Finite Group Databasebeta

← C2×C3×#3077 #10679087

Subgroups — C2×C3×#3077 #10679087

Order 768.

2530 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic, complemented
2C22×C3×C4×D4C211yesmaximal, characteristic
3C3×#3077C2116yesmaximal, direct product factor
4C2×C3×#478C214yesmaximal, complemented
5C2×C3×#476C212yesmaximal, complemented
6C2×C3×#474C212yesmaximal, complemented
7C2×C3×#428C212yesmaximal, complemented
8C2×C3×#423C211yesmaximal, characteristic, complemented
9C2×C3×#418C212yesmaximal
10C2×#3077C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×D4C2211yescharacteristic
12C24×C3×C4—21no—
13C22×C3×M16C2212yes—
14C3×#533—24no—
15C3×#533C2218yescomplemented
16C2×C3×C4⋊Q8C2211yescharacteristic, complemented
17C2×C3×C4⋊Q8C2211yescharacteristic, complemented
18C2×C3×#125C2212yescomplemented
19C2×C3×C22⋊Q8—21no—
20C2×C3×C22⋊Q8—21no—
21C2×C3×C4×Q8—21no—
22C2×C3×C4×Q8—21no—
23C2×C3×C4×D4—21no—
24C2×C3×C4×D4—21no—
25C2×C3×C4×D4—24no—
26C2×C3×C4×D4C2212yes—
27C2×C3×C4×D4C2212yes—
28C2×C3×C4×D4C2218yes—
29C2×C3×#117C2212yes—
30C22×C3×C4⋊C4C2211yescharacteristic
31C22×C3×C22⋊C4—21no—
32C22×C3×C42C2211yescharacteristic
33C3×#478C22132yescomplemented
34C3×#476C22116yescomplemented
35C3×#474C22116yescomplemented
36C3×#428C22116yescomplemented
37C2×C3×#107C2214yescomplemented
38C2×C3×#106C2214yescomplemented
39C3×#423C2218yescomplemented
40C2×C3×C4⋊C8C2212yescomplemented
41C2×C3×C4⋊C8C2212yescomplemented
42C3×#418C22116yes—
43C2×C3×#102C2214yes—
44C2×C3×#102C2214yes—
45C2×#533C2×C311yescharacteristic, complemented
46C22×C4×D4C2×C311yescharacteristic
47#3077C2×C3116yesdirect product factor
48C2×#478C2×C314yescomplemented
49C2×#476C2×C312yescomplemented
50C2×#474C2×C312yescomplemented
51C2×#428C2×C312yescomplemented
52C2×#423C2×C311yescharacteristic, complemented
53C2×#418C2×C312yes—
54C25×C3—21no—
55C22×C3×Q8—21no—
56C22×C3×Q8—21no—
57C22×C3×D4—22no—
58C22×C3×D4—21no—
59C22×C3×D4—24no—
60C22×C3×D4Q818yescomplemented
61C22×C3×D4C2312yes—
62C22×C3×D4C2314yes—
63C23×C3×C4—21no—
64C23×C3×C4—28no—
65C23×C3×C4—22no—
66C23×C3×C4—22no—
67C23×C3×C4—48no—
68C23×C3×C4D411yescharacteristic
69C23×C3×C4D411yescharacteristic
70C23×C3×C4C2311yescharacteristic
71C2×C3×M16—28no—
72C2×C3×M16C2318yes—
73C22×C3×C8C2314yes—
74C3×C4⋊Q8—22no—
75C3×C4⋊Q8—22no—
76C3×C4⋊Q8C2314yescomplemented
77C3×C4⋊Q8C2314yescomplemented
78C3×#125—24no—
79C3×#125C2318yescomplemented
80C3×C22⋊Q8—28no—
81C3×C22⋊Q8—28no—
82C3×C4×Q8—28no—
83C3×C4×Q8—28no—
84C3×C4×D4—28no—
85C3×C4×D4—28no—
86C3×C4×D4—24no—
87C3×C4×D4—24no—
88C3×C4×D4C2318yes—
89C3×C4×D4C2318yes—
90C3×#117—24no—
91C3×#117C2318yes—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4—22no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C4⋊C4C2314yes—
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4—28no—
104C2×C3×C22⋊C4—22no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C42—22no—
107C2×C3×C42D411yescharacteristic
108C2×C3×C42D411yescharacteristic
109C2×C3×C42D414yes—
110C2×C3×C42C2311yescharacteristic
111C2×C3×C42C2311yescharacteristic
112C2×C3×C42C2314yes—
113C3×#107C23116yescomplemented
114C3×#106C23116yescomplemented
115C3×C4⋊C8C2318yescomplemented
116C3×C4⋊C8C2318yescomplemented
117C3×#102C23116yes—
118C3×#102C23116yes—
119C23×D4C22×C311yescharacteristic
120C24×C4—21no—
121C22×M16C22×C312yes—
122#533—24no—
123#533C22×C318yescomplemented
124C2×C4⋊Q8C22×C311yescharacteristic, complemented
125C2×C4⋊Q8C22×C311yescharacteristic, complemented
126C2×#125C22×C312yescomplemented
127C2×C22⋊Q8—21no—
128C2×C22⋊Q8—21no—
129C2×C4×Q8—21no—
130C2×C4×Q8—21no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4—24no—
134C2×C4×D4C22×C312yes—
135C2×C4×D4C22×C318yes—
136C2×C4×D4C22×C312yes—
137C2×#117C22×C312yes—
138C22×C4⋊C4C22×C311yescharacteristic
139C22×C22⋊C4—21no—
140C22×C42C22×C311yescharacteristic
141#478C22×C3132yescomplemented
142#476C22×C3116yescomplemented
143#474C22×C3116yescomplemented
144#428C22×C3116yescomplemented
145C2×#107C22×C314yescomplemented
146C2×#106C22×C314yescomplemented
147#423C22×C318yescomplemented
148C2×C4⋊C8C22×C312yescomplemented
149C2×C4⋊C8C22×C312yescomplemented
150#418C22×C3116yes—
151C2×#102C22×C314yes—
152C2×#102C22×C314yes—
153C24×C3—28no—
154C24×C3—24no—
155C24×C3—22no—
156C24×C3—44no—
157C24×C3—44no—
158C24×C3C2×D411yescharacteristic
159C2×C3×Q8—24no—
160C2×C3×Q8—24no—
161C2×C3×Q8—44no—
162C2×C3×Q8—44no—
163C2×C3×D4—216no—
164C2×C3×D4—28no—
165C2×C3×D4—24no—
166C2×C3×D4—22no—
167C2×C3×D4—24no—
168C2×C3×D4—28no—
169C2×C3×D4C2×Q818yes—
170C2×C3×D4C2×Q8116yescomplemented
171C2×C3×D4C2414yes—
172C22×C3×C4—24no—
173C22×C3×C4—24no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—28no—
178C22×C3×C4—28no—
179C22×C3×C4—42no—
180C22×C3×C4—48no—
181C22×C3×C4—432no—
182C22×C3×C4—44no—
183C22×C3×C4—48no—
184C22×C3×C4C2×D414yes—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D414yes—
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×Q814yes—
193C22×C3×C4#4114yes—
194C22×C3×C4C2412yes—
195C22×C3×C4C2411yescharacteristic
196C3×M16—216no—
197C2×C3×C8—28no—
198C2×C3×C8C2418yes—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—28no—
202C3×C4⋊C4—28no—
203C3×C4⋊C4—24no—
204C3×C4⋊C4—24no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4C2414yes—
208C3×C4⋊C4C2414yes—
209C3×C4⋊C4C2414yes—
210C3×C22⋊C4—28no—
211C3×C22⋊C4—28no—
212C3×C22⋊C4—28no—
213C3×C42—28no—
214C3×C42C2×D414yes—
215C3×C42C2×D414yes—
216C3×C42C2×D412yes—
217C3×C42C2×D412yes—
218C3×C42C2412yes—
219C3×C42C2412yes—
220C25—21no—
221C22×Q8—21no—
222C22×Q8—21no—
223C22×D4—21no—
224C22×D4—22no—
225C22×D4—24no—
226C22×D4C3×Q818yescomplemented
227C22×D4C23×C312yes—
228C22×D4C23×C314yes—
229C23×C4—21no—
230C23×C4—22no—
231C23×C4—28no—
232C23×C4—22no—
233C23×C4—48no—
234C23×C4C3×D411yescharacteristic
235C23×C4C3×D411yescharacteristic
236C23×C4C23×C311yescharacteristic
237C2×M16—28no—
238C2×M16C23×C318yes—
239C22×C8C23×C314yes—
240C4⋊Q8—22no—
241C4⋊Q8—22no—
242C4⋊Q8C23×C314yescomplemented
243C4⋊Q8C23×C314yescomplemented
244#125—24no—
245#125C23×C318yescomplemented
246C22⋊Q8—28no—
247C22⋊Q8—28no—
248C4×Q8—28no—
249C4×Q8—28no—
250C4×D4—28no—
251C4×D4—28no—
252C4×D4—24no—
253C4×D4—24no—
254C4×D4C23×C318yes—
255C4×D4C23×C318yes—
256#117—24no—
257#117C23×C318yes—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4C23×C312yes—
265C2×C4⋊C4C23×C314yes—
266C2×C4⋊C4C23×C312yes—
267C2×C4⋊C4C23×C312yes—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—28no—
272C2×C42—22no—
273C2×C42C3×D411yescharacteristic
274C2×C42C3×D411yescharacteristic
275C2×C42C3×D414yes—
276C2×C42C23×C311yescharacteristic
277C2×C42C23×C311yescharacteristic
278C2×C42C23×C314yes—
279#107C23×C3116yescomplemented
280#106C23×C3116yescomplemented
281C4⋊C8C23×C318yescomplemented
282C4⋊C8C23×C318yescomplemented
283#102C23×C3116yes—
284#102C23×C3116yes—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—28no—
288C23×C3—216no—
289C23×C3—24no—
290C23×C3—48no—
291C23×C3—432no—
292C23×C3—48no—
293C23×C3—44no—
294C23×C3—44no—
295C23×C3C22⋊Q814yes—
296C23×C3C22×D412yes—
297C23×C3C22×D411yescharacteristic
298C3×Q8—48no—
299C3×Q8—48no—
300C3×D4—28no—
301C3×D4—216no—
302C3×D4—24no—
303C3×D4C22×Q818yes—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—28no—
308C2×C3×C4—24no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—44no—
312C2×C3×C4—416no—
313C2×C3×C4—48no—
314C2×C3×C4—44no—
315C2×C3×C4—44no—
316C2×C3×C4—44no—
317C2×C3×C4—416no—
318C2×C3×C4C22⋊Q818yes—
319C2×C3×C4C22⋊Q814yes—
320C2×C3×C4AGL(1,ℤ/8)14yes—
321C2×C3×C4C22×D412yes—
322C2×C3×C4C22×D412yes—
323C2×C3×C4C22×D412yes—
324C2×C3×C4C22×D412yes—
325C2×C3×C4C22×D411yescharacteristic
326C2×C3×C4C22×D412yes—
327C2×C3×C4C22×Q812yes—
328C2×C3×C4C22×Q814yes—
329C2×C3×C4C2×#4112yes—
330C2×C3×C4C2×#4114yes—
331C2×C3×C4C2511yescharacteristic
332C3×C8—28no—
333C24—28no—
334C24—22no—
335C24—24no—
336C24—44no—
337C24—44no—
338C24C2×C3×D411yescharacteristic
339C2×Q8—24no—
340C2×Q8—24no—
341C2×Q8—44no—
342C2×Q8—44no—
343C2×D4—216no—
344C2×D4—28no—
345C2×D4—24no—
346C2×D4—22no—
347C2×D4—24no—
348C2×D4—28no—
349C2×D4C2×C3×Q818yes—
350C2×D4C2×C3×Q8116yescomplemented
351C2×D4C24×C314yes—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—28no—
358C22×C4—28no—
359C22×C4—42no—
360C22×C4—48no—
361C22×C4—432no—
362C22×C4—44no—
363C22×C4—48no—
364C22×C4C2×C3×D411yescharacteristic
365C22×C4C2×C3×D411yescharacteristic
366C22×C4C2×C3×D411yescharacteristic
367C22×C4C2×C3×D414yes—
368C22×C4C2×C3×D414yes—
369C22×C4C2×C3×D411yescharacteristic
370C22×C4C2×C3×D412yes—
371C22×C4C2×C3×D411yescharacteristic
372C22×C4C2×C3×Q814yes—
373C22×C4C3×#4114yes—
374C22×C4C24×C311yescharacteristic
375C22×C4C24×C312yes—
376M16—216no—
377C2×C8—28no—
378C2×C8C24×C318yes—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—28no—
382C4⋊C4—28no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4C24×C314yes—
388C4⋊C4C24×C314yes—
389C4⋊C4C24×C314yes—
390C22⋊C4—28no—
391C22⋊C4—28no—
392C22⋊C4—28no—
393C42—28no—
394C42C2×C3×D414yes—
395C42C2×C3×D414yes—
396C42C2×C3×D412yes—
397C42C2×C3×D412yes—
398C42C24×C312yes—
399C42C24×C312yes—
400C22×C3—28no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—28no—
404C22×C3—432no—
405C22×C3—416no—
406C22×C3—44no—
407C22×C3—44no—
408C22×C3C2×C22⋊Q814yes—
409C22×C3C2×C22⋊Q812yes—
410C22×C3C2×AGL(1,ℤ/8)12yes—
411C22×C3#57512yes—
412C22×C3C23×D411yescharacteristic
413C3×C4—22no—
414C3×C4—22no—
415C3×C4—22no—
416C3×C4—44no—
417C3×C4—44no—
418C3×C4—44no—
419C3×C4C2×C22⋊Q814yes—
420C3×C4C2×C22⋊Q812yes—
421C3×C4C2×AGL(1,ℤ/8)14yes—
422C3×C4C23×Q811yescharacteristic
423C3×C4C22×#4111yescharacteristic
424C23—22no—
425C23—22no—
426C23—24no—
427C23—28no—
428C23—216no—
429C23—48no—
430C23—48no—
431C23—432no—
432C23—44no—
433C23—44no—
434C23C3×C22⋊Q814yes—
435C23C22×C3×D412yes—
436C23C22×C3×D411yescharacteristic
437Q8—48no—
438Q8—48no—
439D4—24no—
440D4—28no—
441D4—216no—
442D4C22×C3×Q818yes—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—28no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—44no—
451C2×C4—416no—
452C2×C4—48no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—416no—
457C2×C4C3×C22⋊Q818yes—
458C2×C4C3×C22⋊Q814yes—
459C2×C4C3×AGL(1,ℤ/8)14yes—
460C2×C4C22×C3×D412yes—
461C2×C4C22×C3×D412yes—
462C2×C4C22×C3×D412yes—
463C2×C4C22×C3×D412yes—
464C2×C4C22×C3×D412yes—
465C2×C4C22×C3×D411yescharacteristic
466C2×C4C22×C3×Q812yes—
467C2×C4C22×C3×Q814yes—
468C2×C4C2×C3×#4112yes—
469C2×C4C2×C3×#4114yes—
470C2×C4C25×C311yescharacteristic
471C8—28no—
472C2×C3—24no—
473C2×C3—48no—
474C2×C3#307714yesdirect product factor
475C2×C3C22×C22⋊Q811yescharacteristic
476C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
477C2×C3C2×#57511yescharacteristic
478C22—22no—
479C22—22no—
480C22—28no—
481C22—28no—
482C22—432no—
483C22—44no—
484C22—44no—
485C22—416no—
486C22C2×C3×C22⋊Q814yes—
487C22C2×C3×C22⋊Q812yes—
488C22C2×C3×AGL(1,ℤ/8)12yes—
489C22C3×#57512yes—
490C22C23×C3×D411yescharacteristic
491C4—22no—
492C4—22no—
493C4—22no—
494C4—44no—
495C4—44no—
496C4—44no—
497C4C2×C3×C22⋊Q814yes—
498C4C2×C3×C22⋊Q812yes—
499C4C2×C3×AGL(1,ℤ/8)14yes—
500C4C23×C3×Q811yescharacteristic
501C4C22×C3×#4111yescharacteristic
502C3C2×#307711yescharacteristic, direct product factor, Sylow
503C2—24no—
504C2—48no—
505C2C3×#307714yesdirect product factor
506C2C22×C3×C22⋊Q811yescharacteristic
507C2C22×C3×AGL(1,ℤ/8)11yescharacteristic
508C2C2×C3×#57511yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC23×C3×D4C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#533C2semidirect
3. C3×#3077C2direct (N × H)
4. C2×C3×#478C2semidirect
5. C2×C3×#476C2semidirect
6. C2×C3×#474C2semidirect
7. C2×C3×#428C2semidirect
8. C2×C3×#423C2semidirect
10. C2×#3077C3direct (N × H)
15. C3×#533C22semidirect
16. C2×C3×C4⋊Q8C22semidirect
17. C2×C3×C4⋊Q8C22semidirect
18. C2×C3×#125C22semidirect
33. C3×#478C22semidirect
34. C3×#476C22semidirect
35. C3×#474C22semidirect
36. C3×#428C22semidirect
37. C2×C3×#107C22semidirect
38. C2×C3×#106C22semidirect
39. C3×#423C22semidirect
40. C2×C3×C4⋊C8C22semidirect
41. C2×C3×C4⋊C8C22semidirect
45. C2×#533C2×C3semidirect
47. #3077C2×C3direct (N × H)
48. C2×#478C2×C3semidirect
49. C2×#476C2×C3semidirect
50. C2×#474C2×C3semidirect
51. C2×#428C2×C3semidirect
52. C2×#423C2×C3semidirect
60. C22×C3×D4Q8semidirect
76. C3×C4⋊Q8C23semidirect
77. C3×C4⋊Q8C23semidirect
79. C3×#125C23semidirect
113. C3×#107C23semidirect
114. C3×#106C23semidirect
115. C3×C4⋊C8C23semidirect
116. C3×C4⋊C8C23semidirect
123. #533C22×C3semidirect
124. C2×C4⋊Q8C22×C3semidirect
125. C2×C4⋊Q8C22×C3semidirect
126. C2×#125C22×C3semidirect
141. #478C22×C3semidirect
142. #476C22×C3semidirect
143. #474C22×C3semidirect
144. #428C22×C3semidirect
145. C2×#107C22×C3semidirect
146. C2×#106C22×C3semidirect
147. #423C22×C3semidirect
148. C2×C4⋊C8C22×C3semidirect
149. C2×C4⋊C8C22×C3semidirect
170. C2×C3×D4C2×Q8semidirect
226. C22×D4C3×Q8semidirect
242. C4⋊Q8C23×C3semidirect
243. C4⋊Q8C23×C3semidirect
245. #125C23×C3semidirect
279. #107C23×C3semidirect
280. #106C23×C3semidirect
281. C4⋊C8C23×C3semidirect
282. C4⋊C8C23×C3semidirect
350. C2×D4C2×C3×Q8semidirect
474. C2×C3#3077direct (N × H)
502. C3C2×#3077direct (N × H)
505. C2C3×#3077direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C22×C3×C4×D42
C3×#30772
C2×C3×#4782
C2×C3×#4762
C2×C3×#4742
C2×C3×#4282
C2×C3×#4232
C2×C3×#4182
C2×#30773