Finite Group Databasebeta

← C3×#20724 #10639331

Subgroups — C3×#20724 #10639331

Order 768.

1490 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#518C211yesmaximal, characteristic
2C3×#2312C212yesmaximal, complemented
3C3×#1819C212yesmaximal, complemented
4C3×#1799C214yesmaximal, complemented
5C3×#1781C212yesmaximal
6C3×C4×#99C214yesmaximal, complemented
7#20724C311yesmaximal, characteristic, direct product factor, Sylow
8C23×C3×D4C2211yescharacteristic
9C3×#518—28no—
10C2×C3×C4×D4C412yescomplemented
11C2×C3×C4×D4C412yescomplemented
12C2×C3×C4×D4C2212yes—
13C2×C3×C4×D4C2212yes—
14C2×C3×#117C412yescomplemented
15C22×C3×C22⋊C4C412yescomplemented
16C22×C3×C22⋊C4C2211yescharacteristic
17C22×C3×C22⋊C4C2211yescharacteristic
18C2×C3×#99C2218yes—
19C3×#388—28no—
20C3×#380C2214yescomplemented
21C3×#379C2214yes—
22C3×C4×C22⋊C4C2214yescomplemented
23C3×#342C2218yescomplemented
24C2×#518C2×C311yescharacteristic
25#2312C2×C312yescomplemented
26#1819C2×C312yescomplemented
27#1799C2×C314yescomplemented
28#1781C2×C312yes—
29C4×#99C2×C314yescomplemented
30C25×C3C2×C411yescharacteristic
31C25×C3C2311yescharacteristic
32C22×C3×D4—24no—
33C22×C3×D4—28no—
34C22×C3×D4C2×C412yes—
35C22×C3×D4C2312yes—
36C23×C3×C4C2×C411yescharacteristic
37C23×C3×C4C2×C412yes—
38C23×C3×C4C2×C412yes—
39C23×C3×C4C2×C411yescharacteristic
40C23×C3×C4C2×C411yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C3×C4×D4—28no—
44C3×C4×D4—28no—
45C3×C4×D4—28no—
46C3×C4×D4—28no—
47C3×#117—28no—
48C2×C3×C4⋊C4—24no—
49C2×C3×C4⋊C4C2×C412yes—
50C2×C3×C4⋊C4C2×C412yes—
51C2×C3×C22⋊C4—24no—
52C2×C3×C22⋊C4—28no—
53C2×C3×C22⋊C4—24no—
54C2×C3×C22⋊C4—24no—
55C2×C3×C22⋊C4C2×C412yes—
56C2×C3×C22⋊C4C2×C412yes—
57C2×C3×C22⋊C4C2×C412yes—
58C2×C3×C22⋊C4C2×C412yes—
59C2×C3×C22⋊C4C2×C412yes—
60C2×C3×C22⋊C4D418yes—
61C2×C3×C22⋊C4C2318yes—
62C2×C3×C22⋊C4C2312yes—
63C2×C3×C42—24no—
64C2×C3×C42C2×C412yes—
65C2×C3×C42C2×C412yes—
66C3×#99—216no—
67C3×#95—24no—
68C3×#95—24no—
69C23×D4C22×C311yescharacteristic
70#518—28no—
71C2×C4×D4C3×C412yescomplemented
72C2×C4×D4C3×C412yescomplemented
73C2×C4×D4C22×C312yes—
74C2×C4×D4C22×C312yes—
75C2×#117C3×C412yescomplemented
76C22×C22⋊C4C3×C412yescomplemented
77C22×C22⋊C4C22×C311yescharacteristic
78C22×C22⋊C4C22×C311yescharacteristic
79C2×#99C22×C318yes—
80#388—28no—
81#380C22×C314yescomplemented
82#379C22×C314yes—
83C4×C22⋊C4C22×C314yescomplemented
84#342C22×C318yescomplemented
85C24×C3—22no—
86C24×C3—24no—
87C24×C3—24no—
88C24×C3—81no—
89C24×C3—81no—
90C24×C3—81no—
91C24×C3—81no—
92C24×C3C22×C411yescharacteristic
93C24×C3C22×C411yescharacteristic
94C24×C3C22×C412yes—
95C24×C3C2×D414yes—
96C24×C3C2411yescharacteristic
97C2×C3×D4—232no—
98C2×C3×D4—24no—
99C2×C3×D4—24no—
100C2×C3×D4—24no—
101C2×C3×D4—24no—
102C2×C3×D4—24no—
103C2×C3×D4#4118yes—
104C22×C3×C4—22no—
105C22×C3×C4—24no—
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4—24no—
109C22×C3×C4—24no—
110C22×C3×C4—28no—
111C22×C3×C4—81no—
112C22×C3×C4—81no—
113C22×C3×C4—82no—
114C22×C3×C4—82no—
115C22×C3×C4—81no—
116C22×C3×C4—81no—
117C22×C3×C4C22×C412yes—
118C22×C3×C4C22×C412yes—
119C22×C3×C4C22×C412yes—
120C22×C3×C4C22×C412yes—
121C22×C3×C4C22×C412yes—
122C22×C3×C4C2×D414yes—
123C22×C3×C4C2×D414yes—
124C3×C4⋊C4—24no—
125C3×C4⋊C4—24no—
126C3×C4⋊C4—84no—
127C3×C22⋊C4—24no—
128C3×C22⋊C4—24no—
129C3×C22⋊C4—24no—
130C3×C22⋊C4—24no—
131C3×C22⋊C4—216no—
132C3×C22⋊C4—84no—
133C3×C22⋊C4—84no—
134C3×C22⋊C4C22⋊C418yes—
135C3×C22⋊C4C22⋊C418yes—
136C3×C42—24no—
137C3×C42—24no—
138C3×C42—84no—
139C25C2×C3×C411yescharacteristic
140C25C23×C311yescharacteristic
141C22×D4—24no—
142C22×D4—28no—
143C22×D4C2×C3×C412yes—
144C22×D4C23×C312yes—
145C23×C4C2×C3×C411yescharacteristic
146C23×C4C2×C3×C412yes—
147C23×C4C2×C3×C412yes—
148C23×C4C2×C3×C411yescharacteristic
149C23×C4C2×C3×C411yescharacteristic
150C23×C4C23×C311yescharacteristic
151C23×C4C23×C311yescharacteristic
152C4×D4—28no—
153C4×D4—28no—
154C4×D4—28no—
155C4×D4—28no—
156#117—28no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4C2×C3×C412yes—
159C2×C4⋊C4C2×C3×C412yes—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—28no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4C2×C3×C412yes—
165C2×C22⋊C4C2×C3×C412yes—
166C2×C22⋊C4C2×C3×C412yes—
167C2×C22⋊C4C2×C3×C412yes—
168C2×C22⋊C4C2×C3×C412yes—
169C2×C22⋊C4C3×D418yes—
170C2×C22⋊C4C23×C318yes—
171C2×C22⋊C4C23×C312yes—
172C2×C42—24no—
173C2×C42C2×C3×C412yes—
174C2×C42C2×C3×C412yes—
175#99—216no—
176#95—24no—
177#95—24no—
178C23×C3—28no—
179C23×C3—22no—
180C23×C3—22no—
181C23×C3—22no—
182C23×C3—22no—
183C23×C3—28no—
184C23×C3—41no—
185C23×C3—41no—
186C23×C3—41no—
187C23×C3—41no—
188C23×C3—41no—
189C23×C3—41no—
190C23×C3—44no—
191C23×C3—44no—
192C23×C3—42no—
193C23×C3—42no—
194C23×C3—82no—
195C23×C3—82no—
196C23×C3—88no—
197C23×C3—88no—
198C23×C3C2×C22⋊C412yes—
199C23×C3C2×C22⋊C412yes—
200C23×C3#11714yes—
201C23×C3C23×C411yescharacteristic
202C23×C3C22×D412yes—
203C23×C3C2×#4114yes—
204C3×D4—216no—
205C3×D4—216no—
206C2×C3×C4—28no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—44no—
212C2×C3×C4—44no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—48no—
218C2×C3×C4—48no—
219C2×C3×C4—48no—
220C2×C3×C4—84no—
221C2×C3×C4—84no—
222C2×C3×C4—82no—
223C2×C3×C4—82no—
224C2×C3×C4C2×C22⋊C414yes—
225C2×C3×C4C2×C22⋊C414yes—
226C2×C3×C4#11714yes—
227C24—22no—
228C24—24no—
229C24—24no—
230C24—81no—
231C24—81no—
232C24—81no—
233C24—81no—
234C24C22×C3×C411yescharacteristic
235C24C22×C3×C411yescharacteristic
236C24C22×C3×C412yes—
237C24C2×C3×D414yes—
238C24C24×C311yescharacteristic
239C2×D4—232no—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—24no—
243C2×D4—24no—
244C2×D4—24no—
245C2×D4C3×#4118yes—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—28no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4C22×C3×C412yes—
260C22×C4C22×C3×C412yes—
261C22×C4C22×C3×C412yes—
262C22×C4C22×C3×C412yes—
263C22×C4C22×C3×C412yes—
264C22×C4C2×C3×D414yes—
265C22×C4C2×C3×D414yes—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—84no—
269C22⋊C4—24no—
270C22⋊C4—24no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—216no—
274C22⋊C4—84no—
275C22⋊C4—84no—
276C22⋊C4C3×C22⋊C418yes—
277C22⋊C4C3×C22⋊C418yes—
278C42—24no—
279C42—24no—
280C42—84no—
281C22×C3—22no—
282C22×C3—22no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—22no—
286C22×C3—22no—
287C22×C3—22no—
288C22×C3—22no—
289C22×C3—22no—
290C22×C3—22no—
291C22×C3—24no—
292C22×C3—44no—
293C22×C3—44no—
294C22×C3—44no—
295C22×C3—44no—
296C22×C3—44no—
297C22×C3—42no—
298C22×C3—42no—
299C22×C3—88no—
300C22×C3—88no—
301C22×C3C22×C22⋊C411yescharacteristic
302C22×C3C2×#11712yes—
303C22×C3#52514yes—
304C3×C4—24no—
305C3×C4—24no—
306C3×C4—24no—
307C3×C4—42no—
308C3×C4—42no—
309C3×C4—88no—
310C23—28no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—28no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—44no—
323C23—44no—
324C23—42no—
325C23—42no—
326C23—82no—
327C23—82no—
328C23—88no—
329C23—88no—
330C23C2×C3×C22⋊C412yes—
331C23C2×C3×C22⋊C412yes—
332C23C3×#11714yes—
333C23C23×C3×C411yescharacteristic
334C23C22×C3×D412yes—
335C23C2×C3×#4114yes—
336D4—216no—
337D4—216no—
338C2×C4—28no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—48no—
350C2×C4—48no—
351C2×C4—48no—
352C2×C4—84no—
353C2×C4—84no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4C2×C3×C22⋊C414yes—
357C2×C4C2×C3×C22⋊C414yes—
358C2×C4C3×#11714yes—
359C2×C3—24no—
360C2×C3—22no—
361C2×C3—42no—
362C2×C3—42no—
363C2×C3—44no—
364C2×C3#228011yescharacteristic
365C2×C3#288311yescharacteristic
366C2×C3#288311yescharacteristic
367C22—22no—
368C22—22no—
369C22—21no—
370C22—21no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—24no—
378C22—44no—
379C22—44no—
380C22—44no—
381C22—44no—
382C22—44no—
383C22—42no—
384C22—42no—
385C22—88no—
386C22—88no—
387C22C22×C3×C22⋊C411yescharacteristic
388C22C2×C3×#11712yes—
389C22C3×#52514yes—
390C4—24no—
391C4—24no—
392C4—24no—
393C4—42no—
394C4—42no—
395C4—88no—
396C3#2072411yescharacteristic, direct product factor, Sylow
397C2—24no—
398C2—22no—
399C2—42no—
400C2—42no—
401C2—44no—
402C2C3×#228011yescharacteristic
403C2C3×#288311yescharacteristic
404C2C3×#288311yescharacteristic
subgroupG/H
center, socleC22×C3C22×C22⋊C4
commutatorC23C23×C3×C4
Frattini, cosocleC24C24×C3
prime radicalC23×C3×D4C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2312C2semidirect
3. C3×#1819C2semidirect
4. C3×#1799C2semidirect
6. C3×C4×#99C2semidirect
7. #20724C3direct (N × H)
10. C2×C3×C4×D4C4semidirect
11. C2×C3×C4×D4C4semidirect
14. C2×C3×#117C4semidirect
15. C22×C3×C22⋊C4C4semidirect
20. C3×#380C22semidirect
22. C3×C4×C22⋊C4C22semidirect
23. C3×#342C22semidirect
25. #2312C2×C3semidirect
26. #1819C2×C3semidirect
27. #1799C2×C3semidirect
29. C4×#99C2×C3semidirect
71. C2×C4×D4C3×C4semidirect
72. C2×C4×D4C3×C4semidirect
75. C2×#117C3×C4semidirect
76. C22×C22⋊C4C3×C4semidirect
81. #380C22×C3semidirect
83. C4×C22⋊C4C22×C3semidirect
84. #342C22×C3semidirect
396. C3#20724direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5182
C3×#23122
C3×#18192
C3×#17992
C3×#17812
C3×C4×#992
#207243