Finite Group Databasebeta

← C3×#10590896 #13157863

Subgroups — C3×#10590896 #13157863

Order 1728.

674 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3746C211yesmaximal, characteristic
2C2×C32×Dic12C211yesmaximal, characteristic
3C3×#63866C212yesmaximal
4C2×C3×#3719C212yesmaximal
5C32×C8×Dic3C211yesmaximal, characteristic
6C3×#5183—31nomaximal
7C3×#4699—31nomaximal
8#10590896C311yesmaximal, characteristic, direct product factor
9C2×C3×#634C2212yes—
10C2×C32×Dic6C2212yes—
11C2×C33×C8C2211yescharacteristic
12C3×#3746—22no—
13C32×C4×Dic3C2211yescharacteristic
14C2×C32×C3⋊C8C2211yescharacteristic
15C32×Dic12—22no—
16C3×Dic3⋊Dic3—24no—
17C3×#3719—24no—
18C2×C32×Q16—31no—
19C2×#3746C2×C311yescharacteristic
20C3×#944—32no—
21C2×C3×#216—32no—
22C2×C3×Dic12—21no—
23C2×C3×Dic12—32no—
24C2×C3×Dic12—31no—
25C2×C3×Dic12—31no—
26C2×C3×Dic12—61no—
27C2×C3×Dic12D311yescharacteristic
28C2×C3×Dic12C2×C311yescharacteristic
29C3×#869—32no—
30#63866C2×C312yes—
31C2×#3719C2×C312yes—
32C32×C4×C8—31no—
33C3×C8×Dic3—21no—
34C3×C8×Dic3D311yescharacteristic
35C3×C8×Dic3C2×C311yescharacteristic
36C2×C33×C4C2311yescharacteristic
37C2×C3×C3⋊Dic3—22no—
38C3×#634—22no—
39C3×#634—22no—
40C2×C32×Dic3—22no—
41C2×C32×Dic3D412yes—
42C32×Dic6—22no—
43C32×Dic6—22no—
44C33×C8D412yes—
45C32×C3⋊C8D412yes—
46C3×#494—91no—
47#5183—31no—
48#4699—31no—
49C2×C32×Q8—32no—
50C2×#634C22×C312yes—
51C2×C3×Dic6—22no—
52C2×C3×Dic6—32no—
53C2×C3×Dic6—32no—
54C2×C3×Dic6—62no—
55C2×C3×Dic6C2×D312yes—
56C2×C3×Dic6C22×C312yes—
57C32×Q16—62no—
58C2×C32×C8—21no—
59C2×C32×C8—21no—
60C2×C32×C8—21no—
61C2×C32×C8—31no—
62C2×C32×C8—41no—
63C2×C32×C8C2×D311yescharacteristic
64C2×C32×C8C2×D311yescharacteristic
65C2×C32×C8C22×C311yescharacteristic
66C32×C42—31no—
67#3746—22no—
68C3×#216—64no—
69C3×C4⋊Dic3—64no—
70C3×Dic3⋊C4—64no—
71C3×C4×Dic3—21no—
72C3×C4×Dic3C2×D311yescharacteristic
73C3×C4×Dic3C22×C311yescharacteristic
74C2×C3×C3⋊C8—21no—
75C2×C3×C3⋊C8C2×D311yescharacteristic
76C2×C3×C3⋊C8C22×C311yescharacteristic
77C3×Dic12—24no—
78C3×Dic12—22no—
79C3×Dic12—22no—
80C3×Dic12—64no—
81C3×Dic12—64no—
82C3×Dic12—62no—
83C3×Dic12—62no—
84Dic3⋊Dic3—24no—
85#3719—24no—
86C22×C33C2×D411yescharacteristic
87C33×C4C2×D411yescharacteristic
88C33×C4C2×D411yescharacteristic
89C3×C3⋊Dic3—42no—
90C32×Dic3—42no—
91C32×Dic3Q1614yescomplemented
92C2×C3×Q16—31no—
93C2×C3×Q16—31no—
94C2×C3×Q16—61no—
95C2×C3×Q16—92no—
96C2×C3×Q16—91no—
97C3×C4⋊Q8—92no—
98#944—32no—
99C2×#216—32no—
100C2×Dic12—32no—
101C2×Dic12—31no—
102C2×Dic12—31no—
103C2×Dic12—61no—
104C2×Dic12C3×D311yescharacteristic
105#869—32no—
106C3×C4×C8—31no—
107C3×C4×C8—31no—
108C3×C4×C8—61no—
109C8×Dic3C3×D311yescharacteristic
110C32×Q8—62no—
111C32×Q8—62no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
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—
122#634—22no—
123#634—22no—
124C2×C3×Dic3—22no—
125C2×C3×Dic3—22no—
126C2×C3×Dic3—22no—
127C2×C3×Dic3—42no—
128C2×C3×Dic3—62no—
129C2×C3×Dic3—62no—
130C2×C3×Dic3—122no—
131C2×C3×Dic3D1212yes—
132C2×C3×Dic3C3×D412yes—
133C3×Dic6—24no—
134C3×Dic6—22no—
135C3×Dic6—22no—
136C3×Dic6—22no—
137C3×Dic6—22no—
138C3×Dic6—42no—
139C3×Dic6—64no—
140C3×Dic6—62no—
141C3×Dic6—62no—
142C3×Dic6—62no—
143C3×Dic6—62no—
144C3×Dic6—122no—
145C32×C8—22no—
146C32×C8—22no—
147C32×C8—22no—
148C32×C8—32no—
149C32×C8—42no—
150C32×C8D1212yes—
151C32×C8#6712yes—
152C32×C8C3×D412yes—
153C3×C3⋊C8—22no—
154C3×C3⋊C8D1212yes—
155C3×C3⋊C8C3×D412yes—
156#494—91noSylow
157C2×C33#12711yescharacteristic
158C2×C33C2×Q1612yes—
159C2×C3×Q8—32no—
160C2×C3×Q8—32no—
161C2×C3×Q8—62no—
162C2×C3×Q8—92no—
163C2×Dic6—32no—
164C2×Dic6—32no—
165C2×Dic6—62no—
166C2×Dic6C2×C3×D312yes—
167C3×Q16—64no—
168C3×Q16—62no—
169C3×Q16—62no—
170C3×Q16—184no—
171C3×Q16—182no—
172C2×C3×C8—21no—
173C2×C3×C8—21no—
174C2×C3×C8—21no—
175C2×C3×C8—31no—
176C2×C3×C8—31no—
177C2×C3×C8—41no—
178C2×C3×C8—61no—
179C2×C3×C8D3211yescharacteristic
180C2×C3×C8C2×C3×D311yescharacteristic
181C2×C3×C8C2×C3×D311yescharacteristic
182C3×C4⋊C4—184no—
183C3×C42—31no—
184C3×C42—31no—
185C3×C42—61no—
186#216—64no—
187C4⋊Dic3—64no—
188Dic3⋊C4—64no—
189C4×Dic3C2×C3×D311yescharacteristic
190C2×C3⋊C8C2×C3×D311yescharacteristic
191Dic12—22no—
192Dic12—64no—
193Dic12—64no—
194Dic12—62no—
195Dic12—62no—
196C22×C32—21no—
197C22×C32—21no—
198C22×C32—21no—
199C22×C32—41no—
200C22×C32C2×D1211yescharacteristic
201C22×C32D3×D411yescharacteristic
202C22×C32C2×C3×D411yescharacteristic
203C32×C4—21no—
204C32×C4—21no—
205C32×C4—21no—
206C32×C4—21no—
207C32×C4—21no—
208C32×C4—21no—
209C32×C4—34no—
210C32×C4—41no—
211C32×C4—41no—
212C32×C4—122no—
213C32×C4C2×D1211yescharacteristic
214C32×C4C2×D1211yescharacteristic
215C32×C4D3×D411yescharacteristic
216C32×C4C2×#6711yescharacteristic
217C32×C4C2×C3×D411yescharacteristic
218C32×C4C2×C3×D411yescharacteristic
219C3⋊Dic3—42no—
220C3×Dic3—24no—
221C3×Dic3—44no—
222C3×Dic3—42no—
223C3×Dic3—42no—
224C3×Dic3—124no—
225C3×Dic3—122no—
226C3×Dic3—122no—
227C3×Dic3Dic1214yescomplemented
228C3×Dic3C3×Q1614yescomplemented
229C2×Q16—31no—
230C2×Q16—92no—
231C2×Q16—91no—
232C4⋊Q8—92no—
233C4×C8—31no—
234C33#49411yescharacteristic, complemented, Sylow
235C3×Q8—64no—
236C3×Q8—62no—
237C3×Q8—62no—
238C3×Q8—62no—
239C3×Q8—62no—
240C3×Q8—122no—
241C3×Q8—182no—
242C3×Q8—182no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—32no—
247C2×C3×C4—32no—
248C2×C3×C4—41no—
249C2×C3×C4—62no—
250C2×C3×C4—62no—
251C2×C3×C4—62no—
252C2×C3×C4—122no—
253C2×C3×C4—182no—
254C2×C3×C4C2×D3211yescharacteristic
255C2×C3×C4C22×C3×D311yescharacteristic
256C2×C3×C4C22×C3×D311yescharacteristic
257C2×Dic3—22no—
258C2×Dic3—62no—
259C2×Dic3—62no—
260C2×Dic3—122no—
261C2×Dic3C3×D1212yes—
262Dic6—22no—
263Dic6—22no—
264Dic6—64no—
265Dic6—62no—
266Dic6—62no—
267Dic6—62no—
268Dic6—62no—
269Dic6—122no—
270C3×C8—22no—
271C3×C8—22no—
272C3×C8—22no—
273C3×C8—32no—
274C3×C8—32no—
275C3×C8—42no—
276C3×C8—62no—
277C3×C8#62612yes—
278C3×C8C3×D1212yes—
279C3×C8C3×#6712yes—
280C3⋊C8C3×D1212yes—
281C2×C32—22no—
282C2×C32—22no—
283C2×C32—22no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32—21no—
287C2×C32—41no—
288C2×C32—42no—
289C2×C32#87411yescharacteristic
290C2×C32C2×Dic1212yes—
291C2×C32D3×Q1612yes—
292C2×C32#94011yescharacteristic
293C2×C32C3×#12711yescharacteristic
294C2×C32C2×C3×Q1612yes—
295C2×Q8—32no—
296C2×Q8—92no—
297Q16—62no—
298Q16—184no—
299Q16—182no—
300C2×C8—31no—
301C2×C8C3×D3211yescharacteristic
302C4⋊C4—184no—
303C42—31no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—41no—
308C22×C3D3×D1211yescharacteristic
309C22×C3C2×C3×D1211yescharacteristic
310C22×C3C3×D3×D411yescharacteristic
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—34no—
318C3×C4—34no—
319C3×C4—41no—
320C3×C4—41no—
321C3×C4—64no—
322C3×C4—124no—
323C3×C4—122no—
324C3×C4—122no—
325C3×C4—362no—
326C3×C4D3×D1211yescharacteristic
327C3×C4C2×#62611yescharacteristic
328C3×C4C2×C3×D1211yescharacteristic
329C3×C4C2×C3×D1211yescharacteristic
330C3×C4C3×D3×D411yescharacteristic
331C3×C4C2×C3×#6711yescharacteristic
332Dic3—42no—
333Dic3—124no—
334Dic3—122no—
335Dic3—122no—
336Dic3C3×Dic1214yescomplemented
337C32—21no—
338C32—21no—
339C32—21no—
340C32—41no—
341C32#469911yescharacteristic, complemented
342C32#518311yescharacteristic, complemented
343C32C3×#49411yescharacteristic, complemented
344Q8—62no—
345Q8—62no—
346Q8—182no—
347Q8—182no—
348C2×C4—32no—
349C2×C4—62no—
350C2×C4—182no—
351C2×C4C2×C3×D3211yescharacteristic
352C8—32no—
353C8C3×#62612yes—
354C2×C3—22no—
355C2×C3—22no—
356C2×C3—22no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—42no—
361C2×C3—41no—
362C2×C3D3×Dic1212yes—
363C2×C3#6391111yescharacteristic
364C2×C3C3×#87411yescharacteristic
365C2×C3C2×C3×Dic1212yes—
366C2×C3C3×D3×Q1612yes—
367C2×C3C3×#94011yescharacteristic
368C22C3×D3×D1211yescharacteristic
369C4—34no—
370C4—122no—
371C4—362no—
372C4C3×D3×D1211yescharacteristic
373C4C2×C3×#62611yescharacteristic
374C3—21no—
375C3—21no—
376C3—21no—
377C3—41no—
378C3#1059089611yescharacteristic, direct product factor
379C3C3×#469911yescharacteristic, complemented
380C3C3×#518311yescharacteristic, complemented
381C2C3×D3×Dic1212yes—
382C2C3×#6391111yescharacteristic
subgroupG/H
centerC22×C3D3×D12
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×C3×C8D32
hypocenterC32C3×#494
LCSG ▷ C2×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. #10590896C3direct (N × H)
91. C32×Dic3Q16semidirect
227. C3×Dic3Dic12semidirect
228. C3×Dic3C3×Q16semidirect
234. C33#494semidirect
336. Dic3C3×Dic12semidirect
341. C32#4699semidirect
342. C32#5183semidirect
343. C32C3×#494semidirect
378. C3#10590896direct (N × H)
379. C3C3×#4699semidirect
380. C3C3×#5183semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37462
C2×C32×Dic122
C3×#638662
C2×C3×#37192
C32×C8×Dic32
C3×#51833
C3×#46993
#105908963