Finite Group Databasebeta

← C3×#2414 #84278

Subgroups — C3×#2414 #84278

Order 384.

704 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#121C211yesmaximal, characteristic, complemented
2C2×C3×#121C211yesmaximal, characteristic, complemented
3C2×C3×C4×D4C211yesmaximal, characteristic, complemented
4C2×C3×#117C211yesmaximal, characteristic, complemented
5C3×#394C212yesmaximal, complemented
6C3×#392C212yesmaximal, complemented
7C3×#390C212yesmaximal, complemented
8C3×#390C211yesmaximal, characteristic, complemented
9C3×#389C211yesmaximal, characteristic, complemented
10C3×#388C212yesmaximal, complemented
11C3×#380C211yesmaximal, characteristic, complemented
12#2414C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×C3×D4C2211yescharacteristic
15C22×C3×D4C2212yes—
16C22×C3×D4C2212yes—
17C23×C3×C4C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C3×#121—24no—
20C3×#121—24no—
21C3×C4×D4—24no—
22C3×#117—24no—
23C2×C3×C4⋊C4C2211yescharacteristic
24C2×C3×C4⋊C4C2211yescharacteristic
25C2×C3×C4⋊C4C2211yescharacteristic
26C2×C3×C4⋊C4C2212yes—
27C2×C3×C4⋊C4C2211yescharacteristic
28C2×C3×C22⋊C4C2212yes—
29C2×C3×C22⋊C4C2212yes—
30C2×C3×C22⋊C4C2212yes—
31C2×C3×C22⋊C4C2211yescharacteristic
32C2×C3×C22⋊C4C2212yes—
33C2×C3×C22⋊C4C2211yescharacteristic
34C2×C3×C22⋊C4C2212yes—
35C2×C3×C22⋊C4C2212yes—
36C2×C3×C42C2212yes—
37C2×C3×C42C2211yescharacteristic
38C3×#95C2212yes—
39C3×#95C2212yes—
40C2×#121C2×C311yescharacteristic, complemented
41C2×#121C2×C311yescharacteristic, complemented
42C2×C4×D4C2×C311yescharacteristic, complemented
43C2×#117C2×C311yescharacteristic, complemented
44#394C2×C312yescomplemented
45#392C2×C312yescomplemented
46#390C2×C312yescomplemented
47#390C2×C311yescharacteristic, complemented
48#389C2×C311yescharacteristic, complemented
49#388C2×C312yescomplemented
50#380C2×C311yescharacteristic, complemented
51C24×C3C2312yes—
52C24×C3C2311yescharacteristic
53C24×C3C2311yescharacteristic
54C2×C3×D4—22no—
55C2×C3×D4—22no—
56C2×C3×D4—24no—
57C2×C3×D4—24no—
58C2×C3×D4—24no—
59C2×C3×D4—22no—
60C2×C3×D4—22no—
61C2×C3×D4—22no—
62C2×C3×D4—22no—
63C2×C3×D4—42no—
64C2×C3×D4—42no—
65C2×C3×D4—42no—
66C22×C3×C4—22no—
67C22×C3×C4—22no—
68C22×C3×C4—22no—
69C22×C3×C4—22no—
70C22×C3×C4—22no—
71C22×C3×C4D414yescomplemented
72C22×C3×C4C2311yescharacteristic
73C22×C3×C4C2312yes—
74C22×C3×C4C2312yes—
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2312yes—
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C3×C4⋊C4—22no—
81C3×C4⋊C4—22no—
82C3×C4⋊C4—22no—
83C3×C4⋊C4—42no—
84C3×C4⋊C4D414yescomplemented
85C3×C22⋊C4—22no—
86C3×C22⋊C4—24no—
87C3×C22⋊C4—22no—
88C3×C22⋊C4—24no—
89C3×C22⋊C4—24no—
90C3×C22⋊C4—24no—
91C3×C22⋊C4—42no—
92C3×C22⋊C4—42no—
93C3×C42—24no—
94C3×C42—22no—
95C22×D4C22×C311yescharacteristic
96C22×D4C22×C311yescharacteristic
97C22×D4C22×C312yes—
98C22×D4C22×C312yes—
99C23×C4C22×C311yescharacteristic
100C23×C4C22×C311yescharacteristic
101#121—24no—
102#121—24no—
103C4×D4—24no—
104#117—24no—
105C2×C4⋊C4C22×C311yescharacteristic
106C2×C4⋊C4C22×C311yescharacteristic
107C2×C4⋊C4C22×C311yescharacteristic
108C2×C4⋊C4C22×C312yes—
109C2×C4⋊C4C22×C311yescharacteristic
110C2×C22⋊C4C22×C312yes—
111C2×C22⋊C4C22×C311yescharacteristic
112C2×C22⋊C4C22×C311yescharacteristic
113C2×C22⋊C4C22×C312yes—
114C2×C22⋊C4C22×C312yes—
115C2×C22⋊C4C22×C312yes—
116C2×C22⋊C4C22×C312yes—
117C2×C22⋊C4C22×C312yes—
118C2×C42C22×C312yes—
119C2×C42C22×C311yescharacteristic
120#95C22×C312yes—
121#95C22×C312yes—
122C23×C3—21no—
123C23×C3—21no—
124C23×C3—21no—
125C23×C3—21no—
126C23×C3—21no—
127C23×C3—21no—
128C23×C3—21no—
129C23×C3—21no—
130C23×C3—21no—
131C23×C3—21no—
132C23×C3—21no—
133C23×C3—21no—
134C23×C3—22no—
135C23×C3—22no—
136C23×C3—22no—
137C23×C3—22no—
138C23×C3—22no—
139C23×C3—22no—
140C23×C3—22no—
141C23×C3C2×D412yes—
142C23×C3#4112yes—
143C23×C3C2411yescharacteristic
144C3×D4—28no—
145C3×D4—48no—
146C3×D4—48no—
147C3×D4—44no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—21no—
151C2×C3×C4—21no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4—28no—
155C2×C3×C4—21no—
156C2×C3×C4—22no—
157C2×C3×C4—21no—
158C2×C3×C4—21no—
159C2×C3×C4—22no—
160C2×C3×C4—22no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—22no—
164C2×C3×C4—22no—
165C2×C3×C4—21no—
166C2×C3×C4—22no—
167C2×C3×C4—44no—
168C2×C3×C4C2×D414yes—
169C2×C3×C4C2×D412yes—
170C2×C3×C4C2×D412yes—
171C2×C3×C4C2×D412yes—
172C2×C3×C4#4112yes—
173C2×C3×C4#4112yes—
174C24C23×C312yes—
175C24C23×C311yescharacteristic
176C24C23×C311yescharacteristic
177C2×D4—22no—
178C2×D4—22no—
179C2×D4—24no—
180C2×D4—24no—
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4C3×D414yescomplemented
195C22×C4C23×C311yescharacteristic
196C22×C4C23×C312yes—
197C22×C4C23×C312yes—
198C22×C4C23×C311yescharacteristic
199C22×C4C23×C312yes—
200C22×C4C23×C311yescharacteristic
201C22×C4C23×C311yescharacteristic
202C22×C4C23×C311yescharacteristic
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—42no—
207C4⋊C4C3×D414yescomplemented
208C22⋊C4—22no—
209C22⋊C4—24no—
210C22⋊C4—22no—
211C22⋊C4—24no—
212C22⋊C4—24no—
213C22⋊C4—24no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C42—24no—
217C42—22no—
218C22×C3—22no—
219C22×C3—22no—
220C22×C3—22no—
221C22×C3—22no—
222C22×C3—22no—
223C22×C3—22no—
224C22×C3—41no—
225C22×C3—41no—
226C22×C3—41no—
227C22×C3—41no—
228C22×C3—41no—
229C22×C3—41no—
230C22×C3—41no—
231C22×C3—41no—
232C22×C3—42no—
233C22×C3—42no—
234C22×C3—42no—
235C22×C3—42no—
236C22×C3—42no—
237C22×C3—42no—
238C22×C3—42no—
239C22×C3C22×D411yescharacteristic
240C22×C3C22×D411yescharacteristic
241C22×C3C2×#4111yescharacteristic
242C22×C3C2×#4111yescharacteristic
243C22×C3C2×#4111yescharacteristic
244C22×C3H(5,𝔽2)11yescharacteristic
245C22×C3H(5,𝔽2)11yescharacteristic
246C3×C4—22no—
247C3×C4—22no—
248C3×C4—24no—
249C3×C4—41no—
250C3×C4—41no—
251C3×C4—42no—
252C3×C4—42no—
253C3×C4#12114yes—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23C2×C3×D412yes—
274C23C3×#4112yes—
275C23C24×C311yescharacteristic
276D4—28no—
277D4—48no—
278D4—48no—
279D4—44no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—28no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—44no—
300C2×C4C2×C3×D414yes—
301C2×C4C2×C3×D412yes—
302C2×C4C2×C3×D412yes—
303C2×C4C2×C3×D412yes—
304C2×C4C3×#4112yes—
305C2×C4C3×#4112yes—
306C2×C3—42no—
307C2×C3—42no—
308C2×C3—82no—
309C2×C3C2×#12111yescharacteristic
310C2×C3#52511yescharacteristic
311C2×C3#53511yescharacteristic
312C2×C3#54611yescharacteristic
313C2×C3#55011yescharacteristic
314C2×C3#55311yescharacteristic
315C2×C3#55511yescharacteristic
316C22—22no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22C22×C3×D411yescharacteristic
338C22C22×C3×D411yescharacteristic
339C22C2×C3×#4111yescharacteristic
340C22C2×C3×#4111yescharacteristic
341C22C2×C3×#4111yescharacteristic
342C22C3×H(5,𝔽2)11yescharacteristic
343C22C3×H(5,𝔽2)11yescharacteristic
344C4—22no—
345C4—22no—
346C4—24no—
347C4—41no—
348C4—42no—
349C4—41no—
350C4—42no—
351C4C3×#12114yes—
352C3#241411yescharacteristic, direct product factor, Sylow
353C2—42no—
354C2—42no—
355C2—82no—
356C2C2×C3×#12111yescharacteristic
357C2C3×#52511yescharacteristic
358C2C3×#53511yescharacteristic
359C2C3×#54611yescharacteristic
360C2C3×#55011yescharacteristic
361C2C3×#55311yescharacteristic
362C2C3×#55511yescharacteristic
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×#121C2semidirect
2. C2×C3×#121C2semidirect
3. C2×C3×C4×D4C2semidirect
4. C2×C3×#117C2semidirect
5. C3×#394C2semidirect
6. C3×#392C2semidirect
7. C3×#390C2semidirect
8. C3×#390C2semidirect
9. C3×#389C2semidirect
10. C3×#388C2semidirect
11. C3×#380C2semidirect
12. #2414C3direct (N × H)
40. C2×#121C2×C3semidirect
41. C2×#121C2×C3semidirect
42. C2×C4×D4C2×C3semidirect
43. C2×#117C2×C3semidirect
44. #394C2×C3semidirect
45. #392C2×C3semidirect
46. #390C2×C3semidirect
47. #390C2×C3semidirect
48. #389C2×C3semidirect
49. #388C2×C3semidirect
50. #380C2×C3semidirect
71. C22×C3×C4D4semidirect
84. C3×C4⋊C4D4semidirect
194. C22×C4C3×D4semidirect
207. C4⋊C4C3×D4semidirect
352. C3#2414direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#1212
C2×C3×C4×D42
C2×C3×#1172
C3×#3942
C3×#3922
C3×#3902
C3×#3892
C3×#3882
C3×#3802
#24143