Finite Group Databasebeta

← C3×#29761 #10648368

Subgroups — C3×#29761 #10648368

Order 768.

910 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#484C211yesmaximal, characteristic, complemented
2C2×C3×#481C211yesmaximal, characteristic
3C3×#2850C211yesmaximal, characteristic
4C2×C3×#389C211yesmaximal, characteristic, complemented
5C3×#2085C212yesmaximal, complemented
6C3×#2067C212yesmaximal, complemented
7C3×#2066C212yesmaximal, complemented
8C3×#2042C212yesmaximal, complemented
9C3×#1874C211yesmaximal, characteristic
10C3×#1872C211yesmaximal, characteristic, complemented
11C3×#1797C211yesmaximal, characteristic, complemented
12#29761C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#124C2211yescharacteristic
14C2×C3×C22⋊Q8C2211yescharacteristic
15C2×C3×#121C2211yescharacteristic
16C22×C3×C4⋊C4—21no—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C42C2211yescharacteristic
20C3×#484—24no—
21C3×#481—24no—
22C2×C3×#107C2212yes—
23C2×C3×#106C2212yes—
24C2×C3×C4⋊C8C2212yes—
25C2×C3×#103C2212yes—
26C2×C3×#103C2212yes—
27C2×C3×#102C2212yes—
28C2×C3×#102C2212yes—
29C2×C3×C22⋊C8C2211yescharacteristic
30C2×C3×C22⋊C8C2211yescharacteristic
31C3×#391C2211yescharacteristic
32C3×#390C2211yescharacteristic
33C3×#389—21no—
34C3×#389—21no—
35C3×#389C2212yes—
36C3×#389C2212yes—
37C3×#388—21no—
38C3×#388—21no—
39C3×#386—21no—
40C3×#386—21no—
41C3×#382C2211yescharacteristic
42C2×C3×#95—21no—
43C3×#340C2212yes—
44C3×#340C2212yes—
45C3×#336C2212yes—
46C2×#484C2×C311yescharacteristic, complemented
47C2×#481C2×C311yescharacteristic
48#2850C2×C311yescharacteristic
49C2×#389C2×C311yescharacteristic, complemented
50#2085C2×C312yescomplemented
51#2067C2×C312yescomplemented
52#2066C2×C312yescomplemented
53#2042C2×C312yescomplemented
54#1874C2×C311yescharacteristic
55#1872C2×C311yescharacteristic, complemented
56#1797C2×C311yescharacteristic, complemented
57C22×C3×Q8C2311yescharacteristic
58C22×C3×D4—21no—
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4—21no—
61C23×C3×C4—21no—
62C23×C3×C4D411yescharacteristic
63C23×C3×C4D411yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C22×C3×C8C2312yes—
66C22×C3×C8C2312yes—
67C3×#124—42no—
68C3×C22⋊Q8—22no—
69C3×C22⋊Q8—22no—
70C3×#121—22no—
71C3×#121—22no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—42no—
79C2×C3×C4⋊C4—44no—
80C2×C3×C4⋊C4D414yes—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C42—42no—
92C2×C3×C42D411yescharacteristic
93C2×C3×C42D411yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C2×C3×C42C2311yescharacteristic
96C3×#107—24no—
97C3×#106—24no—
98C3×C4⋊C8—42no—
99C3×#103—24no—
100C3×#103—42no—
101C3×#102—24no—
102C3×#102—42no—
103C3×C22⋊C8—22no—
104C3×C22⋊C8—22no—
105C3×#95—22no—
106C3×#95—22no—
107C3×#95—21no—
108C3×#95—21no—
109C3×#95—21no—
110C3×#95—21no—
111C2×#124C22×C311yescharacteristic
112C2×C22⋊Q8C22×C311yescharacteristic
113C2×#121C22×C311yescharacteristic
114C22×C4⋊C4—21no—
115C22×C4⋊C4C22×C311yescharacteristic
116C22×C4⋊C4C22×C311yescharacteristic
117C22×C42C22×C311yescharacteristic
118#484—24no—
119#481—24no—
120C2×#107C22×C312yes—
121C2×#106C22×C312yes—
122C2×C4⋊C8C22×C312yes—
123C2×#103C22×C312yes—
124C2×#103C22×C312yes—
125C2×#102C22×C312yes—
126C2×#102C22×C312yes—
127C2×C22⋊C8C22×C311yescharacteristic
128C2×C22⋊C8C22×C311yescharacteristic
129#391C22×C311yescharacteristic
130#390C22×C311yescharacteristic
131#389—21no—
132#389—21no—
133#389C22×C312yes—
134#389C22×C312yes—
135#388—21no—
136#388—21no—
137#386—21no—
138#386—21no—
139#382C22×C311yescharacteristic
140C2×#95—21no—
141#340C22×C312yes—
142#340C22×C312yes—
143#336C22×C312yes—
144C24×C3—21no—
145C24×C3C2×D411yescharacteristic
146C2×C3×Q8—21no—
147C2×C3×Q8—21no—
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C2×C3×D4—21no—
151C2×C3×D4—21no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—42no—
155C2×C3×D4—42no—
156C2×C3×D4—42no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—24no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4—41no—
169C22×C3×C4—41no—
170C22×C3×C4—41no—
171C22×C3×C4—42no—
172C22×C3×C4—42no—
173C22×C3×C4—42no—
174C22×C3×C4—42no—
175C22×C3×C4—42no—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4#4112yes—
183C22×C3×C4C2411yescharacteristic
184C2×C3×C8—22no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C3×C4⋊C4—24no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—42no—
199C3×C4⋊C4—44no—
200C3×C4⋊C4—42no—
201C3×C4⋊C4—44no—
202C3×C4⋊C4C2×D414yes—
203C3×C22⋊C4—42no—
204C3×C22⋊C4—42no—
205C3×C22⋊C4—42no—
206C3×C22⋊C4—42no—
207C3×C22⋊C4—42no—
208C3×C22⋊C4—42no—
209C3×C42—22no—
210C3×C42—22no—
211C3×C42—41no—
212C3×C42—41no—
213C22×Q8C23×C311yescharacteristic
214C22×D4—21no—
215C22×D4C23×C311yescharacteristic
216C23×C4—21no—
217C23×C4—21no—
218C23×C4C3×D411yescharacteristic
219C23×C4C3×D411yescharacteristic
220C23×C4C23×C311yescharacteristic
221C22×C8C23×C312yes—
222C22×C8C23×C312yes—
223#124—42no—
224C22⋊Q8—22no—
225C22⋊Q8—22no—
226#121—22no—
227#121—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4C3×D414yes—
237C2×C4⋊C4C23×C312yes—
238C2×C4⋊C4C23×C311yescharacteristic
239C2×C4⋊C4C23×C311yescharacteristic
240C2×C4⋊C4C23×C312yes—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C42—42no—
248C2×C42C3×D411yescharacteristic
249C2×C42C3×D411yescharacteristic
250C2×C42C23×C311yescharacteristic
251C2×C42C23×C311yescharacteristic
252#107—24no—
253#106—24no—
254C4⋊C8—42no—
255#103—24no—
256#103—42no—
257#102—24no—
258#102—42no—
259C22⋊C8—22no—
260C22⋊C8—22no—
261#95—22no—
262#95—22no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—21no—
270C23×C3—21no—
271C23×C3—41no—
272C23×C3—41no—
273C23×C3—41no—
274C23×C3—41no—
275C23×C3—41no—
276C23×C3—41no—
277C23×C3—41no—
278C23×C3#12112yes—
279C23×C3#13512yes—
280C23×C3#13712yes—
281C23×C3C22×D411yescharacteristic
282C3×Q8—42no—
283C3×Q8—42no—
284C3×D4—42no—
285C3×D4—42no—
286C3×D4—84no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—44no—
304C2×C3×C4—44no—
305C2×C3×C4—48no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4—41no—
314C2×C3×C4—42no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—42no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—84no—
321C2×C3×C4#12112yes—
322C2×C3×C4AGL(1,ℤ/8)12yes—
323C2×C3×C4C22×D411yescharacteristic
324C2×C3×C4C2×#4111yescharacteristic
325C2×C3×C4H(5,𝔽2)12yes—
326C2×C3×C42−1+412yes—
327C3×C8—42no—
328C3×C8—42no—
329C24—21no—
330C24C2×C3×D411yescharacteristic
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4C2×C3×D411yescharacteristic
362C22×C4C2×C3×D411yescharacteristic
363C22×C4C2×C3×D411yescharacteristic
364C22×C4C2×C3×D411yescharacteristic
365C22×C4C2×C3×D412yes—
366C22×C4C2×C3×D411yescharacteristic
367C22×C4C3×#4112yes—
368C22×C4C24×C311yescharacteristic
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C4⋊C4—24no—
376C4⋊C4—22no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—42no—
384C4⋊C4—44no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4C2×C3×D414yes—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C42—22no—
395C42—22no—
396C42—41no—
397C42—41no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—22no—
404C22×C3—81no—
405C22×C3—81no—
406C22×C3—81no—
407C22×C3—81no—
408C22×C3—81no—
409C22×C3—81no—
410C22×C3—81no—
411C22×C3#46414yes—
412C22×C3C2×#12111yescharacteristic
413C22×C3#53411yescharacteristic
414C22×C3#53611yescharacteristic
415C22×C3C2×#13511yescharacteristic
416C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
417C22×C3C2×#13711yescharacteristic
418C22×C3#57511yescharacteristic
419C3×C4—22no—
420C3×C4—22no—
421C3×C4—41no—
422C3×C4—41no—
423C3×C4—41no—
424C3×C4—44no—
425C3×C4—81no—
426C3×C4—82no—
427C3×C4—81no—
428C3×C4—81no—
429C3×C4#53712yes—
430C3×C4#53912yes—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23C3×#12112yes—
443C23C3×#13512yes—
444C23C3×#13712yes—
445C23C22×C3×D411yescharacteristic
446Q8—42no—
447Q8—42no—
448D4—42no—
449D4—42no—
450D4—84no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—24no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—48no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—84no—
485C2×C4C3×#12112yes—
486C2×C4C3×AGL(1,ℤ/8)12yes—
487C2×C4C22×C3×D411yescharacteristic
488C2×C4C2×C3×#4111yescharacteristic
489C2×C4C3×H(5,𝔽2)12yes—
490C2×C4C3×2−1+412yes—
491C8—42no—
492C8—42no—
493C2×C3—24no—
494C2×C3—161no—
495C2×C3#261811yescharacteristic
496C2×C3C2×#46411yescharacteristic
497C2×C3#303611yescharacteristic
498C2×C3#318811yescharacteristic
499C2×C3#319611yescharacteristic
500C2×C3#323211yescharacteristic
501C2×C3#323911yescharacteristic
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22C3×#46414yes—
516C22C2×C3×#12111yescharacteristic
517C22C3×#53411yescharacteristic
518C22C3×#53611yescharacteristic
519C22C2×C3×#13511yescharacteristic
520C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
521C22C2×C3×#13711yescharacteristic
522C22C3×#57511yescharacteristic
523C4—22no—
524C4—22no—
525C4—41no—
526C4—41no—
527C4—41no—
528C4—44no—
529C4—81no—
530C4—82no—
531C4—81no—
532C4—81no—
533C4C3×#53712yes—
534C4C3×#53912yes—
535C3#2976111yescharacteristic, direct product factor, Sylow
536C2—24no—
537C2—161no—
538C2C3×#261811yescharacteristic
539C2C2×C3×#46411yescharacteristic
540C2C3×#303611yescharacteristic
541C2C3×#318811yescharacteristic
542C2C3×#319611yescharacteristic
543C2C3×#323211yescharacteristic
544C2C3×#323911yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#484C2semidirect
4. C2×C3×#389C2semidirect
5. C3×#2085C2semidirect
6. C3×#2067C2semidirect
7. C3×#2066C2semidirect
8. C3×#2042C2semidirect
10. C3×#1872C2semidirect
11. C3×#1797C2semidirect
12. #29761C3direct (N × H)
46. C2×#484C2×C3semidirect
49. C2×#389C2×C3semidirect
50. #2085C2×C3semidirect
51. #2067C2×C3semidirect
52. #2066C2×C3semidirect
53. #2042C2×C3semidirect
55. #1872C2×C3semidirect
56. #1797C2×C3semidirect
535. C3#29761direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4842
C2×C3×#4812
C3×#28502
C2×C3×#3892
C3×#20852
C3×#20672
C3×#20662
C3×#20422
C3×#18742
C3×#18722
C3×#17972
#297613