Finite Group Databasebeta

← C3×#2813 #84677

Subgroups — C3×#2813 #84677

Order 384.

578 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#123C211yesmaximal, characteristic, complemented
2C2×C3×#121C211yesmaximal, characteristic, complemented
3C3×#400C211yesmaximal, characteristic, complemented
4C3×#399C211yesmaximal, characteristic, complemented
5C3×#396C212yesmaximal, complemented
6C3×#394C211yesmaximal, characteristic, complemented
7C3×#394C212yesmaximal, complemented
8C3×#394C212yesmaximal, complemented
9C3×#392C211yesmaximal, characteristic, complemented
10C3×#392C212yesmaximal, complemented
11C3×#380C211yesmaximal, characteristic, complemented
12#2813C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×C3×D4C2212yes—
15C22×C3×D4C2211yescharacteristic
16C22×C3×D4C2212yes—
17C23×C3×C4C2211yescharacteristic
18C3×#123—24no—
19C3×#121—24no—
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×C3×C4⋊C4C2211yescharacteristic
22C2×C3×C4⋊C4C2211yescharacteristic
23C2×C3×C4⋊C4C2212yes—
24C2×C3×C4⋊C4C2212yes—
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×C3×C22⋊C4C2211yescharacteristic
28C2×C3×C22⋊C4C2212yes—
29C2×C3×C22⋊C4C2212yes—
30C2×C3×C22⋊C4C2212yes—
31C2×C3×C22⋊C4C2212yes—
32C2×C3×C22⋊C4C2212yes—
33C2×C3×C22⋊C4C2212yes—
34C3×#95C2211yescharacteristic
35C3×#95C2211yescharacteristic
36C3×#95C2212yes—
37C3×#95C2212yes—
38C2×#123C2×C311yescharacteristic, complemented
39C2×#121C2×C311yescharacteristic, complemented
40#400C2×C311yescharacteristic, complemented
41#399C2×C311yescharacteristic, complemented
42#396C2×C312yescomplemented
43#394C2×C311yescharacteristic, complemented
44#394C2×C312yescomplemented
45#394C2×C312yescomplemented
46#392C2×C311yescharacteristic, complemented
47#392C2×C312yescomplemented
48#380C2×C311yescharacteristic, complemented
49C24×C3C2311yescharacteristic
50C24×C3C2312yes—
51C24×C3C2311yescharacteristic
52C2×C3×D4—24no—
53C2×C3×D4—22no—
54C2×C3×D4—22no—
55C2×C3×D4—42no—
56C2×C3×D4—42no—
57C2×C3×D4—42no—
58C2×C3×D4—42no—
59C2×C3×D4—42no—
60C2×C3×D4—42no—
61C2×C3×D4—42no—
62C22×C3×C4—42no—
63C22×C3×C4D414yescomplemented
64C22×C3×C4C2312yes—
65C22×C3×C4C2311yescharacteristic
66C22×C3×C4C2312yes—
67C22×C3×C4C2311yescharacteristic
68C22×C3×C4C2312yes—
69C22×C3×C4C2311yescharacteristic
70C22×C3×C4C2311yescharacteristic
71C22×C3×C4C2311yescharacteristic
72C3×C4⋊C4—22no—
73C3×C4⋊C4—22no—
74C3×C4⋊C4—22no—
75C3×C4⋊C4—42no—
76C3×C4⋊C4—42no—
77C3×C22⋊C4—22no—
78C3×C22⋊C4—24no—
79C3×C22⋊C4—22no—
80C3×C22⋊C4—22no—
81C3×C22⋊C4—42no—
82C3×C22⋊C4—42no—
83C3×C22⋊C4—42no—
84C3×C22⋊C4—42no—
85C3×C22⋊C4—42no—
86C22×D4C22×C312yes—
87C22×D4C22×C311yescharacteristic
88C22×D4C22×C311yescharacteristic
89C22×D4C22×C312yes—
90C23×C4C22×C311yescharacteristic
91#123—24no—
92#121—24no—
93C2×C4⋊C4C22×C311yescharacteristic
94C2×C4⋊C4C22×C311yescharacteristic
95C2×C4⋊C4C22×C311yescharacteristic
96C2×C4⋊C4C22×C312yes—
97C2×C4⋊C4C22×C312yes—
98C2×C22⋊C4C22×C312yes—
99C2×C22⋊C4C22×C311yescharacteristic
100C2×C22⋊C4C22×C311yescharacteristic
101C2×C22⋊C4C22×C311yescharacteristic
102C2×C22⋊C4C22×C312yes—
103C2×C22⋊C4C22×C312yes—
104C2×C22⋊C4C22×C312yes—
105C2×C22⋊C4C22×C312yes—
106C2×C22⋊C4C22×C312yes—
107#95C22×C311yescharacteristic
108#95C22×C311yescharacteristic
109#95C22×C312yes—
110#95C22×C312yes—
111C23×C3—21no—
112C23×C3—22no—
113C23×C3—22no—
114C23×C3—21no—
115C23×C3—21no—
116C23×C3—22no—
117C23×C3—22no—
118C23×C3—22no—
119C23×C3—22no—
120C23×C3—22no—
121C23×C3—22no—
122C23×C3—22no—
123C23×C3—22no—
124C23×C3—21no—
125C23×C3—21no—
126C23×C3—22no—
127C23×C3C2×D412yes—
128C23×C3C2411yescharacteristic
129C3×D4—44no—
130C3×D4—44no—
131C3×D4—48no—
132C3×D4—48no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—22no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4—22no—
141C2×C3×C4—22no—
142C2×C3×C4—22no—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—22no—
147C2×C3×C4—22no—
148C2×C3×C4—22no—
149C2×C3×C4—22no—
150C2×C3×C4—21no—
151C2×C3×C4—44no—
152C2×C3×C4C2×D412yes—
153C2×C3×C4C2×D412yes—
154C24C23×C311yescharacteristic
155C24C23×C312yes—
156C24C23×C311yescharacteristic
157C2×D4—24no—
158C2×D4—22no—
159C2×D4—22no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C22×C4—42no—
168C22×C4C3×D414yescomplemented
169C22×C4C23×C312yes—
170C22×C4C23×C311yescharacteristic
171C22×C4C23×C312yes—
172C22×C4C23×C311yescharacteristic
173C22×C4C23×C312yes—
174C22×C4C23×C311yescharacteristic
175C22×C4C23×C311yescharacteristic
176C22×C4C23×C311yescharacteristic
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C22⋊C4—22no—
183C22⋊C4—24no—
184C22⋊C4—22no—
185C22⋊C4—22no—
186C22⋊C4—42no—
187C22⋊C4—42no—
188C22⋊C4—42no—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C22×C3—22no—
192C22×C3—22no—
193C22×C3—22no—
194C22×C3—42no—
195C22×C3—42no—
196C22×C3—42no—
197C22×C3—41no—
198C22×C3—42no—
199C22×C3—41no—
200C22×C3—41no—
201C22×C3—42no—
202C22×C3—42no—
203C22×C3—42no—
204C22×C3—42no—
205C22×C3—42no—
206C22×C3—42no—
207C22×C3—42no—
208C22×C3C22×D411yescharacteristic
209C22×C3H(5,𝔽2)12yes—
210C22×C3H(5,𝔽2)12yes—
211C22×C3H(5,𝔽2)11yescharacteristic
212C22×C32−1+411yescharacteristic
213C3×C4—22no—
214C3×C4—22no—
215C3×C4—41no—
216C3×C4—41no—
217C3×C4—42no—
218C3×C4—41no—
219C3×C4—42no—
220C3×C4—42no—
221C23—21no—
222C23—22no—
223C23—22no—
224C23—21no—
225C23—21no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—21no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—21no—
236C23—22no—
237C23C2×C3×D412yes—
238C23C24×C311yescharacteristic
239D4—44no—
240D4—44no—
241D4—48no—
242D4—48no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—44no—
262C2×C4C2×C3×D412yes—
263C2×C4C2×C3×D412yes—
264C2×C3—42no—
265C2×C3—82no—
266C2×C3—81no—
267C2×C3#53411yescharacteristic
268C2×C3#53511yescharacteristic
269C2×C3#53711yescharacteristic
270C2×C3#56012yes—
271C2×C3#56212yes—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—41no—
285C22—42no—
286C22—41no—
287C22—41no—
288C22—42no—
289C22C22×C3×D411yescharacteristic
290C22C3×H(5,𝔽2)11yescharacteristic
291C22C3×H(5,𝔽2)12yes—
292C22C3×H(5,𝔽2)12yes—
293C22C3×2−1+411yescharacteristic
294C4—22no—
295C4—22no—
296C4—42no—
297C4—42no—
298C4—41no—
299C4—42no—
300C4—41no—
301C4—41no—
302C3#281311yescharacteristic, direct product factor, Sylow
303C2—42no—
304C2—81no—
305C2—82no—
306C2C3×#53411yescharacteristic
307C2C3×#53511yescharacteristic
308C2C3×#53711yescharacteristic
309C2C3×#56012yes—
310C2C3×#56212yes—
subgroupG/H
center, socleC23×C3C24
commutator, Frattini, cosocleC23C24×C3
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#123C2semidirect
2. C2×C3×#121C2semidirect
3. C3×#400C2semidirect
4. C3×#399C2semidirect
5. C3×#396C2semidirect
6. C3×#394C2semidirect
7. C3×#394C2semidirect
8. C3×#394C2semidirect
9. C3×#392C2semidirect
10. C3×#392C2semidirect
11. C3×#380C2semidirect
12. #2813C3direct (N × H)
38. C2×#123C2×C3semidirect
39. C2×#121C2×C3semidirect
40. #400C2×C3semidirect
41. #399C2×C3semidirect
42. #396C2×C3semidirect
43. #394C2×C3semidirect
44. #394C2×C3semidirect
45. #394C2×C3semidirect
46. #392C2×C3semidirect
47. #392C2×C3semidirect
48. #380C2×C3semidirect
63. C22×C3×C4D4semidirect
168. C22×C4C3×D4semidirect
302. C3#2813direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#1232
C2×C3×#1212
C3×#4002
C3×#3992
C3×#3962
C3×#3942
C3×#3922
C3×#3802
#28133