Finite Group Databasebeta

← #13155104

Subgroups — #13155104

Order 1728.

272 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 29 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#11723666C211yesmaximal, characteristic
2C3×#63780C211yesmaximal, characteristic
3#11722074C211yesmaximal, characteristic
4#10590060—31nomaximal
5C2×C3×#634C2211yescharacteristic
6C2×#6296C411yescharacteristic
7C3×#3792—21no—
8C3×#3750—21no—
9C3×#3748—21no—
10#88815—21no—
11#88813C411yescharacteristic
12#63840—31no—
13#63839—61no—
14#63780D311yescharacteristic
15#63561—31no—
16#63560—61no—
17C2×C33×C4D411yescharacteristic
18C2×C3×C3⋊Dic3—21no—
19C2×C3×C3⋊Dic3—21no—
20C2×C3×C3⋊Dic3C2×C411yescharacteristic
21C2×C3×C3⋊Dic3D411yescharacteristic
22C3×#634—41no—
23#6296—41no—
24C3×#622—41no—
25C3×C3⋊(C3⋊C8)—41no—
26#4478—91nomaximal
27C2×#634C2×D311yescharacteristic
28C2×C3×Dic6—61no—
29C2×C3×Dic6—61no—
30C2×#627—31no—
31C2×#627—61no—
32#3792—21no—
33#3750—21no—
34#3748—21no—
35C3×Dic3⋊C4—61no—
36C3×Dic3⋊C4—61no—
37C3×Dic3⋊C4—121no—
38C3×#208—61no—
39C3×#208—61no—
40C3×#208—121no—
41#3724—61no—
42Dic3⋊Dic3—61no—
43Dic3⋊Dic3—61no—
44Dic32—31no—
45Dic32—61no—
46C22×C33C22⋊C411yescharacteristic
47C33×C4—21no—
48C3×C3⋊Dic3—22no—
49C3×C3⋊Dic3—21no—
50C3×C3⋊Dic3—21no—
51C3×C3⋊Dic3—41no—
52#944—91no—
53#888—181no—
54C3×#101—91no—
55#836—91no—
56#824—181no—
57C2×C32×C4—21no—
58C2×C32×C4—21no—
59C2×C32×C4—41no—
60C2×C32×C4—41no—
61C2×C32×C4D1211yescharacteristic
62C2×C3⋊Dic3—21no—
63C2×C3⋊Dic3—21no—
64C2×C3⋊Dic3C4×D311yescharacteristic
65C2×C3⋊Dic3#6711yescharacteristic
66#634—41no—
67C2×C3×Dic3—61no—
68C2×C3×Dic3—61no—
69C2×C3×Dic3—61no—
70C2×C3×Dic3—61no—
71C2×C3×Dic3—61no—
72C2×C3×Dic3—61no—
73C2×C3×Dic3—61no—
74C2×C3×Dic3—61no—
75C2×C3×Dic3—121no—
76C2×C3×Dic3—121no—
77C3×Dic6—122no—
78C3×Dic6—121no—
79C3×Dic6—121no—
80#627—122no—
81#627—121no—
82#622—41no—
83C3⋊(C3⋊C8)—41no—
84C3×C3⋊C8—121no—
85C3×C3⋊C8—121no—
86C3×C3⋊C8—122no—
87#356—271noSylow
88C2×C33—21no—
89C2×C33#9911yescharacteristic
90C2×C3×Q8—91no—
91C2×Dic6—61no—
92C2×Dic6—61no—
93C2×Dic6—91no—
94C2×Dic6—181no—
95C3×M16—181no—
96C3×M16—181no—
97C3×C4⋊C4—181no—
98C4⋊Dic3—181no—
99Dic3⋊C4—61no—
100Dic3⋊C4—61no—
101Dic3⋊C4—121no—
102Dic3⋊C4—181no—
103Dic3⋊C4—181no—
104C4×Dic3—91no—
105C4×Dic3—181no—
106#208—61no—
107#208—61no—
108#208—121no—
109C22×C32—21no—
110C22×C32—21no—
111C22×C32—41no—
112C22×C32—41no—
113C22×C32(C2×D3)⋊C411yescharacteristic
114C32×C4—21no—
115C32×C4—21no—
116C32×C4—21no—
117C32×C4—41no—
118C32×C4—41no—
119C32×C4—41no—
120C32×C4—81no—
121C3⋊Dic3—22no—
122C3⋊Dic3—21no—
123C3⋊Dic3—21no—
124C3⋊Dic3—41no—
125C3×Dic3—62no—
126C3×Dic3—62no—
127C3×Dic3—62no—
128C3×Dic3—121no—
129C3×Dic3—121no—
130C3×Dic3—121no—
131C3×Dic3—122no—
132C3×Dic3—121no—
133C3×Dic3—121no—
134C3×Dic3—241no—
135C4⋊Q8—271no—
136#101—91no—
137#101—271no—
138C33#35611yescharacteristic, complemented, Sylow
139C3×Q8—361no—
140C2×C3×C4—21no—
141C2×C3×C4—21no—
142C2×C3×C4—41no—
143C2×C3×C4—41no—
144C2×C3×C4—91no—
145C2×C3×C4—91no—
146C2×C3×C4—181no—
147C2×C3×C4—181no—
148C2×C3×C4GO+(4,𝔽2)11yescharacteristic
149C2×Dic3—61no—
150C2×Dic3—61no—
151C2×Dic3—61no—
152C2×Dic3—61no—
153C2×Dic3—61no—
154C2×Dic3—61no—
155C2×Dic3—61no—
156C2×Dic3—61no—
157C2×Dic3—121no—
158C2×Dic3—121no—
159Dic6—122no—
160Dic6—121no—
161Dic6—121no—
162Dic6—362no—
163Dic6—361no—
164C3×C8—361no—
165C3×C8—361no—
166C3⋊C8—121no—
167C3⋊C8—121no—
168C3⋊C8—122no—
169C2×C32—21no—
170C2×C32—21no—
171C2×C32—21no—
172C2×C32—21no—
173C2×C32—21no—
174C2×C32—41no—
175C2×C32—41no—
176C2×C32—41no—
177C2×C32—41no—
178C2×C32—41no—
179C2×C32—81no—
180C2×C32#80611yescharacteristic
181C2×Q8—91no—
182C2×Q8—271no—
183M16—181no—
184M16—181no—
185C4⋊C4—181no—
186C4⋊C4—541no—
187C42—271no—
188C22×C3—21no—
189C22×C3—21no—
190C22×C3—41no—
191C22×C3—41no—
192C22×C3D32⋊C411yescharacteristic
193C3×C4—21no—
194C3×C4—21no—
195C3×C4—21no—
196C3×C4—41no—
197C3×C4—41no—
198C3×C4—41no—
199C3×C4—81no—
200C3×C4—182no—
201C3×C4—181no—
202C3×C4—181no—
203C3×C4—361no—
204Dic3—62no—
205Dic3—62no—
206Dic3—62no—
207Dic3—121no—
208Dic3—121no—
209Dic3—121no—
210Dic3—122no—
211Dic3—121no—
212Dic3—121no—
213Dic3—241no—
214C32—21no—
215C32—21no—
216C32—41no—
217C32—41no—
218C32#447811yescharacteristic, complemented
219Q8—361no—
220Q8—1081no—
221C2×C4—91no—
222C2×C4—91no—
223C2×C4—181no—
224C2×C4—181no—
225C2×C4#157⋊D311yescharacteristic
226C8—361no—
227C8—361no—
228C2×C3—21no—
229C2×C3—21no—
230C2×C3—21no—
231C2×C3—21no—
232C2×C3—21no—
233C2×C3—41no—
234C2×C3—41no—
235C2×C3—41no—
236C2×C3—41no—
237C2×C3—41no—
238C2×C3—81no—
239C2×C3#6373811yescharacteristic
240C22#8893911yescharacteristic
241C4—21no—
242C4—182no—
243C4—181no—
244C4—181no—
245C4—361no—
246C3—21no—
247C3—21no—
248C3—41no—
249C3—41no—
250C3#1059006011yescharacteristic, complemented
251C2—21no—
252C2#1172279211yescharacteristic
subgroupG/H
centerC2#11722792
commutatorC2×C3×C3⋊Dic3C2×C4
Frattini, hypercenterC2×C4#157⋊D3
FittingC2×C33×C4D4
socleC2×C33#99
prime radicalC22×C33C22⋊C4
cosocleC2×C3×#634C22
hypocenterC33#356
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
138. C33#356semidirect
218. C32#4478semidirect
250. C3#10590060semidirect

Maximal subgroups

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

subgroupindex
#117236662
C3×#637802
#117220742
#105900603
#44789