Finite Group Databasebeta

← C3×#15079 #10633686

Subgroups — C3×#15079 #10633686

Order 768.

1534 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#388C211yesmaximal, characteristic
2C2×C3×#388C212yesmaximal
3C2×C3×#387C212yesmaximal, complemented
4C2×C3×C4×C4⋊C4C211yesmaximal, characteristic, complemented
5C3×#1445C212yesmaximal, complemented
6C3×#1444C211yesmaximal, characteristic, complemented
7C3×#1444C211yesmaximal, characteristic, complemented
8C3×#1439C212yesmaximal
9C3×#1439C211yesmaximal, characteristic
10C3×#1439C211yesmaximal, characteristic
11C3×#1434C211yesmaximal, characteristic, complemented
12#15079C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C4⋊C4C2212yes—
18C22×C3×C22⋊C4C2211yescharacteristic
19C22×C3×C22⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4C2212yes—
21C22×C3×C22⋊C4C2211yescharacteristic
22C22×C3×C22⋊C4C2212yes—
23C22×C3×C42C2212yes—
24C22×C3×C42C2211yescharacteristic
25C3×#388—24no—
26C3×#388—28no—
27C3×#387—28no—
28C3×C4×C4⋊C4—24no—
29C2×C3×#95C2212yes—
30C2×C3×#95C2212yes—
31C2×C3×#95C2212yes—
32C2×C3×#95C2212yes—
33C2×C3×#95C2211yescharacteristic
34C2×C3×#95C2212yes—
35C2×C3×#95C2211yescharacteristic
36C2×C3×#95C2211yescharacteristic
37C2×C3×#95C2211yescharacteristic
38C2×C3×#95C2212yes—
39C2×C3×#95C2212yes—
40C2×#388C2×C311yescharacteristic
41C2×#388C2×C312yes—
42C2×#387C2×C312yescomplemented
43C2×C4×C4⋊C4C2×C311yescharacteristic, complemented
44#1445C2×C312yescomplemented
45#1444C2×C311yescharacteristic, complemented
46#1444C2×C311yescharacteristic, complemented
47#1439C2×C311yescharacteristic
48#1439C2×C312yes—
49#1439C2×C311yescharacteristic
50#1434C2×C311yescharacteristic, complemented
51C25×C3C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2312yes—
56C23×C3×C4C2312yes—
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—24no—
66C2×C3×C4⋊C4—24no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—24no—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—24no—
73C2×C3×C4⋊C4—42no—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4—24no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—24no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—24no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—24no—
84C2×C3×C22⋊C4—24no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—42no—
88C2×C3×C42—22no—
89C2×C3×C42—22no—
90C2×C3×C42—24no—
91C2×C3×C42—24no—
92C2×C3×C42—24no—
93C2×C3×C42D414yes—
94C3×#95—24no—
95C3×#95—24no—
96C3×#95—24no—
97C3×#95—24no—
98C3×#95—24no—
99C3×#95—22no—
100C3×#95—22no—
101C3×#95—22no—
102C3×#95—22no—
103C3×#95—42no—
104C3×#95D418yes—
105C22×C4⋊C4C22×C311yescharacteristic
106C22×C4⋊C4C22×C311yescharacteristic
107C22×C4⋊C4C22×C312yes—
108C22×C4⋊C4C22×C311yescharacteristic
109C22×C4⋊C4C22×C312yes—
110C22×C22⋊C4C22×C312yes—
111C22×C22⋊C4C22×C311yescharacteristic
112C22×C22⋊C4C22×C311yescharacteristic
113C22×C22⋊C4C22×C311yescharacteristic
114C22×C22⋊C4C22×C312yes—
115C22×C42C22×C312yes—
116C22×C42C22×C311yescharacteristic
117#388—24no—
118#388—28no—
119#387—28no—
120C4×C4⋊C4—24no—
121C2×#95C22×C311yescharacteristic
122C2×#95C22×C311yescharacteristic
123C2×#95C22×C312yes—
124C2×#95C22×C312yes—
125C2×#95C22×C312yes—
126C2×#95C22×C312yes—
127C2×#95C22×C311yescharacteristic
128C2×#95C22×C312yes—
129C2×#95C22×C311yescharacteristic
130C2×#95C22×C312yes—
131C2×#95C22×C312yes—
132C24×C3—21no—
133C24×C3—22no—
134C24×C3—22no—
135C24×C3—21no—
136C24×C3—21no—
137C24×C3—22no—
138C24×C3—21no—
139C24×C3—21no—
140C24×C3—22no—
141C24×C3—21no—
142C24×C3#4112yes—
143C24×C3C2411yescharacteristic
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
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—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—22no—
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—22no—
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—21no—
184C22×C3×C4—22no—
185C22×C3×C4—21no—
186C22×C3×C4—22no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—22no—
190C22×C3×C4—21no—
191C22×C3×C4—22no—
192C22×C3×C4—21no—
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D414yes—
195C22×C3×C4C2×D414yes—
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2×D414yes—
198C22×C3×C4C2×D412yes—
199C22×C3×C4#4114yes—
200C22×C3×C4#4112yes—
201C22×C3×C4#4114yes—
202C22×C3×C4#4112yes—
203C22×C3×C4#4112yes—
204C3×C4⋊C4—28no—
205C3×C4⋊C4—28no—
206C3×C4⋊C4—216no—
207C3×C4⋊C4—48no—
208C3×C4⋊C4—44no—
209C3×C22⋊C4—216no—
210C3×C22⋊C4—28no—
211C3×C22⋊C4—48no—
212C3×C22⋊C4—44no—
213C3×C22⋊C4—44no—
214C3×C42—28no—
215C3×C42—48no—
216C25C23×C311yescharacteristic
217C23×C4C23×C311yescharacteristic
218C23×C4C23×C311yescharacteristic
219C23×C4C23×C312yes—
220C23×C4C23×C312yes—
221C23×C4C23×C311yescharacteristic
222C23×C4C23×C311yescharacteristic
223C23×C4C23×C312yes—
224C23×C4C23×C312yes—
225C23×C4C23×C311yescharacteristic
226C23×C4C23×C311yescharacteristic
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—42no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—42no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42C3×D414yes—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—42no—
269#95C3×D418yes—
270C23×C3—22no—
271C23×C3—24no—
272C23×C3—24no—
273C23×C3—22no—
274C23×C3—22no—
275C23×C3—41no—
276C23×C3—41no—
277C23×C3—41no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—41no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—41no—
291C23×C3—42no—
292C23×C3—41no—
293C23×C3C22×D411yescharacteristic
294C23×C3C22×D412yes—
295C23×C3C2×#4111yescharacteristic
296C23×C3C2×#4111yescharacteristic
297C23×C3C2×#4112yes—
298C23×C3C2×#4111yescharacteristic
299C23×C3C2×#4111yescharacteristic
300C23×C3C2×#4112yes—
301C23×C3H(5,𝔽2)11yescharacteristic
302C23×C32−1+412yes—
303C23×C32−1+411yescharacteristic
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—22no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—22no—
313C2×C3×C4—24no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—22no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—41no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—41no—
346C2×C3×C4—42no—
347C2×C3×C4—41no—
348C2×C3×C4—42no—
349C2×C3×C4—41no—
350C2×C3×C4—42no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4#12314yes—
363C2×C3×C4#12714yes—
364C24—21no—
365C24—22no—
366C24—22no—
367C24—21no—
368C24—21no—
369C24—22no—
370C24—21no—
371C24—21no—
372C24—22no—
373C24—21no—
374C24C3×#4112yes—
375C24C24×C311yescharacteristic
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4C2×C3×D412yes—
426C22×C4C2×C3×D414yes—
427C22×C4C2×C3×D414yes—
428C22×C4C2×C3×D412yes—
429C22×C4C2×C3×D414yes—
430C22×C4C2×C3×D412yes—
431C22×C4C3×#4114yes—
432C22×C4C3×#4112yes—
433C22×C4C3×#4114yes—
434C22×C4C3×#4112yes—
435C22×C4C3×#4112yes—
436C4⋊C4—28no—
437C4⋊C4—28no—
438C4⋊C4—216no—
439C4⋊C4—48no—
440C4⋊C4—44no—
441C22⋊C4—216no—
442C22⋊C4—28no—
443C22⋊C4—48no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C42—28no—
447C42—48no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—44no—
451C22×C3—44no—
452C22×C3—42no—
453C22×C3—81no—
454C22×C3—81no—
455C22×C3—82no—
456C22×C3—82no—
457C22×C3—81no—
458C22×C3—81no—
459C22×C3C2×#12311yescharacteristic
460C22×C3#52511yescharacteristic
461C22×C3#52511yescharacteristic
462C22×C3#52511yescharacteristic
463C22×C3#52512yes—
464C22×C3#52512yes—
465C22×C3#52912yes—
466C22×C3#52912yes—
467C22×C3C2×#12711yescharacteristic
468C22×C3#53611yescharacteristic
469C22×C3#53711yescharacteristic
470C22×C3#54112yes—
471C22×C3#54212yes—
472C22×C3#54612yes—
473C22×C3#54712yes—
474C22×C3#54712yes—
475C22×C3#54812yes—
476C22×C3#55212yes—
477C22×C3#55211yescharacteristic
478C22×C3#55311yescharacteristic
479C22×C3#55511yescharacteristic
480C22×C3#55612yes—
481C22×C3#55611yescharacteristic
482C3×C4—24no—
483C3×C4—24no—
484C3×C4—44no—
485C3×C4—42no—
486C3×C4—44no—
487C3×C4—44no—
488C3×C4—42no—
489C3×C4—44no—
490C3×C4—81no—
491C3×C4—81no—
492C23—24no—
493C23—22no—
494C23—24no—
495C23—22no—
496C23—22no—
497C23—41no—
498C23—41no—
499C23—42no—
500C23—41no—
501C23—42no—
502C23—42no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—41no—
513C23—41no—
514C23—42no—
515C23C22×C3×D411yescharacteristic
516C23C22×C3×D412yes—
517C23C2×C3×#4111yescharacteristic
518C23C2×C3×#4111yescharacteristic
519C23C2×C3×#4112yes—
520C23C2×C3×#4111yescharacteristic
521C23C2×C3×#4111yescharacteristic
522C23C2×C3×#4112yes—
523C23C3×H(5,𝔽2)11yescharacteristic
524C23C3×2−1+412yes—
525C23C3×2−1+411yescharacteristic
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
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—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4C3×#12314yes—
585C2×C4C3×#12714yes—
586C2×C3—82no—
587C2×C3#239711yescharacteristic
588C2×C3#246812yes—
589C2×C3#246912yes—
590C2×C3#247412yes—
591C2×C3#247512yes—
592C2×C3#248511yescharacteristic
593C2×C3#250111yescharacteristic
594C2×C3#254011yescharacteristic
595C2×C3#259911yescharacteristic
596C2×C3#263711yescharacteristic
597C2×C3#285311yescharacteristic
598C22—42no—
599C22—44no—
600C22—44no—
601C22—42no—
602C22—42no—
603C22—81no—
604C22—82no—
605C22—82no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22C2×C3×#12311yescharacteristic
610C22C3×#52511yescharacteristic
611C22C3×#52512yes—
612C22C3×#52512yes—
613C22C3×#52511yescharacteristic
614C22C3×#52511yescharacteristic
615C22C3×#52912yes—
616C22C3×#52912yes—
617C22C2×C3×#12711yescharacteristic
618C22C3×#53611yescharacteristic
619C22C3×#53711yescharacteristic
620C22C3×#54112yes—
621C22C3×#54212yes—
622C22C3×#54612yes—
623C22C3×#54712yes—
624C22C3×#54712yes—
625C22C3×#54812yes—
626C22C3×#55212yes—
627C22C3×#55211yescharacteristic
628C22C3×#55311yescharacteristic
629C22C3×#55511yescharacteristic
630C22C3×#55611yescharacteristic
631C22C3×#55612yes—
632C4—24no—
633C4—24no—
634C4—44no—
635C4—42no—
636C4—44no—
637C4—44no—
638C4—42no—
639C4—44no—
640C4—81no—
641C4—81no—
642C3#1507911yescharacteristic, direct product factor, Sylow
643C2—82no—
644C2C3×#239711yescharacteristic
645C2C3×#246812yes—
646C2C3×#246912yes—
647C2C3×#247412yes—
648C2C3×#247512yes—
649C2C3×#248511yescharacteristic
650C2C3×#250111yescharacteristic
651C2C3×#254011yescharacteristic
652C2C3×#259911yescharacteristic
653C2C3×#263711yescharacteristic
654C2C3×#285311yescharacteristic
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. C2×C3×#387C2semidirect
4. C2×C3×C4×C4⋊C4C2semidirect
5. C3×#1445C2semidirect
6. C3×#1444C2semidirect
7. C3×#1444C2semidirect
11. C3×#1434C2semidirect
12. #15079C3direct (N × H)
42. C2×#387C2×C3semidirect
43. C2×C4×C4⋊C4C2×C3semidirect
44. #1445C2×C3semidirect
45. #1444C2×C3semidirect
46. #1444C2×C3semidirect
50. #1434C2×C3semidirect
642. C3#15079direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3882
C2×C3×#3872
C2×C3×C4×C4⋊C42
C3×#14452
C3×#14442
C3×#14392
C3×#14342
#150793