Finite Group Databasebeta

← C3×#15110 #10633717

Subgroups — C3×#15110 #10633717

Order 768.

1334 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#388C211yesmaximal, characteristic
2C2×C3×C4×C4⋊C4C211yesmaximal, characteristic, complemented
3C3×#1443C212yesmaximal, complemented
4C3×#1442C211yesmaximal, characteristic, complemented
5C3×#1442C211yesmaximal, characteristic, complemented
6C3×#1440C212yesmaximal
7C3×#1439C211yesmaximal, characteristic
8C3×#1439C211yesmaximal, characteristic
9C3×#1437C212yesmaximal
10C3×#1436C212yesmaximal, complemented
11C3×#1434C211yesmaximal, characteristic, complemented
12#15110C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C22⋊C4C2211yescharacteristic
17C22×C3×C22⋊C4C2211yescharacteristic
18C22×C3×C22⋊C4C2211yescharacteristic
19C22×C3×C22⋊C4C2212yes—
20C22×C3×C22⋊C4C2212yes—
21C22×C3×C42C2211yescharacteristic
22C22×C3×C42C2212yes—
23C3×#388—24no—
24C3×C4×C4⋊C4—24no—
25C2×C3×#95C2212yes—
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2211yescharacteristic
28C2×C3×#95C2212yes—
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2211yescharacteristic
31C2×C3×#95C2212yes—
32C2×C3×#95C2212yes—
33C2×C3×#95C2212yes—
34C2×C3×#95C2212yes—
35C2×C3×#95C2212yes—
36C2×C3×#95C2212yes—
37C2×C3×#95C2212yes—
38C2×#388C2×C311yescharacteristic
39C2×C4×C4⋊C4C2×C311yescharacteristic, complemented
40#1443C2×C312yescomplemented
41#1442C2×C311yescharacteristic, complemented
42#1442C2×C311yescharacteristic, complemented
43#1440C2×C312yes—
44#1439C2×C311yescharacteristic
45#1439C2×C311yescharacteristic
46#1437C2×C312yes—
47#1436C2×C312yescomplemented
48#1434C2×C311yescharacteristic, complemented
49C25×C3C2311yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C23×C3×C4C2312yes—
52C23×C3×C4C2312yes—
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2312yes—
57C23×C3×C4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—24no—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—24no—
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4—24no—
70C2×C3×C22⋊C4—22no—
71C2×C3×C22⋊C4—22no—
72C2×C3×C22⋊C4—24no—
73C2×C3×C22⋊C4—24no—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—24no—
77C2×C3×C22⋊C4—42no—
78C2×C3×C22⋊C4—42no—
79C2×C3×C22⋊C4—42no—
80C2×C3×C42—22no—
81C2×C3×C42—22no—
82C2×C3×C42—24no—
83C2×C3×C42—42no—
84C2×C3×C42—42no—
85C2×C3×C42D414yes—
86C3×#95—24no—
87C3×#95—22no—
88C3×#95—22no—
89C3×#95—24no—
90C3×#95—24no—
91C3×#95—24no—
92C3×#95—22no—
93C3×#95—22no—
94C3×#95—24no—
95C3×#95—42no—
96C3×#95—42no—
97C3×#95—42no—
98C3×#95—42no—
99C22×C4⋊C4C22×C311yescharacteristic
100C22×C4⋊C4C22×C311yescharacteristic
101C22×C4⋊C4C22×C311yescharacteristic
102C22×C22⋊C4C22×C311yescharacteristic
103C22×C22⋊C4C22×C311yescharacteristic
104C22×C22⋊C4C22×C311yescharacteristic
105C22×C22⋊C4C22×C312yes—
106C22×C22⋊C4C22×C312yes—
107C22×C42C22×C311yescharacteristic
108C22×C42C22×C312yes—
109#388—24no—
110C4×C4⋊C4—24no—
111C2×#95C22×C312yes—
112C2×#95C22×C311yescharacteristic
113C2×#95C22×C311yescharacteristic
114C2×#95C22×C312yes—
115C2×#95C22×C312yes—
116C2×#95C22×C312yes—
117C2×#95C22×C312yes—
118C2×#95C22×C311yescharacteristic
119C2×#95C22×C311yescharacteristic
120C2×#95C22×C312yes—
121C2×#95C22×C312yes—
122C2×#95C22×C312yes—
123C2×#95C22×C312yes—
124C24×C3—21no—
125C24×C3—21no—
126C24×C3—22no—
127C24×C3—22no—
128C24×C3—22no—
129C24×C3—21no—
130C24×C3—21no—
131C24×C3—22no—
132C24×C3—21no—
133C24×C3—21no—
134C24×C3#4112yes—
135C24×C3C2411yescharacteristic
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—21no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
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—21no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
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—21no—
174C22×C3×C4—21no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—21no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—21no—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×D412yes—
190C22×C3×C4#4112yes—
191C22×C3×C4#4114yes—
192C22×C3×C4#4114yes—
193C22×C3×C4#4112yes—
194C22×C3×C4#4112yes—
195C22×C3×C4#4114yes—
196C3×C4⋊C4—28no—
197C3×C4⋊C4—28no—
198C3×C4⋊C4—44no—
199C3×C22⋊C4—28no—
200C3×C22⋊C4—48no—
201C3×C22⋊C4—48no—
202C3×C22⋊C4—44no—
203C3×C22⋊C4—44no—
204C3×C42—28no—
205C3×C42—48no—
206C25C23×C311yescharacteristic
207C23×C4C23×C311yescharacteristic
208C23×C4C23×C311yescharacteristic
209C23×C4C23×C312yes—
210C23×C4C23×C312yes—
211C23×C4C23×C311yescharacteristic
212C23×C4C23×C311yescharacteristic
213C23×C4C23×C312yes—
214C23×C4C23×C312yes—
215C23×C4C23×C311yescharacteristic
216C23×C4C23×C311yescharacteristic
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—24no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42C3×D414yes—
243#95—24no—
244#95—22no—
245#95—22no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—22no—
250#95—22no—
251#95—24no—
252#95—42no—
253#95—42no—
254#95—42no—
255#95—42no—
256C23×C3—22no—
257C23×C3—24no—
258C23×C3—24no—
259C23×C3—22no—
260C23×C3—22no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—41no—
269C23×C3—41no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3—42no—
275C23×C3—41no—
276C23×C3—42no—
277C23×C3—41no—
278C23×C3—42no—
279C23×C3C22×D411yescharacteristic
280C23×C3C2×#4111yescharacteristic
281C23×C3C2×#4111yescharacteristic
282C23×C3C2×#4112yes—
283C23×C3C2×#4111yescharacteristic
284C23×C3C2×#4112yes—
285C23×C3C2×#4111yescharacteristic
286C23×C3C2×#4112yes—
287C23×C3H(5,𝔽2)12yes—
288C23×C3H(5,𝔽2)11yescharacteristic
289C23×C32−1+411yescharacteristic
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—24no—
294C2×C3×C4—24no—
295C2×C3×C4—22no—
296C2×C3×C4—24no—
297C2×C3×C4—24no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—24no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—22no—
305C2×C3×C4—24no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—42no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
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×C4—41no—
320C2×C3×C4—42no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
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—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4#12314yes—
349C2×C3×C4#12414yes—
350C24—21no—
351C24—21no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—21no—
356C24—21no—
357C24—22no—
358C24—21no—
359C24—21no—
360C24C3×#4112yes—
361C24C24×C311yescharacteristic
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4C2×C3×D412yes—
414C22×C4C2×C3×D412yes—
415C22×C4C2×C3×D412yes—
416C22×C4C3×#4112yes—
417C22×C4C3×#4114yes—
418C22×C4C3×#4114yes—
419C22×C4C3×#4112yes—
420C22×C4C3×#4112yes—
421C22×C4C3×#4114yes—
422C4⋊C4—28no—
423C4⋊C4—28no—
424C4⋊C4—44no—
425C22⋊C4—28no—
426C22⋊C4—48no—
427C22⋊C4—48no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C42—28no—
431C42—48no—
432C22×C3—44no—
433C22×C3—44no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3—81no—
438C22×C3—82no—
439C22×C3—82no—
440C22×C3—81no—
441C22×C3—81no—
442C22×C3—81no—
443C22×C3C2×#12311yescharacteristic
444C22×C3#52511yescharacteristic
445C22×C3#52511yescharacteristic
446C22×C3#52511yescharacteristic
447C22×C3C2×#12411yescharacteristic
448C22×C3#52912yes—
449C22×C3#52912yes—
450C22×C3#52912yes—
451C22×C3#52912yes—
452C22×C3#53511yescharacteristic
453C22×C3#53711yescharacteristic
454C22×C3#53812yes—
455C22×C3#53912yes—
456C22×C3#55112yes—
457C22×C3#55211yescharacteristic
458C22×C3#55312yes—
459C22×C3#55312yes—
460C22×C3#55312yes—
461C22×C3#55311yescharacteristic
462C22×C3#55511yescharacteristic
463C22×C3#55612yes—
464C22×C3#55611yescharacteristic
465C22×C3#55912yes—
466C3×C4—24no—
467C3×C4—24no—
468C3×C4—44no—
469C3×C4—44no—
470C3×C4—42no—
471C3×C4—42no—
472C3×C4—81no—
473C3×C4—82no—
474C3×C4—82no—
475C3×C4—81no—
476C23—22no—
477C23—24no—
478C23—24no—
479C23—22no—
480C23—22no—
481C23—42no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—41no—
490C23—41no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—41no—
497C23—41no—
498C23—42no—
499C23C22×C3×D411yescharacteristic
500C23C2×C3×#4112yes—
501C23C2×C3×#4112yes—
502C23C2×C3×#4111yescharacteristic
503C23C2×C3×#4111yescharacteristic
504C23C2×C3×#4112yes—
505C23C2×C3×#4111yescharacteristic
506C23C2×C3×#4111yescharacteristic
507C23C3×H(5,𝔽2)11yescharacteristic
508C23C3×H(5,𝔽2)12yes—
509C23C3×2−1+411yescharacteristic
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4C3×#12314yes—
569C2×C4C3×#12414yes—
570C2×C3—82no—
571C2×C3#239811yescharacteristic
572C2×C3#248511yescharacteristic
573C2×C3#250411yescharacteristic
574C2×C3#257312yes—
575C2×C3#259211yescharacteristic
576C2×C3#259312yes—
577C2×C3#259412yes—
578C2×C3#259512yes—
579C2×C3#259811yescharacteristic
580C2×C3#263611yescharacteristic
581C2×C3#285311yescharacteristic
582C22—42no—
583C22—42no—
584C22—44no—
585C22—44no—
586C22—42no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—82no—
591C22—82no—
592C22—81no—
593C22C2×C3×#12311yescharacteristic
594C22C3×#52511yescharacteristic
595C22C3×#52511yescharacteristic
596C22C3×#52511yescharacteristic
597C22C2×C3×#12411yescharacteristic
598C22C3×#52912yes—
599C22C3×#52912yes—
600C22C3×#52912yes—
601C22C3×#52912yes—
602C22C3×#53511yescharacteristic
603C22C3×#53711yescharacteristic
604C22C3×#53812yes—
605C22C3×#53912yes—
606C22C3×#55112yes—
607C22C3×#55211yescharacteristic
608C22C3×#55311yescharacteristic
609C22C3×#55312yes—
610C22C3×#55312yes—
611C22C3×#55312yes—
612C22C3×#55511yescharacteristic
613C22C3×#55611yescharacteristic
614C22C3×#55612yes—
615C22C3×#55912yes—
616C4—24no—
617C4—24no—
618C4—44no—
619C4—42no—
620C4—42no—
621C4—44no—
622C4—81no—
623C4—81no—
624C4—82no—
625C4—82no—
626C3#1511011yescharacteristic, direct product factor, Sylow
627C2—82no—
628C2C3×#239811yescharacteristic
629C2C3×#248511yescharacteristic
630C2C3×#250411yescharacteristic
631C2C3×#257312yes—
632C2C3×#259211yescharacteristic
633C2C3×#259312yes—
634C2C3×#259412yes—
635C2C3×#259512yes—
636C2C3×#259811yescharacteristic
637C2C3×#263611yescharacteristic
638C2C3×#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
2. C2×C3×C4×C4⋊C4C2semidirect
3. C3×#1443C2semidirect
4. C3×#1442C2semidirect
5. C3×#1442C2semidirect
10. C3×#1436C2semidirect
11. C3×#1434C2semidirect
12. #15110C3direct (N × H)
39. C2×C4×C4⋊C4C2×C3semidirect
40. #1443C2×C3semidirect
41. #1442C2×C3semidirect
42. #1442C2×C3semidirect
47. #1436C2×C3semidirect
48. #1434C2×C3semidirect
626. C3#15110direct (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×C4×C4⋊C42
C3×#14432
C3×#14422
C3×#14402
C3×#14392
C3×#14372
C3×#14362
C3×#14342
#151103