Finite Group Databasebeta

← C3×#29802 #10648409

Subgroups — C3×#29802 #10648409

Order 768.

1002 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#463C211yesmaximal, characteristic, complemented
2C2×C3×#459C211yesmaximal, characteristic, complemented
3C3×#2657C211yesmaximal, characteristic, complemented
4C3×#2319C211yesmaximal, characteristic, complemented
5C3×#2042C212yesmaximal, complemented
6C3×#2041C212yesmaximal, complemented
7C3×#2012C212yesmaximal, complemented
8C3×#2011C212yesmaximal, complemented
9C3×#1983C211yesmaximal, characteristic, complemented
10C3×#1982C211yesmaximal, characteristic, complemented
11C3×#1832C211yesmaximal, characteristic, complemented
12#29802C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2211yescharacteristic
14C22×C3×D8C2211yescharacteristic
15C22×C3×D8C2212yes—
16C2×C3×#127C2211yescharacteristic
17C2×C3×#127C2212yes—
18C2×C3×#124C2211yescharacteristic
19C2×C3×#124C2212yes—
20C2×C3×C4×D4C2211yescharacteristic
21C3×#463—24no—
22C3×#459—24no—
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2212yes—
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2212yes—
29C2×C3×#102C2212yes—
30C2×C3×#102C2212yes—
31C2×C3×#102C2212yes—
32C3×#392—21no—
33C3×#392—21no—
34C3×#392—22no—
35C3×#390C2212yes—
36C3×#389C2212yes—
37C3×#388—22no—
38C3×#381—21no—
39C3×#380—21no—
40C3×C4×C4⋊C4C2211yescharacteristic
41C3×#376C2211yescharacteristic
42C3×#336C2212yes—
43C3×#336C2212yes—
44C2×#463C2×C311yescharacteristic, complemented
45C2×#459C2×C311yescharacteristic, complemented
46#2657C2×C311yescharacteristic, complemented
47#2319C2×C311yescharacteristic, complemented
48#2042C2×C312yescomplemented
49#2041C2×C312yescomplemented
50#2012C2×C312yescomplemented
51#2011C2×C312yescomplemented
52#1983C2×C311yescharacteristic, complemented
53#1982C2×C311yescharacteristic, complemented
54#1832C2×C311yescharacteristic, complemented
55C22×C3×Q8C2311yescharacteristic
56C22×C3×D4—22no—
57C22×C3×D4—22no—
58C22×C3×D4—21no—
59C22×C3×D4—21no—
60C22×C3×D4C2311yescharacteristic
61C22×C3×D4C2312yes—
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—21no—
64C2×C3×SD16—22no—
65C2×C3×SD16—42no—
66C2×C3×D8—24no—
67C2×C3×D8—22no—
68C2×C3×D8—42no—
69C2×C3×D8—42no—
70C2×C3×D8—42no—
71C22×C3×C8C2312yes—
72C22×C3×C8C2311yescharacteristic
73C22×C3×C8C2311yescharacteristic
74C3×#127—22no—
75C3×#127—22no—
76C3×#127—42no—
77C3×#127—42no—
78C3×#124—22no—
79C3×#124—22no—
80C3×#124—42no—
81C3×#124—42no—
82C3×C4×D4—22no—
83C3×C4×D4D414yescomplemented
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4C2312yes—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—22no—
93C2×C3×C22⋊C4—22no—
94C2×C3×C42—21no—
95C2×C3×C42C2312yes—
96C2×C3×C42C2311yescharacteristic
97C3×C4⋊C8—22no—
98C3×C4⋊C8—22no—
99C3×#103—22no—
100C3×#103—42no—
101C3×#102—24no—
102C3×#102—22no—
103C3×#102—42no—
104C3×#102—42no—
105C3×#102—42no—
106C3×#95—21no—
107C3×#95—22no—
108C3×#95D411yescharacteristic
109C3×#95D411yescharacteristic
110C3×#95D412yes—
111C22×SD16C22×C311yescharacteristic
112C22×D8C22×C312yes—
113C22×D8C22×C311yescharacteristic
114C2×#127C22×C311yescharacteristic
115C2×#127C22×C312yes—
116C2×#124C22×C311yescharacteristic
117C2×#124C22×C312yes—
118C2×C4×D4C22×C311yescharacteristic
119#463—24no—
120#459—24no—
121C2×C4⋊C8C22×C311yescharacteristic
122C2×C4⋊C8C22×C311yescharacteristic
123C2×#103C22×C311yescharacteristic
124C2×#103C22×C312yes—
125C2×#102C22×C312yes—
126C2×#102C22×C311yescharacteristic
127C2×#102C22×C312yes—
128C2×#102C22×C312yes—
129C2×#102C22×C312yes—
130#392—21no—
131#392—21no—
132#392—22no—
133#390C22×C312yes—
134#389C22×C312yes—
135#388—22no—
136#381—21no—
137#380—21no—
138C4×C4⋊C4C22×C311yescharacteristic
139#376C22×C311yescharacteristic
140#336C22×C312yes—
141#336C22×C312yes—
142C24×C3—21no—
143C24×C3—21no—
144C24×C3—22no—
145C2×C3×Q8—21no—
146C2×C3×Q8—21no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×D4—21no—
150C2×C3×D4—22no—
151C2×C3×D4—21no—
152C2×C3×D4—21no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—44no—
163C2×C3×D4—44no—
164C2×C3×D4—42no—
165C2×C3×D4—42no—
166C2×C3×D4—81no—
167C2×C3×D4—82no—
168C2×C3×D4—81no—
169C2×C3×D4—82no—
170C2×C3×D4—82no—
171C2×C3×D4—82no—
172C2×C3×D4—82no—
173C2×C3×D4C2×D412yes—
174C22×C3×C4—21no—
175C22×C3×C4—24no—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4—82no—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2411yescharacteristic
184C3×SD16—44no—
185C3×D8—48no—
186C3×D8—44no—
187C2×C3×C8—22no—
188C2×C3×C8—21no—
189C2×C3×C8—21no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—82no—
198C3×C4⋊C4C2×D412yes—
199C3×C22⋊C4—24no—
200C3×C22⋊C4—44no—
201C3×C22⋊C4—44no—
202C3×C22⋊C4—42no—
203C3×C22⋊C4—42no—
204C3×C22⋊C4—82no—
205C3×C22⋊C4—82no—
206C3×C42—22no—
207C3×C42—22no—
208C3×C42—81no—
209C3×C42C2×D412yes—
210C3×C42#4112yes—
211C22×Q8C23×C311yescharacteristic
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—21no—
215C22×D4—21no—
216C22×D4C23×C311yescharacteristic
217C22×D4C23×C312yes—
218C22×D4C23×C311yescharacteristic
219C23×C4—21no—
220C2×SD16—22no—
221C2×SD16—42no—
222C2×D8—24no—
223C2×D8—22no—
224C2×D8—42no—
225C2×D8—42no—
226C2×D8—42no—
227C22×C8C23×C312yes—
228C22×C8C23×C311yescharacteristic
229C22×C8C23×C311yescharacteristic
230#127—22no—
231#127—22no—
232#127—42no—
233#127—42no—
234#124—22no—
235#124—22no—
236#124—42no—
237#124—42no—
238C4×D4—22no—
239C4×D4C3×D414yescomplemented
240C2×C4⋊C4—22no—
241C2×C4⋊C4C23×C312yes—
242C2×C4⋊C4C23×C311yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C42—21no—
251C2×C42C23×C312yes—
252C2×C42C23×C311yescharacteristic
253C4⋊C8—22no—
254C4⋊C8—22no—
255#103—22no—
256#103—42no—
257#102—24no—
258#102—22no—
259#102—42no—
260#102—42no—
261#102—42no—
262#95—21no—
263#95—22no—
264#95C3×D411yescharacteristic
265#95C3×D411yescharacteristic
266#95C3×D412yes—
267C23×C3—22no—
268C23×C3—42no—
269C23×C3—41no—
270C23×C3—42no—
271C23×C3—41no—
272C23×C3—41no—
273C23×C3—42no—
274C23×C3—42no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3—41no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3—41no—
283C23×C3—42no—
284C23×C3C22×D411yescharacteristic
285C3×Q8—41no—
286C3×Q8—41no—
287C3×Q8—42no—
288C3×D4—22no—
289C3×D4—22no—
290C3×D4—41no—
291C3×D4—42no—
292C3×D4—41no—
293C3×D4—42no—
294C3×D4—42no—
295C3×D4—42no—
296C3×D4—42no—
297C3×D4—42no—
298C3×D4—84no—
299C3×D4—84no—
300C3×D4—88no—
301C3×D4—88no—
302C2×C3×C4—21no—
303C2×C3×C4—22no—
304C2×C3×C4—21no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—42no—
313C2×C3×C4—41no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—84no—
319C2×C3×C4#12112yes—
320C2×C3×C4#12112yes—
321C2×C3×C4C22×D411yescharacteristic
322C2×C3×C4C2×#4111yescharacteristic
323C2×C3×C4H(5,𝔽2)12yes—
324C2×C3×C4H(5,𝔽2)12yes—
325C3×C8—42no—
326C3×C8—41no—
327C3×C8—41no—
328C24—21no—
329C24—21no—
330C24—22no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×D4—21no—
336C2×D4—22no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—44no—
349C2×D4—44no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—81no—
353C2×D4—82no—
354C2×D4—81no—
355C2×D4—82no—
356C2×D4—82no—
357C2×D4—82no—
358C2×D4—82no—
359C2×D4C2×C3×D412yes—
360C22×C4—21no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—82no—
365C22×C4C2×C3×D412yes—
366C22×C4C2×C3×D412yes—
367C22×C4C2×C3×D411yescharacteristic
368C22×C4C2×C3×D411yescharacteristic
369C22×C4C24×C311yescharacteristic
370SD16—44no—
371D8—48no—
372D8—44no—
373C2×C8—22no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—82no—
384C4⋊C4C2×C3×D412yes—
385C22⋊C4—24no—
386C22⋊C4—44no—
387C22⋊C4—44no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—82no—
391C22⋊C4—82no—
392C42—22no—
393C42—22no—
394C42—81no—
395C42C2×C3×D412yes—
396C42C3×#4112yes—
397C22×C3—42no—
398C22×C3—42no—
399C22×C3—42no—
400C22×C3—82no—
401C22×C3—82no—
402C22×C3—81no—
403C22×C3—82no—
404C22×C3—82no—
405C22×C3—82no—
406C22×C3—82no—
407C22×C3—82no—
408C22×C3—82no—
409C22×C3—82no—
410C22×C3—82no—
411C22×C3—81no—
412C22×C3—82no—
413C22×C3—81no—
414C22×C3C2×#12111yescharacteristic
415C22×C3#53512yes—
416C22×C3#57611yescharacteristic
417C22×C3#57612yes—
418C22×C3#57711yescharacteristic
419C3×C4—22no—
420C3×C4—22no—
421C3×C4—42no—
422C3×C4—42no—
423C3×C4—42no—
424C3×C4—81no—
425C3×C4—82no—
426C3×C4#53511yescharacteristic
427C3×C4#53511yescharacteristic
428C3×C4#54012yes—
429C23—22no—
430C23—41no—
431C23—42no—
432C23—41no—
433C23—42no—
434C23—41no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—41no—
440C23—42no—
441C23—41no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23C22×C3×D411yescharacteristic
447Q8—41no—
448Q8—41no—
449Q8—42no—
450D4—22no—
451D4—22no—
452D4—41no—
453D4—42no—
454D4—41no—
455D4—42no—
456D4—42no—
457D4—42no—
458D4—42no—
459D4—42no—
460D4—84no—
461D4—84no—
462D4—88no—
463D4—88no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—84no—
481C2×C4C3×#12112yes—
482C2×C4C3×#12112yes—
483C2×C4C22×C3×D411yescharacteristic
484C2×C4C2×C3×#4111yescharacteristic
485C2×C4C3×H(5,𝔽2)12yes—
486C2×C4C3×H(5,𝔽2)12yes—
487C8—42no—
488C8—41no—
489C8—41no—
490C2×C3—82no—
491C2×C3—161no—
492C2×C3—162no—
493C2×C3#261911yescharacteristic
494C2×C3#304111yescharacteristic
495C2×C3#304511yescharacteristic
496C2×C3#322012yes—
497C2×C3#322412yes—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—82no—
502C22—82no—
503C22—81no—
504C22—82no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22—82no—
510C22—82no—
511C22—82no—
512C22—81no—
513C22—82no—
514C22—81no—
515C22C2×C3×#12111yescharacteristic
516C22C3×#53512yes—
517C22C3×#57611yescharacteristic
518C22C3×#57612yes—
519C22C3×#57711yescharacteristic
520C4—22no—
521C4—22no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—81no—
526C4—82no—
527C4C3×#53511yescharacteristic
528C4C3×#53511yescharacteristic
529C4C3×#54012yes—
530C3#2980211yescharacteristic, direct product factor, Sylow
531C2—82no—
532C2—161no—
533C2—162no—
534C2C3×#261911yescharacteristic
535C2C3×#304111yescharacteristic
536C2C3×#304511yescharacteristic
537C2C3×#322012yes—
538C2C3×#322412yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
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×#463C2semidirect
2. C2×C3×#459C2semidirect
3. C3×#2657C2semidirect
4. C3×#2319C2semidirect
5. C3×#2042C2semidirect
6. C3×#2041C2semidirect
7. C3×#2012C2semidirect
8. C3×#2011C2semidirect
9. C3×#1983C2semidirect
10. C3×#1982C2semidirect
11. C3×#1832C2semidirect
12. #29802C3direct (N × H)
44. C2×#463C2×C3semidirect
45. C2×#459C2×C3semidirect
46. #2657C2×C3semidirect
47. #2319C2×C3semidirect
48. #2042C2×C3semidirect
49. #2041C2×C3semidirect
50. #2012C2×C3semidirect
51. #2011C2×C3semidirect
52. #1983C2×C3semidirect
53. #1982C2×C3semidirect
54. #1832C2×C3semidirect
83. C3×C4×D4D4semidirect
239. C4×D4C3×D4semidirect
530. C3#29802direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4632
C2×C3×#4592
C3×#26572
C3×#23192
C3×#20422
C3×#20412
C3×#20122
C3×#20112
C3×#19832
C3×#19822
C3×#18322
#298023