Finite Group Databasebeta

← C3×#10592505 #13162843

Subgroups — C3×#10592505 #13162843

Order 1728.

1116 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×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×C3×#3755C211yesmaximal, characteristic, complemented
3C2×C3×#3744C211yesmaximal, characteristic, complemented
4C3×#64104C211yesmaximal, characteristic
5C33×#103C211yesmaximal, characteristic, complemented
6C3×#63647C211yesmaximal, characteristic, complemented
7C3×#63637C211yesmaximal, characteristic, complemented
8#10592505C311yesmaximal, characteristic, direct product factor
9C3×#4806—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C3×Q8×C3⋊D3—22no—
12C3×Q8×C3⋊D3—22no—
13C2×C3×C4⋊(C3⋊D3)—21no—
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×C4×C3⋊D3—21no—
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C3×#634—21no—
18C2×C3×#634C2211yescharacteristic
19C2×C33×C8C2211yescharacteristic
20C33×C4⋊C4C2211yescharacteristic
21C3×#3755—22no—
22C3×#3752—21no—
23C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
24C3×#3744—22no—
25C2×Q8×C3⋊D3C2×C311yescharacteristic, complemented
26C2×C3×D3×Q8—34no—
27C2×#3755C2×C311yescharacteristic, complemented
28C2×#3744C2×C311yescharacteristic, complemented
29#64104C2×C311yescharacteristic
30C2×C3×#215—34no—
31C2×C3×#204—34no—
32C3×#895—34no—
33C32×#103—24no—
34C32×#103D314yescomplemented
35C32×#103C2×C311yescharacteristic, complemented
36#63647C2×C311yescharacteristic, complemented
37#63637C2×C311yescharacteristic, complemented
38C3×(C2×D3)⋊C8—34no—
39C3×#810—34no—
40C22×C3×C3⋊D3—21no—
41C22×C3×C3⋊D3D411yescharacteristic
42C33×Q8—21no—
43C33×Q8D412yescomplemented
44C2×C33×C4—21no—
45C2×C33×C4—21no—
46C2×C33×C4C2311yescharacteristic
47C2×C3×C3⋊Dic3—21no—
48C2×C3×C3⋊Dic3D411yescharacteristic
49C3×C4⋊(C3⋊D3)—21no—
50C3×C4⋊(C3⋊D3)—21no—
51C3×C4⋊(C3⋊D3)—42no—
52C3×C4×C3⋊D3—24no—
53C3×C4×C3⋊D3—21no—
54C3×C4×C3⋊D3—21no—
55C3×#634—24no—
56C3×#634—21no—
57C3×#634D412yescomplemented
58C33×C8—21no—
59C3×C3⋊(C3⋊C8)—21no—
60C3×#448—91no—
61#4806—34no—
62C2×C32×Q8—24no—
63C2×C32×Q8C2×D314yes—
64C2×C32×Q8C22×C311yescharacteristic
65Q8×C3⋊D3—22no—
66Q8×C3⋊D3—22no—
67C2×C4⋊(C3⋊D3)—21no—
68C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
69C2×C4×C3⋊D3—21no—
70C2×C4×C3⋊D3C22×C311yescharacteristic
71C2×#634—21no—
72C2×#634C22×C311yescharacteristic
73C3×D3×Q8—68no—
74C3×D3×Q8—68no—
75C2×C3×D12—34no—
76C2×C3×D12—64no—
77C2×C3×C4×D3—34no—
78C2×C3×C4×D3—64no—
79C2×C3×Dic6—34no—
80C2×C3×Dic6—64no—
81C2×C32×C8—24no—
82C2×C32×C8C2×D314yes—
83C2×C32×C8C22×C311yescharacteristic
84C32×C4⋊C4—24no—
85C32×C4⋊C4C2×D314yes—
86C32×C4⋊C4C22×C311yescharacteristic
87#3755—22no—
88#3752—21no—
89C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
90#3744—22no—
91C3×#215—68no—
92C3×(C2×D3)⋊C4—64no—
93C2×C3×C3⋊C8—34no—
94C3×#204—68no—
95C22×C33C2×D411yescharacteristic
96C2×C3×C3⋊D3—21no—
97C2×C3×C3⋊D3—21no—
98C2×C3×C3⋊D3—41no—
99C2×C3×C3⋊D3—41no—
100C2×C3×C3⋊D3—41no—
101C2×C3×C3⋊D3SD1612yescomplemented
102C33×C4—22no—
103C33×C4—41no—
104C33×C4C2×D411yescharacteristic
105C33×C4C2×D411yescharacteristic
106C3×C3⋊Dic3—21no—
107C3×C3⋊Dic3—22no—
108C22×C3×Q8—91no—
109C2×D3×Q8—34no—
110C2×C3×SD16—91no—
111C2×C3×SD16—91no—
112C3×#121—91no—
113C2×#215—34no—
114C2×#204—34no—
115#895—34no—
116C3×#103—24no—
117C3×#103—91no—
118C3×#103C3×D314yescomplemented
119C3×#103C3⋊D311yescharacteristic, complemented
120C3×C22⋊C8—91no—
121(C2×D3)⋊C8—34no—
122#810—34no—
123C22×C3⋊D3—21no—
124C22×C3⋊D3C3×D411yescharacteristic
125C22×C3×D3—34no—
126C22×C3×D3—64no—
127C32×Q8—21no—
128C32×Q8—28no—
129C32×Q8—24no—
130C32×Q8—44no—
131C32×Q8D1218yescomplemented
132C32×Q8C3×D412yescomplemented
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—24no—
136C2×C32×C4—24no—
137C2×C32×C4—24no—
138C2×C32×C4—44no—
139C2×C32×C4—44no—
140C2×C32×C4C22×D314yes—
141C2×C32×C4C23×C311yescharacteristic
142C2×C3⋊Dic3—21no—
143C2×C3⋊Dic3C3×D411yescharacteristic
144C4⋊(C3⋊D3)—21no—
145C4⋊(C3⋊D3)—21no—
146C4⋊(C3⋊D3)—42no—
147C4×C3⋊D3—24no—
148C4×C3⋊D3—21no—
149C4×C3⋊D3—21no—
150#634—24no—
151#634—21no—
152#634C3×D412yescomplemented
153C2×C3×Dic3—34no—
154C2×C3×Dic3—64no—
155C3×D12—64no—
156C3×D12—64no—
157C3×D12—128no—
158C3×C4×D3—616no—
159C3×C4×D3—64no—
160C3×C4×D3—64no—
161C3×Dic6—38no—
162C3×Dic6—616no—
163C3×Dic6—64no—
164C32×C8—21no—
165C32×C8—24no—
166C32×C8—44no—
167C3⋊(C3⋊C8)—21no—
168C3×C3⋊C8—64no—
169#448—91noSylow
170C2×C33#12011yescharacteristic
171C2×C33C2×SD1611yescharacteristic
172C2×C33#13711yescharacteristic
173C3×C3⋊D3—22no—
174C3×C3⋊D3—81no—
175C2×C3×Q8—24no—
176C2×C3×Q8—91no—
177C2×C3×Q8—181no—
178C2×C3×Q8—182no—
179C2×C3×Q8—182no—
180C2×C3×Q8C2×C3×D314yes—
181C2×C3×Q8C2×C3⋊D311yescharacteristic
182C2×C3×D4—91no—
183C2×C3×D4—181no—
184C22×C3×C4—91no—
185C22×C3×C4—181no—
186D3×Q8—68no—
187D3×Q8—68no—
188C2×D12—34no—
189C2×D12—64no—
190C2×C4×D3—34no—
191C2×C4×D3—64no—
192C2×Dic6—34no—
193C2×Dic6—64no—
194C3×SD16—182no—
195C3×SD16—182no—
196C2×C3×C8—24no—
197C2×C3×C8—91no—
198C2×C3×C8C2×C3×D314yes—
199C2×C3×C8C2×C3⋊D311yescharacteristic
200C3×C4⋊C4—24no—
201C3×C4⋊C4C2×C3×D314yes—
202C3×C4⋊C4C2×C3⋊D311yescharacteristic
203C3×C22⋊C4—181no—
204#215—68no—
205(C2×D3)⋊C4—64no—
206C2×C3⋊C8—34no—
207#204—68no—
208C22×C32—24no—
209C22×C32D3×D414yes—
210C22×C32C2×C3×D411yescharacteristic
211C2×C3⋊D3—21no—
212C2×C3⋊D3—21no—
213C2×C3⋊D3—41no—
214C2×C3⋊D3—41no—
215C2×C3⋊D3—41no—
216C2×C3⋊D3C3×SD1612yescomplemented
217C2×C3×D3—38no—
218C2×C3×D3—64no—
219C2×C3×D3—64no—
220C2×C3×D3—124no—
221C2×C3×D3—124no—
222C2×C3×D3—124no—
223C32×C4—22no—
224C32×C4—24no—
225C32×C4—24no—
226C32×C4—28no—
227C32×C4—48no—
228C32×C4—41no—
229C32×C4—44no—
230C32×C4—84no—
231C32×C4C2×D1214yes—
232C32×C4D3×D414yes—
233C32×C4C2×C3×D411yescharacteristic
234C32×C4C2×C3×D411yescharacteristic
235C3⋊Dic3—21no—
236C3⋊Dic3—22no—
237C3×Dic3—68no—
238C3×Dic3—64no—
239C22×Q8—91no—
240C2×SD16—91no—
241C2×SD16—91no—
242#121—91no—
243#103—91no—
244#103C3×C3⋊D311yescharacteristic, complemented
245C22⋊C8—91no—
246C33#44811yescharacteristic, complemented, Sylow
247C23×C3—91no—
248C23×C3—181no—
249C22×D3—34no—
250C22×D3—64no—
251C3×Q8—24no—
252C3×Q8—21no—
253C3×Q8—28no—
254C3×Q8—44no—
255C3×Q8—92no—
256C3×Q8—184no—
257C3×Q8—181no—
258C3×Q8C3×D1218yescomplemented
259C3×Q8C4⋊(C3⋊D3)12yescomplemented
260C3×D4—181no—
261C3×D4—181no—
262C3×D4—362no—
263C2×C3×C4—24no—
264C2×C3×C4—21no—
265C2×C3×C4—24no—
266C2×C3×C4—21no—
267C2×C3×C4—24no—
268C2×C3×C4—44no—
269C2×C3×C4—44no—
270C2×C3×C4—91no—
271C2×C3×C4—184no—
272C2×C3×C4—181no—
273C2×C3×C4—181no—
274C2×C3×C4—181no—
275C2×C3×C4C22×C3×D314yes—
276C2×C3×C4C22×C3⋊D311yescharacteristic
277C2×Dic3—34no—
278C2×Dic3—64no—
279D12—64no—
280D12—64no—
281D12—128no—
282C4×D3—616no—
283C4×D3—64no—
284C4×D3—64no—
285Dic6—38no—
286Dic6—616no—
287Dic6—64no—
288C3×C8—24no—
289C3×C8—21no—
290C3×C8—44no—
291C3×C8—181no—
292C3⋊C8—64no—
293C2×C32—24no—
294C2×C32—24no—
295C2×C32—24no—
296C2×C32#88214yes—
297C2×C32D3×SD1614yes—
298C2×C32#91814yes—
299C2×C32C3×#12011yescharacteristic
300C2×C32C2×C3×SD1611yescharacteristic
301C2×C32C3×#13711yescharacteristic
302C3⋊D3—22no—
303C3⋊D3—81no—
304C3×D3—68no—
305C3×D3—244no—
306C2×Q8—91no—
307C2×Q8—181no—
308C2×Q8—182no—
309C2×Q8—182no—
310C2×Q8C2×C3×C3⋊D311yescharacteristic
311C2×D4—91no—
312C2×D4—181no—
313C22×C4—91no—
314C22×C4—181no—
315SD16—182no—
316SD16—182no—
317C2×C8—91no—
318C2×C8C2×C3×C3⋊D311yescharacteristic
319C4⋊C4C2×C3×C3⋊D311yescharacteristic
320C22⋊C4—181no—
321C22×C3—24no—
322C22×C3—92no—
323C22×C3—181no—
324C22×C3—181no—
325C22×C3—361no—
326C22×C3—361no—
327C22×C3—361no—
328C22×C3C3×D3×D414yes—
329C22×C3D4×C3⋊D311yescharacteristic
330C2×D3—38no—
331C2×D3—64no—
332C2×D3—64no—
333C2×D3—124no—
334C2×D3—124no—
335C2×D3—124no—
336C3×C4—28no—
337C3×C4—22no—
338C3×C4—24no—
339C3×C4—24no—
340C3×C4—48no—
341C3×C4—44no—
342C3×C4—41no—
343C3×C4—84no—
344C3×C4—181no—
345C3×C4—182no—
346C3×C4C2×C3×D1214yes—
347C3×C4C3×D3×D414yes—
348C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
349C3×C4D4×C3⋊D311yescharacteristic
350Dic3—68no—
351Dic3—64no—
352C32—24no—
353C32#480614yescomplemented
354C32C3×#44811yescharacteristic, complemented
355C23—91no—
356C23—181no—
357Q8—21no—
358Q8—92no—
359Q8—184no—
360Q8—181no—
361Q8C3×C4⋊(C3⋊D3)12yescomplemented
362D4—181no—
363D4—181no—
364D4—362no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—91no—
368C2×C4—184no—
369C2×C4—181no—
370C2×C4—181no—
371C2×C4—181no—
372C2×C4C22×C3×C3⋊D311yescharacteristic
373C8—21no—
374C8—181no—
375C2×C3—24no—
376C2×C3—24no—
377C2×C3—24no—
378C2×C3—182no—
379C2×C3—721no—
380C2×C3C3×#88214yes—
381C2×C3C3×D3×SD1614yes—
382C2×C3C3×#91814yes—
383C2×C3#6409111yescharacteristic
384C2×C3SD16×C3⋊D311yescharacteristic
385C2×C3#6412711yescharacteristic
386D3—68no—
387D3—244no—
388C22—92no—
389C22—181no—
390C22—181no—
391C22—361no—
392C22—361no—
393C22—361no—
394C22C3×D4×C3⋊D311yescharacteristic
395C4—22no—
396C4—41no—
397C4—181no—
398C4—182no—
399C4C2×C3×C4⋊(C3⋊D3)11yescharacteristic
400C4C3×D4×C3⋊D311yescharacteristic
401C3—24no—
402C3C3×#480614yescomplemented
403C3#1059250511yescharacteristic, direct product factor
404C2—182no—
405C2—721no—
406C2C3×#6409111yescharacteristic
407C2C3×SD16×C3⋊D311yescharacteristic
408C2C3×#6412711yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#103C2
socleC22×C33C2×D4
prime radicalC3×#64104C2
hypercenterC3×#103C3⋊D3
hypocenterC32C3×#448
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

Complements

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

NH ≅ G/Ntype
1. C2×C3×Q8×C3⋊D3C2semidirect
2. C2×C3×#3755C2semidirect
3. C2×C3×#3744C2semidirect
5. C33×#103C2semidirect
6. C3×#63647C2semidirect
7. C3×#63637C2semidirect
8. #10592505C3direct (N × H)
25. C2×Q8×C3⋊D3C2×C3semidirect
27. C2×#3755C2×C3semidirect
28. C2×#3744C2×C3semidirect
34. C32×#103D3semidirect
35. C32×#103C2×C3semidirect
36. #63647C2×C3semidirect
37. #63637C2×C3semidirect
43. C33×Q8D4semidirect
57. C3×#634D4semidirect
101. C2×C3×C3⋊D3SD16semidirect
118. C3×#103C3×D3semidirect
119. C3×#103C3⋊D3semidirect
131. C32×Q8D12semidirect
132. C32×Q8C3×D4semidirect
152. #634C3×D4semidirect
216. C2×C3⋊D3C3×SD16semidirect
244. #103C3×C3⋊D3semidirect
246. C33#448semidirect
258. C3×Q8C3×D12semidirect
259. C3×Q8C4⋊(C3⋊D3)semidirect
353. C32#4806semidirect
354. C32C3×#448semidirect
361. Q8C3×C4⋊(C3⋊D3)semidirect
402. C3C3×#4806semidirect
403. C3#10592505direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
C2×C3×#37552
C2×C3×#37442
C3×#641042
C33×#1032
C3×#636472
C3×#636372
#105925053
C3×#48063