Finite Group Databasebeta

← C3×#25461 #10644068

Subgroups — C3×#25461 #10644068

Order 768.

792 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#501C211yesmaximal, characteristic, complemented
2C2×C3×#476C211yesmaximal, characteristic
3C3×#2566C211yesmaximal, characteristic
4C3×#2365C211yesmaximal, characteristic, complemented
5C3×#2085C211yesmaximal, characteristic, complemented
6C3×#2084C211yesmaximal, characteristic, complemented
7C3×#2067C211yesmaximal, characteristic
8C3×#2065C211yesmaximal, characteristic
9C3×#1989C211yesmaximal, characteristic, complemented
10C3×#1949C211yesmaximal, characteristic, complemented
11C3×#1948C211yesmaximal, characteristic, complemented
12C3×#1940C211yesmaximal, characteristic, complemented
13C3×#1926C211yesmaximal, characteristic
14C3×#1925C211yesmaximal, characteristic
15C3×#1762C211yesmaximal, characteristic
16#25461C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#125C2211yescharacteristic
19C2×C3×C4×Q8—21no—
20C2×C3×C4×D4C2211yescharacteristic
21C3×#501—24no—
22C3×#476—24no—
23C2×C3×#107C2211yescharacteristic
24C2×C3×#107C2211yescharacteristic
25C2×C3×#106C2211yescharacteristic
26C2×C3×#106C2211yescharacteristic
27C2×C3×#106C2211yescharacteristic
28C2×C3×#106C2211yescharacteristic
29C2×C3×C4⋊C8C2211yescharacteristic
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×C4⋊C8C2211yescharacteristic
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2211yescharacteristic
34C2×C3×#102C2211yescharacteristic
35C2×C3×#102C2211yescharacteristic
36C2×C3×#97C2211yescharacteristic
37C3×#396—21no—
38C3×#396—21no—
39C3×#391—21no—
40C3×#390C2211yescharacteristic
41C3×#390C2211yescharacteristic
42C3×#389C2211yescharacteristic
43C3×#389C2211yescharacteristic
44C3×#387—21no—
45C3×#387—21no—
46C3×#385—21no—
47C3×#385—21no—
48C3×#384C2211yescharacteristic
49C3×#382C2211yescharacteristic
50C3×C4×C4⋊C4C2211yescharacteristic
51C3×C4×C4⋊C4C2211yescharacteristic
52C3×C4×C4⋊C4C2211yescharacteristic
53C3×C4×C4⋊C4C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#340C2211yescharacteristic
59C3×#340C2211yescharacteristic
60C3×#336C2211yescharacteristic
61C3×#336C2211yescharacteristic
62C2×#501C2×C311yescharacteristic, complemented
63C2×#476C2×C311yescharacteristic
64#2566C2×C311yescharacteristic
65#2365C2×C311yescharacteristic, complemented
66#2085C2×C311yescharacteristic, complemented
67#2084C2×C311yescharacteristic, complemented
68#2067C2×C311yescharacteristic
69#2065C2×C311yescharacteristic
70#1989C2×C311yescharacteristic, complemented
71#1949C2×C311yescharacteristic, complemented
72#1948C2×C311yescharacteristic, complemented
73#1940C2×C311yescharacteristic, complemented
74#1926C2×C311yescharacteristic
75#1925C2×C311yescharacteristic
76#1762C2×C311yescharacteristic
77C22×C3×Q8—21no—
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×C4⋊Q8—22no—
85C3×C4⋊Q8D414yes—
86C3×#125—22no—
87C3×#125—22no—
88C3×C4×Q8—44no—
89C3×C4×D4—22no—
90C3×C4×D4—22no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4D411yescharacteristic
99C2×C3×C4⋊C4D411yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C42—21no—
113C2×C3×C42—21no—
114C2×C3×C42—21no—
115C2×C3×C42C2311yescharacteristic
116C2×C3×C42C2311yescharacteristic
117C2×C3×C42C2311yescharacteristic
118C3×#107—22no—
119C3×#107—22no—
120C3×#106—22no—
121C3×#106—22no—
122C3×#106—22no—
123C3×#106—22no—
124C3×C4⋊C8—22no—
125C3×C4⋊C8—22no—
126C3×C4⋊C8—22no—
127C3×#102—22no—
128C3×#102—22no—
129C3×#102—22no—
130C3×#102Q814yescomplemented
131C3×#97—22no—
132C3×#95—21no—
133C3×#95—21no—
134C3×#95—21no—
135C3×#95—21no—
136C3×#95—21no—
137C3×#95—21no—
138C3×#95—21no—
139C3×#95D411yescharacteristic
140C3×#95D411yescharacteristic
141C2×C4⋊Q8C22×C311yescharacteristic
142C2×#125C22×C311yescharacteristic
143C2×C4×Q8—21no—
144C2×C4×D4C22×C311yescharacteristic
145#501—24no—
146#476—24no—
147C2×#107C22×C311yescharacteristic
148C2×#107C22×C311yescharacteristic
149C2×#106C22×C311yescharacteristic
150C2×#106C22×C311yescharacteristic
151C2×#106C22×C311yescharacteristic
152C2×#106C22×C311yescharacteristic
153C2×C4⋊C8C22×C311yescharacteristic
154C2×C4⋊C8C22×C311yescharacteristic
155C2×C4⋊C8C22×C311yescharacteristic
156C2×#102C22×C311yescharacteristic
157C2×#102C22×C311yescharacteristic
158C2×#102C22×C311yescharacteristic
159C2×#102C22×C311yescharacteristic
160C2×#97C22×C311yescharacteristic
161#396—21no—
162#396—21no—
163#391—21no—
164#390C22×C311yescharacteristic
165#390C22×C311yescharacteristic
166#389C22×C311yescharacteristic
167#389C22×C311yescharacteristic
168#387—21no—
169#387—21no—
170#385—21no—
171#385—21no—
172#384C22×C311yescharacteristic
173#382C22×C311yescharacteristic
174C4×C4⋊C4C22×C311yescharacteristic
175C4×C4⋊C4C22×C311yescharacteristic
176C4×C4⋊C4C22×C311yescharacteristic
177C4×C4⋊C4C22×C311yescharacteristic
178#340C22×C311yescharacteristic
179#340C22×C311yescharacteristic
180#340C22×C311yescharacteristic
181#340C22×C311yescharacteristic
182#340C22×C311yescharacteristic
183#340C22×C311yescharacteristic
184#336C22×C311yescharacteristic
185#336C22×C311yescharacteristic
186C24×C3—21no—
187C2×C3×Q8—24no—
188C2×C3×Q8—42no—
189C2×C3×Q8—42no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4C2×Q812yes—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—42no—
204C22×C3×C4—81no—
205C22×C3×C4—81no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8C2×Q812yes—
224C2×C3×C8#4112yes—
225C3×C4⋊C4—24no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—41no—
239C3×C4⋊C4—42no—
240C3×C4⋊C4—42no—
241C3×C4⋊C4—42no—
242C3×C4⋊C4—42no—
243C3×C4⋊C4—81no—
244C3×C4⋊C4—81no—
245C3×C4⋊C4C2×D412yes—
246C3×C4⋊C4C2×D412yes—
247C3×C4⋊C4C2×Q812yes—
248C3×C22⋊C4—42no—
249C3×C22⋊C4—42no—
250C3×C22⋊C4—42no—
251C3×C22⋊C4—81no—
252C3×C22⋊C4—81no—
253C3×C42—21no—
254C3×C42—21no—
255C3×C42—21no—
256C3×C42—21no—
257C3×C42—42no—
258C3×C42—42no—
259C3×C42—42no—
260C3×C42C2×D412yes—
261C3×C42#4112yes—
262C22×Q8—21no—
263C22×D4C23×C311yescharacteristic
264C23×C4—21no—
265C22×C8C23×C311yescharacteristic
266C22×C8C23×C311yescharacteristic
267C22×C8C23×C311yescharacteristic
268C22×C8C23×C311yescharacteristic
269C4⋊Q8—22no—
270C4⋊Q8C3×D414yes—
271#125—22no—
272#125—22no—
273C4×Q8—44no—
274C4×D4—22no—
275C4×D4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4C3×D411yescharacteristic
284C2×C4⋊C4C3×D411yescharacteristic
285C2×C4⋊C4C23×C311yescharacteristic
286C2×C4⋊C4C23×C311yescharacteristic
287C2×C4⋊C4C23×C311yescharacteristic
288C2×C4⋊C4C23×C311yescharacteristic
289C2×C4⋊C4C23×C311yescharacteristic
290C2×C4⋊C4C23×C311yescharacteristic
291C2×C4⋊C4C23×C311yescharacteristic
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42C23×C311yescharacteristic
301C2×C42C23×C311yescharacteristic
302C2×C42C23×C311yescharacteristic
303#107—22no—
304#107—22no—
305#106—22no—
306#106—22no—
307#106—22no—
308#106—22no—
309C4⋊C8—22no—
310C4⋊C8—22no—
311C4⋊C8—22no—
312#102—22no—
313#102—22no—
314#102—22no—
315#102C3×Q814yescomplemented
316#97—22no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95C3×D411yescharacteristic
325#95C3×D411yescharacteristic
326C23×C3—22no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3C22×D411yescharacteristic
334C3×Q8—48no—
335C3×D4—22no—
336C3×D4—22no—
337C3×D4—41no—
338C3×D4—41no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—84no—
376C2×C3×C4#12112yes—
377C2×C3×C4#12112yes—
378C2×C3×C4AGL(1,ℤ/8)12yes—
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C22×Q811yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C42−1+411yescharacteristic
384C2×C3×C42−1+411yescharacteristic
385C3×C8—22no—
386C3×C8—22no—
387C3×C8—41no—
388C3×C8—41no—
389C24—21no—
390C2×Q8—24no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4C2×C3×Q812yes—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—42no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4C2×C3×D411yescharacteristic
410C22×C4C2×C3×D411yescharacteristic
411C22×C4C2×C3×D411yescharacteristic
412C22×C4C2×C3×D411yescharacteristic
413C22×C4C2×C3×D411yescharacteristic
414C22×C4C2×C3×D411yescharacteristic
415C22×C4C24×C311yescharacteristic
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8C2×C3×Q812yes—
427C2×C8C3×#4112yes—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—41no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—81no—
447C4⋊C4—81no—
448C4⋊C4C2×C3×D412yes—
449C4⋊C4C2×C3×D412yes—
450C4⋊C4C2×C3×Q812yes—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—81no—
455C22⋊C4—81no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—42no—
461C42—42no—
462C42—42no—
463C42C2×C3×D412yes—
464C42C3×#4112yes—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—81no—
469C22×C3—81no—
470C22×C3—81no—
471C22×C3—81no—
472C22×C3C2×#12111yescharacteristic
473C22×C3#54711yescharacteristic
474C22×C3D4×Q811yescharacteristic
475C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
476C22×C3#57511yescharacteristic
477C22×C3#57711yescharacteristic
478C22×C3#57711yescharacteristic
479C3×C4—22no—
480C3×C4—22no—
481C3×C4—22no—
482C3×C4—41no—
483C3×C4—41no—
484C3×C4—41no—
485C3×C4—42no—
486C3×C4—42no—
487C3×C4—42no—
488C3×C4—81no—
489C3×C4—81no—
490C3×C4—81no—
491C3×C4—81no—
492C3×C4#53311yescharacteristic
493C3×C4#54111yescharacteristic
494C3×C4#54811yescharacteristic
495C3×C4D4×Q811yescharacteristic
496C23—22no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23C22×C3×D411yescharacteristic
504Q8—48no—
505D4—22no—
506D4—22no—
507D4—41no—
508D4—41no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—84no—
546C2×C4C3×#12112yes—
547C2×C4C3×#12112yes—
548C2×C4C3×AGL(1,ℤ/8)12yes—
549C2×C4C22×C3×D411yescharacteristic
550C2×C4C22×C3×Q811yescharacteristic
551C2×C4C2×C3×#4111yescharacteristic
552C2×C4C2×C3×#4111yescharacteristic
553C2×C4C3×2−1+411yescharacteristic
554C2×C4C3×2−1+411yescharacteristic
555C8—22no—
556C8—22no—
557C8—41no—
558C8—41no—
559C2×C3—82no—
560C2×C3#242911yescharacteristic
561C2×C3#303611yescharacteristic
562C2×C3#304311yescharacteristic
563C2×C3#334511yescharacteristic
564C2×C3#335011yescharacteristic
565C2×C3#338411yescharacteristic
566C2×C3#338811yescharacteristic
567C22—42no—
568C22—42no—
569C22—42no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22C2×C3×#12111yescharacteristic
575C22C3×#54711yescharacteristic
576C22C3×D4×Q811yescharacteristic
577C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
578C22C3×#57511yescharacteristic
579C22C3×#57711yescharacteristic
580C22C3×#57711yescharacteristic
581C4—22no—
582C4—22no—
583C4—22no—
584C4—42no—
585C4—42no—
586C4—42no—
587C4—41no—
588C4—41no—
589C4—41no—
590C4—81no—
591C4—81no—
592C4—81no—
593C4—81no—
594C4C3×#53311yescharacteristic
595C4C3×#54111yescharacteristic
596C4C3×#54811yescharacteristic
597C4C3×D4×Q811yescharacteristic
598C3#2546111yescharacteristic, direct product factor, Sylow
599C2—82no—
600C2C3×#242911yescharacteristic
601C2C3×#303611yescharacteristic
602C2C3×#304311yescharacteristic
603C2C3×#334511yescharacteristic
604C2C3×#335011yescharacteristic
605C2C3×#338411yescharacteristic
606C2C3×#338811yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#501C2semidirect
4. C3×#2365C2semidirect
5. C3×#2085C2semidirect
6. C3×#2084C2semidirect
9. C3×#1989C2semidirect
10. C3×#1949C2semidirect
11. C3×#1948C2semidirect
12. C3×#1940C2semidirect
16. #25461C3direct (N × H)
62. C2×#501C2×C3semidirect
65. #2365C2×C3semidirect
66. #2085C2×C3semidirect
67. #2084C2×C3semidirect
70. #1989C2×C3semidirect
71. #1949C2×C3semidirect
72. #1948C2×C3semidirect
73. #1940C2×C3semidirect
130. C3×#102Q8semidirect
315. #102C3×Q8semidirect
598. C3#25461direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5012
C2×C3×#4762
C3×#25662
C3×#23652
C3×#20852
C3×#20842
C3×#20672
C3×#20652
C3×#19892
C3×#19492
C3×#19482
C3×#19402
C3×#19262
C3×#19252
C3×#17622
#254613