Finite Group Databasebeta

← C3×#15119 #10633726

Subgroups — C3×#15119 #10633726

Order 768.

1246 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#391C211yesmaximal, characteristic
2C2×C3×#376C211yesmaximal, characteristic
3C3×#1446C211yesmaximal, characteristic
4C3×#1446C211yesmaximal, characteristic
5C3×#1445C212yesmaximal
6C3×#1444C211yesmaximal, characteristic
7C3×#1443C212yesmaximal
8C3×#1442C211yesmaximal, characteristic
9C3×#1436C212yesmaximal
10C3×#1434C212yesmaximal
11C3×#1434C211yesmaximal, characteristic
12#15119C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×C3×C4⋊C4C2212yes—
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C22×C3×C42C2211yescharacteristic
18C22×C3×C42C2211yescharacteristic
19C3×#391—24no—
20C3×#376—24no—
21C2×C3×#95C2211yescharacteristic
22C2×C3×#95C2211yescharacteristic
23C2×C3×#95C2211yescharacteristic
24C2×C3×#95C2211yescharacteristic
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2212yes—
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2212yes—
30C2×C3×#95C2212yes—
31C2×C3×#95C2212yes—
32C2×C3×#95C2211yescharacteristic
33C2×C3×#95C2212yes—
34C2×C3×#95C2212yes—
35C2×C3×#95C2212yes—
36C2×C3×#95C2212yes—
37C2×C3×#95C2212yes—
38C2×#391C2×C311yescharacteristic
39C2×#376C2×C311yescharacteristic
40#1446C2×C311yescharacteristic
41#1446C2×C311yescharacteristic
42#1445C2×C312yes—
43#1444C2×C311yescharacteristic
44#1443C2×C312yes—
45#1442C2×C311yescharacteristic
46#1436C2×C312yes—
47#1434C2×C311yescharacteristic
48#1434C2×C312yes—
49C22×C3×Q8—22no—
50C22×C3×Q8—24no—
51C22×C3×Q8—22no—
52C22×C3×Q8—22no—
53C22×C3×Q8—42no—
54C23×C3×C4C2312yes—
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2312yes—
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C23×C3×C4C2311yescharacteristic
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—24no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—42no—
69C2×C3×C4⋊C4—42no—
70C2×C3×C42—22no—
71C2×C3×C42—22no—
72C2×C3×C42—22no—
73C2×C3×C42—22no—
74C2×C3×C42—22no—
75C2×C3×C42—22no—
76C2×C3×C42—22no—
77C2×C3×C42D414yes—
78C2×C3×C42Q814yescomplemented
79C3×#95—22no—
80C3×#95—22no—
81C3×#95—22no—
82C3×#95—22no—
83C3×#95—24no—
84C3×#95—24no—
85C3×#95—24no—
86C3×#95—22no—
87C3×#95—24no—
88C3×#95—24no—
89C3×#95—24no—
90C3×#95—24no—
91C3×#95—24no—
92C3×#95—22no—
93C3×#95—42no—
94C3×#95—42no—
95C3×#95—42no—
96C23×Q8C22×C311yescharacteristic
97C22×C4⋊C4C22×C312yes—
98C22×C4⋊C4C22×C311yescharacteristic
99C22×C42C22×C311yescharacteristic
100C22×C42C22×C311yescharacteristic
101C22×C42C22×C311yescharacteristic
102#391—24no—
103#376—24no—
104C2×#95C22×C312yes—
105C2×#95C22×C312yes—
106C2×#95C22×C311yescharacteristic
107C2×#95C22×C312yes—
108C2×#95C22×C312yes—
109C2×#95C22×C312yes—
110C2×#95C22×C311yescharacteristic
111C2×#95C22×C312yes—
112C2×#95C22×C312yes—
113C2×#95C22×C311yescharacteristic
114C2×#95C22×C311yescharacteristic
115C2×#95C22×C312yes—
116C2×#95C22×C311yescharacteristic
117C2×#95C22×C311yescharacteristic
118C2×#95C22×C312yes—
119C2×#95C22×C312yes—
120C2×#95C22×C312yes—
121C24×C3C2411yescharacteristic
122C2×C3×Q8—28no—
123C2×C3×Q8—48no—
124C2×C3×Q8—48no—
125C2×C3×Q8—44no—
126C2×C3×Q8—44no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
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—21no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
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—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×Q812yes—
183C22×C3×C4C2×Q812yes—
184C22×C3×C4C2×Q812yes—
185C22×C3×C4#4112yes—
186C22×C3×C4#4114yes—
187C22×C3×C4#4114yes—
188C22×C3×C4#4112yes—
189C22×C3×C4#4112yes—
190C22×C3×C4#4112yes—
191C22×C3×C4#4112yes—
192C3×C4⋊C4—48no—
193C3×C4⋊C4—44no—
194C3×C42—28no—
195C3×C42—28no—
196C3×C42—28no—
197C22×Q8—22no—
198C22×Q8—24no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—42no—
202C23×C4C23×C312yes—
203C23×C4C23×C312yes—
204C23×C4C23×C311yescharacteristic
205C23×C4C23×C311yescharacteristic
206C23×C4C23×C311yescharacteristic
207C23×C4C23×C311yescharacteristic
208C23×C4C23×C312yes—
209C23×C4C23×C312yes—
210C23×C4C23×C311yescharacteristic
211C23×C4C23×C311yescharacteristic
212C23×C4C23×C311yescharacteristic
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42C3×D414yes—
226C2×C42C3×Q814yescomplemented
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—22no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—22no—
241#95—42no—
242#95—42no—
243#95—42no—
244C23×C3C22×D411yescharacteristic
245C23×C3C22×Q811yescharacteristic
246C23×C3C2×#4111yescharacteristic
247C23×C3C2×#4111yescharacteristic
248C23×C3C2×#4111yescharacteristic
249C23×C3C2×#4112yes—
250C23×C3C2×#4111yescharacteristic
251C23×C3C2×#4111yescharacteristic
252C23×C3C2×#4112yes—
253C23×C3H(5,𝔽2)12yes—
254C23×C32−1+412yes—
255C3×Q8—416no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—24no—
265C2×C3×C4—24no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—24no—
272C2×C3×C4—24no—
273C2×C3×C4—22no—
274C2×C3×C4—24no—
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—42no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—42no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
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—42no—
310C2×C3×C4—42no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4C22⋊Q814yes—
320C2×C3×C4#12314yes—
321C24C24×C311yescharacteristic
322C2×Q8—28no—
323C2×Q8—48no—
324C2×Q8—48no—
325C2×Q8—44no—
326C2×Q8—44no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
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—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
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×C4C2×C3×D412yes—
380C22×C4C2×C3×D412yes—
381C22×C4C2×C3×D412yes—
382C22×C4C2×C3×Q812yes—
383C22×C4C2×C3×Q812yes—
384C22×C4C2×C3×Q812yes—
385C22×C4C3×#4112yes—
386C22×C4C3×#4114yes—
387C22×C4C3×#4114yes—
388C22×C4C3×#4112yes—
389C22×C4C3×#4112yes—
390C22×C4C3×#4112yes—
391C22×C4C3×#4112yes—
392C4⋊C4—48no—
393C4⋊C4—44no—
394C42—28no—
395C42—28no—
396C42—28no—
397C22×C3C2×C22⋊Q811yescharacteristic
398C22×C3C2×#12311yescharacteristic
399C22×C3#52511yescharacteristic
400C22×C3#52511yescharacteristic
401C22×C3#52911yescharacteristic
402C22×C3#52911yescharacteristic
403C22×C3#52912yes—
404C22×C3#53211yescharacteristic
405C22×C3#53311yescharacteristic
406C22×C3#53311yescharacteristic
407C22×C3#53312yes—
408C22×C3#53511yescharacteristic
409C22×C3#53611yescharacteristic
410C22×C3#54212yes—
411C22×C3#54412yes—
412C22×C3#55112yes—
413C22×C3#55112yes—
414C22×C3#55112yes—
415C22×C3#55212yes—
416C22×C3#55212yes—
417C22×C3#55512yes—
418C22×C3#55612yes—
419C22×C3#55612yes—
420C3×C4—24no—
421C3×C4—24no—
422C3×C4—42no—
423C3×C4—42no—
424C3×C4—42no—
425C3×C4—42no—
426C3×C4—44no—
427C3×C4—42no—
428C3×C4—44no—
429C3×C4—82no—
430C3×C4—82no—
431C23C22×C3×D411yescharacteristic
432C23C22×C3×Q811yescharacteristic
433C23C2×C3×#4111yescharacteristic
434C23C2×C3×#4111yescharacteristic
435C23C2×C3×#4111yescharacteristic
436C23C2×C3×#4112yes—
437C23C2×C3×#4111yescharacteristic
438C23C2×C3×#4111yescharacteristic
439C23C2×C3×#4112yes—
440C23C3×H(5,𝔽2)12yes—
441C23C3×2−1+412yes—
442Q8—416no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
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×C4C3×C22⋊Q814yes—
507C2×C4C3×#12314yes—
508C2×C3#239611yescharacteristic
509C2×C3#248211yescharacteristic
510C2×C3#249911yescharacteristic
511C2×C3#252712yes—
512C2×C3#253212yes—
513C2×C3#259911yescharacteristic
514C2×C3#260512yes—
515C2×C3#261011yescharacteristic
516C2×C3#261212yes—
517C2×C3#262811yescharacteristic
518C2×C3#285511yescharacteristic
519C22C2×C3×C22⋊Q811yescharacteristic
520C22C2×C3×#12311yescharacteristic
521C22C3×#52511yescharacteristic
522C22C3×#52511yescharacteristic
523C22C3×#52912yes—
524C22C3×#52911yescharacteristic
525C22C3×#52911yescharacteristic
526C22C3×#53211yescharacteristic
527C22C3×#53312yes—
528C22C3×#53311yescharacteristic
529C22C3×#53311yescharacteristic
530C22C3×#53511yescharacteristic
531C22C3×#53611yescharacteristic
532C22C3×#54212yes—
533C22C3×#54412yes—
534C22C3×#55112yes—
535C22C3×#55112yes—
536C22C3×#55112yes—
537C22C3×#55212yes—
538C22C3×#55212yes—
539C22C3×#55512yes—
540C22C3×#55612yes—
541C22C3×#55612yes—
542C4—24no—
543C4—24no—
544C4—42no—
545C4—42no—
546C4—44no—
547C4—42no—
548C4—44no—
549C4—42no—
550C4—42no—
551C4—82no—
552C4—82no—
553C3#1511911yescharacteristic, direct product factor, Sylow
554C2C3×#239611yescharacteristic
555C2C3×#248211yescharacteristic
556C2C3×#249911yescharacteristic
557C2C3×#252712yes—
558C2C3×#253212yes—
559C2C3×#259911yescharacteristic
560C2C3×#260512yes—
561C2C3×#261011yescharacteristic
562C2C3×#261212yes—
563C2C3×#262811yescharacteristic
564C2C3×#285511yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C3C24
commutator, Frattini, cosocleC24C24×C3
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
12. #15119C3direct (N × H)
78. C2×C3×C42Q8semidirect
226. C2×C42C3×Q8semidirect
553. C3#15119direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3912
C2×C3×#3762
C3×#14462
C3×#14452
C3×#14442
C3×#14432
C3×#14422
C3×#14362
C3×#14342
#151193