Finite Group Databasebeta

← C3×#15100 #10633707

Subgroups — C3×#15100 #10633707

Order 768.

1378 conjugacy classes of subgroups.

The subgroup lattice has 646 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×#1444C211yesmaximal, characteristic, complemented
4C3×#1442C212yesmaximal, complemented
5C3×#1442C211yesmaximal, characteristic, complemented
6C3×#1439C212yesmaximal
7C3×#1439C211yesmaximal, characteristic
8C3×#1439C211yesmaximal, characteristic
9C3×#1437C212yesmaximal
10C3×#1436C212yesmaximal, complemented
11C3×#1434C211yesmaximal, characteristic, complemented
12#15100C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C22⋊C4C2211yescharacteristic
18C22×C3×C22⋊C4C2212yes—
19C22×C3×C22⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4C2212yes—
22C22×C3×C42C2211yescharacteristic
23C22×C3×C42C2212yes—
24C3×#388—24no—
25C3×C4×C4⋊C4—24no—
26C2×C3×#95C2212yes—
27C2×C3×#95C2211yescharacteristic
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2212yes—
30C2×C3×#95C2211yescharacteristic
31C2×C3×#95C2212yes—
32C2×C3×#95C2211yescharacteristic
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#1444C2×C311yescharacteristic, complemented
41#1442C2×C311yescharacteristic, complemented
42#1442C2×C312yescomplemented
43#1439C2×C311yescharacteristic
44#1439C2×C311yescharacteristic
45#1439C2×C312yes—
46#1437C2×C312yes—
47#1436C2×C312yescomplemented
48#1434C2×C311yescharacteristic, complemented
49C25×C3C2311yescharacteristic
50C23×C3×C4C2312yes—
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2312yes—
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×C4⋊C4—42no—
67C2×C3×C4⋊C4—42no—
68C2×C3×C4⋊C4—42no—
69C2×C3×C22⋊C4—22no—
70C2×C3×C22⋊C4—22no—
71C2×C3×C22⋊C4—22no—
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4—24no—
74C2×C3×C22⋊C4—24no—
75C2×C3×C22⋊C4—24no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—42no—
79C2×C3×C22⋊C4—42no—
80C2×C3×C22⋊C4—42no—
81C2×C3×C22⋊C4—42no—
82C2×C3×C22⋊C4Q814yes—
83C2×C3×C42—22no—
84C2×C3×C42—22no—
85C2×C3×C42—24no—
86C2×C3×C42—42no—
87C2×C3×C42—42no—
88C2×C3×C42D414yes—
89C3×#95—24no—
90C3×#95—22no—
91C3×#95—22no—
92C3×#95—22no—
93C3×#95—24no—
94C3×#95—24no—
95C3×#95—24no—
96C3×#95—24no—
97C3×#95—22no—
98C3×#95—42no—
99C3×#95—42no—
100C3×#95—42no—
101C22×C4⋊C4C22×C311yescharacteristic
102C22×C4⋊C4C22×C311yescharacteristic
103C22×C4⋊C4C22×C312yes—
104C22×C4⋊C4C22×C311yescharacteristic
105C22×C22⋊C4C22×C311yescharacteristic
106C22×C22⋊C4C22×C312yes—
107C22×C22⋊C4C22×C311yescharacteristic
108C22×C22⋊C4C22×C312yes—
109C22×C22⋊C4C22×C311yescharacteristic
110C22×C42C22×C311yescharacteristic
111C22×C42C22×C312yes—
112#388—24no—
113C4×C4⋊C4—24no—
114C2×#95C22×C312yes—
115C2×#95C22×C311yescharacteristic
116C2×#95C22×C312yes—
117C2×#95C22×C311yescharacteristic
118C2×#95C22×C311yescharacteristic
119C2×#95C22×C312yes—
120C2×#95C22×C312yes—
121C2×#95C22×C312yes—
122C2×#95C22×C311yescharacteristic
123C2×#95C22×C312yes—
124C2×#95C22×C312yes—
125C2×#95C22×C312yes—
126C24×C3—21no—
127C24×C3—21no—
128C24×C3—22no—
129C24×C3—22no—
130C24×C3—21no—
131C24×C3—21no—
132C24×C3—22no—
133C24×C3—22no—
134C24×C3—21no—
135C24×C3—21no—
136C24×C3C2×Q812yes—
137C24×C3C2411yescharacteristic
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
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—21no—
171C22×C3×C4—21no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
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×C4C2×Q812yes—
191C22×C3×C4C2×Q812yes—
192C22×C3×C4#4114yes—
193C22×C3×C4#4114yes—
194C22×C3×C4#4114yes—
195C22×C3×C4#4112yes—
196C22×C3×C4#4112yes—
197C22×C3×C4#4112yes—
198C3×C4⋊C4—28no—
199C3×C4⋊C4—28no—
200C3×C4⋊C4—44no—
201C3×C4⋊C4—48no—
202C3×C22⋊C4—28no—
203C3×C22⋊C4—48no—
204C3×C22⋊C4—44no—
205C3×C22⋊C4—48no—
206C3×C22⋊C4—44no—
207C3×C42—28no—
208C3×C42—48no—
209C25C23×C311yescharacteristic
210C23×C4C23×C311yescharacteristic
211C23×C4C23×C311yescharacteristic
212C23×C4C23×C312yes—
213C23×C4C23×C311yescharacteristic
214C23×C4C23×C311yescharacteristic
215C23×C4C23×C312yes—
216C23×C4C23×C312yes—
217C23×C4C23×C312yes—
218C23×C4C23×C311yescharacteristic
219C23×C4C23×C311yescharacteristic
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4C3×Q814yes—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—24no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42C3×D414yes—
249#95—24no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—22no—
258#95—42no—
259#95—42no—
260#95—42no—
261C23×C3—22no—
262C23×C3—24no—
263C23×C3—22no—
264C23×C3—24no—
265C23×C3—22no—
266C23×C3—41no—
267C23×C3—41no—
268C23×C3—41no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—42no—
272C23×C3—41no—
273C23×C3—41no—
274C23×C3—41no—
275C23×C3—42no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3—41no—
281C23×C3—41no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3C22×D411yescharacteristic
285C23×C3C22×Q811yescharacteristic
286C23×C3C2×#4111yescharacteristic
287C23×C3C2×#4111yescharacteristic
288C23×C3C2×#4111yescharacteristic
289C23×C3C2×#4112yes—
290C23×C3C2×#4112yes—
291C23×C3C2×#4112yes—
292C23×C3H(5,𝔽2)11yescharacteristic
293C23×C3H(5,𝔽2)11yescharacteristic
294C23×C3H(5,𝔽2)12yes—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—24no—
299C2×C3×C4—24no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—24no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—41no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—41no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
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—41no—
337C2×C3×C4—41no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4C22⋊Q814yes—
354C2×C3×C4#12314yes—
355C24—21no—
356C24—21no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—21no—
361C24—21no—
362C24—22no—
363C24—21no—
364C24—21no—
365C24C2×C3×Q812yes—
366C24C24×C311yescharacteristic
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
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—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4C2×C3×D412yes—
417C22×C4C2×C3×D412yes—
418C22×C4C2×C3×D412yes—
419C22×C4C2×C3×Q812yes—
420C22×C4C2×C3×Q812yes—
421C22×C4C3×#4114yes—
422C22×C4C3×#4114yes—
423C22×C4C3×#4114yes—
424C22×C4C3×#4112yes—
425C22×C4C3×#4112yes—
426C22×C4C3×#4112yes—
427C4⋊C4—28no—
428C4⋊C4—28no—
429C4⋊C4—44no—
430C4⋊C4—48no—
431C22⋊C4—28no—
432C22⋊C4—48no—
433C22⋊C4—44no—
434C22⋊C4—48no—
435C22⋊C4—44no—
436C42—28no—
437C42—48no—
438C22×C3—44no—
439C22×C3—42no—
440C22×C3—44no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3—82no—
444C22×C3—82no—
445C22×C3—81no—
446C22×C3—81no—
447C22×C3—81no—
448C22×C3—81no—
449C22×C3C2×C22⋊Q811yescharacteristic
450C22×C3C2×#12311yescharacteristic
451C22×C3#52511yescharacteristic
452C22×C3#52511yescharacteristic
453C22×C3#52912yes—
454C22×C3#52912yes—
455C22×C3#52912yes—
456C22×C3#53211yescharacteristic
457C22×C3#53312yes—
458C22×C3#53511yescharacteristic
459C22×C3#53511yescharacteristic
460C22×C3#53812yes—
461C22×C3#53812yes—
462C22×C3#55112yes—
463C22×C3#55312yes—
464C22×C3#55312yes—
465C22×C3#55311yescharacteristic
466C22×C3#55311yescharacteristic
467C22×C3#55411yescharacteristic
468C22×C3#55412yes—
469C22×C3#55411yescharacteristic
470C22×C3#55512yes—
471C22×C3#55912yes—
472C3×C4—24no—
473C3×C4—24no—
474C3×C4—42no—
475C3×C4—44no—
476C3×C4—44no—
477C3×C4—44no—
478C3×C4—42no—
479C3×C4—82no—
480C3×C4—81no—
481C3×C4—81no—
482C23—22no—
483C23—24no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—42no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—41no—
501C23—41no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23C22×C3×D411yescharacteristic
506C23C22×C3×Q811yescharacteristic
507C23C2×C3×#4111yescharacteristic
508C23C2×C3×#4112yes—
509C23C2×C3×#4111yescharacteristic
510C23C2×C3×#4111yescharacteristic
511C23C2×C3×#4112yes—
512C23C2×C3×#4112yes—
513C23C3×H(5,𝔽2)12yes—
514C23C3×H(5,𝔽2)11yescharacteristic
515C23C3×H(5,𝔽2)11yescharacteristic
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4C3×C22⋊Q814yes—
575C2×C4C3×#12314yes—
576C2×C3—82no—
577C2×C3#239611yescharacteristic
578C2×C3#248711yescharacteristic
579C2×C3#250011yescharacteristic
580C2×C3#257412yes—
581C2×C3#258011yescharacteristic
582C2×C3#258712yes—
583C2×C3#259312yes—
584C2×C3#259512yes—
585C2×C3#260311yescharacteristic
586C2×C3#263111yescharacteristic
587C2×C3#285311yescharacteristic
588C22—42no—
589C22—42no—
590C22—44no—
591C22—42no—
592C22—44no—
593C22—81no—
594C22—81no—
595C22—82no—
596C22—82no—
597C22—81no—
598C22—81no—
599C22C2×C3×C22⋊Q811yescharacteristic
600C22C2×C3×#12311yescharacteristic
601C22C3×#52511yescharacteristic
602C22C3×#52511yescharacteristic
603C22C3×#52912yes—
604C22C3×#52912yes—
605C22C3×#52912yes—
606C22C3×#53211yescharacteristic
607C22C3×#53312yes—
608C22C3×#53511yescharacteristic
609C22C3×#53511yescharacteristic
610C22C3×#53812yes—
611C22C3×#53812yes—
612C22C3×#55112yes—
613C22C3×#55311yescharacteristic
614C22C3×#55311yescharacteristic
615C22C3×#55312yes—
616C22C3×#55312yes—
617C22C3×#55411yescharacteristic
618C22C3×#55411yescharacteristic
619C22C3×#55412yes—
620C22C3×#55512yes—
621C22C3×#55912yes—
622C4—24no—
623C4—24no—
624C4—44no—
625C4—42no—
626C4—42no—
627C4—44no—
628C4—44no—
629C4—81no—
630C4—81no—
631C4—82no—
632C3#1510011yescharacteristic, direct product factor, Sylow
633C2—82no—
634C2C3×#239611yescharacteristic
635C2C3×#248711yescharacteristic
636C2C3×#250011yescharacteristic
637C2C3×#257412yes—
638C2C3×#258011yescharacteristic
639C2C3×#258712yes—
640C2C3×#259312yes—
641C2C3×#259512yes—
642C2C3×#260311yescharacteristic
643C2C3×#263111yescharacteristic
644C2C3×#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×#1444C2semidirect
4. C3×#1442C2semidirect
5. C3×#1442C2semidirect
10. C3×#1436C2semidirect
11. C3×#1434C2semidirect
12. #15100C3direct (N × H)
39. C2×C4×C4⋊C4C2×C3semidirect
40. #1444C2×C3semidirect
41. #1442C2×C3semidirect
42. #1442C2×C3semidirect
47. #1436C2×C3semidirect
48. #1434C2×C3semidirect
632. C3#15100direct (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×#14442
C3×#14422
C3×#14392
C3×#14372
C3×#14362
C3×#14342
#151003