Finite Group Databasebeta

← C3×#10590807 #13157781

Subgroups — C3×#10590807 #13157781

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×C3×#3746C211yesmaximal, characteristic
2C3×#63844C212yesmaximal
3C32×#818C211yesmaximal, characteristic
4C32×C8×Dic3C211yesmaximal, characteristic
5C3×#63567C212yesmaximal
6C3×#4873—31nomaximal
7C3×C4×Dic12—31nomaximal
8#10590807C311yesmaximal, characteristic, direct product factor
9C2×C3×#634C2212yes—
10C2×C33×C8C2211yescharacteristic
11C3×#3746C414yescomplemented
12C32×C4⋊Dic3C2212yes—
13C32×C4×Dic3C2211yescharacteristic
14C2×C32×C3⋊C8C2211yescharacteristic
15C3×#3724—22no—
16C3×Dic32—22no—
17C2×#3746C2×C311yescharacteristic
18C2×C3×Dic12—31no—
19C2×C3×Dic12—31no—
20C2×C3×Dic12—61no—
21C3×#887—32no—
22C3×C4×Dic6—32no—
23#63844C2×C312yes—
24C32×#107—31no—
25C32×C4×C8—31no—
26C3×#818—21no—
27C3×#818D311yescharacteristic
28C3×#818C2×C311yescharacteristic
29C3×#816—32no—
30C3×C8×Dic3—21no—
31C3×C8×Dic3D311yescharacteristic
32C3×C8×Dic3C2×C311yescharacteristic
33C3×#810—32no—
34#63567C2×C312yes—
35C2×C33×C4C2311yescharacteristic
36C2×C3×C3⋊Dic3—22no—
37C3×#634—22no—
38C3×#634C2×C414yes—
39C2×C32×Dic3—22no—
40C2×C32×Dic3D411yescharacteristic
41C2×C32×Dic3D411yescharacteristic
42C33×C8C2×C412yes—
43C32×C3⋊C8—21no—
44C3×C4×Q16—91no—
45#4873—31no—
46C4×Dic12—31no—
47C2×#634C22×C312yes—
48C2×C3×Dic6—32no—
49C2×C3×Dic6—32no—
50C2×C3×Dic6—62no—
51C2×C32×C8—21no—
52C2×C32×C8—21no—
53C2×C32×C8—21no—
54C2×C32×C8—31no—
55C2×C32×C8—41no—
56C2×C32×C8C2×D311yescharacteristic
57C2×C32×C8C2×D311yescharacteristic
58C2×C32×C8C22×C311yescharacteristic
59C32×C4⋊C4—32no—
60C32×C42—31no—
61#3746C3×C414yescomplemented
62C3×C4⋊Dic3—22no—
63C3×C4⋊Dic3C2×D312yes—
64C3×C4⋊Dic3C22×C312yes—
65C3×Dic3⋊C4—62no—
66C3×Dic3⋊C4—62no—
67C3×C4×Dic3—21no—
68C3×C4×Dic3—62no—
69C3×C4×Dic3—62no—
70C3×C4×Dic3C2×D311yescharacteristic
71C3×C4×Dic3C22×C311yescharacteristic
72C2×C3×C3⋊C8—21no—
73C2×C3×C3⋊C8C2×D311yescharacteristic
74C2×C3×C3⋊C8C22×C311yescharacteristic
75C3×Dic12—34no—
76C3×Dic12—34no—
77C3×Dic12—64no—
78#3724—22no—
79Dic32—22no—
80C22×C33C2×D411yescharacteristic
81C33×C4C22×C411yescharacteristic
82C33×C4#4111yescharacteristic
83C3×C3⋊Dic3—24no—
84C32×Dic3—21no—
85C32×Dic3—42no—
86C32×Dic3Q1612yescomplemented
87C2×C3×Q16—91no—
88C3×C4×Q8—92no—
89C2×Dic12—31no—
90C2×Dic12—31no—
91C2×Dic12—61no—
92#887—32no—
93C4×Dic6—32no—
94C3×#107—31no—
95C3×#107—31no—
96C3×#107—61no—
97C3×#103—92no—
98C3×C4×C8—31no—
99C3×C4×C8—31no—
100C3×C4×C8—61no—
101#818C3×D311yescharacteristic
102#816—32no—
103C8×Dic3C3×D311yescharacteristic
104#810—32no—
105C2×C32×C4—21no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4—31no—
109C2×C32×C4—31no—
110C2×C32×C4—41no—
111C2×C32×C4—62no—
112C2×C32×C4C22×D311yescharacteristic
113C2×C32×C4C22×D311yescharacteristic
114C2×C32×C4C23×C311yescharacteristic
115C2×C3⋊Dic3—22no—
116#634—22no—
117#634C2×C3×C414yes—
118C2×C3×Dic3—22no—
119C2×C3×Dic3—22no—
120C2×C3×Dic3—21no—
121C2×C3×Dic3—21no—
122C2×C3×Dic3—42no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—122no—
126C2×C3×Dic3D1211yescharacteristic
127C2×C3×Dic3D1211yescharacteristic
128C2×C3×Dic3C3×D411yescharacteristic
129C2×C3×Dic3C3×D411yescharacteristic
130C3×Dic6—34no—
131C3×Dic6—34no—
132C3×Dic6—64no—
133C3×Dic6—62no—
134C3×Dic6—62no—
135C3×Dic6—122no—
136C32×C8—22no—
137C32×C8—22no—
138C32×C8—22no—
139C32×C8—42no—
140C32×C8—61no—
141C32×C8C4×D312yes—
142C32×C8C4×D312yes—
143C32×C8C2×C3×C412yes—
144C3×C3⋊C8—22no—
145C3×C3⋊C8—21no—
146C3×C3⋊C8—21no—
147C4×Q16—91noSylow
148C2×C33C4×D411yescharacteristic
149C2×C33C2×Q1611yescharacteristic
150C2×C33#13511yescharacteristic
151C2×C3×Q8—92no—
152C2×Dic6—32no—
153C2×Dic6—32no—
154C2×Dic6—62no—
155C3×Q16—94no—
156C2×C3×C8—21no—
157C2×C3×C8—21no—
158C2×C3×C8—21no—
159C2×C3×C8—31no—
160C2×C3×C8—31no—
161C2×C3×C8—41no—
162C2×C3×C8—61no—
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—31no—
171C3×C42—31no—
172C3×C42—61no—
173C3×C42—182no—
174C4⋊Dic3C2×C3×D312yes—
175Dic3⋊C4—62no—
176Dic3⋊C4—62no—
177C4×Dic3—62no—
178C4×Dic3—62no—
179C4×Dic3C2×C3×D311yescharacteristic
180C2×C3⋊C8C2×C3×D311yescharacteristic
181Dic12—34no—
182Dic12—34no—
183Dic12—64no—
184C22×C32—21no—
185C22×C32—21no—
186C22×C32—21no—
187C22×C32—41no—
188C22×C32C2×D1211yescharacteristic
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—32no—
198C32×C4—41no—
199C32×C4—41no—
200C32×C4—61no—
201C32×C4—122no—
202C32×C4C2×C4×D311yescharacteristic
203C32×C4C2×C4×D311yescharacteristic
204C32×C4#23511yescharacteristic
205C32×C4#23911yescharacteristic
206C32×C4C22×C3×C411yescharacteristic
207C32×C4C3×#4111yescharacteristic
208C3⋊Dic3—24no—
209C3×Dic3—22no—
210C3×Dic3—21no—
211C3×Dic3—21no—
212C3×Dic3—22no—
213C3×Dic3—42no—
214C3×Dic3—42no—
215C3×Dic3—64no—
216C3×Dic3—64no—
217C3×Dic3—82no—
218C3×Dic3—124no—
219C3×Dic3Dic1212yescomplemented
220C3×Dic3C3×Q1612yescomplemented
221C2×Q16—91no—
222C4×Q8—92no—
223#107—31no—
224#103—92no—
225C4×C8—31no—
226C33C4×Q1611yescharacteristic, complemented, Sylow
227C3×Q8—94no—
228C3×Q8—182no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—31no—
233C2×C3×C4—31no—
234C2×C3×C4—31no—
235C2×C3×C4—31no—
236C2×C3×C4—41no—
237C2×C3×C4—61no—
238C2×C3×C4—61no—
239C2×C3×C4—62no—
240C2×C3×C4—62no—
241C2×C3×C4—122no—
242C2×C3×C4—182no—
243C2×C3×C4C2×D3211yescharacteristic
244C2×C3×C4C22×C3×D311yescharacteristic
245C2×C3×C4C22×C3×D311yescharacteristic
246C2×Dic3—22no—
247C2×Dic3—62no—
248C2×Dic3—62no—
249C2×Dic3—122no—
250C2×Dic3C3×D1211yescharacteristic
251C2×Dic3C3×D1211yescharacteristic
252Dic6—34no—
253Dic6—34no—
254Dic6—64no—
255Dic6—62no—
256Dic6—62no—
257Dic6—122no—
258C3×C8—22no—
259C3×C8—22no—
260C3×C8—22no—
261C3×C8—42no—
262C3×C8—62no—
263C3×C8—61no—
264C3×C8—61no—
265C3×C8#62412yes—
266C3×C8C3×C4×D312yes—
267C3×C8C3×C4×D312yes—
268C3⋊C8—21no—
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×C32C4×D1211yescharacteristic
282C2×C32#89311yescharacteristic
283C2×C32#90411yescharacteristic
284C2×C32C2×Dic1211yescharacteristic
285C2×C32#91211yescharacteristic
286C2×C32D3×Q1611yescharacteristic
287C2×C32C3×C4×D411yescharacteristic
288C2×C32C2×C3×Q1611yescharacteristic
289C2×C32C3×#13511yescharacteristic
290C2×Q8—92no—
291Q16—94no—
292C2×C8—31no—
293C2×C8C3×D3211yescharacteristic
294C4⋊C4—32no—
295C4⋊C4—182no—
296C42—31no—
297C42—182no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—41no—
302C22×C3D3×D1211yescharacteristic
303C22×C3C2×C3×D1211yescharacteristic
304C22×C3C3×D3×D411yescharacteristic
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×C4—62no—
317C3×C4—61no—
318C3×C4—61no—
319C3×C4—122no—
320C3×C4—122no—
321C3×C4—184no—
322C3×C4—242no—
323C3×C4#379911yescharacteristic
324C3×C4C2×#62411yescharacteristic
325C3×C4C2×C3×C4×D311yescharacteristic
326C3×C4C2×C3×C4×D311yescharacteristic
327C3×C4C3×#23511yescharacteristic
328C3×C4C3×#23911yescharacteristic
329Dic3—21no—
330Dic3—42no—
331Dic3—64no—
332Dic3—64no—
333Dic3—124no—
334Dic3C3×Dic1212yescomplemented
335C32—21no—
336C32—21no—
337C32—21no—
338C32—41no—
339C32C4×Dic1211yescharacteristic, complemented
340C32#487311yescharacteristic, complemented
341C32C3×C4×Q1611yescharacteristic, complemented
342Q8—94no—
343Q8—182no—
344C2×C4—31no—
345C2×C4—31no—
346C2×C4—62no—
347C2×C4—182no—
348C2×C4C2×C3×D3211yescharacteristic
349C8—61no—
350C8C3×#62412yes—
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×C3D3×Dic1211yescharacteristic
364C2×C3#6380711yescharacteristic
365C2×C3#6389411yescharacteristic
366C2×C3C3×C4×D1211yescharacteristic
367C2×C3C3×#89311yescharacteristic
368C2×C3C3×#90411yescharacteristic
369C2×C3C2×C3×Dic1211yescharacteristic
370C2×C3C3×#91211yescharacteristic
371C2×C3C3×D3×Q1611yescharacteristic
372C22C3×D3×D1211yescharacteristic
373C4—32no—
374C4—61no—
375C4—122no—
376C4—184no—
377C4C3×#379911yescharacteristic
378C4C2×C3×#62411yescharacteristic
379C3—21no—
380C3—21no—
381C3—21no—
382C3—41no—
383C3#1059080711yescharacteristic, direct product factor
384C3C3×C4×Dic1211yescharacteristic, complemented
385C3C3×#487311yescharacteristic, complemented
386C2C3×D3×Dic1211yescharacteristic
387C2C3×#6380711yescharacteristic
388C2C3×#6389411yescharacteristic
subgroupG/H
centerC22×C3D3×D12
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. #10590807C3direct (N × H)
11. C3×#3746C4semidirect
61. #3746C3×C4semidirect
86. C32×Dic3Q16semidirect
219. C3×Dic3Dic12semidirect
220. C3×Dic3C3×Q16semidirect
226. C33C4×Q16semidirect
334. Dic3C3×Dic12semidirect
339. C32C4×Dic12semidirect
340. C32#4873semidirect
341. C32C3×C4×Q16semidirect
383. C3#10590807direct (N × H)
384. C3C3×C4×Dic12semidirect
385. C3C3×#4873semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37462
C3×#638442
C32×#8182
C32×C8×Dic32
C3×#635672
C3×#48733
C3×C4×Dic123
#105908073