Finite Group Databasebeta

← C3×#19084 #10637691

Subgroups — C3×#19084 #10637691

Order 768.

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

normal subgroups only

This group has 824 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×#3556C211yesmaximal, characteristic
2C3×#3165C212yesmaximal
3C3×#3139C211yesmaximal, characteristic
4C3×#1663C218yesmaximal, complemented
5C3×#1526C212yesmaximal
6C3×#1526C211yesmaximal, characteristic
7#19084C311yesmaximal, characteristic, direct product factor, Sylow
8C3×#564C2214yes—
9C3×#557—22no—
10C3×#557—22no—
11C3×#557—22no—
12C3×#541—22no—
13C3×#541—24no—
14C3×#533C2211yescharacteristic
15C3×#533C2212yes—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C3×#501C2218yes—
18C3×#500—22no—
19C3×#498—22no—
20C3×#491C2214yes—
21C3×#487—22no—
22C3×#428—24no—
23C3×#419—22no—
24C3×C4×M16C2211yescharacteristic
25C3×C4×M16C2212yes—
26C3×#330C2218yes—
27C3×#330C2214yes—
28#3556C2×C311yescharacteristic
29#3165C2×C312yes—
30#3139C2×C311yescharacteristic
31#1663C2×C318yescomplemented
32#1526C2×C311yescharacteristic
33#1526C2×C312yes—
34C22×C3×Q8—21no—
35C2×C3×M16—22no—
36C2×C3×M16—21no—
37C3×C4⋊Q8—22no—
38C3×C4⋊Q8—24no—
39C3×C4⋊Q8—22no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8D412yes—
42C3×C4⋊Q8D412yes—
43C3×C4⋊Q8D412yes—
44C3×C4⋊Q8C2312yes—
45C3×#126—28no—
46C3×#125—22no—
47C3×#125—24no—
48C3×#125—28no—
49C3×#125—28no—
50C3×#125—28no—
51C3×#125C2312yes—
52C3×#125C2314yes—
53C3×C22⋊Q8—22no—
54C3×C22⋊Q8—22no—
55C3×C22⋊Q8—22no—
56C3×C4×Q8—24no—
57C3×C4×Q8—24no—
58C3×C4×Q8—24no—
59C3×C4×Q8—22no—
60C3×C4×Q8—22no—
61C3×C4×Q8—22no—
62C3×#117D412yes—
63C3×#117D414yes—
64C2×C3×C4⋊C4—22no—
65C2×C3×C42C2311yescharacteristic
66C3×#107—28no—
67C3×#106—28no—
68C3×#103—28no—
69C3×#97C2312yes—
70C3×#97C2314yes—
71C3×C4×C8—21no—
72C3×C4×C8—22no—
73#564C22×C314yes—
74#557—22no—
75#557—22no—
76#557—22no—
77#541—22no—
78#541—24no—
79#533C22×C311yescharacteristic
80#533C22×C312yes—
81C2×C4⋊Q8C22×C311yescharacteristic
82#501C22×C318yes—
83#500—22no—
84#498—22no—
85#491C22×C314yes—
86#487—22no—
87#428—24no—
88#419—22no—
89C4×M16C22×C311yescharacteristic
90C4×M16C22×C312yes—
91#330C22×C318yes—
92#330C22×C314yes—
93C2×C3×Q8—22no—
94C2×C3×Q8—22no—
95C2×C3×Q8—22no—
96C2×C3×Q8—22no—
97C2×C3×Q8—44no—
98C2×C3×Q8—41no—
99C22×C3×C4—41no—
100C22×C3×C4C2×D412yes—
101C22×C3×C4C2×D411yescharacteristic
102C3×M16—28no—
103C3×M16—42no—
104C2×C3×C8—24no—
105C2×C3×C8—22no—
106C3×C4⋊C4—28no—
107C3×C4⋊C4—28no—
108C3×C4⋊C4—28no—
109C3×C4⋊C4—22no—
110C3×C4⋊C4—22no—
111C3×C4⋊C4—22no—
112C3×C4⋊C4—24no—
113C3×C4⋊C4—24no—
114C3×C4⋊C4—24no—
115C3×C4⋊C4—24no—
116C3×C4⋊C4—42no—
117C3×C4⋊C4C2×D414yes—
118C3×C4⋊C4C2×D414yes—
119C3×C4⋊C4C2×D414yes—
120C3×C22⋊C4—22no—
121C3×C22⋊C4—24no—
122C3×C42—28no—
123C3×C42—22no—
124C3×C42—24no—
125C3×C42C2×D412yes—
126C3×C42C2×D411yescharacteristic
127C3×C42C2411yescharacteristic
128C22×Q8—21no—
129C2×M16—22no—
130C2×M16—21no—
131C4⋊Q8—22no—
132C4⋊Q8—24no—
133C4⋊Q8—22no—
134C4⋊Q8—21no—
135C4⋊Q8C3×D412yes—
136C4⋊Q8C3×D412yes—
137C4⋊Q8C3×D412yes—
138C4⋊Q8C23×C312yes—
139#126—28no—
140#125—22no—
141#125—24no—
142#125—28no—
143#125—28no—
144#125—28no—
145#125C23×C312yes—
146#125C23×C314yes—
147C22⋊Q8—22no—
148C22⋊Q8—22no—
149C22⋊Q8—22no—
150C4×Q8—24no—
151C4×Q8—24no—
152C4×Q8—24no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156#117C3×D412yes—
157#117C3×D414yes—
158C2×C4⋊C4—22no—
159C2×C42C23×C311yescharacteristic
160#107—28no—
161#106—28no—
162#103—28no—
163#97C23×C312yes—
164#97C23×C314yes—
165C4×C8—21no—
166C4×C8—22no—
167C23×C3#12011yescharacteristic
168C3×Q8—44no—
169C3×Q8—44no—
170C3×Q8—44no—
171C3×Q8—44no—
172C3×Q8—82no—
173C2×C3×C4—22no—
174C2×C3×C4—24no—
175C2×C3×C4—24no—
176C2×C3×C4—24no—
177C2×C3×C4—24no—
178C2×C3×C4—42no—
179C2×C3×C4—82no—
180C2×C3×C4#12011yescharacteristic
181C2×C3×C4#12012yes—
182C2×C3×C4C22×D412yes—
183C2×C3×C4C22×D411yescharacteristic
184C3×C8—28no—
185C3×C8—42no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8—22no—
189C2×Q8—22no—
190C2×Q8—44no—
191C2×Q8—41no—
192C22×C4—41no—
193C22×C4C2×C3×D412yes—
194C22×C4C2×C3×D411yescharacteristic
195M16—28no—
196M16—42no—
197C2×C8—24no—
198C2×C8—22no—
199C4⋊C4—28no—
200C4⋊C4—28no—
201C4⋊C4—28no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—24no—
206C4⋊C4—24no—
207C4⋊C4—24no—
208C4⋊C4—24no—
209C4⋊C4—42no—
210C4⋊C4C2×C3×D414yes—
211C4⋊C4C2×C3×D414yes—
212C4⋊C4C2×C3×D414yes—
213C22⋊C4—22no—
214C22⋊C4—24no—
215C42—28no—
216C42—22no—
217C42—24no—
218C42C2×C3×D412yes—
219C42C2×C3×D411yescharacteristic
220C42C24×C311yescharacteristic
221C22×C3—21no—
222C22×C3—22no—
223C22×C3C2×#12011yescharacteristic
224C3×C4—21no—
225C3×C4—22no—
226C3×C4—44no—
227C3×C4—44no—
228C3×C4—44no—
229C3×C4—82no—
230C3×C4#45412yes—
231C3×C4#45614yes—
232C23C3×#12011yescharacteristic
233Q8—44no—
234Q8—44no—
235Q8—44no—
236Q8—44no—
237Q8—82no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—42no—
244C2×C4—82no—
245C2×C4C3×#12011yescharacteristic
246C2×C4C3×#12012yes—
247C2×C4C22×C3×D412yes—
248C2×C4C22×C3×D411yescharacteristic
249C8—28no—
250C8—42no—
251C2×C3—41no—
252C2×C3C2×#45411yescharacteristic
253C2×C3C2×#45612yes—
254C22—21no—
255C22—22no—
256C22C2×C3×#12011yescharacteristic
257C4—21no—
258C4—22no—
259C4—44no—
260C4—44no—
261C4—44no—
262C4—82no—
263C4C3×#45412yes—
264C4C3×#45614yes—
265C3#1908411yescharacteristic, direct product factor, Sylow
266C2—41no—
267C2C2×C3×#45411yescharacteristic
268C2C2×C3×#45612yes—
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
prime radicalC23×C3#120
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
4. C3×#1663C2semidirect
7. #19084C3direct (N × H)
31. #1663C2×C3semidirect
265. C3#19084direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35562
C3×#31652
C3×#31392
C3×#16632
C3×#15262
#190843