Finite Group Databasebeta

← C3×#10590818 #13157792

Subgroups — C3×#10590818 #13157792

Order 1728.

602 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×Dic12C211yesmaximal, characteristic
2C3×#63843C212yesmaximal
3C3×C8×C3⋊Dic3C211yesmaximal, characteristic
4C32×#818C211yesmaximal, characteristic
5C3×#63565C212yesmaximal
6C3×Dic3×Q16—31nomaximal
7C3×#4873—31nomaximal
8#10590818C311yesmaximal, characteristic, direct product factor
9C2×C32×Dic6C2212yes—
10C2×C33×C8C2211yescharacteristic
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
13C32×C4⋊Dic3C2212yes—
14C32×Dic12C414yescomplemented
15C3×Dic3⋊Dic3—22no—
16C3×Dic32—22no—
17C2×C32×Q16—31no—
18C3×Q8×Dic3—32no—
19C2×C3×Dic12—21no—
20C2×C3×Dic12D311yescharacteristic
21C2×C3×Dic12C2×C311yescharacteristic
22C3×#887—32no—
23#63843C2×C312yes—
24C32×#107—31no—
25C8×C3⋊Dic3C2×C311yescharacteristic
26C3×#835—32no—
27C3×#818—21no—
28C3×#818D311yescharacteristic
29C3×#818C2×C311yescharacteristic
30C3×C8×Dic3—31no—
31C3×C8×Dic3—31no—
32C3×C8×Dic3—61no—
33C3×#810—32no—
34#63565C2×C312yes—
35C2×C33×C4C2311yescharacteristic
36C2×C3×C3⋊Dic3D411yescharacteristic
37C2×C3×C3⋊Dic3D411yescharacteristic
38C2×C32×Dic3—22no—
39C2×C32×Dic3—22no—
40C32×Dic6—22no—
41C32×Dic6C2×C414yes—
42C33×C8C2×C412yes—
43C3×C3⋊(C3⋊C8)—21no—
44C3×C4×Q16—91no—
45Dic3×Q16—31no—
46#4873—31no—
47C2×C32×Q8—32no—
48C2×C3×Dic6—22no—
49C2×C3×Dic6C2×D312yes—
50C2×C3×Dic6C22×C312yes—
51C32×Q16—34no—
52C2×C32×C8—21no—
53C2×C32×C8—21no—
54C2×C32×C8—21no—
55C2×C32×C8—41no—
56C2×C32×C8C2×D311yescharacteristic
57C2×C32×C8C2×D311yescharacteristic
58C2×C32×C8C22×C311yescharacteristic
59C32×C4⋊C4—32no—
60C4×C3⋊Dic3C22×C311yescharacteristic
61C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
62C3×C4⋊Dic3—22no—
63C3×C4⋊Dic3—62no—
64C3×C4⋊Dic3C2×D312yes—
65C3×C4⋊Dic3C22×C312yes—
66C3×Dic3⋊C4—62no—
67C3×C4×Dic3—31no—
68C3×C4×Dic3—31no—
69C3×C4×Dic3—62no—
70C3×C4×Dic3—62no—
71C3×C4×Dic3—61no—
72C2×C3×C3⋊C8—31no—
73C2×C3×C3⋊C8—31no—
74C2×C3×C3⋊C8—61no—
75C3×Dic12—24no—
76C3×Dic12Dic314yescomplemented
77C3×Dic12C3×C414yescomplemented
78Dic3⋊Dic3—22no—
79Dic32—22no—
80C22×C33C2×D411yescharacteristic
81C33×C4C22×C411yescharacteristic
82C33×C4#4111yescharacteristic
83C3×C3⋊Dic3—21no—
84C3×C3⋊Dic3Q1612yescomplemented
85C32×Dic3—24no—
86C32×Dic3—42no—
87C2×C3×Q16—31no—
88C2×C3×Q16—31no—
89C2×C3×Q16—61no—
90C3×C4×Q8—92no—
91Q8×Dic3—32no—
92C2×Dic12C3×D311yescharacteristic
93#887—32no—
94C3×#107—31no—
95C3×#107—31no—
96C3×#107—61no—
97C3×#103—92no—
98C3×C4×C8—91no—
99#835—32no—
100#818C3×D311yescharacteristic
101C8×Dic3—31no—
102C8×Dic3—31no—
103C8×Dic3—61no—
104#810—32no—
105C32×Q8—34no—
106C32×Q8—62no—
107C2×C32×C4—21no—
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4—41no—
111C2×C32×C4—62no—
112C2×C32×C4—62no—
113C2×C32×C4C22×D311yescharacteristic
114C2×C32×C4C22×D311yescharacteristic
115C2×C32×C4C23×C311yescharacteristic
116C2×C3⋊Dic3C3×D411yescharacteristic
117C2×C3⋊Dic3C3×D411yescharacteristic
118C2×C3×Dic3—22no—
119C2×C3×Dic3—22no—
120C2×C3×Dic3—22no—
121C2×C3×Dic3—22no—
122C2×C3×Dic3—31no—
123C2×C3×Dic3—31no—
124C2×C3×Dic3—31no—
125C2×C3×Dic3—31no—
126C2×C3×Dic3—42no—
127C2×C3×Dic3—42no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C3×Dic6—22no—
131C3×Dic6—22no—
132C3×Dic6—24no—
133C3×Dic6—42no—
134C3×Dic6C2×Dic314yes—
135C3×Dic6C2×C3×C414yes—
136C32×C8—22no—
137C32×C8—22no—
138C32×C8—22no—
139C32×C8—42no—
140C32×C8C4×D312yes—
141C32×C8C2×Dic312yes—
142C32×C8C2×C3×C412yes—
143C3⋊(C3⋊C8)—21no—
144C3×C3⋊C8—62no—
145C3×C3⋊C8—61no—
146C3×C3⋊C8—61no—
147C4×Q16—91noSylow
148C2×C33C4×D411yescharacteristic
149C2×C33C2×Q1611yescharacteristic
150C2×C33#13511yescharacteristic
151C2×C3×Q8—32no—
152C2×C3×Q8—32no—
153C2×C3×Q8—62no—
154C2×Dic6C2×C3×D312yes—
155C3×Q16—34no—
156C3×Q16—34no—
157C3×Q16—64no—
158C2×C3×C8—21no—
159C2×C3×C8—21no—
160C2×C3×C8—21no—
161C2×C3×C8—41no—
162C2×C3×C8—91no—
163C2×C3×C8D3211yescharacteristic
164C2×C3×C8C2×C3×D311yescharacteristic
165C2×C3×C8C2×C3×D311yescharacteristic
166C3×C4⋊C4—32no—
167C3×C4⋊C4—32no—
168C3×C4⋊C4—62no—
169C3×C4⋊C4—182no—
170C3×C42—91no—
171C3×C42—182no—
172C4⋊Dic3—62no—
173C4⋊Dic3C2×C3×D312yes—
174Dic3⋊C4—62no—
175C4×Dic3—31no—
176C4×Dic3—31no—
177C4×Dic3—61no—
178C4×Dic3—62no—
179C4×Dic3—62no—
180C2×C3⋊C8—31no—
181C2×C3⋊C8—31no—
182C2×C3⋊C8—61no—
183Dic12C3×Dic314yescomplemented
184C22×C32—21no—
185C22×C32—21no—
186C22×C32—21no—
187C22×C32—41no—
188C22×C32D3×D411yescharacteristic
189C22×C32D3×D411yescharacteristic
190C22×C32C2×C3×D411yescharacteristic
191C32×C4—21no—
192C32×C4—21no—
193C32×C4—21no—
194C32×C4—21no—
195C32×C4—21no—
196C32×C4—21no—
197C32×C4—41no—
198C32×C4—41no—
199C32×C4—64no—
200C32×C4—122no—
201C32×C4C2×C4×D311yescharacteristic
202C32×C4#23711yescharacteristic
203C32×C4#23911yescharacteristic
204C32×C4C22×Dic311yescharacteristic
205C32×C4C22×C3×C411yescharacteristic
206C32×C4C3×#4111yescharacteristic
207C3⋊Dic3—21no—
208C3⋊Dic3C3×Q1612yescomplemented
209C3×Dic3—24no—
210C3×Dic3—24no—
211C3×Dic3—32no—
212C3×Dic3—32no—
213C3×Dic3—44no—
214C3×Dic3—42no—
215C3×Dic3—42no—
216C3×Dic3—62no—
217C3×Dic3—62no—
218C3×Dic3—61no—
219C3×Dic3—61no—
220C3×Dic3—82no—
221C2×Q16—31no—
222C4×Q8—92no—
223#107—31no—
224#103—92no—
225C4×C8—91no—
226C33C4×Q1611yescharacteristic, complemented, Sylow
227C3×Q8—34no—
228C3×Q8—34no—
229C3×Q8—64no—
230C3×Q8—62no—
231C3×Q8—62no—
232C3×Q8—122no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—41no—
237C2×C3×C4—62no—
238C2×C3×C4—62no—
239C2×C3×C4—62no—
240C2×C3×C4—62no—
241C2×C3×C4—91no—
242C2×C3×C4—91no—
243C2×C3×C4—122no—
244C2×C3×C4—122no—
245C2×C3×C4C2×D3211yescharacteristic
246C2×C3×C4C22×C3×D311yescharacteristic
247C2×C3×C4C22×C3×D311yescharacteristic
248C2×Dic3—22no—
249C2×Dic3—22no—
250C2×Dic3—31no—
251C2×Dic3—31no—
252C2×Dic3—31no—
253C2×Dic3—31no—
254C2×Dic3—61no—
255C2×Dic3—61no—
256Dic6—22no—
257Dic6C2×C3×Dic314yes—
258C3×C8—22no—
259C3×C8—22no—
260C3×C8—22no—
261C3×C8—42no—
262C3×C8—181no—
263C3×C8D3×Dic312yes—
264C3×C8C3×C4×D312yes—
265C3×C8C2×C3×Dic312yes—
266C3⋊C8—62no—
267C3⋊C8—61no—
268C3⋊C8—61no—
269C2×C32—21no—
270C2×C32—21no—
271C2×C32—21no—
272C2×C32—21no—
273C2×C32—21no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32—21no—
277C2×C32—21no—
278C2×C32—41no—
279C2×C32—41no—
280C2×C32—41no—
281C2×C32#89311yescharacteristic
282C2×C32#91211yescharacteristic
283C2×C32D3×Q1611yescharacteristic
284C2×C32D3×Q1611yescharacteristic
285C2×C32#91911yescharacteristic
286C2×C32D4×Dic311yescharacteristic
287C2×C32C3×C4×D411yescharacteristic
288C2×C32C2×C3×Q1611yescharacteristic
289C2×C32C3×#13511yescharacteristic
290C2×Q8—32no—
291Q16—34no—
292C2×C8—91no—
293C2×C8C3×D3211yescharacteristic
294C4⋊C4—32no—
295C4⋊C4—182no—
296C42—91no—
297C42—182no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—41no—
302C22×C3#380211yescharacteristic
303C22×C3C3×D3×D411yescharacteristic
304C22×C3C3×D3×D411yescharacteristic
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—41no—
312C3×C4—41no—
313C3×C4—64no—
314C3×C4—64no—
315C3×C4—92no—
316C3×C4—124no—
317C3×C4—122no—
318C3×C4—122no—
319C3×C4—181no—
320C3×C4—242no—
321C3×C4#379611yescharacteristic
322C3×C4C2×D3×Dic311yescharacteristic
323C3×C4C2×C3×C4×D311yescharacteristic
324C3×C4C3×#23711yescharacteristic
325C3×C4C3×#23911yescharacteristic
326C3×C4C22×C3×Dic311yescharacteristic
327Dic3—24no—
328Dic3—32no—
329Dic3—32no—
330Dic3—42no—
331Dic3—62no—
332Dic3—62no—
333Dic3—61no—
334Dic3—61no—
335C32—21no—
336C32—21no—
337C32—21no—
338C32—41no—
339C32#487311yescharacteristic, complemented
340C32Dic3×Q1611yescharacteristic, complemented
341C32C3×C4×Q1611yescharacteristic, complemented
342Q8—34no—
343Q8—62no—
344C2×C4—62no—
345C2×C4—62no—
346C2×C4—91no—
347C2×C4—91no—
348C2×C4C2×C3×D3211yescharacteristic
349C8—181no—
350C8C3×D3×Dic312yes—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—41no—
361C2×C3—41no—
362C2×C3—41no—
363C2×C3#6380211yescharacteristic
364C2×C3#6381011yescharacteristic
365C2×C3#6389811yescharacteristic
366C2×C3C3×#89311yescharacteristic
367C2×C3C3×#91211yescharacteristic
368C2×C3C3×D3×Q1611yescharacteristic
369C2×C3C3×D3×Q1611yescharacteristic
370C2×C3C3×#91911yescharacteristic
371C2×C3C3×D4×Dic311yescharacteristic
372C22C3×#380211yescharacteristic
373C4—64no—
374C4—92no—
375C4—122no—
376C4—181no—
377C4C3×#379611yescharacteristic
378C4C2×C3×D3×Dic311yescharacteristic
379C3—21no—
380C3—21no—
381C3—21no—
382C3—41no—
383C3#1059081811yescharacteristic, direct product factor
384C3C3×#487311yescharacteristic, complemented
385C3C3×Dic3×Q1611yescharacteristic, complemented
386C2C3×#6380211yescharacteristic
387C2C3×#6381011yescharacteristic
388C2C3×#6389811yescharacteristic
subgroupG/H
centerC22×C3#3802
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socle, prime radicalC22×C33C2×D4
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×C8D32
hypocenterC32C3×C4×Q16
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
8. #10590818C3direct (N × H)
14. C32×Dic12C4semidirect
76. C3×Dic12Dic3semidirect
77. C3×Dic12C3×C4semidirect
84. C3×C3⋊Dic3Q16semidirect
183. Dic12C3×Dic3semidirect
208. C3⋊Dic3C3×Q16semidirect
226. C33C4×Q16semidirect
339. C32#4873semidirect
340. C32Dic3×Q16semidirect
341. C32C3×C4×Q16semidirect
383. C3#10590818direct (N × H)
384. C3C3×#4873semidirect
385. C3C3×Dic3×Q16semidirect

Maximal subgroups

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

subgroupindex
C2×C32×Dic122
C3×#638432
C3×C8×C3⋊Dic32
C32×#8182
C3×#635652
C3×Dic3×Q163
C3×#48733
#105908183