Finite Group Databasebeta

← C3×#10590718 #13157718

Subgroups — C3×#10590718 #13157718

Order 1728.

702 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
1C3×C4×#625C211yesmaximal, characteristic
2C2×C3×D3×C3⋊C8C212yesmaximal, complemented
3C3×C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C32×(C2×D3)⋊C8C212yesmaximal, complemented
5C3×#63572C211yesmaximal, characteristic, complemented
6C3×#67⋊C8—32nomaximal
7#10590718C311yesmaximal, characteristic, direct product factor
8C2×C32×C4×D3C2212yes—
9C2×C3×#625C411yescharacteristic
10C2×C33×C8C2211yescharacteristic
11C3×C4×C3⋊Dic3C2211yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
13C2×C32×C3⋊C8C2212yes—
14C3×#3724C411yescharacteristic
15C3×(C2×D3)⋊Dic3C412yes—
16C3×D3×C3⋊C8—24no—
17C3×C4×#67—32no—
18C22×C3×C3⋊C8—32no—
19C2×C3×D3×C8—32no—
20C4×#625C2×C311yescharacteristic
21C2×D3×C3⋊C8C2×C312yescomplemented
22C32×C22⋊C8—32no—
23C8×C3⋊Dic3C2×C311yescharacteristic, complemented
24C3×(C2×D3)⋊C8—22no—
25C3×(C2×D3)⋊C8D312yescomplemented
26C3×(C2×D3)⋊C8C2×C312yescomplemented
27C3×Dic3⋊C8—32no—
28C3×C8×Dic3—32no—
29C3×C8×Dic3—61no—
30#63572C2×C311yescharacteristic, complemented
31C22×C32×D3C2×C412yes—
32C2×C33×C4C2311yescharacteristic
33C2×C3×C3⋊Dic3C2×C411yescharacteristic
34C2×C3×C3⋊Dic3C2×C411yescharacteristic
35C2×C32×Dic3C2×C412yes—
36C32×C4×D3—22no—
37C32×C4×D3—22no—
38C3×#625C814yescomplemented
39C33×C8—21no—
40C3×C3⋊(C3⋊C8)D412yescomplemented
41C32×C3⋊C8—22no—
42C3×C8×D4—91no—
43#67⋊C8—32no—
44C22×C32×C4—32no—
45C2×C3×#67—32no—
46C2×C3×C4×D3—22no—
47C2×C3×C4×D3C2×D312yes—
48C2×C3×C4×D3C22×C312yes—
49C2×#625C3×C411yescharacteristic
50C2×C32×C8—21no—
51C2×C32×C8—22no—
52C2×C32×C8—32no—
53C2×C32×C8—41no—
54C2×C32×C8C2×D312yes—
55C2×C32×C8C22×C311yescharacteristic
56C4×C3⋊Dic3C22×C311yescharacteristic
57C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
58C3×#217—32no—
59C3×(C2×D3)⋊C4—32no—
60C3×Dic3⋊C4—32no—
61C3×C4×Dic3—32no—
62C3×C4×Dic3—61no—
63C2×C3×C3⋊C8—22no—
64C2×C3×C3⋊C8—32no—
65C2×C3×C3⋊C8—64no—
66C2×C3×C3⋊C8—61no—
67C2×C3×C3⋊C8C2×D312yes—
68C2×C3×C3⋊C8C22×C312yes—
69C3×D3×C8—64no—
70#3724C3×C411yescharacteristic
71(C2×D3)⋊Dic3C3×C412yes—
72D3×C3⋊C8—24no—
73C22×C33C22×C411yescharacteristic
74C2×C32×D3—22no—
75C2×C32×D3—22no—
76C2×C32×D3C2×C814yescomplemented
77C33×C4C2×D411yescharacteristic
78C33×C4#4111yescharacteristic
79C3×C3⋊Dic3—21no—
80C3×C3⋊Dic3C2×C812yescomplemented
81C32×Dic3—22no—
82C22×C3×C8—92no—
83C3×C4×D4—91no—
84C4×#67—32no—
85C22×C3⋊C8—32no—
86C2×D3×C8—32no—
87C3×C4⋊C8—91no—
88C3×C22⋊C8—32no—
89C3×C22⋊C8—32no—
90C3×C22⋊C8—62no—
91C3×C4×C8—91no—
92(C2×D3)⋊C8C3×D312yescomplemented
93Dic3⋊C8—32no—
94C8×Dic3—32no—
95C8×Dic3—61no—
96C23×C32—32no—
97C22×C3×D3—22no—
98C22×C3×D3C4×D312yes—
99C22×C3×D3C2×C3×C412yes—
100C2×C32×C4—21no—
101C2×C32×C4—22no—
102C2×C32×C4—32no—
103C2×C32×C4—41no—
104C2×C32×C4—62no—
105C2×C32×C4—62no—
106C2×C32×C4C22×D312yes—
107C2×C32×C4C23×C311yescharacteristic
108C2×C3⋊Dic3C2×C3×C411yescharacteristic
109C2×C3⋊Dic3C2×C3×C411yescharacteristic
110C3×#67—38no—
111C2×C3×Dic3—22no—
112C2×C3×Dic3—32no—
113C2×C3×Dic3—32no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3C4×D312yes—
117C2×C3×Dic3C2×C3×C412yes—
118C3×C4×D3—22no—
119C3×C4×D3—22no—
120C3×C4×D3—22no—
121C3×C4×D3—22no—
122C3×C4×D3—42no—
123C3×C4×D3—42no—
124#625C3×C814yescomplemented
125C32×C8—21no—
126C32×C8—22no—
127C32×C8—22no—
128C32×C8—42no—
129C32×C8—42no—
130C32×C8—62no—
131C3⋊(C3⋊C8)C3×D412yescomplemented
132C3×C3⋊C8—22no—
133C3×C3⋊C8—22no—
134C3×C3⋊C8—34no—
135C3×C3⋊C8—42no—
136C3×C3⋊C8—62no—
137C8×D4—91noSylow
138C2×C33C4×D411yescharacteristic
139C2×C33C22×C811yescharacteristic
140C2×C33#13111yescharacteristic
141C32×D3—24no—
142C2×C3×D4—91no—
143C22×C3×C4—32no—
144C22×C3×C4—32no—
145C22×C3×C4—62no—
146C2×#67—32no—
147C2×C4×D3C2×C3×D312yes—
148C2×C3×C8—21no—
149C2×C3×C8—22no—
150C2×C3×C8—32no—
151C2×C3×C8—32no—
152C2×C3×C8—41no—
153C2×C3×C8—62no—
154C2×C3×C8—91no—
155C2×C3×C8—184no—
156C2×C3×C8D3211yescharacteristic
157C2×C3×C8C2×C3×D312yes—
158C3×C4⋊C4—91no—
159C3×C22⋊C4—92no—
160C3×C42—91no—
161#217—32no—
162(C2×D3)⋊C4—32no—
163Dic3⋊C4—32no—
164C4×Dic3—32no—
165C4×Dic3—61no—
166C2×C3⋊C8—32no—
167C2×C3⋊C8—61no—
168C2×C3⋊C8—64no—
169C2×C3⋊C8C2×C3×D312yes—
170D3×C8—64no—
171C22×C32—21no—
172C22×C32—22no—
173C22×C32—34no—
174C22×C32—41no—
175C22×C32—62no—
176C22×C32—62no—
177C22×C32C2×C4×D312yes—
178C22×C32C22×C3×C411yescharacteristic
179C2×C3×D3—22no—
180C2×C3×D3—22no—
181C2×C3×D3—22no—
182C2×C3×D3—22no—
183C2×C3×D3—24no—
184C2×C3×D3—42no—
185C2×C3×D3—42no—
186C2×C3×D3D3×C814yescomplemented
187C2×C3×D3C2×C3×C814yescomplemented
188C32×C4—21no—
189C32×C4—21no—
190C32×C4—22no—
191C32×C4—22no—
192C32×C4—41no—
193C32×C4—41no—
194C32×C4—62no—
195C32×C4D3×D412yes—
196C32×C4#23712yes—
197C32×C4C2×C3×D411yescharacteristic
198C32×C4C3×#4111yescharacteristic
199C3⋊Dic3—21no—
200C3⋊Dic3C2×C3×C812yescomplemented
201C3×Dic3—22no—
202C3×Dic3—22no—
203C3×Dic3—34no—
204C3×Dic3—42no—
205C3×Dic3—62no—
206C3×Dic3—62no—
207C3×Dic3—62no—
208C22×C8—92no—
209C4×D4—91no—
210C4⋊C8—91no—
211C22⋊C8—32no—
212C4×C8—91no—
213C33C8×D411yescharacteristic, complemented, Sylow
214C23×C3—32no—
215C23×C3—32no—
216C23×C3—62no—
217C22×D3C3×C4×D312yes—
218C3×D4—94no—
219C2×C3×C4—22no—
220C2×C3×C4—21no—
221C2×C3×C4—32no—
222C2×C3×C4—32no—
223C2×C3×C4—41no—
224C2×C3×C4—62no—
225C2×C3×C4—62no—
226C2×C3×C4—62no—
227C2×C3×C4—62no—
228C2×C3×C4—62no—
229C2×C3×C4—91no—
230C2×C3×C4—91no—
231C2×C3×C4—122no—
232C2×C3×C4—122no—
233C2×C3×C4C2×D3211yescharacteristic
234C2×C3×C4C22×C3×D312yes—
235#67—38no—
236C2×Dic3—32no—
237C2×Dic3—32no—
238C2×Dic3—61no—
239C2×Dic3—61no—
240C2×Dic3C3×C4×D312yes—
241C4×D3—22no—
242C4×D3—22no—
243C3×C8—22no—
244C3×C8—21no—
245C3×C8—22no—
246C3×C8—42no—
247C3×C8—42no—
248C3×C8—62no—
249C3×C8—62no—
250C3×C8—92no—
251C3×C8—122no—
252C3⋊C8—22no—
253C3⋊C8—34no—
254C3⋊C8—62no—
255C2×C32—22no—
256C2×C32—22no—
257C2×C32—22no—
258C2×C32—21no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—41no—
262C2×C32—41no—
263C2×C32—41no—
264C2×C32—64no—
265C2×C32#67⋊C412yes—
266C2×C32C2×D3×C812yes—
267C2×C32#90712yes—
268C2×C32C3×C4×D411yescharacteristic
269C2×C32C22×C3×C811yescharacteristic
270C2×C32C3×#13111yescharacteristic
271C3×D3—24no—
272C3×D3—24no—
273C3×D3—44no—
274C2×D4—91no—
275C22×C4—32no—
276C2×C8—32no—
277C2×C8—91no—
278C2×C8—184no—
279C2×C8C3×D3211yescharacteristic
280C4⋊C4—91no—
281C22⋊C4—92no—
282C42—91no—
283C22×C3—21no—
284C22×C3—22no—
285C22×C3—34no—
286C22×C3—34no—
287C22×C3—41no—
288C22×C3—64no—
289C22×C3—62no—
290C22×C3—62no—
291C22×C3—62no—
292C22×C3—62no—
293C22×C3—122no—
294C22×C3—122no—
295C22×C3C4×D3211yescharacteristic
296C22×C3C2×C3×C4×D312yes—
297C2×D3—22no—
298C2×D3—22no—
299C2×D3C3×D3×C814yescomplemented
300C3×C4—21no—
301C3×C4—22no—
302C3×C4—21no—
303C3×C4—22no—
304C3×C4—41no—
305C3×C4—41no—
306C3×C4—62no—
307C3×C4—62no—
308C3×C4—92no—
309C3×C4—122no—
310C3×C4—181no—
311C3×C4#380211yescharacteristic
312C3×C4#380511yescharacteristic
313C3×C4C3×D3×D412yes—
314C3×C4C3×#23712yes—
315Dic3—22no—
316Dic3—34no—
317Dic3—62no—
318Dic3—62no—
319Dic3—62no—
320C32—22no—
321C32—21no—
322C32—41no—
323C32#67⋊C812yescomplemented
324C32C3×C8×D411yescharacteristic, complemented
325C23—32no—
326D4—94no—
327C2×C4—32no—
328C2×C4—62no—
329C2×C4—62no—
330C2×C4—91no—
331C2×C4—91no—
332C2×C4C2×C3×D3211yescharacteristic
333C8—21no—
334C8—62no—
335C8—92no—
336C2×C3—21no—
337C2×C3—22no—
338C2×C3—21no—
339C2×C3—22no—
340C2×C3—22no—
341C2×C3—21no—
342C2×C3—41no—
343C2×C3—41no—
344C2×C3—41no—
345C2×C3—64no—
346C2×C3—64no—
347C2×C3—124no—
348C2×C3D32×C811yescharacteristic
349C2×C3#6380511yescharacteristic
350C2×C3#6390211yescharacteristic
351C2×C3C3×#67⋊C412yes—
352C2×C3C2×C3×D3×C812yes—
353C2×C3C3×#90712yes—
354D3—24no—
355C22—34no—
356C22—62no—
357C22—62no—
358C22C3×C4×D3211yescharacteristic
359C4—62no—
360C4—92no—
361C4—181no—
362C4C3×#380211yescharacteristic
363C4C3×#380511yescharacteristic
364C3—22no—
365C3—21no—
366C3—41no—
367C3#1059071811yescharacteristic, direct product factor
368C3C3×#67⋊C812yescomplemented
369C2—64no—
370C2C3×D32×C811yescharacteristic
371C2C3×#6380511yescharacteristic
372C2C3×#6390211yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC2×C32C22×C3×C8
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C22×C4
prime radicalC2×C3×#625C4
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×C8D32
hypocenterC32C3×C8×D4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×D3×C3⋊C8C2semidirect
3. C3×C8×C3⋊Dic3C2semidirect
4. C32×(C2×D3)⋊C8C2semidirect
5. C3×#63572C2semidirect
7. #10590718C3direct (N × H)
21. C2×D3×C3⋊C8C2×C3semidirect
23. C8×C3⋊Dic3C2×C3semidirect
25. C3×(C2×D3)⋊C8D3semidirect
26. C3×(C2×D3)⋊C8C2×C3semidirect
30. #63572C2×C3semidirect
38. C3×#625C8semidirect
40. C3×C3⋊(C3⋊C8)D4semidirect
76. C2×C32×D3C2×C8semidirect
80. C3×C3⋊Dic3C2×C8semidirect
92. (C2×D3)⋊C8C3×D3semidirect
124. #625C3×C8semidirect
131. C3⋊(C3⋊C8)C3×D4semidirect
186. C2×C3×D3D3×C8semidirect
187. C2×C3×D3C2×C3×C8semidirect
200. C3⋊Dic3C2×C3×C8semidirect
213. C33C8×D4semidirect
299. C2×D3C3×D3×C8semidirect
323. C32#67⋊C8semidirect
324. C32C3×C8×D4semidirect
367. C3#10590718direct (N × H)
368. C3C3×#67⋊C8semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#6252
C2×C3×D3×C3⋊C82
C3×C8×C3⋊Dic32
C32×(C2×D3)⋊C82
C3×#635722
C3×#67⋊C83
#105907183