Finite Group Databasebeta

← C3×#3312 #85176

Subgroups — C3×#3312 #85176

Order 384.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#552C211yesmaximal, characteristic
2C3×D4×Q8C211yesmaximal, characteristic
3C3×#491C211yesmaximal, characteristic, complemented
4C3×#485C211yesmaximal, characteristic, complemented
5C3×#483C211yesmaximal, characteristic, complemented
6C3×#479C211yesmaximal, characteristic, complemented
7C3×#470C211yesmaximal, characteristic, complemented
8C3×#467C211yesmaximal, characteristic, complemented
9C3×#462C211yesmaximal, characteristic, complemented
10C3×#451C211yesmaximal, characteristic, complemented
11C3×#451C211yesmaximal, characteristic, complemented
12C3×#441C211yesmaximal, characteristic, complemented
13C3×#435C211yesmaximal, characteristic
14C3×#419C211yesmaximal, characteristic, complemented
15C2×C3×#103C211yesmaximal, characteristic, complemented
16#3312C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×Q8C2211yescharacteristic
18C22×C3×Q8C2211yescharacteristic
19C2×C3×Q16C2211yescharacteristic
20C2×C3×Q16C2211yescharacteristic
21C2×C3×Q16C2211yescharacteristic
22C2×C3×M16C2211yescharacteristic
23C22×C3×C8C2211yescharacteristic
24C3×C4⋊Q8—21no—
25C3×C4⋊Q8C2211yescharacteristic
26C3×#126—21no—
27C3×#125—21no—
28C3×#125C2211yescharacteristic
29C3×#123—21no—
30C3×C22⋊Q8—21no—
31C3×C22⋊Q8—21no—
32C3×C22⋊Q8C2211yescharacteristic
33C3×C22⋊Q8C2211yescharacteristic
34C3×C22⋊Q8C2211yescharacteristic
35C3×C22⋊Q8C2211yescharacteristic
36C3×C4×Q8C2211yescharacteristic
37C3×C4×Q8C2211yescharacteristic
38C3×C4×D4—21no—
39C3×C4×D4C2211yescharacteristic
40C3×#117—21no—
41C3×#117C2211yescharacteristic
42C2×C3×C4⋊C4C2211yescharacteristic
43C3×#107C2211yescharacteristic
44C3×#106C2211yescharacteristic
45C3×#106C2211yescharacteristic
46C3×C4⋊C8C2211yescharacteristic
47C3×#103C2211yescharacteristic
48C3×#103C2211yescharacteristic
49C3×#103C2211yescharacteristic
50C3×#103C2211yescharacteristic
51C3×#103C2211yescharacteristic
52C3×#103C2211yescharacteristic
53C3×#103C2211yescharacteristic
54C3×#103C2211yescharacteristic
55C3×#103C2211yescharacteristic
56C3×#103C2211yescharacteristic
57C3×C22⋊C8C2211yescharacteristic
58C3×C22⋊C8C2211yescharacteristic
59C3×#97C2211yescharacteristic
60#552C2×C311yescharacteristic
61D4×Q8C2×C311yescharacteristic
62#491C2×C311yescharacteristic, complemented
63#485C2×C311yescharacteristic, complemented
64#483C2×C311yescharacteristic, complemented
65#479C2×C311yescharacteristic, complemented
66#470C2×C311yescharacteristic, complemented
67#467C2×C311yescharacteristic, complemented
68#462C2×C311yescharacteristic, complemented
69#451C2×C311yescharacteristic, complemented
70#451C2×C311yescharacteristic, complemented
71#441C2×C311yescharacteristic, complemented
72#435C2×C311yescharacteristic
73#419C2×C311yescharacteristic, complemented
74C2×#103C2×C311yescharacteristic, complemented
75C2×C3×Q8—21no—
76C2×C3×Q8—21no—
77C2×C3×Q8—22no—
78C2×C3×Q8—22no—
79C2×C3×Q8—41no—
80C2×C3×Q8—41no—
81C2×C3×Q8C2311yescharacteristic
82C2×C3×Q8C2311yescharacteristic
83C2×C3×Q8C2311yescharacteristic
84C2×C3×Q8C2311yescharacteristic
85C2×C3×D4D411yescharacteristic
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4C2311yescharacteristic
90C22×C3×C4C2311yescharacteristic
91C3×Q16—22no—
92C3×Q16—41no—
93C3×Q16—41no—
94C3×M16—41no—
95C2×C3×C8—41no—
96C2×C3×C8C2311yescharacteristic
97C2×C3×C8C2311yescharacteristic
98C2×C3×C8C2311yescharacteristic
99C2×C3×C8C2311yescharacteristic
100C3×C4⋊C4—21no—
101C3×C4⋊C4—21no—
102C3×C4⋊C4—21no—
103C3×C4⋊C4—21no—
104C3×C4⋊C4—21no—
105C3×C4⋊C4—21no—
106C3×C4⋊C4—21no—
107C3×C4⋊C4—21no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4—21no—
110C3×C4⋊C4—21no—
111C3×C4⋊C4D411yescharacteristic
112C3×C4⋊C4C2311yescharacteristic
113C3×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
122C3×C22⋊C4D411yescharacteristic
123C3×C42—21no—
124C3×C42—21no—
125C3×C42—21no—
126C3×C42C2311yescharacteristic
127C22×Q8C22×C311yescharacteristic
128C22×Q8C22×C311yescharacteristic
129C2×Q16C22×C311yescharacteristic
130C2×Q16C22×C311yescharacteristic
131C2×Q16C22×C311yescharacteristic
132C2×M16C22×C311yescharacteristic
133C22×C8C22×C311yescharacteristic
134C4⋊Q8—21no—
135C4⋊Q8C22×C311yescharacteristic
136#126—21no—
137#125—21no—
138#125C22×C311yescharacteristic
139#123—21no—
140C22⋊Q8—21no—
141C22⋊Q8—21no—
142C22⋊Q8C22×C311yescharacteristic
143C22⋊Q8C22×C311yescharacteristic
144C22⋊Q8C22×C311yescharacteristic
145C22⋊Q8C22×C311yescharacteristic
146C4×Q8C22×C311yescharacteristic
147C4×Q8C22×C311yescharacteristic
148C4×D4—21no—
149C4×D4C22×C311yescharacteristic
150#117—21no—
151#117C22×C311yescharacteristic
152C2×C4⋊C4C22×C311yescharacteristic
153#107C22×C311yescharacteristic
154#106C22×C311yescharacteristic
155#106C22×C311yescharacteristic
156C4⋊C8C22×C311yescharacteristic
157#103C22×C311yescharacteristic
158#103C22×C311yescharacteristic
159#103C22×C311yescharacteristic
160#103C22×C311yescharacteristic
161#103C22×C311yescharacteristic
162#103C22×C311yescharacteristic
163#103C22×C311yescharacteristic
164#103C22×C311yescharacteristic
165#103C22×C311yescharacteristic
166#103C22×C311yescharacteristic
167C22⋊C8C22×C311yescharacteristic
168C22⋊C8C22×C311yescharacteristic
169#97C22×C311yescharacteristic
170C23×C3C2×D411yescharacteristic
171C23×C3C2×D411yescharacteristic
172C3×Q8—21no—
173C3×Q8—21no—
174C3×Q8—21no—
175C3×Q8—21no—
176C3×Q8—21no—
177C3×Q8—21no—
178C3×Q8—21no—
179C3×Q8—42no—
180C3×Q8—42no—
181C3×Q8#4112yes—
182C3×D4—22no—
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—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4—42no—
198C2×C3×C4C2×D411yescharacteristic
199C2×C3×C4C2×D411yescharacteristic
200C2×C3×C4C2×D411yescharacteristic
201C2×C3×C4C2×D411yescharacteristic
202C2×C3×C4C2411yescharacteristic
203C3×C8—21no—
204C3×C8—21no—
205C3×C8—21no—
206C3×C8—21no—
207C2×Q8—21no—
208C2×Q8—21no—
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8C23×C311yescharacteristic
214C2×Q8C23×C311yescharacteristic
215C2×Q8C23×C311yescharacteristic
216C2×Q8C23×C311yescharacteristic
217C2×D4C3×D411yescharacteristic
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4C23×C311yescharacteristic
222C22×C4C23×C311yescharacteristic
223Q16—22no—
224Q16—41no—
225Q16—41no—
226M16—41no—
227C2×C8—41no—
228C2×C8C23×C311yescharacteristic
229C2×C8C23×C311yescharacteristic
230C2×C8C23×C311yescharacteristic
231C2×C8C23×C311yescharacteristic
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4C3×D411yescharacteristic
244C4⋊C4C23×C311yescharacteristic
245C4⋊C4C23×C311yescharacteristic
246C4⋊C4C23×C311yescharacteristic
247C4⋊C4C23×C311yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—21no—
250C22⋊C4—21no—
251C22⋊C4—21no—
252C22⋊C4—21no—
253C22⋊C4C3×D411yescharacteristic
254C22⋊C4C3×D411yescharacteristic
255C42—21no—
256C42—21no—
257C42—21no—
258C42C23×C311yescharacteristic
259C22×C3—21no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3#13712yes—
265C22×C3C22×D411yescharacteristic
266C3×C4—21no—
267C3×C4—21no—
268C3×C4—22no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—41no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4—41no—
278C3×C4C2×#4111yescharacteristic
279C3×C4H(5,𝔽2)11yescharacteristic
280C23C2×C3×D411yescharacteristic
281C23C2×C3×D411yescharacteristic
282Q8—21no—
283Q8—21no—
284Q8—21no—
285Q8—21no—
286Q8—21no—
287Q8—21no—
288Q8—21no—
289Q8—42no—
290Q8—42no—
291Q8C3×#4112yes—
292D4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4C2×C3×D411yescharacteristic
309C2×C4C2×C3×D411yescharacteristic
310C2×C4C2×C3×D411yescharacteristic
311C2×C4C2×C3×D411yescharacteristic
312C2×C4C24×C311yescharacteristic
313C8—21no—
314C8—21no—
315C8—21no—
316C8—21no—
317C2×C3—22no—
318C2×C3—41no—
319C2×C3#54611yescharacteristic
320C2×C3C2×#13711yescharacteristic
321C2×C3#57811yescharacteristic
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22C3×#13712yes—
328C22C22×C3×D411yescharacteristic
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—22no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4C2×C3×#4111yescharacteristic
342C4C3×H(5,𝔽2)11yescharacteristic
343C3#331211yescharacteristic, direct product factor, Sylow
344C2—22no—
345C2—41no—
346C2C3×#54611yescharacteristic
347C2C2×C3×#13711yescharacteristic
348C2C3×#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×#491C2semidirect
4. C3×#485C2semidirect
5. C3×#483C2semidirect
6. C3×#479C2semidirect
7. C3×#470C2semidirect
8. C3×#467C2semidirect
9. C3×#462C2semidirect
10. C3×#451C2semidirect
11. C3×#451C2semidirect
12. C3×#441C2semidirect
14. C3×#419C2semidirect
15. C2×C3×#103C2semidirect
16. #3312C3direct (N × H)
62. #491C2×C3semidirect
63. #485C2×C3semidirect
64. #483C2×C3semidirect
65. #479C2×C3semidirect
66. #470C2×C3semidirect
67. #467C2×C3semidirect
68. #462C2×C3semidirect
69. #451C2×C3semidirect
70. #451C2×C3semidirect
71. #441C2×C3semidirect
73. #419C2×C3semidirect
74. C2×#103C2×C3semidirect
343. C3#3312direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5522
C3×D4×Q82
C3×#4912
C3×#4852
C3×#4832
C3×#4792
C3×#4702
C3×#4672
C3×#4622
C3×#4512
C3×#4412
C3×#4352
C3×#4192
C2×C3×#1032
#33123