Finite Group Databasebeta

← C3×#10590912 #13157878

Subgroups — C3×#10590912 #13157878

Order 1728.

558 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×ΓL(1,𝔽25)C211yesmaximal, characteristic
2C3×D3×#208C212yesmaximal
3C3×#63646C211yesmaximal, characteristic, complemented
4C32×#819C211yesmaximal, characteristic, complemented
5C3×#63577C212yesmaximal, complemented
6C3×#5125—31nomaximal
7C3×#4893—31nomaximal
8#10590912C311yesmaximal, characteristic, direct product factor
9C2×C32×C4×D3C2211yescharacteristic
10C2×C33×C8C2211yescharacteristic
11C3×#3748C2212yes—
12C32×#208C2212yes—
13C2×C32×C3⋊C8C2211yescharacteristic
14C32×ΓL(1,𝔽25)C414yes—
15C3×#3711—22no—
16C3×D3×C3⋊C8—22no—
17C2×C32×M16—31no—
18C2×C3×#208—32no—
19C3×D3×M16—32no—
20C2×C3×ΓL(1,𝔽25)—21no—
21C2×C3×ΓL(1,𝔽25)D311yescharacteristic
22C2×C3×ΓL(1,𝔽25)C2×C311yescharacteristic
23D3×#208C2×C312yes—
24C32×#108—31no—
25#63646C2×C311yescharacteristic, complemented
26C3×#822—32no—
27C3×#819—21no—
28C3×#819—31no—
29C3×#819—31no—
30C3×#819—32no—
31C3×#819—61no—
32C3×#819D311yescharacteristic, complemented
33C3×#819C2×C311yescharacteristic, complemented
34#63577C2×C312yescomplemented
35C22×C32×D3Q811yescharacteristic
36C2×C33×C4C2311yescharacteristic
37C2×C32×Dic3Q811yescharacteristic
38C32×C4×D3C2×C412yes—
39C32×C4×D3D412yes—
40C33×C8C2×C412yes—
41C3×C3⋊(C3⋊C8)—22no—
42C32×C3⋊C8—22no—
43C32×C3⋊C8C2×C412yes—
44C3×#430—91no—
45#5125—31no—
46#4893—31no—
47C22×C32×C4—31no—
48C2×C3×C4×D3—21no—
49C2×C3×C4×D3C2×D311yescharacteristic
50C2×C3×C4×D3C22×C311yescharacteristic
51C32×M16—34no—
52C32×M16—32no—
53C2×C32×C8—21no—
54C2×C32×C8—21no—
55C2×C32×C8—21no—
56C2×C32×C8—31no—
57C2×C32×C8—41no—
58C2×C32×C8C2×D311yescharacteristic
59C2×C32×C8C2×D311yescharacteristic
60C2×C32×C8C22×C311yescharacteristic
61#3748C22×C312yes—
62C3×#208—22no—
63C3×#208—32no—
64C3×#208—32no—
65C3×#208—62no—
66C3×#208—62no—
67C3×#208C2×D312yes—
68C3×#208C22×C312yes—
69C2×C3×C3⋊C8—21no—
70C2×C3×C3⋊C8—62no—
71C2×C3×C3⋊C8C2×D311yescharacteristic
72C2×C3×C3⋊C8C22×C311yescharacteristic
73C3×ΓL(1,𝔽25)—24no—
74C3×ΓL(1,𝔽25)—62no—
75C3×ΓL(1,𝔽25)Dic314yes—
76C3×ΓL(1,𝔽25)C3×C414yes—
77C3×D3×C8—62no—
78#3711—22no—
79D3×C3⋊C8—22no—
80C22×C33C2×Q811yescharacteristic
81C2×C32×D3—41no—
82C2×C32×D3C4⋊C412yes—
83C33×C4C22×C411yescharacteristic
84C33×C4C2×D411yescharacteristic
85C32×Dic3C4⋊C412yes—
86C2×C3×M16—31no—
87C2×C3×M16—31no—
88C2×C3×M16—61no—
89C2×C3×M16—92no—
90C2×#208—32no—
91D3×M16—32no—
92C2×ΓL(1,𝔽25)C3×D311yescharacteristic
93C3×#108—31no—
94C3×#108—31no—
95C3×#108—61no—
96C3×#108—92no—
97C3×#108—91no—
98#822—32no—
99#819—31no—
100#819—31no—
101#819—32no—
102#819—61no—
103#819C3×D311yescharacteristic, complemented
104C23×C32—31no—
105C22×C3×D3—21no—
106C22×C3×D3Dic611yescharacteristic
107C22×C3×D3C3×Q811yescharacteristic
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4—21no—
111C2×C32×C4—31no—
112C2×C32×C4—32no—
113C2×C32×C4—32no—
114C2×C32×C4—41no—
115C2×C32×C4C22×D311yescharacteristic
116C2×C32×C4C22×D311yescharacteristic
117C2×C32×C4C23×C311yescharacteristic
118C2×C3×Dic3—21no—
119C2×C3×Dic3Dic611yescharacteristic
120C2×C3×Dic3C3×Q811yescharacteristic
121C3×C4×D3—22no—
122C3×C4×D3—22no—
123C3×C4×D3D1212yes—
124C3×C4×D3C2×Dic312yes—
125C3×C4×D3C2×C3×C412yes—
126C3×C4×D3C3×D412yes—
127C32×C8—22no—
128C32×C8—22no—
129C32×C8—22no—
130C32×C8—32no—
131C32×C8—42no—
132C32×C8—62no—
133C32×C8C4×D312yes—
134C32×C8C2×Dic312yes—
135C32×C8C2×C3×C412yes—
136C3⋊(C3⋊C8)—22no—
137C3×C3⋊C8—22no—
138C3×C3⋊C8—22no—
139C3×C3⋊C8—22no—
140C3×C3⋊C8—42no—
141C3×C3⋊C8—62no—
142C3×C3⋊C8—62no—
143C3×C3⋊C8—122no—
144C3×C3⋊C8C2×Dic312yes—
145C3×C3⋊C8C2×C3×C412yes—
146#430—91noSylow
147C2×C33—21no—
148C2×C33C2×C4⋊C411yescharacteristic
149C32×D3—22no—
150C22×C3×C4—31no—
151C22×C3×C4—31no—
152C22×C3×C4—61no—
153C2×C4×D3C2×C3×D311yescharacteristic
154C3×M16—34no—
155C3×M16—34no—
156C3×M16—32no—
157C3×M16—32no—
158C3×M16—64no—
159C3×M16—62no—
160C3×M16—92no—
161C3×M16—182no—
162C2×C3×C8—21no—
163C2×C3×C8—21no—
164C2×C3×C8—21no—
165C2×C3×C8—31no—
166C2×C3×C8—31no—
167C2×C3×C8—41no—
168C2×C3×C8—61no—
169C2×C3×C8—182no—
170C2×C3×C8D3211yescharacteristic
171C2×C3×C8C2×C3×D311yescharacteristic
172C2×C3×C8C2×C3×D311yescharacteristic
173#208—32no—
174#208—32no—
175#208—62no—
176#208—62no—
177#208C2×C3×D312yes—
178C2×C3⋊C8—62no—
179C2×C3⋊C8C2×C3×D311yescharacteristic
180ΓL(1,𝔽25)—62no—
181ΓL(1,𝔽25)C3×Dic314yes—
182D3×C8—62no—
183C22×C32—21no—
184C22×C32—21no—
185C22×C32—21no—
186C22×C32—32no—
187C22×C32—41no—
188C22×C32—121no—
189C22×C32C2×Dic611yescharacteristic
190C22×C32D3×Q811yescharacteristic
191C22×C32C2×C3×Q811yescharacteristic
192C2×C3×D3—22no—
193C2×C3×D3—41no—
194C2×C3×D3—41no—
195C2×C3×D3—42no—
196C2×C3×D3C4⋊Dic312yes—
197C2×C3×D3C3×C4⋊C412yes—
198C32×C4—21no—
199C32×C4—21no—
200C32×C4—21no—
201C32×C4—21no—
202C32×C4—21no—
203C32×C4—21no—
204C32×C4—32no—
205C32×C4—41no—
206C32×C4—41no—
207C32×C4C2×C4×D311yescharacteristic
208C32×C4C2×D1211yescharacteristic
209C32×C4D3×D411yescharacteristic
210C32×C4C22×Dic311yescharacteristic
211C32×C4C22×C3×C411yescharacteristic
212C32×C4C2×C3×D411yescharacteristic
213C3×Dic3—22no—
214C3×Dic3C4⋊Dic312yes—
215C3×Dic3C3×C4⋊C412yes—
216C2×M16—31no—
217C2×M16—92no—
218#108—31no—
219#108—92no—
220#108—91no—
221C33#43011yescharacteristic, complemented, Sylow
222C23×C3—31no—
223C23×C3—31no—
224C23×C3—61no—
225C22×D3C3×Dic611yescharacteristic
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—31no—
230C2×C3×C4—31no—
231C2×C3×C4—32no—
232C2×C3×C4—32no—
233C2×C3×C4—32no—
234C2×C3×C4—32no—
235C2×C3×C4—41no—
236C2×C3×C4—62no—
237C2×C3×C4—62no—
238C2×C3×C4—61no—
239C2×C3×C4C2×D3211yescharacteristic
240C2×C3×C4C22×C3×D311yescharacteristic
241C2×C3×C4C22×C3×D311yescharacteristic
242C2×Dic3C3×Dic611yescharacteristic
243C4×D3C3×D1212yes—
244C4×D3C2×C3×Dic312yes—
245C3×C8—22no—
246C3×C8—22no—
247C3×C8—22no—
248C3×C8—32no—
249C3×C8—32no—
250C3×C8—42no—
251C3×C8—62no—
252C3×C8—62no—
253C3×C8—62no—
254C3×C8—122no—
255C3×C8—182no—
256C3×C8D3×Dic312yes—
257C3×C8C3×C4×D312yes—
258C3×C8C2×C3×Dic312yes—
259C3⋊C8—22no—
260C3⋊C8—62no—
261C3⋊C8—62no—
262C3⋊C8—122no—
263C3⋊C8C2×C3×Dic312yes—
264C2×C32—21no—
265C2×C32—22no—
266C2×C32—21no—
267C2×C32—21no—
268C2×C32—21no—
269C2×C32—21no—
270C2×C32—21no—
271C2×C32—41no—
272C2×C32—41no—
273C2×C32—41no—
274C2×C32—42no—
275C2×C32—62no—
276C2×C32D3×C4⋊C411yescharacteristic
277C2×C32C2×C4⋊Dic311yescharacteristic
278C2×C32C2×C3×C4⋊C411yescharacteristic
279C3×D3—22no—
280C3×D3—22no—
281C3×D3—42no—
282C22×C4—31no—
283M16—34no—
284M16—32no—
285M16—92no—
286M16—182no—
287C2×C8—31no—
288C2×C8—182no—
289C2×C8C3×D3211yescharacteristic
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—32no—
294C22×C3—32no—
295C22×C3—41no—
296C22×C3—62no—
297C22×C3—121no—
298C22×C3—121no—
299C22×C3—122no—
300C22×C3D3×Dic611yescharacteristic
301C22×C3C2×C3×Dic611yescharacteristic
302C22×C3C3×D3×Q811yescharacteristic
303C2×D3—41no—
304C2×D3C3×C4⋊Dic312yes—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—32no—
312C3×C4—32no—
313C3×C4—41no—
314C3×C4—41no—
315C3×C4—62no—
316C3×C4D3×D1211yescharacteristic
317C3×C4C2×D3×Dic311yescharacteristic
318C3×C4C2×C3×C4×D311yescharacteristic
319C3×C4C2×C3×D1211yescharacteristic
320C3×C4C3×D3×D411yescharacteristic
321C3×C4C22×C3×Dic311yescharacteristic
322Dic3C3×C4⋊Dic312yes—
323C32—21no—
324C32—21no—
325C32—21no—
326C32—41no—
327C32#489311yescharacteristic, complemented
328C32#512511yescharacteristic, complemented
329C32C3×#43011yescharacteristic, complemented
330C23—31no—
331C2×C4—31no—
332C2×C4—32no—
333C2×C4—32no—
334C2×C4C2×C3×D3211yescharacteristic
335C8—32no—
336C8—62no—
337C8—182no—
338C8C3×D3×Dic312yes—
339C2×C3—22no—
340C2×C3—21no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—21no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—41no—
347C2×C3—41no—
348C2×C3—42no—
349C2×C3—41no—
350C2×C3—62no—
351C2×C3—62no—
352C2×C3—122no—
353C2×C3D3×C4⋊Dic311yescharacteristic
354C2×C3C3×D3×C4⋊C411yescharacteristic
355C2×C3C2×C3×C4⋊Dic311yescharacteristic
356D3—22no—
357C22—32no—
358C22—121no—
359C22C3×D3×Dic611yescharacteristic
360C4—32no—
361C4C3×D3×D1211yescharacteristic
362C4C2×C3×D3×Dic311yescharacteristic
363C3—21no—
364C3—21no—
365C3—21no—
366C3—41no—
367C3#1059091211yescharacteristic, direct product factor
368C3C3×#489311yescharacteristic, complemented
369C3C3×#512511yescharacteristic, complemented
370C2—21no—
371C2—62no—
372C2C3×D3×C4⋊Dic311yescharacteristic
subgroupG/H
centerC3×C4D3×D12
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC2×C33C2×C4⋊C4
prime radicalC22×C32×D3Q8
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×C8D32
hypocenterC32C3×#430
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
3. C3×#63646C2semidirect
4. C32×#819C2semidirect
5. C3×#63577C2semidirect
8. #10590912C3direct (N × H)
25. #63646C2×C3semidirect
32. C3×#819D3semidirect
33. C3×#819C2×C3semidirect
34. #63577C2×C3semidirect
103. #819C3×D3semidirect
221. C33#430semidirect
327. C32#4893semidirect
328. C32#5125semidirect
329. C32C3×#430semidirect
367. C3#10590912direct (N × H)
368. C3C3×#4893semidirect
369. C3C3×#5125semidirect

Maximal subgroups

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

subgroupindex
C2×C32×ΓL(1,𝔽25)2
C3×D3×#2082
C3×#636462
C32×#8192
C3×#635772
C3×#51253
C3×#48933
#105909123