Finite Group Databasebeta

← C3×#10590257 #13157271

Subgroups — C3×#10590257 #13157271

Order 1728.

566 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 144 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×#64081C211yesmaximal, characteristic
2C32×C4⋊(C3⋊C8)C212yesmaximal, complemented
3C3×#63557C212yesmaximal
4C3×#63555C211yesmaximal, characteristic, complemented
5C3×#63553C211yesmaximal, characteristic, complemented
6C3×#4840—32nomaximal
7#10590257C311yesmaximal, characteristic, direct product factor
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C33×C42C2211yescharacteristic
10C3×#3752C412yes—
11C3×#3750C412yes—
12C3×C4×C3⋊Dic3C2211yescharacteristic
13C2×C32×C3⋊C8C2212yes—
14C2×C32×C3⋊C8C2212yes—
15#64081C2×C311yescharacteristic
16C3×#872—32no—
17C3×#872—61no—
18C32×C4⋊C8—32no—
19C3×(C2×D3)⋊C8—32no—
20C3×(C2×D3)⋊C8—32no—
21C3×#815—32no—
22C3×C8×Dic3—32no—
23C3×C4⋊(C3⋊C8)—22no—
24C3×C4⋊(C3⋊C8)D312yescomplemented
25C3×C4⋊(C3⋊C8)C2×C312yescomplemented
26#63557C2×C312yes—
27#63555C2×C311yescharacteristic, complemented
28#63553C2×C311yescharacteristic, complemented
29C22×C3×C3⋊D3C2×C411yescharacteristic
30C2×C33×C4C2×C412yes—
31C2×C33×C4C2311yescharacteristic
32C2×C3×C3⋊Dic3C2×C412yes—
33C2×C3×C3⋊Dic3C2×C411yescharacteristic
34C3×C4×C3⋊D3—22no—
35C32×C3⋊C8—22no—
36C32×C3⋊C8—22no—
37C3×#433—91no—
38#4840—32no—
39C2×C4×C3⋊D3C22×C311yescharacteristic
40C2×C3×C4×D3—32no—
41C2×C3×C4×D3—61no—
42C2×C32×C8—32no—
43C2×C32×C8—32no—
44C32×C42—21no—
45C32×C42—22no—
46C32×C42—41no—
47C32×C42C2×D312yes—
48C32×C42C22×C311yescharacteristic
49#3752C3×C412yes—
50#3750C3×C412yes—
51C4×C3⋊Dic3C22×C311yescharacteristic
52C3×(C2×D3)⋊C4—32no—
53C3×(C2×D3)⋊C4—32no—
54C3×(C2×D3)⋊C4—62no—
55C3×Dic3⋊C4—32no—
56C3×Dic3⋊C4—32no—
57C3×Dic3⋊C4—62no—
58C3×C4×Dic3—32no—
59C3×C4×Dic3—61no—
60C2×C3×C3⋊C8—22no—
61C2×C3×C3⋊C8—22no—
62C2×C3×C3⋊C8C2×D312yes—
63C2×C3×C3⋊C8C2×D312yes—
64C2×C3×C3⋊C8C22×C312yes—
65C2×C3×C3⋊C8C22×C312yes—
66C22×C33C22×C411yescharacteristic
67C2×C3×C3⋊D3—21no—
68C2×C3×C3⋊D3—22no—
69C33×C4—22no—
70C33×C4#4112yes—
71C3×C3⋊Dic3—21no—
72C3×C3⋊Dic3—22no—
73C3×#117—91no—
74#872—32no—
75#872—61no—
76C3×C4⋊C8—32no—
77C3×C4⋊C8—32no—
78C3×C4⋊C8—62no—
79C3×C22⋊C8—92no—
80C3×#97—91no—
81C3×C4×C8—91no—
82(C2×D3)⋊C8—32no—
83(C2×D3)⋊C8—32no—
84#815—32no—
85C8×Dic3—32no—
86C4⋊(C3⋊C8)C3×D312yescomplemented
87C22×C3⋊D3C2×C3×C411yescharacteristic
88C22×C3×D3—32no—
89C22×C3×D3—61no—
90C2×C32×C4—22no—
91C2×C32×C4—22no—
92C2×C32×C4—22no—
93C2×C32×C4—21no—
94C2×C32×C4—22no—
95C2×C32×C4—42no—
96C2×C32×C4—41no—
97C2×C32×C4C4×D312yes—
98C2×C32×C4C4×D312yes—
99C2×C32×C4C2×C3×C412yes—
100C2×C32×C4C22×D312yes—
101C2×C32×C4C23×C311yescharacteristic
102C2×C3⋊Dic3C2×C3×C412yes—
103C2×C3⋊Dic3C2×C3×C411yescharacteristic
104C4×C3⋊D3—22no—
105C2×C3×Dic3—32no—
106C2×C3×Dic3—32no—
107C2×C3×Dic3—32no—
108C2×C3×Dic3—61no—
109C2×C3×Dic3—62no—
110C3×C4×D3—64no—
111C3×C4×D3—62no—
112C3×C4×D3—62no—
113C32×C8—62no—
114C32×C8—62no—
115C3×C3⋊C8—22no—
116C3×C3⋊C8—22no—
117C3×C3⋊C8—22no—
118C3×C3⋊C8—22no—
119C3×C3⋊C8—42no—
120C3×C3⋊C8—42no—
121#433—91noSylow
122C2×C33#11711yescharacteristic
123C2×C33#13112yes—
124C3×C3⋊D3—41no—
125C22×C3×C4—91no—
126C2×C4×D3—32no—
127C2×C4×D3—61no—
128C2×C3×C8—32no—
129C2×C3×C8—32no—
130C2×C3×C8—32no—
131C2×C3×C8—32no—
132C2×C3×C8—62no—
133C2×C3×C8—62no—
134C3×C4⋊C4—92no—
135C3×C22⋊C4—92no—
136C3×C42—22no—
137C3×C42—21no—
138C3×C42—41no—
139C3×C42—91no—
140C3×C42D3211yescharacteristic
141C3×C42C2×C3×D312yes—
142(C2×D3)⋊C4—32no—
143(C2×D3)⋊C4—32no—
144(C2×D3)⋊C4—62no—
145Dic3⋊C4—32no—
146Dic3⋊C4—32no—
147Dic3⋊C4—62no—
148C4×Dic3—32no—
149C4×Dic3—61no—
150C2×C3⋊C8C2×C3×D312yes—
151C2×C3⋊C8C2×C3×D312yes—
152C22×C32—21no—
153C22×C32—22no—
154C22×C32—41no—
155C22×C32C2×C4×D312yes—
156C22×C32C22×C3×C411yescharacteristic
157C2×C3⋊D3—21no—
158C2×C3⋊D3—22no—
159C2×C3×D3—62no—
160C2×C3×D3—62no—
161C2×C3×D3—62no—
162C2×C3×D3—62no—
163C2×C3×D3—122no—
164C32×C4—22no—
165C32×C4—24no—
166C32×C4—22no—
167C32×C4—22no—
168C32×C4—22no—
169C32×C4—22no—
170C32×C4—22no—
171C32×C4—42no—
172C32×C4—42no—
173C32×C4—42no—
174C32×C4—44no—
175C32×C4#23712yes—
176C32×C4#23912yes—
177C32×C4C3×#4112yes—
178C3⋊Dic3—21no—
179C3⋊Dic3—22no—
180C3×Dic3—62no—
181C3×Dic3—62no—
182C3×Dic3—62no—
183C3×Dic3—62no—
184C3×Dic3—122no—
185#117—91no—
186C4⋊C8—32no—
187C22⋊C8—92no—
188#97—91no—
189C4×C8—91no—
190C33#43311yescharacteristic, complemented, Sylow
191C23×C3—91no—
192C22×D3—32no—
193C22×D3—61no—
194C2×C3×C4—22no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—21no—
199C2×C3×C4—42no—
200C2×C3×C4—41no—
201C2×C3×C4—92no—
202C2×C3×C4—91no—
203C2×C3×C4—182no—
204C2×C3×C4#62412yes—
205C2×C3×C4C3×C4×D312yes—
206C2×C3×C4C3×C4×D312yes—
207C2×C3×C4C2×D3211yescharacteristic
208C2×C3×C4C22×C3×D312yes—
209C2×Dic3—32no—
210C2×Dic3—32no—
211C2×Dic3—32no—
212C2×Dic3—61no—
213C2×Dic3—62no—
214C4×D3—64no—
215C4×D3—62no—
216C4×D3—62no—
217C3×C8—62no—
218C3×C8—62no—
219C3×C8—62no—
220C3×C8—62no—
221C3×C8—122no—
222C3×C8—122no—
223C3⋊C8—22no—
224C3⋊C8—22no—
225C2×C32—22no—
226C2×C32—22no—
227C2×C32—22no—
228C2×C32—22no—
229C2×C32—21no—
230C2×C32—41no—
231C2×C32—42no—
232C2×C32#89212yes—
233C2×C32#90112yes—
234C2×C32#90712yes—
235C2×C32C3×#11711yescharacteristic
236C2×C32C3×#13112yes—
237C3⋊D3—41no—
238C3×D3—122no—
239C3×D3—122no—
240C22×C4—91no—
241C2×C8—32no—
242C2×C8—32no—
243C4⋊C4—92no—
244C22⋊C4—92no—
245C42—91no—
246C42C3×D3211yescharacteristic
247C22×C3—21no—
248C22×C3—22no—
249C22×C3—41no—
250C22×C3—182no—
251C22×C3—181no—
252C22×C3C2×#62411yescharacteristic
253C22×C3C2×C3×C4×D312yes—
254C2×D3—62no—
255C2×D3—62no—
256C2×D3—62no—
257C2×D3—62no—
258C2×D3—122no—
259C3×C4—24no—
260C3×C4—22no—
261C3×C4—22no—
262C3×C4—22no—
263C3×C4—22no—
264C3×C4—22no—
265C3×C4—22no—
266C3×C4—42no—
267C3×C4—42no—
268C3×C4—44no—
269C3×C4—42no—
270C3×C4—182no—
271C3×C4—181no—
272C3×C4#379612yes—
273C3×C4C3×#23712yes—
274C3×C4C3×#23912yes—
275Dic3—62no—
276Dic3—62no—
277Dic3—62no—
278Dic3—62no—
279Dic3—122no—
280C32—21no—
281C32—22no—
282C32—41no—
283C32#484012yescomplemented
284C32C3×#43311yescharacteristic, complemented
285C23—91no—
286C2×C4—92no—
287C2×C4—91no—
288C2×C4—182no—
289C2×C4C3×#62412yes—
290C2×C4C2×C3×D3211yescharacteristic
291C8—62no—
292C8—62no—
293C2×C3—22no—
294C2×C3—22no—
295C2×C3—22no—
296C2×C3—21no—
297C2×C3—22no—
298C2×C3—42no—
299C2×C3—41no—
300C2×C3—361no—
301C2×C3#6381712yes—
302C2×C3#6384911yescharacteristic
303C2×C3C3×#89212yes—
304C2×C3C3×#90112yes—
305C2×C3C3×#90712yes—
306D3—122no—
307D3—122no—
308C22—182no—
309C22—181no—
310C22C2×C3×#62411yescharacteristic
311C4—22no—
312C4—182no—
313C4—181no—
314C4C3×#379612yes—
315C3—21no—
316C3—22no—
317C3—41no—
318C3#1059025711yescharacteristic, direct product factor
319C3C3×#484012yescomplemented
320C2—361no—
321C2C3×#6381712yes—
322C2C3×#6384911yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C22×C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC3×C42D32
hypocenterC32C3×#433
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
2. C32×C4⋊(C3⋊C8)C2semidirect
4. C3×#63555C2semidirect
5. C3×#63553C2semidirect
7. #10590257C3direct (N × H)
24. C3×C4⋊(C3⋊C8)D3semidirect
25. C3×C4⋊(C3⋊C8)C2×C3semidirect
27. #63555C2×C3semidirect
28. #63553C2×C3semidirect
86. C4⋊(C3⋊C8)C3×D3semidirect
190. C33#433semidirect
283. C32#4840semidirect
284. C32C3×#433semidirect
318. C3#10590257direct (N × H)
319. C3C3×#4840semidirect

Maximal subgroups

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

subgroupindex
C3×#640812
C32×C4⋊(C3⋊C8)2
C3×#635572
C3×#635552
C3×#635532
C3×#48403
#105902573