Finite Group Databasebeta

← C3×#18318 #10636925

Subgroups — C3×#18318 #10636925

Order 768.

1172 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#543C211yesmaximal, characteristic, complemented
2C3×#3193C212yesmaximal, complemented
3C3×#2539C211yesmaximal, characteristic
4C3×#1614C214yesmaximal, complemented
5C3×#1596C214yesmaximal, complemented
6C3×#1508C212yesmaximal, complemented
7C3×#1462C211yesmaximal, characteristic, complemented
8#18318C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#543—24no—
10C3×#543—22no—
11C3×#543C2214yes—
12C2×C3×#123—21no—
13C2×C3×C22⋊Q8—22no—
14C2×C3×C22⋊Q8—21no—
15C2×C3×C22⋊Q8—22no—
16C2×C3×C22⋊Q8C2212yes—
17C2×C3×#121C2211yescharacteristic
18C2×C3×#120—21no—
19C22×C3×C22⋊C4—21no—
20C22×C3×C22⋊C4C2211yescharacteristic
21C3×#484C2214yescomplemented
22C3×#480—22no—
23C3×#468—24no—
24C3×#450C2214yescomplemented
25C3×#448—22no—
26C3×#407C2212yes—
27C3×#399C2212yes—
28C3×#392C2212yes—
29C3×#390—22no—
30C3×C4×C22⋊C4C2211yescharacteristic
31C3×#328C2218yescomplemented
32C3×#323C2214yes—
33C2×#543C2×C311yescharacteristic, complemented
34#3193C2×C312yescomplemented
35#2539C2×C311yescharacteristic
36#1614C2×C314yescomplemented
37#1596C2×C314yescomplemented
38#1508C2×C312yescomplemented
39#1462C2×C311yescharacteristic, complemented
40C25×C3D411yescharacteristic
41C22×C3×Q8—21no—
42C22×C3×Q8D412yescomplemented
43C22×C3×D4—21no—
44C22×C3×D4—22no—
45C22×C3×D4C2311yescharacteristic
46C23×C3×C4—21no—
47C23×C3×C4—21no—
48C23×C3×C4D411yescharacteristic
49C23×C3×C4C2311yescharacteristic
50C2×C3×SD16—24no—
51C2×C3×M16—22no—
52C3×#123—44no—
53C3×C22⋊Q8—28no—
54C3×C22⋊Q8—28no—
55C3×C22⋊Q8—24no—
56C3×C22⋊Q8—28no—
57C3×C22⋊Q8—24no—
58C3×C22⋊Q8—22no—
59C3×C22⋊Q8—22no—
60C3×C22⋊Q8—28no—
61C3×C22⋊Q8D414yescomplemented
62C3×C22⋊Q8C2314yes—
63C3×#121—22no—
64C3×#121—22no—
65C3×#120—44no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4D412yescomplemented
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4—24no—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—24no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4D412yes—
83C2×C3×C22⋊C4C2312yes—
84C2×C3×C42—21no—
85C3×#107—24no—
86C3×#103—24no—
87C3×#102—24no—
88C3×C22⋊C8—22no—
89C3×C22⋊C8C2314yes—
90C3×#95C2312yes—
91#543—24no—
92#543—22no—
93#543C22×C314yes—
94C2×#123—21no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8C22×C312yes—
99C2×#121C22×C311yescharacteristic
100C2×#120—21no—
101C22×C22⋊C4—21no—
102C22×C22⋊C4C22×C311yescharacteristic
103#484C22×C314yescomplemented
104#480—22no—
105#468—24no—
106#450C22×C314yescomplemented
107#448—22no—
108#407C22×C312yes—
109#399C22×C312yes—
110#392C22×C312yes—
111#390—22no—
112C4×C22⋊C4C22×C311yescharacteristic
113#328C22×C318yescomplemented
114#323C22×C314yes—
115C24×C3—21no—
116C24×C3—21no—
117C24×C3—24no—
118C24×C3—21no—
119C24×C3—42no—
120C24×C3—81no—
121C24×C3C2×D411yescharacteristic
122C24×C3C2×D412yes—
123C2×C3×Q8—24no—
124C2×C3×Q8—22no—
125C2×C3×Q8—82no—
126C2×C3×Q8—82no—
127C2×C3×Q8C2×D414yes—
128C2×C3×D4—22no—
129C2×C3×D4—44no—
130C2×C3×D4—42no—
131C2×C3×D4—42no—
132C2×C3×D4—41no—
133C2×C3×D4—41no—
134C2×C3×D4—44no—
135C2×C3×D4—81no—
136C2×C3×D4—82no—
137C22×C3×C4—22no—
138C22×C3×C4—21no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—24no—
142C22×C3×C4—24no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
145C22×C3×C4—42no—
146C22×C3×C4—81no—
147C22×C3×C4—82no—
148C22×C3×C4—82no—
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2411yescharacteristic
154C3×SD16—84no—
155C3×M16—82no—
156C2×C3×C8—24no—
157C3×C4⋊C4—24no—
158C3×C4⋊C4—28no—
159C3×C4⋊C4—24no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—24no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—44no—
164C3×C4⋊C4C2×D414yes—
165C3×C22⋊C4—28no—
166C3×C22⋊C4—24no—
167C3×C22⋊C4—24no—
168C3×C22⋊C4—24no—
169C3×C22⋊C4—24no—
170C3×C22⋊C4—24no—
171C3×C22⋊C4—44no—
172C3×C22⋊C4—44no—
173C3×C22⋊C4—42no—
174C3×C42—81no—
175C25C3×D411yescharacteristic
176C22×Q8—21no—
177C22×Q8C3×D412yescomplemented
178C22×D4—21no—
179C22×D4—22no—
180C22×D4C23×C311yescharacteristic
181C23×C4—21no—
182C23×C4—21no—
183C23×C4C3×D411yescharacteristic
184C23×C4C23×C311yescharacteristic
185C2×SD16—24no—
186C2×M16—22no—
187#123—44no—
188C22⋊Q8—28no—
189C22⋊Q8—28no—
190C22⋊Q8—24no—
191C22⋊Q8—28no—
192C22⋊Q8—24no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195C22⋊Q8—28no—
196C22⋊Q8C3×D414yescomplemented
197C22⋊Q8C23×C314yes—
198#121—22no—
199#121—22no—
200#120—44no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4C3×D412yescomplemented
206C2×C4⋊C4C23×C311yescharacteristic
207C2×C22⋊C4—22no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4C3×D412yes—
218C2×C22⋊C4C23×C312yes—
219C2×C42—21no—
220#107—24no—
221#103—24no—
222#102—24no—
223C22⋊C8—22no—
224C22⋊C8C23×C314yes—
225#95C23×C312yes—
226C23×C3—24no—
227C23×C3—24no—
228C23×C3—22no—
229C23×C3—22no—
230C23×C3—22no—
231C23×C3—22no—
232C23×C3—41no—
233C23×C3—42no—
234C23×C3—42no—
235C23×C3—42no—
236C23×C3—41no—
237C23×C3—41no—
238C23×C3—42no—
239C23×C3—42no—
240C23×C3—41no—
241C23×C3—42no—
242C23×C3—81no—
243C23×C3—81no—
244C23×C3—88no—
245C23×C3#12011yescharacteristic
246C23×C3#12011yescharacteristic
247C23×C3#13714yes—
248C23×C3C22×D411yescharacteristic
249C3×Q8—44no—
250C3×Q8—82no—
251C3×Q8—84no—
252C3×D4—44no—
253C3×D4—88no—
254C3×D4—82no—
255C3×D4—82no—
256C2×C3×C4—24no—
257C2×C3×C4—24no—
258C2×C3×C4—22no—
259C2×C3×C4—24no—
260C2×C3×C4—41no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—41no—
264C2×C3×C4—42no—
265C2×C3×C4—44no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—44no—
269C2×C3×C4—42no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—81no—
274C2×C3×C4—84no—
275C2×C3×C4—84no—
276C2×C3×C4#12012yes—
277C2×C3×C4C22×D412yes—
278C3×C8—44no—
279C24—21no—
280C24—21no—
281C24—24no—
282C24—21no—
283C24—42no—
284C24—81no—
285C24C2×C3×D411yescharacteristic
286C24C2×C3×D412yes—
287C2×Q8—24no—
288C2×Q8—22no—
289C2×Q8—82no—
290C2×Q8—82no—
291C2×Q8C2×C3×D414yes—
292C2×D4—22no—
293C2×D4—44no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—44no—
299C2×D4—81no—
300C2×D4—82no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—42no—
310C22×C4—81no—
311C22×C4—82no—
312C22×C4—82no—
313C22×C4C2×C3×D412yes—
314C22×C4C2×C3×D411yescharacteristic
315C22×C4C2×C3×D412yes—
316C22×C4C2×C3×D412yes—
317C22×C4C24×C311yescharacteristic
318SD16—84no—
319M16—82no—
320C2×C8—24no—
321C4⋊C4—24no—
322C4⋊C4—28no—
323C4⋊C4—24no—
324C4⋊C4—22no—
325C4⋊C4—24no—
326C4⋊C4—22no—
327C4⋊C4—44no—
328C4⋊C4C2×C3×D414yes—
329C22⋊C4—28no—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C22⋊C4—24no—
333C22⋊C4—24no—
334C22⋊C4—24no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—42no—
338C42—81no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—24no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—44no—
348C22×C3—44no—
349C22×C3—48no—
350C22×C3—42no—
351C22×C3—42no—
352C22×C3—82no—
353C22×C3—82no—
354C22×C3—82no—
355C22×C3—84no—
356C22×C3—81no—
357C22×C3—81no—
358C22×C3—81no—
359C22×C3#45312yes—
360C22×C3C2×#12011yescharacteristic
361C22×C3C2×#13712yes—
362C3×C4—22no—
363C3×C4—22no—
364C3×C4—44no—
365C3×C4—42no—
366C3×C4—44no—
367C3×C4—44no—
368C3×C4—81no—
369C23—24no—
370C23—24no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—41no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—41no—
380C23—41no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—41no—
385C23—81no—
386C23—81no—
387C23—88no—
388C23C3×#12011yescharacteristic
389C23C3×#12011yescharacteristic
390C23C3×#13714yes—
391C23C22×C3×D411yescharacteristic
392Q8—44no—
393Q8—82no—
394Q8—84no—
395D4—44no—
396D4—88no—
397D4—82no—
398D4—82no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—81no—
417C2×C4—84no—
418C2×C4—84no—
419C2×C4C3×#12012yes—
420C2×C4C22×C3×D412yes—
421C8—44no—
422C2×C3—22no—
423C2×C3—22no—
424C2×C3—41no—
425C2×C3—44no—
426C2×C3—161no—
427C2×C3#300411yescharacteristic
428C2×C3C2×#45311yescharacteristic
429C2×C3#302611yescharacteristic
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—24no—
436C22—21no—
437C22—21no—
438C22—44no—
439C22—44no—
440C22—48no—
441C22—42no—
442C22—42no—
443C22—82no—
444C22—82no—
445C22—82no—
446C22—84no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22C3×#45312yes—
451C22C2×C3×#12011yescharacteristic
452C22C2×C3×#13712yes—
453C4—22no—
454C4—22no—
455C4—44no—
456C4—42no—
457C4—44no—
458C4—44no—
459C4—81no—
460C3#1831811yescharacteristic, direct product factor, Sylow
461C2—22no—
462C2—22no—
463C2—41no—
464C2—44no—
465C2—161no—
466C2C3×#300411yescharacteristic
467C2C2×C3×#45311yescharacteristic
468C2C3×#302611yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2539C2
LCSG ▷ C22×C4 ▷ C22 ▷ 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×#543C2semidirect
2. C3×#3193C2semidirect
4. C3×#1614C2semidirect
5. C3×#1596C2semidirect
6. C3×#1508C2semidirect
7. C3×#1462C2semidirect
8. #18318C3direct (N × H)
21. C3×#484C22semidirect
24. C3×#450C22semidirect
31. C3×#328C22semidirect
33. C2×#543C2×C3semidirect
34. #3193C2×C3semidirect
36. #1614C2×C3semidirect
37. #1596C2×C3semidirect
38. #1508C2×C3semidirect
39. #1462C2×C3semidirect
42. C22×C3×Q8D4semidirect
61. C3×C22⋊Q8D4semidirect
70. C2×C3×C4⋊C4D4semidirect
103. #484C22×C3semidirect
106. #450C22×C3semidirect
113. #328C22×C3semidirect
177. C22×Q8C3×D4semidirect
196. C22⋊Q8C3×D4semidirect
205. C2×C4⋊C4C3×D4semidirect
460. C3#18318direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5432
C3×#31932
C3×#25392
C3×#16142
C3×#15962
C3×#15082
C3×#14622
#183183