Finite Group Databasebeta

← #13155006

Subgroups — #13155006

Order 1728.

374 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 59 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#11724511C211yesmaximal, characteristic
2#11723656C211yesmaximal, characteristic
3C3×#63874C211yesmaximal, characteristic
4#10590054—31nomaximal
5C2×C3×C4×C3⋊D3C2211yescharacteristic
6#88993—21no—
7#88989C412yescomplemented
8#88988—21no—
9C3×#3748—21no—
10C32×C4×Dic3—21no—
11#88809—21no—
12#88808C412yescomplemented
13C3×#3724—21no—
14C3×#3722C412yes—
15#64283—31no—
16C3×#872—61no—
17#63874D311yescharacteristic
18#63818—31no—
19D3×#208—61no—
20C22×C3×C3⋊D3C2×C411yescharacteristic
21C2×C33×C4D411yescharacteristic
22C2×C3×C3⋊Dic3D411yescharacteristic
23C3×C4×C3⋊D3C2×C411yescharacteristic
24C3×C4×C3⋊D3C2×C411yescharacteristic
25C3×C4×C3⋊D3D411yescharacteristic
26C3×C4×C3⋊D3Q811yescharacteristic
27C2×C32×Dic3—22no—
28C3⋊#622—22no—
29C3×C3⋊(C3⋊C8)—22no—
30#4533—91nomaximal
31C2×C4×C3⋊D3C2×D311yescharacteristic
32C2×C3×C4×D3—61no—
33C2×C3×C4×D3—61no—
34#3792—61no—
35#3788—32no—
36#3787—61no—
37C32×C42—61no—
38#3748—21no—
39C3×(C2×D3)⋊C4—62no—
40C3×Dic3⋊C4—62no—
41C3×C4×Dic3—21no—
42C3×C4×Dic3—21no—
43C3×C4×Dic3—41no—
44C3×#208—61no—
45C3×#208—61no—
46C3×#208—121no—
47#3724—21no—
48#3722Dic312yes—
49#3712—61no—
50#3711—62no—
51#3710—32no—
52D3×C3⋊C8—62no—
53C22×C33C22⋊C411yescharacteristic
54C2×C3×C3⋊D3—41no—
55C2×C3×C3⋊D3C4211yescharacteristic
56C2×C3×C3⋊D3C4⋊C411yescharacteristic
57C33×C4C22⋊C411yescharacteristic
58C33×C4C4⋊C411yescharacteristic
59C3×C3⋊Dic3C22⋊C411yescharacteristic
60C3×C3⋊Dic3C4⋊C411yescharacteristic
61C32×Dic3—42no—
62C3×#117—91no—
63C2×#208—91no—
64C2×#208—91no—
65D3×M16—181no—
66#872—61no—
67C22×C3⋊D3C2×Dic311yescharacteristic
68C22×C3×D3—61no—
69C22×C3×D3—61no—
70C2×C32×C4—21no—
71C2×C32×C4—21no—
72C2×C32×C4—41no—
73C2×C32×C4—41no—
74C2×C32×C4—62no—
75C2×C32×C4#6711yescharacteristic
76C2×C3⋊Dic3#6711yescharacteristic
77C4×C3⋊D3Dic611yescharacteristic
78C4×C3⋊D3C4×D311yescharacteristic
79C4×C3⋊D3C4×D311yescharacteristic
80C4×C3⋊D3D1211yescharacteristic
81C2×C3×Dic3—22no—
82C2×C3×Dic3—22no—
83C2×C3×Dic3—42no—
84C2×C3×Dic3—61no—
85C2×C3×Dic3—61no—
86C3×C4×D3—61no—
87C3×C4×D3—61no—
88C3×C4×D3—61no—
89C3×C4×D3—61no—
90C3×C4×D3—61no—
91C3×C4×D3—61no—
92C3×C4×D3—61no—
93C3×C4×D3—61no—
94#622—62no—
95C3⋊(C3⋊C8)—22no—
96C3×C3⋊C8—62no—
97C3×C3⋊C8—62no—
98C3×C3⋊C8—122no—
99#344—271noSylow
100C2×C33—21no—
101C2×C33#9511yescharacteristic
102C3×C3⋊D3—21no—
103C3×C3⋊D3—21no—
104C22×C3×C4—91no—
105C2×C4×D3—61no—
106C2×C4×D3—61no—
107C3×M16—181no—
108C3×C4⋊C4—181no—
109C3×C22⋊C4—92no—
110C3×C42—61no—
111C3×C42—61no—
112C3×C42—121no—
113(C2×D3)⋊C4—62no—
114Dic3⋊C4—62no—
115C4×Dic3—21no—
116#208—61no—
117#208—61no—
118#208—92no—
119#208—121no—
120#208—181no—
121#208—181no—
122C2×C3⋊C8—92no—
123C2×C3⋊C8—181no—
124ΓL(1,𝔽25)—182no—
125D3×C8—182no—
126C22×C32—21no—
127C22×C32—21no—
128C22×C32—41no—
129C22×C32—41no—
130C22×C32#21711yescharacteristic
131C2×C3⋊D3—41no—
132C2×C3⋊D3C4×Dic311yescharacteristic
133C2×C3⋊D3C4⋊Dic311yescharacteristic
134C2×C3×D3—61no—
135C2×C3×D3—61no—
136C2×C3×D3—61no—
137C2×C3×D3—61no—
138C2×C3×D3—122no—
139C2×C3×D3—122no—
140C32×C4—21no—
141C32×C4—21no—
142C32×C4—21no—
143C32×C4—21no—
144C32×C4—41no—
145C32×C4—41no—
146C32×C4—41no—
147C32×C4—41no—
148C32×C4—122no—
149C32×C4Dic3⋊C411yescharacteristic
150C32×C4(C2×D3)⋊C411yescharacteristic
151C3⋊Dic3Dic3⋊C411yescharacteristic
152C3⋊Dic3(C2×D3)⋊C411yescharacteristic
153C3×Dic3—42no—
154C3×Dic3—42no—
155C3×Dic3—44no—
156C3×Dic3—61no—
157C3×Dic3—61no—
158C3×Dic3—61no—
159C3×Dic3—61no—
160C2×M16—271no—
161C2×M16—271no—
162#117—91no—
163C33#34411yescharacteristic, complemented, Sylow
164C23×C3—91no—
165C22×D3—61no—
166C22×D3—61no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—41no—
170C2×C3×C4—41no—
171C2×C3×C4—62no—
172C2×C3×C4—62no—
173C2×C3×C4—91no—
174C2×C3×C4—91no—
175C2×C3×C4—91no—
176C2×C3×C4—91no—
177C2×C3×C4—91no—
178C2×C3×C4—122no—
179C2×C3×C4GO+(4,𝔽2)11yescharacteristic
180C2×Dic3—22no—
181C2×Dic3—61no—
182C2×Dic3—61no—
183C4×D3—61no—
184C4×D3—61no—
185C4×D3—61no—
186C4×D3—61no—
187C4×D3—61no—
188C4×D3—61no—
189C4×D3—61no—
190C4×D3—61no—
191C3×C8—182no—
192C3⋊C8—62no—
193C3⋊C8—62no—
194C3⋊C8—122no—
195C3⋊C8—182no—
196C2×C32—22no—
197C2×C32—21no—
198C2×C32—21no—
199C2×C32—21no—
200C2×C32—41no—
201C2×C32—41no—
202C2×C32—41no—
203C2×C32—42no—
204C2×C32—81no—
205C2×C32C3⋊#9511yescharacteristic
206C3⋊D3—21no—
207C3⋊D3—21no—
208C3×D3—62no—
209C3×D3—121no—
210C3×D3—121no—
211C3×D3—121no—
212C22×C4—91no—
213M16—181no—
214M16—272no—
215M16—541no—
216M16—541no—
217C2×C8—272no—
218C2×C8—541no—
219C4⋊C4—181no—
220C22⋊C4—92no—
221C42—61no—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3—41no—
225C22×C3—41no—
226C22×C3—91no—
227C22×C3—91no—
228C22×C3—361no—
229C22×C3D32⋊C411yescharacteristic
230C2×D3—61no—
231C2×D3—61no—
232C2×D3—61no—
233C2×D3—61no—
234C2×D3—122no—
235C2×D3—122no—
236C3×C4—21no—
237C3×C4—21no—
238C3×C4—21no—
239C3×C4—21no—
240C3×C4—41no—
241C3×C4—41no—
242C3×C4—41no—
243C3×C4—41no—
244C3×C4—91no—
245C3×C4—91no—
246C3×C4—122no—
247C3×C4—122no—
248C3×C4—124no—
249C3×C4D32⋊C411yescharacteristic
250C3×C4#377311yescharacteristic
251Dic3—42no—
252Dic3—61no—
253Dic3—61no—
254Dic3—61no—
255Dic3—61no—
256C32—21no—
257C32—21no—
258C32—41no—
259C32—41no—
260C32#453311yescharacteristic, complemented
261C23—91no—
262C2×C4—62no—
263C2×C4—91no—
264C2×C4—91no—
265C2×C4—91no—
266C2×C4—91no—
267C2×C4—91no—
268C2×C4D32⋊D311yescharacteristic
269C8—182no—
270C8—542no—
271C2×C3—22no—
272C2×C3—21no—
273C2×C3—21no—
274C2×C3—21no—
275C2×C3—41no—
276C2×C3—42no—
277C2×C3—41no—
278C2×C3—41no—
279C2×C3—81no—
280C2×C3—181no—
281C2×C3—181no—
282C2×C3#6373711yescharacteristic
283D3—62no—
284D3—121no—
285D3—121no—
286D3—121no—
287C22—91no—
288C22—91no—
289C22—361no—
290C22D32⋊Dic311yescharacteristic
291C4—91no—
292C4—91no—
293C4—122no—
294C4#8893211yescharacteristic
295C4#8893411yescharacteristic
296C3—21no—
297C3—21no—
298C3—41no—
299C3—41no—
300C3#1059005411yescharacteristic, complemented
301C2—21no—
302C2—181no—
303C2—181no—
304C2#1172276911yescharacteristic
subgroupG/H
centerC4#88932
commutatorC2×C3×C3⋊D3C42
Frattini, hypercenterC2×C4D32⋊D3
FittingC2×C33×C4D4
socleC2×C33#95
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C3×C4×C3⋊D3C22
hypocenterC33#344
LCSG ▷ C2×C3×C3⋊D3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
7. #88989C4semidirect
12. #88808C4semidirect
163. C33#344semidirect
260. C32#4533semidirect
300. C3#10590054semidirect

Maximal subgroups

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

subgroupindex
#117245112
#117236562
C3×#638742
#105900543
#45339