Finite Group Databasebeta

← C2×C3×#1520 #10636172

Subgroups — C2×C3×#1520 #10636172

Order 768.

1522 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C22⋊Q8C211yesmaximal, characteristic
2C22×C3×C22⋊C8C211yesmaximal, characteristic
3C2×C3×#380C211yesmaximal, characteristic
4C3×#1520C218yesmaximal, direct product factor
5C2×C3×#328C214yesmaximal, complemented
6C2×#1520C311yesmaximal, characteristic, direct product factor, Sylow
7C23×C3×Q8C411yescharacteristic, complemented
8C24×C3×C4C2211yescharacteristic
9C23×C3×C8—21no—
10C2×C3×C22⋊Q8—21no—
11C2×C3×C22⋊Q8—22no—
12C2×C3×C22⋊Q8—21no—
13C2×C3×C22⋊Q8—22no—
14C2×C3×C22⋊Q8C412yescomplemented
15C2×C3×C22⋊Q8C414yescomplemented
16C2×C3×C22⋊Q8C2212yes—
17C2×C3×C22⋊Q8C2214yes—
18C22×C3×C4⋊C4—21no—
19C22×C3×C4⋊C4—21no—
20C22×C3×C4⋊C4C411yescharacteristic, complemented
21C22×C3×C22⋊C4—21no—
22C22×C3×C22⋊C4—21no—
23C2×C3×C22⋊C8—21no—
24C2×C3×C22⋊C8—22no—
25C2×C3×C22⋊C8C2212yes—
26C2×C3×C22⋊C8C2214yes—
27C3×#380—22no—
28C3×#380C2214yes—
29C2×C3×#95C2212yes—
30C3×#328C22116yescomplemented
31C22×C22⋊Q8C2×C311yescharacteristic
32C22×C22⋊C8C2×C311yescharacteristic
33C2×#380C2×C311yescharacteristic
34#1520C2×C318yesdirect product factor
35C2×#328C2×C314yescomplemented
36C25×C3D411yescharacteristic
37C22×C3×Q8—22no—
38C22×C3×Q8—21no—
39C22×C3×Q8—44no—
40C22×C3×Q8C2×C412yes—
41C22×C3×Q8C2×C414yescomplemented
42C23×C3×C4—21no—
43C23×C3×C4—21no—
44C23×C3×C4—22no—
45C23×C3×C4—44no—
46C23×C3×C4C2×C411yescharacteristic
47C23×C3×C4C2×C412yes—
48C23×C3×C4C2×C411yescharacteristic
49C23×C3×C4D411yescharacteristic
50C23×C3×C4D411yescharacteristic
51C23×C3×C4D412yes—
52C23×C3×C4D411yescharacteristic
53C23×C3×C4D412yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2312yes—
56C22×C3×C8—24no—
57C22×C3×C8—22no—
58C22×C3×C8—44no—
59C3×C22⋊Q8—24no—
60C3×C22⋊Q8—24no—
61C3×C22⋊Q8—24no—
62C3×C22⋊Q8—22no—
63C3×C22⋊Q8—24no—
64C3×C22⋊Q8—22no—
65C3×C22⋊Q8—24no—
66C3×C22⋊Q8C2×C414yes—
67C3×C22⋊Q8C2×C418yescomplemented
68C3×C22⋊Q8C2314yes—
69C2×C3×C4⋊C4—24no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—24no—
75C2×C3×C4⋊C4—24no—
76C2×C3×C4⋊C4—44no—
77C2×C3×C4⋊C4C2×C412yes—
78C2×C3×C4⋊C4C2×C414yescomplemented
79C2×C3×C22⋊C4—24no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—24no—
83C2×C3×C22⋊C4—24no—
84C2×C3×C22⋊C4—44no—
85C3×C22⋊C8—24no—
86C3×C22⋊C8—22no—
87C3×C22⋊C8C2314yes—
88C3×#95—22no—
89C3×#95C2314yes—
90C23×Q8C3×C411yescharacteristic, complemented
91C24×C4C22×C311yescharacteristic
92C23×C8—21no—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8C3×C412yescomplemented
98C2×C22⋊Q8C3×C414yescomplemented
99C2×C22⋊Q8C22×C312yes—
100C2×C22⋊Q8C22×C314yes—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4C3×C411yescharacteristic, complemented
104C22×C22⋊C4—21no—
105C22×C22⋊C4—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8C22×C312yes—
109C2×C22⋊C8C22×C314yes—
110#380—22no—
111#380C22×C314yes—
112C2×#95C22×C312yes—
113#328C22×C3116yescomplemented
114C24×C3—42no—
115C24×C3—42no—
116C24×C3C22⋊C411yescharacteristic
117C24×C3C22⋊C411yescharacteristic
118C24×C3C22⋊C412yes—
119C24×C3SD1614yes—
120C24×C3Q1614yes—
121C24×C3C2×D411yescharacteristic
122C24×C3C2×D412yes—
123C2×C3×Q8—22no—
124C2×C3×Q8—24no—
125C2×C3×Q8—48no—
126C2×C3×Q8—48no—
127C2×C3×Q8—44no—
128C2×C3×Q8—44no—
129C2×C3×Q8C22×C414yes—
130C22×C3×C4—24no—
131C22×C3×C4—24no—
132C22×C3×C4—24no—
133C22×C3×C4—24no—
134C22×C3×C4—24no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—24no—
138C22×C3×C4—44no—
139C22×C3×C4—44no—
140C22×C3×C4—44no—
141C22×C3×C4—44no—
142C22×C3×C4—42no—
143C22×C3×C4—42no—
144C22×C3×C4—42no—
145C22×C3×C4—44no—
146C22×C3×C4—44no—
147C22×C3×C4—44no—
148C22×C3×C4—44no—
149C22×C3×C4—44no—
150C22×C3×C4C22⋊C412yes—
151C22×C3×C4C22⋊C412yes—
152C22×C3×C4C22⋊C412yes—
153C22×C3×C4C22⋊C411yescharacteristic
154C22×C3×C4C22⋊C412yes—
155C22×C3×C4C22⋊C412yes—
156C22×C3×C4C22⋊C411yescharacteristic
157C22×C3×C4C22×C412yes—
158C22×C3×C4C22×C411yescharacteristic
159C22×C3×C4C22×C411yescharacteristic
160C22×C3×C4C22×C412yes—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2411yescharacteristic
168C2×C3×C8—24no—
169C2×C3×C8—44no—
170C2×C3×C8—48no—
171C3×C4⋊C4—24no—
172C3×C4⋊C4—28no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—24no—
175C3×C4⋊C4—24no—
176C3×C4⋊C4—48no—
177C3×C4⋊C4C22×C414yes—
178C3×C22⋊C4—24no—
179C3×C22⋊C4—28no—
180C3×C22⋊C4—24no—
181C3×C22⋊C4—48no—
182C25C3×D411yescharacteristic
183C22×Q8—21no—
184C22×Q8—22no—
185C22×Q8—44no—
186C22×Q8C2×C3×C412yes—
187C22×Q8C2×C3×C414yescomplemented
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—22no—
191C23×C4—44no—
192C23×C4C2×C3×C411yescharacteristic
193C23×C4C2×C3×C412yes—
194C23×C4C2×C3×C411yescharacteristic
195C23×C4C3×D411yescharacteristic
196C23×C4C3×D411yescharacteristic
197C23×C4C3×D412yes—
198C23×C4C3×D411yescharacteristic
199C23×C4C3×D412yes—
200C23×C4C23×C311yescharacteristic
201C23×C4C23×C312yes—
202C22×C8—22no—
203C22×C8—24no—
204C22×C8—44no—
205C22⋊Q8—24no—
206C22⋊Q8—24no—
207C22⋊Q8—24no—
208C22⋊Q8—22no—
209C22⋊Q8—24no—
210C22⋊Q8—22no—
211C22⋊Q8—24no—
212C22⋊Q8C2×C3×C414yes—
213C22⋊Q8C2×C3×C418yescomplemented
214C22⋊Q8C23×C314yes—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4C2×C3×C412yes—
224C2×C4⋊C4C2×C3×C414yescomplemented
225C2×C22⋊C4—24no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—44no—
231C22⋊C8—24no—
232C22⋊C8—22no—
233C22⋊C8C23×C314yes—
234#95—22no—
235#95C23×C314yes—
236C23×C3—22no—
237C23×C3—22no—
238C23×C3—22no—
239C23×C3—22no—
240C23×C3—22no—
241C23×C3—22no—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3—42no—
245C23×C3—48no—
246C23×C3—48no—
247C23×C3—42no—
248C23×C3#10314yes—
249C23×C3#10314yes—
250C23×C3#10318yes—
251C23×C3C2×C22⋊C411yescharacteristic
252C23×C3C2×C22⋊C412yes—
253C23×C3C2×C22⋊C412yes—
254C23×C3C2×C22⋊C411yescharacteristic
255C23×C3C2×SD1612yes—
256C23×C3C2×SD1614yes—
257C23×C3C2×Q1612yes—
258C23×C3C2×Q1614yes—
259C23×C3C22×D411yescharacteristic
260C3×Q8—44no—
261C3×Q8—44no—
262C3×Q8—48no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—24no—
267C2×C3×C4—24no—
268C2×C3×C4—24no—
269C2×C3×C4—24no—
270C2×C3×C4—24no—
271C2×C3×C4—24no—
272C2×C3×C4—24no—
273C2×C3×C4—24no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—41no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—48no—
283C2×C3×C4—48no—
284C2×C3×C4—44no—
285C2×C3×C4—44no—
286C2×C3×C4—44no—
287C2×C3×C4—44no—
288C2×C3×C4—44no—
289C2×C3×C4—88no—
290C2×C3×C4#9914yes—
291C2×C3×C4C2×C22⋊C411yescharacteristic
292C2×C3×C4C2×C22⋊C412yes—
293C2×C3×C4C2×C22⋊C411yescharacteristic
294C2×C3×C4C2×C22⋊C412yes—
295C2×C3×C4C23×C411yescharacteristic
296C2×C3×C4C22×D411yescharacteristic
297C3×C8—44no—
298C24—42no—
299C24—42no—
300C24C3×C22⋊C411yescharacteristic
301C24C3×C22⋊C411yescharacteristic
302C24C3×C22⋊C412yes—
303C24C3×SD1614yes—
304C24C3×Q1614yes—
305C24C2×C3×D411yescharacteristic
306C24C2×C3×D412yes—
307C2×Q8—22no—
308C2×Q8—24no—
309C2×Q8—48no—
310C2×Q8—48no—
311C2×Q8—44no—
312C2×Q8—44no—
313C2×Q8C22×C3×C414yes—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4C3×C22⋊C411yescharacteristic
335C22×C4C3×C22⋊C412yes—
336C22×C4C3×C22⋊C412yes—
337C22×C4C3×C22⋊C412yes—
338C22×C4C3×C22⋊C412yes—
339C22×C4C3×C22⋊C412yes—
340C22×C4C3×C22⋊C411yescharacteristic
341C22×C4C22×C3×C411yescharacteristic
342C22×C4C22×C3×C412yes—
343C22×C4C22×C3×C411yescharacteristic
344C22×C4C22×C3×C412yes—
345C22×C4C2×C3×D412yes—
346C22×C4C2×C3×D411yescharacteristic
347C22×C4C2×C3×D412yes—
348C22×C4C2×C3×D411yescharacteristic
349C22×C4C2×C3×D412yes—
350C22×C4C2×C3×D411yescharacteristic
351C22×C4C24×C311yescharacteristic
352C2×C8—24no—
353C2×C8—44no—
354C2×C8—48no—
355C4⋊C4—24no—
356C4⋊C4—28no—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—48no—
361C4⋊C4C22×C3×C414yes—
362C22⋊C4—24no—
363C22⋊C4—28no—
364C22⋊C4—24no—
365C22⋊C4—48no—
366C22×C3—22no—
367C22×C3—21no—
368C22×C3—22no—
369C22×C3—21no—
370C22×C3—22no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—22no—
374C22×C3—24no—
375C22×C3—28no—
376C22×C3—28no—
377C22×C3—24no—
378C22×C3—48no—
379C22×C3—44no—
380C22×C3—44no—
381C22×C3C2×#9912yes—
382C22×C3C2×#10314yes—
383C22×C3C2×#10314yes—
384C22×C3C2×#10312yes—
385C22×C3C2×#10312yes—
386C22×C3#42112yes—
387C22×C3C22×C22⋊C411yescharacteristic
388C22×C3C22×SD1611yescharacteristic
389C22×C3C22×Q1611yescharacteristic
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—22no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—44no—
396C3×C4—44no—
397C3×C4—84no—
398C3×C4C2×#9914yes—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—48no—
409C23—48no—
410C23—42no—
411C23C3×#10314yes—
412C23C3×#10314yes—
413C23C3×#10318yes—
414C23C2×C3×C22⋊C411yescharacteristic
415C23C2×C3×C22⋊C412yes—
416C23C2×C3×C22⋊C412yes—
417C23C2×C3×C22⋊C411yescharacteristic
418C23C2×C3×SD1612yes—
419C23C2×C3×SD1614yes—
420C23C2×C3×Q1612yes—
421C23C2×C3×Q1614yes—
422C23C22×C3×D411yescharacteristic
423Q8—44no—
424Q8—44no—
425Q8—48no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—48no—
446C2×C4—48no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—88no—
453C2×C4C3×#9914yes—
454C2×C4C2×C3×C22⋊C411yescharacteristic
455C2×C4C2×C3×C22⋊C412yes—
456C2×C4C2×C3×C22⋊C411yescharacteristic
457C2×C4C2×C3×C22⋊C412yes—
458C2×C4C23×C3×C411yescharacteristic
459C2×C4C22×C3×D411yescharacteristic
460C8—44no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—24no—
464C2×C3—22no—
465C2×C3—24no—
466C2×C3#152014yesdirect product factor
467C2×C3C22×#9911yescharacteristic
468C2×C3C22×#10311yescharacteristic
469C2×C3C2×#42111yescharacteristic
470C22—22no—
471C22—21no—
472C22—22no—
473C22—21no—
474C22—22no—
475C22—21no—
476C22—21no—
477C22—22no—
478C22—24no—
479C22—28no—
480C22—28no—
481C22—24no—
482C22—48no—
483C22—44no—
484C22—44no—
485C22C2×C3×#9912yes—
486C22C2×C3×#10314yes—
487C22C2×C3×#10314yes—
488C22C2×C3×#10312yes—
489C22C2×C3×#10312yes—
490C22C3×#42112yes—
491C22C22×C3×C22⋊C411yescharacteristic
492C22C22×C3×SD1611yescharacteristic
493C22C22×C3×Q1611yescharacteristic
494C4—21no—
495C4—21no—
496C4—21no—
497C4—22no—
498C4—21no—
499C4—44no—
500C4—44no—
501C4—84no—
502C4C2×C3×#9914yes—
503C3C2×#152011yescharacteristic, direct product factor, Sylow
504C2—21no—
505C2—21no—
506C2—24no—
507C2—22no—
508C2—24no—
509C2C3×#152014yesdirect product factor
510C2C22×C3×#9911yescharacteristic
511C2C22×C3×#10311yescharacteristic
512C2C2×C3×#42111yescharacteristic
subgroupG/H
center, socleC23×C3C2×C22⋊C4
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC25×C3D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
4. C3×#1520C2direct (N × H)
5. C2×C3×#328C2semidirect
6. C2×#1520C3direct (N × H)
7. C23×C3×Q8C4semidirect
14. C2×C3×C22⋊Q8C4semidirect
15. C2×C3×C22⋊Q8C4semidirect
20. C22×C3×C4⋊C4C4semidirect
30. C3×#328C22semidirect
34. #1520C2×C3direct (N × H)
35. C2×#328C2×C3semidirect
41. C22×C3×Q8C2×C4semidirect
67. C3×C22⋊Q8C2×C4semidirect
78. C2×C3×C4⋊C4C2×C4semidirect
90. C23×Q8C3×C4semidirect
97. C2×C22⋊Q8C3×C4semidirect
98. C2×C22⋊Q8C3×C4semidirect
103. C22×C4⋊C4C3×C4semidirect
113. #328C22×C3semidirect
187. C22×Q8C2×C3×C4semidirect
213. C22⋊Q8C2×C3×C4semidirect
224. C2×C4⋊C4C2×C3×C4semidirect
466. C2×C3#1520direct (N × H)
503. C3C2×#1520direct (N × H)
509. C2C3×#1520direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊Q82
C22×C3×C22⋊C82
C2×C3×#3802
C3×#15202
C2×C3×#3282
C2×#15203