Finite Group Databasebeta

← C3×#15134 #10633741

Subgroups — C3×#15134 #10633741

Order 768.

2006 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#396C214yesmaximal, complemented
2C2×C3×#390C211yesmaximal, characteristic
3C2×C3×#390C211yesmaximal, characteristic
4C2×C3×#390C211yesmaximal, characteristic
5C2×C3×#389C211yesmaximal, characteristic, complemented
6C2×C3×#384C211yesmaximal, characteristic, complemented
7C2×C3×C4×C4⋊C4C211yesmaximal, characteristic, complemented
8C3×#1439C214yesmaximal, complemented
9C3×C4⋊#95C211yesmaximal, characteristic, complemented
10#15134C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×D4C2211yescharacteristic
12C22×C3×C4⋊C4C2212yes—
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2214yes—
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C4⋊C4C2212yes—
18C22×C3×C22⋊C4C2212yes—
19C22×C3×C22⋊C4C2214yes—
20C22×C3×C22⋊C4C2214yes—
21C22×C3×C22⋊C4C2212yes—
22C22×C3×C42C2211yescharacteristic
23C22×C3×C42C2211yescharacteristic
24C22×C3×C42C2211yescharacteristic
25C3×#396—216no—
26C3×#390—24no—
27C3×#390—24no—
28C3×#390—24no—
29C3×#389—24no—
30C3×#384—24no—
31C3×C4×C4⋊C4—24no—
32C2×C3×#95C2214yes—
33C2×C3×#95C2212yes—
34C2×C3×#95C2212yes—
35C2×#396C2×C314yescomplemented
36C2×#390C2×C311yescharacteristic
37C2×#390C2×C311yescharacteristic
38C2×#390C2×C311yescharacteristic
39C2×#389C2×C311yescharacteristic, complemented
40C2×#384C2×C311yescharacteristic, complemented
41C2×C4×C4⋊C4C2×C311yescharacteristic, complemented
42#1439C2×C314yescomplemented
43C4⋊#95C2×C311yescharacteristic, complemented
44C25×C3C2312yes—
45C22×C3×D4—22no—
46C22×C3×D4—22no—
47C22×C3×D4—22no—
48C22×C3×D4—22no—
49C22×C3×D4—22no—
50C22×C3×D4—22no—
51C22×C3×D4Q814yes—
52C23×C3×C4C2312yes—
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2312yes—
56C23×C3×C4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2312yes—
59C23×C3×C4C2311yescharacteristic
60C2×C3×C4⋊C4—24no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4—28no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—28no—
71C2×C3×C4⋊C4—28no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—44no—
75C2×C3×C4⋊C4D414yes—
76C2×C3×C4⋊C4D418yescomplemented
77C2×C3×C22⋊C4—28no—
78C2×C3×C22⋊C4—24no—
79C2×C3×C22⋊C4—28no—
80C2×C3×C22⋊C4—24no—
81C2×C3×C22⋊C4—28no—
82C2×C3×C22⋊C4—28no—
83C2×C3×C22⋊C4—28no—
84C2×C3×C22⋊C4—28no—
85C2×C3×C22⋊C4—44no—
86C2×C3×C22⋊C4—44no—
87C2×C3×C42—22no—
88C2×C3×C42—22no—
89C2×C3×C42—22no—
90C2×C3×C42—22no—
91C2×C3×C42—22no—
92C2×C3×C42—22no—
93C2×C3×C42—22no—
94C2×C3×C42D414yes—
95C2×C3×C42D414yes—
96C3×#95—24no—
97C3×#95—24no—
98C3×#95—28no—
99C23×D4C22×C311yescharacteristic
100C22×C4⋊C4C22×C312yes—
101C22×C4⋊C4C22×C311yescharacteristic
102C22×C4⋊C4C22×C311yescharacteristic
103C22×C4⋊C4C22×C314yes—
104C22×C4⋊C4C22×C311yescharacteristic
105C22×C4⋊C4C22×C312yes—
106C22×C22⋊C4C22×C312yes—
107C22×C22⋊C4C22×C314yes—
108C22×C22⋊C4C22×C314yes—
109C22×C22⋊C4C22×C312yes—
110C22×C42C22×C311yescharacteristic
111C22×C42C22×C311yescharacteristic
112C22×C42C22×C311yescharacteristic
113#396—216no—
114#390—24no—
115#390—24no—
116#390—24no—
117#389—24no—
118#384—24no—
119C4×C4⋊C4—24no—
120C2×#95C22×C314yes—
121C2×#95C22×C312yes—
122C2×#95C22×C312yes—
123C24×C3—24no—
124C24×C3—22no—
125C24×C3—22no—
126C24×C3—22no—
127C24×C3—24no—
128C24×C3—22no—
129C24×C3—22no—
130C24×C3—22no—
131C24×C3—24no—
132C24×C3—24no—
133C24×C3C2×Q814yes—
134C24×C3C2411yescharacteristic
135C2×C3×D4—28no—
136C2×C3×D4—28no—
137C2×C3×D4—28no—
138C2×C3×D4—44no—
139C2×C3×D4—44no—
140C2×C3×D4—44no—
141C2×C3×D4—44no—
142C22×C3×C4—21no—
143C22×C3×C4—22no—
144C22×C3×C4—21no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—24no—
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—24no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—24no—
164C22×C3×C4—24no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—24no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—24no—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D414yes—
175C22×C3×C4C2×D414yes—
176C22×C3×C4C2×D414yes—
177C22×C3×C4C2×D414yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D414yes—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×Q812yes—
184C22×C3×C4#4114yes—
185C22×C3×C4#4112yes—
186C22×C3×C4#4112yes—
187C22×C3×C4#4112yes—
188C3×C4⋊C4—28no—
189C3×C4⋊C4—216no—
190C3×C4⋊C4—416no—
191C3×C4⋊C4—44no—
192C3×C4⋊C4—44no—
193C3×C4⋊C4—48no—
194C3×C22⋊C4—416no—
195C3×C22⋊C4—48no—
196C3×C22⋊C4—416no—
197C3×C22⋊C4—48no—
198C3×C42—28no—
199C3×C42—28no—
200C3×C42—44no—
201C25C23×C312yes—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4C3×Q814yes—
209C23×C4C23×C312yes—
210C23×C4C23×C312yes—
211C23×C4C23×C312yes—
212C23×C4C23×C312yes—
213C23×C4C23×C311yescharacteristic
214C23×C4C23×C311yescharacteristic
215C23×C4C23×C312yes—
216C23×C4C23×C311yescharacteristic
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4C3×D414yes—
233C2×C4⋊C4C3×D418yescomplemented
234C2×C22⋊C4—28no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42C3×D414yes—
252C2×C42C3×D414yes—
253#95—24no—
254#95—24no—
255#95—28no—
256C23×C3—24no—
257C23×C3—24no—
258C23×C3—24no—
259C23×C3—28no—
260C23×C3—28no—
261C23×C3—44no—
262C23×C3—42no—
263C23×C3—42no—
264C23×C3—42no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—44no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3—44no—
272C23×C3—44no—
273C23×C3—44no—
274C23×C3—44no—
275C23×C3—44no—
276C23×C3—44no—
277C23×C3—44no—
278C23×C3—44no—
279C23×C3C22×D411yescharacteristic
280C23×C3C22×D411yescharacteristic
281C23×C3C22×D411yescharacteristic
282C23×C3C22×D412yes—
283C23×C3C22×Q811yescharacteristic
284C23×C3C2×#4111yescharacteristic
285C23×C3C2×#4111yescharacteristic
286C23×C3C2×#4111yescharacteristic
287C23×C3C2×#4112yes—
288C23×C3H(5,𝔽2)12yes—
289C23×C32−1+412yes—
290C3×D4—416no—
291C2×C3×C4—22no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—22no—
295C2×C3×C4—24no—
296C2×C3×C4—24no—
297C2×C3×C4—24no—
298C2×C3×C4—24no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—22no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—24no—
307C2×C3×C4—28no—
308C2×C3×C4—24no—
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—44no—
317C2×C3×C4—42no—
318C2×C3×C4—44no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—44no—
325C2×C3×C4—42no—
326C2×C3×C4—44no—
327C2×C3×C4—44no—
328C2×C3×C4#12118yes—
329C2×C3×C4#12114yes—
330C2×C3×C4#12114yes—
331C2×C3×C4#12114yes—
332C2×C3×C4C22⋊Q814yes—
333C2×C3×C4C22⋊Q814yes—
334C2×C3×C4C22⋊Q814yes—
335C2×C3×C4#12614yes—
336C2×C3×C4#12714yes—
337C24—24no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—24no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—24no—
346C24—24no—
347C24C2×C3×Q814yes—
348C24C24×C311yescharacteristic
349C2×D4—28no—
350C2×D4—28no—
351C2×D4—28no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4—44no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4C2×C3×D412yes—
388C22×C4C2×C3×D414yes—
389C22×C4C2×C3×D414yes—
390C22×C4C2×C3×D414yes—
391C22×C4C2×C3×D414yes—
392C22×C4C2×C3×D412yes—
393C22×C4C2×C3×D414yes—
394C22×C4C2×C3×D412yes—
395C22×C4C2×C3×D412yes—
396C22×C4C2×C3×D412yes—
397C22×C4C2×C3×Q812yes—
398C22×C4C3×#4114yes—
399C22×C4C3×#4112yes—
400C22×C4C3×#4112yes—
401C22×C4C3×#4112yes—
402C4⋊C4—28no—
403C4⋊C4—216no—
404C4⋊C4—416no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—48no—
408C22⋊C4—416no—
409C22⋊C4—48no—
410C22⋊C4—416no—
411C22⋊C4—48no—
412C42—28no—
413C42—28no—
414C42—44no—
415C22×C3—44no—
416C22×C3—44no—
417C22×C3—48no—
418C22×C3—48no—
419C22×C3—44no—
420C22×C3—82no—
421C22×C3—82no—
422C22×C3—84no—
423C22×C3—84no—
424C22×C3—82no—
425C22×C3—82no—
426C22×C3C2×#12111yescharacteristic
427C22×C3C2×#12112yes—
428C22×C3C2×#12111yescharacteristic
429C22×C3C2×#12111yescharacteristic
430C22×C3C2×C22⋊Q811yescharacteristic
431C22×C3C2×C22⋊Q811yescharacteristic
432C22×C3C2×C22⋊Q811yescharacteristic
433C22×C3C2×#12611yescharacteristic
434C22×C3C2×#12711yescharacteristic
435C22×C3#53211yescharacteristic
436C22×C3#53212yes—
437C22×C3#53511yescharacteristic
438C22×C3#53611yescharacteristic
439C22×C3#53712yes—
440C22×C3#54212yes—
441C22×C3D4212yes—
442C22×C3#54712yes—
443C22×C3#54812yes—
444C22×C3D4×Q812yes—
445C22×C3#55012yes—
446C22×C3#55312yes—
447C22×C3#55412yes—
448C22×C3#55612yes—
449C3×C4—24no—
450C3×C4—28no—
451C3×C4—42no—
452C3×C4—44no—
453C3×C4—44no—
454C3×C4—82no—
455C3×C4—82no—
456C3×C4#39618yescomplemented
457C23—24no—
458C23—24no—
459C23—24no—
460C23—28no—
461C23—28no—
462C23—44no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—44no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23C22×C3×D411yescharacteristic
481C23C22×C3×D411yescharacteristic
482C23C22×C3×D411yescharacteristic
483C23C22×C3×D412yes—
484C23C22×C3×Q811yescharacteristic
485C23C2×C3×#4111yescharacteristic
486C23C2×C3×#4111yescharacteristic
487C23C2×C3×#4111yescharacteristic
488C23C2×C3×#4112yes—
489C23C3×H(5,𝔽2)12yes—
490C23C3×2−1+412yes—
491D4—416no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—24no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4C3×#12118yes—
530C2×C4C3×#12114yes—
531C2×C4C3×#12114yes—
532C2×C4C3×#12114yes—
533C2×C4C3×C22⋊Q814yes—
534C2×C4C3×C22⋊Q814yes—
535C2×C4C3×C22⋊Q814yes—
536C2×C4C3×#12614yes—
537C2×C4C3×#12714yes—
538C2×C3—84no—
539C2×C3C2×#39611yescharacteristic
540C2×C3#241612yes—
541C2×C3#245212yes—
542C2×C3#251212yes—
543C2×C3#253812yes—
544C2×C3#254111yescharacteristic
545C2×C3#254211yescharacteristic
546C2×C3#254511yescharacteristic
547C2×C3#256611yescharacteristic
548C2×C3#262011yescharacteristic
549C2×C3#285611yescharacteristic
550C22—44no—
551C22—48no—
552C22—48no—
553C22—44no—
554C22—44no—
555C22—82no—
556C22—84no—
557C22—84no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22C2×C3×#12111yescharacteristic
562C22C2×C3×#12111yescharacteristic
563C22C2×C3×#12111yescharacteristic
564C22C2×C3×#12112yes—
565C22C2×C3×C22⋊Q811yescharacteristic
566C22C2×C3×C22⋊Q811yescharacteristic
567C22C2×C3×C22⋊Q811yescharacteristic
568C22C2×C3×#12611yescharacteristic
569C22C2×C3×#12711yescharacteristic
570C22C3×#53212yes—
571C22C3×#53211yescharacteristic
572C22C3×#53511yescharacteristic
573C22C3×#53611yescharacteristic
574C22C3×#53712yes—
575C22C3×#54212yes—
576C22C3×D4212yes—
577C22C3×#54712yes—
578C22C3×#54812yes—
579C22C3×D4×Q812yes—
580C22C3×#55012yes—
581C22C3×#55312yes—
582C22C3×#55412yes—
583C22C3×#55612yes—
584C4—28no—
585C4—24no—
586C4—42no—
587C4—44no—
588C4—44no—
589C4—82no—
590C4—82no—
591C4C3×#39618yescomplemented
592C3#1513411yescharacteristic, direct product factor, Sylow
593C2—84no—
594C2C2×C3×#39611yescharacteristic
595C2C3×#241612yes—
596C2C3×#245212yes—
597C2C3×#251212yes—
598C2C3×#253812yes—
599C2C3×#254111yescharacteristic
600C2C3×#254211yescharacteristic
601C2C3×#254511yescharacteristic
602C2C3×#256611yescharacteristic
603C2C3×#262011yescharacteristic
604C2C3×#285611yescharacteristic
subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
prime radicalC23×C3×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#396C2semidirect
5. C2×C3×#389C2semidirect
6. C2×C3×#384C2semidirect
7. C2×C3×C4×C4⋊C4C2semidirect
8. C3×#1439C2semidirect
9. C3×C4⋊#95C2semidirect
10. #15134C3direct (N × H)
35. C2×#396C2×C3semidirect
39. C2×#389C2×C3semidirect
40. C2×#384C2×C3semidirect
41. C2×C4×C4⋊C4C2×C3semidirect
42. #1439C2×C3semidirect
43. C4⋊#95C2×C3semidirect
76. C2×C3×C4⋊C4D4semidirect
233. C2×C4⋊C4C3×D4semidirect
456. C3×C4#396semidirect
591. C4C3×#396semidirect
592. C3#15134direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3962
C2×C3×#3902
C2×C3×#3892
C2×C3×#3842
C2×C3×C4×C4⋊C42
C3×#14392
C3×C4⋊#952
#151343