Finite Group Databasebeta

← C3×#10592482 #13162820

Subgroups — C3×#10592482 #13162820

Order 1728.

1112 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic
2C2×C3×#3754C211yesmaximal, characteristic, complemented
3C2×C3×#3746C211yesmaximal, characteristic, complemented
4C3×#64106C211yesmaximal, characteristic, complemented
5C33×#102C211yesmaximal, characteristic, complemented
6C3×#63647C211yesmaximal, characteristic, complemented
7C3×#63637C211yesmaximal, characteristic, complemented
8#10592482C311yesmaximal, characteristic, direct product factor
9C3×#4783—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C2×C3×#638—21no—
12C22×C3×C3⋊Dic3—21no—
13C3×#3830—22no—
14C3×#3830—22no—
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C2×C3×#634C2211yescharacteristic
17C2×C3×#634C2211yescharacteristic
18C2×C33×C8C2211yescharacteristic
19C33×C4⋊C4C2211yescharacteristic
20C3×#3754—22no—
21C3×#3752—21no—
22C3×C4⋊(C3⋊Dic3)—21no—
23C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
24C3×#3746—22no—
25C2×#3830C2×C311yescharacteristic
26C2×C3×#237—34no—
27C2×#3754C2×C311yescharacteristic, complemented
28C2×#3746C2×C311yescharacteristic, complemented
29#64106C2×C311yescharacteristic, complemented
30C2×C3×#214—34no—
31C2×C3×Dic12—34no—
32C3×#897—34no—
33C32×#102—24no—
34C32×#102D314yescomplemented
35C32×#102C2×C311yescharacteristic, complemented
36#63647C2×C311yescharacteristic, complemented
37#63637C2×C311yescharacteristic, complemented
38C3×(C2×D3)⋊C8—34no—
39C3×#810—34no—
40C23×C33—21no—
41C22×C3×C3⋊D3D411yescharacteristic
42C33×D4—21no—
43C33×D4D412yes—
44C2×C33×C4—21no—
45C2×C33×C4C2311yescharacteristic
46C3×#638—24no—
47C2×C3×C3⋊Dic3—24no—
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C3×C4×C3⋊D3—21no—
52C3×C4×C3⋊D3—21no—
53C3×#634—21no—
54C3×#634—21no—
55C3×#634—21no—
56C3×#634D412yes—
57C33×C8—21no—
58C3×C3⋊(C3⋊C8)—21no—
59C3×#452—91no—
60#4783—34no—
61C2×C32×D4—24no—
62C2×C32×D4C2×D314yes—
63C2×C32×D4C22×C311yescharacteristic
64C2×#638—21no—
65C22×C3⋊Dic3—21no—
66#3830—22no—
67#3830—22no—
68C2×C4×C3⋊D3C22×C311yescharacteristic
69C2×#634C22×C311yescharacteristic
70C2×#634C22×C311yescharacteristic
71C2×C3×#67—64no—
72C22×C3×Dic3—64no—
73C3×#237—68no—
74C3×#237—68no—
75C2×C3×C4×D3—34no—
76C2×C3×Dic6—34no—
77C2×C3×Dic6—34no—
78C2×C32×C8—24no—
79C2×C32×C8C2×D314yes—
80C2×C32×C8C22×C311yescharacteristic
81C32×C4⋊C4—24no—
82C32×C4⋊C4C2×D314yes—
83C32×C4⋊C4C22×C311yescharacteristic
84#3754—22no—
85#3752—21no—
86C4⋊(C3⋊Dic3)—21no—
87C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
88#3746—22no—
89C3×#214—68no—
90C3×(C2×D3)⋊C4—64no—
91C3×C4⋊Dic3—64no—
92C2×C3×C3⋊C8—34no—
93C3×Dic12—68no—
94C22×C33—22no—
95C22×C33—41no—
96C22×C33—41no—
97C22×C33C2×D411yescharacteristic
98C2×C3×C3⋊D3—21no—
99C2×C3×C3⋊D3—21no—
100C2×C3×C3⋊D3—21no—
101C33×C4—41no—
102C33×C4C2×D411yescharacteristic
103C33×C4C2×D411yescharacteristic
104C3×C3⋊Dic3—22no—
105C3×C3⋊Dic3—21no—
106C3×C3⋊Dic3—41no—
107C2×C3×#41—91no—
108C2×#237—34no—
109C2×C3×Q16—91no—
110C2×C3×SD16—91no—
111C3×C22⋊Q8—91no—
112C2×#214—34no—
113C2×Dic12—34no—
114#897—34no—
115C3×#103—91no—
116C3×#102—24no—
117C3×#102C3×D314yescomplemented
118C3×#102C3⋊D311yescharacteristic, complemented
119C3×C22⋊C8—91no—
120(C2×D3)⋊C8—34no—
121#810—34no—
122C23×C32—21no—
123C23×C32—24no—
124C23×C32—44no—
125C22×C3⋊D3C3×D411yescharacteristic
126C22×C3×D3—34no—
127C32×D4—21no—
128C32×D4—28no—
129C32×D4—24no—
130C32×D4—44no—
131C32×D4D1218yes—
132C32×D4C3×D412yes—
133C2×C32×C4—21no—
134C2×C32×C4—24no—
135C2×C32×C4—24no—
136C2×C32×C4—44no—
137C2×C32×C4C22×D314yes—
138C2×C32×C4C23×C311yescharacteristic
139#638—24no—
140C2×C3⋊Dic3—24no—
141C2×C3⋊Dic3—21no—
142C2×C3⋊Dic3—21no—
143C2×C3⋊Dic3C3×D411yescharacteristic
144C4×C3⋊D3—21no—
145C4×C3⋊D3—21no—
146#634—21no—
147#634—21no—
148#634—21no—
149#634C3×D412yes—
150C3×#67—616no—
151C2×C3×Dic3—34no—
152C2×C3×Dic3—616no—
153C2×C3×Dic3—64no—
154C2×C3×Dic3—64no—
155C3×C4×D3—64no—
156C3×C4×D3—64no—
157C3×Dic6—38no—
158C3×Dic6—64no—
159C3×Dic6—64no—
160C3×Dic6—64no—
161C32×C8—21no—
162C32×C8—24no—
163C32×C8—44no—
164C3⋊(C3⋊C8)—21no—
165C3×C3⋊C8—64no—
166#452—91noSylow
167C2×C33—42no—
168C2×C33#12011yescharacteristic
169C2×C33#13511yescharacteristic
170C2×C33#13711yescharacteristic
171C3×C3⋊D3—41no—
172C3×#41—182no—
173C3×#41—182no—
174C2×C3×Q8—91no—
175C2×C3×Q8—91no—
176C2×C3×D4—24no—
177C2×C3×D4—181no—
178C2×C3×D4C2×C3×D314yes—
179C2×C3×D4C2×C3⋊D311yescharacteristic
180C22×C3×C4—91no—
181C22×C3×C4—181no—
182C2×#67—64no—
183C22×Dic3—64no—
184#237—68no—
185#237—68no—
186C2×C4×D3—34no—
187C2×Dic6—34no—
188C2×Dic6—34no—
189C3×Q16—182no—
190C3×SD16—182no—
191C2×C3×C8—24no—
192C2×C3×C8—91no—
193C2×C3×C8C2×C3×D314yes—
194C2×C3×C8C2×C3⋊D311yescharacteristic
195C3×C4⋊C4—24no—
196C3×C4⋊C4—181no—
197C3×C4⋊C4C2×C3×D314yes—
198C3×C4⋊C4C2×C3⋊D311yescharacteristic
199C3×C22⋊C4—181no—
200#214—68no—
201(C2×D3)⋊C4—64no—
202C4⋊Dic3—64no—
203C2×C3⋊C8—34no—
204Dic12—68no—
205C22×C32—22no—
206C22×C32—24no—
207C22×C32—28no—
208C22×C32—48no—
209C22×C32—41no—
210C22×C32—41no—
211C22×C32—44no—
212C22×C32—44no—
213C22×C32—84no—
214C22×C32—84no—
215C22×C32D3×D414yes—
216C22×C32C2×C3×D411yescharacteristic
217C2×C3⋊D3—21no—
218C2×C3⋊D3—21no—
219C2×C3⋊D3—21no—
220C2×C3×D3—64no—
221C2×C3×D3—64no—
222C2×C3×D3—64no—
223C32×C4—24no—
224C32×C4—24no—
225C32×C4—41no—
226C32×C4—44no—
227C32×C4—84no—
228C32×C4C2×D1214yes—
229C32×C4D3×D414yes—
230C32×C4C2×C3×D411yescharacteristic
231C32×C4C2×C3×D411yescharacteristic
232C3⋊Dic3—22no—
233C3⋊Dic3—21no—
234C3⋊Dic3—41no—
235C3×Dic3—68no—
236C3×Dic3—64no—
237C3×Dic3—124no—
238C2×#41—91no—
239C2×Q16—91no—
240C2×SD16—91no—
241C22⋊Q8—91no—
242#103—91no—
243#102C3×C3⋊D311yescharacteristic, complemented
244C22⋊C8—91no—
245C33#45211yescharacteristic, complemented, Sylow
246C23×C3—24no—
247C23×C3—21no—
248C23×C3—44no—
249C23×C3—91no—
250C22×D3—34no—
251C3×Q8—92no—
252C3×Q8—181no—
253C3×Q8—181no—
254C3×Q8—181no—
255C3×D4—24no—
256C3×D4—21no—
257C3×D4—28no—
258C3×D4—44no—
259C3×D4—184no—
260C3×D4C3×D1218yes—
261C3×D4C4⋊(C3⋊D3)12yes—
262C2×C3×C4—24no—
263C2×C3×C4—21no—
264C2×C3×C4—24no—
265C2×C3×C4—44no—
266C2×C3×C4—91no—
267C2×C3×C4—184no—
268C2×C3×C4—181no—
269C2×C3×C4—181no—
270C2×C3×C4—181no—
271C2×C3×C4—181no—
272C2×C3×C4C22×C3×D314yes—
273C2×C3×C4C22×C3⋊D311yescharacteristic
274#67—616no—
275C2×Dic3—34no—
276C2×Dic3—616no—
277C2×Dic3—64no—
278C2×Dic3—64no—
279C4×D3—64no—
280C4×D3—64no—
281Dic6—38no—
282Dic6—64no—
283Dic6—64no—
284Dic6—64no—
285C3×C8—24no—
286C3×C8—21no—
287C3×C8—44no—
288C3×C8—181no—
289C3⋊C8—64no—
290C2×C32—24no—
291C2×C32—24no—
292C2×C32—24no—
293C2×C32—42no—
294C2×C32—48no—
295C2×C32—88no—
296C2×C32#88214yes—
297C2×C32#91214yes—
298C2×C32#91514yes—
299C2×C32C3×#12011yescharacteristic
300C2×C32C3×#13511yescharacteristic
301C2×C32C3×#13711yescharacteristic
302C3⋊D3—41no—
303C3×D3—124no—
304#41—182no—
305#41—182no—
306C2×Q8—91no—
307C2×Q8—91no—
308C2×D4—181no—
309C2×D4C2×C3×C3⋊D311yescharacteristic
310C22×C4—91no—
311C22×C4—181no—
312Q16—182no—
313SD16—182no—
314C2×C8—91no—
315C2×C8C2×C3×C3⋊D311yescharacteristic
316C4⋊C4—181no—
317C4⋊C4C2×C3×C3⋊D311yescharacteristic
318C22⋊C4—181no—
319C22×C3—28no—
320C22×C3—22no—
321C22×C3—24no—
322C22×C3—48no—
323C22×C3—44no—
324C22×C3—44no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—84no—
328C22×C3—84no—
329C22×C3—181no—
330C22×C3—181no—
331C22×C3—181no—
332C22×C3C3×D3×D414yes—
333C22×C3D4×C3⋊D311yescharacteristic
334C2×D3—64no—
335C2×D3—64no—
336C2×D3—64no—
337C3×C4—24no—
338C3×C4—24no—
339C3×C4—44no—
340C3×C4—41no—
341C3×C4—84no—
342C3×C4—181no—
343C3×C4—182no—
344C3×C4—361no—
345C3×C4C2×C3×D1214yes—
346C3×C4C3×D3×D414yes—
347C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
348C3×C4D4×C3⋊D311yescharacteristic
349Dic3—68no—
350Dic3—64no—
351Dic3—124no—
352C32—24no—
353C32#478314yescomplemented
354C32C3×#45211yescharacteristic, complemented
355C23—21no—
356C23—91no—
357Q8—92no—
358Q8—181no—
359Q8—181no—
360Q8—181no—
361D4—21no—
362D4—184no—
363D4C3×C4⋊(C3⋊D3)12yes—
364C2×C4—21no—
365C2×C4—91no—
366C2×C4—184no—
367C2×C4—181no—
368C2×C4—181no—
369C2×C4—181no—
370C2×C4—181no—
371C2×C4C22×C3×C3⋊D311yescharacteristic
372C8—21no—
373C8—181no—
374C2×C3—24no—
375C2×C3—24no—
376C2×C3—24no—
377C2×C3—48no—
378C2×C3—42no—
379C2×C3—88no—
380C2×C3—361no—
381C2×C3C3×#88214yes—
382C2×C3C3×#91214yes—
383C2×C3C3×#91514yes—
384C2×C3#6409111yescharacteristic
385C2×C3#6412111yescharacteristic
386C2×C3#6412411yescharacteristic
387D3—124no—
388C22—22no—
389C22—41no—
390C22—41no—
391C22—181no—
392C22—181no—
393C22—181no—
394C22C3×D4×C3⋊D311yescharacteristic
395C4—41no—
396C4—181no—
397C4—182no—
398C4—361no—
399C4C2×C3×C4⋊(C3⋊D3)11yescharacteristic
400C4C3×D4×C3⋊D311yescharacteristic
401C3—24no—
402C3C3×#478314yescomplemented
403C3#1059248211yescharacteristic, direct product factor
404C2—42no—
405C2—361no—
406C2C3×#6409111yescharacteristic
407C2C3×#6412111yescharacteristic
408C2C3×#6412411yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#102C2
socleC22×C33C2×D4
prime radicalC2×C3×#3830C2
hypercenterC3×#102C3⋊D3
hypocenterC32C3×#452
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#102

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3754C2semidirect
3. C2×C3×#3746C2semidirect
4. C3×#64106C2semidirect
5. C33×#102C2semidirect
6. C3×#63647C2semidirect
7. C3×#63637C2semidirect
8. #10592482C3direct (N × H)
27. C2×#3754C2×C3semidirect
28. C2×#3746C2×C3semidirect
29. #64106C2×C3semidirect
34. C32×#102D3semidirect
35. C32×#102C2×C3semidirect
36. #63647C2×C3semidirect
37. #63637C2×C3semidirect
117. C3×#102C3×D3semidirect
118. C3×#102C3⋊D3semidirect
243. #102C3×C3⋊D3semidirect
245. C33#452semidirect
353. C32#4783semidirect
354. C32C3×#452semidirect
402. C3C3×#4783semidirect
403. C3#10592482direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C2×C3×#37542
C2×C3×#37462
C3×#641062
C33×#1022
C3×#636472
C3×#636372
#105924823
C3×#47833