Finite Group Databasebeta

← C3×C3⋊D3×#100 #13162782

Subgroups — C3×C3⋊D3×#100 #13162782

Order 1728.

1300 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1300 conjugacy classes of subgroups — too many to draw. Only its 202 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×M16×C3⋊D3C212yesmaximal, complemented
3C33×#100C211yesmaximal, characteristic, complemented
4C3×#63660C211yesmaximal, characteristic, complemented
5C3×#63650C212yesmaximal, complemented
6C3⋊D3×#100C311yesmaximal, characteristic, direct product factor
7C3×D3×#100—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
15C33×M16C2212yescomplemented
16C3×#3748C2212yescomplemented
17C3×#3743—22no—
18C3×C8×C3⋊D3—22no—
19C2×D4×C3⋊D3C2×C311yescharacteristic
20C2×C3×D3×D4—34no—
21M16×C3⋊D3C2×C312yescomplemented
22C3×D3×M16—38no—
23C32×#100—24no—
24C32×#100D314yescomplemented
25C32×#100C2×C311yescharacteristic, complemented
26#63660C2×C311yescharacteristic, complemented
27#63650C2×C312yescomplemented
28C3×#833—34no—
29C3×#823—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—
43C33×C8—22no—
44C3×C3⋊(C3⋊C8)—22no—
45C2×C3×#100—91no—
46D3×#100—34no—
47C23×C3⋊D3C3×C412yes—
48C23×C3×D3—38no—
49C2×C32×D4—24no—
50C2×C32×D4C2×D314yes—
51C2×C32×D4C22×C311yescharacteristic
52C2×#638C3×C412yes—
53D4×C3⋊D3—22no—
54D4×C3⋊D3—22no—
55C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
56C2×C4×C3⋊D3C22×C311yescharacteristic
57C2×C3×#67—38no—
58C3×D3×D4—68no—
59C3×D3×D4—68no—
60C2×C3×D12—34no—
61C2×C3×C4×D3—34no—
62C32×M16—28no—
63C32×M16C2×D318yescomplemented
64C32×M16C22×C312yescomplemented
65#3748C22×C312yescomplemented
66#3743—22no—
67C8×C3⋊D3—22no—
68C3×#208—38no—
69C3×ΓL(1,𝔽25)—68no—
70C3×D3×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×C3×M16—92no—
86D3×M16—38no—
87C3×#100—24no—
88C3×#100—91no—
89C3×#100—92no—
90C3×#100C3×D314yescomplemented
91C3×#100C3⋊D311yescharacteristic, direct product factor
92#833—34no—
93#823—38no—
94C23×C32—28no—
95C23×C32C4×D318yes—
96C23×C32C2×C3×C412yes—
97C22×C3⋊D3—22no—
98C22×C3⋊D3—22no—
99C22×C3⋊D3—44no—
100C22×C3⋊D3C2×C3×C411yescharacteristic
101C22×C3⋊D3C2×C3×C412yes—
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×C4C23×C311yescharacteristic
113#638—24no—
114C2×C3⋊Dic3C2×C3×C411yescharacteristic
115C4⋊(C3⋊D3)—22no—
116C4×C3⋊D3C3×D412yes—
117C4×C3⋊D3C3×D412yes—
118C3×#67—616no—
119C2×C3×Dic3—34no—
120C3×D12—68no—
121C3×C4×D3—38no—
122C3×C4×D3—38no—
123C32×C8—22no—
124C32×C8—28no—
125C32×C8—48no—
126C3⋊(C3⋊C8)—22no—
127C3×C3⋊C8—68no—
128C2×#100—91noSylow
129C2×C33—21no—
130C2×C33—42no—
131C2×C33C2×C22⋊C411yescharacteristic
132C3×C3⋊D3—21no—
133C3×C3⋊D3—42no—
134C3×C3⋊D3#10012yesdirect product factor
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×D4C2×C3×D314yes—
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—28no—
151C3×M16—92no—
152C3×M16—182no—
153C3×M16C2×C3×D318yescomplemented
154C3×M16C2×C3⋊D312yescomplemented
155C2×C3×C8—182no—
156#208—38no—
157ΓL(1,𝔽25)—68no—
158D3×C8—68no—
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×C3×C411yescharacteristic
168C2×C3⋊D3—22no—
169C2×C3⋊D3—22no—
170C2×C3⋊D3—44no—
171C2×C3⋊D3—44no—
172C2×C3⋊D3—42no—
173C2×C3⋊D3C3×C22⋊C411yescharacteristic
174C2×C3⋊D3C3×C22⋊C411yescharacteristic
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×C3×D412yes—
185C3⋊Dic3C3×C22⋊C412yes—
186C3×Dic3—38no—
187C22×D4—91no—
188C2×M16—92no—
189#100—91no—
190#100—92no—
191#100C3×C3⋊D311yescharacteristic, direct product factor
192C33C2×#10011yescharacteristic, complemented, Sylow
193C23×C3—28no—
194C23×C3—91no—
195C23×C3—92no—
196C23×C3—182no—
197C23×C3—182no—
198C23×C3—364no—
199C23×C3C3×C4×D318yes—
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×C4C22×C3×D314yes—
216C2×C3×C4C22×C3⋊D311yescharacteristic
217#67—616no—
218C2×Dic3—34no—
219D12—68no—
220C4×D3—38no—
221C4×D3—38no—
222C3×C8—28no—
223C3×C8—22no—
224C3×C8—48no—
225C3×C8—182no—
226C3⋊C8—68no—
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×C3×C22⋊C411yescharacteristic
236C3⋊D3—21no—
237C3⋊D3—42no—
238C3⋊D3C3×#10012yesdirect 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×D4C2×C3×C3⋊D311yescharacteristic
248C22×C4—91no—
249M16—92no—
250M16—182no—
251M16C2×C3×C3⋊D312yescomplemented
252C2×C8—182no—
253C22×C3—28no—
254C22×C3—22no—
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×C3×C4×D314yes—
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×C4C3×D3×D418yes—
279C3×C4D4×C3⋊D312yes—
280Dic3—38no—
281C32—24no—
282C32D3×#10014yescomplemented
283C32C2×C3×#10011yescharacteristic, complemented
284C23—91no—
285C23—92no—
286C23—182no—
287C23—182no—
288C23—364no—
289C23C3×C4×C3⋊D312yes—
290D4—22no—
291D4—184no—
292D4—182no—
293C2×C4—91no—
294C2×C4—92no—
295C2×C4—92no—
296C2×C4C22×C3×C3⋊D311yescharacteristic
297C8—22no—
298C8—182no—
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×C3C3×D3×C22⋊C414yes—
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×C3×C4×C3⋊D311yescharacteristic
324C4—92no—
325C4C3×D4×C3⋊D312yes—
326C3—24no—
327C3C3×D3×#10014yescomplemented
328C3C3⋊D3×#10011yescharacteristic, direct product factor
329C2—21no—
330C2—42no—
331C2—92no—
332C2—181no—
333C2—362no—
334C2C3×C22⋊C4×C3⋊D311yescharacteristic
subgroupG/H
centerC2×C3C22⋊C4×C3⋊D3
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#100C2
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×D4×C3⋊D3C2
cosocleC2×C32×C4C23×C3
hypercenterC3×#100C3⋊D3
hypocenterC32C2×C3×#100
LCSG ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#100

Complements

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

NH ≅ G/Ntype
2. C3×M16×C3⋊D3C2semidirect
3. C33×#100C2semidirect
4. C3×#63660C2semidirect
5. C3×#63650C2semidirect
6. C3⋊D3×#100C3direct (N × H)
15. C33×M16C22semidirect
16. C3×#3748C22semidirect
21. M16×C3⋊D3C2×C3semidirect
24. C32×#100D3semidirect
25. C32×#100C2×C3semidirect
26. #63660C2×C3semidirect
27. #63650C2×C3semidirect
63. C32×M16C2×D3semidirect
64. C32×M16C22×C3semidirect
65. #3748C22×C3semidirect
90. C3×#100C3×D3semidirect
91. C3×#100C3⋊D3direct (N × H)
134. C3×C3⋊D3#100direct (N × H)
153. C3×M16C2×C3×D3semidirect
154. C3×M16C2×C3⋊D3semidirect
191. #100C3×C3⋊D3direct (N × H)
192. C33C2×#100semidirect
238. C3⋊D3C3×#100direct (N × H)
251. M16C2×C3×C3⋊D3semidirect
282. C32D3×#100semidirect
283. C32C2×C3×#100semidirect
327. C3C3×D3×#100semidirect
328. C3C3⋊D3×#100direct (N × H)

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×M16×C3⋊D32
C33×#1002
C3×#636602
C3×#636502
C3⋊D3×#1003
C3×D3×#1003