Finite Group Databasebeta

← C3×#19075 #10637682

Subgroups — C3×#19075 #10637682

Order 768.

1116 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 1116 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, complemented
2C3×#3151C212yesmaximal, complemented
3C3×#3138C211yesmaximal, characteristic, complemented
4C3×#1656C218yesmaximal, complemented
5C3×#1526C211yesmaximal, characteristic, complemented
6C3×#1525C212yesmaximal, complemented
7#19075C311yesmaximal, characteristic, direct product factor, Sylow
8C2×C3×AGL(1,ℤ/8)—24no—
9C3×#562C2214yescomplemented
10C3×#554—22no—
11C3×#550—22no—
12C3×#547—22no—
13C3×#540—22no—
14C3×#537—24no—
15C3×#533C2211yescharacteristic
16C3×#532C2212yes—
17C2×C3×#127C2211yescharacteristic
18C3×#496C2218yes—
19C3×#493—22no—
20C3×#492—22no—
21C3×#490C2214yes—
22C3×#486—22no—
23C3×#418—22no—
24C3×C4×M16C2211yescharacteristic
25C3×C4×M16C2212yes—
26C3×#330C2214yescomplemented
27C3×#329C2218yescomplemented
28#3548C2×C311yescharacteristic, complemented
29#3151C2×C312yescomplemented
30#3138C2×C311yescharacteristic, complemented
31#1656C2×C318yescomplemented
32#1526C2×C311yescharacteristic, complemented
33#1525C2×C312yescomplemented
34C2×C3×#41—24no—
35C2×C3×#41D414yes—
36C22×C3×D4—22no—
37C22×C3×D4—21no—
38C3×AGL(1,ℤ/8)—416no—
39C2×C3×SD16—28no—
40C2×C3×D8—28no—
41C2×C3×M16—22no—
42C2×C3×M16—21no—
43C3×C4⋊Q8D412yes—
44C3×C4⋊Q8D412yes—
45C3×#127—22no—
46C3×#127—21no—
47C3×#127—22no—
48C3×#127D412yes—
49C3×#127C2312yes—
50C3×#125—22no—
51C3×#125C2312yes—
52C3×#124C2314yes—
53C3×#123—28no—
54C3×C22⋊Q8—24no—
55C3×C22⋊Q8—28no—
56C3×C22⋊Q8—22no—
57C3×#121—28no—
58C3×#121—24no—
59C3×#121—28no—
60C3×#121—22no—
61C3×#121—22no—
62C3×C4×Q8—22no—
63C3×C4×D4—24no—
64C3×C4×D4—22no—
65C3×C4×D4—22no—
66C3×#117D412yes—
67C2×C3×C4⋊C4—24no—
68C2×C3×C42C2311yescharacteristic
69C3×#102—28no—
70C3×#97C2312yes—
71C3×#97C2314yes—
72C3×C4×C8—21no—
73C3×C4×C8—22no—
74C2×AGL(1,ℤ/8)—24no—
75#562C22×C314yescomplemented
76#554—22no—
77#550—22no—
78#547—22no—
79#540—22no—
80#537—24no—
81#533C22×C311yescharacteristic
82#532C22×C312yes—
83C2×#127C22×C311yescharacteristic
84#496C22×C318yes—
85#493—22no—
86#492—22no—
87#490C22×C314yes—
88#486—22no—
89#418—22no—
90C4×M16C22×C311yescharacteristic
91C4×M16C22×C312yes—
92#330C22×C314yescomplemented
93#329C22×C318yescomplemented
94C24×C3—41no—
95C3×#41—48no—
96C3×#41—88no—
97C2×C3×Q8—22no—
98C2×C3×Q8C2×D414yes—
99C2×C3×D4—28no—
100C2×C3×D4—24no—
101C2×C3×D4—22no—
102C2×C3×D4—22no—
103C2×C3×D4—22no—
104C2×C3×D4—24no—
105C2×C3×D4—48no—
106C2×C3×D4—44no—
107C2×C3×D4—42no—
108C2×C3×D4—41no—
109C2×C3×D4C2×D414yes—
110C22×C3×C4—28no—
111C22×C3×C4—24no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D412yes—
114C3×SD16—416no—
115C3×D8—416no—
116C3×M16—28no—
117C3×M16—42no—
118C2×C3×C8—24no—
119C2×C3×C8—22no—
120C3×C4⋊C4—28no—
121C3×C4⋊C4—22no—
122C3×C4⋊C4—22no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—24no—
125C3×C4⋊C4C2×D414yes—
126C3×C22⋊C4—28no—
127C3×C22⋊C4—22no—
128C3×C22⋊C4—24no—
129C3×C22⋊C4—24no—
130C3×C42—22no—
131C3×C42C2×D411yescharacteristic
132C3×C42C2×D412yes—
133C3×C42C2411yescharacteristic
134C2×#41—24no—
135C2×#41C3×D414yes—
136C22×D4—22no—
137C22×D4—21no—
138AGL(1,ℤ/8)—416no—
139C2×SD16—28no—
140C2×D8—28no—
141C2×M16—22no—
142C2×M16—21no—
143C4⋊Q8C3×D412yes—
144C4⋊Q8C3×D412yes—
145#127—22no—
146#127—21no—
147#127—22no—
148#127C3×D412yes—
149#127C23×C312yes—
150#125—22no—
151#125C23×C312yes—
152#124C23×C314yes—
153#123—28no—
154C22⋊Q8—24no—
155C22⋊Q8—28no—
156C22⋊Q8—22no—
157#121—28no—
158#121—24no—
159#121—28no—
160#121—22no—
161#121—22no—
162C4×Q8—22no—
163C4×D4—24no—
164C4×D4—22no—
165C4×D4—22no—
166#117C3×D412yes—
167C2×C4⋊C4—24no—
168C2×C42C23×C311yescharacteristic
169#102—28no—
170#97C23×C312yes—
171#97C23×C314yes—
172C4×C8—21no—
173C4×C8—22no—
174C23×C3—24no—
175C23×C3—42no—
176C23×C3—84no—
177C23×C3—82no—
178C23×C3#12011yescharacteristic
179C3×Q8—44no—
180C3×Q8—44no—
181C3×D4—44no—
182C3×D4—48no—
183C3×D4—44no—
184C3×D4—44no—
185C3×D4—44no—
186C3×D4—84no—
187C3×D4—88no—
188C3×D4—84no—
189C3×D4—82no—
190C2×C3×C4—24no—
191C2×C3×C4—22no—
192C2×C3×C4—24no—
193C2×C3×C4—24no—
194C2×C3×C4—48no—
195C2×C3×C4—88no—
196C2×C3×C4#12012yes—
197C2×C3×C4#12011yescharacteristic
198C2×C3×C4C22×D411yescharacteristic
199C2×C3×C4C22×D412yes—
200C3×C8—28no—
201C3×C8—42no—
202C24—41no—
203#41—48no—
204#41—88no—
205C2×Q8—22no—
206C2×Q8C2×C3×D414yes—
207C2×D4—28no—
208C2×D4—24no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—22no—
212C2×D4—24no—
213C2×D4—48no—
214C2×D4—44no—
215C2×D4—42no—
216C2×D4—41no—
217C2×D4C2×C3×D414yes—
218C22×C4—28no—
219C22×C4—24no—
220C22×C4C2×C3×D411yescharacteristic
221C22×C4C2×C3×D412yes—
222SD16—416no—
223D8—416no—
224M16—28no—
225M16—42no—
226C2×C8—24no—
227C2×C8—22no—
228C4⋊C4—28no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—24no—
233C4⋊C4C2×C3×D414yes—
234C22⋊C4—28no—
235C22⋊C4—22no—
236C22⋊C4—24no—
237C22⋊C4—24no—
238C42—22no—
239C42C2×C3×D411yescharacteristic
240C42C2×C3×D412yes—
241C42C24×C311yescharacteristic
242C22×C3—22no—
243C22×C3—21no—
244C22×C3—44no—
245C22×C3—44no—
246C22×C3—44no—
247C22×C3—84no—
248C22×C3—82no—
249C22×C3—164no—
250C22×C3C2×#12011yescharacteristic
251C3×C4—22no—
252C3×C4—21no—
253C3×C4—44no—
254C3×C4—44no—
255C3×C4#45314yes—
256C3×C4#45412yes—
257C23—24no—
258C23—42no—
259C23—84no—
260C23—82no—
261C23C3×#12011yescharacteristic
262Q8—44no—
263Q8—44no—
264D4—44no—
265D4—48no—
266D4—44no—
267D4—44no—
268D4—44no—
269D4—84no—
270D4—88no—
271D4—84no—
272D4—82no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—48no—
278C2×C4—88no—
279C2×C4C3×#12012yes—
280C2×C4C3×#12011yescharacteristic
281C2×C4C22×C3×D411yescharacteristic
282C2×C4C22×C3×D412yes—
283C8—28no—
284C8—42no—
285C2×C3—41no—
286C2×C3—84no—
287C2×C3—162no—
288C2×C3C2×#45312yes—
289C2×C3C2×#45411yescharacteristic
290C22—22no—
291C22—21no—
292C22—44no—
293C22—44no—
294C22—44no—
295C22—84no—
296C22—82no—
297C22—164no—
298C22C2×C3×#12011yescharacteristic
299C4—22no—
300C4—21no—
301C4—44no—
302C4—44no—
303C4C3×#45314yes—
304C4C3×#45412yes—
305C3#1907511yescharacteristic, direct product factor, Sylow
306C2—41no—
307C2—84no—
308C2—162no—
309C2C2×C3×#45312yes—
310C2C2×C3×#45411yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
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
1. C3×#3548C2semidirect
2. C3×#3151C2semidirect
3. C3×#3138C2semidirect
4. C3×#1656C2semidirect
5. C3×#1526C2semidirect
6. C3×#1525C2semidirect
7. #19075C3direct (N × H)
9. C3×#562C22semidirect
26. C3×#330C22semidirect
27. C3×#329C22semidirect
28. #3548C2×C3semidirect
29. #3151C2×C3semidirect
30. #3138C2×C3semidirect
31. #1656C2×C3semidirect
32. #1526C2×C3semidirect
33. #1525C2×C3semidirect
75. #562C22×C3semidirect
92. #330C22×C3semidirect
93. #329C22×C3semidirect
305. C3#19075direct (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×#31512
C3×#31382
C3×#16562
C3×#15262
C3×#15252
#190753