Finite Group Databasebeta

← C3×#36166 #10654773

Subgroups — C3×#36166 #10654773

Order 768.

1866 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2863C211yesmaximal, characteristic, complemented
2C3×#2852C212yesmaximal, complemented
3C3×#2514C213yesmaximal, complemented
4C3×#2510C216yesmaximal, complemented
5C3×#2495C213yesmaximal, complemented
6C3×#2493C216yesmaximal, complemented
7C3×#2349C213yesmaximal, complemented
8C3×#2339C213yesmaximal
9C3×C4×#126C211yesmaximal, characteristic, complemented
10C2×C3×#383C212yesmaximal, complemented
11C3×C42×D4C211yesmaximal, characteristic, complemented
12#36166C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C2×C3×#126C2211yescharacteristic
15C2×C3×C4×D4C2213yes—
16C2×C3×C4×D4C2213yes—
17C22×C3×C42C2212yes—
18C3×#401C2216yes—
19C3×#400C2216yes—
20C3×#400C2216yes—
21C3×#397C2216yes—
22C3×#391C2213yes—
23C3×#391C2213yes—
24C3×#389C2213yes—
25C3×#389C2213yes—
26C3×#388C2216yes—
27C3×#388C2213yes—
28C3×#387C2213yes—
29C3×#387C2216yes—
30C3×#387C2216yes—
31C3×#386C2216yes—
32C3×#386C2216yes—
33C3×#385C2216yes—
34C3×#385C2216yes—
35C3×#383C2216yes—
36C3×#383C2216yes—
37C3×#383C2211yescharacteristic
38C3×#383C2212yes—
39C3×#382C2213yes—
40C3×#381C2216yes—
41C3×#380C2216yes—
42C3×#380C2213yes—
43C3×C4×C4⋊C4C2213yes—
44C3×C4×C4⋊C4C2213yes—
45C3×C4×C22⋊C4C2216yes—
46C3×C4×C22⋊C4C2213yes—
47C2×C3×#95C2216yes—
48C2×C3×#95C2216yes—
49C3×C43C2211yescharacteristic
50#2863C2×C311yescharacteristic, complemented
51#2852C2×C312yescomplemented
52#2514C2×C313yescomplemented
53#2510C2×C316yescomplemented
54#2495C2×C313yescomplemented
55#2493C2×C316yescomplemented
56#2349C2×C313yescomplemented
57#2339C2×C313yes—
58C4×#126C2×C311yescharacteristic, complemented
59C2×#383C2×C312yescomplemented
60C42×D4C2×C311yescharacteristic, complemented
61C2×C3×#41—212no—
62C22×C3×Q8C2311yescharacteristic
63C22×C3×D4C2312yes—
64C22×C3×D4C2311yescharacteristic
65C23×C3×C4C2312yes—
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2316yes—
68C23×C3×C4C2316yes—
69C3×#126—24no—
70C3×C4×D4—212no—
71C3×C4×D4—212no—
72C2×C3×C4⋊C4C2313yes—
73C2×C3×C4⋊C4C2313yes—
74C2×C3×C4⋊C4C2313yes—
75C2×C3×C4⋊C4C2313yes—
76C2×C3×C4⋊C4C2316yes—
77C2×C3×C4⋊C4C2316yes—
78C2×C3×C22⋊C4C2316yes—
79C2×C3×C22⋊C4C2313yes—
80C2×C3×C22⋊C4C2316yes—
81C2×C3×C22⋊C4C2313yes—
82C2×C3×C22⋊C4C2316yes—
83C2×C3×C42—28no—
84C2×C3×C42C2313yes—
85C2×C3×C42C2313yes—
86C2×C3×C42C2316yes—
87C2×C3×C42C2313yes—
88C2×C3×C42C2311yescharacteristic
89C3×#95C2313yes—
90C3×#95C2313yes—
91C3×#95C2316yes—
92C3×#95C2316yes—
93C3×#95C2316yes—
94C3×#95C2316yes—
95C3×#95C2316yes—
96C3×#95C2316yes—
97C3×#95C2316yes—
98C3×#95C2313yes—
99C3×#95C2313yes—
100C3×#95C2316yes—
101C3×#95C2316yes—
102C3×#95C2316yes—
103C22×#41C22×C311yescharacteristic
104C2×#126C22×C311yescharacteristic
105C2×C4×D4C22×C313yes—
106C2×C4×D4C22×C313yes—
107C22×C42C22×C312yes—
108#401C22×C316yes—
109#400C22×C316yes—
110#400C22×C316yes—
111#397C22×C316yes—
112#391C22×C313yes—
113#391C22×C313yes—
114#389C22×C313yes—
115#389C22×C313yes—
116#388C22×C313yes—
117#388C22×C316yes—
118#387C22×C313yes—
119#387C22×C316yes—
120#387C22×C316yes—
121#386C22×C316yes—
122#386C22×C316yes—
123#385C22×C316yes—
124#385C22×C316yes—
125#383C22×C311yescharacteristic
126#383C22×C312yes—
127#383C22×C316yes—
128#383C22×C316yes—
129#382C22×C313yes—
130#381C22×C316yes—
131#380C22×C313yes—
132#380C22×C316yes—
133C4×C4⋊C4C22×C313yes—
134C4×C4⋊C4C22×C313yes—
135C4×C22⋊C4C22×C313yes—
136C4×C22⋊C4C22×C316yes—
137C2×#95C22×C316yes—
138C2×#95C22×C316yes—
139C43C22×C311yescharacteristic
140C24×C3C2411yescharacteristic
141C24×C3C2412yes—
142C3×#41—416no—
143C2×C3×Q8—26no—
144C2×C3×D4—212no—
145C2×C3×D4#41112yes—
146C22×C3×C4—212no—
147C22×C3×C4—212no—
148C22×C3×C4—212no—
149C22×C3×C4—212no—
150C22×C3×C4—26no—
151C22×C3×C4—46no—
152C22×C3×C4—46no—
153C22×C3×C4—46no—
154C22×C3×C4C2411yescharacteristic
155C22×C3×C4C2412yes—
156C22×C3×C4C2413yes—
157C22×C3×C4C2416yes—
158C22×C3×C4C2416yes—
159C22×C3×C4C2413yes—
160C22×C3×C4C2413yes—
161C22×C3×C4C2413yes—
162C22×C3×C4C2411yescharacteristic
163C3×C4⋊C4—26no—
164C3×C4⋊C4—26no—
165C3×C4⋊C4—26no—
166C3×C4⋊C4—43no—
167C3×C4⋊C4—46no—
168C3×C4⋊C4—46no—
169C3×C22⋊C4—212no—
170C3×C22⋊C4—212no—
171C3×C22⋊C4—26no—
172C3×C22⋊C4—43no—
173C3×C22⋊C4—46no—
174C3×C42—212no—
175C3×C42—26no—
176C3×C42—26no—
177C3×C42—43no—
178C3×C42#4114yescomplemented
179C2×#41—212no—
180C22×Q8C23×C311yescharacteristic
181C22×D4C23×C312yes—
182C22×D4C23×C311yescharacteristic
183C23×C4C23×C312yes—
184C23×C4C23×C311yescharacteristic
185C23×C4C23×C316yes—
186C23×C4C23×C316yes—
187#126—24no—
188C4×D4—212no—
189C4×D4—212no—
190C2×C4⋊C4C23×C313yes—
191C2×C4⋊C4C23×C313yes—
192C2×C4⋊C4C23×C313yes—
193C2×C4⋊C4C23×C313yes—
194C2×C4⋊C4C23×C316yes—
195C2×C4⋊C4C23×C316yes—
196C2×C22⋊C4C23×C316yes—
197C2×C22⋊C4C23×C316yes—
198C2×C22⋊C4C23×C313yes—
199C2×C22⋊C4C23×C313yes—
200C2×C22⋊C4C23×C316yes—
201C2×C42—28no—
202C2×C42C23×C313yes—
203C2×C42C23×C313yes—
204C2×C42C23×C316yes—
205C2×C42C23×C313yes—
206C2×C42C23×C311yescharacteristic
207#95C23×C313yes—
208#95C23×C313yes—
209#95C23×C316yes—
210#95C23×C316yes—
211#95C23×C316yes—
212#95C23×C316yes—
213#95C23×C316yes—
214#95C23×C316yes—
215#95C23×C316yes—
216#95C23×C313yes—
217#95C23×C313yes—
218#95C23×C316yes—
219#95C23×C316yes—
220#95C23×C316yes—
221C23×C3—22no—
222C23×C3—26no—
223C23×C3—21no—
224C23×C3—23no—
225C23×C3—23no—
226C23×C3C2×#41112yes—
227C23×C3C2511yescharacteristic
228C3×Q8—44no—
229C3×D4—48no—
230C3×D4#126116yescomplemented
231C2×C3×C4—23no—
232C2×C3×C4—23no—
233C2×C3×C4—26no—
234C2×C3×C4—23no—
235C2×C3×C4—23no—
236C2×C3×C4—23no—
237C2×C3×C4—23no—
238C2×C3×C4—26no—
239C2×C3×C4—26no—
240C2×C3×C4—23no—
241C2×C3×C4—26no—
242C2×C3×C4—23no—
243C2×C3×C4—23no—
244C2×C3×C4—26no—
245C2×C3×C4—26no—
246C2×C3×C4—23no—
247C2×C3×C4—23no—
248C2×C3×C4—26no—
249C2×C3×C4—23no—
250C2×C3×C4—424no—
251C2×C3×C4—48no—
252C2×C3×C4—44no—
253C2×C3×C4—812no—
254C2×C3×C4C2×#4116yes—
255C2×C3×C4C2×#4116yes—
256C24C24×C311yescharacteristic
257C24C24×C312yes—
258#41—416no—
259C2×Q8—26no—
260C2×D4—212no—
261C2×D4C3×#41112yes—
262C22×C4—212no—
263C22×C4—212no—
264C22×C4—212no—
265C22×C4—212no—
266C22×C4—26no—
267C22×C4—46no—
268C22×C4—46no—
269C22×C4—46no—
270C22×C4C24×C311yescharacteristic
271C22×C4C24×C312yes—
272C22×C4C24×C313yes—
273C22×C4C24×C316yes—
274C22×C4C24×C316yes—
275C22×C4C24×C313yes—
276C22×C4C24×C313yes—
277C22×C4C24×C313yes—
278C22×C4C24×C311yescharacteristic
279C4⋊C4—26no—
280C4⋊C4—26no—
281C4⋊C4—26no—
282C4⋊C4—43no—
283C4⋊C4—46no—
284C4⋊C4—46no—
285C22⋊C4—212no—
286C22⋊C4—212no—
287C22⋊C4—26no—
288C22⋊C4—43no—
289C22⋊C4—46no—
290C42—212no—
291C42—26no—
292C42—26no—
293C42—43no—
294C42C3×#4114yescomplemented
295C22×C3—212no—
296C22×C3—212no—
297C22×C3—41no—
298C22×C3—43no—
299C22×C3—43no—
300C22×C3C2×#12618yes—
301C22×C3C22×#4113yes—
302C22×C3C22×#4111yescharacteristic
303C22×C3#58513yes—
304C3×C4—26no—
305C3×C4—41no—
306C3×C4—42no—
307C3×C4—43no—
308C3×C4—46no—
309C3×C4—43no—
310C3×C4—46no—
311C3×C4—43no—
312C3×C4C2×#12614yes—
313C23—22no—
314C23—26no—
315C23—21no—
316C23—23no—
317C23—23no—
318C23C2×C3×#41112yes—
319C23C25×C311yescharacteristic
320Q8—44no—
321D4—48no—
322D4C3×#126116yescomplemented
323C2×C4—23no—
324C2×C4—23no—
325C2×C4—26no—
326C2×C4—23no—
327C2×C4—23no—
328C2×C4—23no—
329C2×C4—23no—
330C2×C4—26no—
331C2×C4—26no—
332C2×C4—23no—
333C2×C4—26no—
334C2×C4—23no—
335C2×C4—23no—
336C2×C4—26no—
337C2×C4—26no—
338C2×C4—23no—
339C2×C4—23no—
340C2×C4—26no—
341C2×C4—23no—
342C2×C4—424no—
343C2×C4—48no—
344C2×C4—44no—
345C2×C4—812no—
346C2×C4C2×C3×#4116yes—
347C2×C4C2×C3×#4116yes—
348C2×C3—28no—
349C2×C3—81no—
350C2×C3C22×#12611yescharacteristic
351C2×C3#347513yes—
352C2×C3#352113yes—
353C22—212no—
354C22—212no—
355C22—41no—
356C22—43no—
357C22—43no—
358C22C2×C3×#12618yes—
359C22C22×C3×#4113yes—
360C22C22×C3×#4111yescharacteristic
361C22C3×#58513yes—
362C4—26no—
363C4—41no—
364C4—42no—
365C4—46no—
366C4—43no—
367C4—43no—
368C4—46no—
369C4—43no—
370C4C2×C3×#12614yes—
371C3#3616611yescharacteristic, direct product factor, Sylow
372C2—28no—
373C2—81no—
374C2C22×C3×#12611yescharacteristic
375C2C3×#347513yes—
376C2C3×#352113yes—
subgroupG/H
center, socleC23×C3C25
commutator, Frattini, cosocleC23C25×C3
prime radicalC22×C3×#41C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×#2863C2semidirect
2. C3×#2852C2semidirect
3. C3×#2514C2semidirect
4. C3×#2510C2semidirect
5. C3×#2495C2semidirect
6. C3×#2493C2semidirect
7. C3×#2349C2semidirect
9. C3×C4×#126C2semidirect
10. C2×C3×#383C2semidirect
11. C3×C42×D4C2semidirect
12. #36166C3direct (N × H)
50. #2863C2×C3semidirect
51. #2852C2×C3semidirect
52. #2514C2×C3semidirect
53. #2510C2×C3semidirect
54. #2495C2×C3semidirect
55. #2493C2×C3semidirect
56. #2349C2×C3semidirect
58. C4×#126C2×C3semidirect
59. C2×#383C2×C3semidirect
60. C42×D4C2×C3semidirect
178. C3×C42#41semidirect
230. C3×D4#126semidirect
294. C42C3×#41semidirect
322. D4C3×#126semidirect
371. C3#36166direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28632
C3×#28522
C3×#25142
C3×#25102
C3×#24952
C3×#24932
C3×#23492
C3×#23392
C3×C4×#1262
C2×C3×#3832
C3×C42×D42
#361663