Finite Group Databasebeta

← C2×C3×#1466 #10635966

Subgroups — C2×C3×#1466 #10635966

Order 768.

1650 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D4C211yesmaximal, characteristic
2C2×C3×#432C212yesmaximal, complemented
3C3×#1466C218yesmaximal, direct product factor
4C2×C3×#323C214yesmaximal, complemented
5C2×#1466C311yesmaximal, characteristic, direct product factor, Sylow
6C23×C3×D4C411yescharacteristic
7C24×C3×C4C412yes—
8C2×C3×C4×D4—22no—
9C2×C3×C4×D4—22no—
10C2×C3×C4×D4—24no—
11C2×C3×C4×D4C414yes—
12C2×C3×C4×D4C2214yes—
13C22×C3×C4⋊C4C411yescharacteristic
14C22×C3×C22⋊C4C2212yes—
15C22×C3×C42C2211yescharacteristic
16C3×#432—24no—
17C3×#432C2218yescomplemented
18C2×C3×C4⋊C8—22no—
19C2×C3×C22⋊C8C2214yes—
20C2×C3×#97—22no—
21C3×#323C22116yescomplemented
22C22×C4×D4C2×C311yescharacteristic
23C2×#432C2×C312yescomplemented
24#1466C2×C318yesdirect product factor
25C2×#323C2×C314yescomplemented
26C25×C3C2×C412yes—
27C22×C3×D4—28no—
28C22×C3×D4—22no—
29C22×C3×D4—22no—
30C22×C3×D4C2×C414yes—
31C23×C3×C4—24no—
32C23×C3×C4—22no—
33C23×C3×C4—22no—
34C23×C3×C4—48no—
35C23×C3×C4C2×C411yescharacteristic
36C23×C3×C4C2×C411yescharacteristic
37C23×C3×C4C2×C412yes—
38C23×C3×C4C2×C418yes—
39C23×C3×C4C2311yescharacteristic
40C22×C3×C8—24no—
41C3×C4×D4—28no—
42C3×C4×D4—28no—
43C3×C4×D4—28no—
44C3×C4×D4—28no—
45C2×C3×C4⋊C4—22no—
46C2×C3×C4⋊C4—22no—
47C2×C3×C4⋊C4C2×C414yes—
48C2×C3×C22⋊C4—24no—
49C2×C3×C22⋊C4—24no—
50C2×C3×C22⋊C4C2×C414yes—
51C2×C3×C22⋊C4C2314yes—
52C2×C3×C42C2×C412yes—
53C2×C3×C42D412yes—
54C2×C3×C42D412yes—
55C2×C3×C42D414yes—
56C2×C3×C42C2312yes—
57C3×C4⋊C8—28no—
58C3×C22⋊C8—24no—
59C3×C22⋊C8C2318yes—
60C3×#97—28no—
61C23×D4C3×C411yescharacteristic
62C24×C4C3×C412yes—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—24no—
66C2×C4×D4C3×C414yes—
67C2×C4×D4C22×C314yes—
68C22×C4⋊C4C3×C411yescharacteristic
69C22×C22⋊C4C22×C312yes—
70C22×C42C22×C311yescharacteristic
71#432—24no—
72#432C22×C318yescomplemented
73C2×C4⋊C8—22no—
74C2×C22⋊C8C22×C314yes—
75C2×#97—22no—
76#323C22×C3116yescomplemented
77C24×C3—22no—
78C24×C3—22no—
79C24×C3—24no—
80C24×C3—44no—
81C24×C3—44no—
82C24×C3M1618yes—
83C24×C3C22×C411yescharacteristic
84C24×C3C22×C414yes—
85C2×C3×D4—216no—
86C2×C3×D4—216no—
87C2×C3×D4—216no—
88C2×C3×D4—28no—
89C22×C3×C4—24no—
90C22×C3×C4—24no—
91C22×C3×C4—22no—
92C22×C3×C4—22no—
93C22×C3×C4—28no—
94C22×C3×C4—28no—
95C22×C3×C4—42no—
96C22×C3×C4—48no—
97C22×C3×C4—416no—
98C22×C3×C4—416no—
99C22×C3×C4—48no—
100C22×C3×C4—44no—
101C22×C3×C4C22⋊C412yes—
102C22×C3×C4C22⋊C412yes—
103C22×C3×C4C22⋊C412yes—
104C22×C3×C4C22⋊C411yescharacteristic
105C22×C3×C4C22⋊C411yescharacteristic
106C22×C3×C4C22×C412yes—
107C22×C3×C4C22×C412yes—
108C22×C3×C4C22×C414yes—
109C22×C3×C4C22×C411yescharacteristic
110C22×C3×C4C2×D412yes—
111C22×C3×C4C2×D412yes—
112C22×C3×C4C2411yescharacteristic
113C2×C3×C8—28no—
114C2×C3×C8—48no—
115C3×C4⋊C4—28no—
116C3×C22⋊C4—28no—
117C3×C22⋊C4—28no—
118C3×C42C22⋊C414yes—
119C3×C42C22⋊C414yes—
120C3×C42C2×D414yes—
121C3×C42C2×D414yes—
122C25C2×C3×C412yes—
123C22×D4—28no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4C2×C3×C414yes—
127C23×C4—24no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—48no—
131C23×C4C2×C3×C411yescharacteristic
132C23×C4C2×C3×C411yescharacteristic
133C23×C4C2×C3×C412yes—
134C23×C4C2×C3×C418yes—
135C23×C4C23×C311yescharacteristic
136C22×C8—24no—
137C4×D4—28no—
138C4×D4—28no—
139C4×D4—28no—
140C4×D4—28no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4C2×C3×C414yes—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4C2×C3×C414yes—
147C2×C22⋊C4C23×C314yes—
148C2×C42C2×C3×C412yes—
149C2×C42C3×D412yes—
150C2×C42C3×D412yes—
151C2×C42C3×D414yes—
152C2×C42C23×C312yes—
153C4⋊C8—28no—
154C22⋊C8—24no—
155C22⋊C8C23×C318yes—
156#97—28no—
157C23×C3—22no—
158C23×C3—22no—
159C23×C3—24no—
160C23×C3—24no—
161C23×C3—28no—
162C23×C3—28no—
163C23×C3—44no—
164C23×C3—416no—
165C23×C3—44no—
166C23×C3—44no—
167C23×C3—44no—
168C23×C3—416no—
169C23×C3—44no—
170C23×C3—44no—
171C23×C3C2×C22⋊C411yescharacteristic
172C23×C3C2×C22⋊C411yescharacteristic
173C23×C3C2×M1614yes—
174C23×C3C2×M1618yes—
175C23×C3C23×C411yescharacteristic
176C3×D4—232no—
177C2×C3×C4—24no—
178C2×C3×C4—24no—
179C2×C3×C4—24no—
180C2×C3×C4—24no—
181C2×C3×C4—22no—
182C2×C3×C4—24no—
183C2×C3×C4—24no—
184C2×C3×C4—24no—
185C2×C3×C4—41no—
186C2×C3×C4—42no—
187C2×C3×C4—41no—
188C2×C3×C4—416no—
189C2×C3×C4—44no—
190C2×C3×C4—44no—
191C2×C3×C4—416no—
192C2×C3×C4—44no—
193C2×C3×C4#10014yes—
194C2×C3×C4C2×C22⋊C412yes—
195C2×C3×C4C2×C22⋊C414yes—
196C2×C3×C4C2×C22⋊C412yes—
197C2×C3×C4C2×C22⋊C412yes—
198C2×C3×C4C22×D412yes—
199C3×C8—48no—
200C24—22no—
201C24—22no—
202C24—24no—
203C24—44no—
204C24—44no—
205C24C3×M1618yes—
206C24C22×C3×C411yescharacteristic
207C24C22×C3×C414yes—
208C2×D4—216no—
209C2×D4—216no—
210C2×D4—216no—
211C2×D4—28no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—28no—
217C22×C4—28no—
218C22×C4—42no—
219C22×C4—48no—
220C22×C4—416no—
221C22×C4—416no—
222C22×C4—48no—
223C22×C4—44no—
224C22×C4C3×C22⋊C412yes—
225C22×C4C3×C22⋊C412yes—
226C22×C4C3×C22⋊C411yescharacteristic
227C22×C4C3×C22⋊C412yes—
228C22×C4C3×C22⋊C411yescharacteristic
229C22×C4C22×C3×C412yes—
230C22×C4C22×C3×C412yes—
231C22×C4C22×C3×C414yes—
232C22×C4C22×C3×C411yescharacteristic
233C22×C4C2×C3×D412yes—
234C22×C4C2×C3×D412yes—
235C22×C4C24×C311yescharacteristic
236C2×C8—28no—
237C2×C8—48no—
238C4⋊C4—28no—
239C22⋊C4—28no—
240C22⋊C4—28no—
241C42C3×C22⋊C414yes—
242C42C3×C22⋊C414yes—
243C42C2×C3×D414yes—
244C42C2×C3×D414yes—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—28no—
250C22×C3—28no—
251C22×C3—48no—
252C22×C3—48no—
253C22×C3—416no—
254C22×C3—44no—
255C22×C3—44no—
256C22×C3—416no—
257C22×C3#40712yes—
258C22×C3#40712yes—
259C22×C3#41012yes—
260C22×C3C2×#10012yes—
261C22×C3C22×C22⋊C411yescharacteristic
262C22×C3C22×M1612yes—
263C3×C4—24no—
264C3×C4—22no—
265C3×C4—22no—
266C3×C4—22no—
267C3×C4—44no—
268C3×C4C2×#10014yes—
269C23—22no—
270C23—22no—
271C23—24no—
272C23—28no—
273C23—28no—
274C23—24no—
275C23—44no—
276C23—416no—
277C23—44no—
278C23—44no—
279C23—44no—
280C23—416no—
281C23—44no—
282C23—44no—
283C23C2×C3×C22⋊C411yescharacteristic
284C23C2×C3×C22⋊C411yescharacteristic
285C23C2×C3×M1614yes—
286C23C2×C3×M1618yes—
287C23C23×C3×C411yescharacteristic
288D4—232no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—416no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—416no—
304C2×C4—44no—
305C2×C4C3×#10014yes—
306C2×C4C2×C3×C22⋊C412yes—
307C2×C4C2×C3×C22⋊C414yes—
308C2×C4C2×C3×C22⋊C412yes—
309C2×C4C2×C3×C22⋊C412yes—
310C2×C4C22×C3×D412yes—
311C8—48no—
312C2×C3—22no—
313C2×C3—22no—
314C2×C3—48no—
315C2×C3#146614yesdirect product factor
316C2×C3C2×#40711yescharacteristic
317C2×C3C2×#41011yescharacteristic
318C2×C3C22×#10011yescharacteristic
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—28no—
324C22—28no—
325C22—48no—
326C22—48no—
327C22—416no—
328C22—44no—
329C22—44no—
330C22—416no—
331C22C3×#40712yes—
332C22C3×#40712yes—
333C22C3×#41012yes—
334C22C2×C3×#10012yes—
335C22C22×C3×C22⋊C411yescharacteristic
336C22C22×C3×M1612yes—
337C4—24no—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—44no—
342C4C2×C3×#10014yes—
343C3C2×#146611yescharacteristic, direct product factor, Sylow
344C2—22no—
345C2—22no—
346C2—48no—
347C2C3×#146614yesdirect product factor
348C2C2×C3×#40711yescharacteristic
349C2C2×C3×#41011yescharacteristic
350C2C22×C3×#10011yescharacteristic
subgroupG/H
center, socleC23×C3C2×C22⋊C4
commutatorC23C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC23×C3×D4C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#432C2semidirect
3. C3×#1466C2direct (N × H)
4. C2×C3×#323C2semidirect
5. C2×#1466C3direct (N × H)
17. C3×#432C22semidirect
21. C3×#323C22semidirect
23. C2×#432C2×C3semidirect
24. #1466C2×C3direct (N × H)
25. C2×#323C2×C3semidirect
72. #432C22×C3semidirect
76. #323C22×C3semidirect
315. C2×C3#1466direct (N × H)
343. C3C2×#1466direct (N × H)
347. C2C3×#1466direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×D42
C2×C3×#4322
C3×#14662
C2×C3×#3232
C2×#14663