Finite Group Databasebeta

← C3×#3121 #84985

Subgroups — C3×#3121 #84985

Order 384.

370 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#542C211yesmaximal, characteristic
2C3×#519C211yesmaximal, characteristic, complemented
3C3×#485C211yesmaximal, characteristic, complemented
4C3×#482C211yesmaximal, characteristic
5C3×#477C211yesmaximal, characteristic, complemented
6C3×#475C211yesmaximal, characteristic, complemented
7C3×#464C211yesmaximal, characteristic, complemented
8C3×#464C211yesmaximal, characteristic, complemented
9C3×#451C211yesmaximal, characteristic, complemented
10C3×#448C211yesmaximal, characteristic
11C3×#441C211yesmaximal, characteristic, complemented
12C3×#440C211yesmaximal, characteristic, complemented
13C3×C4×Q16C211yesmaximal, characteristic, complemented
14C3×C4×SD16C211yesmaximal, characteristic, complemented
15C3×#433C211yesmaximal, characteristic, complemented
16#3121C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×Q8C2211yescharacteristic
18C2×C3×Q16C2211yescharacteristic
19C2×C3×Q16C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic
21C2×C3×SD16C2211yescharacteristic
22C3×C4⋊Q8C2211yescharacteristic
23C3×#126—21no—
24C3×#124—21no—
25C3×#124C2211yescharacteristic
26C3×#123—21no—
27C3×C22⋊Q8—21no—
28C3×C22⋊Q8C2211yescharacteristic
29C3×#121C2211yescharacteristic
30C3×C4×Q8—21no—
31C3×C4×Q8—21no—
32C3×C4×Q8C2211yescharacteristic
33C3×C4×Q8C2211yescharacteristic
34C3×C4×Q8C2211yescharacteristic
35C3×C4×Q8C2211yescharacteristic
36C3×C4×Q8C2211yescharacteristic
37C3×C4×D4C2211yescharacteristic
38C3×#117—21no—
39C3×#117—21no—
40C3×#117C2211yescharacteristic
41C3×#117C2211yescharacteristic
42C3×#107C2211yescharacteristic
43C3×#107C2211yescharacteristic
44C3×#106C2211yescharacteristic
45C3×#106C2211yescharacteristic
46C3×C4⋊C8C2211yescharacteristic
47C3×C4⋊C8C2211yescharacteristic
48C3×#103C2211yescharacteristic
49C3×#103C2211yescharacteristic
50C3×#103C2211yescharacteristic
51C3×#103C2211yescharacteristic
52C3×#103C2211yescharacteristic
53C3×#103C2211yescharacteristic
54C3×#102C2211yescharacteristic
55C3×#102C2211yescharacteristic
56C3×C22⋊C8C2211yescharacteristic
57C3×C22⋊C8C2211yescharacteristic
58C3×#97C2211yescharacteristic
59C3×C4×C8C2211yescharacteristic
60#542C2×C311yescharacteristic
61#519C2×C311yescharacteristic, complemented
62#485C2×C311yescharacteristic, complemented
63#482C2×C311yescharacteristic
64#477C2×C311yescharacteristic, complemented
65#475C2×C311yescharacteristic, complemented
66#464C2×C311yescharacteristic, complemented
67#464C2×C311yescharacteristic, complemented
68#451C2×C311yescharacteristic, complemented
69#448C2×C311yescharacteristic
70#441C2×C311yescharacteristic, complemented
71#440C2×C311yescharacteristic, complemented
72C4×Q16C2×C311yescharacteristic, complemented
73C4×SD16C2×C311yescharacteristic, complemented
74#433C2×C311yescharacteristic, complemented
75C2×C3×Q8—22no—
76C2×C3×Q8—22no—
77C2×C3×Q8—21no—
78C2×C3×Q8—21no—
79C2×C3×Q8C2311yescharacteristic
80C2×C3×Q8C2311yescharacteristic
81C2×C3×Q8C2311yescharacteristic
82C2×C3×D4—21no—
83C2×C3×D4C2311yescharacteristic
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4C2311yescharacteristic
87C3×Q16—22no—
88C3×Q16—22no—
89C3×SD16—22no—
90C3×SD16—22no—
91C2×C3×C8C2311yescharacteristic
92C2×C3×C8C2311yescharacteristic
93C2×C3×C8C2311yescharacteristic
94C2×C3×C8C2311yescharacteristic
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⋊C4D411yescharacteristic
103C3×C4⋊C4D411yescharacteristic
104C3×C4⋊C4C2311yescharacteristic
105C3×C4⋊C4C2311yescharacteristic
106C3×C4⋊C4C2311yescharacteristic
107C3×C4⋊C4C2311yescharacteristic
108C3×C22⋊C4—21no—
109C3×C22⋊C4—21no—
110C3×C22⋊C4—21no—
111C3×C22⋊C4—21no—
112C3×C22⋊C4—21no—
113C3×C22⋊C4—21no—
114C3×C22⋊C4D411yescharacteristic
115C3×C22⋊C4D411yescharacteristic
116C3×C42—21no—
117C3×C42—21no—
118C3×C42—21no—
119C3×C42—21no—
120C3×C42—21no—
121C3×C42—21no—
122C3×C42C2311yescharacteristic
123C3×C42C2311yescharacteristic
124C22×Q8C22×C311yescharacteristic
125C2×Q16C22×C311yescharacteristic
126C2×Q16C22×C311yescharacteristic
127C2×SD16C22×C311yescharacteristic
128C2×SD16C22×C311yescharacteristic
129C4⋊Q8C22×C311yescharacteristic
130#126—21no—
131#124—21no—
132#124C22×C311yescharacteristic
133#123—21no—
134C22⋊Q8—21no—
135C22⋊Q8C22×C311yescharacteristic
136#121C22×C311yescharacteristic
137C4×Q8—21no—
138C4×Q8—21no—
139C4×Q8C22×C311yescharacteristic
140C4×Q8C22×C311yescharacteristic
141C4×Q8C22×C311yescharacteristic
142C4×Q8C22×C311yescharacteristic
143C4×Q8C22×C311yescharacteristic
144C4×D4C22×C311yescharacteristic
145#117—21no—
146#117—21no—
147#117C22×C311yescharacteristic
148#117C22×C311yescharacteristic
149#107C22×C311yescharacteristic
150#107C22×C311yescharacteristic
151#106C22×C311yescharacteristic
152#106C22×C311yescharacteristic
153C4⋊C8C22×C311yescharacteristic
154C4⋊C8C22×C311yescharacteristic
155#103C22×C311yescharacteristic
156#103C22×C311yescharacteristic
157#103C22×C311yescharacteristic
158#103C22×C311yescharacteristic
159#103C22×C311yescharacteristic
160#103C22×C311yescharacteristic
161#102C22×C311yescharacteristic
162#102C22×C311yescharacteristic
163C22⋊C8C22×C311yescharacteristic
164C22⋊C8C22×C311yescharacteristic
165#97C22×C311yescharacteristic
166C4×C8C22×C311yescharacteristic
167C23×C3—21no—
168C23×C3C2×D411yescharacteristic
169C3×Q8—24no—
170C3×Q8—21no—
171C3×Q8—21no—
172C3×Q8—21no—
173C3×Q8—21no—
174C3×Q8—81no—
175C3×Q8#4112yes—
176C3×Q8#4112yes—
177C3×D4—21no—
178C3×D4—21no—
179C3×D4—81no—
180C2×C3×C4—24no—
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—21no—
189C2×C3×C4—21no—
190C2×C3×C4—81no—
191C2×C3×C4C2×D411yescharacteristic
192C2×C3×C4C2×D411yescharacteristic
193C2×C3×C4C2×D411yescharacteristic
194C2×C3×C4C2×D411yescharacteristic
195C2×C3×C4C2×D411yescharacteristic
196C2×C3×C4C2411yescharacteristic
197C3×C8—21no—
198C3×C8—21no—
199C3×C8—21no—
200C3×C8—21no—
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—21no—
204C2×Q8—21no—
205C2×Q8C23×C311yescharacteristic
206C2×Q8C23×C311yescharacteristic
207C2×Q8C23×C311yescharacteristic
208C2×D4—21no—
209C2×D4C23×C311yescharacteristic
210C22×C4—21no—
211C22×C4—21no—
212C22×C4C23×C311yescharacteristic
213Q16—22no—
214Q16—22no—
215SD16—22no—
216SD16—22no—
217C2×C8C23×C311yescharacteristic
218C2×C8C23×C311yescharacteristic
219C2×C8C23×C311yescharacteristic
220C2×C8C23×C311yescharacteristic
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⋊C4C3×D411yescharacteristic
241C22⋊C4C3×D411yescharacteristic
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42C23×C311yescharacteristic
249C42C23×C311yescharacteristic
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—41no—
254C22×C3—41no—
255C22×C3—41no—
256C22×C3C22×D411yescharacteristic
257C3×C4—22no—
258C3×C4—22no—
259C3×C4—21no—
260C3×C4—21no—
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—21no—
264C3×C4—41no—
265C3×C4—41no—
266C3×C4—41no—
267C3×C4—41no—
268C3×C4—41no—
269C3×C4C2×#4111yescharacteristic
270C3×C4C2×#4111yescharacteristic
271C23—21no—
272C23C2×C3×D411yescharacteristic
273Q8—24no—
274Q8—21no—
275Q8—21no—
276Q8—21no—
277Q8—21no—
278Q8—81no—
279Q8C3×#4112yes—
280Q8C3×#4112yes—
281D4—21no—
282D4—21no—
283D4—81no—
284C2×C4—24no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—81no—
295C2×C4C2×C3×D411yescharacteristic
296C2×C4C2×C3×D411yescharacteristic
297C2×C4C2×C3×D411yescharacteristic
298C2×C4C2×C3×D411yescharacteristic
299C2×C4C2×C3×D411yescharacteristic
300C2×C4C24×C311yescharacteristic
301C8—21no—
302C8—21no—
303C8—21no—
304C8—21no—
305C2×C3—41no—
306C2×C3—81no—
307C2×C3#52511yescharacteristic
308C2×C3#57711yescharacteristic
309C2×C3#57811yescharacteristic
310C22—21no—
311C22—21no—
312C22—21no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22C22×C3×D411yescharacteristic
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—22no—
322C4—22no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4C2×C3×#4111yescharacteristic
330C4C2×C3×#4111yescharacteristic
331C3#312111yescharacteristic, direct product factor, Sylow
332C2—41no—
333C2—81no—
334C2C3×#52511yescharacteristic
335C2C3×#57711yescharacteristic
336C2C3×#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×#519C2semidirect
3. C3×#485C2semidirect
5. C3×#477C2semidirect
6. C3×#475C2semidirect
7. C3×#464C2semidirect
8. C3×#464C2semidirect
9. C3×#451C2semidirect
11. C3×#441C2semidirect
12. C3×#440C2semidirect
13. C3×C4×Q16C2semidirect
14. C3×C4×SD16C2semidirect
15. C3×#433C2semidirect
16. #3121C3direct (N × H)
61. #519C2×C3semidirect
62. #485C2×C3semidirect
64. #477C2×C3semidirect
65. #475C2×C3semidirect
66. #464C2×C3semidirect
67. #464C2×C3semidirect
68. #451C2×C3semidirect
70. #441C2×C3semidirect
71. #440C2×C3semidirect
72. C4×Q16C2×C3semidirect
73. C4×SD16C2×C3semidirect
74. #433C2×C3semidirect
331. C3#3121direct (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×#5192
C3×#4852
C3×#4822
C3×#4772
C3×#4752
C3×#4642
C3×#4512
C3×#4482
C3×#4412
C3×#4402
C3×C4×Q162
C3×C4×SD162
C3×#4332
#31213