Finite Group Databasebeta

← C3×#23925 #10642532

Subgroups — C3×#23925 #10642532

Order 768.

2080 conjugacy classes of subgroups; only the 302 normal subgroups are shown.

normal subgroups only

This group has 2080 conjugacy classes of subgroups — too many to draw. Only its 302 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
1C2×C3×#535C212yesmaximal, complemented
2C2×C3×#447C214yesmaximal, complemented
3C3×#1999C214yesmaximal, complemented
4C3×#1875C214yesmaximal, complemented
5C3×#1780C211yesmaximal, characteristic, complemented
6#23925C311yesmaximal, characteristic, direct product factor, Sylow
7C22×C3×#41C2211yescharacteristic
8C23×C3×D4C2212yescomplemented
9C22×C3×D8C2214yes—
10C3×#535—216no—
11C2×C3×#127—22no—
12C2×C3×#124—22no—
13C2×C3×#121C2214yes—
14C2×C3×#121C2214yescomplemented
15C2×C3×#120—22no—
16C2×C3×#120—22no—
17C2×C3×#117C2212yescomplemented
18C3×#447—216no—
19C2×C3×#102C2218yescomplemented
20C2×C3×C22⋊C8C2214yescomplemented
21C2×C3×C22⋊C8C2212yescomplemented
22C3×#340C2214yescomplemented
23C2×#535C2×C312yescomplemented
24C2×#447C2×C314yescomplemented
25#1999C2×C314yescomplemented
26#1875C2×C314yescomplemented
27#1780C2×C311yescharacteristic, complemented
28C25×C3—22no—
29C2×C3×#41—28no—
30C2×C3×#41—42no—
31C22×C3×Q8D411yescharacteristic
32C22×C3×D4—24no—
33C22×C3×D4—24no—
34C22×C3×D4—28no—
35C22×C3×D4—22no—
36C22×C3×D4—28no—
37C22×C3×D4—44no—
38C22×C3×D4D411yescharacteristic
39C22×C3×D4D412yes—
40C22×C3×D4C2314yes—
41C23×C3×C4C2311yescharacteristic
42C23×C3×C4C2312yes—
43C2×C3×D8—216no—
44C2×C3×D8—44no—
45C22×C3×C8C2314yes—
46C3×#127—216no—
47C3×#124—216no—
48C3×#121—28no—
49C3×#121—28no—
50C3×#121—28no—
51C3×#121D4116yescomplemented
52C3×#120—216no—
53C3×#120—216no—
54C3×#117—28no—
55C2×C3×C4⋊C4C2314yes—
56C2×C3×C22⋊C4—22no—
57C2×C3×C22⋊C4—24no—
58C2×C3×C22⋊C4—24no—
59C2×C3×C42—22no—
60C3×#102—216no—
61C3×C22⋊C8—28no—
62C3×C22⋊C8—42no—
63C22×#41C22×C311yescharacteristic
64C23×D4C22×C312yescomplemented
65C22×D8C22×C314yes—
66#535—216no—
67C2×#127—22no—
68C2×#124—22no—
69C2×#121C22×C314yes—
70C2×#121C22×C314yescomplemented
71C2×#120—22no—
72C2×#120—22no—
73C2×#117C22×C312yescomplemented
74#447—216no—
75C2×#102C22×C318yescomplemented
76C2×C22⋊C8C22×C314yescomplemented
77C2×C22⋊C8C22×C312yescomplemented
78#340C22×C314yescomplemented
79C24×C3—28no—
80C24×C3—24no—
81C24×C3—44no—
82C24×C3—44no—
83C24×C3—88no—
84C24×C3C2×D411yescharacteristic
85C24×C3C2×D412yes—
86C3×#41—416no—
87C2×C3×Q8—24no—
88C2×C3×Q8—41no—
89C2×C3×D4—28no—
90C2×C3×D4—24no—
91C2×C3×D4—24no—
92C2×C3×D4—216no—
93C2×C3×D4—216no—
94C2×C3×D4—24no—
95C2×C3×D4—28no—
96C2×C3×D4—24no—
97C2×C3×D4—28no—
98C2×C3×D4—432no—
99C2×C3×D4—42no—
100C2×C3×D4—41no—
101C2×C3×D4—44no—
102C2×C3×D4—44no—
103C2×C3×D4—88no—
104C2×C3×D4—88no—
105C2×C3×D4C2×D418yes—
106C22×C3×C4—24no—
107C22×C3×C4—24no—
108C22×C3×C4—24no—
109C22×C3×C4—24no—
110C22×C3×C4—22no—
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D412yes—
113C22×C3×C4C2×D418yes—
114C22×C3×C4C2411yescharacteristic
115C3×D8—416no—
116C2×C3×C8—28no—
117C2×C3×C8—24no—
118C3×C4⋊C4—24no—
119C3×C4⋊C4C2×D418yes—
120C3×C22⋊C4—28no—
121C3×C22⋊C4—216no—
122C3×C22⋊C4—216no—
123C3×C42—28no—
124C25—22no—
125C2×#41—28no—
126C2×#41—42no—
127C22×Q8C3×D411yescharacteristic
128C22×D4—24no—
129C22×D4—24no—
130C22×D4—28no—
131C22×D4—22no—
132C22×D4—28no—
133C22×D4—44no—
134C22×D4C3×D411yescharacteristic
135C22×D4C3×D412yes—
136C22×D4C23×C314yes—
137C23×C4C23×C311yescharacteristic
138C23×C4C23×C312yes—
139C2×D8—216no—
140C2×D8—44no—
141C22×C8C23×C314yes—
142#127—216no—
143#124—216no—
144#121—28no—
145#121—28no—
146#121—28no—
147#121C3×D4116yescomplemented
148#120—216no—
149#120—216no—
150#117—28no—
151C2×C4⋊C4C23×C314yes—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C42—22no—
156#102—216no—
157C22⋊C8—28no—
158C22⋊C8—42no—
159C23×C3—22no—
160C23×C3—22no—
161C23×C3—28no—
162C23×C3—28no—
163C23×C3—22no—
164C23×C3—21no—
165C23×C3—24no—
166C23×C3—416no—
167C23×C3—44no—
168C23×C3—44no—
169C23×C3—48no—
170C23×C3—48no—
171C23×C3—816no—
172C23×C3—816no—
173C23×C3—816no—
174C23×C3#12014yes—
175C23×C3C22×D411yescharacteristic
176C3×Q8—44no—
177C3×D4—216no—
178C3×D4—48no—
179C3×D4—416no—
180C3×D4—44no—
181C3×D4—48no—
182C3×D4—816no—
183C3×D4—816no—
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—216no—
187C2×C3×C4—28no—
188C2×C3×C4—22no—
189C2×C3×C4—21no—
190C2×C3×C4—48no—
191C2×C3×C4—44no—
192C2×C3×C4#12014yes—
193C2×C3×C4C22×D414yes—
194C2×C3×C4H(5,𝔽2)12yes—
195C3×C8—44no—
196C24—28no—
197C24—24no—
198C24—44no—
199C24—44no—
200C24—88no—
201C24C2×C3×D411yescharacteristic
202C24C2×C3×D412yes—
203#41—416no—
204C2×Q8—24no—
205C2×Q8—41no—
206C2×D4—28no—
207C2×D4—24no—
208C2×D4—24no—
209C2×D4—216no—
210C2×D4—216no—
211C2×D4—24no—
212C2×D4—28no—
213C2×D4—24no—
214C2×D4—28no—
215C2×D4—432no—
216C2×D4—42no—
217C2×D4—41no—
218C2×D4—44no—
219C2×D4—44no—
220C2×D4—88no—
221C2×D4—88no—
222C2×D4C2×C3×D418yes—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4C2×C3×D411yescharacteristic
229C22×C4C2×C3×D412yes—
230C22×C4C2×C3×D418yes—
231C22×C4C24×C311yescharacteristic
232D8—416no—
233C2×C8—28no—
234C2×C8—24no—
235C4⋊C4—24no—
236C4⋊C4C2×C3×D418yes—
237C22⋊C4—28no—
238C22⋊C4—216no—
239C22⋊C4—216no—
240C42—28no—
241C22×C3—24no—
242C22×C3—28no—
243C22×C3—44no—
244C22×C3—44no—
245C22×C3—41no—
246C22×C3—48no—
247C22×C3—42no—
248C22×C3—44no—
249C22×C3—48no—
250C22×C3—48no—
251C22×C3—832no—
252C22×C3—88no—
253C22×C3—88no—
254C22×C3C2×#12012yes—
255C22×C3#53411yescharacteristic
256C22×C3#57614yes—
257C3×C4—24no—
258C3×C4—41no—
259C3×C4—48no—
260C3×C4D4214yes—
261C23—28no—
262C23—22no—
263C23—22no—
264C23—24no—
265C23—28no—
266C23—22no—
267C23—21no—
268C23—416no—
269C23—48no—
270C23—48no—
271C23—44no—
272C23—44no—
273C23—816no—
274C23—816no—
275C23—816no—
276C23C3×#12014yes—
277C23C22×C3×D411yescharacteristic
278Q8—44no—
279D4—216no—
280D4—48no—
281D4—416no—
282D4—44no—
283D4—48no—
284D4—816no—
285D4—816no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—216no—
289C2×C4—28no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—48no—
293C2×C4—44no—
294C2×C4C3×#12014yes—
295C2×C4C22×C3×D414yes—
296C2×C4C3×H(5,𝔽2)12yes—
297C8—44no—
298C2×C3—44no—
299C2×C3—81no—
300C2×C3—88no—
301C2×C3#240311yescharacteristic
302C2×C3#300814yes—
303C2×C3#319912yes—
304C22—24no—
305C22—28no—
306C22—44no—
307C22—44no—
308C22—41no—
309C22—48no—
310C22—42no—
311C22—44no—
312C22—48no—
313C22—48no—
314C22—832no—
315C22—88no—
316C22—88no—
317C22C2×C3×#12012yes—
318C22C3×#53411yescharacteristic
319C22C3×#57614yes—
320C4—24no—
321C4—41no—
322C4—48no—
323C4C3×D4214yes—
324C3#2392511yescharacteristic, direct product factor, Sylow
325C2—44no—
326C2—81no—
327C2—88no—
328C2C3×#240311yescharacteristic
329C2C3×#300814yes—
330C2C3×#319912yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×#41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#535C2semidirect
2. C2×C3×#447C2semidirect
3. C3×#1999C2semidirect
4. C3×#1875C2semidirect
5. C3×#1780C2semidirect
6. #23925C3direct (N × H)
8. C23×C3×D4C22semidirect
14. C2×C3×#121C22semidirect
17. C2×C3×#117C22semidirect
19. C2×C3×#102C22semidirect
20. C2×C3×C22⋊C8C22semidirect
21. C2×C3×C22⋊C8C22semidirect
22. C3×#340C22semidirect
23. C2×#535C2×C3semidirect
24. C2×#447C2×C3semidirect
25. #1999C2×C3semidirect
26. #1875C2×C3semidirect
27. #1780C2×C3semidirect
51. C3×#121D4semidirect
64. C23×D4C22×C3semidirect
70. C2×#121C22×C3semidirect
73. C2×#117C22×C3semidirect
75. C2×#102C22×C3semidirect
76. C2×C22⋊C8C22×C3semidirect
77. C2×C22⋊C8C22×C3semidirect
78. #340C22×C3semidirect
147. #121C3×D4semidirect
324. C3#23925direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5352
C2×C3×#4472
C3×#19992
C3×#18752
C3×#17802
#239253