Finite Group Databasebeta

← C3×#3306 #85170

Subgroups — C3×#3306 #85170

Order 384.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#547C211yesmaximal, characteristic, complemented
2C3×#546C211yesmaximal, characteristic, complemented
3C3×#489C211yesmaximal, characteristic, complemented
4C3×#483C211yesmaximal, characteristic, complemented
5C3×#481C211yesmaximal, characteristic, complemented
6C3×#476C211yesmaximal, characteristic, complemented
7C3×#469C211yesmaximal, characteristic, complemented
8C3×#465C211yesmaximal, characteristic, complemented
9C3×#463C211yesmaximal, characteristic, complemented
10C3×#449C211yesmaximal, characteristic, complemented
11C3×#447C211yesmaximal, characteristic, complemented
12C3×#442C211yesmaximal, characteristic, complemented
13C3×#435C211yesmaximal, characteristic, complemented
14C3×#417C211yesmaximal, characteristic, complemented
15C2×C3×#102C211yesmaximal, characteristic, complemented
16#3306C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×#41—21no—
18C2×C3×#41C2211yescharacteristic
19C22×C3×D4C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic, complemented
21C2×C3×D8C2211yescharacteristic
22C2×C3×D8C2211yescharacteristic
23C2×C3×M16C2211yescharacteristic, complemented
24C22×C3×C8C2211yescharacteristic
25C3×C4⋊Q8C2211yescharacteristic, complemented
26C3×#124C2211yescharacteristic
27C3×#123—21no—
28C3×#123—21no—
29C3×#123—21no—
30C3×C22⋊Q8—21no—
31C3×C22⋊Q8C2211yescharacteristic
32C3×#121—21no—
33C3×#121C2211yescharacteristic
34C3×#121C2211yescharacteristic
35C3×#121C2211yescharacteristic, complemented
36C3×#120—21no—
37C3×C4×D4C2211yescharacteristic
38C3×C4×D4C2211yescharacteristic
39C3×C4×D4C2211yescharacteristic, complemented
40C2×C3×C4⋊C4C2211yescharacteristic
41C2×C3×C4⋊C4C2211yescharacteristic, complemented
42C2×C3×C22⋊C4—21no—
43C3×#106C2211yescharacteristic, complemented
44C3×#106C2211yescharacteristic, 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×#97C2211yescharacteristic, complemented
60#547C2×C311yescharacteristic, complemented
61#546C2×C311yescharacteristic, complemented
62#489C2×C311yescharacteristic, complemented
63#483C2×C311yescharacteristic, complemented
64#481C2×C311yescharacteristic, complemented
65#476C2×C311yescharacteristic, complemented
66#469C2×C311yescharacteristic, complemented
67#465C2×C311yescharacteristic, complemented
68#463C2×C311yescharacteristic, complemented
69#449C2×C311yescharacteristic, complemented
70#447C2×C311yescharacteristic, complemented
71#442C2×C311yescharacteristic, complemented
72#435C2×C311yescharacteristic, complemented
73#417C2×C311yescharacteristic, complemented
74C2×#102C2×C311yescharacteristic, complemented
75C24×C3—21no—
76C3×#41—22no—
77C3×#41—41no—
78C3×#41—44no—
79C2×C3×Q8—21no—
80C2×C3×Q8C2311yescharacteristic
81C2×C3×D4—22no—
82C2×C3×D4—21no—
83C2×C3×D4—21no—
84C2×C3×D4—21no—
85C2×C3×D4—21no—
86C2×C3×D4—21no—
87C2×C3×D4—41no—
88C2×C3×D4D411yescharacteristic
89C2×C3×D4C2311yescharacteristic
90C2×C3×D4C2311yescharacteristic
91C2×C3×D4C2311yescharacteristic
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4C2311yescharacteristic
98C22×C3×C4C2311yescharacteristic
99C3×SD16—41no—
100C3×D8—22no—
101C3×D8—41no—
102C3×M16—41no—
103C2×C3×C8—41no—
104C2×C3×C8C2311yescharacteristic
105C2×C3×C8C2311yescharacteristic
106C2×C3×C8C2311yescharacteristic
107C2×C3×C8C2311yescharacteristic
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⋊C4—21no—
124C3×C22⋊C4—21no—
125C3×C22⋊C4D411yescharacteristic
126C3×C22⋊C4D411yescharacteristic
127C3×C42C2311yescharacteristic
128C2×#41—21no—
129C2×#41C22×C311yescharacteristic
130C22×D4C22×C311yescharacteristic
131C2×SD16C22×C311yescharacteristic, complemented
132C2×D8C22×C311yescharacteristic
133C2×D8C22×C311yescharacteristic
134C2×M16C22×C311yescharacteristic, complemented
135C22×C8C22×C311yescharacteristic
136C4⋊Q8C22×C311yescharacteristic, complemented
137#124C22×C311yescharacteristic
138#123—21no—
139#123—21no—
140#123—21no—
141C22⋊Q8—21no—
142C22⋊Q8C22×C311yescharacteristic
143#121—21no—
144#121C22×C311yescharacteristic
145#121C22×C311yescharacteristic
146#121C22×C311yescharacteristic, complemented
147#120—21no—
148C4×D4C22×C311yescharacteristic
149C4×D4C22×C311yescharacteristic
150C4×D4C22×C311yescharacteristic, complemented
151C2×C4⋊C4C22×C311yescharacteristic
152C2×C4⋊C4C22×C311yescharacteristic, complemented
153C2×C22⋊C4—21no—
154#106C22×C311yescharacteristic, complemented
155#106C22×C311yescharacteristic, complemented
156#106C22×C311yescharacteristic, complemented
157C4⋊C8C22×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
167#102C22×C311yescharacteristic, complemented
168C22⋊C8C22×C311yescharacteristic
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—81no—
177C23×C3C2×D411yescharacteristic
178C23×C3C2×D411yescharacteristic
179C3×Q8—21no—
180C3×Q8—21no—
181C3×Q8—42no—
182C3×D4—22no—
183C3×D4—21no—
184C3×D4—21no—
185C3×D4—21no—
186C3×D4—21no—
187C3×D4—21no—
188C3×D4—42no—
189C3×D4—42no—
190C3×D4—42no—
191C3×D4—42no—
192C3×D4—81no—
193C3×D4#4112yescomplemented
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—21no—
202C2×C3×C4—21no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4—81no—
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—
218#41—22no—
219#41—41no—
220#41—44no—
221C2×Q8—21no—
222C2×Q8C23×C311yescharacteristic
223C2×D4—22no—
224C2×D4—21no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—21no—
229C2×D4—41no—
230C2×D4C3×D411yescharacteristic
231C2×D4C23×C311yescharacteristic
232C2×D4C23×C311yescharacteristic
233C2×D4C23×C311yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4C23×C311yescharacteristic
240C22×C4C23×C311yescharacteristic
241SD16—41no—
242D8—22no—
243D8—41no—
244M16—41no—
245C2×C8—41no—
246C2×C8C23×C311yescharacteristic
247C2×C8C23×C311yescharacteristic
248C2×C8C23×C311yescharacteristic
249C2×C8C23×C311yescharacteristic
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⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4C3×D411yescharacteristic
268C22⋊C4C3×D411yescharacteristic
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×C3AGL(1,ℤ/8)12yes—
286C22×C3C22×D411yescharacteristic
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—41no—
292C3×C4—41no—
293C3×C4—41no—
294C3×C4—41no—
295C3×C4—41no—
296C3×C4C2×#4111yescharacteristic
297C3×C4H(5,𝔽2)11yescharacteristic
298C23—21no—
299C23—21no—
300C23—21no—
301C23—42no—
302C23—42no—
303C23—81no—
304C23C2×C3×D411yescharacteristic
305C23C2×C3×D411yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—42no—
309D4—22no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—21no—
315D4—42no—
316D4—42no—
317D4—42no—
318D4—42no—
319D4—81no—
320D4C3×#4112yescomplemented
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—81no—
335C2×C4C2×C3×D411yescharacteristic
336C2×C4C2×C3×D411yescharacteristic
337C2×C4C2×C3×D411yescharacteristic
338C2×C4C2×C3×D411yescharacteristic
339C2×C4C24×C311yescharacteristic
340C8—21no—
341C8—21no—
342C8—21no—
343C8—21no—
344C2×C3—22no—
345C2×C3—41no—
346C2×C3—42no—
347C2×C3—81no—
348C2×C3—81no—
349C2×C3#54611yescharacteristic
350C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
351C2×C3#57711yescharacteristic
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—84no—
367C22C3×AGL(1,ℤ/8)12yes—
368C22C22×C3×D411yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4C2×C3×#4111yescharacteristic
379C4C3×H(5,𝔽2)11yescharacteristic
380C3#330611yescharacteristic, direct product factor, Sylow
381C2—22no—
382C2—41no—
383C2—42no—
384C2—81no—
385C2—81no—
386C2C3×#54611yescharacteristic
387C2C2×C3×AGL(1,ℤ/8)11yescharacteristic
388C2C3×#57711yescharacteristic
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×#547C2semidirect
2. C3×#546C2semidirect
3. C3×#489C2semidirect
4. C3×#483C2semidirect
5. C3×#481C2semidirect
6. C3×#476C2semidirect
7. C3×#469C2semidirect
8. C3×#465C2semidirect
9. C3×#463C2semidirect
10. C3×#449C2semidirect
11. C3×#447C2semidirect
12. C3×#442C2semidirect
13. C3×#435C2semidirect
14. C3×#417C2semidirect
15. C2×C3×#102C2semidirect
16. #3306C3direct (N × H)
20. C2×C3×SD16C22semidirect
23. C2×C3×M16C22semidirect
25. C3×C4⋊Q8C22semidirect
35. C3×#121C22semidirect
39. C3×C4×D4C22semidirect
41. C2×C3×C4⋊C4C22semidirect
43. C3×#106C22semidirect
44. C3×#106C22semidirect
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×#97C22semidirect
60. #547C2×C3semidirect
61. #546C2×C3semidirect
62. #489C2×C3semidirect
63. #483C2×C3semidirect
64. #481C2×C3semidirect
65. #476C2×C3semidirect
66. #469C2×C3semidirect
67. #465C2×C3semidirect
68. #463C2×C3semidirect
69. #449C2×C3semidirect
70. #447C2×C3semidirect
71. #442C2×C3semidirect
72. #435C2×C3semidirect
73. #417C2×C3semidirect
74. C2×#102C2×C3semidirect
131. C2×SD16C22×C3semidirect
134. C2×M16C22×C3semidirect
136. C4⋊Q8C22×C3semidirect
146. #121C22×C3semidirect
150. C4×D4C22×C3semidirect
152. C2×C4⋊C4C22×C3semidirect
154. #106C22×C3semidirect
155. #106C22×C3semidirect
156. #106C22×C3semidirect
157. C4⋊C8C22×C3semidirect
158. #103C22×C3semidirect
159. #103C22×C3semidirect
160. #103C22×C3semidirect
167. #102C22×C3semidirect
169. C22⋊C8C22×C3semidirect
170. #97C22×C3semidirect
193. C3×D4#41semidirect
320. D4C3×#41semidirect
380. C3#3306direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5472
C3×#5462
C3×#4892
C3×#4832
C3×#4812
C3×#4762
C3×#4692
C3×#4652
C3×#4632
C3×#4492
C3×#4472
C3×#4422
C3×#4352
C3×#4172
C2×C3×#1022
#33063