Finite Group Databasebeta

← C3×#3516 #87400

Subgroups — C3×#3516 #87400

Order 384.

1012 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#557C211yesmaximal, characteristic, complemented
2C3×#556C213yesmaximal, complemented
3C3×#554C213yesmaximal, complemented
4C3×#553C213yesmaximal, complemented
5C3×#552C213yesmaximal, complemented
6C3×D4×Q8C211yesmaximal, characteristic, complemented
7C3×#548C211yesmaximal, characteristic, complemented
8C3×#548C211yesmaximal, characteristic, complemented
9C3×#542C213yesmaximal, complemented
10C3×#541C213yesmaximal, complemented
11C3×#529C213yesmaximal, complemented
12C3×#529C213yesmaximal, complemented
13C3×#520C211yesmaximal, characteristic, complemented
14C3×#519C211yesmaximal, characteristic, complemented
15C2×C3×C4×Q8C211yesmaximal, characteristic, complemented
16#3516C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×#41C2211yescharacteristic
18C22×C3×Q8C2211yescharacteristic
19C22×C3×Q8C2211yescharacteristic
20C3×C4⋊Q8C2213yes—
21C3×C4⋊Q8C2213yes—
22C3×#126C2216yes—
23C3×#126C2213yes—
24C3×#126C2213yes—
25C3×#126C2216yes—
26C3×#125C2213yes—
27C3×#125C2213yes—
28C3×#125C2216yes—
29C3×#125C2213yes—
30C3×#124C2213yes—
31C3×#124C2213yes—
32C3×#124C2213yes—
33C3×#123C2211yescharacteristic
34C3×#123C2213yes—
35C3×#123C2216yes—
36C3×C22⋊Q8C2213yes—
37C3×C22⋊Q8C2213yes—
38C3×C22⋊Q8C2213yes—
39C3×C22⋊Q8C2213yes—
40C3×C22⋊Q8C2213yes—
41C3×C22⋊Q8C2216yes—
42C3×#121C2213yes—
43C3×#121C2213yes—
44C3×#121C2213yes—
45C3×C4×Q8C2211yescharacteristic
46C3×C4×Q8C2213yes—
47C3×C4×Q8C2211yescharacteristic
48C3×C4×Q8C2213yes—
49C3×C4×Q8C2213yes—
50C3×C4×Q8C2211yescharacteristic
51C3×C4×Q8C2213yes—
52C3×C4×Q8C2211yescharacteristic
53C3×C4×Q8C2213yes—
54C3×C4×D4C2213yes—
55C3×C4×D4C2213yes—
56C3×C4×D4C2213yes—
57C3×C4×D4C2213yes—
58C3×C4×D4C2213yes—
59C3×C4×D4C2216yes—
60C3×#117C2213yes—
61C3×#117C2213yes—
62C3×#117C2213yes—
63C3×#117C2216yes—
64C2×C3×C4⋊C4C2213yes—
65C2×C3×C4⋊C4C2213yes—
66C2×C3×C42C2213yes—
67#557C2×C311yescharacteristic, complemented
68#556C2×C313yescomplemented
69#554C2×C313yescomplemented
70#553C2×C313yescomplemented
71#552C2×C313yescomplemented
72D4×Q8C2×C311yescharacteristic, complemented
73#548C2×C311yescharacteristic, complemented
74#548C2×C311yescharacteristic, complemented
75#542C2×C313yescomplemented
76#541C2×C313yescomplemented
77#529C2×C313yescomplemented
78#529C2×C313yescomplemented
79#520C2×C311yescharacteristic, complemented
80#519C2×C311yescharacteristic, complemented
81C2×C4×Q8C2×C311yescharacteristic, complemented
82C3×#41—24no—
83C2×C3×Q8—24no—
84C2×C3×Q8—24no—
85C2×C3×Q8C2313yes—
86C2×C3×Q8C2313yes—
87C2×C3×Q8C2313yes—
88C2×C3×Q8C2311yescharacteristic
89C2×C3×D4C2313yes—
90C2×C3×D4C2311yescharacteristic
91C2×C3×D4C2313yes—
92C2×C3×D4C2313yes—
93C22×C3×C4C2311yescharacteristic
94C22×C3×C4C2313yes—
95C22×C3×C4C2313yes—
96C22×C3×C4C2313yes—
97C22×C3×C4C2313yes—
98C22×C3×C4C2316yes—
99C3×C4⋊C4C2313yes—
100C3×C4⋊C4C2313yes—
101C3×C4⋊C4C2313yes—
102C3×C4⋊C4C2313yes—
103C3×C4⋊C4C2316yes—
104C3×C4⋊C4C2311yescharacteristic
105C3×C4⋊C4C2313yes—
106C3×C4⋊C4C2313yes—
107C3×C4⋊C4C2313yes—
108C3×C4⋊C4C2313yes—
109C3×C4⋊C4C2316yes—
110C3×C4⋊C4C2313yes—
111C3×C4⋊C4C2313yes—
112C3×C4⋊C4C2316yes—
113C3×C4⋊C4C2313yes—
114C3×C4⋊C4C2311yescharacteristic
115C3×C4⋊C4C2313yes—
116C3×C22⋊C4C2313yes—
117C3×C22⋊C4C2311yescharacteristic
118C3×C22⋊C4C2313yes—
119C3×C22⋊C4C2313yes—
120C3×C22⋊C4C2313yes—
121C3×C22⋊C4C2316yes—
122C3×C22⋊C4C2311yescharacteristic
123C3×C22⋊C4C2313yes—
124C3×C22⋊C4C2311yescharacteristic
125C3×C22⋊C4C2316yes—
126C3×C22⋊C4C2313yes—
127C3×C22⋊C4C2313yes—
128C3×C42C2313yes—
129C3×C42C2313yes—
130C3×C42C2313yes—
131C3×C42C2313yes—
132C3×C42C2316yes—
133C3×C42C2313yes—
134C3×C42C2313yes—
135C2×#41C22×C311yescharacteristic
136C22×Q8C22×C311yescharacteristic
137C22×Q8C22×C311yescharacteristic
138C4⋊Q8C22×C313yes—
139C4⋊Q8C22×C313yes—
140#126C22×C316yes—
141#126C22×C313yes—
142#126C22×C313yes—
143#126C22×C316yes—
144#125C22×C313yes—
145#125C22×C313yes—
146#125C22×C316yes—
147#125C22×C313yes—
148#124C22×C313yes—
149#124C22×C313yes—
150#124C22×C313yes—
151#123C22×C311yescharacteristic
152#123C22×C313yes—
153#123C22×C316yes—
154C22⋊Q8C22×C313yes—
155C22⋊Q8C22×C313yes—
156C22⋊Q8C22×C313yes—
157C22⋊Q8C22×C313yes—
158C22⋊Q8C22×C313yes—
159C22⋊Q8C22×C316yes—
160#121C22×C313yes—
161#121C22×C313yes—
162#121C22×C313yes—
163C4×Q8C22×C311yescharacteristic
164C4×Q8C22×C313yes—
165C4×Q8C22×C311yescharacteristic
166C4×Q8C22×C313yes—
167C4×Q8C22×C313yes—
168C4×Q8C22×C311yescharacteristic
169C4×Q8C22×C313yes—
170C4×Q8C22×C311yescharacteristic
171C4×Q8C22×C313yes—
172C4×D4C22×C313yes—
173C4×D4C22×C313yes—
174C4×D4C22×C313yes—
175C4×D4C22×C313yes—
176C4×D4C22×C313yes—
177C4×D4C22×C316yes—
178#117C22×C313yes—
179#117C22×C313yes—
180#117C22×C313yes—
181#117C22×C316yes—
182C2×C4⋊C4C22×C313yes—
183C2×C4⋊C4C22×C313yes—
184C2×C42C22×C313yes—
185C23×C3C2411yescharacteristic
186C23×C3C2413yes—
187C23×C3C2411yescharacteristic
188C3×Q8—26no—
189C3×Q8—26no—
190C3×Q8—43no—
191C3×Q8#4114yes—
192C3×D4—26no—
193C3×D4—22no—
194C3×D4—26no—
195C3×D4—43no—
196C2×C3×C4—22no—
197C2×C3×C4—26no—
198C2×C3×C4—26no—
199C2×C3×C4—26no—
200C2×C3×C4—46no—
201C2×C3×C4—43no—
202C2×C3×C4C2411yescharacteristic
203C2×C3×C4C2411yescharacteristic
204C2×C3×C4C2413yes—
205C2×C3×C4C2411yescharacteristic
206C2×C3×C4C2413yes—
207C2×C3×C4C2411yescharacteristic
208C2×C3×C4C2413yes—
209C2×C3×C4C2413yes—
210C2×C3×C4C2413yes—
211C2×C3×C4C2413yes—
212C2×C3×C4C2411yescharacteristic
213C2×C3×C4C2413yes—
214#41—24no—
215C2×Q8—24no—
216C2×Q8—24no—
217C2×Q8C23×C313yes—
218C2×Q8C23×C313yes—
219C2×Q8C23×C313yes—
220C2×Q8C23×C311yescharacteristic
221C2×D4C23×C313yes—
222C2×D4C23×C311yescharacteristic
223C2×D4C23×C313yes—
224C2×D4C23×C313yes—
225C22×C4C23×C311yescharacteristic
226C22×C4C23×C313yes—
227C22×C4C23×C313yes—
228C22×C4C23×C313yes—
229C22×C4C23×C313yes—
230C22×C4C23×C316yes—
231C4⋊C4C23×C313yes—
232C4⋊C4C23×C313yes—
233C4⋊C4C23×C313yes—
234C4⋊C4C23×C313yes—
235C4⋊C4C23×C316yes—
236C4⋊C4C23×C311yescharacteristic
237C4⋊C4C23×C313yes—
238C4⋊C4C23×C313yes—
239C4⋊C4C23×C313yes—
240C4⋊C4C23×C313yes—
241C4⋊C4C23×C316yes—
242C4⋊C4C23×C313yes—
243C4⋊C4C23×C313yes—
244C4⋊C4C23×C316yes—
245C4⋊C4C23×C313yes—
246C4⋊C4C23×C311yescharacteristic
247C4⋊C4C23×C313yes—
248C22⋊C4C23×C313yes—
249C22⋊C4C23×C311yescharacteristic
250C22⋊C4C23×C313yes—
251C22⋊C4C23×C313yes—
252C22⋊C4C23×C313yes—
253C22⋊C4C23×C316yes—
254C22⋊C4C23×C311yescharacteristic
255C22⋊C4C23×C313yes—
256C22⋊C4C23×C311yescharacteristic
257C22⋊C4C23×C316yes—
258C22⋊C4C23×C313yes—
259C22⋊C4C23×C313yes—
260C42C23×C313yes—
261C42C23×C313yes—
262C42C23×C313yes—
263C42C23×C313yes—
264C42C23×C316yes—
265C42C23×C313yes—
266C42C23×C313yes—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—23no—
271C22×C3—21no—
272C22×C3—23no—
273C22×C3—23no—
274C22×C3—21no—
275C22×C32−1+412yes—
276C22×C3C2511yescharacteristic
277C3×C4—21no—
278C3×C4—23no—
279C3×C4—23no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—23no—
283C3×C4—21no—
284C3×C4—23no—
285C3×C4—23no—
286C3×C4—23no—
287C3×C4—21no—
288C3×C4C2×#4116yes—
289C23C24×C311yescharacteristic
290C23C24×C313yes—
291C23C24×C311yescharacteristic
292Q8—26no—
293Q8—26no—
294Q8—43no—
295Q8C3×#4114yes—
296D4—26no—
297D4—22no—
298D4—26no—
299D4—43no—
300C2×C4—22no—
301C2×C4—26no—
302C2×C4—26no—
303C2×C4—26no—
304C2×C4—46no—
305C2×C4—43no—
306C2×C4C24×C311yescharacteristic
307C2×C4C24×C311yescharacteristic
308C2×C4C24×C313yes—
309C2×C4C24×C311yescharacteristic
310C2×C4C24×C313yes—
311C2×C4C24×C311yescharacteristic
312C2×C4C24×C313yes—
313C2×C4C24×C313yes—
314C2×C4C24×C313yes—
315C2×C4C24×C313yes—
316C2×C4C24×C311yescharacteristic
317C2×C4C24×C313yes—
318C2×C3—22no—
319C2×C3—41no—
320C2×C3—43no—
321C2×C3C22×#4111yescharacteristic
322C2×C3C2×2−1+411yescharacteristic
323C2×C3#58511yescharacteristic
324C22—21no—
325C22—21no—
326C22—21no—
327C22—23no—
328C22—21no—
329C22—23no—
330C22—23no—
331C22—21no—
332C22C3×2−1+412yes—
333C22C25×C311yescharacteristic
334C4—21no—
335C4—23no—
336C4—23no—
337C4—21no—
338C4—21no—
339C4—23no—
340C4—21no—
341C4—23no—
342C4—23no—
343C4—23no—
344C4—21no—
345C4C2×C3×#4116yes—
346C3#351611yescharacteristic, direct product factor, Sylow
347C2—22no—
348C2—41no—
349C2—43no—
350C2C22×C3×#4111yescharacteristic
351C2C2×C3×2−1+411yescharacteristic
352C2C3×#58511yescharacteristic
subgroupG/H
center, socleC22×C3C25
commutator, Frattini, cosocleC22C25×C3
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#557C2semidirect
2. C3×#556C2semidirect
3. C3×#554C2semidirect
4. C3×#553C2semidirect
5. C3×#552C2semidirect
6. C3×D4×Q8C2semidirect
7. C3×#548C2semidirect
8. C3×#548C2semidirect
9. C3×#542C2semidirect
10. C3×#541C2semidirect
11. C3×#529C2semidirect
12. C3×#529C2semidirect
13. C3×#520C2semidirect
14. C3×#519C2semidirect
15. C2×C3×C4×Q8C2semidirect
16. #3516C3direct (N × H)
67. #557C2×C3semidirect
68. #556C2×C3semidirect
69. #554C2×C3semidirect
70. #553C2×C3semidirect
71. #552C2×C3semidirect
72. D4×Q8C2×C3semidirect
73. #548C2×C3semidirect
74. #548C2×C3semidirect
75. #542C2×C3semidirect
76. #541C2×C3semidirect
77. #529C2×C3semidirect
78. #529C2×C3semidirect
79. #520C2×C3semidirect
80. #519C2×C3semidirect
81. C2×C4×Q8C2×C3semidirect
346. C3#3516direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5572
C3×#5562
C3×#5542
C3×#5532
C3×#5522
C3×D4×Q82
C3×#5482
C3×#5422
C3×#5412
C3×#5292
C3×#5202
C3×#5192
C2×C3×C4×Q82
#35163