Finite Group Databasebeta

← C3×#3315 #85179

Subgroups — C3×#3315 #85179

Order 384.

382 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#552C211yesmaximal, characteristic, complemented
2C3×#547C211yesmaximal, characteristic
3C3×#491C211yesmaximal, characteristic, complemented
4C3×#485C211yesmaximal, characteristic, complemented
5C3×#484C211yesmaximal, characteristic, complemented
6C3×#478C211yesmaximal, characteristic, complemented
7C3×#470C211yesmaximal, characteristic, complemented
8C3×#467C211yesmaximal, characteristic, complemented
9C3×#462C211yesmaximal, characteristic, complemented
10C3×#452C211yesmaximal, characteristic, complemented
11C3×#452C211yesmaximal, characteristic, complemented
12C3×#440C211yesmaximal, characteristic, complemented
13C3×#435C211yesmaximal, characteristic, complemented
14C3×#417C211yesmaximal, characteristic, complemented
15C3×#416C211yesmaximal, characteristic, complemented
16#3315C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×C3×#41C2211yescharacteristic
19C2×C3×Q16C2211yescharacteristic
20C2×C3×Q16C2211yescharacteristic
21C2×C3×SD16C2211yescharacteristic
22C2×C3×M16C2211yescharacteristic
23C22×C3×C8C2211yescharacteristic
24C3×C4⋊Q8C2211yescharacteristic
25C3×#126—21no—
26C3×#125—21no—
27C3×#125C2211yescharacteristic
28C3×#123—21no—
29C3×#123—21no—
30C3×#123—21no—
31C3×C22⋊Q8C2211yescharacteristic
32C3×C22⋊Q8C2211yescharacteristic
33C3×C22⋊Q8C2211yescharacteristic
34C3×C22⋊Q8C2211yescharacteristic
35C3×#121—21no—
36C3×C4×Q8C2211yescharacteristic
37C3×C4×D4C2211yescharacteristic
38C3×C4×D4C2211yescharacteristic
39C3×#117—21no—
40C3×#117C2211yescharacteristic
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4C2211yescharacteristic
43C3×#107C2211yescharacteristic
44C3×#107C2211yescharacteristic
45C3×#106C2211yescharacteristic
46C3×C4⋊C8C2211yescharacteristic
47C3×#103C2211yescharacteristic
48C3×#103C2211yescharacteristic
49C3×#103C2211yescharacteristic
50C3×#103C2211yescharacteristic
51C3×#103C2211yescharacteristic
52C3×#103C2211yescharacteristic
53C3×#103C2211yescharacteristic
54C3×#102C2211yescharacteristic
55C3×#102C2211yescharacteristic
56C3×#102C2211yescharacteristic
57C3×C22⋊C8C2211yescharacteristic
58C3×C22⋊C8C2211yescharacteristic
59C3×#97C2211yescharacteristic
60#552C2×C311yescharacteristic, complemented
61#547C2×C311yescharacteristic
62#491C2×C311yescharacteristic, complemented
63#485C2×C311yescharacteristic, complemented
64#484C2×C311yescharacteristic, complemented
65#478C2×C311yescharacteristic, complemented
66#470C2×C311yescharacteristic, complemented
67#467C2×C311yescharacteristic, complemented
68#462C2×C311yescharacteristic, complemented
69#452C2×C311yescharacteristic, complemented
70#452C2×C311yescharacteristic, complemented
71#440C2×C311yescharacteristic, complemented
72#435C2×C311yescharacteristic, complemented
73#417C2×C311yescharacteristic, complemented
74#416C2×C311yescharacteristic, complemented
75C3×#41—22no—
76C3×#41—22no—
77C3×#41—41no—
78C3×#41—41no—
79C2×C3×Q8C2311yescharacteristic
80C2×C3×Q8C2311yescharacteristic
81C2×C3×Q8C2311yescharacteristic
82C2×C3×D4—21no—
83C2×C3×D4—21no—
84C2×C3×D4D411yescharacteristic
85C2×C3×D4C2311yescharacteristic
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4C2311yescharacteristic
91C22×C3×C4C2311yescharacteristic
92C3×Q16—41no—
93C3×Q16—41no—
94C3×SD16—22no—
95C3×M16—41no—
96C2×C3×C8—41no—
97C2×C3×C8C2311yescharacteristic
98C2×C3×C8C2311yescharacteristic
99C2×C3×C8C2311yescharacteristic
100C2×C3×C8C2311yescharacteristic
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⋊C4—21no—
122C3×C22⋊C4D411yescharacteristic
123C3×C22⋊C4D411yescharacteristic
124C3×C42—21no—
125C3×C42—21no—
126C3×C42C2311yescharacteristic
127C2×#41C22×C311yescharacteristic
128C2×#41C22×C311yescharacteristic
129C2×Q16C22×C311yescharacteristic
130C2×Q16C22×C311yescharacteristic
131C2×SD16C22×C311yescharacteristic
132C2×M16C22×C311yescharacteristic
133C22×C8C22×C311yescharacteristic
134C4⋊Q8C22×C311yescharacteristic
135#126—21no—
136#125—21no—
137#125C22×C311yescharacteristic
138#123—21no—
139#123—21no—
140#123—21no—
141C22⋊Q8C22×C311yescharacteristic
142C22⋊Q8C22×C311yescharacteristic
143C22⋊Q8C22×C311yescharacteristic
144C22⋊Q8C22×C311yescharacteristic
145#121—21no—
146C4×Q8C22×C311yescharacteristic
147C4×D4C22×C311yescharacteristic
148C4×D4C22×C311yescharacteristic
149#117—21no—
150#117C22×C311yescharacteristic
151C2×C4⋊C4—21no—
152C2×C4⋊C4C22×C311yescharacteristic
153#107C22×C311yescharacteristic
154#107C22×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#102C22×C311yescharacteristic
165#102C22×C311yescharacteristic
166#102C22×C311yescharacteristic
167C22⋊C8C22×C311yescharacteristic
168C22⋊C8C22×C311yescharacteristic
169#97C22×C311yescharacteristic
170C23×C3—21no—
171C23×C3C2×D411yescharacteristic
172C23×C3C2×D411yescharacteristic
173C3×Q8—21no—
174C3×Q8—21no—
175C3×Q8—21no—
176C3×Q8—21no—
177C3×Q8—21no—
178C3×Q8—21no—
179C3×D4—21no—
180C3×D4—41no—
181C3×D4—42no—
182C3×D4—42no—
183C3×D4#4112yes—
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×C4—42no—
199C2×C3×C4C2×D411yescharacteristic
200C2×C3×C4C2×D411yescharacteristic
201C2×C3×C4C2×D411yescharacteristic
202C2×C3×C4C2×D411yescharacteristic
203C2×C3×C4C2411yescharacteristic
204C3×C8—21no—
205C3×C8—21no—
206C3×C8—21no—
207C3×C8—21no—
208#41—22no—
209#41—22no—
210#41—41no—
211#41—41no—
212C2×Q8C23×C311yescharacteristic
213C2×Q8C23×C311yescharacteristic
214C2×Q8C23×C311yescharacteristic
215C2×D4—21no—
216C2×D4—21no—
217C2×D4C3×D411yescharacteristic
218C2×D4C23×C311yescharacteristic
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4C23×C311yescharacteristic
224C22×C4C23×C311yescharacteristic
225Q16—41no—
226Q16—41no—
227SD16—22no—
228M16—41no—
229C2×C8—41no—
230C2×C8C23×C311yescharacteristic
231C2×C8C23×C311yescharacteristic
232C2×C8C23×C311yescharacteristic
233C2×C8C23×C311yescharacteristic
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⋊C4—21no—
244C4⋊C4C3×D411yescharacteristic
245C4⋊C4C23×C311yescharacteristic
246C4⋊C4C23×C311yescharacteristic
247C4⋊C4C23×C311yescharacteristic
248C4⋊C4C23×C311yescharacteristic
249C22⋊C4—21no—
250C22⋊C4—21no—
251C22⋊C4—21no—
252C22⋊C4—21no—
253C22⋊C4—21no—
254C22⋊C4—21no—
255C22⋊C4C3×D411yescharacteristic
256C22⋊C4C3×D411yescharacteristic
257C42—21no—
258C42—21no—
259C42C23×C311yescharacteristic
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—22no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3C22×D411yescharacteristic
270C3×C4—21no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—41no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4—41no—
278C3×C4—41no—
279C3×C4—41no—
280C3×C4—41no—
281C3×C4C2×#4111yescharacteristic
282C3×C4H(5,𝔽2)11yescharacteristic
283C23—21no—
284C23C2×C3×D411yescharacteristic
285C23C2×C3×D411yescharacteristic
286Q8—21no—
287Q8—21no—
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—21no—
292D4—21no—
293D4—41no—
294D4—42no—
295D4—42no—
296D4C3×#4112yes—
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—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4C2×C3×D411yescharacteristic
313C2×C4C2×C3×D411yescharacteristic
314C2×C4C2×C3×D411yescharacteristic
315C2×C4C2×C3×D411yescharacteristic
316C2×C4C24×C311yescharacteristic
317C8—21no—
318C8—21no—
319C8—21no—
320C8—21no—
321C2×C3—41no—
322C2×C3—41no—
323C2×C3—42no—
324C2×C3#54611yescharacteristic
325C2×C3#57511yescharacteristic
326C2×C3#57811yescharacteristic
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—22no—
334C22—41no—
335C22—41no—
336C22C22×C3×D411yescharacteristic
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4C2×C3×#4111yescharacteristic
349C4C3×H(5,𝔽2)11yescharacteristic
350C3#331511yescharacteristic, direct product factor, Sylow
351C2—41no—
352C2—41no—
353C2—42no—
354C2C3×#54611yescharacteristic
355C2C3×#57511yescharacteristic
356C2C3×#57811yescharacteristic
subgroupG/H
center, socleC22×C3C22×D4
commutator, Frattini, cosocleC2×C4C24×C3
prime radicalC3×#547C2
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
1. C3×#552C2semidirect
3. C3×#491C2semidirect
4. C3×#485C2semidirect
5. C3×#484C2semidirect
6. C3×#478C2semidirect
7. C3×#470C2semidirect
8. C3×#467C2semidirect
9. C3×#462C2semidirect
10. C3×#452C2semidirect
11. C3×#452C2semidirect
12. C3×#440C2semidirect
13. C3×#435C2semidirect
14. C3×#417C2semidirect
15. C3×#416C2semidirect
16. #3315C3direct (N × H)
60. #552C2×C3semidirect
62. #491C2×C3semidirect
63. #485C2×C3semidirect
64. #484C2×C3semidirect
65. #478C2×C3semidirect
66. #470C2×C3semidirect
67. #467C2×C3semidirect
68. #462C2×C3semidirect
69. #452C2×C3semidirect
70. #452C2×C3semidirect
71. #440C2×C3semidirect
72. #435C2×C3semidirect
73. #417C2×C3semidirect
74. #416C2×C3semidirect
350. C3#3315direct (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×#5472
C3×#4912
C3×#4852
C3×#4842
C3×#4782
C3×#4702
C3×#4672
C3×#4622
C3×#4522
C3×#4402
C3×#4352
C3×#4172
C3×#4162
#33153