Finite Group Databasebeta

← C2×C3×#63578 #13158170

Subgroups — C2×C3×#63578 #13158170

Order 1728.

690 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
2C2×C32×#208C212yesmaximal
3C3×#63578C214yesmaximal, direct product factor
4C2×C3×#822—32nomaximal
5C2×#63578C311yesmaximal, characteristic, direct product factor
6C22×C33×C4C2211yescharacteristic
7C2×C3×C3⋊(C3⋊C8)C412yes—
8C2×C3×C3⋊(C3⋊C8)C412yes—
9C2×C3×C3⋊(C3⋊C8)C2212yes—
10C32×#208—22no—
11C32×#208C2214yes—
12C2×C32×C3⋊C8—22no—
13C2×C32×M16—32no—
14C22×C3⋊(C3⋊C8)C2×C311yescharacteristic
15C2×C3×#208—22no—
16C2×C3×#208D312yes—
17C2×C3×#208C2×C312yes—
18C22×C3×C3⋊C8—32no—
19C22×C3×C3⋊C8—61no—
20C3×#822—38no—
21#63578C2×C314yesdirect product factor
22C23×C33Q811yescharacteristic
23C2×C33×C4C2×C411yescharacteristic
24C2×C33×C4C2×C411yescharacteristic
25C2×C33×C4D411yescharacteristic
26C2×C33×C4D411yescharacteristic
27C2×C33×C4Q811yescharacteristic
28C2×C33×C4C2311yescharacteristic
29C3×C3⋊(C3⋊C8)C2×C414yes—
30C32×C3⋊C8—24no—
31C2×C3×#108—91no—
32C2×#822—32no—
33C22×C32×C4—21no—
34C22×C32×C4—22no—
35C22×C32×C4—41no—
36C22×C32×C4C2×D312yes—
37C22×C32×C4C22×C311yescharacteristic
38C32×M16—34no—
39C32×M16—62no—
40C2×C32×C8—62no—
41C2×C3⋊(C3⋊C8)C3×C412yes—
42C2×C3⋊(C3⋊C8)C3×C412yes—
43C2×C3⋊(C3⋊C8)C22×C312yes—
44C3×#208—24no—
45C3×#208—24no—
46C3×#208—22no—
47C3×#208—22no—
48C3×#208C2×D314yes—
49C3×#208C22×C314yes—
50C2×C3×C3⋊C8—24no—
51C2×C3×C3⋊C8—22no—
52C2×C3×C3⋊C8—22no—
53C2×C3×C3⋊C8—34no—
54C2×C3×C3⋊C8—34no—
55C2×C3×C3⋊C8—34no—
56C2×C3×C3⋊C8—62no—
57C2×C3×C3⋊C8—62no—
58C2×C3×C3⋊C8—62no—
59C22×C33—22no—
60C22×C33C4⋊C411yescharacteristic
61C22×C33C4⋊C411yescharacteristic
62C22×C33C2×Q811yescharacteristic
63C33×C4C4⋊C411yescharacteristic
64C33×C4C4⋊C411yescharacteristic
65C33×C4C22×C411yescharacteristic
66C33×C4C2×D411yescharacteristic
67C2×C3×M16—32no—
68C2×C3×M16—32no—
69C2×C3×M16—62no—
70C22×C3×C8—91no—
71C2×#208C3×D312yes—
72C22×C3⋊C8—32no—
73C22×C3⋊C8—61no—
74C3×#108—94no—
75#822—38no—
76C23×C32—21no—
77C23×C32—22no—
78C23×C32—41no—
79C23×C32Dic612yes—
80C23×C32C3×Q811yescharacteristic
81C2×C32×C4—21no—
82C2×C32×C4—21no—
83C2×C32×C4—21no—
84C2×C32×C4—21no—
85C2×C32×C4—21no—
86C2×C32×C4—22no—
87C2×C32×C4—22no—
88C2×C32×C4—22no—
89C2×C32×C4—22no—
90C2×C32×C4—22no—
91C2×C32×C4—21no—
92C2×C32×C4—22no—
93C2×C32×C4—41no—
94C2×C32×C4—41no—
95C2×C32×C4—41no—
96C2×C32×C4—41no—
97C2×C32×C4—41no—
98C2×C32×C4—41no—
99C2×C32×C4Dic612yes—
100C2×C32×C4C4×D312yes—
101C2×C32×C4C4×D312yes—
102C2×C32×C4#6712yes—
103C2×C32×C4#6712yes—
104C2×C32×C4C2×C3×C411yescharacteristic
105C2×C32×C4C2×C3×C411yescharacteristic
106C2×C32×C4C3×D411yescharacteristic
107C2×C32×C4C3×D411yescharacteristic
108C2×C32×C4C3×Q811yescharacteristic
109C2×C32×C4C22×D312yes—
110C2×C32×C4C23×C311yescharacteristic
111C32×C8—64no—
112C3⋊(C3⋊C8)C2×C3×C414yes—
113C3×C3⋊C8—28no—
114C3×C3⋊C8—24no—
115C3×C3⋊C8—24no—
116C3×C3⋊C8—38no—
117C3×C3⋊C8—64no—
118C2×#108—91noSylow
119C2×C33—21no—
120C2×C33—21no—
121C2×C33#10812yescomplemented
122C2×C33C2×C4⋊C411yescharacteristic
123C22×C3×C4—21no—
124C22×C3×C4—22no—
125C22×C3×C4—41no—
126C22×C3×C4D3211yescharacteristic
127C22×C3×C4C2×C3×D312yes—
128C3×M16—34no—
129C3×M16—34no—
130C3×M16—64no—
131C3×M16—64no—
132C3×M16—62no—
133C3×M16—62no—
134C2×C3×C8—64no—
135C2×C3×C8—62no—
136C2×C3×C8—62no—
137C2×C3×C8—92no—
138C2×C3×C8—92no—
139C2×C3×C8—92no—
140#208—22no—
141#208C2×C3×D314yes—
142C2×C3⋊C8—22no—
143C2×C3⋊C8—34no—
144C2×C3⋊C8—34no—
145C2×C3⋊C8—34no—
146C2×C3⋊C8—62no—
147C2×C3⋊C8—62no—
148C2×C3⋊C8—62no—
149C22×C32—22no—
150C22×C32—24no—
151C22×C32—24no—
152C22×C32—21no—
153C22×C32—21no—
154C22×C32—22no—
155C22×C32—22no—
156C22×C32—21no—
157C22×C32—22no—
158C22×C32—41no—
159C22×C32—41no—
160C22×C32—41no—
161C22×C32—44no—
162C22×C32—44no—
163C22×C32Dic3⋊C412yes—
164C22×C32Dic3⋊C412yes—
165C22×C32C3×C4⋊C411yescharacteristic
166C22×C32C3×C4⋊C411yescharacteristic
167C22×C32C2×Dic612yes—
168C22×C32C2×C3×Q811yescharacteristic
169C32×C4—21no—
170C32×C4—21no—
171C32×C4—22no—
172C32×C4—22no—
173C32×C4—21no—
174C32×C4—21no—
175C32×C4—22no—
176C32×C4—22no—
177C32×C4—41no—
178C32×C4—41no—
179C32×C4—41no—
180C32×C4—41no—
181C32×C4Dic3⋊C412yes—
182C32×C4Dic3⋊C412yes—
183C32×C4C3×C4⋊C411yescharacteristic
184C32×C4C3×C4⋊C411yescharacteristic
185C32×C4C2×C4×D312yes—
186C32×C4C2×#6712yes—
187C32×C4C22×C3×C411yescharacteristic
188C32×C4C2×C3×D411yescharacteristic
189C2×M16—32no—
190C22×C8—91no—
191#108—94no—
192C33C2×#10811yescharacteristic, complemented, Sylow
193C23×C3—21no—
194C23×C3—22no—
195C23×C3—41no—
196C23×C3#62711yescharacteristic
197C23×C3C3×Dic612yes—
198C2×C3×C4—22no—
199C2×C3×C4—22no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—22no—
203C2×C3×C4—22no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—22no—
207C2×C3×C4—21no—
208C2×C3×C4—22no—
209C2×C3×C4—21no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4#62411yescharacteristic
217C2×C3×C4#62411yescharacteristic
218C2×C3×C4#62511yescharacteristic
219C2×C3×C4#62511yescharacteristic
220C2×C3×C4#62711yescharacteristic
221C2×C3×C4C3×Dic612yes—
222C2×C3×C4C3×C4×D312yes—
223C2×C3×C4C3×C4×D312yes—
224C2×C3×C4C3×#6712yes—
225C2×C3×C4C3×#6712yes—
226C2×C3×C4C2×D3211yescharacteristic
227C2×C3×C4C22×C3×D312yes—
228C3×C8—68no—
229C3×C8—64no—
230C3×C8—64no—
231C3×C8—94no—
232C3⋊C8—24no—
233C3⋊C8—38no—
234C3⋊C8—64no—
235C2×C32—21no—
236C2×C32—21no—
237C2×C32—22no—
238C2×C32—22no—
239C2×C32—22no—
240C2×C32—22no—
241C2×C32—22no—
242C2×C32—22no—
243C2×C32—24no—
244C2×C32—21no—
245C2×C32—42no—
246C2×C32—41no—
247C2×C32—42no—
248C2×C32—42no—
249C2×C32—42no—
250C2×C32—42no—
251C2×C32#82214yescomplemented
252C2×C32C3×#10812yescomplemented
253C2×C32C2×Dic3⋊C412yes—
254C2×C32C2×C3×C4⋊C411yescharacteristic
255C22×C4C3×D3211yescharacteristic
256M16—34no—
257M16—62no—
258C2×C8—62no—
259C2×C8—92no—
260C2×C8—92no—
261C2×C8—92no—
262C22×C3—24no—
263C22×C3—22no—
264C22×C3—24no—
265C22×C3—21no—
266C22×C3—22no—
267C22×C3—22no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—22no—
271C22×C3—44no—
272C22×C3—44no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3#372411yescharacteristic
277C22×C3#372411yescharacteristic
278C22×C3C3×Dic3⋊C412yes—
279C22×C3C3×Dic3⋊C412yes—
280C22×C3C2×#62711yescharacteristic
281C22×C3C2×C3×Dic612yes—
282C3×C4—21no—
283C3×C4—22no—
284C3×C4—22no—
285C3×C4—21no—
286C3×C4—22no—
287C3×C4—22no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—41no—
291C3×C4—41no—
292C3×C4—41no—
293C3×C4—41no—
294C3×C4#372411yescharacteristic
295C3×C4#372411yescharacteristic
296C3×C4C3×Dic3⋊C412yes—
297C3×C4C3×Dic3⋊C412yes—
298C3×C4C2×#62411yescharacteristic
299C3×C4C2×#62511yescharacteristic
300C3×C4C2×C3×C4×D312yes—
301C3×C4C2×C3×#6712yes—
302C32—21no—
303C32—22no—
304C32—41no—
305C32C2×#82212yescomplemented
306C32C2×C3×#10811yescharacteristic, complemented
307C23C3×#62711yescharacteristic
308C2×C4C3×#62411yescharacteristic
309C2×C4C3×#62411yescharacteristic
310C2×C4C3×#62511yescharacteristic
311C2×C4C3×#62511yescharacteristic
312C2×C4C3×#62711yescharacteristic
313C2×C4C2×C3×D3211yescharacteristic
314C8—64no—
315C8—94no—
316C2×C3—22no—
317C2×C3—22no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—22no—
321C2×C3—22no—
322C2×C3—22no—
323C2×C3—24no—
324C2×C3—22no—
325C2×C3—21no—
326C2×C3—42no—
327C2×C3—42no—
328C2×C3—42no—
329C2×C3—42no—
330C2×C3—42no—
331C2×C3—41no—
332C2×C3#6357812yesdirect product factor
333C2×C3C3×#82214yescomplemented
334C2×C3C2×#372411yescharacteristic
335C2×C3C2×C3×Dic3⋊C412yes—
336C22—22no—
337C22C3×#372411yescharacteristic
338C22C3×#372411yescharacteristic
339C22C2×C3×#62711yescharacteristic
340C4C3×#372411yescharacteristic
341C4C3×#372411yescharacteristic
342C4C2×C3×#62411yescharacteristic
343C4C2×C3×#62511yescharacteristic
344C3—21no—
345C3—22no—
346C3—41no—
347C3C2×#6357811yescharacteristic, direct product factor
348C3C2×C3×#82212yescomplemented
349C2—21no—
350C2—21no—
351C2C3×#6357812yesdirect product factor
352C2C2×C3×#372411yescharacteristic
subgroupG/H
centerC2×C3×C4#625
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C4⋊C4
prime radicalC23×C33Q8
cosocleC2×C32×C4C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C2×C3×#108
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
3. C3×#63578C2direct (N × H)
5. C2×#63578C3direct (N × H)
21. #63578C2×C3direct (N × H)
121. C2×C33#108semidirect
192. C33C2×#108semidirect
251. C2×C32#822semidirect
252. C2×C32C3×#108semidirect
305. C32C2×#822semidirect
306. C32C2×C3×#108semidirect
332. C2×C3#63578direct (N × H)
333. C2×C3C3×#822semidirect
347. C3C2×#63578direct (N × H)
348. C3C2×C3×#822semidirect
351. C2C3×#63578direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C3⋊(C3⋊C8)2
C2×C32×#2082
C3×#635782
C2×C3×#8223
C2×#635783