Finite Group Databasebeta

← C3×SD16×C3⋊Dic3 #13163199

Subgroups — C3×SD16×C3⋊Dic3 #13163199

Order 1728.

1004 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
1C2×C33×SD16C211yesmaximal, characteristic
2C3×Q8×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C3×D4×C3⋊Dic3C211yesmaximal, characteristic
4C3×#63662C211yesmaximal, characteristic, complemented
5C3×#63659C211yesmaximal, characteristic
6C3×#63644C211yesmaximal, characteristic, complemented
7C3×C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
8SD16×C3⋊Dic3C311yesmaximal, characteristic, direct product factor
9C3×Dic3×SD16—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C33×D4C2211yescharacteristic
12C22×C3×C3⋊Dic3—21no—
13C33×SD16C414yescomplemented
14C2×C33×C8C2211yescharacteristic
15C3×#3757—21no—
16C3×C4⋊(C3⋊Dic3)—21no—
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C3×C4×C3⋊Dic3—21no—
20C3×C4×C3⋊Dic3C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C2×C32×SD16—24no—
23C2×C32×SD16D314yes—
24C2×C32×SD16C2×C311yescharacteristic
25Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
26D4×C3⋊Dic3C2×C311yescharacteristic
27C3×Q8×Dic3—34no—
28C3×D4×Dic3—34no—
29#63662C2×C311yescharacteristic, complemented
30#63659C2×C311yescharacteristic
31#63644C2×C311yescharacteristic, complemented
32C8×C3⋊Dic3C2×C311yescharacteristic, complemented
33C3×#835—34no—
34C3×#832—34no—
35C3×#817—34no—
36C3×C8×Dic3—34no—
37C23×C33—21no—
38C33×Q8—21no—
39C33×Q8C2×C412yescomplemented
40C33×D4—21no—
41C33×D4C2×C412yes—
42C2×C33×C4—21no—
43C2×C33×C4C2311yescharacteristic
44C2×C3×C3⋊Dic3—21no—
45C2×C3×C3⋊Dic3—21no—
46C2×C3×C3⋊Dic3—42no—
47C2×C3×C3⋊Dic3D411yescharacteristic
48C2×C3×C3⋊Dic3D411yescharacteristic
49C33×C8C2×C412yescomplemented
50C3×C3⋊(C3⋊C8)—21no—
51C3×C4×SD16—91no—
52Dic3×SD16—34no—
53C2×C32×Q8—24no—
54C2×C32×Q8C2×D314yes—
55C2×C32×Q8C22×C311yescharacteristic
56C2×C32×D4—24no—
57C2×C32×D4C2×D314yes—
58C2×C32×D4C22×C311yescharacteristic
59C22×C3⋊Dic3—21no—
60C22×C3×Dic3—64no—
61C32×SD16—216no—
62C32×SD16Dic3116yescomplemented
63C32×SD16C3×C414yescomplemented
64C2×C32×C8—24no—
65C2×C32×C8C2×D314yes—
66C2×C32×C8C22×C311yescharacteristic
67#3757—21no—
68C4⋊(C3⋊Dic3)—21no—
69C4⋊(C3⋊Dic3)C22×C311yescharacteristic
70C4⋊(C3⋊Dic3)C22×C311yescharacteristic
71C4×C3⋊Dic3—21no—
72C4×C3⋊Dic3C22×C311yescharacteristic
73C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
74C3×#217—64no—
75C3×C4⋊Dic3—34no—
76C3×C4⋊Dic3—34no—
77C3×C4⋊Dic3—64no—
78C3×C4×Dic3—34no—
79C3×C4×Dic3—64no—
80C2×C3×C3⋊C8—34no—
81C22×C33—22no—
82C22×C33—41no—
83C22×C33—41no—
84C22×C33C2×D411yescharacteristic
85C33×C4—22no—
86C33×C4C22×C411yescharacteristic
87C33×C4#4111yescharacteristic
88C3×C3⋊Dic3—21no—
89C3×C3⋊Dic3—41no—
90C3×C3⋊Dic3—41no—
91C3×C3⋊Dic3SD1612yesdirect product factor
92C2×C3×SD16—24no—
93C2×C3×SD16C3×D314yes—
94C2×C3×SD16C3⋊D311yescharacteristic
95C3×C4×Q8—91no—
96C3×C4×D4—91no—
97Q8×Dic3—34no—
98D4×Dic3—34no—
99C3×#106—91no—
100C3×#103—91no—
101C3×#102—91no—
102C3×C4×C8—91no—
103#835—34no—
104#832—34no—
105#817—34no—
106C8×Dic3—34no—
107C23×C32—21no—
108C23×C32—24no—
109C23×C32—44no—
110C32×Q8—21no—
111C32×Q8—28no—
112C32×Q8—24no—
113C32×Q8—44no—
114C32×Q8C2×Dic318yescomplemented
115C32×Q8C2×C3×C412yescomplemented
116C32×D4—21no—
117C32×D4—28no—
118C32×D4—24no—
119C32×D4—44no—
120C32×D4C2×Dic318yes—
121C32×D4C2×C3×C412yes—
122C2×C32×C4—21no—
123C2×C32×C4—24no—
124C2×C32×C4—24no—
125C2×C32×C4—44no—
126C2×C32×C4C22×D314yes—
127C2×C32×C4C23×C311yescharacteristic
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3—42no—
131C2×C3⋊Dic3C3×D411yescharacteristic
132C2×C3⋊Dic3C3×D411yescharacteristic
133C2×C3×Dic3—34no—
134C2×C3×Dic3—34no—
135C2×C3×Dic3—64no—
136C2×C3×Dic3—64no—
137C2×C3×Dic3—128no—
138C32×C8—28no—
139C32×C8C2×Dic318yescomplemented
140C32×C8C2×C3×C412yescomplemented
141C3⋊(C3⋊C8)—21no—
142C3×C3⋊C8—64no—
143C4×SD16—91noSylow
144C2×C33—42no—
145C2×C33C4×D411yescharacteristic
146C2×C33C2×SD1611yescharacteristic
147C2×C33#13511yescharacteristic
148C2×C3×Q8—24no—
149C2×C3×Q8C2×C3×D314yes—
150C2×C3×Q8C2×C3⋊D311yescharacteristic
151C2×C3×D4—24no—
152C2×C3×D4C2×C3×D314yes—
153C2×C3×D4C2×C3⋊D311yescharacteristic
154C22×C3×C4—181no—
155C22×Dic3—64no—
156C3×SD16—216no—
157C3×SD16C3×Dic3116yescomplemented
158C3×SD16C3⋊Dic314yesdirect product factor
159C2×C3×C8—24no—
160C2×C3×C8—91no—
161C2×C3×C8C2×C3×D314yes—
162C2×C3×C8C2×C3⋊D311yescharacteristic
163C3×C4⋊C4—91no—
164C3×C4⋊C4—91no—
165C3×C4⋊C4—181no—
166C3×C22⋊C4—181no—
167C3×C42—91no—
168C3×C42—181no—
169#217—64no—
170C4⋊Dic3—34no—
171C4⋊Dic3—34no—
172C4⋊Dic3—64no—
173C4×Dic3—34no—
174C4×Dic3—64no—
175C2×C3⋊C8—34no—
176C22×C32—22no—
177C22×C32—24no—
178C22×C32—28no—
179C22×C32—48no—
180C22×C32—41no—
181C22×C32—41no—
182C22×C32—44no—
183C22×C32—44no—
184C22×C32—84no—
185C22×C32—84no—
186C22×C32D3×D414yes—
187C22×C32C2×C3×D411yescharacteristic
188C32×C4—22no—
189C32×C4—24no—
190C32×C4—24no—
191C32×C4—28no—
192C32×C4—48no—
193C32×C4#23714yes—
194C32×C4C22×Dic314yes—
195C32×C4C22×C3×C411yescharacteristic
196C32×C4C3×#4111yescharacteristic
197C3⋊Dic3—21no—
198C3⋊Dic3—41no—
199C3⋊Dic3—41no—
200C3⋊Dic3C3×SD1612yesdirect product factor
201C3×Dic3—38no—
202C3×Dic3—64no—
203C3×Dic3—124no—
204C3×Dic3—124no—
205C2×SD16C3×C3⋊D311yescharacteristic
206C4×Q8—91no—
207C4×D4—91no—
208#106—91no—
209#103—91no—
210#102—91no—
211C4×C8—91no—
212C33C4×SD1611yescharacteristic, complemented, Sylow
213C23×C3—24no—
214C23×C3—21no—
215C23×C3—44no—
216C3×Q8—24no—
217C3×Q8—21no—
218C3×Q8—28no—
219C3×Q8—44no—
220C3×Q8C2×C3×Dic318yescomplemented
221C3×Q8C2×C3⋊Dic312yescomplemented
222C3×D4—24no—
223C3×D4—21no—
224C3×D4—28no—
225C3×D4—44no—
226C3×D4C2×C3×Dic318yes—
227C3×D4C2×C3⋊Dic312yes—
228C2×C3×C4—24no—
229C2×C3×C4—21no—
230C2×C3×C4—24no—
231C2×C3×C4—44no—
232C2×C3×C4—91no—
233C2×C3×C4—91no—
234C2×C3×C4—181no—
235C2×C3×C4—181no—
236C2×C3×C4—362no—
237C2×C3×C4C22×C3×D314yes—
238C2×C3×C4C22×C3⋊D311yescharacteristic
239C2×Dic3—34no—
240C2×Dic3—34no—
241C2×Dic3—64no—
242C2×Dic3—64no—
243C2×Dic3—128no—
244C3×C8—28no—
245C3×C8—181no—
246C3×C8C2×C3×Dic318yescomplemented
247C3×C8C2×C3⋊Dic312yescomplemented
248C3⋊C8—64no—
249C2×C32—24no—
250C2×C32—24no—
251C2×C32—24no—
252C2×C32—42no—
253C2×C32—48no—
254C2×C32—88no—
255C2×C32D3×SD1614yes—
256C2×C32#91614yes—
257C2×C32D4×Dic314yes—
258C2×C32C3×C4×D411yescharacteristic
259C2×C32C2×C3×SD1611yescharacteristic
260C2×C32C3×#13511yescharacteristic
261C2×Q8C2×C3×C3⋊D311yescharacteristic
262C2×D4C2×C3×C3⋊D311yescharacteristic
263C22×C4—181no—
264SD16C3×C3⋊Dic314yesdirect product factor
265C2×C8—91no—
266C2×C8C2×C3×C3⋊D311yescharacteristic
267C4⋊C4—91no—
268C4⋊C4—91no—
269C4⋊C4—181no—
270C22⋊C4—181no—
271C42—91no—
272C42—181no—
273C22×C3—28no—
274C22×C3—22no—
275C22×C3—24no—
276C22×C3—48no—
277C22×C3—44no—
278C22×C3—44no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—84no—
282C22×C3—84no—
283C22×C3C3×D3×D414yes—
284C22×C3D4×C3⋊D311yescharacteristic
285C3×C4—28no—
286C3×C4—22no—
287C3×C4—24no—
288C3×C4—24no—
289C3×C4—48no—
290C3×C4—92no—
291C3×C4—181no—
292C3×C4—361no—
293C3×C4—361no—
294C3×C4C3×#23714yes—
295C3×C4C22×C3×Dic314yes—
296C3×C4#383011yescharacteristic
297C3×C4C22×C3⋊Dic311yescharacteristic
298Dic3—38no—
299Dic3—64no—
300Dic3—124no—
301Dic3—124no—
302C32—24no—
303C32Dic3×SD1614yescomplemented
304C32C3×C4×SD1611yescharacteristic, complemented
305C23—21no—
306Q8—21no—
307Q8C2×C3×C3⋊Dic312yescomplemented
308D4—21no—
309D4C2×C3×C3⋊Dic312yes—
310C2×C4—21no—
311C2×C4—91no—
312C2×C4—91no—
313C2×C4—181no—
314C2×C4—181no—
315C2×C4—362no—
316C2×C4C22×C3×C3⋊D311yescharacteristic
317C8—181no—
318C8C2×C3×C3⋊Dic312yescomplemented
319C2×C3—24no—
320C2×C3—24no—
321C2×C3—24no—
322C2×C3—48no—
323C2×C3—42no—
324C2×C3—88no—
325C2×C3C3×D3×SD1614yes—
326C2×C3C3×#91614yes—
327C2×C3C3×D4×Dic314yes—
328C2×C3SD16×C3⋊D311yescharacteristic
329C2×C3#6412511yescharacteristic
330C2×C3D4×C3⋊Dic311yescharacteristic
331C22—22no—
332C22—41no—
333C22—41no—
334C22C3×D4×C3⋊D311yescharacteristic
335C4—22no—
336C4—92no—
337C4—181no—
338C4—361no—
339C4—361no—
340C4C3×#383011yescharacteristic
341C4C22×C3×C3⋊Dic311yescharacteristic
342C3—24no—
343C3C3×Dic3×SD1614yescomplemented
344C3SD16×C3⋊Dic311yescharacteristic, direct product factor
345C2—42no—
346C2C3×SD16×C3⋊D311yescharacteristic
347C2C3×#6412511yescharacteristic
348C2C3×D4×C3⋊Dic311yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C22×C3×C3⋊D3
FittingC2×C33×SD16C2
socleC22×C33C2×D4
prime radicalC2×C33×D4C22
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×SD16C3⋊D3
hypocenterC32C3×C4×SD16
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×SD16

Complements

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

NH ≅ G/Ntype
2. C3×Q8×C3⋊Dic3C2semidirect
4. C3×#63662C2semidirect
6. C3×#63644C2semidirect
7. C3×C8×C3⋊Dic3C2semidirect
8. SD16×C3⋊Dic3C3direct (N × H)
13. C33×SD16C4semidirect
25. Q8×C3⋊Dic3C2×C3semidirect
29. #63662C2×C3semidirect
31. #63644C2×C3semidirect
32. C8×C3⋊Dic3C2×C3semidirect
39. C33×Q8C2×C4semidirect
49. C33×C8C2×C4semidirect
62. C32×SD16Dic3semidirect
63. C32×SD16C3×C4semidirect
91. C3×C3⋊Dic3SD16direct (N × H)
114. C32×Q8C2×Dic3semidirect
115. C32×Q8C2×C3×C4semidirect
139. C32×C8C2×Dic3semidirect
140. C32×C8C2×C3×C4semidirect
157. C3×SD16C3×Dic3semidirect
158. C3×SD16C3⋊Dic3direct (N × H)
200. C3⋊Dic3C3×SD16direct (N × H)
212. C33C4×SD16semidirect
220. C3×Q8C2×C3×Dic3semidirect
221. C3×Q8C2×C3⋊Dic3semidirect
246. C3×C8C2×C3×Dic3semidirect
247. C3×C8C2×C3⋊Dic3semidirect
264. SD16C3×C3⋊Dic3direct (N × H)
303. C32Dic3×SD16semidirect
304. C32C3×C4×SD16semidirect
307. Q8C2×C3×C3⋊Dic3semidirect
318. C8C2×C3×C3⋊Dic3semidirect
343. C3C3×Dic3×SD16semidirect
344. C3SD16×C3⋊Dic3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×SD162
C3×Q8×C3⋊Dic32
C3×D4×C3⋊Dic32
C3×#636622
C3×#636592
C3×#636442
C3×C8×C3⋊Dic32
SD16×C3⋊Dic33
C3×Dic3×SD163