Finite Group Databasebeta

← C3×#10592405 #13162746

Subgroups — C3×#10592405 #13162746

Order 1728.

904 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3743C211yesmaximal, characteristic, complemented
2C2×C3×C8×C3⋊D3C211yesmaximal, characteristic, complemented
3C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
4C33×#97C211yesmaximal, characteristic, complemented
5C3×#63647C211yesmaximal, characteristic, complemented
6C3×#63647C211yesmaximal, characteristic, complemented
7C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592405C311yesmaximal, characteristic, direct product factor
9C3×#4706—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C411yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C2×C33×C8C2211yescharacteristic
15C33×C42C2211yescharacteristic
16C3×#3752C411yescharacteristic
17C3×#3752C411yescharacteristic
18C3×C4⋊(C3⋊Dic3)C411yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C3×#3743—22no—
22C3×C8×C3⋊D3—22no—
23C2×#3743C2×C311yescharacteristic, complemented
24C2×C8×C3⋊D3C2×C311yescharacteristic, complemented
25C4×C4⋊(C3⋊D3)C2×C311yescharacteristic
26C2×C3×ΓL(1,𝔽25)—34no—
27C2×C3×D3×C8—34no—
28C3×C4×D12—34no—
29C32×#97—24no—
30C32×#97D314yescomplemented
31C32×#97C2×C311yescharacteristic, complemented
32#63647C2×C311yescharacteristic, complemented
33#63647C2×C311yescharacteristic, complemented
34C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
35C3×(C2×D3)⋊C8—34no—
36C3×(C2×D3)⋊C8—34no—
37C3×C4⋊(C3⋊C8)—34no—
38C22×C3×C3⋊D3C2×C411yescharacteristic
39C22×C3×C3⋊D3C2×C411yescharacteristic
40C2×C33×C4C2×C411yescharacteristic
41C2×C33×C4C2×C411yescharacteristic
42C2×C33×C4C2311yescharacteristic
43C2×C3×C3⋊Dic3C2×C411yescharacteristic
44C2×C3×C3⋊Dic3C2×C411yescharacteristic
45C3×C4⋊(C3⋊D3)—22no—
46C3×C4×C3⋊D3—21no—
47C3×C4×C3⋊D3—21no—
48C3×C4×C3⋊D3—21no—
49C3×C4×C3⋊D3—21no—
50C33×C8—21no—
51C33×C8D412yescomplemented
52C3×C3⋊(C3⋊C8)—21no—
53C3×C3⋊(C3⋊C8)—21no—
54C3×#435—91no—
55#4706—34no—
56C2×C4⋊(C3⋊D3)C3×C411yescharacteristic
57C2×C4×C3⋊D3C22×C311yescharacteristic
58C2×C4×C3⋊D3C22×C311yescharacteristic
59C2×C3×D12—34no—
60C2×C3×C4×D3—34no—
61C2×C3×C4×D3—34no—
62C2×C32×C8—24no—
63C2×C32×C8—24no—
64C2×C32×C8C2×D314yes—
65C2×C32×C8C2×D314yes—
66C2×C32×C8C22×C311yescharacteristic
67C2×C32×C8C22×C311yescharacteristic
68C32×C42—24no—
69C32×C42C2×D314yes—
70C32×C42C22×C311yescharacteristic
71#3752C3×C411yescharacteristic
72#3752C3×C411yescharacteristic
73C4⋊(C3⋊Dic3)C3×C411yescharacteristic
74C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
75C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
76#3743—22no—
77C8×C3⋊D3—22no—
78C3×(C2×D3)⋊C4—34no—
79C3×(C2×D3)⋊C4—34no—
80C3×C4⋊Dic3—34no—
81C2×C3×C3⋊C8—34no—
82C2×C3×C3⋊C8—34no—
83C3×ΓL(1,𝔽25)—68no—
84C3×D3×C8—68no—
85C22×C33C22×C411yescharacteristic
86C2×C3×C3⋊D3—21no—
87C2×C3×C3⋊D3—21no—
88C2×C3×C3⋊D3—21no—
89C2×C3×C3⋊D3—21no—
90C2×C3×C3⋊D3—21no—
91C2×C3×C3⋊D3M1612yescomplemented
92C33×C4—21no—
93C33×C4—21no—
94C33×C4C2×D411yescharacteristic
95C33×C4#4111yescharacteristic
96C3×C3⋊Dic3—21no—
97C3×C3⋊Dic3—21no—
98C2×C3×M16—91no—
99C22×C3×C8—91no—
100C3×C4×D4—91no—
101C2×ΓL(1,𝔽25)—34no—
102C2×D3×C8—34no—
103C4×D12—34no—
104C3×C4⋊C8—91no—
105C3×C22⋊C8—91no—
106C3×C22⋊C8—91no—
107C3×#97—24no—
108C3×#97C3×D314yescomplemented
109C3×#97C3⋊D311yescharacteristic, complemented
110(C2×D3)⋊C8—34no—
111(C2×D3)⋊C8—34no—
112C4⋊(C3⋊C8)—34no—
113C22×C3⋊D3C2×C3×C411yescharacteristic
114C22×C3⋊D3C2×C3×C411yescharacteristic
115C22×C3×D3—34no—
116C22×C3×D3—34no—
117C2×C32×C4—24no—
118C2×C32×C4—24no—
119C2×C32×C4—24no—
120C2×C32×C4C4×D314yes—
121C2×C32×C4C4×D314yes—
122C2×C32×C4C2×C3×C411yescharacteristic
123C2×C32×C4C2×C3×C411yescharacteristic
124C2×C32×C4C22×D314yes—
125C2×C32×C4C23×C311yescharacteristic
126C2×C3⋊Dic3C2×C3×C411yescharacteristic
127C2×C3⋊Dic3C2×C3×C411yescharacteristic
128C4⋊(C3⋊D3)—22no—
129C4×C3⋊D3—21no—
130C4×C3⋊D3—21no—
131C4×C3⋊D3—21no—
132C4×C3⋊D3—21no—
133C2×C3×Dic3—34no—
134C2×C3×Dic3—34no—
135C3×D12—68no—
136C3×C4×D3—64no—
137C3×C4×D3—64no—
138C3×C4×D3—64no—
139C3×C4×D3—64no—
140C32×C8—28no—
141C32×C8—24no—
142C32×C8—21no—
143C32×C8—28no—
144C32×C8D1218yescomplemented
145C32×C8C3×D412yescomplemented
146C3⋊(C3⋊C8)—21no—
147C3⋊(C3⋊C8)—21no—
148C3×C3⋊C8—64no—
149C3×C3⋊C8—64no—
150#435—91noSylow
151C2×C33C4×D411yescharacteristic
152C2×C33C2×M1611yescharacteristic
153C2×C33#13111yescharacteristic
154C3×C3⋊D3—22no—
155C3×C3⋊D3—41no—
156C2×C3×D4—91no—
157C22×C3×C4—91no—
158C22×C3×C4—91no—
159C2×D12—34no—
160C2×C4×D3—34no—
161C2×C4×D3—34no—
162C3×M16—182no—
163C2×C3×C8—24no—
164C2×C3×C8—24no—
165C2×C3×C8—91no—
166C2×C3×C8—91no—
167C2×C3×C8—182no—
168C2×C3×C8C2×C3×D314yes—
169C2×C3×C8C2×C3×D314yes—
170C2×C3×C8C2×C3⋊D311yescharacteristic
171C2×C3×C8C2×C3⋊D311yescharacteristic
172C3×C4⋊C4—91no—
173C3×C22⋊C4—91no—
174C3×C22⋊C4—91no—
175C3×C42—24no—
176C3×C42C2×C3×D314yes—
177C3×C42C2×C3⋊D311yescharacteristic
178(C2×D3)⋊C4—34no—
179(C2×D3)⋊C4—34no—
180C4⋊Dic3—34no—
181C2×C3⋊C8—34no—
182C2×C3⋊C8—34no—
183ΓL(1,𝔽25)—68no—
184D3×C8—68no—
185C22×C32—24no—
186C22×C32C2×C4×D314yes—
187C22×C32C22×C3×C411yescharacteristic
188C2×C3⋊D3—21no—
189C2×C3⋊D3—21no—
190C2×C3⋊D3—21no—
191C2×C3⋊D3—21no—
192C2×C3⋊D3—21no—
193C2×C3⋊D3C3×M1612yescomplemented
194C2×C3×D3—38no—
195C2×C3×D3—64no—
196C2×C3×D3—64no—
197C2×C3×D3—64no—
198C2×C3×D3—64no—
199C2×C3×D3—64no—
200C32×C4—21no—
201C32×C4—21no—
202C32×C4—24no—
203C32×C4—24no—
204C32×C4—24no—
205C32×C4—24no—
206C32×C4—44no—
207C32×C4—44no—
208C32×C4C2×D1214yes—
209C32×C4#23514yes—
210C32×C4C2×C3×D411yescharacteristic
211C32×C4C3×#4111yescharacteristic
212C3⋊Dic3—21no—
213C3⋊Dic3—21no—
214C3×Dic3—64no—
215C3×Dic3—64no—
216C2×M16—91no—
217C22×C8—91no—
218C4×D4—91no—
219C4⋊C8—91no—
220C22⋊C8—91no—
221C22⋊C8—91no—
222#97C3×C3⋊D311yescharacteristic, complemented
223C33#43511yescharacteristic, complemented, Sylow
224C23×C3—91no—
225C23×C3—91no—
226C22×D3—34no—
227C22×D3—34no—
228C3×D4—182no—
229C2×C3×C4—24no—
230C2×C3×C4—24no—
231C2×C3×C4—24no—
232C2×C3×C4—91no—
233C2×C3×C4—91no—
234C2×C3×C4—181no—
235C2×C3×C4—181no—
236C2×C3×C4—181no—
237C2×C3×C4—181no—
238C2×C3×C4C3×C4×D314yes—
239C2×C3×C4C3×C4×D314yes—
240C2×C3×C4C4×C3⋊D311yescharacteristic
241C2×C3×C4C4×C3⋊D311yescharacteristic
242C2×C3×C4C22×C3×D314yes—
243C2×C3×C4C22×C3⋊D311yescharacteristic
244C2×Dic3—34no—
245C2×Dic3—34no—
246D12—68no—
247C4×D3—64no—
248C4×D3—64no—
249C4×D3—64no—
250C4×D3—64no—
251C3×C8—28no—
252C3×C8—24no—
253C3×C8—21no—
254C3×C8—28no—
255C3×C8—181no—
256C3×C8—181no—
257C3×C8C3×D1218yescomplemented
258C3×C8C4⋊(C3⋊D3)12yescomplemented
259C3⋊C8—64no—
260C3⋊C8—64no—
261C2×C32—24no—
262C2×C32—24no—
263C2×C32—24no—
264C2×C32C4×D1214yes—
265C2×C32D3×M1614yes—
266C2×C32#90714yes—
267C2×C32C3×C4×D411yescharacteristic
268C2×C32C2×C3×M1611yescharacteristic
269C2×C32C3×#13111yescharacteristic
270C3⋊D3—22no—
271C3⋊D3—41no—
272C3×D3—68no—
273C3×D3—124no—
274C2×D4—91no—
275C22×C4—91no—
276C22×C4—91no—
277M16—182no—
278C2×C8—91no—
279C2×C8—91no—
280C2×C8—182no—
281C2×C8C2×C3×C3⋊D311yescharacteristic
282C2×C8C2×C3×C3⋊D311yescharacteristic
283C4⋊C4—91no—
284C22⋊C4—91no—
285C22⋊C4—91no—
286C42C2×C3×C3⋊D311yescharacteristic
287C22×C3—24no—
288C22×C3—92no—
289C22×C3—181no—
290C22×C3—181no—
291C22×C3—181no—
292C22×C3—181no—
293C22×C3—181no—
294C22×C3C2×C3×C4×D314yes—
295C22×C3C2×C4×C3⋊D311yescharacteristic
296C2×D3—38no—
297C2×D3—64no—
298C2×D3—64no—
299C2×D3—64no—
300C2×D3—64no—
301C2×D3—64no—
302C3×C4—24no—
303C3×C4—24no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—24no—
307C3×C4—24no—
308C3×C4—44no—
309C3×C4—44no—
310C3×C4—181no—
311C3×C4—181no—
312C3×C4C2×C3×D1214yes—
313C3×C4C3×#23514yes—
314C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
315C3×C4#382811yescharacteristic
316Dic3—64no—
317Dic3—64no—
318C32—24no—
319C32#470614yescomplemented
320C32C3×#43511yescharacteristic, complemented
321C23—91no—
322C23—91no—
323D4—182no—
324C2×C4—91no—
325C2×C4—91no—
326C2×C4—181no—
327C2×C4—181no—
328C2×C4—181no—
329C2×C4—181no—
330C2×C4C3×C4×C3⋊D311yescharacteristic
331C2×C4C3×C4×C3⋊D311yescharacteristic
332C2×C4C22×C3×C3⋊D311yescharacteristic
333C8—21no—
334C8—181no—
335C8—181no—
336C8C3×C4⋊(C3⋊D3)12yescomplemented
337C2×C3—24no—
338C2×C3—24no—
339C2×C3—24no—
340C2×C3—182no—
341C2×C3—361no—
342C2×C3C3×C4×D1214yes—
343C2×C3C3×D3×M1614yes—
344C2×C3C3×#90714yes—
345C2×C3C4×C4⋊(C3⋊D3)11yescharacteristic
346C2×C3M16×C3⋊D311yescharacteristic
347C2×C3#6411611yescharacteristic
348D3—68no—
349D3—124no—
350C22—92no—
351C22—181no—
352C22—181no—
353C22—181no—
354C22—181no—
355C22—181no—
356C22C2×C3×C4×C3⋊D311yescharacteristic
357C4—21no—
358C4—21no—
359C4—181no—
360C4—181no—
361C4C2×C3×C4⋊(C3⋊D3)11yescharacteristic
362C4C3×#382811yescharacteristic
363C3—24no—
364C3C3×#470614yescomplemented
365C3#1059240511yescharacteristic, direct product factor
366C2—182no—
367C2—361no—
368C2C3×C4×C4⋊(C3⋊D3)11yescharacteristic
369C2C3×M16×C3⋊D311yescharacteristic
370C2C3×#6411611yescharacteristic
subgroupG/H
centerC2×C3×C4C22×C3⋊D3
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#97C2
socleC22×C33C22×C4
prime radicalC2×C3×C4⋊(C3⋊D3)C4
cosocleC2×C32×C4C23×C3
hypercenterC3×#97C3⋊D3
hypocenterC32C3×#435
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C3×#97

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3743C2semidirect
2. C2×C3×C8×C3⋊D3C2semidirect
4. C33×#97C2semidirect
5. C3×#63647C2semidirect
6. C3×#63647C2semidirect
7. C3×C4⋊(C3⋊(C3⋊C8))C2semidirect
8. #10592405C3direct (N × H)
23. C2×#3743C2×C3semidirect
24. C2×C8×C3⋊D3C2×C3semidirect
30. C32×#97D3semidirect
31. C32×#97C2×C3semidirect
32. #63647C2×C3semidirect
33. #63647C2×C3semidirect
34. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
51. C33×C8D4semidirect
91. C2×C3×C3⋊D3M16semidirect
108. C3×#97C3×D3semidirect
109. C3×#97C3⋊D3semidirect
144. C32×C8D12semidirect
145. C32×C8C3×D4semidirect
193. C2×C3⋊D3C3×M16semidirect
222. #97C3×C3⋊D3semidirect
223. C33#435semidirect
257. C3×C8C3×D12semidirect
258. C3×C8C4⋊(C3⋊D3)semidirect
319. C32#4706semidirect
320. C32C3×#435semidirect
336. C8C3×C4⋊(C3⋊D3)semidirect
364. C3C3×#4706semidirect
365. C3#10592405direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37432
C2×C3×C8×C3⋊D32
C3×C4×C4⋊(C3⋊D3)2
C33×#972
C3×#636472
C3×C4⋊(C3⋊(C3⋊C8))2
#105924053
C3×#47063