Finite Group Databasebeta

← C3×#25465 #10644072

Subgroups — C3×#25465 #10644072

Order 768.

812 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#501C211yesmaximal, characteristic, complemented
2C2×C3×#478C211yesmaximal, characteristic
3C3×#2542C211yesmaximal, characteristic
4C3×#2370C211yesmaximal, characteristic, complemented
5C3×#2084C211yesmaximal, characteristic, complemented
6C3×#2083C211yesmaximal, characteristic, complemented
7C3×#2067C211yesmaximal, characteristic
8C3×#2065C211yesmaximal, characteristic
9C3×#1987C211yesmaximal, characteristic, complemented
10C3×#1948C211yesmaximal, characteristic, complemented
11C3×#1947C211yesmaximal, characteristic, complemented
12C3×#1940C211yesmaximal, characteristic, complemented
13C3×#1926C211yesmaximal, characteristic
14C3×#1925C211yesmaximal, characteristic
15C3×#1762C211yesmaximal, characteristic
16#25465C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#125C2211yescharacteristic
19C2×C3×C22⋊Q8—21no—
20C2×C3×C22⋊Q8—21no—
21C2×C3×C4×D4C2211yescharacteristic
22C3×#501—24no—
23C3×#478—24no—
24C2×C3×#107C2211yescharacteristic
25C2×C3×#107C2211yescharacteristic
26C2×C3×#107C2211yescharacteristic
27C2×C3×#106C2211yescharacteristic
28C2×C3×#106C2211yescharacteristic
29C2×C3×#106C2211yescharacteristic
30C2×C3×C4⋊C8C2211yescharacteristic
31C2×C3×C4⋊C8C2211yescharacteristic
32C2×C3×C4⋊C8C2211yescharacteristic
33C2×C3×#102C2211yescharacteristic
34C2×C3×#102C2211yescharacteristic
35C2×C3×#102C2211yescharacteristic
36C2×C3×#102C2211yescharacteristic
37C2×C3×#97C2211yescharacteristic
38C3×#399—21no—
39C3×#399—21no—
40C3×#390C2211yescharacteristic
41C3×#390C2211yescharacteristic
42C3×#389—21no—
43C3×#389C2211yescharacteristic
44C3×#389C2211yescharacteristic
45C3×#387—21no—
46C3×#387—21no—
47C3×#384C2211yescharacteristic
48C3×#384C2211yescharacteristic
49C3×C4×C4⋊C4—21no—
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×#478C2×C311yescharacteristic
64#2542C2×C311yescharacteristic
65#2370C2×C311yescharacteristic, complemented
66#2084C2×C311yescharacteristic, complemented
67#2083C2×C311yescharacteristic, complemented
68#2067C2×C311yescharacteristic
69#2065C2×C311yescharacteristic
70#1987C2×C311yescharacteristic, complemented
71#1948C2×C311yescharacteristic, complemented
72#1947C2×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⋊Q8—22no—
86C3×#125—22no—
87C3×#125D414yes—
88C3×C22⋊Q8—44no—
89C3×C22⋊Q8—44no—
90C3×C4×D4—22no—
91C3×C4×D4—22no—
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⋊C4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4D411yescharacteristic
102C2×C3×C4⋊C4D411yescharacteristic
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4—21no—
114C2×C3×C22⋊C4—21no—
115C2×C3×C42—21no—
116C2×C3×C42—21no—
117C2×C3×C42—21no—
118C2×C3×C42C2311yescharacteristic
119C2×C3×C42C2311yescharacteristic
120C2×C3×C42C2311yescharacteristic
121C3×#107—22no—
122C3×#107—22no—
123C3×#107—22no—
124C3×#106—22no—
125C3×#106—22no—
126C3×#106—22no—
127C3×C4⋊C8—22no—
128C3×C4⋊C8—22no—
129C3×C4⋊C8—22no—
130C3×#102—22no—
131C3×#102—22no—
132C3×#102—22no—
133C3×#102Q814yescomplemented
134C3×#97—22no—
135C3×#95—21no—
136C3×#95—21no—
137C3×#95—21no—
138C3×#95—21no—
139C3×#95—21no—
140C3×#95D411yescharacteristic
141C3×#95D411yescharacteristic
142C2×C4⋊Q8C22×C311yescharacteristic
143C2×#125C22×C311yescharacteristic
144C2×C22⋊Q8—21no—
145C2×C22⋊Q8—21no—
146C2×C4×D4C22×C311yescharacteristic
147#501—24no—
148#478—24no—
149C2×#107C22×C311yescharacteristic
150C2×#107C22×C311yescharacteristic
151C2×#107C22×C311yescharacteristic
152C2×#106C22×C311yescharacteristic
153C2×#106C22×C311yescharacteristic
154C2×#106C22×C311yescharacteristic
155C2×C4⋊C8C22×C311yescharacteristic
156C2×C4⋊C8C22×C311yescharacteristic
157C2×C4⋊C8C22×C311yescharacteristic
158C2×#102C22×C311yescharacteristic
159C2×#102C22×C311yescharacteristic
160C2×#102C22×C311yescharacteristic
161C2×#102C22×C311yescharacteristic
162C2×#97C22×C311yescharacteristic
163#399—21no—
164#399—21no—
165#390C22×C311yescharacteristic
166#390C22×C311yescharacteristic
167#389—21no—
168#389C22×C311yescharacteristic
169#389C22×C311yescharacteristic
170#387—21no—
171#387—21no—
172#384C22×C311yescharacteristic
173#384C22×C311yescharacteristic
174C4×C4⋊C4—21no—
175C4×C4⋊C4C22×C311yescharacteristic
176C4×C4⋊C4C22×C311yescharacteristic
177C4×C4⋊C4C22×C311yescharacteristic
178C4×C4⋊C4C22×C311yescharacteristic
179#340C22×C311yescharacteristic
180#340C22×C311yescharacteristic
181#340C22×C311yescharacteristic
182#340C22×C311yescharacteristic
183#340C22×C311yescharacteristic
184#340C22×C311yescharacteristic
185#336C22×C311yescharacteristic
186#336C22×C311yescharacteristic
187C24×C3—21no—
188C2×C3×Q8—42no—
189C2×C3×Q8—42no—
190C2×C3×Q8—42no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4C2×Q812yes—
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—21no—
204C22×C3×C4—42no—
205C22×C3×C4—42no—
206C22×C3×C4—42no—
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2411yescharacteristic
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×C8—21no—
224C2×C3×C8C2×Q812yes—
225C2×C3×C8#4112yes—
226C3×C4⋊C4—24no—
227C3×C4⋊C4—24no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—22no—
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—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—42no—
242C3×C4⋊C4—42no—
243C3×C4⋊C4—42no—
244C3×C4⋊C4—42no—
245C3×C4⋊C4—42no—
246C3×C4⋊C4—42no—
247C3×C4⋊C4—81no—
248C3×C4⋊C4C2×D412yes—
249C3×C4⋊C4C2×D412yes—
250C3×C4⋊C4C2×Q812yes—
251C3×C22⋊C4—42no—
252C3×C22⋊C4—42no—
253C3×C22⋊C4—42no—
254C3×C22⋊C4—42no—
255C3×C22⋊C4—42no—
256C3×C42—21no—
257C3×C42—21no—
258C3×C42—21no—
259C3×C42—21no—
260C3×C42—42no—
261C3×C42—81no—
262C3×C42—81no—
263C3×C42C2×D412yes—
264C3×C42#4112yes—
265C22×Q8—21no—
266C22×D4C23×C311yescharacteristic
267C23×C4—21no—
268C22×C8C23×C311yescharacteristic
269C22×C8C23×C311yescharacteristic
270C22×C8C23×C311yescharacteristic
271C22×C8C23×C311yescharacteristic
272C4⋊Q8—22no—
273C4⋊Q8—22no—
274#125—22no—
275#125C3×D414yes—
276C22⋊Q8—44no—
277C22⋊Q8—44no—
278C4×D4—22no—
279C4×D4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4C3×D411yescharacteristic
290C2×C4⋊C4C3×D411yescharacteristic
291C2×C4⋊C4C23×C311yescharacteristic
292C2×C4⋊C4C23×C311yescharacteristic
293C2×C4⋊C4C23×C311yescharacteristic
294C2×C4⋊C4C23×C311yescharacteristic
295C2×C4⋊C4C23×C311yescharacteristic
296C2×C4⋊C4C23×C311yescharacteristic
297C2×C4⋊C4C23×C311yescharacteristic
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42C23×C311yescharacteristic
307C2×C42C23×C311yescharacteristic
308C2×C42C23×C311yescharacteristic
309#107—22no—
310#107—22no—
311#107—22no—
312#106—22no—
313#106—22no—
314#106—22no—
315C4⋊C8—22no—
316C4⋊C8—22no—
317C4⋊C8—22no—
318#102—22no—
319#102—22no—
320#102—22no—
321#102C3×Q814yescomplemented
322#97—22no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95C3×D411yescharacteristic
329#95C3×D411yescharacteristic
330C23×C3—22no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3—41no—
336C23×C3—41no—
337C23×C3C22×D411yescharacteristic
338C3×Q8—84no—
339C3×D4—22no—
340C3×D4—22no—
341C3×D4—41no—
342C3×D4—41no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
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—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—84no—
379C2×C3×C4#12112yes—
380C2×C3×C4#12112yes—
381C2×C3×C4AGL(1,ℤ/8)12yes—
382C2×C3×C4AGL(1,ℤ/8)12yes—
383C2×C3×C4C22×D411yescharacteristic
384C2×C3×C4C22×Q811yescharacteristic
385C2×C3×C4C2×#4111yescharacteristic
386C2×C3×C4C2×#4111yescharacteristic
387C2×C3×C42−1+411yescharacteristic
388C2×C3×C42−1+411yescharacteristic
389C3×C8—22no—
390C3×C8—22no—
391C3×C8—41no—
392C3×C8—41no—
393C24—21no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4C2×C3×Q812yes—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4C2×C3×D411yescharacteristic
414C22×C4C2×C3×D411yescharacteristic
415C22×C4C2×C3×D411yescharacteristic
416C22×C4C2×C3×D411yescharacteristic
417C22×C4C2×C3×D411yescharacteristic
418C22×C4C2×C3×D411yescharacteristic
419C22×C4C24×C311yescharacteristic
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8C2×C3×Q812yes—
431C2×C8C3×#4112yes—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—81no—
454C4⋊C4C2×C3×D412yes—
455C4⋊C4C2×C3×D412yes—
456C4⋊C4C2×C3×Q812yes—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—42no—
467C42—81no—
468C42—81no—
469C42C2×C3×D412yes—
470C42C3×#4112yes—
471C22×C3—42no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—81no—
475C22×C3—81no—
476C22×C3—81no—
477C22×C3—81no—
478C22×C3C2×#12111yescharacteristic
479C22×C3#54711yescharacteristic
480C22×C3D4×Q811yescharacteristic
481C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
482C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
483C22×C3#57711yescharacteristic
484C22×C3#57811yescharacteristic
485C3×C4—22no—
486C3×C4—22no—
487C3×C4—22no—
488C3×C4—22no—
489C3×C4—41no—
490C3×C4—41no—
491C3×C4—42no—
492C3×C4—42no—
493C3×C4—42no—
494C3×C4—81no—
495C3×C4—81no—
496C3×C4—81no—
497C3×C4—81no—
498C3×C4#53311yescharacteristic
499C3×C4#54111yescharacteristic
500C3×C4#54811yescharacteristic
501C3×C4D4×Q811yescharacteristic
502C23—22no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23C22×C3×D411yescharacteristic
510Q8—84no—
511D4—22no—
512D4—22no—
513D4—41no—
514D4—41no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
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—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—84no—
551C2×C4C3×#12112yes—
552C2×C4C3×#12112yes—
553C2×C4C3×AGL(1,ℤ/8)12yes—
554C2×C4C3×AGL(1,ℤ/8)12yes—
555C2×C4C22×C3×D411yescharacteristic
556C2×C4C22×C3×Q811yescharacteristic
557C2×C4C2×C3×#4111yescharacteristic
558C2×C4C2×C3×#4111yescharacteristic
559C2×C4C3×2−1+411yescharacteristic
560C2×C4C3×2−1+411yescharacteristic
561C8—22no—
562C8—22no—
563C8—41no—
564C8—41no—
565C2×C3—82no—
566C2×C3#242911yescharacteristic
567C2×C3#303811yescharacteristic
568C2×C3#304611yescharacteristic
569C2×C3#334311yescharacteristic
570C2×C3#334511yescharacteristic
571C2×C3#338411yescharacteristic
572C2×C3#338611yescharacteristic
573C22—42no—
574C22—42no—
575C22—42no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22C2×C3×#12111yescharacteristic
581C22C3×#54711yescharacteristic
582C22C3×D4×Q811yescharacteristic
583C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
584C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
585C22C3×#57711yescharacteristic
586C22C3×#57811yescharacteristic
587C4—22no—
588C4—22no—
589C4—22no—
590C4—22no—
591C4—42no—
592C4—42no—
593C4—42no—
594C4—41no—
595C4—41no—
596C4—81no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4C3×#53311yescharacteristic
601C4C3×#54111yescharacteristic
602C4C3×#54811yescharacteristic
603C4C3×D4×Q811yescharacteristic
604C3#2546511yescharacteristic, direct product factor, Sylow
605C2—82no—
606C2C3×#242911yescharacteristic
607C2C3×#303811yescharacteristic
608C2C3×#304611yescharacteristic
609C2C3×#334311yescharacteristic
610C2C3×#334511yescharacteristic
611C2C3×#338411yescharacteristic
612C2C3×#338611yescharacteristic
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×#2370C2semidirect
5. C3×#2084C2semidirect
6. C3×#2083C2semidirect
9. C3×#1987C2semidirect
10. C3×#1948C2semidirect
11. C3×#1947C2semidirect
12. C3×#1940C2semidirect
16. #25465C3direct (N × H)
62. C2×#501C2×C3semidirect
65. #2370C2×C3semidirect
66. #2084C2×C3semidirect
67. #2083C2×C3semidirect
70. #1987C2×C3semidirect
71. #1948C2×C3semidirect
72. #1947C2×C3semidirect
73. #1940C2×C3semidirect
133. C3×#102Q8semidirect
321. #102C3×Q8semidirect
604. C3#25465direct (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×#4782
C3×#25422
C3×#23702
C3×#20842
C3×#20832
C3×#20672
C3×#20652
C3×#19872
C3×#19482
C3×#19472
C3×#19402
C3×#19262
C3×#19252
C3×#17622
#254653