Finite Group Databasebeta

← C3×#3258 #85122

Subgroups — C3×#3258 #85122

Order 384.

334 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#542C211yesmaximal, characteristic
2C3×#541C211yesmaximal, characteristic, complemented
3C3×#479C211yesmaximal, characteristic, complemented
4C3×#478C211yesmaximal, characteristic, complemented
5C3×#477C211yesmaximal, characteristic, complemented
6C3×#475C211yesmaximal, characteristic, complemented
7C3×#470C211yesmaximal, characteristic, complemented
8C3×#468C211yesmaximal, characteristic
9C3×#467C211yesmaximal, characteristic, complemented
10C3×#465C211yesmaximal, characteristic
11C3×#464C211yesmaximal, characteristic, complemented
12C3×#464C211yesmaximal, characteristic, complemented
13C3×#462C211yesmaximal, characteristic, complemented
14C3×#461C211yesmaximal, characteristic, complemented
15C3×#424C211yesmaximal, characteristic, complemented
16#3258C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×Q16C2211yescharacteristic
18C2×C3×Q16C2211yescharacteristic
19C2×C3×SD16C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic
21C2×C3×M16C2211yescharacteristic
22C22×C3×C8C2211yescharacteristic
23C3×C4⋊Q8C2211yescharacteristic
24C3×C4⋊Q8C2211yescharacteristic
25C3×#126—21no—
26C3×#126—21no—
27C3×#126—21no—
28C3×#125—21no—
29C3×#125—21no—
30C3×#125C2211yescharacteristic
31C3×#124—21no—
32C3×#124C2211yescharacteristic
33C3×#123—21no—
34C3×C22⋊Q8—21no—
35C3×C22⋊Q8C2211yescharacteristic
36C3×C22⋊Q8C2211yescharacteristic
37C3×C22⋊Q8C2211yescharacteristic
38C3×#121C2211yescharacteristic
39C3×C4×Q8C2211yescharacteristic
40C3×C4×Q8C2211yescharacteristic
41C3×C4×Q8C2211yescharacteristic
42C3×C4×D4C2211yescharacteristic
43C3×#117C2211yescharacteristic
44C3×#107C2211yescharacteristic
45C3×#107C2211yescharacteristic
46C3×#106C2211yescharacteristic
47C3×#106C2211yescharacteristic
48C3×C4⋊C8C2211yescharacteristic
49C3×C4⋊C8C2211yescharacteristic
50C3×C4⋊C8C2211yescharacteristic
51C3×C4⋊C8C2211yescharacteristic
52C3×#103C2211yescharacteristic
53C3×#103C2211yescharacteristic
54C3×#103C2211yescharacteristic
55C3×#103C2211yescharacteristic
56C3×#103C2211yescharacteristic
57C3×#103C2211yescharacteristic
58C3×#102C2211yescharacteristic
59C3×#102C2211yescharacteristic
60#542C2×C311yescharacteristic
61#541C2×C311yescharacteristic, complemented
62#479C2×C311yescharacteristic, complemented
63#478C2×C311yescharacteristic, complemented
64#477C2×C311yescharacteristic, complemented
65#475C2×C311yescharacteristic, complemented
66#470C2×C311yescharacteristic, complemented
67#468C2×C311yescharacteristic
68#467C2×C311yescharacteristic, complemented
69#465C2×C311yescharacteristic
70#464C2×C311yescharacteristic, complemented
71#464C2×C311yescharacteristic, complemented
72#462C2×C311yescharacteristic, complemented
73#461C2×C311yescharacteristic, complemented
74#424C2×C311yescharacteristic, complemented
75C2×C3×Q8—21no—
76C2×C3×Q8C2311yescharacteristic
77C2×C3×Q8C2311yescharacteristic
78C2×C3×Q8C2311yescharacteristic
79C2×C3×D4—21no—
80C2×C3×D4C2311yescharacteristic
81C22×C3×C4—21no—
82C22×C3×C4C2311yescharacteristic
83C3×Q16—41no—
84C3×Q16—41no—
85C3×SD16—41no—
86C3×SD16—41no—
87C3×M16—41no—
88C2×C3×C8—41no—
89C2×C3×C8C2311yescharacteristic
90C2×C3×C8C2311yescharacteristic
91C2×C3×C8C2311yescharacteristic
92C2×C3×C8C2311yescharacteristic
93C3×C4⋊C4—21no—
94C3×C4⋊C4—21no—
95C3×C4⋊C4—21no—
96C3×C4⋊C4—21no—
97C3×C4⋊C4—21no—
98C3×C4⋊C4—21no—
99C3×C4⋊C4—21no—
100C3×C4⋊C4—21no—
101C3×C4⋊C4—21no—
102C3×C4⋊C4—21no—
103C3×C4⋊C4—21no—
104C3×C4⋊C4D411yescharacteristic
105C3×C4⋊C4D411yescharacteristic
106C3×C4⋊C4C2311yescharacteristic
107C3×C4⋊C4C2311yescharacteristic
108C3×C4⋊C4C2311yescharacteristic
109C3×C4⋊C4C2311yescharacteristic
110C3×C22⋊C4—21no—
111C3×C22⋊C4—21no—
112C3×C22⋊C4—21no—
113C3×C22⋊C4—21no—
114C3×C22⋊C4—21no—
115C3×C22⋊C4—21no—
116C3×C22⋊C4—21no—
117C3×C22⋊C4D411yescharacteristic
118C3×C22⋊C4D411yescharacteristic
119C3×C42—21no—
120C3×C42—21no—
121C3×C42—21no—
122C3×C42C2311yescharacteristic
123C3×C42C2311yescharacteristic
124C2×Q16C22×C311yescharacteristic
125C2×Q16C22×C311yescharacteristic
126C2×SD16C22×C311yescharacteristic
127C2×SD16C22×C311yescharacteristic
128C2×M16C22×C311yescharacteristic
129C22×C8C22×C311yescharacteristic
130C4⋊Q8C22×C311yescharacteristic
131C4⋊Q8C22×C311yescharacteristic
132#126—21no—
133#126—21no—
134#126—21no—
135#125—21no—
136#125—21no—
137#125C22×C311yescharacteristic
138#124—21no—
139#124C22×C311yescharacteristic
140#123—21no—
141C22⋊Q8—21no—
142C22⋊Q8C22×C311yescharacteristic
143C22⋊Q8C22×C311yescharacteristic
144C22⋊Q8C22×C311yescharacteristic
145#121C22×C311yescharacteristic
146C4×Q8C22×C311yescharacteristic
147C4×Q8C22×C311yescharacteristic
148C4×Q8C22×C311yescharacteristic
149C4×D4C22×C311yescharacteristic
150#117C22×C311yescharacteristic
151#107C22×C311yescharacteristic
152#107C22×C311yescharacteristic
153#106C22×C311yescharacteristic
154#106C22×C311yescharacteristic
155C4⋊C8C22×C311yescharacteristic
156C4⋊C8C22×C311yescharacteristic
157C4⋊C8C22×C311yescharacteristic
158C4⋊C8C22×C311yescharacteristic
159#103C22×C311yescharacteristic
160#103C22×C311yescharacteristic
161#103C22×C311yescharacteristic
162#103C22×C311yescharacteristic
163#103C22×C311yescharacteristic
164#103C22×C311yescharacteristic
165#102C22×C311yescharacteristic
166#102C22×C311yescharacteristic
167C23×C3—21no—
168C23×C3C2×D411yescharacteristic
169C3×Q8—21no—
170C3×Q8—21no—
171C3×Q8—21no—
172C3×Q8—21no—
173C3×Q8—21no—
174C3×Q8—21no—
175C3×Q8—81no—
176C3×D4—21no—
177C3×D4—21no—
178C3×D4—81no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—81no—
189C2×C3×C4C2×D411yescharacteristic
190C2×C3×C4C2×D411yescharacteristic
191C2×C3×C4C2×D411yescharacteristic
192C2×C3×C4C2×D411yescharacteristic
193C2×C3×C4C2×D411yescharacteristic
194C2×C3×C4C2411yescharacteristic
195C3×C8—21no—
196C3×C8—21no—
197C3×C8—21no—
198C3×C8—21no—
199C2×Q8—21no—
200C2×Q8C23×C311yescharacteristic
201C2×Q8C23×C311yescharacteristic
202C2×Q8C23×C311yescharacteristic
203C2×D4—21no—
204C2×D4C23×C311yescharacteristic
205C22×C4—21no—
206C22×C4C23×C311yescharacteristic
207Q16—41no—
208Q16—41no—
209SD16—41no—
210SD16—41no—
211M16—41no—
212C2×C8—41no—
213C2×C8C23×C311yescharacteristic
214C2×C8C23×C311yescharacteristic
215C2×C8C23×C311yescharacteristic
216C2×C8C23×C311yescharacteristic
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4C3×D411yescharacteristic
229C4⋊C4C3×D411yescharacteristic
230C4⋊C4C23×C311yescharacteristic
231C4⋊C4C23×C311yescharacteristic
232C4⋊C4C23×C311yescharacteristic
233C4⋊C4C23×C311yescharacteristic
234C22⋊C4—21no—
235C22⋊C4—21no—
236C22⋊C4—21no—
237C22⋊C4—21no—
238C22⋊C4—21no—
239C22⋊C4—21no—
240C22⋊C4—21no—
241C22⋊C4C3×D411yescharacteristic
242C22⋊C4C3×D411yescharacteristic
243C42—21no—
244C42—21no—
245C42—21no—
246C42C23×C311yescharacteristic
247C42C23×C311yescharacteristic
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3—41no—
252C22×C3—41no—
253C22×C3—41no—
254C22×C3C22×D411yescharacteristic
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—21no—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—41no—
261C3×C4—41no—
262C3×C4—41no—
263C3×C4—41no—
264C3×C4—41no—
265C3×C4—41no—
266C3×C4—41no—
267C3×C4H(5,𝔽2)11yescharacteristic
268C3×C42−1+411yescharacteristic
269C23—21no—
270C23C2×C3×D411yescharacteristic
271Q8—21no—
272Q8—21no—
273Q8—21no—
274Q8—21no—
275Q8—21no—
276Q8—21no—
277Q8—81no—
278D4—21no—
279D4—21no—
280D4—81no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—81no—
291C2×C4C2×C3×D411yescharacteristic
292C2×C4C2×C3×D411yescharacteristic
293C2×C4C2×C3×D411yescharacteristic
294C2×C4C2×C3×D411yescharacteristic
295C2×C4C2×C3×D411yescharacteristic
296C2×C4C24×C311yescharacteristic
297C8—21no—
298C8—21no—
299C8—21no—
300C8—21no—
301C2×C3—41no—
302C2×C3—81no—
303C2×C3#53711yescharacteristic
304C2×C3#57711yescharacteristic
305C2×C3#57811yescharacteristic
306C22—21no—
307C22—21no—
308C22—21no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22C22×C3×D411yescharacteristic
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4C3×H(5,𝔽2)11yescharacteristic
326C4C3×2−1+411yescharacteristic
327C3#325811yescharacteristic, direct product factor, Sylow
328C2—41no—
329C2—81no—
330C2C3×#53711yescharacteristic
331C2C3×#57711yescharacteristic
332C2C3×#57811yescharacteristic
subgroupG/H
center, socleC22×C3C22×D4
commutator, Frattini, cosocleC2×C4C24×C3
prime radicalC3×#121C22
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
2. C3×#541C2semidirect
3. C3×#479C2semidirect
4. C3×#478C2semidirect
5. C3×#477C2semidirect
6. C3×#475C2semidirect
7. C3×#470C2semidirect
9. C3×#467C2semidirect
11. C3×#464C2semidirect
12. C3×#464C2semidirect
13. C3×#462C2semidirect
14. C3×#461C2semidirect
15. C3×#424C2semidirect
16. #3258C3direct (N × H)
61. #541C2×C3semidirect
62. #479C2×C3semidirect
63. #478C2×C3semidirect
64. #477C2×C3semidirect
65. #475C2×C3semidirect
66. #470C2×C3semidirect
68. #467C2×C3semidirect
70. #464C2×C3semidirect
71. #464C2×C3semidirect
72. #462C2×C3semidirect
73. #461C2×C3semidirect
74. #424C2×C3semidirect
327. C3#3258direct (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×#5412
C3×#4792
C3×#4782
C3×#4772
C3×#4752
C3×#4702
C3×#4682
C3×#4672
C3×#4652
C3×#4642
C3×#4622
C3×#4612
C3×#4242
#32583