Finite Group Databasebeta

← C3×#82508 #11868798

Subgroups — C3×#82508 #11868798

Order 1152.

987 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×C4×#41C211yesmaximal, characteristic
2C2×C3×#869C211yesmaximal, characteristic, complemented
3C3×#5232C212yesmaximal
4C3×#5021C212yesmaximal, complemented
5C3×#5020C212yesmaximal, complemented
6C3×#5008C212yesmaximal, complemented
7C3×#5007C212yesmaximal, complemented
8C3×#4995C212yesmaximal, complemented
9C3×#4924C211yesmaximal, characteristic, complemented
10C3×#3079—31nomaximal
11#82508C311yesmaximal, characteristic, direct product factor
12C2×C32×#41C2211yescharacteristic
13C22×C3×Dic6—21no—
14C32×C4×Q8C2211yescharacteristic
15C32×C4×Q8C2211yescharacteristic
16C32×C4×D4—21no—
17C32×C4×D4—21no—
18C32×C4×D4C2211yescharacteristic
19C32×C4×D4C2211yescharacteristic
20C32×#117—21no—
21C32×#117C2211yescharacteristic
22C2×C32×C42—21no—
23C2×C32×C42C2211yescharacteristic
24C2×C3×C4⋊Dic3—21no—
25C2×C3×C4⋊Dic3C2212yes—
26C2×C3×#208C2212yes—
27C3×#869—21no—
28C3×#869—21no—
29C3×#869C2212yes—
30C3×#869C2212yes—
31C3×#835C2214yes—
32C3×#832C2214yes—
33C3×#808C2214yes—
34C3×#807C2214yes—
35C3×C4⋊(C3⋊C8)C2212yes—
36C3×C4⋊(C3⋊C8)C2212yes—
37C2×C3×C4⋊Q8—31no—
38C3×C4×#41—21no—
39C3×C4×#41D311yescharacteristic
40C3×C4×#41C2×C311yescharacteristic
41C2×#869C2×C311yescharacteristic, complemented
42C3×#477—32no—
43C3×#476—32no—
44C3×#475—32no—
45C3×#474—32no—
46C3×#428—32no—
47C3×#423—31no—
48C3×#417—32no—
49#5232C2×C312yes—
50#5021C2×C312yescomplemented
51#5020C2×C312yescomplemented
52#5008C2×C312yescomplemented
53#5007C2×C312yescomplemented
54#4995C2×C312yescomplemented
55#4924C2×C311yescharacteristic, complemented
56C32×#41—22no—
57C32×#41Q814yescomplemented
58C2×C32×Q8C2311yescharacteristic
59C2×C32×D4—21no—
60C2×C32×D4C2311yescharacteristic
61C22×C32×C4—21no—
62C22×C32×C4—21no—
63C22×C32×C4—21no—
64C22×C32×C4D411yescharacteristic
65C22×C32×C4D411yescharacteristic
66C22×C32×C4C2311yescharacteristic
67C22×C3×Dic3—22no—
68C2×C3×Dic6—22no—
69C2×C3×Dic6—22no—
70C2×C3×Dic6—44no—
71C32×C4⋊C4—21no—
72C32×C4⋊C4—21no—
73C32×C4⋊C4C2311yescharacteristic
74C32×C4⋊C4C2311yescharacteristic
75C32×C22⋊C4—21no—
76C32×C22⋊C4—21no—
77C32×C22⋊C4—21no—
78C32×C42—21no—
79C32×C42—21no—
80C32×C42—21no—
81C32×C42D411yescharacteristic
82C32×C42D411yescharacteristic
83C32×C42C2311yescharacteristic
84C32×C42C2311yescharacteristic
85C3×C4⋊Dic3—22no—
86C3×C4⋊Dic3—22no—
87C3×C4⋊Dic3—22no—
88C3×C4⋊Dic3C2314yes—
89C3×#208—24no—
90C2×C3×C3⋊C8C2314yes—
91#3079—31noSylow
92C2×C3×#41—21no—
93C2×C3×#41C2×D311yescharacteristic
94C2×C3×#41C22×C311yescharacteristic
95C22×C3×Q8—61no—
96C22×Dic6—21no—
97C2×C3×M16—32no—
98C3×C4⋊Q8—32no—
99C3×C4⋊Q8—32no—
100C3×C4⋊Q8—61no—
101C3×C4⋊Q8—61no—
102C3×C4×Q8—21no—
103C3×C4×Q8—21no—
104C3×C4×Q8C2×D311yescharacteristic
105C3×C4×Q8C2×D311yescharacteristic
106C3×C4×Q8C22×C311yescharacteristic
107C3×C4×Q8C22×C311yescharacteristic
108C3×C4×D4—22no—
109C3×C4×D4—22no—
110C3×C4×D4—21no—
111C3×C4×D4—21no—
112C3×C4×D4—21no—
113C3×C4×D4—21no—
114C3×C4×D4—21no—
115C3×C4×D4—21no—
116C3×C4×D4C2×D311yescharacteristic
117C3×C4×D4C2×D311yescharacteristic
118C3×C4×D4C22×C311yescharacteristic
119C3×C4×D4C22×C311yescharacteristic
120C3×#117—22no—
121C3×#117—21no—
122C3×#117—21no—
123C3×#117—21no—
124C3×#117C2×D311yescharacteristic
125C3×#117C22×C311yescharacteristic
126C2×C3×C4⋊C4—32no—
127C2×C3×C4⋊C4—61no—
128C2×C3×C42—22no—
129C2×C3×C42—21no—
130C2×C3×C42—21no—
131C2×C3×C42—21no—
132C2×C3×C42C2×D311yescharacteristic
133C2×C3×C42C22×C311yescharacteristic
134C2×C4⋊Dic3—21no—
135C2×C4⋊Dic3C22×C312yes—
136C2×#208C22×C312yes—
137#869—21no—
138#869—21no—
139#869C22×C312yes—
140#869C22×C312yes—
141C3×#107—34no—
142C3×#106—34no—
143C3×C4⋊C8—32no—
144C3×C4⋊C8—32no—
145C3×#103—34no—
146C3×#102—34no—
147#835C22×C314yes—
148#832C22×C314yes—
149#808C22×C314yes—
150#807C22×C314yes—
151C4⋊(C3⋊C8)C22×C312yes—
152C4⋊(C3⋊C8)C22×C312yes—
153C23×C32—21no—
154C23×C32C2×D411yescharacteristic
155C32×Q8—21no—
156C32×Q8C2×Q812yescomplemented
157C32×D4—24no—
158C32×D4—21no—
159C32×D4C2×Q812yescomplemented
160C2×C32×C4—21no—
161C2×C32×C4—24no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—22no—
165C2×C32×C4—22no—
166C2×C32×C4—42no—
167C2×C32×C4—42no—
168C2×C32×C4C2×D411yescharacteristic
169C2×C32×C4C2×D411yescharacteristic
170C2×C32×C4C2×D411yescharacteristic
171C2×C32×C4C2×D411yescharacteristic
172C2×C32×C4C2×D411yescharacteristic
173C2×C32×C4C2×Q812yes—
174C2×C32×C4#4112yes—
175C2×C32×C4C2411yescharacteristic
176C2×C3×Dic3—24no—
177C2×C3×Dic3—44no—
178C3×Dic6—44no—
179C3×Dic6—44no—
180C3×C3⋊C8—24no—
181C2×C4⋊Q8—31no—
182C4×#41C3×D311yescharacteristic
183#477—32no—
184#476—32no—
185#475—32no—
186#474—32no—
187#428—32no—
188#423—31no—
189#417—32no—
190C3×#41—24no—
191C3×#41—22no—
192C3×#41—22no—
193C3×#41—24no—
194C3×#41Dic614yescomplemented
195C3×#41C3×Q814yescomplemented
196C2×C3×Q8—21no—
197C2×C3×Q8—62no—
198C2×C3×Q8—62no—
199C2×C3×Q8—124no—
200C2×C3×Q8C22×D311yescharacteristic
201C2×C3×Q8C23×C311yescharacteristic
202C2×C3×D4—22no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4C22×D311yescharacteristic
207C2×C3×D4C23×C311yescharacteristic
208C22×C3×C4—22no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—22no—
212C22×C3×C4—22no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—62no—
221C22×C3×C4#6711yescharacteristic
222C22×C3×C4#6711yescharacteristic
223C22×C3×C4C3×D411yescharacteristic
224C22×C3×C4C3×D411yescharacteristic
225C22×C3×C4C22×D311yescharacteristic
226C22×C3×C4C23×C311yescharacteristic
227C22×Dic3—22no—
228C2×Dic6—22no—
229C2×Dic6—22no—
230C2×Dic6—44no—
231C3×M16—64no—
232C2×C3×C8—34no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—22no—
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—34no—
242C3×C4⋊C4—62no—
243C3×C4⋊C4—62no—
244C3×C4⋊C4—62no—
245C3×C4⋊C4C22×D311yescharacteristic
246C3×C4⋊C4C22×D311yescharacteristic
247C3×C4⋊C4C23×C311yescharacteristic
248C3×C4⋊C4C23×C311yescharacteristic
249C3×C22⋊C4—22no—
250C3×C22⋊C4—22no—
251C3×C22⋊C4—22no—
252C3×C22⋊C4—21no—
253C3×C22⋊C4—21no—
254C3×C22⋊C4—21no—
255C3×C22⋊C4—21no—
256C3×C22⋊C4—21no—
257C3×C22⋊C4—21no—
258C3×C42—22no—
259C3×C42—22no—
260C3×C42—22no—
261C3×C42—21no—
262C3×C42—21no—
263C3×C42—21no—
264C3×C42—21no—
265C3×C42—21no—
266C3×C42—21no—
267C3×C42—21no—
268C3×C42—21no—
269C3×C42—21no—
270C3×C42—21no—
271C3×C42#6711yescharacteristic
272C3×C42#6711yescharacteristic
273C3×C42C3×D411yescharacteristic
274C3×C42C3×D411yescharacteristic
275C3×C42C22×D311yescharacteristic
276C3×C42C22×D311yescharacteristic
277C3×C42C23×C311yescharacteristic
278C3×C42C23×C311yescharacteristic
279C4⋊Dic3—22no—
280C4⋊Dic3—22no—
281C4⋊Dic3—22no—
282C4⋊Dic3C23×C314yes—
283#208—24no—
284C2×C3⋊C8C23×C314yes—
285C22×C32—22no—
286C22×C32—21no—
287C22×C32—21no—
288C22×C32—41no—
289C22×C32—41no—
290C22×C32C22⋊Q812yes—
291C22×C32C22×D411yescharacteristic
292C32×C4—22no—
293C32×C4—21no—
294C32×C4—21no—
295C32×C4—41no—
296C32×C4—41no—
297C32×C4C22⋊Q812yes—
298C32×C4AGL(1,ℤ/8)12yes—
299C32×C4#13712yes—
300C32×C4C22×Q811yescharacteristic
301C32×C4C2×#4111yescharacteristic
302C3×Dic3—44no—
303C2×#41C2×C3×D311yescharacteristic
304C22×Q8—61no—
305C2×M16—32no—
306C4⋊Q8—32no—
307C4⋊Q8—32no—
308C4⋊Q8—61no—
309C4⋊Q8—61no—
310C4×Q8C2×C3×D311yescharacteristic
311C4×Q8C2×C3×D311yescharacteristic
312C4×D4—21no—
313C4×D4—21no—
314C4×D4C2×C3×D311yescharacteristic
315C4×D4C2×C3×D311yescharacteristic
316#117—21no—
317#117C2×C3×D311yescharacteristic
318C2×C4⋊C4—32no—
319C2×C4⋊C4—61no—
320C2×C42—21no—
321C2×C42C2×C3×D311yescharacteristic
322#107—34no—
323#106—34no—
324C4⋊C8—32no—
325C4⋊C8—32no—
326#103—34no—
327#102—34no—
328C23×C3—22no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3—21no—
332C23×C3C2×#6711yescharacteristic
333C23×C3C2×C3×D411yescharacteristic
334C3×Q8—22no—
335C3×Q8—21no—
336C3×Q8—21no—
337C3×Q8—22no—
338C3×Q8—124no—
339C3×Q8—124no—
340C3×Q8C2×Dic612yescomplemented
341C3×Q8C2×C3×Q812yescomplemented
342C3×D4—28no—
343C3×D4—22no—
344C3×D4—24no—
345C3×D4—24no—
346C3×D4—21no—
347C3×D4—21no—
348C3×D4—22no—
349C3×D4C2×Dic612yescomplemented
350C3×D4C2×C3×Q812yescomplemented
351C2×C3×C4—22no—
352C2×C3×C4—28no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—24no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—24no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—22no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—42no—
376C2×C3×C4—42no—
377C2×C3×C4—44no—
378C2×C3×C4—44no—
379C2×C3×C4—42no—
380C2×C3×C4—42no—
381C2×C3×C4—42no—
382C2×C3×C4—42no—
383C2×C3×C4—64no—
384C2×C3×C4—124no—
385C2×C3×C4C2×Dic612yes—
386C2×C3×C4#23512yes—
387C2×C3×C4C2×#6711yescharacteristic
388C2×C3×C4C2×#6711yescharacteristic
389C2×C3×C4C2×#6711yescharacteristic
390C2×C3×C4C2×#6711yescharacteristic
391C2×C3×C4C2×#6711yescharacteristic
392C2×C3×C4C2×C3×D411yescharacteristic
393C2×C3×C4C2×C3×D411yescharacteristic
394C2×C3×C4C2×C3×D411yescharacteristic
395C2×C3×C4C2×C3×D411yescharacteristic
396C2×C3×C4C2×C3×D411yescharacteristic
397C2×C3×C4C2×C3×Q812yes—
398C2×C3×C4C3×#4112yes—
399C2×C3×C4C23×D311yescharacteristic
400C2×C3×C4C24×C311yescharacteristic
401C2×Dic3—24no—
402C2×Dic3—44no—
403Dic6—44no—
404Dic6—44no—
405C3×C8—64no—
406C3⋊C8—24no—
407C2×C32—22no—
408C2×C32—42no—
409C2×C32C2×C22⋊Q811yescharacteristic
410C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
411C2×C32C2×#13711yescharacteristic
412#41—22no—
413#41C3×Dic614yescomplemented
414C2×Q8—62no—
415C2×Q8—62no—
416C2×Q8—124no—
417C2×Q8C22×C3×D311yescharacteristic
418C2×D4—21no—
419C2×D4C22×C3×D311yescharacteristic
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—62no—
424C22×C4C3×#6711yescharacteristic
425C22×C4C3×#6711yescharacteristic
426C22×C4C22×C3×D311yescharacteristic
427M16—64no—
428C2×C8—34no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—34no—
432C4⋊C4—62no—
433C4⋊C4—62no—
434C4⋊C4—62no—
435C4⋊C4C22×C3×D311yescharacteristic
436C4⋊C4C22×C3×D311yescharacteristic
437C22⋊C4—21no—
438C22⋊C4—21no—
439C22⋊C4—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42C3×#6711yescharacteristic
444C42C3×#6711yescharacteristic
445C42C22×C3×D311yescharacteristic
446C42C22×C3×D311yescharacteristic
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—24no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—22no—
455C22×C3—21no—
456C22×C3—41no—
457C22×C3—41no—
458C22×C3—42no—
459C22×C3—42no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3#92412yes—
465C22×C3C3×C22⋊Q812yes—
466C22×C3C22×#6711yescharacteristic
467C22×C3C22×C3×D411yescharacteristic
468C3×C4—24no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—22no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—22no—
475C3×C4—22no—
476C3×C4—22no—
477C3×C4—22no—
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—41no—
481C3×C4—41no—
482C3×C4—42no—
483C3×C4—42no—
484C3×C4—41no—
485C3×C4—41no—
486C3×C4—41no—
487C3×C4—41no—
488C3×C4—124no—
489C3×C4#92412yes—
490C3×C4#94912yes—
491C3×C4#95112yes—
492C3×C4C3×C22⋊Q812yes—
493C3×C4C3×AGL(1,ℤ/8)12yes—
494C3×C4C3×#13712yes—
495C3×C4C22×Dic611yescharacteristic
496C3×C4C2×#23511yescharacteristic
497C3×C4C22×C3×Q811yescharacteristic
498C3×C4C2×C3×#4111yescharacteristic
499Dic3—44no—
500C32#307911yescharacteristic, complemented, Sylow
501C23—21no—
502C23C2×C3×#6711yescharacteristic
503Q8—21no—
504Q8—124no—
505Q8—124no—
506Q8C2×C3×Dic612yescomplemented
507D4—24no—
508D4—21no—
509D4C2×C3×Dic612yescomplemented
510C2×C4—21no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—64no—
519C2×C4—124no—
520C2×C4C2×C3×Dic612yes—
521C2×C4C3×#23512yes—
522C2×C4C2×C3×#6711yescharacteristic
523C2×C4C2×C3×#6711yescharacteristic
524C2×C4C2×C3×#6711yescharacteristic
525C2×C4C2×C3×#6711yescharacteristic
526C2×C4C2×C3×#6711yescharacteristic
527C2×C4C23×C3×D311yescharacteristic
528C8—64no—
529C2×C3—22no—
530C2×C3—22no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—42no—
535C2×C3—44no—
536C2×C3—42no—
537C2×C3—42no—
538C2×C3C2×#92411yescharacteristic
539C2×C3C2×#94911yescharacteristic
540C2×C3C2×#95111yescharacteristic
541C2×C3C2×C3×C22⋊Q811yescharacteristic
542C2×C3C2×C3×AGL(1,ℤ/8)11yescharacteristic
543C2×C3C2×C3×#13711yescharacteristic
544C22—21no—
545C22—21no—
546C22—22no—
547C22—41no—
548C22—41no—
549C22C3×#92412yes—
550C22C22×C3×#6711yescharacteristic
551C4—22no—
552C4—21no—
553C4—21no—
554C4—41no—
555C4—41no—
556C4—124no—
557C4C3×#92412yes—
558C4C3×#94912yes—
559C4C3×#95112yes—
560C4C22×C3×Dic611yescharacteristic
561C4C2×C3×#23511yescharacteristic
562C3—21no—
563C3#8250811yescharacteristic, direct product factor
564C3C3×#307911yescharacteristic, complemented
565C2—22no—
566C2—42no—
567C2C2×C3×#92411yescharacteristic
568C2C2×C3×#94911yescharacteristic
569C2C2×C3×#95111yescharacteristic
subgroupG/H
centerC22×C3C22×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×C4×#41C2
socleC22×C32C22×D4
prime radicalC2×C32×#41C22
hypercenterC3×C4×#41D3
hypocenterC3C3×#3079
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ C3×C4×#41

Complements

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

NH ≅ G/Ntype
2. C2×C3×#869C2semidirect
4. C3×#5021C2semidirect
5. C3×#5020C2semidirect
6. C3×#5008C2semidirect
7. C3×#5007C2semidirect
8. C3×#4995C2semidirect
9. C3×#4924C2semidirect
11. #82508C3direct (N × H)
41. C2×#869C2×C3semidirect
50. #5021C2×C3semidirect
51. #5020C2×C3semidirect
52. #5008C2×C3semidirect
53. #5007C2×C3semidirect
54. #4995C2×C3semidirect
55. #4924C2×C3semidirect
57. C32×#41Q8semidirect
156. C32×Q8C2×Q8semidirect
159. C32×D4C2×Q8semidirect
194. C3×#41Dic6semidirect
195. C3×#41C3×Q8semidirect
340. C3×Q8C2×Dic6semidirect
341. C3×Q8C2×C3×Q8semidirect
349. C3×D4C2×Dic6semidirect
350. C3×D4C2×C3×Q8semidirect
413. #41C3×Dic6semidirect
500. C32#3079semidirect
506. Q8C2×C3×Dic6semidirect
509. D4C2×C3×Dic6semidirect
563. C3#82508direct (N × H)
564. C3C3×#3079semidirect

Maximal subgroups

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

subgroupindex
C32×C4×#412
C2×C3×#8692
C3×#52322
C3×#50212
C3×#50202
C3×#50082
C3×#50072
C3×#49952
C3×#49242
C3×#30793
#825083