Finite Group Databasebeta

← C3×#23921 #10642528

Subgroups — C3×#23921 #10642528

Order 768.

1908 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C22⋊Q8C212yesmaximal
2C2×C3×#451C214yesmaximal, complemented
3C3×#2001C214yesmaximal, complemented
4C3×#1872C214yesmaximal, complemented
5C3×#1779C211yesmaximal, characteristic
6#23921C311yesmaximal, characteristic, direct product factor, Sylow
7C23×C3×Q8C2212yes—
8C24×C3×C4C2211yescharacteristic
9C22×C3×Q16C2214yescomplemented
10C2×C3×C22⋊Q8—22no—
11C2×C3×C22⋊Q8—22no—
12C2×C3×C22⋊Q8—216no—
13C2×C3×C22⋊Q8C2214yes—
14C2×C3×C22⋊Q8C2214yes—
15C22×C3×C4⋊C4—22no—
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C22⋊C4—22no—
18C3×#451—216no—
19C2×C3×#103C2218yescomplemented
20C2×C3×C22⋊C8C2214yes—
21C2×C3×C22⋊C8C2212yes—
22C3×#340C2214yescomplemented
23C22×C22⋊Q8C2×C312yes—
24C2×#451C2×C314yescomplemented
25#2001C2×C314yescomplemented
26#1872C2×C314yescomplemented
27#1779C2×C311yescharacteristic
28C25×C3D411yescharacteristic
29C22×C3×Q8—28no—
30C22×C3×Q8—22no—
31C22×C3×Q8—28no—
32C22×C3×Q8—44no—
33C22×C3×Q8C2314yes—
34C23×C3×C4—22no—
35C23×C3×C4—22no—
36C23×C3×C4—28no—
37C23×C3×C4—42no—
38C23×C3×C4D412yes—
39C23×C3×C4D411yescharacteristic
40C23×C3×C4C2311yescharacteristic
41C23×C3×C4C2312yes—
42C2×C3×Q16—216no—
43C2×C3×Q16—44no—
44C22×C3×C8C2314yes—
45C3×C22⋊Q8—216no—
46C3×C22⋊Q8—216no—
47C3×C22⋊Q8—28no—
48C3×C22⋊Q8—28no—
49C3×C22⋊Q8—28no—
50C3×C22⋊Q8D4116yes—
51C2×C3×C4⋊C4—216no—
52C2×C3×C4⋊C4—24no—
53C2×C3×C4⋊C4—24no—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4—28no—
56C2×C3×C4⋊C4C2314yes—
57C2×C3×C22⋊C4—216no—
58C2×C3×C22⋊C4—24no—
59C2×C3×C22⋊C4—24no—
60C3×#103—216no—
61C3×C22⋊C8—28no—
62C3×C22⋊C8—42no—
63C23×Q8C22×C312yes—
64C24×C4C22×C311yescharacteristic
65C22×Q16C22×C314yescomplemented
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—216no—
69C2×C22⋊Q8C22×C314yes—
70C2×C22⋊Q8C22×C314yes—
71C22×C4⋊C4—22no—
72C22×C4⋊C4C22×C312yes—
73C22×C22⋊C4—22no—
74#451—216no—
75C2×#103C22×C318yescomplemented
76C2×C22⋊C8C22×C314yes—
77C2×C22⋊C8C22×C312yes—
78#340C22×C314yescomplemented
79C24×C3—24no—
80C24×C3—24no—
81C24×C3—41no—
82C24×C3—41no—
83C24×C3Q1614yescomplemented
84C24×C3C2×D411yescharacteristic
85C24×C3C2×D412yes—
86C2×C3×Q8—28no—
87C2×C3×Q8—24no—
88C2×C3×Q8—28no—
89C2×C3×Q8—28no—
90C2×C3×Q8—432no—
91C2×C3×Q8—44no—
92C2×C3×Q8—44no—
93C2×C3×Q8C2×D418yes—
94C22×C3×C4—24no—
95C22×C3×C4—24no—
96C22×C3×C4—28no—
97C22×C3×C4—28no—
98C22×C3×C4—24no—
99C22×C3×C4—24no—
100C22×C3×C4—24no—
101C22×C3×C4—24no—
102C22×C3×C4—24no—
103C22×C3×C4—24no—
104C22×C3×C4—22no—
105C22×C3×C4—41no—
106C22×C3×C4—42no—
107C22×C3×C4—416no—
108C22×C3×C4—44no—
109C22×C3×C4—44no—
110C22×C3×C4—44no—
111C22×C3×C4—44no—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4C2×D412yes—
114C22×C3×C4C2×D418yes—
115C22×C3×C4C2411yescharacteristic
116C3×Q16—416no—
117C2×C3×C8—28no—
118C2×C3×C8—24no—
119C3×C4⋊C4—28no—
120C3×C4⋊C4—216no—
121C3×C4⋊C4—216no—
122C3×C4⋊C4—24no—
123C3×C4⋊C4C2×D418yes—
124C3×C22⋊C4—216no—
125C3×C22⋊C4—216no—
126C25C3×D411yescharacteristic
127C22×Q8—28no—
128C22×Q8—22no—
129C22×Q8—28no—
130C22×Q8—44no—
131C22×Q8C23×C314yes—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—28no—
135C23×C4—42no—
136C23×C4C3×D412yes—
137C23×C4C3×D411yescharacteristic
138C23×C4C23×C311yescharacteristic
139C23×C4C23×C312yes—
140C2×Q16—216no—
141C2×Q16—44no—
142C22×C8C23×C314yes—
143C22⋊Q8—216no—
144C22⋊Q8—216no—
145C22⋊Q8—28no—
146C22⋊Q8—28no—
147C22⋊Q8—28no—
148C22⋊Q8C3×D4116yes—
149C2×C4⋊C4—216no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4C23×C314yes—
155C2×C22⋊C4—216no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158#103—216no—
159C22⋊C8—28no—
160C22⋊C8—42no—
161C23×C3—22no—
162C23×C3—22no—
163C23×C3—22no—
164C23×C3—22no—
165C23×C3—22no—
166C23×C3—21no—
167C23×C3—22no—
168C23×C3—21no—
169C23×C3—216no—
170C23×C3—44no—
171C23×C3—44no—
172C23×C3—44no—
173C23×C3—48no—
174C23×C3#12014yes—
175C23×C3C2×Q1612yes—
176C23×C3C2×Q1614yes—
177C23×C3#13714yes—
178C23×C3C22×D411yescharacteristic
179C3×Q8—216no—
180C3×Q8—416no—
181C3×Q8—48no—
182C2×C3×C4—22no—
183C2×C3×C4—22no—
184C2×C3×C4—216no—
185C2×C3×C4—28no—
186C2×C3×C4—28no—
187C2×C3×C4—22no—
188C2×C3×C4—21no—
189C2×C3×C4—44no—
190C2×C3×C4—48no—
191C2×C3×C4—416no—
192C2×C3×C4—416no—
193C2×C3×C4—44no—
194C2×C3×C4—44no—
195C2×C3×C4—44no—
196C2×C3×C4—44no—
197C2×C3×C4—44no—
198C2×C3×C4#12014yes—
199C2×C3×C4C22×D414yes—
200C2×C3×C4H(5,𝔽2)12yes—
201C3×C8—44no—
202C24—24no—
203C24—24no—
204C24—41no—
205C24—41no—
206C24C3×Q1614yescomplemented
207C24C2×C3×D411yescharacteristic
208C24C2×C3×D412yes—
209C2×Q8—28no—
210C2×Q8—24no—
211C2×Q8—28no—
212C2×Q8—28no—
213C2×Q8—432no—
214C2×Q8—44no—
215C2×Q8—44no—
216C2×Q8C2×C3×D418yes—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—28no—
220C22×C4—28no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—416no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4C2×C3×D411yescharacteristic
236C22×C4C2×C3×D412yes—
237C22×C4C2×C3×D418yes—
238C22×C4C24×C311yescharacteristic
239Q16—416no—
240C2×C8—28no—
241C2×C8—24no—
242C4⋊C4—28no—
243C4⋊C4—216no—
244C4⋊C4—216no—
245C4⋊C4—24no—
246C4⋊C4C2×C3×D418yes—
247C22⋊C4—216no—
248C22⋊C4—216no—
249C22×C3—24no—
250C22×C3—24no—
251C22×C3—24no—
252C22×C3—24no—
253C22×C3—24no—
254C22×C3—24no—
255C22×C3—22no—
256C22×C3—24no—
257C22×C3—416no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—42no—
261C22×C3#45118yescomplemented
262C22×C3C2×#12012yes—
263C22×C3#53411yescharacteristic
264C22×C3C22×Q1611yescharacteristic
265C22×C3C2×#13712yes—
266C22×C3#57511yescharacteristic
267C3×C4—24no—
268C3×C4—41no—
269C3×C4—48no—
270C3×C4—48no—
271C3×C4D4214yes—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—21no—
278C23—216no—
279C23—22no—
280C23—21no—
281C23—44no—
282C23—44no—
283C23—44no—
284C23—48no—
285C23C3×#12014yes—
286C23C2×C3×Q1612yes—
287C23C2×C3×Q1614yes—
288C23C3×#13714yes—
289C23C22×C3×D411yescharacteristic
290Q8—216no—
291Q8—416no—
292Q8—48no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—216no—
296C2×C4—28no—
297C2×C4—28no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—44no—
301C2×C4—48no—
302C2×C4—416no—
303C2×C4—416no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4C3×#12014yes—
310C2×C4C22×C3×D414yes—
311C2×C4C3×H(5,𝔽2)12yes—
312C8—44no—
313C2×C3—28no—
314C2×C3—42no—
315C2×C3#240311yescharacteristic
316C2×C3C2×#45112yes—
317C2×C3#300712yes—
318C2×C3#318911yescharacteristic
319C2×C3#319511yescharacteristic
320C22—24no—
321C22—24no—
322C22—24no—
323C22—24no—
324C22—24no—
325C22—24no—
326C22—22no—
327C22—24no—
328C22—416no—
329C22—41no—
330C22—41no—
331C22—42no—
332C22C3×#45118yescomplemented
333C22C2×C3×#12012yes—
334C22C3×#53411yescharacteristic
335C22C22×C3×Q1611yescharacteristic
336C22C2×C3×#13712yes—
337C22C3×#57511yescharacteristic
338C4—24no—
339C4—41no—
340C4—48no—
341C4—48no—
342C4C3×D4214yes—
343C3#2392111yescharacteristic, direct product factor, Sylow
344C2—28no—
345C2—42no—
346C2C3×#240311yescharacteristic
347C2C2×C3×#45112yes—
348C2C3×#300712yes—
349C2C3×#318911yescharacteristic
350C2C3×#319511yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC25×C3D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#451C2semidirect
3. C3×#2001C2semidirect
4. C3×#1872C2semidirect
6. #23921C3direct (N × H)
9. C22×C3×Q16C22semidirect
19. C2×C3×#103C22semidirect
22. C3×#340C22semidirect
24. C2×#451C2×C3semidirect
25. #2001C2×C3semidirect
26. #1872C2×C3semidirect
65. C22×Q16C22×C3semidirect
75. C2×#103C22×C3semidirect
78. #340C22×C3semidirect
83. C24×C3Q16semidirect
206. C24C3×Q16semidirect
261. C22×C3#451semidirect
332. C22C3×#451semidirect
343. C3#23921direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊Q82
C2×C3×#4512
C3×#20012
C3×#18722
C3×#17792
#239213