Finite Group Databasebeta

← C2×C3×#1467 #10635967

Subgroups — C2×C3×#1467 #10635967

Order 768.

1218 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×Q8C211yesmaximal, characteristic
2C2×C3×#432C212yesmaximal
3C3×#1467C218yesmaximal, direct product factor
4C2×C3×#324C214yesmaximal
5C2×#1467C311yesmaximal, characteristic, direct product factor, Sylow
6C23×C3×Q8C411yescharacteristic
7C2×C3×C4×Q8—22no—
8C2×C3×C4×Q8—22no—
9C2×C3×C4×Q8—24no—
10C2×C3×C4×Q8C414yes—
11C2×C3×C4×Q8C2214yes—
12C22×C3×C4⋊C4C411yescharacteristic
13C22×C3×C4⋊C4C2212yes—
14C22×C3×C42C412yes—
15C22×C3×C42C2211yescharacteristic
16C3×#432—24no—
17C3×#432C2218yes—
18C2×C3×C4⋊C8—22no—
19C2×C3×C22⋊C8C2214yes—
20C2×C3×#97—22no—
21C3×#324C22116yes—
22C22×C4×Q8C2×C311yescharacteristic
23C2×#432C2×C312yes—
24#1467C2×C318yesdirect product factor
25C2×#324C2×C314yes—
26C22×C3×Q8—22no—
27C22×C3×Q8—22no—
28C22×C3×Q8—28no—
29C22×C3×Q8C2×C414yes—
30C23×C3×C4C2×C411yescharacteristic
31C23×C3×C4C2×C411yescharacteristic
32C23×C3×C4C2×C412yes—
33C23×C3×C4C2×C412yes—
34C23×C3×C4C2311yescharacteristic
35C22×C3×C8—24no—
36C3×C4×Q8—28no—
37C3×C4×Q8—28no—
38C3×C4×Q8—28no—
39C3×C4×Q8—28no—
40C2×C3×C4⋊C4—22no—
41C2×C3×C4⋊C4—22no—
42C2×C3×C4⋊C4—24no—
43C2×C3×C4⋊C4—24no—
44C2×C3×C4⋊C4C2×C414yes—
45C2×C3×C4⋊C4C2×C414yes—
46C2×C3×C4⋊C4C2314yes—
47C2×C3×C42—24no—
48C2×C3×C42—22no—
49C2×C3×C42—22no—
50C2×C3×C42C2×C418yes—
51C2×C3×C42C2×C412yes—
52C2×C3×C42D412yes—
53C2×C3×C42D412yes—
54C2×C3×C42D414yes—
55C2×C3×C42C2312yes—
56C3×C4⋊C8—28no—
57C3×C22⋊C8—24no—
58C3×C22⋊C8C2318yes—
59C3×#97—28no—
60C23×Q8C3×C411yescharacteristic
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—24no—
64C2×C4×Q8C3×C414yes—
65C2×C4×Q8C22×C314yes—
66C22×C4⋊C4C3×C411yescharacteristic
67C22×C4⋊C4C22×C312yes—
68C22×C42C3×C412yes—
69C22×C42C22×C311yescharacteristic
70#432—24no—
71#432C22×C318yes—
72C2×C4⋊C8—22no—
73C2×C22⋊C8C22×C314yes—
74C2×#97—22no—
75#324C22×C3116yes—
76C24×C3C22×C411yescharacteristic
77C2×C3×Q8—216no—
78C2×C3×Q8—216no—
79C2×C3×Q8—28no—
80C2×C3×Q8—216no—
81C22×C3×C4—22no—
82C22×C3×C4—22no—
83C22×C3×C4—24no—
84C22×C3×C4—24no—
85C22×C3×C4—24no—
86C22×C3×C4—22no—
87C22×C3×C4—22no—
88C22×C3×C4—42no—
89C22×C3×C4—44no—
90C22×C3×C4C22⋊C412yes—
91C22×C3×C4C22⋊C412yes—
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4C22⋊C412yes—
94C22×C3×C4C22⋊C411yescharacteristic
95C22×C3×C4M1618yes—
96C22×C3×C4C22×C412yes—
97C22×C3×C4C22×C412yes—
98C22×C3×C4C22×C414yes—
99C22×C3×C4C22×C414yes—
100C22×C3×C4C22×C411yescharacteristic
101C22×C3×C4C2×D412yes—
102C22×C3×C4C2×D412yes—
103C22×C3×C4C2411yescharacteristic
104C2×C3×C8—28no—
105C2×C3×C8—48no—
106C3×C4⋊C4—28no—
107C3×C4⋊C4—28no—
108C3×C4⋊C4—28no—
109C3×C42—28no—
110C3×C42—28no—
111C3×C42C22⋊C414yes—
112C3×C42C22⋊C414yes—
113C3×C42C2×D414yes—
114C3×C42C2×D414yes—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—28no—
118C22×Q8C2×C3×C414yes—
119C23×C4C2×C3×C411yescharacteristic
120C23×C4C2×C3×C411yescharacteristic
121C23×C4C2×C3×C412yes—
122C23×C4C2×C3×C412yes—
123C23×C4C23×C311yescharacteristic
124C22×C8—24no—
125C4×Q8—28no—
126C4×Q8—28no—
127C4×Q8—28no—
128C4×Q8—28no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4C2×C3×C414yes—
134C2×C4⋊C4C2×C3×C414yes—
135C2×C4⋊C4C23×C314yes—
136C2×C42—22no—
137C2×C42—24no—
138C2×C42—22no—
139C2×C42C2×C3×C418yes—
140C2×C42C2×C3×C412yes—
141C2×C42C3×D412yes—
142C2×C42C3×D412yes—
143C2×C42C3×D414yes—
144C2×C42C23×C312yes—
145C4⋊C8—28no—
146C22⋊C8—24no—
147C22⋊C8C23×C318yes—
148#97—28no—
149C23×C3—22no—
150C23×C3—22no—
151C23×C3C2×C22⋊C411yescharacteristic
152C23×C3C2×C22⋊C411yescharacteristic
153C23×C3C2×M1614yes—
154C23×C3C23×C411yescharacteristic
155C3×Q8—232no—
156C2×C3×C4—24no—
157C2×C3×C4—24no—
158C2×C3×C4—24no—
159C2×C3×C4—24no—
160C2×C3×C4—24no—
161C2×C3×C4—28no—
162C2×C3×C4—28no—
163C2×C3×C4—24no—
164C2×C3×C4—22no—
165C2×C3×C4—24no—
166C2×C3×C4—24no—
167C2×C3×C4—24no—
168C2×C3×C4—41no—
169C2×C3×C4—42no—
170C2×C3×C4—41no—
171C2×C3×C4—44no—
172C2×C3×C4—44no—
173C2×C3×C4—44no—
174C2×C3×C4#10114yes—
175C2×C3×C4C2×C22⋊C412yes—
176C2×C3×C4C2×C22⋊C412yes—
177C2×C3×C4C2×C22⋊C414yes—
178C2×C3×C4C2×C22⋊C412yes—
179C2×C3×C4C2×M1618yes—
180C2×C3×C4C22×D412yes—
181C3×C8—48no—
182C24C22×C3×C411yescharacteristic
183C2×Q8—216no—
184C2×Q8—216no—
185C2×Q8—28no—
186C2×Q8—216no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—42no—
195C22×C4—44no—
196C22×C4C3×C22⋊C411yescharacteristic
197C22×C4C3×C22⋊C412yes—
198C22×C4C3×C22⋊C412yes—
199C22×C4C3×C22⋊C412yes—
200C22×C4C3×C22⋊C411yescharacteristic
201C22×C4C3×M1618yes—
202C22×C4C22×C3×C412yes—
203C22×C4C22×C3×C412yes—
204C22×C4C22×C3×C414yes—
205C22×C4C22×C3×C414yes—
206C22×C4C22×C3×C411yescharacteristic
207C22×C4C2×C3×D412yes—
208C22×C4C2×C3×D412yes—
209C22×C4C24×C311yescharacteristic
210C2×C8—28no—
211C2×C8—48no—
212C4⋊C4—28no—
213C4⋊C4—28no—
214C4⋊C4—28no—
215C42—28no—
216C42—28no—
217C42C3×C22⋊C414yes—
218C42C3×C22⋊C414yes—
219C42C2×C3×D414yes—
220C42C2×C3×D414yes—
221C22×C3—21no—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3—21no—
225C22×C3—28no—
226C22×C3#40712yes—
227C22×C3#40712yes—
228C22×C3#41012yes—
229C22×C3C2×#10112yes—
230C22×C3C22×C22⋊C411yescharacteristic
231C22×C3C22×M1612yes—
232C3×C4—24no—
233C3×C4—28no—
234C3×C4—22no—
235C3×C4—22no—
236C3×C4—22no—
237C3×C4—44no—
238C3×C4C2×#10114yes—
239C23—22no—
240C23—22no—
241C23C2×C3×C22⋊C411yescharacteristic
242C23C2×C3×C22⋊C411yescharacteristic
243C23C2×C3×M1614yes—
244C23C23×C3×C411yescharacteristic
245Q8—232no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—28no—
251C2×C4—28no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4C3×#10114yes—
265C2×C4C2×C3×C22⋊C412yes—
266C2×C4C2×C3×C22⋊C412yes—
267C2×C4C2×C3×C22⋊C414yes—
268C2×C4C2×C3×C22⋊C412yes—
269C2×C4C2×C3×M1618yes—
270C2×C4C22×C3×D412yes—
271C8—48no—
272C2×C3—22no—
273C2×C3—22no—
274C2×C3#146714yesdirect product factor
275C2×C3C2×#40711yescharacteristic
276C2×C3C2×#41011yescharacteristic
277C2×C3C22×#10111yescharacteristic
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—28no—
283C22C3×#40712yes—
284C22C3×#40712yes—
285C22C3×#41012yes—
286C22C2×C3×#10112yes—
287C22C22×C3×C22⋊C411yescharacteristic
288C22C22×C3×M1612yes—
289C4—24no—
290C4—28no—
291C4—22no—
292C4—22no—
293C4—22no—
294C4—44no—
295C4C2×C3×#10114yes—
296C3C2×#146711yescharacteristic, direct product factor, Sylow
297C2—22no—
298C2—22no—
299C2C3×#146714yesdirect product factor
300C2C2×C3×#40711yescharacteristic
301C2C2×C3×#41011yescharacteristic
302C2C22×C3×#10111yescharacteristic
subgroupG/H
center, socleC23×C3C2×C22⋊C4
commutatorC23C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C22×C4
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
3. C3×#1467C2direct (N × H)
5. C2×#1467C3direct (N × H)
24. #1467C2×C3direct (N × H)
274. C2×C3#1467direct (N × H)
296. C3C2×#1467direct (N × H)
299. C2C3×#1467direct (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×Q82
C2×C3×#4322
C3×#14672
C2×C3×#3242
C2×#14673