Finite Group Databasebeta

← C3×#3224 #85088

Subgroups — C3×#3224 #85088

Order 384.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#542C211yesmaximal, characteristic, complemented
2C3×#535C211yesmaximal, characteristic, complemented
3C3×#496C211yesmaximal, characteristic, complemented
4C3×#495C211yesmaximal, characteristic, complemented
5C3×#489C211yesmaximal, characteristic, complemented
6C3×#486C211yesmaximal, characteristic, complemented
7C3×#464C211yesmaximal, characteristic, complemented
8C3×#463C211yesmaximal, characteristic, complemented
9C3×#460C211yesmaximal, characteristic, complemented
10C3×#459C211yesmaximal, characteristic, complemented
11C3×#452C211yesmaximal, characteristic, complemented
12C3×#450C211yesmaximal, characteristic, complemented
13C3×#449C211yesmaximal, characteristic, complemented
14C3×#447C211yesmaximal, characteristic, complemented
15C3×#433C211yesmaximal, characteristic, complemented
16#3224C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×#41C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C2×C3×Q16C2211yescharacteristic, complemented
20C2×C3×SD16C2211yescharacteristic
21C2×C3×SD16C2211yescharacteristic
22C2×C3×SD16C2211yescharacteristic, complemented
23C2×C3×SD16C2211yescharacteristic, complemented
24C2×C3×D8C2211yescharacteristic
25C2×C3×D8C2211yescharacteristic
26C2×C3×D8C2211yescharacteristic, complemented
27C3×C4⋊Q8C2211yescharacteristic, complemented
28C3×#127C2211yescharacteristic
29C3×#127C2211yescharacteristic
30C3×#126—21no—
31C3×#124—21no—
32C3×#124C2211yescharacteristic
33C3×#124C2211yescharacteristic
34C3×#124C2211yescharacteristic, complemented
35C3×#123—21no—
36C3×C22⋊Q8—21no—
37C3×C22⋊Q8C2211yescharacteristic
38C3×#121—21no—
39C3×#121—21no—
40C3×#121C2211yescharacteristic
41C3×#120—21no—
42C3×#120—21no—
43C3×C4×Q8C2211yescharacteristic, complemented
44C3×C4×D4C2211yescharacteristic, complemented
45C3×#117C2211yescharacteristic
46C3×C4⋊C8C2211yescharacteristic, complemented
47C3×C4⋊C8C2211yescharacteristic, 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×#97C2211yescharacteristic, complemented
59C3×C4×C8C2211yescharacteristic, complemented
60#542C2×C311yescharacteristic, complemented
61#535C2×C311yescharacteristic, complemented
62#496C2×C311yescharacteristic, complemented
63#495C2×C311yescharacteristic, complemented
64#489C2×C311yescharacteristic, complemented
65#486C2×C311yescharacteristic, complemented
66#464C2×C311yescharacteristic, complemented
67#463C2×C311yescharacteristic, complemented
68#460C2×C311yescharacteristic, complemented
69#459C2×C311yescharacteristic, complemented
70#452C2×C311yescharacteristic, complemented
71#450C2×C311yescharacteristic, complemented
72#449C2×C311yescharacteristic, complemented
73#447C2×C311yescharacteristic, complemented
74#433C2×C311yescharacteristic, complemented
75C24×C3—21no—
76C3×#41—41no—
77C3×#41—41no—
78C2×C3×Q8—21no—
79C2×C3×Q8C2311yescharacteristic
80C2×C3×Q8C2311yescharacteristic
81C2×C3×D4—21no—
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—41no—
89C2×C3×D4—41no—
90C2×C3×D4C2311yescharacteristic
91C2×C3×D4C2311yescharacteristic
92C2×C3×D4C2311yescharacteristic
93C2×C3×D4C2311yescharacteristic
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4C2311yescharacteristic
97C3×Q16—41no—
98C3×SD16—41no—
99C3×SD16—41no—
100C3×SD16—41no—
101C3×SD16—41no—
102C3×D8—41no—
103C3×D8—41no—
104C3×D8—41no—
105C2×C3×C8C2311yescharacteristic
106C2×C3×C8C2311yescharacteristic
107C2×C3×C8C2311yescharacteristic
108C2×C3×C8C2311yescharacteristic
109C3×C4⋊C4—21no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4—21no—
112C3×C4⋊C4D411yescharacteristic
113C3×C4⋊C4D411yescharacteristic
114C3×C4⋊C4C2311yescharacteristic
115C3×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
124C3×C22⋊C4—21no—
125C3×C22⋊C4D411yescharacteristic
126C3×C22⋊C4D411yescharacteristic
127C3×C42—21no—
128C3×C42C2311yescharacteristic
129C3×C42C2311yescharacteristic
130C2×#41C22×C311yescharacteristic
131C22×D4C22×C311yescharacteristic
132C2×Q16C22×C311yescharacteristic, complemented
133C2×SD16C22×C311yescharacteristic
134C2×SD16C22×C311yescharacteristic
135C2×SD16C22×C311yescharacteristic, complemented
136C2×SD16C22×C311yescharacteristic, complemented
137C2×D8C22×C311yescharacteristic
138C2×D8C22×C311yescharacteristic
139C2×D8C22×C311yescharacteristic, complemented
140C4⋊Q8C22×C311yescharacteristic, complemented
141#127C22×C311yescharacteristic
142#127C22×C311yescharacteristic
143#126—21no—
144#124—21no—
145#124C22×C311yescharacteristic
146#124C22×C311yescharacteristic
147#124C22×C311yescharacteristic, complemented
148#123—21no—
149C22⋊Q8—21no—
150C22⋊Q8C22×C311yescharacteristic
151#121—21no—
152#121—21no—
153#121C22×C311yescharacteristic
154#120—21no—
155#120—21no—
156C4×Q8C22×C311yescharacteristic, complemented
157C4×D4C22×C311yescharacteristic, complemented
158#117C22×C311yescharacteristic
159C4⋊C8C22×C311yescharacteristic, complemented
160C4⋊C8C22×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
171#97C22×C311yescharacteristic, complemented
172C4×C8C22×C311yescharacteristic, complemented
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—21no—
177C23×C3—42no—
178C23×C3—81no—
179C23×C3—81no—
180C23×C3C2×D411yescharacteristic
181C3×Q8—21no—
182C3×Q8—21no—
183C3×Q8—21no—
184C3×Q8—21no—
185C3×Q8—81no—
186C3×D4—21no—
187C3×D4—21no—
188C3×D4—21no—
189C3×D4—21no—
190C3×D4—21no—
191C3×D4—21no—
192C3×D4—21no—
193C3×D4—21no—
194C3×D4—42no—
195C3×D4—42no—
196C3×D4—81no—
197C3×D4—81no—
198C3×D4—81no—
199C3×D4—81no—
200C3×D4—81no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—42no—
208C2×C3×C4—81no—
209C2×C3×C4C2×D411yescharacteristic
210C2×C3×C4C2×D411yescharacteristic
211C2×C3×C4C2×D411yescharacteristic
212C2×C3×C4C2×D411yescharacteristic
213C2×C3×C4C2×D411yescharacteristic
214C2×C3×C4C2411yescharacteristic
215C3×C8—21no—
216C3×C8—21no—
217C3×C8—21no—
218C3×C8—21no—
219C24—21no—
220#41—41no—
221#41—41no—
222C2×Q8—21no—
223C2×Q8C23×C311yescharacteristic
224C2×Q8C23×C311yescharacteristic
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4—21no—
231C2×D4—21no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4C23×C311yescharacteristic
235C2×D4C23×C311yescharacteristic
236C2×D4C23×C311yescharacteristic
237C2×D4C23×C311yescharacteristic
238C22×C4—21no—
239C22×C4—21no—
240C22×C4C23×C311yescharacteristic
241Q16—41no—
242SD16—41no—
243SD16—41no—
244SD16—41no—
245SD16—41no—
246D8—41no—
247D8—41no—
248D8—41no—
249C2×C8C23×C311yescharacteristic
250C2×C8C23×C311yescharacteristic
251C2×C8C23×C311yescharacteristic
252C2×C8C23×C311yescharacteristic
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4C3×D411yescharacteristic
257C4⋊C4C3×D411yescharacteristic
258C4⋊C4C23×C311yescharacteristic
259C4⋊C4C23×C311yescharacteristic
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4C3×D411yescharacteristic
270C22⋊C4C3×D411yescharacteristic
271C42—21no—
272C42C23×C311yescharacteristic
273C42C23×C311yescharacteristic
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3—41no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—84no—
290C22×C3C22×D411yescharacteristic
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—41no—
297C3×C4—41no—
298C3×C4—41no—
299C3×C4—41no—
300C3×C4H(5,𝔽2)11yescharacteristic
301C3×C4H(5,𝔽2)11yescharacteristic
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—42no—
307C23—81no—
308C23—81no—
309C23C2×C3×D411yescharacteristic
310Q8—21no—
311Q8—21no—
312Q8—21no—
313Q8—21no—
314Q8—81no—
315D4—21no—
316D4—21no—
317D4—21no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—42no—
325D4—81no—
326D4—81no—
327D4—81no—
328D4—81no—
329D4—81no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—81no—
338C2×C4C2×C3×D411yescharacteristic
339C2×C4C2×C3×D411yescharacteristic
340C2×C4C2×C3×D411yescharacteristic
341C2×C4C2×C3×D411yescharacteristic
342C2×C4C2×C3×D411yescharacteristic
343C2×C4C24×C311yescharacteristic
344C8—21no—
345C8—21no—
346C8—21no—
347C8—21no—
348C2×C3—41no—
349C2×C3—81no—
350C2×C3—81no—
351C2×C3—81no—
352C2×C3—81no—
353C2×C3#53511yescharacteristic
354C2×C3#57611yescharacteristic
355C2×C3#57711yescharacteristic
356C22—21no—
357C22—21no—
358C22—21no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—84no—
372C22C22×C3×D411yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4C3×H(5,𝔽2)11yescharacteristic
383C4C3×H(5,𝔽2)11yescharacteristic
384C3#322411yescharacteristic, direct product factor, Sylow
385C2—41no—
386C2—81no—
387C2—81no—
388C2—81no—
389C2—81no—
390C2C3×#53511yescharacteristic
391C2C3×#57611yescharacteristic
392C2C3×#57711yescharacteristic
subgroupG/H
center, socleC22×C3C22×D4
commutator, Frattini, cosocleC2×C4C24×C3
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#542C2semidirect
2. C3×#535C2semidirect
3. C3×#496C2semidirect
4. C3×#495C2semidirect
5. C3×#489C2semidirect
6. C3×#486C2semidirect
7. C3×#464C2semidirect
8. C3×#463C2semidirect
9. C3×#460C2semidirect
10. C3×#459C2semidirect
11. C3×#452C2semidirect
12. C3×#450C2semidirect
13. C3×#449C2semidirect
14. C3×#447C2semidirect
15. C3×#433C2semidirect
16. #3224C3direct (N × H)
19. C2×C3×Q16C22semidirect
22. C2×C3×SD16C22semidirect
23. C2×C3×SD16C22semidirect
26. C2×C3×D8C22semidirect
27. C3×C4⋊Q8C22semidirect
34. C3×#124C22semidirect
43. C3×C4×Q8C22semidirect
44. C3×C4×D4C22semidirect
46. C3×C4⋊C8C22semidirect
47. C3×C4⋊C8C22semidirect
48. C3×#103C22semidirect
49. C3×#103C22semidirect
54. C3×#102C22semidirect
55. C3×#102C22semidirect
58. C3×#97C22semidirect
59. C3×C4×C8C22semidirect
60. #542C2×C3semidirect
61. #535C2×C3semidirect
62. #496C2×C3semidirect
63. #495C2×C3semidirect
64. #489C2×C3semidirect
65. #486C2×C3semidirect
66. #464C2×C3semidirect
67. #463C2×C3semidirect
68. #460C2×C3semidirect
69. #459C2×C3semidirect
70. #452C2×C3semidirect
71. #450C2×C3semidirect
72. #449C2×C3semidirect
73. #447C2×C3semidirect
74. #433C2×C3semidirect
132. C2×Q16C22×C3semidirect
135. C2×SD16C22×C3semidirect
136. C2×SD16C22×C3semidirect
139. C2×D8C22×C3semidirect
140. C4⋊Q8C22×C3semidirect
147. #124C22×C3semidirect
156. C4×Q8C22×C3semidirect
157. C4×D4C22×C3semidirect
159. C4⋊C8C22×C3semidirect
160. C4⋊C8C22×C3semidirect
161. #103C22×C3semidirect
162. #103C22×C3semidirect
167. #102C22×C3semidirect
168. #102C22×C3semidirect
171. #97C22×C3semidirect
172. C4×C8C22×C3semidirect
384. C3#3224direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5422
C3×#5352
C3×#4962
C3×#4952
C3×#4892
C3×#4862
C3×#4642
C3×#4632
C3×#4602
C3×#4592
C3×#4522
C3×#4502
C3×#4492
C3×#4472
C3×#4332
#32243