Finite Group Databasebeta

← C3×#23928 #10642535

Subgroups — C3×#23928 #10642535

Order 768.

1696 conjugacy classes of subgroups; only the 302 normal subgroups are shown.

normal subgroups only

This group has 1696 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×#536C212yesmaximal
2C2×C3×#451C214yesmaximal, complemented
3C3×#2001C214yesmaximal, complemented
4C3×#1873C214yesmaximal, complemented
5C3×#1780C211yesmaximal, characteristic
6#23928C311yesmaximal, characteristic, direct product factor, Sylow
7C22×C3×#41C2211yescharacteristic
8C23×C3×Q8C2212yes—
9C22×C3×Q16C2214yes—
10C3×#536—216no—
11C2×C3×C4⋊Q8—22no—
12C2×C3×#124—22no—
13C2×C3×#123—22no—
14C2×C3×#123—22no—
15C2×C3×C22⋊Q8C2214yes—
16C2×C3×C22⋊Q8C2214yes—
17C2×C3×#117C2212yes—
18C3×#451—216no—
19C2×C3×#103C2218yes—
20C2×C3×C22⋊C8C2214yes—
21C2×C3×C22⋊C8C2212yes—
22C3×#340C2214yes—
23C2×#536C2×C312yes—
24C2×#451C2×C314yescomplemented
25#2001C2×C314yescomplemented
26#1873C2×C314yescomplemented
27#1780C2×C311yescharacteristic
28C2×C3×#41—28no—
29C2×C3×#41—42no—
30C22×C3×Q8—28no—
31C22×C3×Q8—22no—
32C22×C3×Q8—28no—
33C22×C3×Q8—44no—
34C22×C3×Q8D411yescharacteristic
35C22×C3×Q8C2314yes—
36C22×C3×D4D411yescharacteristic
37C22×C3×D4D412yes—
38C23×C3×C4—22no—
39C23×C3×C4C2311yescharacteristic
40C23×C3×C4C2312yes—
41C2×C3×Q16—216no—
42C2×C3×Q16—44no—
43C22×C3×C8C2314yes—
44C3×C4⋊Q8—216no—
45C3×#124—216no—
46C3×#123—216no—
47C3×#123—216no—
48C3×C22⋊Q8—28no—
49C3×C22⋊Q8—28no—
50C3×C22⋊Q8—28no—
51C3×C22⋊Q8D4116yes—
52C3×#117—28no—
53C2×C3×C4⋊C4—24no—
54C2×C3×C4⋊C4—24no—
55C2×C3×C4⋊C4C2314yes—
56C2×C3×C22⋊C4—22no—
57C2×C3×C22⋊C4—24no—
58C2×C3×C22⋊C4—24no—
59C2×C3×C42—22no—
60C3×#103—216no—
61C3×C22⋊C8—28no—
62C3×C22⋊C8—42no—
63C22×#41C22×C311yescharacteristic
64C23×Q8C22×C312yes—
65C22×Q16C22×C314yes—
66#536—216no—
67C2×C4⋊Q8—22no—
68C2×#124—22no—
69C2×#123—22no—
70C2×#123—22no—
71C2×C22⋊Q8C22×C314yes—
72C2×C22⋊Q8C22×C314yes—
73C2×#117C22×C312yes—
74#451—216no—
75C2×#103C22×C318yes—
76C2×C22⋊C8C22×C314yes—
77C2×C22⋊C8C22×C312yes—
78#340C22×C314yes—
79C24×C3C2×D411yescharacteristic
80C24×C3C2×D412yes—
81C3×#41—416no—
82C2×C3×Q8—28no—
83C2×C3×Q8—24no—
84C2×C3×Q8—24no—
85C2×C3×Q8—28no—
86C2×C3×Q8—28no—
87C2×C3×Q8—41no—
88C2×C3×Q8—432no—
89C2×C3×Q8—44no—
90C2×C3×Q8—44no—
91C2×C3×Q8C2×D418yes—
92C2×C3×D4—24no—
93C2×C3×D4—24no—
94C2×C3×D4—24no—
95C2×C3×D4—42no—
96C2×C3×D4—41no—
97C22×C3×C4—24no—
98C22×C3×C4—24no—
99C22×C3×C4—28no—
100C22×C3×C4—24no—
101C22×C3×C4—24no—
102C22×C3×C4—24no—
103C22×C3×C4—22no—
104C22×C3×C4—44no—
105C22×C3×C4—44no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D412yes—
108C22×C3×C4C2×D418yes—
109C22×C3×C4C2411yescharacteristic
110C3×Q16—416no—
111C2×C3×C8—28no—
112C2×C3×C8—24no—
113C3×C4⋊C4—216no—
114C3×C4⋊C4—216no—
115C3×C4⋊C4—24no—
116C3×C4⋊C4C2×D418yes—
117C3×C22⋊C4—28no—
118C3×C22⋊C4—216no—
119C3×C22⋊C4—216no—
120C3×C42—28no—
121C2×#41—28no—
122C2×#41—42no—
123C22×Q8—28no—
124C22×Q8—22no—
125C22×Q8—28no—
126C22×Q8—44no—
127C22×Q8C3×D411yescharacteristic
128C22×Q8C23×C314yes—
129C22×D4C3×D411yescharacteristic
130C22×D4C3×D412yes—
131C23×C4—22no—
132C23×C4C23×C311yescharacteristic
133C23×C4C23×C312yes—
134C2×Q16—216no—
135C2×Q16—44no—
136C22×C8C23×C314yes—
137C4⋊Q8—216no—
138#124—216no—
139#123—216no—
140#123—216no—
141C22⋊Q8—28no—
142C22⋊Q8—28no—
143C22⋊Q8—28no—
144C22⋊Q8C3×D4116yes—
145#117—28no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4C23×C314yes—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—24no—
152C2×C42—22no—
153#103—216no—
154C22⋊C8—28no—
155C22⋊C8—42no—
156C23×C3—22no—
157C23×C3—22no—
158C23×C3—28no—
159C23×C3—22no—
160C23×C3—21no—
161C23×C3—24no—
162C23×C3#12014yes—
163C23×C3C22×D411yescharacteristic
164C3×Q8—216no—
165C3×Q8—44no—
166C3×Q8—416no—
167C3×Q8—48no—
168C3×D4—48no—
169C3×D4—44no—
170C2×C3×C4—22no—
171C2×C3×C4—22no—
172C2×C3×C4—216no—
173C2×C3×C4—28no—
174C2×C3×C4—28no—
175C2×C3×C4—22no—
176C2×C3×C4—21no—
177C2×C3×C4—416no—
178C2×C3×C4—44no—
179C2×C3×C4—44no—
180C2×C3×C4—48no—
181C2×C3×C4—44no—
182C2×C3×C4#12014yes—
183C2×C3×C4C22×D414yes—
184C2×C3×C4H(5,𝔽2)12yes—
185C3×C8—44no—
186C24C2×C3×D411yescharacteristic
187C24C2×C3×D412yes—
188#41—416no—
189C2×Q8—28no—
190C2×Q8—24no—
191C2×Q8—24no—
192C2×Q8—28no—
193C2×Q8—28no—
194C2×Q8—41no—
195C2×Q8—432no—
196C2×Q8—44no—
197C2×Q8—44no—
198C2×Q8C2×C3×D418yes—
199C2×D4—24no—
200C2×D4—24no—
201C2×D4—24no—
202C2×D4—42no—
203C2×D4—41no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—28no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—22no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4C2×C3×D411yescharacteristic
214C22×C4C2×C3×D412yes—
215C22×C4C2×C3×D418yes—
216C22×C4C24×C311yescharacteristic
217Q16—416no—
218C2×C8—28no—
219C2×C8—24no—
220C4⋊C4—216no—
221C4⋊C4—216no—
222C4⋊C4—24no—
223C4⋊C4C2×C3×D418yes—
224C22⋊C4—28no—
225C22⋊C4—216no—
226C22⋊C4—216no—
227C42—28no—
228C22×C3—24no—
229C22×C3—28no—
230C22×C3—44no—
231C22×C3—44no—
232C22×C3—41no—
233C22×C3—42no—
234C22×C3—44no—
235C22×C3C2×#12012yes—
236C22×C3#53411yescharacteristic
237C22×C3#57814yes—
238C3×C4—24no—
239C3×C4—41no—
240C3×C4—48no—
241C3×C4—48no—
242C3×C4D4214yes—
243C23—28no—
244C23—22no—
245C23—22no—
246C23—24no—
247C23—22no—
248C23—21no—
249C23C3×#12014yes—
250C23C22×C3×D411yescharacteristic
251Q8—216no—
252Q8—44no—
253Q8—416no—
254Q8—48no—
255D4—48no—
256D4—44no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—216no—
260C2×C4—28no—
261C2×C4—28no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—416no—
265C2×C4—48no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4C3×#12014yes—
270C2×C4C22×C3×D414yes—
271C2×C4C3×H(5,𝔽2)12yes—
272C8—44no—
273C2×C3—44no—
274C2×C3—81no—
275C2×C3#240311yescharacteristic
276C2×C3#301114yes—
277C2×C3#320212yes—
278C22—24no—
279C22—28no—
280C22—44no—
281C22—44no—
282C22—41no—
283C22—42no—
284C22—44no—
285C22C2×C3×#12012yes—
286C22C3×#53411yescharacteristic
287C22C3×#57814yes—
288C4—24no—
289C4—41no—
290C4—48no—
291C4—48no—
292C4C3×D4214yes—
293C3#2392811yescharacteristic, direct product factor, Sylow
294C2—44no—
295C2—81no—
296C2C3×#240311yescharacteristic
297C2C3×#301114yes—
298C2C3×#320212yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×#41C22
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
2. C2×C3×#451C2semidirect
3. C3×#2001C2semidirect
4. C3×#1873C2semidirect
6. #23928C3direct (N × H)
24. C2×#451C2×C3semidirect
25. #2001C2×C3semidirect
26. #1873C2×C3semidirect
293. C3#23928direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5362
C2×C3×#4512
C3×#20012
C3×#18732
C3×#17802
#239283