Finite Group Databasebeta

← C3⋊D3×#833 #13161040

Subgroups — C3⋊D3×#833 #13161040

Order 1728.

1300 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1300 conjugacy classes of subgroups — too many to draw. Only its 176 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
1C2×C3×D4×C3⋊D3C211yesmaximal, characteristic
2C3⋊D3×#208C212yesmaximal, complemented
3#11722203C211yesmaximal, characteristic, complemented
4C32×#833C211yesmaximal, characteristic, complemented
5#11722026C212yesmaximal, complemented
6C3⋊D3×#100—31nomaximal
7D3×#833—34nomaximal
8C23×C3×C3⋊D3C412yes—
9C2×C33×D4C2211yescharacteristic
10C2×C3×#638C412yes—
11C3×D4×C3⋊D3—22no—
12C3×D4×C3⋊D3—22no—
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15#88855C2212yescomplemented
16C32×#208C2212yescomplemented
17#88787—22no—
18C3⋊D3×C3⋊C8—22no—
19C2×D4×C3⋊D3D311yescharacteristic
20C2×C3×D3×D4—34no—
21M16×C3⋊D3—32no—
22D3×#208—38no—
23C32×#100—31no—
24#63660—31no—
25#63660—34no—
26#63660—64no—
27#63650—32no—
28C3×#833D314yescomplemented
29#63558—38no—
30C23×C33C2×C412yes—
31C22×C3×C3⋊D3—22no—
32C22×C3×C3⋊D3—22no—
33C22×C3×C3⋊D3—44no—
34C22×C3×C3⋊D3C2×C411yescharacteristic
35C22×C3×C3⋊D3C2×C412yes—
36C33×D4—22no—
37C2×C33×C4C2311yescharacteristic
38C3×#638—24no—
39C2×C3×C3⋊Dic3C2×C411yescharacteristic
40C3×C4⋊(C3⋊D3)—22no—
41C3×C4×C3⋊D3D412yes—
42C3×C4×C3⋊D3D412yes—
43C3⋊(C3⋊(C3⋊C8))—22no—
44C32×C3⋊C8—22no—
45C2×#833—91no—
46D3×#100—94no—
47C23×C3⋊D3Dic312yes—
48C23×C3×D3—38no—
49C2×C32×D4—24no—
50C2×C32×D4C2×D314yes—
51C2×C32×D4C2×D311yescharacteristic
52C2×#638Dic312yes—
53D4×C3⋊D3—22no—
54D4×C3⋊D3—22no—
55C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
56C2×C4×C3⋊D3C2×D311yescharacteristic
57C2×C3×#67—38no—
58C3×D3×D4—68no—
59C3×D3×D4—68no—
60C2×C3×D12—34no—
61C2×C3×C4×D3—34no—
62C32×M16—32no—
63#3748—32no—
64#3748—38no—
65#3748—68no—
66#3743—62no—
67C8×C3⋊D3—62no—
68C3×#208C2×D318yescomplemented
69#3711—68no—
70D3×C3⋊C8—68no—
71C22×C33—22no—
72C22×C33—42no—
73C22×C33C22×C411yescharacteristic
74C2×C3×C3⋊D3—22no—
75C2×C3×C3⋊D3—22no—
76C2×C3×C3⋊D3—44no—
77C2×C3×C3⋊D3—44no—
78C2×C3×C3⋊D3—42no—
79C2×C3×C3⋊D3C22⋊C411yescharacteristic
80C2×C3×C3⋊D3C22⋊C411yescharacteristic
81C33×C4C2×D412yes—
82C3×C3⋊Dic3C22⋊C412yes—
83C22×C3×D4—91no—
84C2×D3×D4—34no—
85C2×#208—92no—
86D3×M16—98no—
87C3×#100—34no—
88#833—94no—
89#833—91no—
90#833—92no—
91#833—184no—
92#833C3⋊D311yescharacteristic, direct product factor
93#823—98no—
94C23×C32—28no—
95C23×C32C4×D318yes—
96C23×C32C2×Dic312yes—
97C22×C3⋊D3—22no—
98C22×C3⋊D3—22no—
99C22×C3⋊D3—44no—
100C22×C3⋊D3C2×Dic311yescharacteristic
101C22×C3⋊D3C2×Dic312yes—
102C22×C3×D3—34no—
103C22×C3×D3—38no—
104C22×C3×D3—68no—
105C22×C3×D3—68no—
106C22×C3×D3—1216no—
107C32×D4—22no—
108C32×D4—28no—
109C32×D4—48no—
110C2×C32×C4—24no—
111C2×C32×C4C22×D314yes—
112C2×C32×C4C22×D311yescharacteristic
113#638—24no—
114C2×C3⋊Dic3C2×Dic311yescharacteristic
115C4⋊(C3⋊D3)—22no—
116C4×C3⋊D3#6712yes—
117C4×C3⋊D3#6712yes—
118C3×#67—616no—
119C2×C3×Dic3—34no—
120C3×D12—68no—
121C3×C4×D3—38no—
122C3×C4×D3—38no—
123C32×C8—62no—
124C3⋊(C3⋊C8)—62no—
125C3⋊(C3⋊C8)—68no—
126C3⋊(C3⋊C8)—128no—
127C3×C3⋊C8—28no—
128C2×#100—271noSylow
129C2×C33—21no—
130C2×C33—42no—
131C2×C33C2×C22⋊C411yescharacteristic
132C3×C3⋊D3—21no—
133C3×C3⋊D3—42no—
134C3×C3⋊D3#10012yescomplemented
135C24×C3—92no—
136C23×D3—38no—
137C2×C3×D4—24no—
138C2×C3×D4—92no—
139C2×C3×D4—91no—
140C2×C3×D4—182no—
141C2×C3×D4—182no—
142C2×C3×D4D3214yes—
143C2×C3×D4C2×C3⋊D311yescharacteristic
144C22×C3×C4—91no—
145C2×#67—38no—
146D3×D4—68no—
147D3×D4—68no—
148C2×D12—34no—
149C2×C4×D3—34no—
150C3×M16—38no—
151#208—98no—
152#208—92no—
153#208—188no—
154#208—182no—
155#208C2×C3⋊D312yescomplemented
156C2×C3⋊C8—182no—
157ΓL(1,𝔽25)—188no—
158D3×C8—188no—
159C22×C32—22no—
160C22×C32—24no—
161C22×C32—28no—
162C22×C32—48no—
163C22×C32—42no—
164C22×C32—48no—
165C22×C32—88no—
166C22×C32C2×C4×D314yes—
167C22×C32C22×Dic311yescharacteristic
168C2×C3⋊D3—22no—
169C2×C3⋊D3—22no—
170C2×C3⋊D3—44no—
171C2×C3⋊D3—44no—
172C2×C3⋊D3—42no—
173C2×C3⋊D3#21711yescharacteristic
174C2×C3⋊D3#21711yescharacteristic
175C2×C3×D3—34no—
176C2×C3×D3—34no—
177C2×C3×D3—68no—
178C2×C3×D3—68no—
179C2×C3×D3—1216no—
180C2×C3×D3—1216no—
181C2×C3×D3—128no—
182C32×C4—28no—
183C32×C4D3×D418yes—
184C32×C4C2×#6712yes—
185C3⋊Dic3#21712yes—
186C3×Dic3—38no—
187C22×D4—91no—
188C2×M16—272no—
189#100—31no—
190#100—272no—
191#100—271no—
192C33C2×#10011yescharacteristic, complemented, Sylow
193C23×C3—28no—
194C23×C3—91no—
195C23×C3—92no—
196C23×C3—182no—
197C23×C3—182no—
198C23×C3—364no—
199C23×C3D3×Dic318yes—
200C23×C3C4×C3⋊D312yes—
201C22×D3—34no—
202C22×D3—38no—
203C22×D3—68no—
204C22×D3—68no—
205C22×D3—1216no—
206C3×D4—28no—
207C3×D4—22no—
208C3×D4—48no—
209C3×D4—184no—
210C3×D4—182no—
211C2×C3×C4—24no—
212C2×C3×C4—91no—
213C2×C3×C4—92no—
214C2×C3×C4—92no—
215C2×C3×C4C2×D3214yes—
216C2×C3×C4C22×C3⋊D311yescharacteristic
217#67—616no—
218C2×Dic3—34no—
219D12—68no—
220C4×D3—38no—
221C4×D3—38no—
222C3×C8—68no—
223C3⋊C8—22no—
224C3⋊C8—188no—
225C3⋊C8—182no—
226C3⋊C8—368no—
227C2×C32—21no—
228C2×C32—24no—
229C2×C32—24no—
230C2×C32—44no—
231C2×C32—42no—
232C2×C32—48no—
233C2×C32—88no—
234C2×C32D3×C22⋊C414yes—
235C2×C32C2×#21711yescharacteristic
236C3⋊D3—21no—
237C3⋊D3—42no—
238C3⋊D3#83312yesdirect product factor
239C3×D3—38no—
240C3×D3—64no—
241C3×D3—128no—
242C24—92no—
243C2×D4—92no—
244C2×D4—91no—
245C2×D4—182no—
246C2×D4—182no—
247C2×D4D3×C3⋊D311yescharacteristic
248C22×C4—91no—
249M16—32no—
250M16—272no—
251M16—542no—
252C2×C8—542no—
253C22×C3—22no—
254C22×C3—28no—
255C22×C3—24no—
256C22×C3—48no—
257C22×C3—48no—
258C22×C3—42no—
259C22×C3—88no—
260C22×C3—91no—
261C22×C3—91no—
262C22×C3—182no—
263C22×C3—182no—
264C22×C3—362no—
265C22×C3—364no—
266C22×C3—364no—
267C22×C3C2×D3×Dic314yes—
268C22×C3C2×C4×C3⋊D311yescharacteristic
269C2×D3—34no—
270C2×D3—34no—
271C2×D3—68no—
272C2×D3—68no—
273C2×D3—1216no—
274C2×D3—1216no—
275C2×D3—128no—
276C3×C4—28no—
277C3×C4—92no—
278C3×C4D3×#6718yes—
279C3×C4D4×C3⋊D312yes—
280Dic3—38no—
281C32—24no—
282C32D3×#10014yescomplemented
283C32C2×#83311yescharacteristic, complemented
284C23—91no—
285C23—92no—
286C23—182no—
287C23—182no—
288C23—364no—
289C23Dic3×C3⋊D312yes—
290D4—22no—
291D4—184no—
292D4—182no—
293C2×C4—91no—
294C2×C4—92no—
295C2×C4—92no—
296C2×C4C2×D3×C3⋊D311yescharacteristic
297C8—62no—
298C8—542no—
299C2×C3—24no—
300C2×C3—21no—
301C2×C3—24no—
302C2×C3—44no—
303C2×C3—48no—
304C2×C3—42no—
305C2×C3—88no—
306C2×C3—92no—
307C2×C3—181no—
308C2×C3—362no—
309C2×C3D3×#21714yes—
310C2×C3C22⋊C4×C3⋊D311yescharacteristic
311D3—38no—
312D3—64no—
313D3—128no—
314C22—22no—
315C22—42no—
316C22—91no—
317C22—91no—
318C22—182no—
319C22—182no—
320C22—362no—
321C22—364no—
322C22—364no—
323C22C2×Dic3×C3⋊D311yescharacteristic
324C4—92no—
325C4C3⋊D3×#6712yes—
326C3—24no—
327C3D3×#83314yescomplemented
328C3C3⋊D3×#10011yescharacteristic, complemented
329C2—21no—
330C2—42no—
331C2—92no—
332C2—181no—
333C2—362no—
334C2C3⋊D3×#21711yescharacteristic
subgroupG/H
centerC2C3⋊D3×#217
commutatorC22×C33C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×D4×C3⋊D3C2
cosocleC2×C33×C4C23
hypercenterC2×D4D3×C3⋊D3
hypocenterC33C2×#100
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C3⋊D3×#208C2semidirect
3. #11722203C2semidirect
4. C32×#833C2semidirect
5. #11722026C2semidirect
15. #88855C22semidirect
16. C32×#208C22semidirect
28. C3×#833D3semidirect
68. C3×#208C2×D3semidirect
92. #833C3⋊D3direct (N × H)
134. C3×C3⋊D3#100semidirect
155. #208C2×C3⋊D3semidirect
192. C33C2×#100semidirect
238. C3⋊D3#833direct (N × H)
282. C32D3×#100semidirect
283. C32C2×#833semidirect
327. C3D3×#833semidirect
328. C3C3⋊D3×#100semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×C3⋊D32
C3⋊D3×#2082
#117222032
C32×#8332
#117220262
C3⋊D3×#1003
D3×#8333