Finite Group Databasebeta

← C3×#3297 #85161

Subgroups — C3×#3297 #85161

Order 384.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#553C211yesmaximal, characteristic, complemented
2C3×#546C211yesmaximal, characteristic, complemented
3C3×#488C211yesmaximal, characteristic, complemented
4C3×#484C211yesmaximal, characteristic, complemented
5C3×#482C211yesmaximal, characteristic, complemented
6C3×#478C211yesmaximal, characteristic, complemented
7C3×#466C211yesmaximal, characteristic, complemented
8C3×#465C211yesmaximal, characteristic, complemented
9C3×#463C211yesmaximal, characteristic, complemented
10C3×#449C211yesmaximal, characteristic, complemented
11C3×#447C211yesmaximal, characteristic, complemented
12C3×C4×D8C211yesmaximal, characteristic, complemented
13C3×C8×D4C211yesmaximal, characteristic, complemented
14C3×#416C211yesmaximal, characteristic, complemented
15C2×C3×#102C211yesmaximal, characteristic, complemented
16#3297C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×#41C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C2×C3×SD16C2211yescharacteristic, complemented
20C2×C3×D8C2211yescharacteristic
21C2×C3×D8C2211yescharacteristic
22C22×C3×C8C2211yescharacteristic
23C22×C3×C8C2211yescharacteristic, complemented
24C3×#126—21no—
25C3×#125C2211yescharacteristic, complemented
26C3×#124C2211yescharacteristic
27C3×#123—21no—
28C3×#123—21no—
29C3×C22⋊Q8C2211yescharacteristic
30C3×#121—21no—
31C3×#121—21no—
32C3×#121C2211yescharacteristic
33C3×#121C2211yescharacteristic
34C3×#121C2211yescharacteristic, complemented
35C3×#120—21no—
36C3×C4×D4—21no—
37C3×C4×D4C2211yescharacteristic
38C3×C4×D4C2211yescharacteristic
39C3×C4×D4C2211yescharacteristic, complemented
40C3×#117C2211yescharacteristic, complemented
41C2×C3×C4⋊C4C2211yescharacteristic
42C2×C3×C22⋊C4—21no—
43C3×#107C2211yescharacteristic, complemented
44C3×#107C2211yescharacteristic, complemented
45C3×#106C2211yescharacteristic, complemented
46C3×C4⋊C8C2211yescharacteristic, complemented
47C3×#103C2211yescharacteristic, complemented
48C3×#103C2211yescharacteristic, complemented
49C3×#103C2211yescharacteristic, complemented
50C3×#102C2211yescharacteristic
51C3×#102C2211yescharacteristic
52C3×#102C2211yescharacteristic
53C3×#102C2211yescharacteristic
54C3×#102C2211yescharacteristic
55C3×#102C2211yescharacteristic
56C3×#102C2211yescharacteristic, complemented
57C3×C22⋊C8C2211yescharacteristic
58C3×C22⋊C8C2211yescharacteristic, complemented
59C3×C4×C8C2211yescharacteristic, complemented
60#553C2×C311yescharacteristic, complemented
61#546C2×C311yescharacteristic, complemented
62#488C2×C311yescharacteristic, complemented
63#484C2×C311yescharacteristic, complemented
64#482C2×C311yescharacteristic, complemented
65#478C2×C311yescharacteristic, complemented
66#466C2×C311yescharacteristic, complemented
67#465C2×C311yescharacteristic, complemented
68#463C2×C311yescharacteristic, complemented
69#449C2×C311yescharacteristic, complemented
70#447C2×C311yescharacteristic, complemented
71C4×D8C2×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×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—41no—
86C2×C3×D4D411yescharacteristic
87C2×C3×D4C2311yescharacteristic
88C2×C3×D4C2311yescharacteristic
89C2×C3×D4C2311yescharacteristic
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4C2311yescharacteristic
95C22×C3×C4C2311yescharacteristic
96C3×SD16—41no—
97C3×D8—22no—
98C3×D8—41no—
99C2×C3×C8—41no—
100C2×C3×C8—41no—
101C2×C3×C8C2311yescharacteristic
102C2×C3×C8C2311yescharacteristic
103C2×C3×C8C2311yescharacteristic
104C2×C3×C8C2311yescharacteristic
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⋊C4—21no—
123C3×C22⋊C4D411yescharacteristic
124C3×C22⋊C4D411yescharacteristic
125C3×C42—21no—
126C3×C42C2311yescharacteristic
127C2×#41C22×C311yescharacteristic
128C22×D4C22×C311yescharacteristic
129C2×SD16C22×C311yescharacteristic, complemented
130C2×D8C22×C311yescharacteristic
131C2×D8C22×C311yescharacteristic
132C22×C8C22×C311yescharacteristic
133C22×C8C22×C311yescharacteristic, complemented
134#126—21no—
135#125C22×C311yescharacteristic, complemented
136#124C22×C311yescharacteristic
137#123—21no—
138#123—21no—
139C22⋊Q8C22×C311yescharacteristic
140#121—21no—
141#121—21no—
142#121C22×C311yescharacteristic
143#121C22×C311yescharacteristic
144#121C22×C311yescharacteristic, complemented
145#120—21no—
146C4×D4—21no—
147C4×D4C22×C311yescharacteristic
148C4×D4C22×C311yescharacteristic
149C4×D4C22×C311yescharacteristic, complemented
150#117C22×C311yescharacteristic, complemented
151C2×C4⋊C4C22×C311yescharacteristic
152C2×C22⋊C4—21no—
153#107C22×C311yescharacteristic, complemented
154#107C22×C311yescharacteristic, complemented
155#106C22×C311yescharacteristic, complemented
156C4⋊C8C22×C311yescharacteristic, complemented
157#103C22×C311yescharacteristic, complemented
158#103C22×C311yescharacteristic, complemented
159#103C22×C311yescharacteristic, complemented
160#102C22×C311yescharacteristic
161#102C22×C311yescharacteristic
162#102C22×C311yescharacteristic
163#102C22×C311yescharacteristic
164#102C22×C311yescharacteristic
165#102C22×C311yescharacteristic
166#102C22×C311yescharacteristic, complemented
167C22⋊C8C22×C311yescharacteristic
168C22⋊C8C22×C311yescharacteristic, complemented
169C4×C8C22×C311yescharacteristic, complemented
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—42no—
174C23×C3—42no—
175C23×C3—81no—
176C23×C3C2×D411yescharacteristic
177C23×C3C2×D411yescharacteristic
178C3×Q8—21no—
179C3×Q8—21no—
180C3×D4—22no—
181C3×D4—21no—
182C3×D4—21no—
183C3×D4—21no—
184C3×D4—21no—
185C3×D4—21no—
186C3×D4—42no—
187C3×D4—42no—
188C3×D4—42no—
189C3×D4—81no—
190C3×D4—81no—
191C3×D4#4112yes—
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—21no—
200C2×C3×C4—21no—
201C2×C3×C4—42no—
202C2×C3×C4—81no—
203C2×C3×C4—81no—
204C2×C3×C4—81no—
205C2×C3×C4C2×D411yescharacteristic
206C2×C3×C4C2×D411yescharacteristic
207C2×C3×C4C2×D411yescharacteristic
208C2×C3×C4C2×D411yescharacteristic
209C2×C3×C4C2411yescharacteristic
210C3×C8—21no—
211C3×C8—21no—
212C3×C8—21no—
213C3×C8—21no—
214C24—21no—
215#41—22no—
216#41—41no—
217C2×Q8C23×C311yescharacteristic
218C2×D4—22no—
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—21no—
222C2×D4—21no—
223C2×D4—21no—
224C2×D4—41no—
225C2×D4C3×D411yescharacteristic
226C2×D4C23×C311yescharacteristic
227C2×D4C23×C311yescharacteristic
228C2×D4C23×C311yescharacteristic
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4C23×C311yescharacteristic
234C22×C4C23×C311yescharacteristic
235SD16—41no—
236D8—22no—
237D8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8C23×C311yescharacteristic
241C2×C8C23×C311yescharacteristic
242C2×C8C23×C311yescharacteristic
243C2×C8C23×C311yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4C3×D411yescharacteristic
249C4⋊C4C23×C311yescharacteristic
250C4⋊C4C23×C311yescharacteristic
251C4⋊C4C23×C311yescharacteristic
252C4⋊C4C23×C311yescharacteristic
253C22⋊C4—21no—
254C22⋊C4—21no—
255C22⋊C4—21no—
256C22⋊C4—21no—
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4C3×D411yescharacteristic
263C22⋊C4C3×D411yescharacteristic
264C42—21no—
265C42C23×C311yescharacteristic
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—22no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—84no—
281C22×C3#13512yes—
282C22×C3C22×D411yescharacteristic
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—41no—
288C3×C4—41no—
289C3×C4—41no—
290C3×C4—41no—
291C3×C4—41no—
292C3×C4C2×#4111yescharacteristic
293C3×C4H(5,𝔽2)11yescharacteristic
294C23—21no—
295C23—21no—
296C23—21no—
297C23—42no—
298C23—42no—
299C23—81no—
300C23C2×C3×D411yescharacteristic
301C23C2×C3×D411yescharacteristic
302Q8—21no—
303Q8—21no—
304D4—22no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—21no—
309D4—21no—
310D4—42no—
311D4—42no—
312D4—42no—
313D4—81no—
314D4—81no—
315D4C3×#4112yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—42no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4C2×C3×D411yescharacteristic
330C2×C4C2×C3×D411yescharacteristic
331C2×C4C2×C3×D411yescharacteristic
332C2×C4C2×C3×D411yescharacteristic
333C2×C4C24×C311yescharacteristic
334C8—21no—
335C8—21no—
336C8—21no—
337C8—21no—
338C2×C3—22no—
339C2×C3—41no—
340C2×C3—42no—
341C2×C3—81no—
342C2×C3—81no—
343C2×C3#54611yescharacteristic
344C2×C3C2×#13511yescharacteristic
345C2×C3#57611yescharacteristic
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—22no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—84no—
361C22C3×#13512yes—
362C22C22×C3×D411yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4C2×C3×#4111yescharacteristic
373C4C3×H(5,𝔽2)11yescharacteristic
374C3#329711yescharacteristic, direct product factor, Sylow
375C2—22no—
376C2—41no—
377C2—42no—
378C2—81no—
379C2—81no—
380C2C3×#54611yescharacteristic
381C2C2×C3×#13511yescharacteristic
382C2C3×#57611yescharacteristic
subgroupG/H
center, socleC22×C3C22×D4
commutator, Frattini, cosocleC2×C4C24×C3
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×#553C2semidirect
2. C3×#546C2semidirect
3. C3×#488C2semidirect
4. C3×#484C2semidirect
5. C3×#482C2semidirect
6. C3×#478C2semidirect
7. C3×#466C2semidirect
8. C3×#465C2semidirect
9. C3×#463C2semidirect
10. C3×#449C2semidirect
11. C3×#447C2semidirect
12. C3×C4×D8C2semidirect
13. C3×C8×D4C2semidirect
14. C3×#416C2semidirect
15. C2×C3×#102C2semidirect
16. #3297C3direct (N × H)
19. C2×C3×SD16C22semidirect
23. C22×C3×C8C22semidirect
25. C3×#125C22semidirect
34. C3×#121C22semidirect
39. C3×C4×D4C22semidirect
40. C3×#117C22semidirect
43. C3×#107C22semidirect
44. C3×#107C22semidirect
45. C3×#106C22semidirect
46. C3×C4⋊C8C22semidirect
47. C3×#103C22semidirect
48. C3×#103C22semidirect
49. C3×#103C22semidirect
56. C3×#102C22semidirect
58. C3×C22⋊C8C22semidirect
59. C3×C4×C8C22semidirect
60. #553C2×C3semidirect
61. #546C2×C3semidirect
62. #488C2×C3semidirect
63. #484C2×C3semidirect
64. #482C2×C3semidirect
65. #478C2×C3semidirect
66. #466C2×C3semidirect
67. #465C2×C3semidirect
68. #463C2×C3semidirect
69. #449C2×C3semidirect
70. #447C2×C3semidirect
71. C4×D8C2×C3semidirect
72. C8×D4C2×C3semidirect
73. #416C2×C3semidirect
74. C2×#102C2×C3semidirect
129. C2×SD16C22×C3semidirect
133. C22×C8C22×C3semidirect
135. #125C22×C3semidirect
144. #121C22×C3semidirect
149. C4×D4C22×C3semidirect
150. #117C22×C3semidirect
153. #107C22×C3semidirect
154. #107C22×C3semidirect
155. #106C22×C3semidirect
156. C4⋊C8C22×C3semidirect
157. #103C22×C3semidirect
158. #103C22×C3semidirect
159. #103C22×C3semidirect
166. #102C22×C3semidirect
168. C22⋊C8C22×C3semidirect
169. C4×C8C22×C3semidirect
374. C3#3297direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5532
C3×#5462
C3×#4882
C3×#4842
C3×#4822
C3×#4782
C3×#4662
C3×#4652
C3×#4632
C3×#4492
C3×#4472
C3×C4×D82
C3×C8×D42
C3×#4162
C2×C3×#1022
#32973