Finite Group Databasebeta

← C3×#3198 #85062

Subgroups — C3×#3198 #85062

Order 384.

422 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#537C211yesmaximal, characteristic, complemented
2C3×#535C211yesmaximal, characteristic, complemented
3C3×#485C211yesmaximal, characteristic, complemented
4C3×#482C211yesmaximal, characteristic, complemented
5C3×#481C211yesmaximal, characteristic, complemented
6C3×#480C211yesmaximal, characteristic, complemented
7C3×#469C211yesmaximal, characteristic, complemented
8C3×#468C211yesmaximal, characteristic, complemented
9C3×#466C211yesmaximal, characteristic, complemented
10C3×#465C211yesmaximal, characteristic, complemented
11C3×#450C211yesmaximal, characteristic, complemented
12C3×#449C211yesmaximal, characteristic, complemented
13C3×#449C211yesmaximal, characteristic, complemented
14C3×#447C211yesmaximal, characteristic, complemented
15C3×#408C211yesmaximal, characteristic, complemented
16#3198C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×C3×#41C2211yescharacteristic, complemented
19C22×C3×D4C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic
21C2×C3×SD16C2211yescharacteristic
22C2×C3×D8C2211yescharacteristic
23C2×C3×D8C2211yescharacteristic
24C2×C3×M16C2211yescharacteristic, complemented
25C22×C3×C8C2211yescharacteristic, complemented
26C3×#127—21no—
27C3×#124—21no—
28C3×C22⋊Q8—21no—
29C3×C22⋊Q8C2211yescharacteristic
30C3×C22⋊Q8C2211yescharacteristic, complemented
31C3×#121—21no—
32C3×#121—21no—
33C3×#121—21no—
34C3×#121C2211yescharacteristic
35C3×#121C2211yescharacteristic
36C3×#121C2211yescharacteristic
37C3×#121C2211yescharacteristic
38C3×#121C2211yescharacteristic
39C3×#121C2211yescharacteristic, complemented
40C3×#120—21no—
41C3×#120—21no—
42C3×#117C2211yescharacteristic
43C2×C3×C4⋊C4C2211yescharacteristic, complemented
44C3×#107C2211yescharacteristic, complemented
45C3×#107C2211yescharacteristic, complemented
46C3×#106C2211yescharacteristic, complemented
47C3×#106C2211yescharacteristic, complemented
48C3×#103C2211yescharacteristic, complemented
49C3×#103C2211yescharacteristic, complemented
50C3×#102C2211yescharacteristic
51C3×#102C2211yescharacteristic
52C3×#102C2211yescharacteristic
53C3×#102C2211yescharacteristic
54C3×#102C2211yescharacteristic, complemented
55C3×#102C2211yescharacteristic, complemented
56C3×C22⋊C8C2211yescharacteristic
57C3×C22⋊C8C2211yescharacteristic
58C3×C22⋊C8C2211yescharacteristic, complemented
59C3×C22⋊C8C2211yescharacteristic, complemented
60#537C2×C311yescharacteristic, complemented
61#535C2×C311yescharacteristic, complemented
62#485C2×C311yescharacteristic, complemented
63#482C2×C311yescharacteristic, complemented
64#481C2×C311yescharacteristic, complemented
65#480C2×C311yescharacteristic, complemented
66#469C2×C311yescharacteristic, complemented
67#468C2×C311yescharacteristic, complemented
68#466C2×C311yescharacteristic, complemented
69#465C2×C311yescharacteristic, complemented
70#450C2×C311yescharacteristic, complemented
71#449C2×C311yescharacteristic, complemented
72#449C2×C311yescharacteristic, complemented
73#447C2×C311yescharacteristic, complemented
74#408C2×C311yescharacteristic, complemented
75C24×C3—21no—
76C3×#41—41no—
77C3×#41—41no—
78C3×#41—41no—
79C3×#41—41no—
80C2×C3×Q8D411yescharacteristic
81C2×C3×Q8C2311yescharacteristic
82C2×C3×D4—21no—
83C2×C3×D4—21no—
84C2×C3×D4—21no—
85C2×C3×D4—21no—
86C2×C3×D4—21no—
87C2×C3×D4—21no—
88C2×C3×D4—21no—
89C2×C3×D4—21no—
90C2×C3×D4—41no—
91C2×C3×D4—41no—
92C2×C3×D4D411yescharacteristic
93C2×C3×D4D411yescharacteristic
94C2×C3×D4D411yescharacteristic
95C2×C3×D4C2311yescharacteristic
96C2×C3×D4C2311yescharacteristic
97C2×C3×D4C2311yescharacteristic
98C22×C3×C4—21no—
99C22×C3×C4—21no—
100C22×C3×C4C2311yescharacteristic
101C22×C3×C4C2311yescharacteristic
102C22×C3×C4C2311yescharacteristic
103C3×SD16—41no—
104C3×SD16—41no—
105C3×D8—41no—
106C3×D8—41no—
107C3×M16—41no—
108C2×C3×C8—41no—
109C2×C3×C8C2311yescharacteristic
110C2×C3×C8C2311yescharacteristic
111C2×C3×C8C2311yescharacteristic
112C2×C3×C8C2311yescharacteristic
113C3×C4⋊C4—21no—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—21no—
116C3×C4⋊C4C2311yescharacteristic
117C3×C4⋊C4C2311yescharacteristic
118C3×C4⋊C4C2311yescharacteristic
119C3×C4⋊C4C2311yescharacteristic
120C3×C22⋊C4—21no—
121C3×C22⋊C4—21no—
122C3×C22⋊C4—21no—
123C3×C22⋊C4—21no—
124C3×C22⋊C4—21no—
125C3×C22⋊C4—21no—
126C3×C22⋊C4—21no—
127C3×C22⋊C4—21no—
128C3×C22⋊C4—21no—
129C3×C42—21no—
130C2×#41C22×C311yescharacteristic
131C2×#41C22×C311yescharacteristic, complemented
132C22×D4C22×C311yescharacteristic
133C2×SD16C22×C311yescharacteristic
134C2×SD16C22×C311yescharacteristic
135C2×D8C22×C311yescharacteristic
136C2×D8C22×C311yescharacteristic
137C2×M16C22×C311yescharacteristic, complemented
138C22×C8C22×C311yescharacteristic, complemented
139#127—21no—
140#124—21no—
141C22⋊Q8—21no—
142C22⋊Q8C22×C311yescharacteristic
143C22⋊Q8C22×C311yescharacteristic, complemented
144#121—21no—
145#121—21no—
146#121—21no—
147#121C22×C311yescharacteristic
148#121C22×C311yescharacteristic
149#121C22×C311yescharacteristic
150#121C22×C311yescharacteristic
151#121C22×C311yescharacteristic
152#121C22×C311yescharacteristic, complemented
153#120—21no—
154#120—21no—
155#117C22×C311yescharacteristic
156C2×C4⋊C4C22×C311yescharacteristic, complemented
157#107C22×C311yescharacteristic, complemented
158#107C22×C311yescharacteristic, complemented
159#106C22×C311yescharacteristic, complemented
160#106C22×C311yescharacteristic, complemented
161#103C22×C311yescharacteristic, complemented
162#103C22×C311yescharacteristic, complemented
163#102C22×C311yescharacteristic
164#102C22×C311yescharacteristic
165#102C22×C311yescharacteristic
166#102C22×C311yescharacteristic
167#102C22×C311yescharacteristic, complemented
168#102C22×C311yescharacteristic, complemented
169C22⋊C8C22×C311yescharacteristic
170C22⋊C8C22×C311yescharacteristic
171C22⋊C8C22×C311yescharacteristic, complemented
172C22⋊C8C22×C311yescharacteristic, complemented
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—42no—
177C23×C3—81no—
178C23×C3—81no—
179C23×C3C2×D411yescharacteristic
180C23×C3C2×D411yescharacteristic
181C23×C3C2×D411yescharacteristic
182C3×Q8—21no—
183C3×Q8—21no—
184C3×Q8—41no—
185C3×D4—21no—
186C3×D4—21no—
187C3×D4—21no—
188C3×D4—21no—
189C3×D4—21no—
190C3×D4—21no—
191C3×D4—41no—
192C3×D4—41no—
193C3×D4—41no—
194C3×D4—42no—
195C3×D4—42no—
196C3×D4—81no—
197C3×D4—81no—
198C3×D4—81no—
199C3×D4—81no—
200C3×D4—81no—
201C3×D4—81no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—42no—
214C2×C3×C4—81no—
215C2×C3×C4C2×D411yescharacteristic
216C2×C3×C4C2×D411yescharacteristic
217C2×C3×C4C2×D411yescharacteristic
218C2×C3×C4C2411yescharacteristic
219C3×C8—21no—
220C3×C8—21no—
221C3×C8—21no—
222C3×C8—21no—
223C24—21no—
224#41—41no—
225#41—41no—
226#41—41no—
227#41—41no—
228C2×Q8C3×D411yescharacteristic
229C2×Q8C23×C311yescharacteristic
230C2×D4—21no—
231C2×D4—21no—
232C2×D4—21no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4C3×D411yescharacteristic
241C2×D4C3×D411yescharacteristic
242C2×D4C3×D411yescharacteristic
243C2×D4C23×C311yescharacteristic
244C2×D4C23×C311yescharacteristic
245C2×D4C23×C311yescharacteristic
246C22×C4—21no—
247C22×C4—21no—
248C22×C4C23×C311yescharacteristic
249C22×C4C23×C311yescharacteristic
250C22×C4C23×C311yescharacteristic
251SD16—41no—
252SD16—41no—
253D8—41no—
254D8—41no—
255M16—41no—
256C2×C8—41no—
257C2×C8C23×C311yescharacteristic
258C2×C8C23×C311yescharacteristic
259C2×C8C23×C311yescharacteristic
260C2×C8C23×C311yescharacteristic
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4C23×C311yescharacteristic
265C4⋊C4C23×C311yescharacteristic
266C4⋊C4C23×C311yescharacteristic
267C4⋊C4C23×C311yescharacteristic
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C42—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—84no—
297C22×C3C22×D411yescharacteristic
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—41no—
302C3×C4—41no—
303C3×C4—41no—
304C3×C4—41no—
305C3×C4—41no—
306C3×C4H(5,𝔽2)11yescharacteristic
307C3×C4H(5,𝔽2)11yescharacteristic
308C23—21no—
309C23—21no—
310C23—21no—
311C23—42no—
312C23—81no—
313C23—81no—
314C23C2×C3×D411yescharacteristic
315C23C2×C3×D411yescharacteristic
316C23C2×C3×D411yescharacteristic
317Q8—21no—
318Q8—21no—
319Q8—41no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—21no—
325D4—21no—
326D4—41no—
327D4—41no—
328D4—41no—
329D4—42no—
330D4—42no—
331D4—81no—
332D4—81no—
333D4—81no—
334D4—81no—
335D4—81no—
336D4—81no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—42no—
349C2×C4—81no—
350C2×C4C2×C3×D411yescharacteristic
351C2×C4C2×C3×D411yescharacteristic
352C2×C4C2×C3×D411yescharacteristic
353C2×C4C24×C311yescharacteristic
354C8—21no—
355C8—21no—
356C8—21no—
357C8—21no—
358C2×C3—41no—
359C2×C3—41no—
360C2×C3—41no—
361C2×C3—81no—
362C2×C3—81no—
363C2×C3—81no—
364C2×C3#53411yescharacteristic
365C2×C3#57611yescharacteristic
366C2×C3#57711yescharacteristic
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—84no—
386C22C22×C3×D411yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4C3×H(5,𝔽2)11yescharacteristic
396C4C3×H(5,𝔽2)11yescharacteristic
397C3#319811yescharacteristic, direct product factor, Sylow
398C2—41no—
399C2—41no—
400C2—41no—
401C2—81no—
402C2—81no—
403C2—81no—
404C2C3×#53411yescharacteristic
405C2C3×#57611yescharacteristic
406C2C3×#57711yescharacteristic
subgroupG/H
center, socleC22×C3C22×D4
commutator, Frattini, cosocleC2×C4C24×C3
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×#41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#537C2semidirect
2. C3×#535C2semidirect
3. C3×#485C2semidirect
4. C3×#482C2semidirect
5. C3×#481C2semidirect
6. C3×#480C2semidirect
7. C3×#469C2semidirect
8. C3×#468C2semidirect
9. C3×#466C2semidirect
10. C3×#465C2semidirect
11. C3×#450C2semidirect
12. C3×#449C2semidirect
13. C3×#449C2semidirect
14. C3×#447C2semidirect
15. C3×#408C2semidirect
16. #3198C3direct (N × H)
18. C2×C3×#41C22semidirect
24. C2×C3×M16C22semidirect
25. C22×C3×C8C22semidirect
30. C3×C22⋊Q8C22semidirect
39. C3×#121C22semidirect
43. C2×C3×C4⋊C4C22semidirect
44. C3×#107C22semidirect
45. C3×#107C22semidirect
46. C3×#106C22semidirect
47. C3×#106C22semidirect
48. C3×#103C22semidirect
49. C3×#103C22semidirect
54. C3×#102C22semidirect
55. C3×#102C22semidirect
58. C3×C22⋊C8C22semidirect
59. C3×C22⋊C8C22semidirect
60. #537C2×C3semidirect
61. #535C2×C3semidirect
62. #485C2×C3semidirect
63. #482C2×C3semidirect
64. #481C2×C3semidirect
65. #480C2×C3semidirect
66. #469C2×C3semidirect
67. #468C2×C3semidirect
68. #466C2×C3semidirect
69. #465C2×C3semidirect
70. #450C2×C3semidirect
71. #449C2×C3semidirect
72. #449C2×C3semidirect
73. #447C2×C3semidirect
74. #408C2×C3semidirect
131. C2×#41C22×C3semidirect
137. C2×M16C22×C3semidirect
138. C22×C8C22×C3semidirect
143. C22⋊Q8C22×C3semidirect
152. #121C22×C3semidirect
156. C2×C4⋊C4C22×C3semidirect
157. #107C22×C3semidirect
158. #107C22×C3semidirect
159. #106C22×C3semidirect
160. #106C22×C3semidirect
161. #103C22×C3semidirect
162. #103C22×C3semidirect
167. #102C22×C3semidirect
168. #102C22×C3semidirect
171. C22⋊C8C22×C3semidirect
172. C22⋊C8C22×C3semidirect
397. C3#3198direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5372
C3×#5352
C3×#4852
C3×#4822
C3×#4812
C3×#4802
C3×#4692
C3×#4682
C3×#4662
C3×#4652
C3×#4502
C3×#4492
C3×#4472
C3×#4082
#31983