Finite Group Databasebeta

← C3×#15730 #10634337

Subgroups — C3×#15730 #10634337

Order 768.

1474 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#399C211yesmaximal, characteristic
2C2×C3×#385C211yesmaximal, characteristic
3C3×#1441C212yesmaximal, complemented
4C3×#1441C212yesmaximal, complemented
5C3×#1441C212yesmaximal, complemented
6C3×#1439C212yesmaximal
7C3×#1437C211yesmaximal, characteristic
8C3×#1436C212yesmaximal, complemented
9C3×C22⋊#95C212yesmaximal
10#15730C311yesmaximal, characteristic, direct product factor, Sylow
11C24×C3×C4C2211yescharacteristic
12C22×C3×C4⋊C4C2212yes—
13C22×C3×C4⋊C4C2212yes—
14C22×C3×C4⋊C4C2212yes—
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C22⋊C4C2211yescharacteristic
17C22×C3×C22⋊C4C2212yes—
18C22×C3×C22⋊C4C2211yescharacteristic
19C22×C3×C22⋊C4C2212yes—
20C22×C3×C42C2211yescharacteristic
21C3×#399—24no—
22C3×#385—24no—
23C2×C3×#95C2211yescharacteristic
24C2×C3×#95C2212yes—
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2212yes—
28C2×C3×#95C2212yes—
29C2×C3×#95C2212yes—
30C2×C3×#95C2212yes—
31C2×C3×#95C2211yescharacteristic
32C2×C3×#95C2212yes—
33C2×C3×#95C2212yes—
34C2×#399C2×C311yescharacteristic
35C2×#385C2×C311yescharacteristic
36#1441C2×C312yescomplemented
37#1441C2×C312yescomplemented
38#1441C2×C312yescomplemented
39#1439C2×C312yes—
40#1437C2×C311yescharacteristic
41#1436C2×C312yescomplemented
42C22⋊#95C2×C312yes—
43C25×C3C2311yescharacteristic
44C23×C3×C4—22no—
45C23×C3×C4—24no—
46C23×C3×C4—41no—
47C23×C3×C4—41no—
48C23×C3×C4—42no—
49C23×C3×C4C2312yes—
50C23×C3×C4C2311yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2312yes—
53C23×C3×C4C2312yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2312yes—
56C23×C3×C4C2312yes—
57C2×C3×C4⋊C4—24no—
58C2×C3×C4⋊C4—24no—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4—24no—
61C2×C3×C4⋊C4—24no—
62C2×C3×C4⋊C4—24no—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4—42no—
65C2×C3×C4⋊C4—42no—
66C2×C3×C4⋊C4—42no—
67C2×C3×C4⋊C4—41no—
68C2×C3×C4⋊C4—41no—
69C2×C3×C22⋊C4—22no—
70C2×C3×C22⋊C4—24no—
71C2×C3×C22⋊C4—24no—
72C2×C3×C22⋊C4—24no—
73C2×C3×C22⋊C4—41no—
74C2×C3×C22⋊C4—41no—
75C2×C3×C22⋊C4—42no—
76C2×C3×C22⋊C4—42no—
77C2×C3×C22⋊C4—42no—
78C2×C3×C22⋊C4D414yes—
79C2×C3×C22⋊C4Q818yes—
80C2×C3×C42—41no—
81C2×C3×C42—42no—
82C3×#95—24no—
83C3×#95—22no—
84C3×#95—24no—
85C3×#95—24no—
86C3×#95—41no—
87C3×#95—42no—
88C3×#95—42no—
89C3×#95—42no—
90C3×#95—42no—
91C3×#95—42no—
92C3×#95—42no—
93C24×C4C22×C311yescharacteristic
94C22×C4⋊C4C22×C312yes—
95C22×C4⋊C4C22×C312yes—
96C22×C4⋊C4C22×C312yes—
97C22×C4⋊C4C22×C311yescharacteristic
98C22×C22⋊C4C22×C311yescharacteristic
99C22×C22⋊C4C22×C312yes—
100C22×C22⋊C4C22×C311yescharacteristic
101C22×C22⋊C4C22×C312yes—
102C22×C42C22×C311yescharacteristic
103#399—24no—
104#385—24no—
105C2×#95C22×C311yescharacteristic
106C2×#95C22×C312yes—
107C2×#95C22×C312yes—
108C2×#95C22×C312yes—
109C2×#95C22×C312yes—
110C2×#95C22×C312yes—
111C2×#95C22×C312yes—
112C2×#95C22×C312yes—
113C2×#95C22×C312yes—
114C2×#95C22×C311yescharacteristic
115C2×#95C22×C312yes—
116C24×C3—22no—
117C24×C3—22no—
118C24×C3—22no—
119C24×C3—22no—
120C24×C3—21no—
121C24×C3—21no—
122C24×C3—22no—
123C24×C3C2×D412yes—
124C24×C3C2×Q814yes—
125C24×C3C2411yescharacteristic
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—44no—
172C22×C3×C4—48no—
173C22×C3×C4—48no—
174C22×C3×C4—44no—
175C22×C3×C4—44no—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×Q814yes—
179C22×C3×C4C2×Q814yes—
180C22×C3×C4#4114yes—
181C22×C3×C4#4114yes—
182C22×C3×C4#4114yes—
183C3×C4⋊C4—48no—
184C3×C4⋊C4—48no—
185C3×C4⋊C4—48no—
186C3×C4⋊C4—44no—
187C3×C22⋊C4—28no—
188C3×C22⋊C4—44no—
189C3×C22⋊C4—48no—
190C3×C22⋊C4—48no—
191C3×C42—44no—
192C25C23×C311yescharacteristic
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—41no—
196C23×C4—41no—
197C23×C4—42no—
198C23×C4C23×C312yes—
199C23×C4C23×C311yescharacteristic
200C23×C4C23×C311yescharacteristic
201C23×C4C23×C312yes—
202C23×C4C23×C312yes—
203C23×C4C23×C312yes—
204C23×C4C23×C312yes—
205C23×C4C23×C312yes—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4C3×D414yes—
228C2×C22⋊C4C3×Q818yes—
229C2×C42—41no—
230C2×C42—42no—
231#95—24no—
232#95—22no—
233#95—24no—
234#95—24no—
235#95—41no—
236#95—42no—
237#95—42no—
238#95—42no—
239#95—42no—
240#95—42no—
241#95—42no—
242C23×C3—22no—
243C23×C3—24no—
244C23×C3—22no—
245C23×C3—24no—
246C23×C3—24no—
247C23×C3—24no—
248C23×C3—24no—
249C23×C3—24no—
250C23×C3—24no—
251C23×C3—24no—
252C23×C3—42no—
253C23×C3—42no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3—41no—
257C23×C3—42no—
258C23×C3—42no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3C4⋊Q814yes—
263C23×C3C22×D411yescharacteristic
264C23×C3C22×Q812yes—
265C23×C3C2×#4112yes—
266C23×C3C2×#4112yes—
267C23×C3C2×#4112yes—
268C23×C3H(5,𝔽2)11yescharacteristic
269C23×C3H(5,𝔽2)11yescharacteristic
270C23×C3H(5,𝔽2)12yes—
271C23×C3H(5,𝔽2)12yes—
272C2×C3×C4—24no—
273C2×C3×C4—24no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—24no—
277C2×C3×C4—24no—
278C2×C3×C4—24no—
279C2×C3×C4—24no—
280C2×C3×C4—24no—
281C2×C3×C4—24no—
282C2×C3×C4—24no—
283C2×C3×C4—24no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—24no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—24no—
290C2×C3×C4—24no—
291C2×C3×C4—416no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—41no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—42no—
315C2×C3×C4—41no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—21no—
330C24—21no—
331C24—22no—
332C24C2×C3×D412yes—
333C24C2×C3×Q814yes—
334C24C24×C311yescharacteristic
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—44no—
381C22×C4—48no—
382C22×C4—48no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4C2×C3×D412yes—
386C22×C4C2×C3×D412yes—
387C22×C4C2×C3×Q814yes—
388C22×C4C2×C3×Q814yes—
389C22×C4C3×#4114yes—
390C22×C4C3×#4114yes—
391C22×C4C3×#4114yes—
392C4⋊C4—48no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—44no—
396C22⋊C4—28no—
397C22⋊C4—44no—
398C22⋊C4—48no—
399C22⋊C4—48no—
400C42—44no—
401C22×C3—24no—
402C22×C3—28no—
403C22×C3—44no—
404C22×C3—42no—
405C22×C3—42no—
406C22×C3—44no—
407C22×C3—44no—
408C22×C3—44no—
409C22×C3—44no—
410C22×C3#52511yescharacteristic
411C22×C3#52912yes—
412C22×C3C2×C4⋊Q811yescharacteristic
413C22×C3#53312yes—
414C22×C3#53411yescharacteristic
415C22×C3#53812yes—
416C22×C3#54012yes—
417C22×C3#54012yes—
418C22×C3#54312yes—
419C22×C3#54612yes—
420C22×C3#54612yes—
421C22×C3#55112yes—
422C22×C3#55111yescharacteristic
423C22×C3#55312yes—
424C22×C3#55412yes—
425C22×C3#55412yes—
426C22×C3#55412yes—
427C22×C3#55412yes—
428C22×C3#55911yescharacteristic
429C22×C3#56011yescharacteristic
430C22×C3#56011yescharacteristic
431C3×C4—42no—
432C3×C4—44no—
433C3×C4—42no—
434C3×C4—44no—
435C3×C4—44no—
436C3×C4—44no—
437C3×C4—44no—
438C3×C4—82no—
439C23—22no—
440C23—24no—
441C23—22no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—41no—
457C23—42no—
458C23—42no—
459C23C3×C4⋊Q814yes—
460C23C22×C3×D411yescharacteristic
461C23C22×C3×Q812yes—
462C23C2×C3×#4112yes—
463C23C2×C3×#4112yes—
464C23C2×C3×#4112yes—
465C23C3×H(5,𝔽2)11yescharacteristic
466C23C3×H(5,𝔽2)11yescharacteristic
467C23C3×H(5,𝔽2)12yes—
468C23C3×H(5,𝔽2)12yes—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—416no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C3—44no—
523C2×C3#245712yes—
524C2×C3#252212yes—
525C2×C3#254912yes—
526C2×C3#258311yescharacteristic
527C2×C3#258912yes—
528C2×C3#269711yescharacteristic
529C2×C3#270911yescharacteristic
530C2×C3#274812yes—
531C2×C3#278012yes—
532C22—28no—
533C22—24no—
534C22—42no—
535C22—42no—
536C22—44no—
537C22—44no—
538C22—44no—
539C22—44no—
540C22—44no—
541C22C3×#52511yescharacteristic
542C22C3×#52912yes—
543C22C2×C3×C4⋊Q811yescharacteristic
544C22C3×#53312yes—
545C22C3×#53411yescharacteristic
546C22C3×#53812yes—
547C22C3×#54012yes—
548C22C3×#54012yes—
549C22C3×#54312yes—
550C22C3×#54612yes—
551C22C3×#54612yes—
552C22C3×#55111yescharacteristic
553C22C3×#55112yes—
554C22C3×#55312yes—
555C22C3×#55412yes—
556C22C3×#55412yes—
557C22C3×#55412yes—
558C22C3×#55412yes—
559C22C3×#55911yescharacteristic
560C22C3×#56011yescharacteristic
561C22C3×#56011yescharacteristic
562C4—44no—
563C4—42no—
564C4—42no—
565C4—44no—
566C4—44no—
567C4—44no—
568C4—44no—
569C4—82no—
570C3#1573011yescharacteristic, direct product factor, Sylow
571C2—44no—
572C2C3×#245712yes—
573C2C3×#252212yes—
574C2C3×#254912yes—
575C2C3×#258311yescharacteristic
576C2C3×#258912yes—
577C2C3×#269711yescharacteristic
578C2C3×#270911yescharacteristic
579C2C3×#274812yes—
580C2C3×#278012yes—
subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
prime radicalC25×C3C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#1441C2semidirect
4. C3×#1441C2semidirect
5. C3×#1441C2semidirect
8. C3×#1436C2semidirect
10. #15730C3direct (N × H)
36. #1441C2×C3semidirect
37. #1441C2×C3semidirect
38. #1441C2×C3semidirect
41. #1436C2×C3semidirect
570. C3#15730direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3992
C2×C3×#3852
C3×#14412
C3×#14392
C3×#14372
C3×#14362
C3×C22⋊#952
#157303