Finite Group Databasebeta

← C3×#23918 #10642525

Subgroups — C3×#23918 #10642525

Order 768.

2596 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#121C212yesmaximal, complemented
2C2×C3×#447C214yesmaximal, complemented
3C3×#1999C214yesmaximal, complemented
4C3×#1874C214yesmaximal, complemented
5C3×#1779C211yesmaximal, characteristic, complemented
6#23918C311yesmaximal, characteristic, direct product factor, Sylow
7C23×C3×D4—22no—
8C23×C3×D4C2212yescomplemented
9C24×C3×C4C2211yescharacteristic
10C22×C3×D8C2214yescomplemented
11C2×C3×#121—22no—
12C2×C3×#121—22no—
13C2×C3×#121—216no—
14C2×C3×#121C2214yes—
15C2×C3×#121C2214yescomplemented
16C22×C3×C4⋊C4C2212yescomplemented
17C22×C3×C22⋊C4—22no—
18C3×#447—216no—
19C2×C3×#102C2218yescomplemented
20C2×C3×C22⋊C8C2214yescomplemented
21C2×C3×C22⋊C8C2212yescomplemented
22C3×#340C2214yescomplemented
23C22×#121C2×C312yescomplemented
24C2×#447C2×C314yescomplemented
25#1999C2×C314yescomplemented
26#1874C2×C314yescomplemented
27#1779C2×C311yescharacteristic, complemented
28C25×C3—22no—
29C25×C3D411yescharacteristic
30C22×C3×D4—216no—
31C22×C3×D4—24no—
32C22×C3×D4—24no—
33C22×C3×D4—28no—
34C22×C3×D4—22no—
35C22×C3×D4—28no—
36C22×C3×D4—44no—
37C22×C3×D4—48no—
38C22×C3×D4—48no—
39C22×C3×D4C2314yes—
40C23×C3×C4—22no—
41C23×C3×C4—28no—
42C23×C3×C4—42no—
43C23×C3×C4D412yes—
44C23×C3×C4D411yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C23×C3×C4C2312yes—
47C2×C3×D8—216no—
48C2×C3×D8—44no—
49C22×C3×C8C2314yes—
50C3×#121—216no—
51C3×#121—216no—
52C3×#121—28no—
53C3×#121—28no—
54C3×#121—28no—
55C3×#121D4116yescomplemented
56C2×C3×C4⋊C4—22no—
57C2×C3×C4⋊C4—28no—
58C2×C3×C4⋊C4C2314yes—
59C2×C3×C22⋊C4—216no—
60C2×C3×C22⋊C4—24no—
61C2×C3×C22⋊C4—24no—
62C3×#102—216no—
63C3×C22⋊C8—28no—
64C3×C22⋊C8—42no—
65C23×D4—22no—
66C23×D4C22×C312yescomplemented
67C24×C4C22×C311yescharacteristic
68C22×D8C22×C314yescomplemented
69C2×#121—22no—
70C2×#121—22no—
71C2×#121—216no—
72C2×#121C22×C314yes—
73C2×#121C22×C314yescomplemented
74C22×C4⋊C4C22×C312yescomplemented
75C22×C22⋊C4—22no—
76#447—216no—
77C2×#102C22×C318yescomplemented
78C2×C22⋊C8C22×C314yescomplemented
79C2×C22⋊C8C22×C312yescomplemented
80#340C22×C314yescomplemented
81C24×C3—24no—
82C24×C3—24no—
83C24×C3—28no—
84C24×C3—24no—
85C24×C3—41no—
86C24×C3—41no—
87C24×C3—44no—
88C24×C3—44no—
89C24×C3—44no—
90C24×C3—44no—
91C24×C3—44no—
92C24×C3—44no—
93C24×C3D814yescomplemented
94C24×C3C2×D411yescharacteristic
95C24×C3C2×D412yes—
96C2×C3×D4—28no—
97C2×C3×D4—216no—
98C2×C3×D4—216no—
99C2×C3×D4—28no—
100C2×C3×D4—24no—
101C2×C3×D4—28no—
102C2×C3×D4—432no—
103C2×C3×D4—44no—
104C2×C3×D4—464no—
105C2×C3×D4—48no—
106C2×C3×D4—48no—
107C2×C3×D4—48no—
108C2×C3×D4—48no—
109C2×C3×D4—44no—
110C2×C3×D4C2×D418yescomplemented
111C22×C3×C4—24no—
112C22×C3×C4—24no—
113C22×C3×C4—28no—
114C22×C3×C4—24no—
115C22×C3×C4—24no—
116C22×C3×C4—24no—
117C22×C3×C4—24no—
118C22×C3×C4—24no—
119C22×C3×C4—22no—
120C22×C3×C4—41no—
121C22×C3×C4—42no—
122C22×C3×C4—416no—
123C22×C3×C4—44no—
124C22×C3×C4—44no—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D412yes—
127C22×C3×C4C2×D418yes—
128C22×C3×C4C2411yescharacteristic
129C3×D8—416no—
130C2×C3×C8—28no—
131C2×C3×C8—24no—
132C3×C4⋊C4—28no—
133C3×C4⋊C4—24no—
134C3×C4⋊C4C2×D418yescomplemented
135C3×C22⋊C4—216no—
136C3×C22⋊C4—216no—
137C25—22no—
138C25C3×D411yescharacteristic
139C22×D4—216no—
140C22×D4—24no—
141C22×D4—24no—
142C22×D4—28no—
143C22×D4—22no—
144C22×D4—28no—
145C22×D4—44no—
146C22×D4—48no—
147C22×D4—48no—
148C22×D4C23×C314yes—
149C23×C4—22no—
150C23×C4—28no—
151C23×C4—42no—
152C23×C4C3×D412yes—
153C23×C4C3×D411yescharacteristic
154C23×C4C23×C311yescharacteristic
155C23×C4C23×C312yes—
156C2×D8—216no—
157C2×D8—44no—
158C22×C8C23×C314yes—
159#121—216no—
160#121—216no—
161#121—28no—
162#121—28no—
163#121—28no—
164#121C3×D4116yescomplemented
165C2×C4⋊C4—22no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4C23×C314yes—
168C2×C22⋊C4—216no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171#102—216no—
172C22⋊C8—28no—
173C22⋊C8—42no—
174C23×C3—22no—
175C23×C3—22no—
176C23×C3—22no—
177C23×C3—22no—
178C23×C3—22no—
179C23×C3—21no—
180C23×C3—22no—
181C23×C3—21no—
182C23×C3—216no—
183C23×C3—28no—
184C23×C3—44no—
185C23×C3—44no—
186C23×C3—44no—
187C23×C3—48no—
188C23×C3—416no—
189C23×C3—416no—
190C23×C3—416no—
191C23×C3—44no—
192C23×C3—44no—
193C23×C3—44no—
194C23×C3—44no—
195C23×C3—44no—
196C23×C3—44no—
197C23×C3—88no—
198C23×C3—88no—
199C23×C3—88no—
200C23×C3—88no—
201C23×C3#12014yes—
202C23×C3C2×D812yes—
203C23×C3C2×D814yes—
204C23×C3AGL(1,ℤ/8)14yes—
205C23×C3C22×D411yescharacteristic
206C3×D4—216no—
207C3×D4—416no—
208C3×D4—432no—
209C3×D4—432no—
210C3×D4—48no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—216no—
214C2×C3×C4—22no—
215C2×C3×C4—28no—
216C2×C3×C4—21no—
217C2×C3×C4—44no—
218C2×C3×C4—48no—
219C2×C3×C4—416no—
220C2×C3×C4—44no—
221C2×C3×C4—44no—
222C2×C3×C4—44no—
223C2×C3×C4#12014yes—
224C2×C3×C4C22×D414yes—
225C2×C3×C4H(5,𝔽2)12yes—
226C3×C8—44no—
227C24—24no—
228C24—24no—
229C24—28no—
230C24—24no—
231C24—41no—
232C24—41no—
233C24—44no—
234C24—44no—
235C24—44no—
236C24—44no—
237C24—44no—
238C24—44no—
239C24C3×D814yescomplemented
240C24C2×C3×D411yescharacteristic
241C24C2×C3×D412yes—
242C2×D4—28no—
243C2×D4—216no—
244C2×D4—216no—
245C2×D4—28no—
246C2×D4—24no—
247C2×D4—28no—
248C2×D4—432no—
249C2×D4—44no—
250C2×D4—464no—
251C2×D4—48no—
252C2×D4—48no—
253C2×D4—48no—
254C2×D4—48no—
255C2×D4—44no—
256C2×D4C2×C3×D418yescomplemented
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—28no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—416no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4C2×C3×D411yescharacteristic
272C22×C4C2×C3×D412yes—
273C22×C4C2×C3×D418yes—
274C22×C4C24×C311yescharacteristic
275D8—416no—
276C2×C8—28no—
277C2×C8—24no—
278C4⋊C4—28no—
279C4⋊C4—24no—
280C4⋊C4C2×C3×D418yescomplemented
281C22⋊C4—216no—
282C22⋊C4—216no—
283C22×C3—24no—
284C22×C3—24no—
285C22×C3—24no—
286C22×C3—24no—
287C22×C3—24no—
288C22×C3—24no—
289C22×C3—22no—
290C22×C3—24no—
291C22×C3—416no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—42no—
295C22×C3—48no—
296C22×C3—48no—
297C22×C3—48no—
298C22×C3—832no—
299C22×C3—84no—
300C22×C3—84no—
301C22×C3—84no—
302C22×C3—84no—
303C22×C3#44718yescomplemented
304C22×C3C2×#12012yes—
305C22×C3#53411yescharacteristic
306C22×C3C22×D811yescharacteristic
307C22×C3C2×AGL(1,ℤ/8)12yes—
308C22×C3#57511yescharacteristic
309C3×C4—24no—
310C3×C4—41no—
311C3×C4—48no—
312C3×C4D4214yes—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—21no—
319C23—216no—
320C23—22no—
321C23—21no—
322C23—28no—
323C23—44no—
324C23—44no—
325C23—44no—
326C23—48no—
327C23—416no—
328C23—416no—
329C23—416no—
330C23—44no—
331C23—44no—
332C23—44no—
333C23—44no—
334C23—44no—
335C23—44no—
336C23—88no—
337C23—88no—
338C23—88no—
339C23—88no—
340C23C3×#12014yes—
341C23C2×C3×D812yes—
342C23C2×C3×D814yes—
343C23C3×AGL(1,ℤ/8)14yes—
344C23C22×C3×D411yescharacteristic
345D4—216no—
346D4—416no—
347D4—432no—
348D4—432no—
349D4—48no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—216no—
353C2×C4—22no—
354C2×C4—28no—
355C2×C4—21no—
356C2×C4—44no—
357C2×C4—48no—
358C2×C4—416no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4C3×#12014yes—
363C2×C4C22×C3×D414yes—
364C2×C4C3×H(5,𝔽2)12yes—
365C8—44no—
366C2×C3—28no—
367C2×C3—42no—
368C2×C3—88no—
369C2×C3#240311yescharacteristic
370C2×C3C2×#44712yes—
371C2×C3#300612yes—
372C2×C3#318611yescharacteristic
373C2×C3#319211yescharacteristic
374C22—24no—
375C22—24no—
376C22—24no—
377C22—24no—
378C22—24no—
379C22—24no—
380C22—22no—
381C22—24no—
382C22—416no—
383C22—41no—
384C22—41no—
385C22—42no—
386C22—48no—
387C22—48no—
388C22—48no—
389C22—832no—
390C22—84no—
391C22—84no—
392C22—84no—
393C22—84no—
394C22C3×#44718yescomplemented
395C22C2×C3×#12012yes—
396C22C3×#53411yescharacteristic
397C22C22×C3×D811yescharacteristic
398C22C2×C3×AGL(1,ℤ/8)12yes—
399C22C3×#57511yescharacteristic
400C4—24no—
401C4—41no—
402C4—48no—
403C4C3×D4214yes—
404C3#2391811yescharacteristic, direct product factor, Sylow
405C2—28no—
406C2—42no—
407C2—88no—
408C2C3×#240311yescharacteristic
409C2C2×C3×#44712yes—
410C2C3×#300612yes—
411C2C3×#318611yescharacteristic
412C2C3×#319211yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
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
1. C22×C3×#121C2semidirect
2. C2×C3×#447C2semidirect
3. C3×#1999C2semidirect
4. C3×#1874C2semidirect
5. C3×#1779C2semidirect
6. #23918C3direct (N × H)
8. C23×C3×D4C22semidirect
10. C22×C3×D8C22semidirect
15. C2×C3×#121C22semidirect
16. C22×C3×C4⋊C4C22semidirect
19. C2×C3×#102C22semidirect
20. C2×C3×C22⋊C8C22semidirect
21. C2×C3×C22⋊C8C22semidirect
22. C3×#340C22semidirect
23. C22×#121C2×C3semidirect
24. C2×#447C2×C3semidirect
25. #1999C2×C3semidirect
26. #1874C2×C3semidirect
27. #1779C2×C3semidirect
55. C3×#121D4semidirect
66. C23×D4C22×C3semidirect
68. C22×D8C22×C3semidirect
73. C2×#121C22×C3semidirect
74. C22×C4⋊C4C22×C3semidirect
77. C2×#102C22×C3semidirect
78. C2×C22⋊C8C22×C3semidirect
79. C2×C22⋊C8C22×C3semidirect
80. #340C22×C3semidirect
93. C24×C3D8semidirect
110. C2×C3×D4C2×D4semidirect
134. C3×C4⋊C4C2×D4semidirect
164. #121C3×D4semidirect
239. C24C3×D8semidirect
256. C2×D4C2×C3×D4semidirect
280. C4⋊C4C2×C3×D4semidirect
303. C22×C3#447semidirect
394. C22C3×#447semidirect
404. C3#23918direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1212
C2×C3×#4472
C3×#19992
C3×#18742
C3×#17792
#239183