Finite Group Databasebeta

← C3×#2407 #84271

Subgroups — C3×#2407 #84271

Order 384.

908 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#41C211yesmaximal, characteristic, complemented
2C2×C3×C22⋊Q8C211yesmaximal, characteristic, complemented
3C2×C3×#121C211yesmaximal, characteristic, complemented
4C2×C3×C4×D4C211yesmaximal, characteristic, complemented
5C3×#394C212yesmaximal, complemented
6C3×#392C212yesmaximal, complemented
7C3×#391C211yesmaximal, characteristic, complemented
8C3×#390C211yesmaximal, characteristic, complemented
9C3×#386C212yesmaximal, complemented
10C3×#385C212yesmaximal, complemented
11C3×#380C211yesmaximal, characteristic, complemented
12#2407C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#41—24no—
14C2×C3×#41—24no—
15C2×C3×#41—24no—
16C22×C3×Q8C2211yescharacteristic
17C22×C3×D4C2212yes—
18C22×C3×D4C2211yescharacteristic
19C22×C3×D4C2212yes—
20C22×C3×D4C2211yescharacteristic
21C23×C3×C4C2212yes—
22C23×C3×C4C2211yescharacteristic
23C23×C3×C4C2211yescharacteristic
24C3×C22⋊Q8—24no—
25C3×#121—24no—
26C3×C4×D4—24no—
27C2×C3×C4⋊C4C2211yescharacteristic
28C2×C3×C4⋊C4C2211yescharacteristic
29C2×C3×C4⋊C4C2211yescharacteristic
30C2×C3×C4⋊C4C2212yes—
31C2×C3×C22⋊C4C2212yes—
32C2×C3×C22⋊C4C2212yes—
33C2×C3×C22⋊C4C2211yescharacteristic
34C2×C3×C22⋊C4C2212yes—
35C2×C3×C22⋊C4C2211yescharacteristic
36C2×C3×C22⋊C4C2212yes—
37C2×C3×C22⋊C4C2212yes—
38C2×C3×C42C2211yescharacteristic
39C3×#95C2212yes—
40C3×#95C2212yes—
41C3×#95C2212yes—
42C22×#41C2×C311yescharacteristic, complemented
43C2×C22⋊Q8C2×C311yescharacteristic, complemented
44C2×#121C2×C311yescharacteristic, complemented
45C2×C4×D4C2×C311yescharacteristic, complemented
46#394C2×C312yescomplemented
47#392C2×C312yescomplemented
48#391C2×C311yescharacteristic, complemented
49#390C2×C311yescharacteristic, complemented
50#386C2×C312yescomplemented
51#385C2×C312yescomplemented
52#380C2×C311yescharacteristic, complemented
53C24×C3C2312yes—
54C24×C3C2311yescharacteristic
55C24×C3C2311yescharacteristic
56C3×#41—232no—
57C2×C3×Q8—22no—
58C2×C3×Q8—22no—
59C2×C3×Q8D414yescomplemented
60C2×C3×D4—22no—
61C2×C3×D4—24no—
62C2×C3×D4—24no—
63C2×C3×D4—22no—
64C2×C3×D4—24no—
65C2×C3×D4—24no—
66C2×C3×D4—24no—
67C2×C3×D4—22no—
68C2×C3×D4—22no—
69C2×C3×D4—22no—
70C2×C3×D4—42no—
71C2×C3×D4D414yescomplemented
72C22×C3×C4—24no—
73C22×C3×C4—24no—
74C22×C3×C4—24no—
75C22×C3×C4—22no—
76C22×C3×C4—22no—
77C22×C3×C4—22no—
78C22×C3×C4—22no—
79C22×C3×C4—22no—
80C22×C3×C4D414yescomplemented
81C22×C3×C4C2312yes—
82C22×C3×C4C2311yescharacteristic
83C22×C3×C4C2312yes—
84C22×C3×C4C2311yescharacteristic
85C22×C3×C4C2311yescharacteristic
86C22×C3×C4C2312yes—
87C22×C3×C4C2311yescharacteristic
88C22×C3×C4C2311yescharacteristic
89C3×C4⋊C4—22no—
90C3×C4⋊C4—22no—
91C3×C4⋊C4—22no—
92C3×C4⋊C4—24no—
93C3×C22⋊C4—22no—
94C3×C22⋊C4—24no—
95C3×C22⋊C4—22no—
96C3×C22⋊C4—24no—
97C3×C22⋊C4—24no—
98C3×C22⋊C4—42no—
99C3×C22⋊C4—42no—
100C3×C42—22no—
101C2×#41—24no—
102C2×#41—24no—
103C2×#41—24no—
104C22×Q8C22×C311yescharacteristic
105C22×D4C22×C311yescharacteristic
106C22×D4C22×C312yes—
107C22×D4C22×C312yes—
108C22×D4C22×C311yescharacteristic
109C23×C4C22×C311yescharacteristic
110C23×C4C22×C312yes—
111C23×C4C22×C311yescharacteristic
112C22⋊Q8—24no—
113#121—24no—
114C4×D4—24no—
115C2×C4⋊C4C22×C311yescharacteristic
116C2×C4⋊C4C22×C311yescharacteristic
117C2×C4⋊C4C22×C311yescharacteristic
118C2×C4⋊C4C22×C312yes—
119C2×C22⋊C4C22×C311yescharacteristic
120C2×C22⋊C4C22×C311yescharacteristic
121C2×C22⋊C4C22×C312yes—
122C2×C22⋊C4C22×C312yes—
123C2×C22⋊C4C22×C312yes—
124C2×C22⋊C4C22×C312yes—
125C2×C22⋊C4C22×C312yes—
126C2×C42C22×C311yescharacteristic
127#95C22×C312yes—
128#95C22×C312yes—
129#95C22×C312yes—
130C23×C3—21no—
131C23×C3—21no—
132C23×C3—22no—
133C23×C3—21no—
134C23×C3—22no—
135C23×C3—21no—
136C23×C3—21no—
137C23×C3—22no—
138C23×C3—21no—
139C23×C3—21no—
140C23×C3—21no—
141C23×C3—22no—
142C23×C3—21no—
143C23×C3—22no—
144C23×C3—21no—
145C23×C3—21no—
146C23×C3—22no—
147C23×C3—21no—
148C23×C3C2×D414yes—
149C23×C3C2×D412yes—
150C23×C3#4112yescomplemented
151C23×C3C2411yescharacteristic
152C3×Q8—28no—
153C3×D4—28no—
154C3×D4—216no—
155C3×D4—28no—
156C3×D4—48no—
157C2×C3×C4—216no—
158C2×C3×C4—21no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—22no—
162C2×C3×C4—22no—
163C2×C3×C4—21no—
164C2×C3×C4—28no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—22no—
168C2×C3×C4—21no—
169C2×C3×C4—22no—
170C2×C3×C4—21no—
171C2×C3×C4—22no—
172C2×C3×C4—22no—
173C2×C3×C4—21no—
174C2×C3×C4—22no—
175C2×C3×C4—22no—
176C2×C3×C4—21no—
177C2×C3×C4—44no—
178C2×C3×C4C2×D412yes—
179C2×C3×C4C2×D414yes—
180C2×C3×C4C2×D412yes—
181C2×C3×C4C2×D412yes—
182C2×C3×C4C2×D412yes—
183C2×C3×C4#4112yescomplemented
184C24C23×C312yes—
185C24C23×C311yescharacteristic
186C24C23×C311yescharacteristic
187#41—232no—
188C2×Q8—22no—
189C2×Q8—22no—
190C2×Q8C3×D414yescomplemented
191C2×D4—22no—
192C2×D4—24no—
193C2×D4—24no—
194C2×D4—22no—
195C2×D4—24no—
196C2×D4—24no—
197C2×D4—24no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—42no—
202C2×D4C3×D414yescomplemented
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4C3×D414yescomplemented
212C22×C4C23×C312yes—
213C22×C4C23×C311yescharacteristic
214C22×C4C23×C312yes—
215C22×C4C23×C311yescharacteristic
216C22×C4C23×C311yescharacteristic
217C22×C4C23×C312yes—
218C22×C4C23×C311yescharacteristic
219C22×C4C23×C311yescharacteristic
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—24no—
224C22⋊C4—22no—
225C22⋊C4—24no—
226C22⋊C4—22no—
227C22⋊C4—24no—
228C22⋊C4—24no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C42—22no—
232C22×C3—24no—
233C22×C3—22no—
234C22×C3—22no—
235C22×C3—22no—
236C22×C3—22no—
237C22×C3—22no—
238C22×C3—24no—
239C22×C3—24no—
240C22×C3—22no—
241C22×C3—41no—
242C22×C3—42no—
243C22×C3—41no—
244C22×C3—42no—
245C22×C3—41no—
246C22×C3—41no—
247C22×C3—41no—
248C22×C3—41no—
249C22×C3—42no—
250C22×C3—41no—
251C22×C3—42no—
252C22×C3—41no—
253C22×C3C22×D411yescharacteristic
254C22×C3C22×D411yescharacteristic
255C22×C3C22×D411yescharacteristic
256C22×C3C2×#4111yescharacteristic
257C22×C3C2×#4111yescharacteristic
258C22×C3H(5,𝔽2)11yescharacteristic
259C22×C32−1+411yescharacteristic
260C3×C4—22no—
261C3×C4—24no—
262C3×C4—22no—
263C3×C4—41no—
264C3×C4—41no—
265C3×C4—42no—
266C3×C4—42no—
267C3×C4#12014yes—
268C23—21no—
269C23—21no—
270C23—22no—
271C23—21no—
272C23—22no—
273C23—21no—
274C23—21no—
275C23—22no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—22no—
280C23—21no—
281C23—22no—
282C23—21no—
283C23—21no—
284C23—22no—
285C23—21no—
286C23C2×C3×D414yes—
287C23C2×C3×D412yes—
288C23C3×#4112yescomplemented
289C23C24×C311yescharacteristic
290Q8—28no—
291D4—216no—
292D4—28no—
293D4—28no—
294D4—48no—
295C2×C4—216no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—28no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—44no—
316C2×C4C2×C3×D412yes—
317C2×C4C2×C3×D414yes—
318C2×C4C2×C3×D412yes—
319C2×C4C2×C3×D412yes—
320C2×C4C2×C3×D412yes—
321C2×C4C3×#4112yescomplemented
322C2×C3—42no—
323C2×C3—44no—
324C2×C3—42no—
325C2×C3C2×#12011yescharacteristic
326C2×C3#52511yescharacteristic
327C2×C3#53711yescharacteristic
328C2×C3#54611yescharacteristic
329C2×C3#54711yescharacteristic
330C2×C3#54811yescharacteristic
331C2×C3#55011yescharacteristic
332C22—24no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—24no—
339C22—24no—
340C22—22no—
341C22—41no—
342C22—42no—
343C22—41no—
344C22—42no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—42no—
350C22—41no—
351C22—42no—
352C22—41no—
353C22C22×C3×D411yescharacteristic
354C22C22×C3×D411yescharacteristic
355C22C22×C3×D411yescharacteristic
356C22C2×C3×#4111yescharacteristic
357C22C2×C3×#4111yescharacteristic
358C22C3×H(5,𝔽2)11yescharacteristic
359C22C3×2−1+411yescharacteristic
360C4—22no—
361C4—24no—
362C4—22no—
363C4—41no—
364C4—41no—
365C4—42no—
366C4—42no—
367C4C3×#12014yes—
368C3#240711yescharacteristic, direct product factor, Sylow
369C2—42no—
370C2—44no—
371C2—42no—
372C2C2×C3×#12011yescharacteristic
373C2C3×#52511yescharacteristic
374C2C3×#53711yescharacteristic
375C2C3×#54611yescharacteristic
376C2C3×#54711yescharacteristic
377C2C3×#54811yescharacteristic
378C2C3×#55011yescharacteristic
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. C22×C3×#41C2semidirect
2. C2×C3×C22⋊Q8C2semidirect
3. C2×C3×#121C2semidirect
4. C2×C3×C4×D4C2semidirect
5. C3×#394C2semidirect
6. C3×#392C2semidirect
7. C3×#391C2semidirect
8. C3×#390C2semidirect
9. C3×#386C2semidirect
10. C3×#385C2semidirect
11. C3×#380C2semidirect
12. #2407C3direct (N × H)
42. C22×#41C2×C3semidirect
43. C2×C22⋊Q8C2×C3semidirect
44. C2×#121C2×C3semidirect
45. C2×C4×D4C2×C3semidirect
46. #394C2×C3semidirect
47. #392C2×C3semidirect
48. #391C2×C3semidirect
49. #390C2×C3semidirect
50. #386C2×C3semidirect
51. #385C2×C3semidirect
52. #380C2×C3semidirect
59. C2×C3×Q8D4semidirect
71. C2×C3×D4D4semidirect
80. C22×C3×C4D4semidirect
150. C23×C3#41semidirect
183. C2×C3×C4#41semidirect
190. C2×Q8C3×D4semidirect
202. C2×D4C3×D4semidirect
211. C22×C4C3×D4semidirect
288. C23C3×#41semidirect
321. C2×C4C3×#41semidirect
368. C3#2407direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#412
C2×C3×C22⋊Q82
C2×C3×#1212
C2×C3×C4×D42
C3×#3942
C3×#3922
C3×#3912
C3×#3902
C3×#3862
C3×#3852
C3×#3802
#24073