Finite Group Databasebeta

← #13145741

Subgroups — #13145741

Order 1728.

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

normal subgroups only

This group has 610 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×#64149C211yesmaximal, characteristic
2C3×#63661C212yesmaximal, complemented
3#10589179—31nomaximal
4#10589178—32nomaximal
5C2×C33×D4C2211yescharacteristic
6C2×C3×#638—22no—
7C2×C3×C4⋊(C3⋊D3)C411yescharacteristic, complemented
8C3×#3757—22no—
9C3×C4×C3⋊Dic3C411yescharacteristic, complemented
10#64149D311yescharacteristic
11C3×#940—32no—
12C3×#940—61no—
13#63661D312yescomplemented
14C3×#834—32no—
15C3×#834—32no—
16C3×#834—62no—
17C23×C33D412yes—
18C22×C3×C3⋊D3—21no—
19C33×D4—41no—
20C2×C33×C4C2×C411yescharacteristic
21C3×#638—44no—
22C2×C3×C3⋊Dic3—21no—
23C2×C3×C3⋊Dic3—42no—
24C3×C4⋊(C3⋊D3)—41no—
25#4545—91no—
26#4474—92no—
27C2×C32×D4—21no—
28C2×C32×D4—22no—
29C2×C32×D4—41no—
30C2×C32×D4C2×D312yes—
31C2×C32×D4C2×D311yescharacteristic
32C2×#638—22no—
33C2×C4⋊(C3⋊D3)Dic311yescharacteristic, complemented
34C2×C3×#67—64no—
35C2×C3×#67—62no—
36C2×C3×#67—62no—
37C2×C3×D12—32no—
38C2×C3×D12—61no—
39#3757—22no—
40C4×C3⋊Dic3Dic311yescharacteristic, complemented
41C3×#217—64no—
42C3×#217—62no—
43C3×#217—62no—
44C3×C4×Dic3—32no—
45C3×C4×Dic3—61no—
46C22×C33—22no—
47C22×C33—42no—
48C22×C33C22⋊C411yescharacteristic
49C2×C3×C3⋊D3—42no—
50C2×C3×C3⋊D3—41no—
51C33×C4—21no—
52C3×C3⋊Dic3—22no—
53C3×C3⋊Dic3—82no—
54C3×#127—91no—
55#940—32no—
56#940—61no—
57C3×#99—92no—
58#834—32no—
59#834—32no—
60#834—62no—
61C23×C32—22no—
62C23×C32—22no—
63C23×C32—22no—
64C23×C32—42no—
65C23×C32D1212yes—
66C23×C32#6712yes—
67C23×C32#6712yes—
68C22×C3⋊D3—21no—
69C22×C3×D3—62no—
70C22×C3×D3—62no—
71C32×D4—41no—
72C32×D4—42no—
73C32×D4—42no—
74C32×D4—44no—
75C32×D4—44no—
76C2×C32×C4—22no—
77C2×C32×C4—21no—
78C2×C32×C4—41no—
79C2×C32×C4C4×D312yes—
80C2×C32×C4C2×Dic311yescharacteristic
81#638—44no—
82C2×C3⋊Dic3—21no—
83C2×C3⋊Dic3—42no—
84C4⋊(C3⋊D3)—41no—
85C3×#67—124no—
86C3×#67—124no—
87C3×#67—128no—
88C2×C3×Dic3—62no—
89C2×C3×Dic3—62no—
90C2×C3×Dic3—124no—
91C2×C3×Dic3—122no—
92C2×C3×Dic3—122no—
93C3×D12—122no—
94C3×D12—122no—
95#353—271noSylow
96C2×C33—21no—
97C2×C33—42no—
98C2×C33#9911yescharacteristic
99C3×C3⋊D3—81no—
100C2×C3×D4—22no—
101C2×C3×D4—21no—
102C2×C3×D4—41no—
103C2×C3×D4—91no—
104C2×C3×D4—182no—
105C2×C3×D4D3212yes—
106C2×C3×D4D3211yescharacteristic
107C2×#67—64no—
108C2×#67—62no—
109C2×#67—62no—
110C2×D12—32no—
111C2×D12—61no—
112C3×C22⋊C4—182no—
113C3×C42—91no—
114#217—64no—
115#217—62no—
116#217—62no—
117C4×Dic3—32no—
118C4×Dic3—61no—
119C22×C32—22no—
120C22×C32—24no—
121C22×C32—22no—
122C22×C32—22no—
123C22×C32—22no—
124C22×C32—21no—
125C22×C32—41no—
126C22×C32—42no—
127C22×C32—42no—
128C22×C32—44no—
129C22×C32—42no—
130C22×C32—42no—
131C22×C32—42no—
132C22×C32—44no—
133C22×C32—82no—
134C22×C32—82no—
135C22×C32—84no—
136C22×C32(C2×D3)⋊C412yes—
137C22×C32#21711yescharacteristic
138C2×C3⋊D3—42no—
139C2×C3⋊D3—41no—
140C2×C3×D3—122no—
141C2×C3×D3—124no—
142C2×C3×D3—122no—
143C2×C3×D3—122no—
144C2×C3×D3—122no—
145C32×C4—21no—
146C32×C4—22no—
147C32×C4—22no—
148C32×C4—42no—
149C32×C4—42no—
150C3⋊Dic3—22no—
151C3⋊Dic3—82no—
152C3×Dic3—64no—
153C3×Dic3—64no—
154C3×Dic3—244no—
155C3×Dic3—242no—
156C3×Dic3—242no—
157#127—91no—
158#99—92no—
159C33#35311yescharacteristic, complemented, Sylow
160C23×C3—22no—
161C23×C3—22no—
162C23×C3—22no—
163C23×C3—42no—
164C23×C3—181no—
165C23×C3#62512yes—
166C23×C3#62612yes—
167C23×C3#62612yes—
168C22×D3—62no—
169C22×D3—62no—
170C3×D4—42no—
171C3×D4—41no—
172C3×D4—42no—
173C3×D4—44no—
174C3×D4—44no—
175C3×D4—361no—
176C3×D4—364no—
177C2×C3×C4—21no—
178C2×C3×C4—22no—
179C2×C3×C4—41no—
180C2×C3×C4—181no—
181C2×C3×C4—362no—
182C2×C3×C4D3×Dic312yes—
183C2×C3×C4#62411yescharacteristic
184#67—124no—
185#67—124no—
186#67—128no—
187C2×Dic3—62no—
188C2×Dic3—62no—
189C2×Dic3—124no—
190C2×Dic3—122no—
191C2×Dic3—122no—
192D12—122no—
193D12—122no—
194C2×C32—21no—
195C2×C32—22no—
196C2×C32—22no—
197C2×C32—22no—
198C2×C32—21no—
199C2×C32—41no—
200C2×C32—42no—
201C2×C32—42no—
202C2×C32—42no—
203C2×C32—44no—
204C2×C32—42no—
205C2×C32—42no—
206C2×C32—82no—
207C2×C32—82no—
208C2×C32—84no—
209C2×C32#80612yes—
210C2×C32#83411yescharacteristic
211C3⋊D3—81no—
212C3×D3—242no—
213C3×D3—242no—
214C2×D4—91no—
215C2×D4—182no—
216C2×D4#111311yescharacteristic
217C22⋊C4—182no—
218C42—91no—
219C22×C3—24no—
220C22×C3—22no—
221C22×C3—22no—
222C22×C3—22no—
223C22×C3—21no—
224C22×C3—22no—
225C22×C3—42no—
226C22×C3—42no—
227C22×C3—44no—
228C22×C3—42no—
229C22×C3—42no—
230C22×C3—42no—
231C22×C3—44no—
232C22×C3—41no—
233C22×C3—82no—
234C22×C3—82no—
235C22×C3—84no—
236C22×C3—362no—
237C22×C3—361no—
238C22×C3(C2×D3)⋊Dic312yes—
239C22×C3#372211yescharacteristic
240C2×D3—122no—
241C2×D3—124no—
242C2×D3—122no—
243C2×D3—122no—
244C2×D3—122no—
245C3×C4—22no—
246C3×C4—22no—
247C3×C4—21no—
248C3×C4—42no—
249C3×C4—42no—
250C3×C4—182no—
251C3×C4—722no—
252Dic3—64no—
253Dic3—64no—
254Dic3—244no—
255Dic3—242no—
256Dic3—242no—
257C32—22no—
258C32—21no—
259C32—41no—
260C32#447412yescomplemented
261C32#454511yescharacteristic, complemented
262C23—181no—
263C23#629512yes—
264D4—41no—
265D4—361no—
266D4—364no—
267C2×C4—181no—
268C2×C4—362no—
269C2×C4#629411yescharacteristic
270C2×C3—22no—
271C2×C3—22no—
272C2×C3—21no—
273C2×C3—21no—
274C2×C3—22no—
275C2×C3—42no—
276C2×C3—42no—
277C2×C3—44no—
278C2×C3—42no—
279C2×C3—42no—
280C2×C3—42no—
281C2×C3—41no—
282C2×C3—82no—
283C2×C3—82no—
284C2×C3—84no—
285C2×C3—721no—
286C2×C3#6358012yes—
287C2×C3#6358111yescharacteristic
288D3—242no—
289D3—242no—
290C22—22no—
291C22—42no—
292C22—362no—
293C22—361no—
294C22#8881411yescharacteristic
295C4—21no—
296C4—182no—
297C4—722no—
298C3—21no—
299C3—22no—
300C3—41no—
301C3#1058917812yescomplemented
302C3#1058917911yescharacteristic, complemented
303C2—21no—
304C2—42no—
305C2—721no—
306C2#1172208311yescharacteristic
subgroupG/H
centerC2#11722083
commutatorC2×C33×C4C2×C4
Frattini, hypercenterC2×D4#1113
Fitting, cosocleC2×C33×D4C22
socleC2×C33#99
prime radicalC3×#64149C2
hypocenterC33#353
LCSG ▷ C2×C33×C4 ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C3×#63661C2semidirect
7. C2×C3×C4⋊(C3⋊D3)C4semidirect
9. C3×C4×C3⋊Dic3C4semidirect
13. #63661D3semidirect
33. C2×C4⋊(C3⋊D3)Dic3semidirect
40. C4×C3⋊Dic3Dic3semidirect
159. C33#353semidirect
260. C32#4474semidirect
261. C32#4545semidirect
301. C3#10589178semidirect
302. C3#10589179semidirect

Maximal subgroups

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

subgroupindex
C3×#641492
C3×#636612
#105891793
#105891783