescharacteristic
96C2×Dic6—32no—
97C2×Dic6—61no—
98C3×M16—182no—
99C3×C4⋊C4—182no—
100C3×C42—91no—
101Dic3⋊C4—64no—
102Dic3⋊C4—62no—
103Dic3⋊C4—62no—
104C4×Dic3—32no—
105C4×Dic3—61no—
106#208—64no—
107#208—62no—
108#208—62no—
109C22×C32—22no—
110C22×C32—21no—
111C22×C32—41no—
112C22×C32(C2×D3)⋊C412yes—
113C22×C32#21711yescharacteristic
114C32×C4—22no—
115C32×C4—21no—
116C32×C4—24no—
117C32×C4—22no—
118C32×C4—22no—
119C32×C4—22no—
120C32×C4—22no—
121C32×C4—42no—
122C32×C4—42no—
123C32×C4—42no—
124C32×C4—44no—
125C32×C4—42no—
126C3⋊Dic3—22no—
127C3⋊Dic3—41no—
128C3×Dic3—64no—
129C3×Dic3—64no—
130C3×Dic3—122no—
131C3×Dic3—122no—
132C4⋊Q8—91no—
133#101—92no—
134C33#35611yescharacteristic, complemented, Sylow
135C3×Q8—42no—
136C3×Q8—41no—
137C3×Q8—44no—
138C3×Q8—42no—
139C3×Q8—44no—
140C3×Q8—361no—
141C2×C3×C4—21no—
142C2×C3×C4—22no—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—41no—
147C2×C3×C4—42no—
148C2×C3×C4—181no—
149C2×C3×C4—181no—
150C2×C3×C4D3×Dic312yes—
151C2×C3×C4#62411yescharacteristic
152C2×C3×C4#62512yes—
153C2×C3×C4#62612yes—
154C2×C3×C4#62612yes—
155C2×Dic3—62no—
156C2×Dic3—62no—
157C2×Dic3—62no—
158C2×Dic3—62no—
159Dic6—122no—
160Dic6—122no—
161C3×C8—362no—
162C3⋊C8—124no—
163C3⋊C8—122no—
164C3⋊C8—122no—
165C2×C32—21no—
166C2×C32—22no—
167C2×C32—22no—
168C2×C32—22no—
169C2×C32—21no—
170C2×C32—41no—
171C2×C32—42no—
172C2×C32—42no—
173C2×C32#80612yes—
174C2×C32#83411yescharacteristic
175C2×Q8—91no—
176C2×Q8#111311yescharacteristic
177M16—182no—
178C4⋊C4—182no—
179C42—91no—
180C22×C3—21no—
181C22×C3—22no—
182C22×C3—41no—
183C22×C3(C2×D3)⋊Dic312yes—
184C22×C3#372211yescharacteristic
185C3×C4—24no—
186C3×C4—22no—
187C3×C4—22no—
188C3×C4—22no—
189C3×C4—22no—
190C3×C4—22no—
191C3×C4—21no—
192C3×C4—42no—
193C3×C4—42no—
194C3×C4—42no—
195C3×C4—44no—
196C3×C4—42no—
197C3×C4—182no—
198C3×C4—361no—
199Dic3—64no—
200Dic3—64no—
201Dic3—122no—
202Dic3—122no—
203C32—22no—
204C32—21no—
205C32—41no—
206C32#447812yescomplemented
207C32#454811yescharacteristic, complemented
208Q8—41no—
209Q8—361no—
210C2×C4—181no—
211C2×C4—181no—
212C2×C4#629411yescharacteristic
213C2×C4#629512yes—
214C8—362no—
215C2×C3—22no—
216C2×C3—22no—
217C2×C3—21no—
218C2×C3—21no—
219C2×C3—22no—
220C2×C3—42no—
221C2×C3—42no—
222C2×C3—41no—
223C2×C3#6358012yes—
224C2×C3#6358111yescharacteristic
225C22#8881411yescharacteristic
226C4—22no—
227C4—21no—
228C4—182no—
229C4—361no—
230C3—21no—
231C3—22no—
232C3—41no—
233C3#1058918412yescomplemented
234C3#1058918511yescharacteristic, complemented
235C2—21no—
236C2#1172208311yescharacteristic
subgroupG/H
centerC2#11722083
commutatorC2×C33×C4C2×C4
Frattini, hypercenterC2×Q8#1113
Fitting, cosocleC2×C33×Q8C22
socleC2×C33#99
prime radicalC22×C33C22⋊C4
hypocenterC33#356
LCSG ▷ C2×C33×C4 ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
134. C33#356semidirect
206. C32#4478semidirect
207. C32#4548semidirect
233. C3#10589184semidirect
234. C3#10589185semidirect

Maximal subgroups

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

subgroupindex
C3×#641532
C3×#636632
#105891853
#105891843