Finite Group Databasebeta

← C3×#33039 #10651646

Subgroups — C3×#33039 #10651646

Order 768.

884 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#479C211yesmaximal, characteristic, complemented
2C2×C3×#460C211yesmaximal, characteristic
3C3×#2641C211yesmaximal, characteristic
4C3×#2276C211yesmaximal, characteristic, complemented
5C3×#1989C211yesmaximal, characteristic, complemented
6C3×#1982C211yesmaximal, characteristic
7C3×#1926C212yesmaximal, complemented
8C3×#1925C212yesmaximal, complemented
9C3×#1880C212yesmaximal, complemented
10C3×#1880C212yesmaximal, complemented
11C3×#1832C211yesmaximal, characteristic, complemented
12#33039C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2212yes—
14C2×C3×#127C2211yescharacteristic
15C2×C3×#125C2211yescharacteristic
16C2×C3×C4×Q8C2212yes—
17C2×C3×C4×Q8C2211yescharacteristic
18C3×#479—24no—
19C3×#460—24no—
20C2×C3×#107C2211yescharacteristic
21C2×C3×#106C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#102C2212yes—
27C2×C3×#102C2212yes—
28C2×C3×#102C2212yes—
29C3×#394—21no—
30C3×#394—22no—
31C3×#394—21no—
32C3×#390C2212yes—
33C3×#390C2212yes—
34C3×C4×C4⋊C4—22no—
35C3×C4×C4⋊C4—21no—
36C3×C4×C4⋊C4C2211yescharacteristic
37C3×C4×C4⋊C4C2212yes—
38C3×#376—21no—
39C3×#376C2211yescharacteristic
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#336C2212yes—
43C3×#336C2212yes—
44C2×#479C2×C311yescharacteristic, complemented
45C2×#460C2×C311yescharacteristic
46#2641C2×C311yescharacteristic
47#2276C2×C311yescharacteristic, complemented
48#1989C2×C311yescharacteristic, complemented
49#1982C2×C311yescharacteristic
50#1926C2×C312yescomplemented
51#1925C2×C312yescomplemented
52#1880C2×C312yescomplemented
53#1880C2×C312yescomplemented
54#1832C2×C311yescharacteristic, complemented
55C22×C3×Q8C2311yescharacteristic
56C22×C3×D4—21no—
57C22×C3×D4—21no—
58C22×C3×D4C2312yes—
59C2×C3×SD16—24no—
60C2×C3×SD16—24no—
61C2×C3×SD16—24no—
62C22×C3×C8C2312yes—
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C3×#127—22no—
66C3×#127—22no—
67C3×#125—22no—
68C3×#125—22no—
69C3×C4×Q8—24no—
70C3×C4×Q8—24no—
71C3×C4×Q8—22no—
72C3×C4×Q8D414yes—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C42—22no—
86C2×C3×C42—21no—
87C2×C3×C42—22no—
88C2×C3×C42—21no—
89C2×C3×C42C2312yes—
90C2×C3×C42C2311yescharacteristic
91C3×#107—22no—
92C3×#106—22no—
93C3×C4⋊C8—22no—
94C3×C4⋊C8—22no—
95C3×#103—22no—
96C3×#103—22no—
97C3×#102—24no—
98C3×#102—24no—
99C3×#102—24no—
100C3×#95—22no—
101C3×#95—21no—
102C3×#95D411yescharacteristic
103C3×#95D412yes—
104C3×#95D411yescharacteristic
105C22×SD16C22×C312yes—
106C2×#127C22×C311yescharacteristic
107C2×#125C22×C311yescharacteristic
108C2×C4×Q8C22×C312yes—
109C2×C4×Q8C22×C311yescharacteristic
110#479—24no—
111#460—24no—
112C2×#107C22×C311yescharacteristic
113C2×#106C22×C311yescharacteristic
114C2×C4⋊C8C22×C311yescharacteristic
115C2×C4⋊C8C22×C311yescharacteristic
116C2×#103C22×C311yescharacteristic
117C2×#103C22×C311yescharacteristic
118C2×#102C22×C312yes—
119C2×#102C22×C312yes—
120C2×#102C22×C312yes—
121#394—21no—
122#394—22no—
123#394—21no—
124#390C22×C312yes—
125#390C22×C312yes—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4C22×C312yes—
129C4×C4⋊C4C22×C311yescharacteristic
130#376—21no—
131#376C22×C311yescharacteristic
132#340C22×C312yes—
133#340C22×C312yes—
134#336C22×C312yes—
135#336C22×C312yes—
136C24×C3—22no—
137C2×C3×Q8—21no—
138C2×C3×Q8—22no—
139C2×C3×Q8C2×D412yes—
140C2×C3×Q8#4114yes—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4—42no—
148C2×C3×D4—42no—
149C2×C3×D4—81no—
150C2×C3×D4—82no—
151C2×C3×D4—81no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2411yescharacteristic
162C3×SD16—48no—
163C2×C3×C8—22no—
164C2×C3×C8—21no—
165C2×C3×C8—21no—
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C2×C3×C8—22no—
169C2×C3×C8—22no—
170C3×C4⋊C4—28no—
171C3×C4⋊C4—24no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—42no—
179C3×C4⋊C4—42no—
180C3×C4⋊C4C2×D412yes—
181C3×C4⋊C4#4114yes—
182C3×C22⋊C4—82no—
183C3×C22⋊C4—82no—
184C3×C22⋊C4—82no—
185C3×C22⋊C4—82no—
186C3×C42—28no—
187C3×C42—24no—
188C3×C42—22no—
189C3×C42—22no—
190C3×C42—81no—
191C3×C42—82no—
192C3×C42C2×D412yes—
193C3×C42#4112yes—
194C22×Q8C23×C311yescharacteristic
195C22×D4—21no—
196C22×D4—21no—
197C22×D4C23×C312yes—
198C2×SD16—24no—
199C2×SD16—24no—
200C2×SD16—24no—
201C22×C8C23×C312yes—
202C22×C8C23×C311yescharacteristic
203C22×C8C23×C311yescharacteristic
204#127—22no—
205#127—22no—
206#125—22no—
207#125—22no—
208C4×Q8—24no—
209C4×Q8—24no—
210C4×Q8—22no—
211C4×Q8C3×D414yes—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4C23×C311yescharacteristic
217C2×C4⋊C4C23×C311yescharacteristic
218C2×C4⋊C4C23×C311yescharacteristic
219C2×C4⋊C4C23×C312yes—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C42—22no—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42—21no—
228C2×C42C23×C312yes—
229C2×C42C23×C311yescharacteristic
230#107—22no—
231#106—22no—
232C4⋊C8—22no—
233C4⋊C8—22no—
234#103—22no—
235#103—22no—
236#102—24no—
237#102—24no—
238#102—24no—
239#95—22no—
240#95—21no—
241#95C3×D411yescharacteristic
242#95C3×D412yes—
243#95C3×D411yescharacteristic
244C23×C3—42no—
245C23×C3—42no—
246C23×C3—42no—
247C23×C3—42no—
248C23×C3—42no—
249C23×C3—42no—
250C23×C3—42no—
251C23×C3C22×D411yescharacteristic
252C3×Q8—24no—
253C3×Q8—22no—
254C3×Q8#12414yes—
255C3×D4—42no—
256C3×D4—42no—
257C3×D4—42no—
258C3×D4—42no—
259C3×D4—84no—
260C3×D4—84no—
261C2×C3×C4—22no—
262C2×C3×C4—24no—
263C2×C3×C4—21no—
264C2×C3×C4—22no—
265C2×C3×C4—21no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—24no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—42no—
275C2×C3×C4—41no—
276C2×C3×C4—42no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—42no—
282C2×C3×C4—41no—
283C2×C3×C4#12112yes—
284C2×C3×C4#12112yes—
285C2×C3×C4C22×D411yescharacteristic
286C2×C3×C4C2×#4112yes—
287C2×C3×C4C2×#4111yescharacteristic
288C2×C3×C4C2×#4112yes—
289C3×C8—42no—
290C3×C8—41no—
291C3×C8—41no—
292C24—22no—
293C2×Q8—21no—
294C2×Q8—22no—
295C2×Q8C2×C3×D412yes—
296C2×Q8C3×#4114yes—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—81no—
306C2×D4—82no—
307C2×D4—81no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4C2×C3×D412yes—
314C22×C4C2×C3×D412yes—
315C22×C4C2×C3×D411yescharacteristic
316C22×C4C2×C3×D411yescharacteristic
317C22×C4C24×C311yescharacteristic
318SD16—48no—
319C2×C8—22no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C4⋊C4—28no—
327C4⋊C4—24no—
328C4⋊C4—21no—
329C4⋊C4—22no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4C2×C3×D412yes—
337C4⋊C4C3×#4114yes—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C42—28no—
343C42—24no—
344C42—22no—
345C42—22no—
346C42—81no—
347C42—82no—
348C42C2×C3×D412yes—
349C42C3×#4112yes—
350C22×C3—82no—
351C22×C3—82no—
352C22×C3—82no—
353C22×C3—82no—
354C22×C3—82no—
355C22×C3—82no—
356C22×C3—82no—
357C22×C3C2×#12111yescharacteristic
358C22×C3#52512yes—
359C22×C3#57611yescharacteristic
360C22×C3#57712yes—
361C22×C3#57711yescharacteristic
362C3×C4—24no—
363C3×C4—22no—
364C3×C4—22no—
365C3×C4—42no—
366C3×C4—44no—
367C3×C4—42no—
368C3×C4—42no—
369C3×C4—81no—
370C3×C4—81no—
371C3×C4C2×#12411yescharacteristic
372C3×C4#52912yes—
373C3×C4#53211yescharacteristic
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23C22×C3×D411yescharacteristic
382Q8—24no—
383Q8—22no—
384Q8C3×#12414yes—
385D4—42no—
386D4—42no—
387D4—42no—
388D4—42no—
389D4—84no—
390D4—84no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4C3×#12112yes—
414C2×C4C3×#12112yes—
415C2×C4C22×C3×D411yescharacteristic
416C2×C4C2×C3×#4112yes—
417C2×C4C2×C3×#4111yescharacteristic
418C2×C4C2×C3×#4112yes—
419C8—42no—
420C8—41no—
421C8—41no—
422C2×C3—162no—
423C2×C3#285011yescharacteristic
424C2×C3#304311yescharacteristic
425C2×C3#304511yescharacteristic
426C2×C3#311912yes—
427C2×C3#312612yes—
428C22—82no—
429C22—82no—
430C22—82no—
431C22—82no—
432C22—82no—
433C22—82no—
434C22—82no—
435C22C2×C3×#12111yescharacteristic
436C22C3×#52512yes—
437C22C3×#57611yescharacteristic
438C22C3×#57712yes—
439C22C3×#57711yescharacteristic
440C4—22no—
441C4—22no—
442C4—24no—
443C4—42no—
444C4—42no—
445C4—44no—
446C4—42no—
447C4—81no—
448C4—81no—
449C4C2×C3×#12411yescharacteristic
450C4C3×#52912yes—
451C4C3×#53211yescharacteristic
452C3#3303911yescharacteristic, direct product factor, Sylow
453C2—162no—
454C2C3×#285011yescharacteristic
455C2C3×#304311yescharacteristic
456C2C3×#304511yescharacteristic
457C2C3×#311912yes—
458C2C3×#312612yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#127C22
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×#479C2semidirect
4. C3×#2276C2semidirect
5. C3×#1989C2semidirect
7. C3×#1926C2semidirect
8. C3×#1925C2semidirect
9. C3×#1880C2semidirect
10. C3×#1880C2semidirect
11. C3×#1832C2semidirect
12. #33039C3direct (N × H)
44. C2×#479C2×C3semidirect
47. #2276C2×C3semidirect
48. #1989C2×C3semidirect
50. #1926C2×C3semidirect
51. #1925C2×C3semidirect
52. #1880C2×C3semidirect
53. #1880C2×C3semidirect
54. #1832C2×C3semidirect
452. C3#33039direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4792
C2×C3×#4602
C3×#26412
C3×#22762
C3×#19892
C3×#19822
C3×#19262
C3×#19252
C3×#18802
C3×#18322
#330393