Finite Group Databasebeta

← C3×#10590706 #13157707

Subgroups — C3×#10590706 #13157707

Order 1728.

626 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×D3×C8C211yesmaximal, characteristic
2C3×#63841C212yesmaximal
3C3×#63644C211yesmaximal, characteristic, complemented
4C32×#817C211yesmaximal, characteristic, complemented
5C3×#63575C212yesmaximal, complemented
6C3×#5106—31nomaximal
7C3×#4860—31nomaximal
8#10590706C311yesmaximal, characteristic, direct product factor
9C2×C32×C4×D3C2211yescharacteristic
10C2×C33×C8C2211yescharacteristic
11C3×C4⋊(C3⋊Dic3)C2212yes—
12C32×C4⋊Dic3C2212yes—
13C2×C32×C3⋊C8C2211yescharacteristic
14C32×D3×C8C414yescomplemented
15C3×(C2×D3)⋊Dic3—22no—
16C3×Dic32—22no—
17C22×C32×C8—31no—
18C3×#926—32no—
19C2×C3×D3×C8—21no—
20C2×C3×D3×C8D311yescharacteristic
21C2×C3×D3×C8C2×C311yescharacteristic
22C3×#892—32no—
23#63841C2×C312yes—
24C32×#106—31no—
25#63644C2×C311yescharacteristic, complemented
26C3×#818—32no—
27C3×#817—21no—
28C3×#817—31no—
29C3×#817—31no—
30C3×#817—61no—
31C3×#817D311yescharacteristic, complemented
32C3×#817C2×C311yescharacteristic, complemented
33C3×#807—32no—
34#63575C2×C312yescomplemented
35C22×C32×D3D411yescharacteristic
36C2×C33×C4C2311yescharacteristic
37C2×C3×C3⋊Dic3—22no—
38C2×C32×Dic3—22no—
39C2×C32×Dic3D411yescharacteristic
40C32×C4×D3C2×C412yes—
41C32×C4×D3Q812yes—
42C33×C8C2×C412yes—
43C32×C3⋊C8C2×C412yes—
44C3×#427—91no—
45#5106—31no—
46#4860—31no—
47C22×C32×C4—31no—
48C2×C3×C4×D3—21no—
49C2×C3×C4×D3C2×D311yescharacteristic
50C2×C3×C4×D3C22×C311yescharacteristic
51C2×C32×C8—21no—
52C2×C32×C8—21no—
53C2×C32×C8—21no—
54C2×C32×C8—31no—
55C2×C32×C8—34no—
56C2×C32×C8—41no—
57C2×C32×C8C2×D311yescharacteristic
58C2×C32×C8C2×D311yescharacteristic
59C2×C32×C8C22×C311yescharacteristic
60C32×C4⋊C4—32no—
61C4⋊(C3⋊Dic3)C22×C312yes—
62C3×#217—62no—
63C3×(C2×D3)⋊C4—62no—
64C3×C4⋊Dic3—22no—
65C3×C4⋊Dic3—32no—
66C3×C4⋊Dic3—32no—
67C3×C4⋊Dic3—62no—
68C3×C4⋊Dic3C2×D312yes—
69C3×C4⋊Dic3C22×C312yes—
70C3×C4×Dic3—62no—
71C3×C4×Dic3—62no—
72C2×C3×C3⋊C8—21no—
73C2×C3×C3⋊C8C2×D311yescharacteristic
74C2×C3×C3⋊C8C22×C311yescharacteristic
75C3×D3×C8—24no—
76C3×D3×C8Dic314yescomplemented
77C3×D3×C8C3×C414yescomplemented
78(C2×D3)⋊Dic3—22no—
79Dic32—22no—
80C22×C33C2×D411yescharacteristic
81C2×C32×D3—22no—
82C2×C32×D3C4⋊C412yes—
83C33×C4C22×C411yescharacteristic
84C33×C4C2×Q811yescharacteristic
85C3×C3⋊Dic3—42no—
86C32×Dic3—42no—
87C32×Dic3C4⋊C412yes—
88C22×C3×C8—31no—
89C22×C3×C8—31no—
90C22×C3×C8—61no—
91C3×#117—92no—
92#926—32no—
93C2×D3×C8C3×D311yescharacteristic
94#892—32no—
95C3×#107—92no—
96C3×#106—31no—
97C3×#106—31no—
98C3×#106—61no—
99C3×#106—91no—
100#818—32no—
101#817—31no—
102#817—31no—
103#817—61no—
104#817C3×D311yescharacteristic, complemented
105#807—32no—
106C23×C32—31no—
107C22×C3×D3—21no—
108C22×C3×D3D1211yescharacteristic
109C22×C3×D3C3×D411yescharacteristic
110C2×C32×C4—21no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—31no—
114C2×C32×C4—32no—
115C2×C32×C4—32no—
116C2×C32×C4—41no—
117C2×C32×C4—62no—
118C2×C32×C4C22×D311yescharacteristic
119C2×C32×C4C22×D311yescharacteristic
120C2×C32×C4C23×C311yescharacteristic
121C2×C3⋊Dic3—22no—
122C2×C3×Dic3—22no—
123C2×C3×Dic3—22no—
124C2×C3×Dic3—21no—
125C2×C3×Dic3—42no—
126C2×C3×Dic3—62no—
127C2×C3×Dic3—62no—
128C2×C3×Dic3—122no—
129C2×C3×Dic3D1211yescharacteristic
130C2×C3×Dic3C3×D411yescharacteristic
131C3×C4×D3—22no—
132C3×C4×D3—22no—
133C3×C4×D3Dic612yes—
134C3×C4×D3C2×Dic312yes—
135C3×C4×D3C2×C3×C412yes—
136C3×C4×D3C3×Q812yes—
137C32×C8—22no—
138C32×C8—22no—
139C32×C8—22no—
140C32×C8—32no—
141C32×C8—42no—
142C32×C8C4×D312yes—
143C32×C8C2×Dic312yes—
144C32×C8C2×C3×C412yes—
145C3×C3⋊C8—22no—
146C3×C3⋊C8C2×Dic312yes—
147C3×C3⋊C8C2×C3×C412yes—
148#427—91noSylow
149C2×C33C2×C4⋊C411yescharacteristic
150C2×C33#13512yes—
151C32×D3—22no—
152C22×C3×C4—31no—
153C22×C3×C4—31no—
154C22×C3×C4—61no—
155C2×C4×D3C2×C3×D311yescharacteristic
156C2×C3×C8—21no—
157C2×C3×C8—21no—
158C2×C3×C8—21no—
159C2×C3×C8—31no—
160C2×C3×C8—31no—
161C2×C3×C8—34no—
162C2×C3×C8—34no—
163C2×C3×C8—41no—
164C2×C3×C8—64no—
165C2×C3×C8—61no—
166C2×C3×C8D3211yescharacteristic
167C2×C3×C8C2×C3×D311yescharacteristic
168C2×C3×C8C2×C3×D311yescharacteristic
169C3×C4⋊C4—32no—
170C3×C4⋊C4—32no—
171C3×C4⋊C4—62no—
172C3×C4⋊C4—92no—
173C3×C22⋊C4—182no—
174C3×C42—182no—
175#217—62no—
176(C2×D3)⋊C4—62no—
177C4⋊Dic3—32no—
178C4⋊Dic3—32no—
179C4⋊Dic3—62no—
180C4⋊Dic3C2×C3×D312yes—
181C4×Dic3—62no—
182C4×Dic3—62no—
183C2×C3⋊C8C2×C3×D311yescharacteristic
184D3×C8C3×Dic314yescomplemented
185C22×C32—21no—
186C22×C32—21no—
187C22×C32—21no—
188C22×C32—32no—
189C22×C32—41no—
190C22×C32—62no—
191C22×C32C2×D1211yescharacteristic
192C22×C32D3×D411yescharacteristic
193C22×C32C2×C3×D411yescharacteristic
194C2×C3×D3—24no—
195C2×C3×D3—22no—
196C2×C3×D3—22no—
197C2×C3×D3—22no—
198C2×C3×D3C4⋊Dic312yes—
199C2×C3×D3C3×C4⋊C412yes—
200C32×C4—21no—
201C32×C4—21no—
202C32×C4—21no—
203C32×C4—21no—
204C32×C4—21no—
205C32×C4—21no—
206C32×C4—32no—
207C32×C4—41no—
208C32×C4—41no—
209C32×C4—122no—
210C32×C4C2×Dic611yescharacteristic
211C32×C4C2×C4×D311yescharacteristic
212C32×C4D3×Q811yescharacteristic
213C32×C4C22×Dic311yescharacteristic
214C32×C4C22×C3×C411yescharacteristic
215C32×C4C2×C3×Q811yescharacteristic
216C3⋊Dic3—42no—
217C3×Dic3—22no—
218C3×Dic3—42no—
219C3×Dic3—42no—
220C3×Dic3—82no—
221C3×Dic3—122no—
222C3×Dic3—122no—
223C3×Dic3—242no—
224C3×Dic3C4⋊Dic312yes—
225C3×Dic3C3×C4⋊C412yes—
226C22×C8—31no—
227#117—92no—
228#107—92no—
229#106—31no—
230#106—91no—
231C33#42711yescharacteristic, complemented, Sylow
232C23×C3—31no—
233C23×C3—31no—
234C23×C3—61no—
235C22×D3C3×D1211yescharacteristic
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—31no—
240C2×C3×C4—31no—
241C2×C3×C4—32no—
242C2×C3×C4—32no—
243C2×C3×C4—32no—
244C2×C3×C4—32no—
245C2×C3×C4—41no—
246C2×C3×C4—62no—
247C2×C3×C4—62no—
248C2×C3×C4—61no—
249C2×C3×C4—62no—
250C2×C3×C4—62no—
251C2×C3×C4—122no—
252C2×C3×C4—182no—
253C2×C3×C4C2×D3211yescharacteristic
254C2×C3×C4C22×C3×D311yescharacteristic
255C2×C3×C4C22×C3×D311yescharacteristic
256C2×Dic3—22no—
257C2×Dic3—62no—
258C2×Dic3—62no—
259C2×Dic3—122no—
260C2×Dic3C3×D1211yescharacteristic
261C4×D3C3×Dic612yes—
262C4×D3C2×C3×Dic312yes—
263C3×C8—22no—
264C3×C8—22no—
265C3×C8—22no—
266C3×C8—32no—
267C3×C8—32no—
268C3×C8—42no—
269C3×C8—62no—
270C3×C8D3×Dic312yes—
271C3×C8C3×C4×D312yes—
272C3×C8C2×C3×Dic312yes—
273C3⋊C8C2×C3×Dic312yes—
274C2×C32—21no—
275C2×C32—22no—
276C2×C32—22no—
277C2×C32—21no—
278C2×C32—22no—
279C2×C32—21no—
280C2×C32—41no—
281C2×C32—42no—
282C2×C32—62no—
283C2×C32D3×C4⋊C411yescharacteristic
284C2×C32#90412yes—
285C2×C32#91612yes—
286C2×C32C2×C4⋊Dic311yescharacteristic
287C2×C32C2×C3×C4⋊C411yescharacteristic
288C2×C32C3×#13512yes—
289C3×D3—24no—
290C3×D3—22no—
291C3×D3—22no—
292C22×C4—31no—
293C2×C8—31no—
294C2×C8—34no—
295C2×C8C3×D3211yescharacteristic
296C4⋊C4—32no—
297C4⋊C4—92no—
298C22⋊C4—182no—
299C42—182no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—32no—
304C22×C3—32no—
305C22×C3—41no—
306C22×C3—62no—
307C22×C3—64no—
308C22×C3—62no—
309C22×C3—62no—
310C22×C3D3×D1211yescharacteristic
311C22×C3C2×C3×D1211yescharacteristic
312C22×C3C3×D3×D411yescharacteristic
313C2×D3—22no—
314C2×D3C3×C4⋊Dic312yes—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—32no—
322C3×C4—32no—
323C3×C4—41no—
324C3×C4—41no—
325C3×C4—62no—
326C3×C4—122no—
327C3×C4—122no—
328C3×C4—242no—
329C3×C4—362no—
330C3×C4D3×Dic611yescharacteristic
331C3×C4C2×D3×Dic311yescharacteristic
332C3×C4C2×C3×Dic611yescharacteristic
333C3×C4C2×C3×C4×D311yescharacteristic
334C3×C4C3×D3×Q811yescharacteristic
335C3×C4C22×C3×Dic311yescharacteristic
336Dic3—42no—
337Dic3—122no—
338Dic3—122no—
339Dic3—242no—
340Dic3C3×C4⋊Dic312yes—
341C32—21no—
342C32—21no—
343C32—21no—
344C32—41no—
345C32#486011yescharacteristic, complemented
346C32#510611yescharacteristic, complemented
347C32C3×#42711yescharacteristic, complemented
348C23—31no—
349C2×C4—31no—
350C2×C4—32no—
351C2×C4—32no—
352C2×C4—62no—
353C2×C4—182no—
354C2×C4C2×C3×D3211yescharacteristic
355C8—32no—
356C8C3×D3×Dic312yes—
357C2×C3—22no—
358C2×C3—22no—
359C2×C3—22no—
360C2×C3—21no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—42no—
364C2×C3—41no—
365C2×C3—64no—
366C2×C3—62no—
367C2×C3—62no—
368C2×C3#6380612yes—
369C2×C3D3×C4⋊Dic311yescharacteristic
370C2×C3C3×D3×C4⋊C411yescharacteristic
371C2×C3C3×#90412yes—
372C2×C3C3×#91612yes—
373C2×C3C2×C3×C4⋊Dic311yescharacteristic
374D3—22no—
375C22—32no—
376C22—62no—
377C22C3×D3×D1211yescharacteristic
378C4—32no—
379C4—122no—
380C4—362no—
381C4C3×D3×Dic611yescharacteristic
382C4C2×C3×D3×Dic311yescharacteristic
383C3—21no—
384C3—21no—
385C3—21no—
386C3—41no—
387C3#1059070611yescharacteristic, direct product factor
388C3C3×#486011yescharacteristic, complemented
389C3C3×#510611yescharacteristic, complemented
390C2—62no—
391C2C3×#6380612yes—
392C2C3×D3×C4⋊Dic311yescharacteristic
subgroupG/H
centerC22×C3D3×D12
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×C8D32
hypocenterC32C3×#427
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
3. C3×#63644C2semidirect
4. C32×#817C2semidirect
5. C3×#63575C2semidirect
8. #10590706C3direct (N × H)
14. C32×D3×C8C4semidirect
25. #63644C2×C3semidirect
31. C3×#817D3semidirect
32. C3×#817C2×C3semidirect
34. #63575C2×C3semidirect
76. C3×D3×C8Dic3semidirect
77. C3×D3×C8C3×C4semidirect
104. #817C3×D3semidirect
184. D3×C8C3×Dic3semidirect
231. C33#427semidirect
345. C32#4860semidirect
346. C32#5106semidirect
347. C32C3×#427semidirect
387. C3#10590706direct (N × H)
388. C3C3×#4860semidirect
389. C3C3×#5106semidirect

Maximal subgroups

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

subgroupindex
C2×C32×D3×C82
C3×#638412
C3×#636442
C32×#8172
C3×#635752
C3×#51063
C3×#48603
#105907063