Finite Group Databasebeta

← C3×#33036 #10651643

Subgroups — C3×#33036 #10651643

Order 768.

918 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#474C211yesmaximal, characteristic, complemented
2C2×C3×#463C211yesmaximal, characteristic
3C3×#2640C211yesmaximal, characteristic, complemented
4C3×#2275C211yesmaximal, characteristic, complemented
5C3×#1987C211yesmaximal, characteristic, complemented
6C3×#1983C211yesmaximal, characteristic, complemented
7C3×#1928C212yesmaximal, complemented
8C3×#1925C212yesmaximal, complemented
9C3×#1879C212yesmaximal
10C3×#1877C212yesmaximal, complemented
11C3×#1832C211yesmaximal, characteristic, complemented
12#33036C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2211yescharacteristic
14C22×C3×D8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#124C2211yescharacteristic
17C2×C3×C4×D4C2212yes—
18C2×C3×C4×D4C2211yescharacteristic
19C3×#474—24no—
20C3×#463—24no—
21C2×C3×#107C2212yes—
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#103C2212yes—
25C2×C3×#103C2211yescharacteristic
26C2×C3×#102C2212yes—
27C2×C3×#102C2212yes—
28C2×C3×#102C2211yescharacteristic
29C3×#397—22no—
30C3×#393—21no—
31C3×#393—21no—
32C3×#391C2212yes—
33C3×#390C2212yes—
34C3×C4×C4⋊C4C2212yes—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×C4×C22⋊C4—21no—
37C3×C4×C22⋊C4—22no—
38C3×#376C2211yescharacteristic
39C2×C3×#95—21no—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#336C2212yes—
43C3×#336C2212yes—
44C2×#474C2×C311yescharacteristic, complemented
45C2×#463C2×C311yescharacteristic
46#2640C2×C311yescharacteristic, complemented
47#2275C2×C311yescharacteristic, complemented
48#1987C2×C311yescharacteristic, complemented
49#1983C2×C311yescharacteristic, complemented
50#1928C2×C312yescomplemented
51#1925C2×C312yescomplemented
52#1879C2×C312yes—
53#1877C2×C312yescomplemented
54#1832C2×C311yescharacteristic, complemented
55C22×C3×Q8—21no—
56C22×C3×Q8C2311yescharacteristic
57C22×C3×D4C2311yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—22no—
60C23×C3×C4—21no—
61C2×C3×SD16—22no—
62C2×C3×SD16—24no—
63C2×C3×D8—22no—
64C2×C3×D8—24no—
65C22×C3×C8C2311yescharacteristic
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2312yes—
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8—22no—
70C3×#124—22no—
71C3×#124—22no—
72C3×C4×D4—24no—
73C3×C4×D4—24no—
74C3×C4×D4—22no—
75C3×C4×D4D414yes—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C42—22no—
87C2×C3×C42—21no—
88C2×C3×C42C2312yes—
89C2×C3×C42C2311yescharacteristic
90C3×#107—24no—
91C3×C4⋊C8—22no—
92C3×C4⋊C8—22no—
93C3×#103—22no—
94C3×#103—24no—
95C3×#102—24no—
96C3×#102—22no—
97C3×#102—24no—
98C3×#95—22no—
99C3×#95—21no—
100C3×#95—22no—
101C3×#95—22no—
102C3×#95D411yescharacteristic
103C3×#95D411yescharacteristic
104C3×#95D412yes—
105C22×SD16C22×C311yescharacteristic
106C22×D8C22×C311yescharacteristic
107C2×C4⋊Q8C22×C311yescharacteristic
108C2×#124C22×C311yescharacteristic
109C2×C4×D4C22×C312yes—
110C2×C4×D4C22×C311yescharacteristic
111#474—24no—
112#463—24no—
113C2×#107C22×C312yes—
114C2×C4⋊C8C22×C311yescharacteristic
115C2×C4⋊C8C22×C311yescharacteristic
116C2×#103C22×C312yes—
117C2×#103C22×C311yescharacteristic
118C2×#102C22×C312yes—
119C2×#102C22×C312yes—
120C2×#102C22×C311yescharacteristic
121#397—22no—
122#393—21no—
123#393—21no—
124#391C22×C312yes—
125#390C22×C312yes—
126C4×C4⋊C4C22×C312yes—
127C4×C4⋊C4C22×C311yescharacteristic
128C4×C22⋊C4—21no—
129C4×C22⋊C4—22no—
130#376C22×C311yescharacteristic
131C2×#95—21no—
132#340C22×C312yes—
133#340C22×C312yes—
134#336C22×C312yes—
135#336C22×C312yes—
136C24×C3—21no—
137C24×C3—21no—
138C2×C3×Q8—21no—
139C2×C3×Q8—22no—
140C2×C3×Q8—21no—
141C2×C3×Q8—22no—
142C2×C3×Q8—42no—
143C2×C3×Q8—82no—
144C2×C3×D4—21no—
145C2×C3×D4—21no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—21no—
149C2×C3×D4—22no—
150C2×C3×D4C2×D412yes—
151C2×C3×D4#4114yes—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—28no—
155C22×C3×C4—24no—
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4—82no—
159C22×C3×C4—82no—
160C22×C3×C4—82no—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2411yescharacteristic
166C3×SD16—44no—
167C3×D8—44no—
168C2×C3×C8—22no—
169C2×C3×C8—21no—
170C2×C3×C8—21no—
171C2×C3×C8—22no—
172C2×C3×C8—22no—
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—42no—
180C3×C4⋊C4C2×D412yes—
181C3×C4⋊C4#4114yes—
182C3×C22⋊C4—28no—
183C3×C22⋊C4—24no—
184C3×C22⋊C4—42no—
185C3×C22⋊C4—42no—
186C3×C22⋊C4—82no—
187C3×C22⋊C4—82no—
188C3×C42—22no—
189C3×C42—22no—
190C3×C42—81no—
191C3×C42—82no—
192C3×C42C2×D412yes—
193C3×C42#4112yes—
194C22×Q8—21no—
195C22×Q8C23×C311yescharacteristic
196C22×D4C23×C311yescharacteristic
197C22×D4C23×C311yescharacteristic
198C23×C4—22no—
199C23×C4—21no—
200C2×SD16—22no—
201C2×SD16—24no—
202C2×D8—22no—
203C2×D8—24no—
204C22×C8C23×C311yescharacteristic
205C22×C8C23×C311yescharacteristic
206C22×C8C23×C312yes—
207C4⋊Q8—22no—
208C4⋊Q8—22no—
209#124—22no—
210#124—22no—
211C4×D4—24no—
212C4×D4—24no—
213C4×D4—22no—
214C4×D4C3×D414yes—
215C2×C4⋊C4—21no—
216C2×C4⋊C4C23×C312yes—
217C2×C4⋊C4C23×C311yescharacteristic
218C2×C4⋊C4C23×C312yes—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C42—22no—
226C2×C42—21no—
227C2×C42C23×C312yes—
228C2×C42C23×C311yescharacteristic
229#107—24no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232#103—22no—
233#103—24no—
234#102—24no—
235#102—22no—
236#102—24no—
237#95—22no—
238#95—21no—
239#95—22no—
240#95—22no—
241#95C3×D411yescharacteristic
242#95C3×D411yescharacteristic
243#95C3×D412yes—
244C23×C3—22no—
245C23×C3—24no—
246C23×C3—42no—
247C23×C3—41no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3—42no—
251C23×C3—42no—
252C23×C3—41no—
253C23×C3—41no—
254C23×C3C22×D411yescharacteristic
255C3×Q8—41no—
256C3×Q8—42no—
257C3×Q8—41no—
258C3×Q8—84no—
259C3×D4—24no—
260C3×D4—22no—
261C3×D4—41no—
262C3×D4—42no—
263C3×D4—41no—
264C3×D4#12414yes—
265C2×C3×C4—22no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—24no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—41no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—84no—
285C2×C3×C4—88no—
286C2×C3×C4#12112yes—
287C2×C3×C4#12112yes—
288C2×C3×C4C22×D411yescharacteristic
289C2×C3×C4C2×#4112yes—
290C2×C3×C4C2×#4111yescharacteristic
291C2×C3×C4C2×#4112yes—
292C3×C8—41no—
293C3×C8—41no—
294C3×C8—42no—
295C24—21no—
296C24—21no—
297C2×Q8—21no—
298C2×Q8—22no—
299C2×Q8—21no—
300C2×Q8—22no—
301C2×Q8—42no—
302C2×Q8—82no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—21no—
308C2×D4—22no—
309C2×D4C2×C3×D412yes—
310C2×D4C3×#4114yes—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—28no—
314C22×C4—24no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4C2×C3×D412yes—
321C22×C4C2×C3×D412yes—
322C22×C4C2×C3×D411yescharacteristic
323C22×C4C2×C3×D411yescharacteristic
324C22×C4C24×C311yescharacteristic
325SD16—44no—
326D8—44no—
327C2×C8—22no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—42no—
339C4⋊C4C2×C3×D412yes—
340C4⋊C4C3×#4114yes—
341C22⋊C4—28no—
342C22⋊C4—24no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C42—22no—
348C42—22no—
349C42—81no—
350C42—82no—
351C42C2×C3×D412yes—
352C42C3×#4112yes—
353C22×C3—24no—
354C22×C3—44no—
355C22×C3—44no—
356C22×C3—42no—
357C22×C3—42no—
358C22×C3—81no—
359C22×C3—82no—
360C22×C3—82no—
361C22×C3—81no—
362C22×C3—81no—
363C22×C3C2×#12111yescharacteristic
364C22×C3#52512yes—
365C22×C3#57612yes—
366C22×C3#57711yescharacteristic
367C22×C3#57811yescharacteristic
368C3×C4—22no—
369C3×C4—22no—
370C3×C4—42no—
371C3×C4—44no—
372C3×C4—42no—
373C3×C4—42no—
374C3×C4—81no—
375C3×C4—82no—
376C3×C4C2×#12411yescharacteristic
377C3×C4#52912yes—
378C3×C4#53211yescharacteristic
379C23—22no—
380C23—24no—
381C23—42no—
382C23—41no—
383C23—41no—
384C23—42no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—42no—
389C23C22×C3×D411yescharacteristic
390Q8—41no—
391Q8—42no—
392Q8—41no—
393Q8—84no—
394D4—24no—
395D4—22no—
396D4—41no—
397D4—42no—
398D4—41no—
399D4C3×#12414yes—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—84no—
420C2×C4—88no—
421C2×C4C3×#12112yes—
422C2×C4C3×#12112yes—
423C2×C4C22×C3×D411yescharacteristic
424C2×C4C2×C3×#4112yes—
425C2×C4C2×C3×#4111yescharacteristic
426C2×C4C2×C3×#4112yes—
427C8—41no—
428C8—41no—
429C8—42no—
430C2×C3—44no—
431C2×C3—161no—
432C2×C3#285011yescharacteristic
433C2×C3#304111yescharacteristic
434C2×C3#304611yescharacteristic
435C2×C3#311812yes—
436C2×C3#312412yes—
437C22—24no—
438C22—42no—
439C22—42no—
440C22—44no—
441C22—44no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—82no—
446C22—82no—
447C22C2×C3×#12111yescharacteristic
448C22C3×#52512yes—
449C22C3×#57612yes—
450C22C3×#57711yescharacteristic
451C22C3×#57811yescharacteristic
452C4—22no—
453C4—22no—
454C4—42no—
455C4—42no—
456C4—44no—
457C4—42no—
458C4—82no—
459C4—81no—
460C4C2×C3×#12411yescharacteristic
461C4C3×#52912yes—
462C4C3×#53211yescharacteristic
463C3#3303611yescharacteristic, direct product factor, Sylow
464C2—44no—
465C2—161no—
466C2C3×#285011yescharacteristic
467C2C3×#304111yescharacteristic
468C2C3×#304611yescharacteristic
469C2C3×#311812yes—
470C2C3×#312412yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×#376 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#474C2semidirect
3. C3×#2640C2semidirect
4. C3×#2275C2semidirect
5. C3×#1987C2semidirect
6. C3×#1983C2semidirect
7. C3×#1928C2semidirect
8. C3×#1925C2semidirect
10. C3×#1877C2semidirect
11. C3×#1832C2semidirect
12. #33036C3direct (N × H)
44. C2×#474C2×C3semidirect
46. #2640C2×C3semidirect
47. #2275C2×C3semidirect
48. #1987C2×C3semidirect
49. #1983C2×C3semidirect
50. #1928C2×C3semidirect
51. #1925C2×C3semidirect
53. #1877C2×C3semidirect
54. #1832C2×C3semidirect
463. C3#33036direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4742
C2×C3×#4632
C3×#26402
C3×#22752
C3×#19872
C3×#19832
C3×#19282
C3×#19252
C3×#18792
C3×#18772
C3×#18322
#330363