Finite Group Databasebeta

← C3×#10593410 #13166772

Subgroups — C3×#10593410 #13166772

Order 1728.

576 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 118 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×C4×#157C211yesmaximal, characteristic
2C2×C3×#3773C211yesmaximal, characteristic
3C2×C3×#3773C211yesmaximal, characteristic
4C3×#63883C212yesmaximal
5C3×#63737C212yesmaximal
6#10593410C311yesmaximal, characteristic, direct product factor
7C22×C3×#157C2211yescharacteristic
8C22×C3×#157C2211yescharacteristic
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C3×#624C2212yes—
11C2×C3×#624C2212yes—
12C3×C4×#157—22no—
13C3×#3773—22no—
14C3×#3773C414yescomplemented
15C32×Dic3⋊C4—22no—
16C3×Dic3⋊Dic3—22no—
17C2×C4×#157C2×C311yescharacteristic
18C2×#3773C2×C311yescharacteristic
19C2×#3773C2×C311yescharacteristic
20C3×D3×C4⋊C4—62no—
21#63883C2×C312yes—
22#63737C2×C312yes—
23C22×C3×C3⋊D3C2311yescharacteristic
24C2×C3×#157—21no—
25C2×C3×#157—21no—
26C2×C3×#157C2×C412yes—
27C2×C3×#157D412yes—
28C2×C3×#157Q812yes—
29C2×C3×#157Q812yes—
30C2×C33×C4D411yescharacteristic
31C2×C3×C3⋊Dic3D411yescharacteristic
32C3×C4×C3⋊D3—21no—
33C3×C4×C3⋊D3—21no—
34C2×C32×Dic3—22no—
35C2×C32×Dic3—22no—
36C3×#624—22no—
37C3×#624—22no—
38C3×#624—22no—
39C3×#624C2×C414yes—
40C3×#389—91nomaximal
41C22×#157C22×C311yescharacteristic
42C22×#157C22×C311yescharacteristic
43C2×C4×C3⋊D3C22×C311yescharacteristic
44C2×C3×C4×D3—62no—
45C2×C3×C4×D3—62no—
46C2×C3×C4×D3—62no—
47C2×#624C22×C312yes—
48C2×#624C22×C312yes—
49C4×#157—22no—
50#3773—22no—
51#3773C3×C414yescomplemented
52C32×C4⋊C4—62no—
53C3×C4⋊Dic3—62no—
54C3×Dic3⋊C4—22no—
55C3×Dic3⋊C4—22no—
56C3×Dic3⋊C4—42no—
57C3×Dic3⋊C4—62no—
58Dic3⋊Dic3—22no—
59C22×C33C2×D411yescharacteristic
60C2×C3×C3⋊D3C22×C411yescharacteristic
61C2×C3×C3⋊D3C2×D411yescharacteristic
62C2×C3×C3⋊D3C2×Q811yescharacteristic
63C2×C3×C3⋊D3C2×Q811yescharacteristic
64C2×C3×C3⋊D3#4111yescharacteristic
65C2×C3×C3⋊D3#4111yescharacteristic
66C3×#157—22no—
67C3×#157C4⋊C414yescomplemented
68C33×C4—21no—
69C3×C3⋊Dic3—21no—
70C32×Dic3—24no—
71C32×Dic3—42no—
72C2×C3×C4⋊C4—92no—
73C2×C3×C4⋊C4—91no—
74C2×C3×C4⋊C4—91no—
75C2×C3×C42—91no—
76D3×C4⋊C4—62no—
77C3×#95—92no—
78C22×C3⋊D3C23×C311yescharacteristic
79C22×C3×D3—62no—
80C2×#157—21no—
81C2×#157—21no—
82C2×#157C2×C3×C412yes—
83C2×#157C3×D412yes—
84C2×#157C3×Q812yes—
85C2×#157C3×Q812yes—
86C2×C32×C4—22no—
87C2×C32×C4—42no—
88C2×C32×C4—62no—
89C2×C32×C4—62no—
90C2×C32×C4C3×D411yescharacteristic
91C2×C3⋊Dic3C3×D411yescharacteristic
92C4×C3⋊D3—21no—
93C4×C3⋊D3—21no—
94C2×C3×Dic3—22no—
95C2×C3×Dic3—22no—
96C2×C3×Dic3—22no—
97C2×C3×Dic3—22no—
98C2×C3×Dic3—42no—
99C2×C3×Dic3—42no—
100C2×C3×Dic3—62no—
101C3×C4×D3—64no—
102C3×C4×D3—64no—
103C3×C4×D3—122no—
104C3×C4×D3—122no—
105C3×C4×D3—122no—
106C3×C4×D3—122no—
107#624—22no—
108#624—22no—
109#624—22no—
110#624C2×C3×C414yes—
111#389—91noSylow
112C2×C33C4×D411yescharacteristic
113C2×C33#12111yescharacteristic
114C2×C33C4⋊Q811yescharacteristic
115C3×C3⋊D3C2×C4⋊C411yescharacteristic
116C3×C3⋊D3C4×Q811yescharacteristic
117C3×C3⋊D3C22⋊Q811yescharacteristic
118C3×C3⋊D3#12511yescharacteristic
119C22×C3×C4—92no—
120C22×C3×C4—92no—
121C22×C3×C4—91no—
122C22×C3×C4—91no—
123C22×C3×C4—91no—
124C2×C4×D3—62no—
125C2×C4×D3—62no—
126C2×C4×D3—62no—
127C3×C4⋊C4—62no—
128C3×C4⋊C4—62no—
129C3×C4⋊C4—94no—
130C3×C4⋊C4—122no—
131C3×C4⋊C4—182no—
132C3×C4⋊C4—182no—
133C3×C42—182no—
134C4⋊Dic3—62no—
135Dic3⋊C4—22no—
136Dic3⋊C4—62no—
137C22×C32—22no—
138C22×C32—42no—
139C22×C32C2×C3×D411yescharacteristic
140C2×C3⋊D3C22×C3×C411yescharacteristic
141C2×C3⋊D3C2×C3×D411yescharacteristic
142C2×C3⋊D3C2×C3×Q811yescharacteristic
143C2×C3⋊D3C2×C3×Q811yescharacteristic
144C2×C3⋊D3C3×#4111yescharacteristic
145C2×C3⋊D3C3×#4111yescharacteristic
146C2×C3×D3—62no—
147C2×C3×D3—62no—
148C2×C3×D3—62no—
149C2×C3×D3—62no—
150C2×C3×D3—62no—
151C2×C3×D3—62no—
152#157—22no—
153#157C3×C4⋊C414yescomplemented
154C32×C4—21no—
155C32×C4—42no—
156C32×C4—64no—
157C32×C4—82no—
158C32×C4—122no—
159C3⋊Dic3—21no—
160C3×Dic3—24no—
161C3×Dic3—24no—
162C3×Dic3—44no—
163C3×Dic3—42no—
164C3×Dic3—42no—
165C3×Dic3—82no—
166C3×Dic3—122no—
167C2×C4⋊C4—92no—
168C2×C4⋊C4—91no—
169C2×C4⋊C4—91no—
170C2×C42—91no—
171#95—92no—
172C33#38911yescharacteristic, complemented, Sylow
173C23×C3—91no—
174C22×D3—62no—
175C2×C3×C4—22no—
176C2×C3×C4—42no—
177C2×C3×C4—62no—
178C2×C3×C4—62no—
179C2×C3×C4—62no—
180C2×C3×C4—62no—
181C2×C3×C4—94no—
182C2×C3×C4—92no—
183C2×C3×C4—92no—
184C2×C3×C4—92no—
185C2×C3×C4—92no—
186C2×C3×C4—91no—
187C2×C3×C4—122no—
188C2×C3×C4—122no—
189C2×C3×C4—181no—
190C2×C3×C4—181no—
191C2×C3×C4—182no—
192C2×C3×C4—182no—
193C2×C3×C4—182no—
194C2×C3×C4—181no—
195C2×C3×C4—181no—
196C2×C3×C4GO+(4,𝔽2)11yescharacteristic
197C2×Dic3—22no—
198C2×Dic3—22no—
199C2×Dic3—62no—
200C4×D3—64no—
201C4×D3—64no—
202C4×D3—122no—
203C4×D3—122no—
204C4×D3—122no—
205C4×D3—122no—
206C2×C32—22no—
207C2×C32—22no—
208C2×C32—22no—
209C2×C32—42no—
210C2×C32—42no—
211C2×C32—42no—
212C2×C32C3×C4×D411yescharacteristic
213C2×C32C3×#12111yescharacteristic
214C2×C32C3×C4⋊Q811yescharacteristic
215C3⋊D3C2×C3×C4⋊C411yescharacteristic
216C3⋊D3C3×C4×Q811yescharacteristic
217C3⋊D3C3×C22⋊Q811yescharacteristic
218C3⋊D3C3×#12511yescharacteristic
219C3×D3—62no—
220C3×D3—62no—
221C3×D3—62no—
222C3×D3—62no—
223C22×C4—92no—
224C22×C4—92no—
225C22×C4—91no—
226C22×C4—91no—
227C22×C4—91no—
228C4⋊C4—62no—
229C4⋊C4—94no—
230C4⋊C4—182no—
231C4⋊C4—182no—
232C42—182no—
233C22×C3—22no—
234C22×C3—42no—
235C22×C3—91no—
236C22×C3—91no—
237C22×C3—91no—
238C22×C3—91no—
239C22×C3—91no—
240C22×C3—91no—
241C22×C3C2×GO+(4,𝔽2)11yescharacteristic
242C2×D3—62no—
243C2×D3—62no—
244C2×D3—62no—
245C2×D3—62no—
246C2×D3—62no—
247C2×D3—62no—
248C3×C4—21no—
249C3×C4—42no—
250C3×C4—64no—
251C3×C4—64no—
252C3×C4—82no—
253C3×C4—94no—
254C3×C4—124no—
255C3×C4—122no—
256C3×C4—122no—
257C3×C4—182no—
258C3×C4—181no—
259C3×C4—242no—
260Dic3—24no—
261Dic3—42no—
262Dic3—122no—
263C32—22no—
264C32—42no—
265C32C3×#38911yescharacteristic, complemented
266C23—91no—
267C2×C4—62no—
268C2×C4—62no—
269C2×C4—94no—
270C2×C4—92no—
271C2×C4—92no—
272C2×C4—92no—
273C2×C4—92no—
274C2×C4—91no—
275C2×C4—181no—
276C2×C4—181no—
277C2×C4—182no—
278C2×C4—182no—
279C2×C4—182no—
280C2×C4—181no—
281C2×C4—181no—
282C2×C4C3×GO+(4,𝔽2)11yescharacteristic
283C2×C3—22no—
284C2×C3—22no—
285C2×C3—22no—
286C2×C3—42no—
287C2×C3—42no—
288C2×C3—42no—
289C2×C3—91no—
290C2×C3—91no—
291C2×C3—91no—
292C2×C3—91no—
293C2×C3#6422211yescharacteristic
294C2×C3C4×GO+(4,𝔽2)11yescharacteristic
295C2×C3#6424211yescharacteristic
296D3—62no—
297D3—62no—
298D3—62no—
299D3—62no—
300C22—91no—
301C22—91no—
302C22—91no—
303C22—91no—
304C22—91no—
305C22—91no—
306C22C2×C3×GO+(4,𝔽2)11yescharacteristic
307C4—21no—
308C4—64no—
309C4—94no—
310C4—122no—
311C4—182no—
312C4—181no—
313C3—22no—
314C3—42no—
315C3#1059341011yescharacteristic, direct product factor
316C2—91no—
317C2—91no—
318C2—91no—
319C2—91no—
320C2C3×#6422211yescharacteristic
321C2C3×C4×GO+(4,𝔽2)11yescharacteristic
322C2C3×#6424211yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutatorC2×C3⋊D3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3C23
cosocleC22×C3⋊D3C23×C3
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#389
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
6. #10593410C3direct (N × H)
14. C3×#3773C4semidirect
51. #3773C3×C4semidirect
67. C3×#157C4⋊C4semidirect
153. #157C3×C4⋊C4semidirect
172. C33#389semidirect
265. C32C3×#389semidirect
315. C3#10593410direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×#1572
C2×C3×#37732
C3×#638832
C3×#637372
#105934103
C3×#3899