Finite Group Databasebeta

← C3×#15103 #10633710

Subgroups — C3×#15103 #10633710

Order 768.

1462 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#400C211yesmaximal, characteristic, complemented
2C2×C3×#399C211yesmaximal, characteristic
3C2×C3×#388C211yesmaximal, characteristic
4C2×C3×C4×C4⋊C4C211yesmaximal, characteristic, complemented
5C3×#1441C211yesmaximal, characteristic, complemented
6C3×#1441C212yesmaximal, complemented
7C3×#1439C211yesmaximal, characteristic
8C3×#1439C212yesmaximal
9C3×#1437C212yesmaximal
10C3×#1436C212yesmaximal, complemented
11C3×C4⋊#95C211yesmaximal, characteristic, complemented
12#15103C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C22⋊C4C2211yescharacteristic
19C22×C3×C22⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4C2212yes—
21C22×C3×C22⋊C4C2211yescharacteristic
22C22×C3×C22⋊C4C2212yes—
23C22×C3×C42C2211yescharacteristic
24C22×C3×C42C2212yes—
25C3×#400—24no—
26C3×#399—24no—
27C3×#388—24no—
28C3×C4×C4⋊C4—24no—
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2211yescharacteristic
31C2×C3×#95C2212yes—
32C2×C3×#95C2212yes—
33C2×C3×#95C2212yes—
34C2×C3×#95C2211yescharacteristic
35C2×C3×#95C2211yescharacteristic
36C2×C3×#95C2212yes—
37C2×C3×#95C2212yes—
38C2×C3×#95C2212yes—
39C2×C3×#95C2212yes—
40C2×#400C2×C311yescharacteristic, complemented
41C2×#399C2×C311yescharacteristic
42C2×#388C2×C311yescharacteristic
43C2×C4×C4⋊C4C2×C311yescharacteristic, complemented
44#1441C2×C311yescharacteristic, complemented
45#1441C2×C312yescomplemented
46#1439C2×C311yescharacteristic
47#1439C2×C312yes—
48#1437C2×C312yes—
49#1436C2×C312yescomplemented
50C4⋊#95C2×C311yescharacteristic, complemented
51C25×C3C2311yescharacteristic
52C23×C3×C4C2312yes—
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2312yes—
58C23×C3×C4C2312yes—
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—24no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—24no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—42no—
74C2×C3×C4⋊C4—42no—
75C2×C3×C4⋊C4D414yes—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—24no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—24no—
83C2×C3×C22⋊C4—24no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—24no—
86C2×C3×C22⋊C4—42no—
87C2×C3×C22⋊C4—42no—
88C2×C3×C22⋊C4—42no—
89C2×C3×C22⋊C4Q814yes—
90C2×C3×C42—24no—
91C2×C3×C42—24no—
92C2×C3×C42—42no—
93C2×C3×C42—42no—
94C2×C3×C42D414yes—
95C3×#95—24no—
96C3×#95—22no—
97C3×#95—22no—
98C3×#95—22no—
99C3×#95—24no—
100C3×#95—22no—
101C3×#95—24no—
102C3×#95—24no—
103C3×#95—24no—
104C3×#95—42no—
105C3×#95—42no—
106C22×C4⋊C4C22×C311yescharacteristic
107C22×C4⋊C4C22×C311yescharacteristic
108C22×C4⋊C4C22×C312yes—
109C22×C4⋊C4C22×C312yes—
110C22×C4⋊C4C22×C311yescharacteristic
111C22×C22⋊C4C22×C311yescharacteristic
112C22×C22⋊C4C22×C311yescharacteristic
113C22×C22⋊C4C22×C312yes—
114C22×C22⋊C4C22×C311yescharacteristic
115C22×C22⋊C4C22×C312yes—
116C22×C42C22×C311yescharacteristic
117C22×C42C22×C312yes—
118#400—24no—
119#399—24no—
120#388—24no—
121C4×C4⋊C4—24no—
122C2×#95C22×C311yescharacteristic
123C2×#95C22×C311yescharacteristic
124C2×#95C22×C312yes—
125C2×#95C22×C312yes—
126C2×#95C22×C312yes—
127C2×#95C22×C311yescharacteristic
128C2×#95C22×C312yes—
129C2×#95C22×C312yes—
130C2×#95C22×C311yescharacteristic
131C2×#95C22×C312yes—
132C2×#95C22×C312yes—
133C24×C3—21no—
134C24×C3—21no—
135C24×C3—21no—
136C24×C3—22no—
137C24×C3—22no—
138C24×C3—21no—
139C24×C3—22no—
140C24×C3—22no—
141C24×C3—21no—
142C24×C3—21no—
143C24×C3C2×Q812yes—
144C24×C3C2411yescharacteristic
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
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—21no—
166C22×C3×C4—21no—
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—21no—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—21no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—21no—
187C22×C3×C4—22no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4C2×D414yes—
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2×D412yes—
198C22×C3×C4C2×Q812yes—
199C22×C3×C4C2×Q812yes—
200C22×C3×C4#4114yes—
201C22×C3×C4#4114yes—
202C22×C3×C4#4112yes—
203C22×C3×C4#4112yes—
204C22×C3×C4#4114yes—
205C3×C4⋊C4—28no—
206C3×C4⋊C4—28no—
207C3×C4⋊C4—48no—
208C3×C4⋊C4—48no—
209C3×C4⋊C4—44no—
210C3×C22⋊C4—28no—
211C3×C22⋊C4—48no—
212C3×C22⋊C4—48no—
213C3×C22⋊C4—44no—
214C3×C22⋊C4—44no—
215C3×C42—28no—
216C3×C42—48no—
217C25C23×C311yescharacteristic
218C23×C4C23×C311yescharacteristic
219C23×C4C23×C312yes—
220C23×C4C23×C311yescharacteristic
221C23×C4C23×C312yes—
222C23×C4C23×C311yescharacteristic
223C23×C4C23×C312yes—
224C23×C4C23×C312yes—
225C23×C4C23×C311yescharacteristic
226C23×C4C23×C311yescharacteristic
227C23×C4C23×C311yescharacteristic
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4C3×D414yes—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4C3×Q814yes—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42C3×D414yes—
261#95—24no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—24no—
266#95—22no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—42no—
271#95—42no—
272C23×C3—22no—
273C23×C3—24no—
274C23×C3—22no—
275C23×C3—22no—
276C23×C3—24no—
277C23×C3—41no—
278C23×C3—41no—
279C23×C3—41no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3—41no—
283C23×C3—42no—
284C23×C3—41no—
285C23×C3—41no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—41no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—41no—
294C23×C3—42no—
295C23×C3C22×D411yescharacteristic
296C23×C3C22×D411yescharacteristic
297C23×C3C22×Q811yescharacteristic
298C23×C3C2×#4111yescharacteristic
299C23×C3C2×#4111yescharacteristic
300C23×C3C2×#4112yes—
301C23×C3C2×#4112yes—
302C23×C3C2×#4112yes—
303C23×C3H(5,𝔽2)12yes—
304C23×C3H(5,𝔽2)11yescharacteristic
305C23×C32−1+411yescharacteristic
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—22no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—22no—
328C2×C3×C4—24no—
329C2×C3×C4—22no—
330C2×C3×C4—41no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—41no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—41no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—41no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—41no—
354C2×C3×C4—42no—
355C2×C3×C4—41no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4#12114yes—
364C2×C3×C4C22⋊Q814yes—
365C2×C3×C4#12314yes—
366C24—21no—
367C24—21no—
368C24—21no—
369C24—22no—
370C24—22no—
371C24—21no—
372C24—22no—
373C24—22no—
374C24—21no—
375C24—21no—
376C24C2×C3×Q812yes—
377C24C24×C311yescharacteristic
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4C2×C3×D414yes—
427C22×C4C2×C3×D412yes—
428C22×C4C2×C3×D412yes—
429C22×C4C2×C3×D412yes—
430C22×C4C2×C3×D412yes—
431C22×C4C2×C3×Q812yes—
432C22×C4C2×C3×Q812yes—
433C22×C4C3×#4114yes—
434C22×C4C3×#4114yes—
435C22×C4C3×#4112yes—
436C22×C4C3×#4112yes—
437C22×C4C3×#4114yes—
438C4⋊C4—28no—
439C4⋊C4—28no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—44no—
443C22⋊C4—28no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C42—28no—
449C42—48no—
450C22×C3—44no—
451C22×C3—42no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3—44no—
455C22×C3—82no—
456C22×C3—81no—
457C22×C3—81no—
458C22×C3—81no—
459C22×C3—81no—
460C22×C3—82no—
461C22×C3C2×#12111yescharacteristic
462C22×C3C2×C22⋊Q811yescharacteristic
463C22×C3C2×#12311yescharacteristic
464C22×C3#52511yescharacteristic
465C22×C3#52912yes—
466C22×C3#52912yes—
467C22×C3#53212yes—
468C22×C3#53211yescharacteristic
469C22×C3#53312yes—
470C22×C3#53511yescharacteristic
471C22×C3#53711yescharacteristic
472C22×C3#53812yes—
473C22×C3#53912yes—
474C22×C3#54612yes—
475C22×C3D4×Q811yescharacteristic
476C22×C3#55011yescharacteristic
477C22×C3#55112yes—
478C22×C3#55212yes—
479C22×C3#55312yes—
480C22×C3#55411yescharacteristic
481C22×C3#55412yes—
482C22×C3#55611yescharacteristic
483C22×C3#55912yes—
484C3×C4—24no—
485C3×C4—24no—
486C3×C4—24no—
487C3×C4—44no—
488C3×C4—42no—
489C3×C4—44no—
490C3×C4—44no—
491C3×C4—81no—
492C3×C4—81no—
493C3×C4—82no—
494C23—22no—
495C23—24no—
496C23—22no—
497C23—24no—
498C23—22no—
499C23—42no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—42no—
504C23—42no—
505C23—41no—
506C23—42no—
507C23—41no—
508C23—41no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—41no—
517C23C22×C3×D411yescharacteristic
518C23C22×C3×D411yescharacteristic
519C23C22×C3×Q811yescharacteristic
520C23C2×C3×#4111yescharacteristic
521C23C2×C3×#4112yes—
522C23C2×C3×#4111yescharacteristic
523C23C2×C3×#4112yes—
524C23C2×C3×#4112yes—
525C23C3×H(5,𝔽2)12yes—
526C23C3×H(5,𝔽2)11yescharacteristic
527C23C3×2−1+411yescharacteristic
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4C3×#12114yes—
586C2×C4C3×C22⋊Q814yes—
587C2×C4C3×#12314yes—
588C2×C3—82no—
589C2×C3#239611yescharacteristic
590C2×C3#248711yescharacteristic
591C2×C3#249811yescharacteristic
592C2×C3#253811yescharacteristic
593C2×C3#254411yescharacteristic
594C2×C3#255412yes—
595C2×C3#255612yes—
596C2×C3#258912yes—
597C2×C3#259612yes—
598C2×C3#263611yescharacteristic
599C2×C3#285411yescharacteristic
600C22—42no—
601C22—42no—
602C22—42no—
603C22—44no—
604C22—44no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—82no—
609C22—82no—
610C22—81no—
611C22C2×C3×#12111yescharacteristic
612C22C2×C3×C22⋊Q811yescharacteristic
613C22C2×C3×#12311yescharacteristic
614C22C3×#52511yescharacteristic
615C22C3×#52912yes—
616C22C3×#52912yes—
617C22C3×#53211yescharacteristic
618C22C3×#53212yes—
619C22C3×#53312yes—
620C22C3×#53511yescharacteristic
621C22C3×#53711yescharacteristic
622C22C3×#53812yes—
623C22C3×#53912yes—
624C22C3×#54612yes—
625C22C3×D4×Q811yescharacteristic
626C22C3×#55011yescharacteristic
627C22C3×#55112yes—
628C22C3×#55212yes—
629C22C3×#55312yes—
630C22C3×#55412yes—
631C22C3×#55411yescharacteristic
632C22C3×#55611yescharacteristic
633C22C3×#55912yes—
634C4—24no—
635C4—24no—
636C4—24no—
637C4—44no—
638C4—42no—
639C4—44no—
640C4—44no—
641C4—81no—
642C4—82no—
643C4—81no—
644C3#1510311yescharacteristic, direct product factor, Sylow
645C2—82no—
646C2C3×#239611yescharacteristic
647C2C3×#248711yescharacteristic
648C2C3×#249811yescharacteristic
649C2C3×#253811yescharacteristic
650C2C3×#254411yescharacteristic
651C2C3×#255412yes—
652C2C3×#255612yes—
653C2C3×#258912yes—
654C2C3×#259612yes—
655C2C3×#263611yescharacteristic
656C2C3×#285411yescharacteristic
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
1. C2×C3×#400C2semidirect
4. C2×C3×C4×C4⋊C4C2semidirect
5. C3×#1441C2semidirect
6. C3×#1441C2semidirect
10. C3×#1436C2semidirect
11. C3×C4⋊#95C2semidirect
12. #15103C3direct (N × H)
40. C2×#400C2×C3semidirect
43. C2×C4×C4⋊C4C2×C3semidirect
44. #1441C2×C3semidirect
45. #1441C2×C3semidirect
49. #1436C2×C3semidirect
50. C4⋊#95C2×C3semidirect
644. C3#15103direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4002
C2×C3×#3992
C2×C3×#3882
C2×C3×C4×C4⋊C42
C3×#14412
C3×#14392
C3×#14372
C3×#14362
C3×C4⋊#952
#151033