Finite Group Databasebeta

← C3×#3349 #85213

Subgroups — C3×#3349 #85213

Order 384.

336 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#556C211yesmaximal, characteristic
2C3×#552C211yesmaximal, characteristic
3C3×#501C211yesmaximal, characteristic, complemented
4C3×#485C211yesmaximal, characteristic, complemented
5C3×#485C211yesmaximal, characteristic, complemented
6C3×#485C211yesmaximal, characteristic, complemented
7C3×#484C211yesmaximal, characteristic, complemented
8C3×#479C211yesmaximal, characteristic, complemented
9C3×#477C211yesmaximal, characteristic, complemented
10C3×#470C211yesmaximal, characteristic, complemented
11C3×#467C211yesmaximal, characteristic, complemented
12C3×#441C211yesmaximal, characteristic, complemented
13C3×#435C211yesmaximal, characteristic
14C3×#428C211yesmaximal, characteristic, complemented
15C3×#427C211yesmaximal, characteristic, complemented
16#3349C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×Q16C2211yescharacteristic
18C2×C3×M16C2211yescharacteristic
19C22×C3×C8C2211yescharacteristic
20C3×C4⋊Q8C2211yescharacteristic
21C3×#126—21no—
22C3×#126—21no—
23C3×#126—21no—
24C3×#125—21no—
25C3×#125C2211yescharacteristic
26C3×#124—21no—
27C3×#123—21no—
28C3×C22⋊Q8C2211yescharacteristic
29C3×C22⋊Q8C2211yescharacteristic
30C3×C22⋊Q8C2211yescharacteristic
31C3×C22⋊Q8C2211yescharacteristic
32C3×C4×Q8—21no—
33C3×C4×Q8C2211yescharacteristic
34C3×C4×Q8C2211yescharacteristic
35C3×C4×D4C2211yescharacteristic
36C3×#117—21no—
37C3×#117C2211yescharacteristic
38C3×#117C2211yescharacteristic
39C3×#117C2211yescharacteristic
40C2×C3×C4⋊C4C2211yescharacteristic
41C3×#107C2211yescharacteristic
42C3×#107C2211yescharacteristic
43C3×#107C2211yescharacteristic
44C3×#107C2211yescharacteristic
45C3×#107C2211yescharacteristic
46C3×#106C2211yescharacteristic
47C3×#106C2211yescharacteristic
48C3×#106C2211yescharacteristic
49C3×#106C2211yescharacteristic
50C3×C4⋊C8C2211yescharacteristic
51C3×#103C2211yescharacteristic
52C3×#103C2211yescharacteristic
53C3×#103C2211yescharacteristic
54C3×#103C2211yescharacteristic
55C3×#103C2211yescharacteristic
56C3×#103C2211yescharacteristic
57C3×C22⋊C8C2211yescharacteristic
58C3×C22⋊C8C2211yescharacteristic
59C3×#97C2211yescharacteristic
60#556C2×C311yescharacteristic
61#552C2×C311yescharacteristic
62#501C2×C311yescharacteristic, complemented
63#485C2×C311yescharacteristic, complemented
64#485C2×C311yescharacteristic, complemented
65#485C2×C311yescharacteristic, complemented
66#484C2×C311yescharacteristic, complemented
67#479C2×C311yescharacteristic, complemented
68#477C2×C311yescharacteristic, complemented
69#470C2×C311yescharacteristic, complemented
70#467C2×C311yescharacteristic, complemented
71#441C2×C311yescharacteristic, complemented
72#435C2×C311yescharacteristic
73#428C2×C311yescharacteristic, complemented
74#427C2×C311yescharacteristic, complemented
75C2×C3×Q8—21no—
76C2×C3×Q8C2311yescharacteristic
77C2×C3×Q8C2311yescharacteristic
78C2×C3×D4D411yescharacteristic
79C22×C3×C4—21no—
80C22×C3×C4C2311yescharacteristic
81C22×C3×C4C2311yescharacteristic
82C3×Q16—22no—
83C3×M16—22no—
84C2×C3×C8—22no—
85C2×C3×C8C2311yescharacteristic
86C2×C3×C8C2311yescharacteristic
87C2×C3×C8C2311yescharacteristic
88C2×C3×C8C2311yescharacteristic
89C3×C4⋊C4—21no—
90C3×C4⋊C4—21no—
91C3×C4⋊C4—21no—
92C3×C4⋊C4—21no—
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⋊C4D411yescharacteristic
100C3×C4⋊C4C2311yescharacteristic
101C3×C4⋊C4C2311yescharacteristic
102C3×C4⋊C4C2311yescharacteristic
103C3×C4⋊C4C2311yescharacteristic
104C3×C4⋊C4C2311yescharacteristic
105C3×C4⋊C4C2311yescharacteristic
106C3×C22⋊C4—21no—
107C3×C22⋊C4—21no—
108C3×C22⋊C4—21no—
109C3×C22⋊C4—21no—
110C3×C22⋊C4—21no—
111C3×C22⋊C4—21no—
112C3×C22⋊C4—21no—
113C3×C22⋊C4D411yescharacteristic
114C3×C22⋊C4D411yescharacteristic
115C3×C42—21no—
116C3×C42—21no—
117C3×C42—21no—
118C3×C42—21no—
119C3×C42—21no—
120C3×C42C2311yescharacteristic
121C2×Q16C22×C311yescharacteristic
122C2×M16C22×C311yescharacteristic
123C22×C8C22×C311yescharacteristic
124C4⋊Q8C22×C311yescharacteristic
125#126—21no—
126#126—21no—
127#126—21no—
128#125—21no—
129#125C22×C311yescharacteristic
130#124—21no—
131#123—21no—
132C22⋊Q8C22×C311yescharacteristic
133C22⋊Q8C22×C311yescharacteristic
134C22⋊Q8C22×C311yescharacteristic
135C22⋊Q8C22×C311yescharacteristic
136C4×Q8—21no—
137C4×Q8C22×C311yescharacteristic
138C4×Q8C22×C311yescharacteristic
139C4×D4C22×C311yescharacteristic
140#117—21no—
141#117C22×C311yescharacteristic
142#117C22×C311yescharacteristic
143#117C22×C311yescharacteristic
144C2×C4⋊C4C22×C311yescharacteristic
145#107C22×C311yescharacteristic
146#107C22×C311yescharacteristic
147#107C22×C311yescharacteristic
148#107C22×C311yescharacteristic
149#107C22×C311yescharacteristic
150#106C22×C311yescharacteristic
151#106C22×C311yescharacteristic
152#106C22×C311yescharacteristic
153#106C22×C311yescharacteristic
154C4⋊C8C22×C311yescharacteristic
155#103C22×C311yescharacteristic
156#103C22×C311yescharacteristic
157#103C22×C311yescharacteristic
158#103C22×C311yescharacteristic
159#103C22×C311yescharacteristic
160#103C22×C311yescharacteristic
161C22⋊C8C22×C311yescharacteristic
162C22⋊C8C22×C311yescharacteristic
163#97C22×C311yescharacteristic
164C23×C3C2×D411yescharacteristic
165C23×C3C2×D411yescharacteristic
166C3×Q8—21no—
167C3×Q8—21no—
168C3×Q8—21no—
169C3×Q8—21no—
170C3×Q8—42no—
171C3×D4—41no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—42no—
185C2×C3×C4C2×D411yescharacteristic
186C2×C3×C4C2×D411yescharacteristic
187C2×C3×C4C2×D411yescharacteristic
188C2×C3×C4C2×D411yescharacteristic
189C2×C3×C4C2411yescharacteristic
190C3×C8—21no—
191C3×C8—21no—
192C3×C8—21no—
193C3×C8#4112yes—
194C2×Q8—21no—
195C2×Q8C23×C311yescharacteristic
196C2×Q8C23×C311yescharacteristic
197C2×D4C3×D411yescharacteristic
198C22×C4—21no—
199C22×C4C23×C311yescharacteristic
200C22×C4C23×C311yescharacteristic
201Q16—22no—
202M16—22no—
203C2×C8—22no—
204C2×C8C23×C311yescharacteristic
205C2×C8C23×C311yescharacteristic
206C2×C8C23×C311yescharacteristic
207C2×C8C23×C311yescharacteristic
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4C3×D411yescharacteristic
219C4⋊C4C23×C311yescharacteristic
220C4⋊C4C23×C311yescharacteristic
221C4⋊C4C23×C311yescharacteristic
222C4⋊C4C23×C311yescharacteristic
223C4⋊C4C23×C311yescharacteristic
224C4⋊C4C23×C311yescharacteristic
225C22⋊C4—21no—
226C22⋊C4—21no—
227C22⋊C4—21no—
228C22⋊C4—21no—
229C22⋊C4—21no—
230C22⋊C4—21no—
231C22⋊C4—21no—
232C22⋊C4C3×D411yescharacteristic
233C22⋊C4C3×D411yescharacteristic
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42C23×C311yescharacteristic
240C22×C3—21no—
241C22×C3—21no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3C22×D411yescharacteristic
247C3×C4—21no—
248C3×C4—21no—
249C3×C4—21no—
250C3×C4—21no—
251C3×C4—41no—
252C3×C4—41no—
253C3×C4—41no—
254C3×C4—41no—
255C3×C4—41no—
256C3×C4—41no—
257C3×C4—41no—
258C3×C4—41no—
259C3×C4C2×#4111yescharacteristic
260C3×C42−1+411yescharacteristic
261C23C2×C3×D411yescharacteristic
262C23C2×C3×D411yescharacteristic
263Q8—21no—
264Q8—21no—
265Q8—21no—
266Q8—21no—
267Q8—42no—
268D4—41no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—42no—
282C2×C4C2×C3×D411yescharacteristic
283C2×C4C2×C3×D411yescharacteristic
284C2×C4C2×C3×D411yescharacteristic
285C2×C4C2×C3×D411yescharacteristic
286C2×C4C24×C311yescharacteristic
287C8—21no—
288C8—21no—
289C8—21no—
290C8C3×#4112yes—
291C2×C3—41no—
292C2×C3—41no—
293C2×C3#54711yescharacteristic
294C2×C3#57511yescharacteristic
295C2×C3#57811yescharacteristic
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22C22×C3×D411yescharacteristic
303C4—21no—
304C4—21no—
305C4—21no—
306C4—21no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4C2×C3×#4111yescharacteristic
316C4C3×2−1+411yescharacteristic
317C3#334911yescharacteristic, direct product factor, Sylow
318C2—41no—
319C2—41no—
320C2C3×#54711yescharacteristic
321C2C3×#57511yescharacteristic
322C2C3×#57811yescharacteristic
subgroupG/H
center, socleC22×C3C22×D4
commutator, Frattini, cosocleC2×C4C24×C3
prime radicalC2×C3×D4D4
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
3. C3×#501C2semidirect
4. C3×#485C2semidirect
5. C3×#485C2semidirect
6. C3×#485C2semidirect
7. C3×#484C2semidirect
8. C3×#479C2semidirect
9. C3×#477C2semidirect
10. C3×#470C2semidirect
11. C3×#467C2semidirect
12. C3×#441C2semidirect
14. C3×#428C2semidirect
15. C3×#427C2semidirect
16. #3349C3direct (N × H)
62. #501C2×C3semidirect
63. #485C2×C3semidirect
64. #485C2×C3semidirect
65. #485C2×C3semidirect
66. #484C2×C3semidirect
67. #479C2×C3semidirect
68. #477C2×C3semidirect
69. #470C2×C3semidirect
70. #467C2×C3semidirect
71. #441C2×C3semidirect
73. #428C2×C3semidirect
74. #427C2×C3semidirect
317. C3#3349direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5562
C3×#5522
C3×#5012
C3×#4852
C3×#4842
C3×#4792
C3×#4772
C3×#4702
C3×#4672
C3×#4412
C3×#4352
C3×#4282
C3×#4272
#33493