Finite Group Databasebeta

← C3⋊Dic3×#214 #13160924

Subgroups — C3⋊Dic3×#214 #13160924

Order 1728.

932 conjugacy classes of subgroups; only the 220 normal subgroups are shown.

normal subgroups only

This group has 932 conjugacy classes of subgroups — too many to draw. Only its 220 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D4×C3⋊Dic3C211yesmaximal, characteristic
2C2×C32×#214C211yesmaximal, characteristic
3Dic6×C3⋊Dic3C211yesmaximal, characteristic, complemented
4#11722202C211yesmaximal, characteristic
5#11722052C211yesmaximal, characteristic, complemented
6#11722037C211yesmaximal, characteristic, complemented
7C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
8SD16×C3⋊Dic3—31nomaximal
9Dic3×#214—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C22×C3×C3⋊Dic3—21no—
12C2×C32×Dic6C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×#3757—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×#214C414yescomplemented
19C2×C32×C3⋊C8C2211yescharacteristic
20Dic3⋊(C3⋊Dic3)—21no—
21Dic3×C3⋊Dic3—21no—
22C2×C32×SD16—31no—
23Q8×C3⋊Dic3—31no—
24D4×C3⋊Dic3D311yescharacteristic
25C3×D4×Dic3—34no—
26C2×C3×#214D314yes—
27Dic3×Dic6—34no—
28#63662—31no—
29#63659—31no—
30#63659—34no—
31#63659—64no—
32#63644—31no—
33C8×C3⋊Dic3—31no—
34#63573—34no—
35#63565—34no—
36Dic3×C3⋊C8—34no—
37C23×C33—21no—
38C33×D4—21no—
39C33×D4C2×C412yes—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42C2×C3×C3⋊Dic3—21no—
43C2×C3×C3⋊Dic3—42no—
44C2×C3×C3⋊Dic3D411yescharacteristic
45C2×C3×C3⋊Dic3D411yescharacteristic
46C2×C32×Dic3—21no—
47C32×Dic6—21no—
48C32×Dic6C2×C412yescomplemented
49C3⋊(C3⋊(C3⋊C8))—21no—
50C32×C3⋊C8C2×C412yescomplemented
51Dic3×SD16—94no—
52C4×#214—91no—
53C2×C32×Q8—31no—
54C2×C32×D4—24no—
55C2×C32×D4C2×D311yescharacteristic
56C2×C32×D4C2×D314yes—
57C22×C3⋊Dic3—21no—
58C22×C3×Dic3—64no—
59C2×C3×Dic6C2×D314yes—
60C32×SD16—34no—
61C2×C32×C8—31no—
62#3757—21no—
63C4⋊(C3⋊Dic3)—31no—
64C4⋊(C3⋊Dic3)—34no—
65C4⋊(C3⋊Dic3)—61no—
66C4⋊(C3⋊Dic3)—64no—
67C4⋊(C3⋊Dic3)C2×D311yescharacteristic
68C4×C3⋊Dic3—61no—
69C4×C3⋊Dic3C2×D311yescharacteristic
70C2×C3⋊(C3⋊C8)—31no—
71C2×C3⋊(C3⋊C8)—34no—
72C2×C3⋊(C3⋊C8)—64no—
73C3×#217—64no—
74C3×#214Dic3116yescomplemented
75C3×C4⋊Dic3—34no—
76C3×C4×Dic3—34no—
77C2×C3×C3⋊C8C2×D314yes—
78Dic3⋊Dic3—64no—
79Dic32—64no—
80C22×C33—22no—
81C22×C33—41no—
82C22×C33—41no—
83C22×C33C2×D411yescharacteristic
84C33×C4C22×C411yescharacteristic
85C33×C4#4111yescharacteristic
86C3⋊(C3⋊Dic3)—41no—
87C3×C3⋊Dic3—21no—
88C3×C3⋊Dic3—41no—
89C3×C3⋊Dic3SD1612yescomplemented
90C32×Dic3—22no—
91C2×C3×SD16—34no—
92C3×C4×D4—91no—
93Q8×Dic3—94no—
94D4×Dic3—34no—
95C2×#214C3⋊D311yescharacteristic
96C4×Dic6—91no—
97#835—94no—
98#832—94no—
99#832—91no—
100#832—184no—
101#817—94no—
102C8×Dic3—94no—
103#810—91no—
104#808—91no—
105C4×C3⋊C8—91no—
106C23×C32—21no—
107C23×C32—24no—
108C23×C32—44no—
109C32×Q8—32no—
110C32×Q8—61no—
111C32×D4—21no—
112C32×D4—28no—
113C32×D4—24no—
114C32×D4—44no—
115C32×D4C4×D312yes—
116C32×D4C2×Dic318yes—
117C2×C32×C4—24no—
118C2×C32×C4—61no—
119C2×C32×C4C22×D314yes—
120C2×C32×C4C22×D311yescharacteristic
121C2×C3⋊Dic3—21no—
122C2×C3⋊Dic3—42no—
123C2×C3⋊Dic3—61no—
124C2×C3⋊Dic3—64no—
125C2×C3⋊Dic3—124no—
126C2×C3⋊Dic3#6711yescharacteristic
127C2×C3⋊Dic3#6711yescharacteristic
128C2×C3×Dic3—24no—
129C2×C3×Dic3—34no—
130C2×C3×Dic3—34no—
131C2×C3×Dic3—64no—
132C2×C3×Dic3—128no—
133C3×Dic6—24no—
134C3×Dic6C2×Dic318yescomplemented
135C32×C8—32no—
136C3⋊(C3⋊C8)—61no—
137C3⋊(C3⋊C8)—64no—
138C3⋊(C3⋊C8)—124no—
139C3×C3⋊C8C2×Dic318yescomplemented
140C4×SD16—271noSylow
141C2×C33—42no—
142C2×C33C4×D411yescharacteristic
143C2×C33C2×SD1611yescharacteristic
144C2×C33#13511yescharacteristic
145C2×C3×Q8—34no—
146C2×C3×D4—24no—
147C2×C3×D4D3214yes—
148C2×C3×D4C2×C3⋊D311yescharacteristic
149C22×C3×C4—181no—
150C22×Dic3—64no—
151C2×Dic6C2×C3⋊D311yescharacteristic
152C3×SD16—316no—
153C2×C3×C8—34no—
154C3×C4⋊C4—91no—
155C3×C22⋊C4—181no—
156C3×C42—91no—
157#217—64no—
158#214C3⋊Dic314yesdirect product factor
159C4⋊Dic3—34no—
160C4⋊Dic3—94no—
161C4⋊Dic3—91no—
162C4⋊Dic3—184no—
163C4⋊Dic3—184no—
164Dic3⋊C4—181no—
165C4×Dic3—34no—
166C4×Dic3—181no—
167C4×Dic3—184no—
168C2×C3⋊C8—94no—
169C2×C3⋊C8—91no—
170C2×C3⋊C8—184no—
171C2×C3⋊C8C2×C3⋊D311yescharacteristic
172C22×C32—22no—
173C22×C32—24no—
174C22×C32—28no—
175C22×C32—48no—
176C22×C32—41no—
177C22×C32—41no—
178C22×C32—44no—
179C22×C32—44no—
180C22×C32—84no—
181C22×C32—84no—
182C22×C32D3×D414yes—
183C22×C32C2×#6711yescharacteristic
184C32×C4—24no—
185C32×C4—24no—
186C32×C4—62no—
187C32×C4C2×C4×D311yescharacteristic
188C32×C4#23511yescharacteristic
189C32×C4#23714yes—
190C32×C4C22×Dic314yes—
191C3⋊Dic3—21no—
192C3⋊Dic3—41no—
193C3⋊Dic3—121no—
194C3⋊Dic3—124no—
195C3⋊Dic3—244no—
196C3⋊Dic3#21412yesdirect product factor
197C3×Dic3—28no—
198C3×Dic3—38no—
199C3×Dic3—64no—
200C3×Dic3—124no—
201C2×SD16—31no—
202C4×Q8—271no—
203C4×D4—91no—
204#106—271no—
205#103—271no—
206#102—271no—
207C4×C8—271no—
208C33C4×SD1611yescharacteristic, complemented, Sylow
209C23×C3—24no—
210C23×C3—21no—
211C23×C3—44no—
212C3×Q8—38no—
213C3×Q8—64no—
214C3×D4—24no—
215C3×D4—21no—
216C3×D4—28no—
217C3×D4—44no—
218C3×D4D3×Dic318yes—
219C3×D4C2×C3⋊Dic312yes—
220C2×C3×C4—24no—
221C2×C3×C4—64no—
222C2×C3×C4—91no—
223C2×C3×C4—91no—
224C2×C3×C4—181no—
225C2×C3×C4—362no—
226C2×C3×C4C2×D3214yes—
227C2×C3×C4C22×C3⋊D311yescharacteristic
228C2×Dic3—21no—
229C2×Dic3—34no—
230C2×Dic3—34no—
231C2×Dic3—64no—
232C2×Dic3—128no—
233C2×Dic3—184no—
234C2×Dic3—181no—
235C2×Dic3—364no—
236Dic6—21no—
237Dic6C2×C3⋊Dic312yescomplemented
238C3×C8—38no—
239C3⋊C8—184no—
240C3⋊C8—181no—
241C3⋊C8—364no—
242C3⋊C8C2×C3⋊Dic312yescomplemented
243C2×C32—24no—
244C2×C32—24no—
245C2×C32—24no—
246C2×C32—42no—
247C2×C32—48no—
248C2×C32—88no—
249C2×C32D3×SD1614yes—
250C2×C32#91614yes—
251C2×C32C4×#6711yescharacteristic
252C2×C32C2×#21411yescharacteristic
253C2×C32D4×Dic314yes—
254C2×C32#95011yescharacteristic
255C2×Q8—31no—
256C2×D4D3×C3⋊D311yescharacteristic
257C22×C4—181no—
258SD16—34no—
259C2×C8—31no—
260C2×C8—271no—
261C4⋊C4—91no—
262C4⋊C4—271no—
263C4⋊C4—541no—
264C22⋊C4—181no—
265C42—91no—
266C42—541no—
267C22×C3—22no—
268C22×C3—28no—
269C22×C3—24no—
270C22×C3—48no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—44no—
274C22×C3—44no—
275C22×C3—84no—
276C22×C3—84no—
277C22×C3D3×#6714yes—
278C22×C3D4×C3⋊D311yescharacteristic
279C3×C4—24no—
280C3×C4—24no—
281C3×C4—68no—
282C3×C4—92no—
283C3×C4—181no—
284C3×C4—361no—
285C3×C4C2×D3×Dic314yes—
286C3×C4#380414yes—
287C3×C4#383011yescharacteristic
288C3×C4C22×C3⋊Dic311yescharacteristic
289Dic3—22no—
290Dic3—38no—
291Dic3—64no—
292Dic3—124no—
293Dic3—361no—
294Dic3—364no—
295Dic3—724no—
296C32—24no—
297C32C4×#21411yescharacteristic, complemented
298C32Dic3×SD1614yescomplemented
299C23—21no—
300Q8—32no—
301Q8—61no—
302D4—21no—
303D4D3×C3⋊Dic312yes—
304C2×C4—61no—
305C2×C4—91no—
306C2×C4—91no—
307C2×C4—181no—
308C2×C4—362no—
309C2×C4—541no—
310C2×C4C2×D3×C3⋊D311yescharacteristic
311C8—32no—
312C8—541no—
313C2×C3—24no—
314C2×C3—24no—
315C2×C3—24no—
316C2×C3—42no—
317C2×C3—48no—
318C2×C3—88no—
319C2×C3D3×#21414yes—
320C2×C3#6393514yes—
321C2×C3Dic3×#6714yes—
322C2×C3SD16×C3⋊D311yescharacteristic
323C2×C3#6412511yescharacteristic
324C2×C3D4×C3⋊Dic311yescharacteristic
325C22—22no—
326C22—41no—
327C22—41no—
328C22C3⋊D3×#6711yescharacteristic
329C4—62no—
330C4—92no—
331C4—181no—
332C4—361no—
333C4—1081no—
334C4C2×D3×C3⋊Dic311yescharacteristic
335C4#8902811yescharacteristic
336C3—24no—
337C3Dic3×#21414yescomplemented
338C3SD16×C3⋊Dic311yescharacteristic, complemented
339C2—42no—
340C2C3⋊D3×#21411yescharacteristic
341C2#1172356111yescharacteristic
342C2#67×C3⋊Dic311yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
Fitting, prime radicalC2×C33×D4C22
socleC22×C33C2×D4
cosocleC2×C33×C4C23
hypercenterC2×D4D3×C3⋊D3
hypocenterC33C4×SD16
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
3. Dic6×C3⋊Dic3C2semidirect
5. #11722052C2semidirect
6. #11722037C2semidirect
7. C3⋊C8×C3⋊Dic3C2semidirect
18. C32×#214C4semidirect
48. C32×Dic6C2×C4semidirect
50. C32×C3⋊C8C2×C4semidirect
74. C3×#214Dic3semidirect
89. C3×C3⋊Dic3SD16semidirect
134. C3×Dic6C2×Dic3semidirect
139. C3×C3⋊C8C2×Dic3semidirect
158. #214C3⋊Dic3direct (N × H)
196. C3⋊Dic3#214direct (N × H)
208. C33C4×SD16semidirect
237. Dic6C2×C3⋊Dic3semidirect
242. C3⋊C8C2×C3⋊Dic3semidirect
297. C32C4×#214semidirect
298. C32Dic3×SD16semidirect
337. C3Dic3×#214semidirect
338. C3SD16×C3⋊Dic3semidirect

Maximal subgroups

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

subgroupindex
C3×D4×C3⋊Dic32
C2×C32×#2142
Dic6×C3⋊Dic32
#117222022
#117220522
#117220372
C3⋊C8×C3⋊Dic32
SD16×C3⋊Dic33
Dic3×#2143