Finite Group Databasebeta

← C3×#19080 #10637687

Subgroups — C3×#19080 #10637687

Order 768.

936 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 936 conjugacy classes of subgroups — too many to draw. Only its 230 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3548C211yesmaximal, characteristic
2C3×#3141C212yesmaximal, complemented
3C3×#3139C211yesmaximal, characteristic, complemented
4C3×#1662C218yesmaximal, complemented
5C3×#1526C211yesmaximal, characteristic, complemented
6C3×#1525C212yesmaximal, complemented
7#19080C311yesmaximal, characteristic, direct product factor, Sylow
8C3×#562C2214yes—
9C3×#554—22no—
10C3×#550—22no—
11C3×#547—22no—
12C3×#540—22no—
13C3×#537—24no—
14C3×#533C2211yescharacteristic
15C3×#532C2212yes—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C3×#491C2214yes—
18C3×#489C2218yes—
19C3×#487—22no—
20C3×#487—22no—
21C3×#486—22no—
22C3×#419—22no—
23C3×#417—24no—
24C3×C4×M16C2211yescharacteristic
25C3×C4×M16C2212yes—
26C3×#330C2214yes—
27C3×#329C2218yes—
28#3548C2×C311yescharacteristic
29#3141C2×C312yescomplemented
30#3139C2×C311yescharacteristic, complemented
31#1662C2×C318yescomplemented
32#1526C2×C311yescharacteristic, complemented
33#1525C2×C312yescomplemented
34C2×C3×#41—24no—
35C2×C3×#41D414yes—
36C22×C3×Q8—21no—
37C2×C3×M16—22no—
38C2×C3×M16—21no—
39C3×C4⋊Q8—22no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8D412yes—
42C3×C4⋊Q8D412yes—
43C3×C4⋊Q8C2312yes—
44C3×#127—22no—
45C3×#127D412yes—
46C3×#125—22no—
47C3×#125C2312yes—
48C3×#124C2314yes—
49C3×#123—28no—
50C3×C22⋊Q8—24no—
51C3×C22⋊Q8—28no—
52C3×C22⋊Q8—22no—
53C3×#121—28no—
54C3×#121—24no—
55C3×#121—28no—
56C3×#121—22no—
57C3×#121—22no—
58C3×C4×Q8—22no—
59C3×C4×D4—24no—
60C3×C4×D4—22no—
61C3×C4×D4—22no—
62C3×#117D412yes—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C42C2311yescharacteristic
66C3×#103—28no—
67C3×#103—28no—
68C3×#102—28no—
69C3×#97C2312yes—
70C3×#97C2314yes—
71C3×C4×C8—22no—
72C3×C4×C8—21no—
73#562C22×C314yes—
74#554—22no—
75#550—22no—
76#547—22no—
77#540—22no—
78#537—24no—
79#533C22×C311yescharacteristic
80#532C22×C312yes—
81C2×C4⋊Q8C22×C311yescharacteristic
82#491C22×C314yes—
83#489C22×C318yes—
84#487—22no—
85#487—22no—
86#486—22no—
87#419—22no—
88#417—24no—
89C4×M16C22×C311yescharacteristic
90C4×M16C22×C312yes—
91#330C22×C314yes—
92#329C22×C318yes—
93C3×#41—48no—
94C3×#41—88no—
95C2×C3×Q8—22no—
96C2×C3×Q8—22no—
97C2×C3×Q8—44no—
98C2×C3×Q8—41no—
99C2×C3×Q8C2×D414yes—
100C2×C3×D4—28no—
101C2×C3×D4—24no—
102C2×C3×D4—22no—
103C2×C3×D4—22no—
104C2×C3×D4C2×D414yes—
105C22×C3×C4—28no—
106C22×C3×C4—24no—
107C22×C3×C4—41no—
108C22×C3×C4C2×D412yes—
109C22×C3×C4C2×D411yescharacteristic
110C3×M16—42no—
111C3×M16—44no—
112C2×C3×C8—24no—
113C2×C3×C8—22no—
114C3×C4⋊C4—28no—
115C3×C4⋊C4—22no—
116C3×C4⋊C4—22no—
117C3×C4⋊C4—22no—
118C3×C4⋊C4—24no—
119C3×C4⋊C4—24no—
120C3×C4⋊C4—42no—
121C3×C4⋊C4C2×D414yes—
122C3×C22⋊C4—28no—
123C3×C22⋊C4—22no—
124C3×C22⋊C4—24no—
125C3×C22⋊C4—24no—
126C3×C42—22no—
127C3×C42C2×D411yescharacteristic
128C3×C42C2×D412yes—
129C3×C42C2411yescharacteristic
130C2×#41—24no—
131C2×#41C3×D414yes—
132C22×Q8—21no—
133C2×M16—22no—
134C2×M16—21no—
135C4⋊Q8—22no—
136C4⋊Q8—21no—
137C4⋊Q8C3×D412yes—
138C4⋊Q8C3×D412yes—
139C4⋊Q8C23×C312yes—
140#127—22no—
141#127C3×D412yes—
142#125—22no—
143#125C23×C312yes—
144#124C23×C314yes—
145#123—28no—
146C22⋊Q8—24no—
147C22⋊Q8—28no—
148C22⋊Q8—22no—
149#121—28no—
150#121—24no—
151#121—28no—
152#121—22no—
153#121—22no—
154C4×Q8—22no—
155C4×D4—24no—
156C4×D4—22no—
157C4×D4—22no—
158#117C3×D412yes—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C42C23×C311yescharacteristic
162#103—28no—
163#103—28no—
164#102—28no—
165#97C23×C312yes—
166#97C23×C314yes—
167C4×C8—22no—
168C4×C8—21no—
169C23×C3—24no—
170C23×C3#12011yescharacteristic
171C3×Q8—44no—
172C3×Q8—44no—
173C3×Q8—44no—
174C3×Q8—82no—
175C3×D4—44no—
176C3×D4—44no—
177C3×D4—44no—
178C3×D4—84no—
179C3×D4—88no—
180C2×C3×C4—22no—
181C2×C3×C4—24no—
182C2×C3×C4—24no—
183C2×C3×C4—24no—
184C2×C3×C4—48no—
185C2×C3×C4—42no—
186C2×C3×C4—88no—
187C2×C3×C4—82no—
188C2×C3×C4#12011yescharacteristic
189C2×C3×C4#12012yes—
190C2×C3×C4C22×D411yescharacteristic
191C2×C3×C4C22×D412yes—
192C3×C8—44no—
193C3×C8—42no—
194#41—48no—
195#41—88no—
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8—44no—
199C2×Q8—41no—
200C2×Q8C2×C3×D414yes—
201C2×D4—28no—
202C2×D4—24no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4C2×C3×D414yes—
206C22×C4—28no—
207C22×C4—24no—
208C22×C4—41no—
209C22×C4C2×C3×D412yes—
210C22×C4C2×C3×D411yescharacteristic
211M16—42no—
212M16—44no—
213C2×C8—24no—
214C2×C8—22no—
215C4⋊C4—28no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—24no—
220C4⋊C4—24no—
221C4⋊C4—42no—
222C4⋊C4C2×C3×D414yes—
223C22⋊C4—28no—
224C22⋊C4—22no—
225C22⋊C4—24no—
226C22⋊C4—24no—
227C42—22no—
228C42C2×C3×D411yescharacteristic
229C42C2×C3×D412yes—
230C42C24×C311yescharacteristic
231C22×C3—22no—
232C22×C3—21no—
233C22×C3—44no—
234C22×C3—44no—
235C22×C3—44no—
236C22×C3C2×#12011yescharacteristic
237C3×C4—22no—
238C3×C4—21no—
239C3×C4—44no—
240C3×C4—44no—
241C3×C4—82no—
242C3×C4#45412yes—
243C3×C4#45514yes—
244C23—24no—
245C23C3×#12011yescharacteristic
246Q8—44no—
247Q8—44no—
248Q8—44no—
249Q8—82no—
250D4—44no—
251D4—44no—
252D4—44no—
253D4—84no—
254D4—88no—
255C2×C4—22no—
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—48no—
260C2×C4—42no—
261C2×C4—88no—
262C2×C4—82no—
263C2×C4C3×#12011yescharacteristic
264C2×C4C3×#12012yes—
265C2×C4C22×C3×D411yescharacteristic
266C2×C4C22×C3×D412yes—
267C8—44no—
268C8—42no—
269C2×C3—41no—
270C2×C3—84no—
271C2×C3C2×#45512yes—
272C2×C3C2×#45411yescharacteristic
273C22—22no—
274C22—21no—
275C22—44no—
276C22—44no—
277C22—44no—
278C22C2×C3×#12011yescharacteristic
279C4—22no—
280C4—21no—
281C4—44no—
282C4—44no—
283C4—82no—
284C4C3×#45412yes—
285C4C3×#45514yes—
286C3#1908011yescharacteristic, direct product factor, Sylow
287C2—41no—
288C2—84no—
289C2C2×C3×#45512yes—
290C2C2×C3×#45411yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
prime radicalC3×#3548C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3141C2semidirect
3. C3×#3139C2semidirect
4. C3×#1662C2semidirect
5. C3×#1526C2semidirect
6. C3×#1525C2semidirect
7. #19080C3direct (N × H)
29. #3141C2×C3semidirect
30. #3139C2×C3semidirect
31. #1662C2×C3semidirect
32. #1526C2×C3semidirect
33. #1525C2×C3semidirect
286. C3#19080direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35482
C3×#31412
C3×#31392
C3×#16622
C3×#15262
C3×#15252
#190803