Finite Group Databasebeta

← C2×C3×#1512 #10636160

Subgroups — C2×C3×#1512 #10636160

Order 768.

1098 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#529C211yesmaximal, characteristic
2C2×C3×#432C211yesmaximal, characteristic, complemented
3C2×C3×#382C211yesmaximal, characteristic, complemented
4C3×#1512C218yesmaximal, direct product factor
5C2×C3×#328C212yesmaximal, complemented
6C2×C3×#327C212yesmaximal
7C2×#1512C311yesmaximal, characteristic, direct product factor, Sylow
8C3×#529—22no—
9C3×#529—22no—
10C3×#529C414yescomplemented
11C3×#529C2214yes—
12C2×C3×#126—21no—
13C2×C3×#125C411yescharacteristic, complemented
14C2×C3×#124C411yescharacteristic, complemented
15C2×C3×#123—21no—
16C2×C3×C22⋊Q8C2211yescharacteristic
17C2×C3×#121C2211yescharacteristic
18C2×C3×C4×Q8C411yescharacteristic, complemented
19C2×C3×C4×D4—21no—
20C2×C3×C4×D4C411yescharacteristic, complemented
21C2×C3×#117—21no—
22C22×C3×C4⋊C4—21no—
23C22×C3×C42C2211yescharacteristic
24C3×#432—22no—
25C3×#432C2214yescomplemented
26C2×C3×C4⋊C8—21no—
27C2×C3×C22⋊C8C2212yes—
28C2×C3×#97—21no—
29C3×#382—22no—
30C3×#382C2214yescomplemented
31C2×C3×#95—21no—
32C2×C3×#95C2212yes—
33C3×#328C2218yescomplemented
34C3×#327C2218yes—
35C2×#529C2×C311yescharacteristic
36C2×#432C2×C311yescharacteristic, complemented
37C2×#382C2×C311yescharacteristic, complemented
38#1512C2×C318yesdirect product factor
39C2×#328C2×C312yescomplemented
40C2×#327C2×C312yes—
41C22×C3×Q8C2×C411yescharacteristic
42C22×C3×D4—21no—
43C22×C3×D4C2×C411yescharacteristic
44C23×C3×C4—21no—
45C23×C3×C4—22no—
46C23×C3×C4D411yescharacteristic
47C23×C3×C4D411yescharacteristic
48C23×C3×C4C2311yescharacteristic
49C22×C3×C8—22no—
50C3×#126—24no—
51C3×#126—24no—
52C3×#125—22no—
53C3×#125C2×C414yescomplemented
54C3×#124—22no—
55C3×#124C2×C414yescomplemented
56C3×#123—24no—
57C3×#123—24no—
58C3×C22⋊Q8—21no—
59C3×C22⋊Q8—21no—
60C3×C22⋊Q8C2×C412yes—
61C3×C22⋊Q8C2312yes—
62C3×#121—21no—
63C3×#121—21no—
64C3×#121C2×C412yes—
65C3×#121C2312yes—
66C3×C4×Q8—22no—
67C3×C4×Q8C2×C414yescomplemented
68C3×C4×D4—24no—
69C3×C4×D4—24no—
70C3×C4×D4—22no—
71C3×C4×D4C2×C414yescomplemented
72C3×#117—24no—
73C3×#117—24no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—24no—
79C2×C3×C4⋊C4—44no—
80C2×C3×C4⋊C4C2×C411yescharacteristic
81C2×C3×C4⋊C4C2×C411yescharacteristic
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C42—21no—
88C2×C3×C42C2×C411yescharacteristic
89C2×C3×C42C2×C411yescharacteristic
90C2×C3×C42C2×C412yes—
91C2×C3×C42D412yes—
92C2×C3×C42D411yescharacteristic
93C2×C3×C42D411yescharacteristic
94C2×C3×C42D412yes—
95C2×C3×C42C2312yes—
96C3×C4⋊C8—24no—
97C3×C22⋊C8—22no—
98C3×C22⋊C8C2314yes—
99C3×#97—24no—
100C3×#95—24no—
101C3×#95—22no—
102C3×#95C2314yes—
103#529—22no—
104#529—22no—
105#529C3×C414yescomplemented
106#529C22×C314yes—
107C2×#126—21no—
108C2×#125C3×C411yescharacteristic, complemented
109C2×#124C3×C411yescharacteristic, complemented
110C2×#123—21no—
111C2×C22⋊Q8C22×C311yescharacteristic
112C2×#121C22×C311yescharacteristic
113C2×C4×Q8C3×C411yescharacteristic, complemented
114C2×C4×D4—21no—
115C2×C4×D4C3×C411yescharacteristic, complemented
116C2×#117—21no—
117C22×C4⋊C4—21no—
118C22×C42C22×C311yescharacteristic
119#432—22no—
120#432C22×C314yescomplemented
121C2×C4⋊C8—21no—
122C2×C22⋊C8C22×C312yes—
123C2×#97—21no—
124#382—22no—
125#382C22×C314yescomplemented
126C2×#95—21no—
127C2×#95C22×C312yes—
128#328C22×C318yescomplemented
129#327C22×C318yes—
130C24×C3—21no—
131C24×C3C2×D411yescharacteristic
132C2×C3×Q8—21no—
133C2×C3×Q8—42no—
134C2×C3×Q8C22×C412yes—
135C2×C3×D4—22no—
136C2×C3×D4—22no—
137C2×C3×D4—21no—
138C2×C3×D4—42no—
139C2×C3×D4—42no—
140C2×C3×D4—42no—
141C2×C3×D4C22×C412yes—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—24no—
148C22×C3×C4—24no—
149C22×C3×C4—42no—
150C22×C3×C4—42no—
151C22×C3×C4—44no—
152C22×C3×C4—44no—
153C22×C3×C4—44no—
154C22×C3×C4—44no—
155C22×C3×C4C22⋊C412yes—
156C22×C3×C4C22⋊C411yescharacteristic
157C22×C3×C4C22⋊C411yescharacteristic
158C22×C3×C4C22⋊C411yescharacteristic
159C22×C3×C4C22⋊C411yescharacteristic
160C22×C3×C4C22⋊C412yes—
161C22×C3×C4C22×C411yescharacteristic
162C22×C3×C4C22×C411yescharacteristic
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2411yescharacteristic
169C2×C3×C8—24no—
170C2×C3×C8—44no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—24no—
173C3×C4⋊C4—24no—
174C3×C4⋊C4—24no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—48no—
179C3×C4⋊C4C22×C412yes—
180C3×C4⋊C4C22×C412yes—
181C3×C22⋊C4—22no—
182C3×C22⋊C4—22no—
183C3×C22⋊C4—22no—
184C3×C22⋊C4—24no—
185C3×C22⋊C4—24no—
186C3×C22⋊C4—24no—
187C3×C22⋊C4—22no—
188C3×C42—24no—
189C3×C42C22⋊C412yes—
190C3×C42C22⋊C412yes—
191C3×C42C22⋊C412yes—
192C3×C42C22⋊C412yes—
193C3×C42C22×C412yes—
194C3×C42C22×C412yes—
195C3×C42C2×D412yes—
196C3×C42C2×D412yes—
197C22×Q8C2×C3×C411yescharacteristic
198C22×D4—21no—
199C22×D4C2×C3×C411yescharacteristic
200C23×C4—21no—
201C23×C4—22no—
202C23×C4C3×D411yescharacteristic
203C23×C4C3×D411yescharacteristic
204C23×C4C23×C311yescharacteristic
205C22×C8—22no—
206#126—24no—
207#126—24no—
208#125—22no—
209#125C2×C3×C414yescomplemented
210#124—22no—
211#124C2×C3×C414yescomplemented
212#123—24no—
213#123—24no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216C22⋊Q8C2×C3×C412yes—
217C22⋊Q8C23×C312yes—
218#121—21no—
219#121—21no—
220#121C2×C3×C412yes—
221#121C23×C312yes—
222C4×Q8—22no—
223C4×Q8C2×C3×C414yescomplemented
224C4×D4—24no—
225C4×D4—24no—
226C4×D4—22no—
227C4×D4C2×C3×C414yescomplemented
228#117—24no—
229#117—24no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4C2×C3×C411yescharacteristic
237C2×C4⋊C4C2×C3×C411yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C42—21no—
244C2×C42C2×C3×C411yescharacteristic
245C2×C42C2×C3×C411yescharacteristic
246C2×C42C2×C3×C412yes—
247C2×C42C3×D412yes—
248C2×C42C3×D411yescharacteristic
249C2×C42C3×D411yescharacteristic
250C2×C42C3×D412yes—
251C2×C42C23×C312yes—
252C4⋊C8—24no—
253C22⋊C8—22no—
254C22⋊C8C23×C314yes—
255#97—24no—
256#95—24no—
257#95—22no—
258#95C23×C314yes—
259C23×C3—22no—
260C23×C3—22no—
261C23×C3—22no—
262C23×C3—42no—
263C23×C3—42no—
264C23×C3—42no—
265C23×C3C2×C22⋊C411yescharacteristic
266C23×C3C2×C22⋊C411yescharacteristic
267C23×C3AGL(1,ℤ/8)12yes—
268C23×C3#13712yes—
269C23×C3C22×D411yescharacteristic
270C3×Q8—42no—
271C3×Q8—42no—
272C3×D4—42no—
273C3×D4—42no—
274C3×D4—42no—
275C3×D4—42no—
276C3×D4—44no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—24no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—42no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—42no—
294C2×C3×C4—41no—
295C2×C3×C4—48no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—44no—
300C2×C3×C4—44no—
301C2×C3×C4—44no—
302C2×C3×C4—88no—
303C2×C3×C4C2×C22⋊C412yes—
304C2×C3×C4C2×C22⋊C411yescharacteristic
305C2×C3×C4C2×C22⋊C411yescharacteristic
306C2×C3×C4C2×C22⋊C412yes—
307C2×C3×C4C2×C22⋊C412yes—
308C2×C3×C4C2×C22⋊C412yes—
309C2×C3×C4C23×C411yescharacteristic
310C2×C3×C4C22×D411yescharacteristic
311C3×C8—44no—
312C24—21no—
313C24C2×C3×D411yescharacteristic
314C2×Q8—21no—
315C2×Q8—42no—
316C2×Q8C22×C3×C412yes—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—21no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4C22×C3×C412yes—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4C3×C22⋊C411yescharacteristic
338C22×C4C3×C22⋊C411yescharacteristic
339C22×C4C3×C22⋊C411yescharacteristic
340C22×C4C3×C22⋊C411yescharacteristic
341C22×C4C3×C22⋊C412yes—
342C22×C4C3×C22⋊C412yes—
343C22×C4C22×C3×C411yescharacteristic
344C22×C4C22×C3×C411yescharacteristic
345C22×C4C2×C3×D412yes—
346C22×C4C2×C3×D411yescharacteristic
347C22×C4C2×C3×D411yescharacteristic
348C22×C4C2×C3×D412yes—
349C22×C4C2×C3×D411yescharacteristic
350C22×C4C24×C311yescharacteristic
351C2×C8—24no—
352C2×C8—44no—
353C4⋊C4—22no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—22no—
360C4⋊C4—48no—
361C4⋊C4C22×C3×C412yes—
362C4⋊C4C22×C3×C412yes—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—24no—
369C22⋊C4—22no—
370C42—24no—
371C42C3×C22⋊C412yes—
372C42C3×C22⋊C412yes—
373C42C3×C22⋊C412yes—
374C42C3×C22⋊C412yes—
375C42C22×C3×C412yes—
376C42C22×C3×C412yes—
377C42C2×C3×D412yes—
378C42C2×C3×D412yes—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—24no—
384C22×C3—24no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3—42no—
388C22×C3—84no—
389C22×C3#41012yes—
390C22×C3#41712yes—
391C22×C3#41712yes—
392C22×C3#42112yes—
393C22×C3C22×C22⋊C411yescharacteristic
394C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
395C22×C3C2×#13711yescharacteristic
396C3×C4—21no—
397C3×C4—21no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—22no—
401C3×C4—22no—
402C3×C4—22no—
403C3×C4—22no—
404C3×C4—42no—
405C3×C4—42no—
406C3×C4—42no—
407C3×C4—84no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23C2×C3×C22⋊C411yescharacteristic
415C23C2×C3×C22⋊C411yescharacteristic
416C23C3×AGL(1,ℤ/8)12yes—
417C23C3×#13712yes—
418C23C22×C3×D411yescharacteristic
419Q8—42no—
420Q8—42no—
421D4—42no—
422D4—42no—
423D4—42no—
424D4—42no—
425D4—44no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—48no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—88no—
452C2×C4C2×C3×C22⋊C412yes—
453C2×C4C2×C3×C22⋊C411yescharacteristic
454C2×C4C2×C3×C22⋊C411yescharacteristic
455C2×C4C2×C3×C22⋊C412yes—
456C2×C4C2×C3×C22⋊C412yes—
457C2×C4C2×C3×C22⋊C412yes—
458C2×C4C23×C3×C411yescharacteristic
459C2×C4C22×C3×D411yescharacteristic
460C8—44no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—22no—
464C2×C3—82no—
465C2×C3#151214yesdirect product factor
466C2×C3C2×#41011yescharacteristic
467C2×C3C2×#41711yescharacteristic
468C2×C3C2×#42111yescharacteristic
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—24no—
474C22—24no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—84no—
479C22C3×#41012yes—
480C22C3×#41712yes—
481C22C3×#41712yes—
482C22C3×#42112yes—
483C22C22×C3×C22⋊C411yescharacteristic
484C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
485C22C2×C3×#13711yescharacteristic
486C4—21no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—22no—
491C4—22no—
492C4—22no—
493C4—22no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—84no—
498C3C2×#151211yescharacteristic, direct product factor, Sylow
499C2—21no—
500C2—21no—
501C2—22no—
502C2—82no—
503C2C3×#151214yesdirect product factor
504C2C2×C3×#41011yescharacteristic
505C2C2×C3×#41711yescharacteristic
506C2C2×C3×#42111yescharacteristic
subgroupG/H
center, socleC23×C3C2×C22⋊C4
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#432C2semidirect
3. C2×C3×#382C2semidirect
4. C3×#1512C2direct (N × H)
5. C2×C3×#328C2semidirect
7. C2×#1512C3direct (N × H)
10. C3×#529C4semidirect
13. C2×C3×#125C4semidirect
14. C2×C3×#124C4semidirect
18. C2×C3×C4×Q8C4semidirect
20. C2×C3×C4×D4C4semidirect
25. C3×#432C22semidirect
30. C3×#382C22semidirect
33. C3×#328C22semidirect
36. C2×#432C2×C3semidirect
37. C2×#382C2×C3semidirect
38. #1512C2×C3direct (N × H)
39. C2×#328C2×C3semidirect
53. C3×#125C2×C4semidirect
55. C3×#124C2×C4semidirect
67. C3×C4×Q8C2×C4semidirect
71. C3×C4×D4C2×C4semidirect
105. #529C3×C4semidirect
108. C2×#125C3×C4semidirect
109. C2×#124C3×C4semidirect
113. C2×C4×Q8C3×C4semidirect
115. C2×C4×D4C3×C4semidirect
120. #432C22×C3semidirect
125. #382C22×C3semidirect
128. #328C22×C3semidirect
209. #125C2×C3×C4semidirect
211. #124C2×C3×C4semidirect
223. C4×Q8C2×C3×C4semidirect
227. C4×D4C2×C3×C4semidirect
465. C2×C3#1512direct (N × H)
498. C3C2×#1512direct (N × H)
503. C2C3×#1512direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5292
C2×C3×#4322
C2×C3×#3822
C3×#15122
C2×C3×#3282
C2×C3×#3272
C2×#15123