Finite Group Databasebeta

← #13145728

Subgroups — #13145728

Order 1728.

558 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 103 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×#3750C211yesmaximal, characteristic
2C3×#63657C212yesmaximal
3#10589148—31nomaximal
4#10589147—32nomaximal
5C22×C33×C4C2211yescharacteristic
6C22×C3×C3⋊Dic3C411yescharacteristic
7C22×C3×C3⋊Dic3C411yescharacteristic
8C3×#3750—22no—
9C2×C3×C3⋊(C3⋊C8)—22no—
10C2×#3750D311yescharacteristic
11C2×C3×Dic3⋊C4—32no—
12C2×C3×Dic3⋊C4—61no—
13#63657D312yes—
14C3×#830—32no—
15C3×#830—32no—
16C3×#830—62no—
17C23×C33C2×C411yescharacteristic
18C2×C33×C4—41no—
19C2×C33×C4D412yes—
20C2×C3×C3⋊Dic3—21no—
21C2×C3×C3⋊Dic3—21no—
22C2×C3×C3⋊Dic3—21no—
23C2×C3×C3⋊Dic3—41no—
24C2×C3×C3⋊Dic3C812yescomplemented
25C3×C3⋊(C3⋊C8)—42no—
26#4528—91no—
27#4469—92no—
28C22×C32×C4—21no—
29C22×C32×C4—22no—
30C22×C32×C4—41no—
31C22×C32×C4C2×D312yes—
32C22×C32×C4C2×D311yescharacteristic
33C22×C3⋊Dic3Dic311yescharacteristic
34C22×C3⋊Dic3Dic311yescharacteristic
35C22×C3×Dic3—32no—
36C22×C3×Dic3—32no—
37C22×C3×Dic3—61no—
38C22×C3×Dic3—61no—
39#3750—22no—
40C2×C3⋊(C3⋊C8)—22no—
41C3×Dic3⋊C4—64no—
42C3×Dic3⋊C4—62no—
43C3×Dic3⋊C4—62no—
44C2×C3×C3⋊C8—64no—
45C2×C3×C3⋊C8—62no—
46C2×C3×C3⋊C8—62no—
47C22×C33—21no—
48C22×C33—21no—
49C22×C33C22⋊C411yescharacteristic
50C22×C33C2×C811yescharacteristic
51C22×C33M1611yescharacteristic
52C33×C4—22no—
53C3×C3⋊Dic3—22no—
54C3×C3⋊Dic3—41no—
55C2×C3×C4⋊C4—91no—
56C2×Dic3⋊C4—32no—
57C2×Dic3⋊C4—61no—
58C3×C22⋊C8—92no—
59#830—32no—
60#830—32no—
61#830—62no—
62C23×C32—22no—
63C23×C32—21no—
64C23×C32—41no—
65C23×C32C4×D312yes—
66C23×C32C2×Dic311yescharacteristic
67C2×C32×C4—22no—
68C2×C32×C4—22no—
69C2×C32×C4—22no—
70C2×C32×C4—42no—
71C2×C32×C4—42no—
72C2×C32×C4—41no—
73C2×C32×C4—44no—
74C2×C32×C4—42no—
75C2×C32×C4—44no—
76C2×C32×C4D1212yes—
77C2×C32×C4#6712yes—
78C2×C32×C4#6712yes—
79C2×C3⋊Dic3—21no—
80C2×C3⋊Dic3—21no—
81C2×C3⋊Dic3—21no—
82C2×C3⋊Dic3—41no—
83C2×C3⋊Dic3C3⋊C812yescomplemented
84C2×C3×Dic3—34no—
85C2×C3×Dic3—62no—
86C2×C3×Dic3—62no—
87C2×C3×Dic3—62no—
88C2×C3×Dic3—62no—
89C2×C3×Dic3—62no—
90C2×C3×Dic3—62no—
91C2×C3×Dic3—62no—
92C2×C3×Dic3—122no—
93C2×C3×Dic3—122no—
94C3⋊(C3⋊C8)—42no—
95C3×C3⋊C8—124no—
96C3×C3⋊C8—122no—
97C3×C3⋊C8—122no—
98#324—271noSylow
99C2×C33—21no—
100C2×C33—21no—
101C2×C33C22⋊C811yescharacteristic
102C2×C33#9911yescharacteristic
103C2×C33#10111yescharacteristic
104C22×C3×C4—22no—
105C22×C3×C4—21no—
106C22×C3×C4—41no—
107C22×C3×C4—91no—
108C22×C3×C4—91no—
109C22×C3×C4D3212yes—
110C22×C3×C4D3211yescharacteristic
111C22×Dic3—32no—
112C22×Dic3—32no—
113C22×Dic3—61no—
114C22×Dic3—61no—
115C2×C3×C8—182no—
116C3×C4⋊C4—182no—
117Dic3⋊C4—64no—
118Dic3⋊C4—62no—
119Dic3⋊C4—62no—
120C2×C3⋊C8—64no—
121C2×C3⋊C8—62no—
122C2×C3⋊C8—62no—
123C22×C32—21no—
124C22×C32—21no—
125C22×C32—22no—
126C22×C32—22no—
127C22×C32—22no—
128C22×C32—22no—
129C22×C32—22no—
130C22×C32—21no—
131C22×C32—22no—
132C22×C32—21no—
133C22×C32—22no—
134C22×C32—21no—
135C22×C32—41no—
136C22×C32—41no—
137C22×C32—41no—
138C22×C32—42no—
139C22×C32—42no—
140C22×C32—42no—
141C22×C32—42no—
142C22×C32D3×C812yes—
143C22×C32ΓL(1,𝔽25)12yes—
144C22×C32C2×C3⋊C811yescharacteristic
145C22×C32#20811yescharacteristic
146C22×C32(C2×D3)⋊C412yes—
147C22×C32#21711yescharacteristic
148C32×C4—22no—
149C32×C4—24no—
150C32×C4—22no—
151C32×C4—22no—
152C32×C4—42no—
153C32×C4—42no—
154C32×C4—44no—
155C3⋊Dic3—22no—
156C3⋊Dic3—41no—
157C3×Dic3—64no—
158C3×Dic3—64no—
159C3×Dic3—122no—
160C3×Dic3—122no—
161C2×C4⋊C4—91no—
162C22⋊C8—92no—
163C33#32411yescharacteristic, complemented, Sylow
164C23×C3—21no—
165C23×C3—22no—
166C23×C3—41no—
167C23×C3D3×Dic312yes—
168C23×C3#62411yescharacteristic
169C2×C3×C4—22no—
170C2×C3×C4—22no—
171C2×C3×C4—22no—
172C2×C3×C4—42no—
173C2×C3×C4—41no—
174C2×C3×C4—44no—
175C2×C3×C4—42no—
176C2×C3×C4—44no—
177C2×C3×C4—42no—
178C2×C3×C4—92no—
179C2×C3×C4—181no—
180C2×C3×C4—181no—
181C2×C3×C4—181no—
182C2×C3×C4—361no—
183C2×C3×C4#62512yes—
184C2×C3×C4#62612yes—
185C2×C3×C4#62612yes—
186C2×Dic3—34no—
187C2×Dic3—62no—
188C2×Dic3—62no—
189C2×Dic3—62no—
190C2×Dic3—62no—
191C2×Dic3—62no—
192C2×Dic3—62no—
193C2×Dic3—62no—
194C2×Dic3—122no—
195C2×Dic3—122no—
196C3×C8—362no—
197C3⋊C8—124no—
198C3⋊C8—122no—
199C3⋊C8—122no—
200C2×C32—21no—
201C2×C32—21no—
202C2×C32—22no—
203C2×C32—22no—
204C2×C32—22no—
205C2×C32—22no—
206C2×C32—22no—
207C2×C32—21no—
208C2×C32—22no—
209C2×C32—21no—
210C2×C32—22no—
211C2×C32—21no—
212C2×C32—41no—
213C2×C32—41no—
214C2×C32—41no—
215C2×C32—42no—
216C2×C32—42no—
217C2×C32—42no—
218C2×C32—42no—
219C2×C32#80612yes—
220C2×C32(C2×D3)⋊C812yes—
221C2×C32#82412yes—
222C2×C32#83011yescharacteristic
223C2×C32#83411yescharacteristic
224C2×C32#83611yescharacteristic
225C22×C4—91no—
226C22×C4—91no—
227C22×C4#111311yescharacteristic
228C2×C8—182no—
229C4⋊C4—182no—
230C22×C3—22no—
231C22×C3—22no—
232C22×C3—22no—
233C22×C3—21no—
234C22×C3—22no—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3—22no—
238C22×C3—21no—
239C22×C3—22no—
240C22×C3—21no—
241C22×C3—22no—
242C22×C3—42no—
243C22×C3—42no—
244C22×C3—42no—
245C22×C3—42no—
246C22×C3—41no—
247C22×C3—41no—
248C22×C3—41no—
249C22×C3D3×C3⋊C812yes—
250C22×C3#371011yescharacteristic
251C22×C3#371112yes—
252C22×C3#371211yescharacteristic
253C22×C3(C2×D3)⋊Dic312yes—
254C22×C3#372211yescharacteristic
255C3×C4—24no—
256C3×C4—22no—
257C3×C4—22no—
258C3×C4—22no—
259C3×C4—42no—
260C3×C4—42no—
261C3×C4—44no—
262C3×C4—182no—
263C3×C4—361no—
264Dic3—64no—
265Dic3—64no—
266Dic3—122no—
267Dic3—122no—
268C32—22no—
269C32—21no—
270C32—41no—
271C32#446912yescomplemented
272C32#452811yescharacteristic, complemented
273C23#629411yescharacteristic
274C2×C4—41no—
275C2×C4—92no—
276C2×C4—181no—
277C2×C4—181no—
278C2×C4—181no—
279C2×C4—361no—
280C2×C4#629512yes—
281C8—362no—
282C2×C3—22no—
283C2×C3—22no—
284C2×C3—22no—
285C2×C3—22no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3—22no—
290C2×C3—21no—
291C2×C3—22no—
292C2×C3—21no—
293C2×C3—22no—
294C2×C3—42no—
295C2×C3—42no—
296C2×C3—42no—
297C2×C3—42no—
298C2×C3—41no—
299C2×C3—41no—
300C2×C3—41no—
301C2×C3(C2×D3)⋊(C3⋊C8)12yes—
302C2×C3#6355711yescharacteristic
303C2×C3#6356012yes—
304C2×C3#6356111yescharacteristic
305C2×C3#6358012yes—
306C2×C3#6358111yescharacteristic
307C22—21no—
308C22—21no—
309C22#8880811yescharacteristic
310C22#8880911yescharacteristic
311C22#8881411yescharacteristic
312C4—22no—
313C4—182no—
314C4—361no—
315C3—21no—
316C3—22no—
317C3—41no—
318C3#1058914712yescomplemented
319C3#1058914811yescharacteristic, complemented
320C2—21no—
321C2—21no—
322C2#1172207211yescharacteristic
323C2#1172207411yescharacteristic
324C2#1172208311yescharacteristic
subgroupG/H
centerC22#88814
commutatorC22×C33C2×C8
Frattini, hypercenterC22×C4#1113
Fitting, cosocleC22×C33×C4C22
socleC22×C33C22⋊C4
prime radicalC23×C33C2×C4
hypocenterC33#324
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
24. C2×C3×C3⋊Dic3C8semidirect
83. C2×C3⋊Dic3C3⋊C8semidirect
163. C33#324semidirect
271. C32#4469semidirect
272. C32#4528semidirect
318. C3#10589147semidirect
319. C3#10589148semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37502
C3×#636572
#105891483
#105891473