Finite Group Databasebeta

← C3×#32990 #10651597

Subgroups — C3×#32990 #10651597

Order 768.

1574 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#518C211yesmaximal, characteristic, complemented
2C3×#3094C212yesmaximal, complemented
3C3×#2433C211yesmaximal, characteristic
4C3×#1887C214yesmaximal, complemented
5C3×#1856C214yesmaximal, complemented
6C3×#1826C212yesmaximal, complemented
7C3×#1783C211yesmaximal, characteristic, complemented
8#32990C311yesmaximal, characteristic, direct product factor, Sylow
9C23×C3×D4C2211yescharacteristic
10C2×C3×#127C2212yes—
11C2×C3×#121C2211yescharacteristic
12C2×C3×#120—22no—
13C3×#518—22no—
14C3×#518—24no—
15C3×#518C2214yes—
16C2×C3×C4×D4—22no—
17C2×C3×C4×D4—22no—
18C2×C3×#117C2212yes—
19C22×C3×C22⋊C4—22no—
20C3×#482—24no—
21C3×#481—22no—
22C3×#480—22no—
23C3×#428C2214yescomplemented
24C3×#418C2214yescomplemented
25C2×C3×#100C2214yes—
26C3×#407C2212yes—
27C3×#392—22no—
28C3×#390C2212yes—
29C3×C4×C22⋊C4C2211yescharacteristic
30C3×#339C2218yescomplemented
31C2×#518C2×C311yescharacteristic, complemented
32#3094C2×C312yescomplemented
33#2433C2×C311yescharacteristic
34#1887C2×C314yescomplemented
35#1856C2×C314yescomplemented
36#1826C2×C312yescomplemented
37#1783C2×C311yescharacteristic, complemented
38C25×C3D412yes—
39C22×C3×D4—28no—
40C22×C3×D4—21no—
41C22×C3×D4—24no—
42C22×C3×D4—22no—
43C22×C3×D4—22no—
44C22×C3×D4D412yes—
45C22×C3×D4C2311yescharacteristic
46C22×C3×D4C2312yes—
47C23×C3×C4—22no—
48C23×C3×C4—22no—
49C23×C3×C4C2311yescharacteristic
50C2×C3×M16C2314yes—
51C3×#127—28no—
52C3×#121—22no—
53C3×#121—22no—
54C3×#120—48no—
55C3×C4×D4—28no—
56C3×C4×D4—28no—
57C3×C4×D4—28no—
58C3×C4×D4—28no—
59C3×#117—22no—
60C3×#117—22no—
61C3×#117D414yescomplemented
62C3×#117C2314yes—
63C2×C3×C4⋊C4D412yescomplemented
64C2×C3×C4⋊C4D412yescomplemented
65C2×C3×C4⋊C4C2311yescharacteristic
66C2×C3×C22⋊C4—24no—
67C2×C3×C22⋊C4—28no—
68C2×C3×C22⋊C4—21no—
69C2×C3×C22⋊C4—22no—
70C2×C3×C22⋊C4—28no—
71C2×C3×C22⋊C4—24no—
72C2×C3×C22⋊C4—21no—
73C2×C3×C42—22no—
74C2×C3×C42C2312yes—
75C3×#107—24no—
76C3×#106—24no—
77C3×#102—24no—
78C3×#102—24no—
79C3×#100—28no—
80C3×C22⋊C8—22no—
81C3×C22⋊C8—22no—
82C3×#95—21no—
83C23×D4C22×C311yescharacteristic
84C2×#127C22×C312yes—
85C2×#121C22×C311yescharacteristic
86C2×#120—22no—
87#518—22no—
88#518—24no—
89#518C22×C314yes—
90C2×C4×D4—22no—
91C2×C4×D4—22no—
92C2×#117C22×C312yes—
93C22×C22⋊C4—22no—
94#482—24no—
95#481—22no—
96#480—22no—
97#428C22×C314yescomplemented
98#418C22×C314yescomplemented
99C2×#100C22×C314yes—
100#407C22×C312yes—
101#392—22no—
102#390C22×C312yes—
103C4×C22⋊C4C22×C311yescharacteristic
104#339C22×C318yescomplemented
105C24×C3—22no—
106C24×C3—22no—
107C24×C3—28no—
108C24×C3—21no—
109C24×C3—44no—
110C24×C3—82no—
111C24×C3C2×D411yescharacteristic
112C24×C3C2×D414yes—
113C2×C3×D4—216no—
114C2×C3×D4—216no—
115C2×C3×D4—28no—
116C2×C3×D4—24no—
117C2×C3×D4—24no—
118C2×C3×D4—22no—
119C2×C3×D4—24no—
120C2×C3×D4—24no—
121C2×C3×D4—48no—
122C2×C3×D4—44no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C2×C3×D4—48no—
126C2×C3×D4—84no—
127C2×C3×D4#4118yes—
128C22×C3×C4—24no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—28no—
133C22×C3×C4—28no—
134C22×C3×C4—21no—
135C22×C3×C4—84no—
136C22×C3×C4—84no—
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D412yes—
139C22×C3×C4C2×D412yes—
140C22×C3×C4C2411yescharacteristic
141C3×M16—28no—
142C2×C3×C8—24no—
143C3×C4⋊C4—22no—
144C3×C4⋊C4—22no—
145C3×C4⋊C4—22no—
146C3×C4⋊C4C2×D414yes—
147C3×C4⋊C4C2×D414yes—
148C3×C22⋊C4—216no—
149C3×C22⋊C4—24no—
150C3×C22⋊C4—24no—
151C3×C22⋊C4—24no—
152C3×C22⋊C4—48no—
153C3×C22⋊C4—42no—
154C3×C42—24no—
155C3×C42—24no—
156C3×C42—24no—
157C25C3×D412yes—
158C22×D4—28no—
159C22×D4—21no—
160C22×D4—24no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4C3×D412yes—
164C22×D4C23×C311yescharacteristic
165C22×D4C23×C312yes—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4C23×C311yescharacteristic
169C2×M16C23×C314yes—
170#127—28no—
171#121—22no—
172#121—22no—
173#120—48no—
174C4×D4—28no—
175C4×D4—28no—
176C4×D4—28no—
177C4×D4—28no—
178#117—22no—
179#117—22no—
180#117C3×D414yescomplemented
181#117C23×C314yes—
182C2×C4⋊C4C3×D412yescomplemented
183C2×C4⋊C4C3×D412yescomplemented
184C2×C4⋊C4C23×C311yescharacteristic
185C2×C22⋊C4—24no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—21no—
192C2×C42—22no—
193C2×C42C23×C312yes—
194#107—24no—
195#106—24no—
196#102—24no—
197#102—24no—
198#100—28no—
199C22⋊C8—22no—
200C22⋊C8—22no—
201#95—21no—
202C23×C3—28no—
203C23×C3—28no—
204C23×C3—24no—
205C23×C3—22no—
206C23×C3—22no—
207C23×C3—24no—
208C23×C3—41no—
209C23×C3—44no—
210C23×C3—44no—
211C23×C3—44no—
212C23×C3—42no—
213C23×C3—42no—
214C23×C3—44no—
215C23×C3—41no—
216C23×C3—42no—
217C23×C3—82no—
218C23×C3—82no—
219C23×C3—816no—
220C23×C3#12011yescharacteristic
221C23×C3#12011yescharacteristic
222C23×C3#12318yes—
223C23×C3C22×D411yescharacteristic
224C3×D4—216no—
225C3×D4—216no—
226C3×D4—44no—
227C3×D4—816no—
228C3×D4—82no—
229C3×D4—82no—
230C2×C3×C4—22no—
231C2×C3×C4—24no—
232C2×C3×C4—24no—
233C2×C3×C4—24no—
234C2×C3×C4—24no—
235C2×C3×C4—24no—
236C2×C3×C4—44no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—44no—
240C2×C3×C4—42no—
241C2×C3×C4—41no—
242C2×C3×C4—88no—
243C2×C3×C4—88no—
244C2×C3×C4#12012yes—
245C2×C3×C4C22×D412yes—
246C2×C3×C4C2×#4114yes—
247C3×C8—44no—
248C24—22no—
249C24—22no—
250C24—28no—
251C24—21no—
252C24—44no—
253C24—82no—
254C24C2×C3×D411yescharacteristic
255C24C2×C3×D414yes—
256C2×D4—216no—
257C2×D4—216no—
258C2×D4—28no—
259C2×D4—24no—
260C2×D4—24no—
261C2×D4—22no—
262C2×D4—24no—
263C2×D4—24no—
264C2×D4—48no—
265C2×D4—44no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—48no—
269C2×D4—84no—
270C2×D4C3×#4118yes—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—28no—
276C22×C4—28no—
277C22×C4—21no—
278C22×C4—84no—
279C22×C4—84no—
280C22×C4C2×C3×D411yescharacteristic
281C22×C4C2×C3×D412yes—
282C22×C4C2×C3×D412yes—
283C22×C4C24×C311yescharacteristic
284M16—28no—
285C2×C8—24no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4C2×C3×D414yes—
290C4⋊C4C2×C3×D414yes—
291C22⋊C4—216no—
292C22⋊C4—24no—
293C22⋊C4—24no—
294C22⋊C4—24no—
295C22⋊C4—48no—
296C22⋊C4—42no—
297C42—24no—
298C42—24no—
299C42—24no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—21no—
305C22×C3—28no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—44no—
309C22×C3—48no—
310C22×C3—416no—
311C22×C3—44no—
312C22×C3—44no—
313C22×C3—84no—
314C22×C3—84no—
315C22×C3—88no—
316C22×C3—81no—
317C22×C3—81no—
318C22×C3—81no—
319C22×C3#45312yes—
320C22×C3C2×#12011yescharacteristic
321C22×C3C2×#12312yes—
322C3×C4—22no—
323C3×C4—44no—
324C3×C4—44no—
325C3×C4—44no—
326C3×C4—81no—
327C3×C4#52514yes—
328C23—28no—
329C23—28no—
330C23—22no—
331C23—24no—
332C23—22no—
333C23—24no—
334C23—41no—
335C23—44no—
336C23—44no—
337C23—44no—
338C23—42no—
339C23—42no—
340C23—44no—
341C23—41no—
342C23—42no—
343C23—82no—
344C23—82no—
345C23—816no—
346C23C3×#12011yescharacteristic
347C23C3×#12011yescharacteristic
348C23C3×#12318yes—
349C23C22×C3×D411yescharacteristic
350D4—216no—
351D4—216no—
352D4—44no—
353D4—816no—
354D4—82no—
355D4—82no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—88no—
369C2×C4—88no—
370C2×C4C3×#12012yes—
371C2×C4C22×C3×D412yes—
372C2×C4C2×C3×#4114yes—
373C8—44no—
374C2×C3—22no—
375C2×C3—22no—
376C2×C3—41no—
377C2×C3—48no—
378C2×C3—161no—
379C2×C3#284911yescharacteristic
380C2×C3C2×#45311yescharacteristic
381C2×C3#301911yescharacteristic
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—28no—
390C22—44no—
391C22—48no—
392C22—416no—
393C22—44no—
394C22—44no—
395C22—84no—
396C22—84no—
397C22—88no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22C3×#45312yes—
402C22C2×C3×#12011yescharacteristic
403C22C2×C3×#12312yes—
404C4—22no—
405C4—44no—
406C4—44no—
407C4—44no—
408C4—81no—
409C4C3×#52514yes—
410C3#3299011yescharacteristic, direct product factor, Sylow
411C2—22no—
412C2—22no—
413C2—41no—
414C2—48no—
415C2—161no—
416C2C3×#284911yescharacteristic
417C2C2×C3×#45311yescharacteristic
418C2C3×#301911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2433C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#518C2semidirect
2. C3×#3094C2semidirect
4. C3×#1887C2semidirect
5. C3×#1856C2semidirect
6. C3×#1826C2semidirect
7. C3×#1783C2semidirect
8. #32990C3direct (N × H)
23. C3×#428C22semidirect
24. C3×#418C22semidirect
30. C3×#339C22semidirect
31. C2×#518C2×C3semidirect
32. #3094C2×C3semidirect
34. #1887C2×C3semidirect
35. #1856C2×C3semidirect
36. #1826C2×C3semidirect
37. #1783C2×C3semidirect
61. C3×#117D4semidirect
63. C2×C3×C4⋊C4D4semidirect
64. C2×C3×C4⋊C4D4semidirect
97. #428C22×C3semidirect
98. #418C22×C3semidirect
104. #339C22×C3semidirect
180. #117C3×D4semidirect
182. C2×C4⋊C4C3×D4semidirect
183. C2×C4⋊C4C3×D4semidirect
410. C3#32990direct (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×#30942
C3×#24332
C3×#18872
C3×#18562
C3×#18262
C3×#17832
#329903