Finite Group Databasebeta

← C3×#3296 #85160

Subgroups — C3×#3296 #85160

Order 384.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#552C211yesmaximal, characteristic, complemented
2C3×#546C211yesmaximal, characteristic
3C3×#488C211yesmaximal, characteristic, complemented
4C3×#483C211yesmaximal, characteristic, complemented
5C3×#482C211yesmaximal, characteristic, complemented
6C3×#478C211yesmaximal, characteristic, complemented
7C3×#467C211yesmaximal, characteristic, complemented
8C3×#465C211yesmaximal, characteristic, complemented
9C3×#464C211yesmaximal, characteristic, complemented
10C3×#452C211yesmaximal, characteristic, complemented
11C3×#450C211yesmaximal, characteristic, complemented
12C3×C4×SD16C211yesmaximal, characteristic, complemented
13C3×C8×D4C211yesmaximal, characteristic, complemented
14C3×#416C211yesmaximal, characteristic, complemented
15C2×C3×#102C211yesmaximal, characteristic, complemented
16#3296C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×#41C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C2×C3×Q16C2211yescharacteristic, complemented
20C2×C3×SD16C2211yescharacteristic
21C2×C3×SD16C2211yescharacteristic
22C22×C3×C8C2211yescharacteristic
23C22×C3×C8C2211yescharacteristic, complemented
24C3×#126—21no—
25C3×#125—21no—
26C3×#125C2211yescharacteristic, complemented
27C3×#124C2211yescharacteristic
28C3×#123—21no—
29C3×#123—21no—
30C3×C22⋊Q8C2211yescharacteristic
31C3×C22⋊Q8C2211yescharacteristic
32C3×C22⋊Q8C2211yescharacteristic, complemented
33C3×#121—21no—
34C3×#121C2211yescharacteristic
35C3×#120—21no—
36C3×C4×Q8C2211yescharacteristic, complemented
37C3×C4×D4C2211yescharacteristic
38C3×C4×D4C2211yescharacteristic
39C3×#117—21no—
40C3×#117C2211yescharacteristic, complemented
41C2×C3×C4⋊C4C2211yescharacteristic
42C2×C3×C22⋊C4—21no—
43C3×#107C2211yescharacteristic, complemented
44C3×#106C2211yescharacteristic, complemented
45C3×#106C2211yescharacteristic, complemented
46C3×C4⋊C8C2211yescharacteristic, complemented
47C3×#103C2211yescharacteristic, complemented
48C3×#103C2211yescharacteristic, complemented
49C3×#103C2211yescharacteristic, complemented
50C3×#103C2211yescharacteristic, complemented
51C3×#102C2211yescharacteristic
52C3×#102C2211yescharacteristic
53C3×#102C2211yescharacteristic
54C3×#102C2211yescharacteristic
55C3×#102C2211yescharacteristic
56C3×#102C2211yescharacteristic
57C3×C22⋊C8C2211yescharacteristic
58C3×C22⋊C8C2211yescharacteristic, complemented
59C3×C4×C8C2211yescharacteristic, complemented
60#552C2×C311yescharacteristic, complemented
61#546C2×C311yescharacteristic
62#488C2×C311yescharacteristic, complemented
63#483C2×C311yescharacteristic, complemented
64#482C2×C311yescharacteristic, complemented
65#478C2×C311yescharacteristic, complemented
66#467C2×C311yescharacteristic, complemented
67#465C2×C311yescharacteristic, complemented
68#464C2×C311yescharacteristic, complemented
69#452C2×C311yescharacteristic, complemented
70#450C2×C311yescharacteristic, complemented
71C4×SD16C2×C311yescharacteristic, complemented
72C8×D4C2×C311yescharacteristic, complemented
73#416C2×C311yescharacteristic, complemented
74C2×#102C2×C311yescharacteristic, complemented
75C24×C3—21no—
76C3×#41—22no—
77C3×#41—41no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×Q8C2311yescharacteristic
80C2×C3×D4—22no—
81C2×C3×D4—21no—
82C2×C3×D4—21no—
83C2×C3×D4—21no—
84C2×C3×D4—41no—
85C2×C3×D4D411yescharacteristic
86C2×C3×D4C2311yescharacteristic
87C2×C3×D4C2311yescharacteristic
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4C2311yescharacteristic
92C22×C3×C4C2311yescharacteristic
93C3×Q16—41no—
94C3×SD16—22no—
95C3×SD16—41no—
96C2×C3×C8—41no—
97C2×C3×C8—41no—
98C2×C3×C8C2311yescharacteristic
99C2×C3×C8C2311yescharacteristic
100C2×C3×C8C2311yescharacteristic
101C2×C3×C8C2311yescharacteristic
102C3×C4⋊C4—21no—
103C3×C4⋊C4—21no—
104C3×C4⋊C4—21no—
105C3×C4⋊C4—21no—
106C3×C4⋊C4—21no—
107C3×C4⋊C4—21no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4D411yescharacteristic
110C3×C4⋊C4C2311yescharacteristic
111C3×C4⋊C4C2311yescharacteristic
112C3×C4⋊C4C2311yescharacteristic
113C3×C4⋊C4C2311yescharacteristic
114C3×C22⋊C4—21no—
115C3×C22⋊C4—21no—
116C3×C22⋊C4—21no—
117C3×C22⋊C4—21no—
118C3×C22⋊C4—21no—
119C3×C22⋊C4—21no—
120C3×C22⋊C4—21no—
121C3×C22⋊C4—21no—
122C3×C22⋊C4D411yescharacteristic
123C3×C22⋊C4D411yescharacteristic
124C3×C42—21no—
125C3×C42—21no—
126C3×C42C2311yescharacteristic
127C2×#41C22×C311yescharacteristic
128C22×D4C22×C311yescharacteristic
129C2×Q16C22×C311yescharacteristic, complemented
130C2×SD16C22×C311yescharacteristic
131C2×SD16C22×C311yescharacteristic
132C22×C8C22×C311yescharacteristic
133C22×C8C22×C311yescharacteristic, complemented
134#126—21no—
135#125—21no—
136#125C22×C311yescharacteristic, complemented
137#124C22×C311yescharacteristic
138#123—21no—
139#123—21no—
140C22⋊Q8C22×C311yescharacteristic
141C22⋊Q8C22×C311yescharacteristic
142C22⋊Q8C22×C311yescharacteristic, complemented
143#121—21no—
144#121C22×C311yescharacteristic
145#120—21no—
146C4×Q8C22×C311yescharacteristic, complemented
147C4×D4C22×C311yescharacteristic
148C4×D4C22×C311yescharacteristic
149#117—21no—
150#117C22×C311yescharacteristic, complemented
151C2×C4⋊C4C22×C311yescharacteristic
152C2×C22⋊C4—21no—
153#107C22×C311yescharacteristic, complemented
154#106C22×C311yescharacteristic, complemented
155#106C22×C311yescharacteristic, complemented
156C4⋊C8C22×C311yescharacteristic, complemented
157#103C22×C311yescharacteristic, complemented
158#103C22×C311yescharacteristic, complemented
159#103C22×C311yescharacteristic, complemented
160#103C22×C311yescharacteristic, complemented
161#102C22×C311yescharacteristic
162#102C22×C311yescharacteristic
163#102C22×C311yescharacteristic
164#102C22×C311yescharacteristic
165#102C22×C311yescharacteristic
166#102C22×C311yescharacteristic
167C22⋊C8C22×C311yescharacteristic
168C22⋊C8C22×C311yescharacteristic, complemented
169C4×C8C22×C311yescharacteristic, complemented
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—42no—
173C23×C3—42no—
174C23×C3—81no—
175C23×C3C2×D411yescharacteristic
176C23×C3C2×D411yescharacteristic
177C3×Q8—21no—
178C3×Q8—21no—
179C3×Q8—21no—
180C3×Q8—21no—
181C3×D4—22no—
182C3×D4—21no—
183C3×D4—21no—
184C3×D4—21no—
185C3×D4—42no—
186C3×D4—42no—
187C3×D4—81no—
188C3×D4#4112yes—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—81no—
202C2×C3×C4C2×D411yescharacteristic
203C2×C3×C4C2×D411yescharacteristic
204C2×C3×C4C2×D411yescharacteristic
205C2×C3×C4C2×D411yescharacteristic
206C2×C3×C4C2411yescharacteristic
207C3×C8—21no—
208C3×C8—21no—
209C3×C8—21no—
210C3×C8—21no—
211C24—21no—
212#41—22no—
213#41—41no—
214C2×Q8C23×C311yescharacteristic
215C2×Q8C23×C311yescharacteristic
216C2×D4—22no—
217C2×D4—21no—
218C2×D4—21no—
219C2×D4—21no—
220C2×D4—41no—
221C2×D4C3×D411yescharacteristic
222C2×D4C23×C311yescharacteristic
223C2×D4C23×C311yescharacteristic
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4C23×C311yescharacteristic
228C22×C4C23×C311yescharacteristic
229Q16—41no—
230SD16—22no—
231SD16—41no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8C23×C311yescharacteristic
235C2×C8C23×C311yescharacteristic
236C2×C8C23×C311yescharacteristic
237C2×C8C23×C311yescharacteristic
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4C3×D411yescharacteristic
246C4⋊C4C23×C311yescharacteristic
247C4⋊C4C23×C311yescharacteristic
248C4⋊C4C23×C311yescharacteristic
249C4⋊C4C23×C311yescharacteristic
250C22⋊C4—21no—
251C22⋊C4—21no—
252C22⋊C4—21no—
253C22⋊C4—21no—
254C22⋊C4—21no—
255C22⋊C4—21no—
256C22⋊C4—21no—
257C22⋊C4—21no—
258C22⋊C4C3×D411yescharacteristic
259C22⋊C4C3×D411yescharacteristic
260C42—21no—
261C42—21no—
262C42C23×C311yescharacteristic
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—22no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—84no—
275C22×C3#13512yes—
276C22×C3C22×D411yescharacteristic
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—41no—
282C3×C4—41no—
283C3×C4—41no—
284C3×C4—41no—
285C3×C4—41no—
286C3×C4—41no—
287C3×C4C2×#4111yescharacteristic
288C3×C4H(5,𝔽2)11yescharacteristic
289C23—21no—
290C23—21no—
291C23—42no—
292C23—42no—
293C23—81no—
294C23C2×C3×D411yescharacteristic
295C23C2×C3×D411yescharacteristic
296Q8—21no—
297Q8—21no—
298Q8—21no—
299Q8—21no—
300D4—22no—
301D4—21no—
302D4—21no—
303D4—21no—
304D4—42no—
305D4—42no—
306D4—81no—
307D4C3×#4112yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—81no—
321C2×C4C2×C3×D411yescharacteristic
322C2×C4C2×C3×D411yescharacteristic
323C2×C4C2×C3×D411yescharacteristic
324C2×C4C2×C3×D411yescharacteristic
325C2×C4C24×C311yescharacteristic
326C8—21no—
327C8—21no—
328C8—21no—
329C8—21no—
330C2×C3—22no—
331C2×C3—41no—
332C2×C3—42no—
333C2×C3—81no—
334C2×C3#54611yescharacteristic
335C2×C3C2×#13511yescharacteristic
336C2×C3#57711yescharacteristic
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—22no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—84no—
349C22C3×#13512yes—
350C22C22×C3×D411yescharacteristic
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4C2×C3×#4111yescharacteristic
362C4C3×H(5,𝔽2)11yescharacteristic
363C3#329611yescharacteristic, direct product factor, Sylow
364C2—22no—
365C2—41no—
366C2—42no—
367C2—81no—
368C2C3×#54611yescharacteristic
369C2C2×C3×#13511yescharacteristic
370C2C3×#57711yescharacteristic
subgroupG/H
center, socleC22×C3C22×D4
commutator, Frattini, cosocleC2×C4C24×C3
prime radicalC3×#546C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#552C2semidirect
3. C3×#488C2semidirect
4. C3×#483C2semidirect
5. C3×#482C2semidirect
6. C3×#478C2semidirect
7. C3×#467C2semidirect
8. C3×#465C2semidirect
9. C3×#464C2semidirect
10. C3×#452C2semidirect
11. C3×#450C2semidirect
12. C3×C4×SD16C2semidirect
13. C3×C8×D4C2semidirect
14. C3×#416C2semidirect
15. C2×C3×#102C2semidirect
16. #3296C3direct (N × H)
19. C2×C3×Q16C22semidirect
23. C22×C3×C8C22semidirect
26. C3×#125C22semidirect
32. C3×C22⋊Q8C22semidirect
36. C3×C4×Q8C22semidirect
40. C3×#117C22semidirect
43. C3×#107C22semidirect
44. C3×#106C22semidirect
45. C3×#106C22semidirect
46. C3×C4⋊C8C22semidirect
47. C3×#103C22semidirect
48. C3×#103C22semidirect
49. C3×#103C22semidirect
50. C3×#103C22semidirect
58. C3×C22⋊C8C22semidirect
59. C3×C4×C8C22semidirect
60. #552C2×C3semidirect
62. #488C2×C3semidirect
63. #483C2×C3semidirect
64. #482C2×C3semidirect
65. #478C2×C3semidirect
66. #467C2×C3semidirect
67. #465C2×C3semidirect
68. #464C2×C3semidirect
69. #452C2×C3semidirect
70. #450C2×C3semidirect
71. C4×SD16C2×C3semidirect
72. C8×D4C2×C3semidirect
73. #416C2×C3semidirect
74. C2×#102C2×C3semidirect
129. C2×Q16C22×C3semidirect
133. C22×C8C22×C3semidirect
136. #125C22×C3semidirect
142. C22⋊Q8C22×C3semidirect
146. C4×Q8C22×C3semidirect
150. #117C22×C3semidirect
153. #107C22×C3semidirect
154. #106C22×C3semidirect
155. #106C22×C3semidirect
156. C4⋊C8C22×C3semidirect
157. #103C22×C3semidirect
158. #103C22×C3semidirect
159. #103C22×C3semidirect
160. #103C22×C3semidirect
168. C22⋊C8C22×C3semidirect
169. C4×C8C22×C3semidirect
363. C3#3296direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5522
C3×#5462
C3×#4882
C3×#4832
C3×#4822
C3×#4782
C3×#4672
C3×#4652
C3×#4642
C3×#4522
C3×#4502
C3×C4×SD162
C3×C8×D42
C3×#4162
C2×C3×#1022
#32963