Finite Group Databasebeta

← C3×#3310 #85174

Subgroups — C3×#3310 #85174

Order 384.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#552C211yesmaximal, characteristic, complemented
2C3×D42C211yesmaximal, characteristic
3C3×#490C211yesmaximal, characteristic, complemented
4C3×#482C211yesmaximal, characteristic, complemented
5C3×#480C211yesmaximal, characteristic, complemented
6C3×#478C211yesmaximal, characteristic, complemented
7C3×#468C211yesmaximal, characteristic, complemented
8C3×#465C211yesmaximal, characteristic, complemented
9C3×#460C211yesmaximal, characteristic, complemented
10C3×#450C211yesmaximal, characteristic, complemented
11C3×#450C211yesmaximal, characteristic, complemented
12C3×#440C211yesmaximal, characteristic, complemented
13C3×#435C211yesmaximal, characteristic, complemented
14C3×#418C211yesmaximal, characteristic, complemented
15C2×C3×#102C211yesmaximal, characteristic, complemented
16#3310C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×D4—21no—
18C22×C3×D4C2211yescharacteristic
19C22×C3×D4C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic
21C2×C3×SD16C2211yescharacteristic, complemented
22C2×C3×SD16C2211yescharacteristic, complemented
23C2×C3×M16C2211yescharacteristic, complemented
24C22×C3×C8C2211yescharacteristic, complemented
25C3×#127C2211yescharacteristic
26C3×#126—21no—
27C3×#125—21no—
28C3×#125C2211yescharacteristic, complemented
29C3×#123—21no—
30C3×C22⋊Q8C2211yescharacteristic, complemented
31C3×C22⋊Q8C2211yescharacteristic, complemented
32C3×#121—21no—
33C3×#121C2211yescharacteristic
34C3×#121C2211yescharacteristic
35C3×#120—21no—
36C3×#120—21no—
37C3×C4×Q8C2211yescharacteristic, complemented
38C3×C4×D4C2211yescharacteristic
39C3×C4×D4C2211yescharacteristic
40C3×#117—21no—
41C3×#117C2211yescharacteristic, complemented
42C2×C3×C4⋊C4C2211yescharacteristic, complemented
43C3×#107C2211yescharacteristic, complemented
44C3×#107C2211yescharacteristic, complemented
45C3×#106C2211yescharacteristic, complemented
46C3×C4⋊C8C2211yescharacteristic, complemented
47C3×#103C2211yescharacteristic, complemented
48C3×#102C2211yescharacteristic
49C3×#102C2211yescharacteristic
50C3×#102C2211yescharacteristic
51C3×#102C2211yescharacteristic, complemented
52C3×#102C2211yescharacteristic, complemented
53C3×#102C2211yescharacteristic, complemented
54C3×#102C2211yescharacteristic, complemented
55C3×#102C2211yescharacteristic, complemented
56C3×#102C2211yescharacteristic, complemented
57C3×C22⋊C8C2211yescharacteristic, complemented
58C3×C22⋊C8C2211yescharacteristic, complemented
59C3×#97C2211yescharacteristic, complemented
60#552C2×C311yescharacteristic, complemented
61D42C2×C311yescharacteristic
62#490C2×C311yescharacteristic, complemented
63#482C2×C311yescharacteristic, complemented
64#480C2×C311yescharacteristic, complemented
65#478C2×C311yescharacteristic, complemented
66#468C2×C311yescharacteristic, complemented
67#465C2×C311yescharacteristic, complemented
68#460C2×C311yescharacteristic, complemented
69#450C2×C311yescharacteristic, complemented
70#450C2×C311yescharacteristic, complemented
71#440C2×C311yescharacteristic, complemented
72#435C2×C311yescharacteristic, complemented
73#418C2×C311yescharacteristic, complemented
74C2×#102C2×C311yescharacteristic, complemented
75C24×C3—21no—
76C24×C3—21no—
77C2×C3×Q8C2311yescharacteristic
78C2×C3×D4—22no—
79C2×C3×D4—22no—
80C2×C3×D4—21no—
81C2×C3×D4—21no—
82C2×C3×D4—21no—
83C2×C3×D4—21no—
84C2×C3×D4—21no—
85C2×C3×D4—21no—
86C2×C3×D4—41no—
87C2×C3×D4—41no—
88C2×C3×D4—44no—
89C2×C3×D4D411yescharacteristic
90C2×C3×D4C2311yescharacteristic
91C2×C3×D4C2311yescharacteristic
92C2×C3×D4C2311yescharacteristic
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4C2311yescharacteristic
96C22×C3×C4C2311yescharacteristic
97C3×SD16—22no—
98C3×SD16—41no—
99C3×SD16—41no—
100C3×M16—41no—
101C2×C3×C8—41no—
102C2×C3×C8C2311yescharacteristic
103C2×C3×C8C2311yescharacteristic
104C2×C3×C8C2311yescharacteristic
105C2×C3×C8C2311yescharacteristic
106C3×C4⋊C4—21no—
107C3×C4⋊C4—21no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4D411yescharacteristic
113C3×C4⋊C4C2311yescharacteristic
114C3×C4⋊C4C2311yescharacteristic
115C3×C4⋊C4C2311yescharacteristic
116C3×C4⋊C4C2311yescharacteristic
117C3×C22⋊C4—21no—
118C3×C22⋊C4—21no—
119C3×C22⋊C4—21no—
120C3×C22⋊C4—21no—
121C3×C22⋊C4—21no—
122C3×C22⋊C4—21no—
123C3×C22⋊C4D411yescharacteristic
124C3×C22⋊C4D411yescharacteristic
125C3×C42—21no—
126C3×C42—21no—
127C3×C42C2311yescharacteristic
128C22×D4—21no—
129C22×D4C22×C311yescharacteristic
130C22×D4C22×C311yescharacteristic
131C2×SD16C22×C311yescharacteristic
132C2×SD16C22×C311yescharacteristic, complemented
133C2×SD16C22×C311yescharacteristic, complemented
134C2×M16C22×C311yescharacteristic, complemented
135C22×C8C22×C311yescharacteristic, complemented
136#127C22×C311yescharacteristic
137#126—21no—
138#125—21no—
139#125C22×C311yescharacteristic, complemented
140#123—21no—
141C22⋊Q8C22×C311yescharacteristic, complemented
142C22⋊Q8C22×C311yescharacteristic, complemented
143#121—21no—
144#121C22×C311yescharacteristic
145#121C22×C311yescharacteristic
146#120—21no—
147#120—21no—
148C4×Q8C22×C311yescharacteristic, complemented
149C4×D4C22×C311yescharacteristic
150C4×D4C22×C311yescharacteristic
151#117—21no—
152#117C22×C311yescharacteristic, complemented
153C2×C4⋊C4C22×C311yescharacteristic, complemented
154#107C22×C311yescharacteristic, complemented
155#107C22×C311yescharacteristic, complemented
156#106C22×C311yescharacteristic, complemented
157C4⋊C8C22×C311yescharacteristic, complemented
158#103C22×C311yescharacteristic, complemented
159#102C22×C311yescharacteristic
160#102C22×C311yescharacteristic
161#102C22×C311yescharacteristic
162#102C22×C311yescharacteristic, complemented
163#102C22×C311yescharacteristic, complemented
164#102C22×C311yescharacteristic, complemented
165#102C22×C311yescharacteristic, complemented
166#102C22×C311yescharacteristic, complemented
167#102C22×C311yescharacteristic, complemented
168C22⋊C8C22×C311yescharacteristic, complemented
169C22⋊C8C22×C311yescharacteristic, complemented
170#97C22×C311yescharacteristic, complemented
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—21no—
174C23×C3—42no—
175C23×C3—42no—
176C23×C3—42no—
177C23×C3—42no—
178C23×C3—81no—
179C23×C3—81no—
180C23×C3C2×D411yescharacteristic
181C23×C3C2×D411yescharacteristic
182C3×Q8—21no—
183C3×Q8—21no—
184C3×D4—22no—
185C3×D4—21no—
186C3×D4—21no—
187C3×D4—21no—
188C3×D4—21no—
189C3×D4—21no—
190C3×D4—42no—
191C3×D4—42no—
192C3×D4—42no—
193C3×D4—42no—
194C3×D4—42no—
195C3×D4—81no—
196C3×D4#4112yes—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—42no—
207C2×C3×C4—42no—
208C2×C3×C4C2×D411yescharacteristic
209C2×C3×C4C2×D411yescharacteristic
210C2×C3×C4C2×D411yescharacteristic
211C2×C3×C4C2×D411yescharacteristic
212C2×C3×C4C2411yescharacteristic
213C3×C8—21no—
214C3×C8—21no—
215C3×C8—21no—
216C3×C8—21no—
217C24—21no—
218C24—21no—
219C2×Q8C23×C311yescharacteristic
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—21no—
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—44no—
231C2×D4C3×D411yescharacteristic
232C2×D4C23×C311yescharacteristic
233C2×D4C23×C311yescharacteristic
234C2×D4C23×C311yescharacteristic
235C22×C4—21no—
236C22×C4—21no—
237C22×C4C23×C311yescharacteristic
238C22×C4C23×C311yescharacteristic
239SD16—22no—
240SD16—41no—
241SD16—41no—
242M16—41no—
243C2×C8—41no—
244C2×C8C23×C311yescharacteristic
245C2×C8C23×C311yescharacteristic
246C2×C8C23×C311yescharacteristic
247C2×C8C23×C311yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4C3×D411yescharacteristic
255C4⋊C4C23×C311yescharacteristic
256C4⋊C4C23×C311yescharacteristic
257C4⋊C4C23×C311yescharacteristic
258C4⋊C4C23×C311yescharacteristic
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4C3×D411yescharacteristic
266C22⋊C4C3×D411yescharacteristic
267C42—21no—
268C42—21no—
269C42C23×C311yescharacteristic
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—22no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3—84no—
285C22×C3—84no—
286C22×C3AGL(1,ℤ/8)12yes—
287C22×C3C22×D411yescharacteristic
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—41no—
293C3×C4—41no—
294C3×C4—41no—
295C3×C4—41no—
296C3×C4—41no—
297C3×C4C2×#4111yescharacteristic
298C3×C4H(5,𝔽2)11yescharacteristic
299C23—21no—
300C23—21no—
301C23—21no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—81no—
307C23—81no—
308C23C2×C3×D411yescharacteristic
309C23C2×C3×D411yescharacteristic
310Q8—21no—
311Q8—21no—
312D4—22no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—21no—
318D4—42no—
319D4—42no—
320D4—42no—
321D4—42no—
322D4—42no—
323D4—81no—
324D4C3×#4112yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4C2×C3×D411yescharacteristic
337C2×C4C2×C3×D411yescharacteristic
338C2×C4C2×C3×D411yescharacteristic
339C2×C4C2×C3×D411yescharacteristic
340C2×C4C24×C311yescharacteristic
341C8—21no—
342C8—21no—
343C8—21no—
344C8—21no—
345C2×C3—22no—
346C2×C3—41no—
347C2×C3—42no—
348C2×C3—81no—
349C2×C3—81no—
350C2×C3#54611yescharacteristic
351C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
352C2×C3#57711yescharacteristic
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—84no—
368C22—84no—
369C22C3×AGL(1,ℤ/8)12yes—
370C22C22×C3×D411yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4C2×C3×#4111yescharacteristic
381C4C3×H(5,𝔽2)11yescharacteristic
382C3#331011yescharacteristic, direct product factor, Sylow
383C2—22no—
384C2—41no—
385C2—42no—
386C2—81no—
387C2—81no—
388C2C3×#54611yescharacteristic
389C2C2×C3×AGL(1,ℤ/8)11yescharacteristic
390C2C3×#57711yescharacteristic
subgroupG/H
center, socleC22×C3C22×D4
commutator, Frattini, cosocleC2×C4C24×C3
prime radicalC3×D42C2
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×#490C2semidirect
4. C3×#482C2semidirect
5. C3×#480C2semidirect
6. C3×#478C2semidirect
7. C3×#468C2semidirect
8. C3×#465C2semidirect
9. C3×#460C2semidirect
10. C3×#450C2semidirect
11. C3×#450C2semidirect
12. C3×#440C2semidirect
13. C3×#435C2semidirect
14. C3×#418C2semidirect
15. C2×C3×#102C2semidirect
16. #3310C3direct (N × H)
21. C2×C3×SD16C22semidirect
22. C2×C3×SD16C22semidirect
23. C2×C3×M16C22semidirect
24. C22×C3×C8C22semidirect
28. C3×#125C22semidirect
30. C3×C22⋊Q8C22semidirect
31. C3×C22⋊Q8C22semidirect
37. C3×C4×Q8C22semidirect
41. C3×#117C22semidirect
42. C2×C3×C4⋊C4C22semidirect
43. C3×#107C22semidirect
44. C3×#107C22semidirect
45. C3×#106C22semidirect
46. C3×C4⋊C8C22semidirect
47. C3×#103C22semidirect
51. C3×#102C22semidirect
52. C3×#102C22semidirect
53. C3×#102C22semidirect
54. C3×#102C22semidirect
55. C3×#102C22semidirect
56. C3×#102C22semidirect
57. C3×C22⋊C8C22semidirect
58. C3×C22⋊C8C22semidirect
59. C3×#97C22semidirect
60. #552C2×C3semidirect
62. #490C2×C3semidirect
63. #482C2×C3semidirect
64. #480C2×C3semidirect
65. #478C2×C3semidirect
66. #468C2×C3semidirect
67. #465C2×C3semidirect
68. #460C2×C3semidirect
69. #450C2×C3semidirect
70. #450C2×C3semidirect
71. #440C2×C3semidirect
72. #435C2×C3semidirect
73. #418C2×C3semidirect
74. C2×#102C2×C3semidirect
132. C2×SD16C22×C3semidirect
133. C2×SD16C22×C3semidirect
134. C2×M16C22×C3semidirect
135. C22×C8C22×C3semidirect
139. #125C22×C3semidirect
141. C22⋊Q8C22×C3semidirect
142. C22⋊Q8C22×C3semidirect
148. C4×Q8C22×C3semidirect
152. #117C22×C3semidirect
153. C2×C4⋊C4C22×C3semidirect
154. #107C22×C3semidirect
155. #107C22×C3semidirect
156. #106C22×C3semidirect
157. C4⋊C8C22×C3semidirect
158. #103C22×C3semidirect
162. #102C22×C3semidirect
163. #102C22×C3semidirect
164. #102C22×C3semidirect
165. #102C22×C3semidirect
166. #102C22×C3semidirect
167. #102C22×C3semidirect
168. C22⋊C8C22×C3semidirect
169. C22⋊C8C22×C3semidirect
170. #97C22×C3semidirect
382. C3#3310direct (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×D422
C3×#4902
C3×#4822
C3×#4802
C3×#4782
C3×#4682
C3×#4652
C3×#4602
C3×#4502
C3×#4402
C3×#4352
C3×#4182
C2×C3×#1022
#33103