Finite Group Databasebeta

← C2×C3×#3112 #10679151

Subgroups — C2×C3×#3112 #10679151

Order 768.

2398 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic
2C22×C3×C4×Q8C211yesmaximal, characteristic, complemented
3C3×#3112C2116yesmaximal, direct product factor
4C2×C3×#480C212yesmaximal, complemented
5C2×C3×#477C212yesmaximal, complemented
6C2×C3×#460C212yesmaximal, complemented
7C2×C3×#448C212yesmaximal, complemented
8C2×C3×#440C214yesmaximal, complemented
9C2×C3×#432C211yesmaximal, characteristic, complemented
10C2×#3112C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41—21no—
12C23×C3×Q8C2211yescharacteristic
13C22×C3×SD16C2214yes—
14C3×#532—24no—
15C3×#532C2218yes—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#127C2211yescharacteristic
18C2×C3×#124—21no—
19C2×C3×#121—21no—
20C2×C3×#121C2212yes—
21C2×C3×C4×Q8—21no—
22C2×C3×C4×Q8—21no—
23C2×C3×C4×Q8—24no—
24C2×C3×C4×Q8C2212yes—
25C2×C3×C4×Q8C2212yes—
26C2×C3×C4×Q8C2218yescomplemented
27C2×C3×C4×D4—21no—
28C2×C3×C4×D4C2212yes—
29C22×C3×C4⋊C4—21no—
30C22×C3×C4⋊C4C2211yescharacteristic
31C22×C3×C42—21no—
32C22×C3×C42C2211yescharacteristic
33C3×#480C22116yescomplemented
34C3×#477C22116yescomplemented
35C3×#460C22116yescomplemented
36C3×#448C22116yescomplemented
37C3×#440C22132yescomplemented
38C3×#432C2218yescomplemented
39C2×C3×#107C2214yes—
40C2×C3×C4⋊C8C2212yes—
41C2×C3×#103C2214yes—
42C2×C3×#102C2214yes—
43C2×C3×C22⋊C8C2212yes—
44C2×C3×#97C2212yes—
45C2×#532C2×C311yescharacteristic
46C22×C4×Q8C2×C311yescharacteristic, complemented
47#3112C2×C3116yesdirect product factor
48C2×#480C2×C312yescomplemented
49C2×#477C2×C312yescomplemented
50C2×#460C2×C312yescomplemented
51C2×#448C2×C312yescomplemented
52C2×#440C2×C314yescomplemented
53C2×#432C2×C311yescharacteristic, complemented
54C2×C3×#41—28no—
55C2×C3×#41—48no—
56C22×C3×Q8—28no—
57C22×C3×Q8—22no—
58C22×C3×Q8—21no—
59C22×C3×Q8—21no—
60C22×C3×Q8C2314yes—
61C22×C3×Q8C2312yes—
62C22×C3×D4—22no—
63C22×C3×D4—21no—
64C22×C3×D4—21no—
65C22×C3×D4C2312yes—
66C23×C3×C4—21no—
67C23×C3×C4—21no—
68C23×C3×C4—22no—
69C23×C3×C4D411yescharacteristic
70C23×C3×C4D411yescharacteristic
71C23×C3×C4C2311yescharacteristic
72C2×C3×SD16—216no—
73C2×C3×SD16C23116yes—
74C22×C3×C8C2314yes—
75C3×C4⋊Q8—22no—
76C3×C4⋊Q8C2314yes—
77C3×#127—22no—
78C3×#127C2314yes—
79C3×#124—28no—
80C3×#121—28no—
81C3×#121—24no—
82C3×#121C2318yes—
83C3×C4×Q8—24no—
84C3×C4×Q8—24no—
85C3×C4×Q8—28no—
86C3×C4×Q8—28no—
87C3×C4×Q8C2318yes—
88C3×C4×Q8C2318yes—
89C3×C4×D4—28no—
90C3×C4×D4—24no—
91C3×C4×D4C2318yes—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4—28no—
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⋊C4C2314yes—
101C2×C3×C22⋊C4—22no—
102C2×C3×C22⋊C4—22no—
103C2×C3×C42—22no—
104C2×C3×C42—22no—
105C2×C3×C42—28no—
106C2×C3×C42D411yescharacteristic
107C2×C3×C42D411yescharacteristic
108C2×C3×C42D414yes—
109C2×C3×C42C2311yescharacteristic
110C2×C3×C42C2311yescharacteristic
111C2×C3×C42C2314yes—
112C3×#107C23116yes—
113C3×C4⋊C8C2318yes—
114C3×#103C23116yes—
115C3×#102C23116yes—
116C3×C22⋊C8C2318yes—
117C3×#97C2318yes—
118C22×#41—21no—
119C23×Q8C22×C311yescharacteristic
120C22×SD16C22×C314yes—
121#532—24no—
122#532C22×C318yes—
123C2×C4⋊Q8C22×C311yescharacteristic
124C2×#127C22×C311yescharacteristic
125C2×#124—21no—
126C2×#121—21no—
127C2×#121C22×C312yes—
128C2×C4×Q8—21no—
129C2×C4×Q8—21no—
130C2×C4×Q8—24no—
131C2×C4×Q8C22×C312yes—
132C2×C4×Q8C22×C312yes—
133C2×C4×Q8C22×C318yescomplemented
134C2×C4×D4—21no—
135C2×C4×D4C22×C312yes—
136C22×C4⋊C4—21no—
137C22×C4⋊C4C22×C311yescharacteristic
138C22×C42—21no—
139C22×C42C22×C311yescharacteristic
140#480C22×C3116yescomplemented
141#477C22×C3116yescomplemented
142#460C22×C3116yescomplemented
143#448C22×C3116yescomplemented
144#440C22×C3132yescomplemented
145#432C22×C318yescomplemented
146C2×#107C22×C314yes—
147C2×C4⋊C8C22×C312yes—
148C2×#103C22×C314yes—
149C2×#102C22×C314yes—
150C2×C22⋊C8C22×C312yes—
151C2×#97C22×C312yes—
152C24×C3—22no—
153C24×C3C2×D411yescharacteristic
154C3×#41—432no—
155C2×C3×Q8—216no—
156C2×C3×Q8—28no—
157C2×C3×Q8—216no—
158C2×C3×Q8—24no—
159C2×C3×Q8—22no—
160C2×C3×Q8—24no—
161C2×C3×Q8—24no—
162C2×C3×Q8—44no—
163C2×C3×Q8#4118yescomplemented
164C2×C3×Q8C2414yes—
165C2×C3×D4—28no—
166C2×C3×D4—22no—
167C2×C3×D4—24no—
168C2×C3×D4—24no—
169C2×C3×D4—28no—
170C2×C3×D4—44no—
171C2×C3×D4—48no—
172C2×C3×D4—44no—
173C2×C3×D4C2414yes—
174C22×C3×C4—28no—
175C22×C3×C4—24no—
176C22×C3×C4—24no—
177C22×C3×C4—22no—
178C22×C3×C4—24no—
179C22×C3×C4—24no—
180C22×C3×C4—24no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—28no—
184C22×C3×C4—48no—
185C22×C3×C4—44no—
186C22×C3×C4C2×D414yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D414yes—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2412yes—
195C22×C3×C4C2411yescharacteristic
196C3×SD16—232no—
197C2×C3×C8—28no—
198C2×C3×C8C2418yes—
199C3×C4⋊C4—28no—
200C3×C4⋊C4—28no—
201C3×C4⋊C4—24no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—24no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4C2414yes—
206C3×C4⋊C4C2414yes—
207C3×C22⋊C4—28no—
208C3×C22⋊C4—28no—
209C3×C42—28no—
210C3×C42—28no—
211C3×C42C2×D412yes—
212C3×C42C2×D414yes—
213C3×C42C2×D414yes—
214C3×C42C2×D412yes—
215C3×C42C2412yes—
216C3×C42C2412yes—
217C2×#41—28no—
218C2×#41—48no—
219C22×Q8—28no—
220C22×Q8—21no—
221C22×Q8—22no—
222C22×Q8—21no—
223C22×Q8C23×C314yes—
224C22×Q8C23×C312yes—
225C22×D4—22no—
226C22×D4—21no—
227C22×D4—21no—
228C22×D4C23×C312yes—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—22no—
232C23×C4C3×D411yescharacteristic
233C23×C4C3×D411yescharacteristic
234C23×C4C23×C311yescharacteristic
235C2×SD16—216no—
236C2×SD16C23×C3116yes—
237C22×C8C23×C314yes—
238C4⋊Q8—22no—
239C4⋊Q8C23×C314yes—
240#127—22no—
241#127C23×C314yes—
242#124—28no—
243#121—28no—
244#121—24no—
245#121C23×C318yes—
246C4×Q8—24no—
247C4×Q8—24no—
248C4×Q8—28no—
249C4×Q8—28no—
250C4×Q8C23×C318yes—
251C4×Q8C23×C318yes—
252C4×D4—28no—
253C4×D4—24no—
254C4×D4C23×C318yes—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4C23×C312yes—
262C2×C4⋊C4C23×C314yes—
263C2×C4⋊C4C23×C312yes—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—28no—
269C2×C42C3×D411yescharacteristic
270C2×C42C3×D411yescharacteristic
271C2×C42C3×D414yes—
272C2×C42C23×C311yescharacteristic
273C2×C42C23×C311yescharacteristic
274C2×C42C23×C314yes—
275#107C23×C3116yes—
276C4⋊C8C23×C318yes—
277#103C23×C3116yes—
278#102C23×C3116yes—
279C22⋊C8C23×C318yes—
280#97C23×C318yes—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—24no—
284C23×C3—44no—
285C23×C3—44no—
286C23×C3—44no—
287C23×C3AGL(1,ℤ/8)14yes—
288C23×C3C22×D412yes—
289C23×C3C22×D411yescharacteristic
290C3×Q8—216no—
291C3×Q8—28no—
292C3×Q8—24no—
293C3×Q8—48no—
294C3×Q8C2×#4118yescomplemented
295C3×D4—28no—
296C3×D4—48no—
297C3×D4—416no—
298C3×D4—44no—
299C3×D4—48no—
300C2×C3×C4—216no—
301C2×C3×C4—28no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—28no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—28no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—416no—
314C2×C3×C4—48no—
315C2×C3×C4—44no—
316C2×C3×C4—44no—
317C2×C3×C4#13714yes—
318C2×C3×C4C22×D412yes—
319C2×C3×C4C22×D412yes—
320C2×C3×C4C22×D412yes—
321C2×C3×C4C22×D412yes—
322C2×C3×C4C22×D411yescharacteristic
323C2×C3×C4C22×D412yes—
324C2×C3×C4C2×#4114yes—
325C2×C3×C4C2511yescharacteristic
326C3×C8—28no—
327C24—22no—
328C24C2×C3×D411yescharacteristic
329#41—432no—
330C2×Q8—216no—
331C2×Q8—28no—
332C2×Q8—216no—
333C2×Q8—24no—
334C2×Q8—22no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—44no—
338C2×Q8C3×#4118yescomplemented
339C2×Q8C24×C314yes—
340C2×D4—28no—
341C2×D4—22no—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—28no—
345C2×D4—44no—
346C2×D4—48no—
347C2×D4—44no—
348C2×D4C24×C314yes—
349C22×C4—28no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—28no—
359C22×C4—48no—
360C22×C4—44no—
361C22×C4C2×C3×D411yescharacteristic
362C22×C4C2×C3×D414yes—
363C22×C4C2×C3×D411yescharacteristic
364C22×C4C2×C3×D412yes—
365C22×C4C2×C3×D411yescharacteristic
366C22×C4C2×C3×D411yescharacteristic
367C22×C4C2×C3×D414yes—
368C22×C4C2×C3×D411yescharacteristic
369C22×C4C24×C312yes—
370C22×C4C24×C311yescharacteristic
371SD16—232no—
372C2×C8—28no—
373C2×C8C24×C318yes—
374C4⋊C4—28no—
375C4⋊C4—28no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4C24×C314yes—
381C4⋊C4C24×C314yes—
382C22⋊C4—28no—
383C22⋊C4—28no—
384C42—28no—
385C42—28no—
386C42C2×C3×D412yes—
387C42C2×C3×D414yes—
388C42C2×C3×D414yes—
389C42C2×C3×D412yes—
390C42C24×C312yes—
391C42C24×C312yes—
392C22×C3—28no—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—44no—
396C22×C3—44no—
397C22×C3—44no—
398C22×C3—88no—
399C22×C3#52512yes—
400C22×C3C2×AGL(1,ℤ/8)14yes—
401C22×C3C2×AGL(1,ℤ/8)12yes—
402C22×C3C2×#13712yes—
403C22×C3C23×D411yescharacteristic
404C3×C4—28no—
405C3×C4—22no—
406C3×C4—22no—
407C3×C4—22no—
408C3×C4—22no—
409C3×C4—44no—
410C3×C4—44no—
411C3×C4#52512yes—
412C3×C4C2×#13714yes—
413C3×C4C22×#4112yes—
414C23—22no—
415C23—22no—
416C23—24no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23C3×AGL(1,ℤ/8)14yes—
421C23C22×C3×D412yes—
422C23C22×C3×D411yescharacteristic
423Q8—216no—
424Q8—24no—
425Q8—28no—
426Q8—48no—
427Q8C2×C3×#4118yescomplemented
428D4—28no—
429D4—48no—
430D4—416no—
431D4—48no—
432D4—44no—
433C2×C4—216no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—28no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—28no—
440C2×C4—22no—
441C2×C4—28no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—416no—
447C2×C4—44no—
448C2×C4—48no—
449C2×C4—44no—
450C2×C4C3×#13714yes—
451C2×C4C22×C3×D412yes—
452C2×C4C22×C3×D412yes—
453C2×C4C22×C3×D412yes—
454C2×C4C22×C3×D412yes—
455C2×C4C22×C3×D412yes—
456C2×C4C22×C3×D411yescharacteristic
457C2×C4C2×C3×#4114yes—
458C2×C4C25×C311yescharacteristic
459C8—28no—
460C2×C3—24no—
461C2×C3—84no—
462C2×C3#311214yesdirect product factor
463C2×C3C2×#52511yescharacteristic
464C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
465C2×C3C22×#13711yescharacteristic
466C22—22no—
467C22—22no—
468C22—28no—
469C22—44no—
470C22—44no—
471C22—44no—
472C22—88no—
473C22C3×#52512yes—
474C22C2×C3×AGL(1,ℤ/8)14yes—
475C22C2×C3×AGL(1,ℤ/8)12yes—
476C22C2×C3×#13712yes—
477C22C23×C3×D411yescharacteristic
478C4—28no—
479C4—22no—
480C4—22no—
481C4—22no—
482C4—22no—
483C4—44no—
484C4—44no—
485C4C3×#52512yes—
486C4C2×C3×#13714yes—
487C4C22×C3×#4112yes—
488C3C2×#311211yescharacteristic, direct product factor, Sylow
489C2—24no—
490C2—84no—
491C2C3×#311214yesdirect product factor
492C2C2×C3×#52511yescharacteristic
493C2C22×C3×AGL(1,ℤ/8)11yescharacteristic
494C2C22×C3×#13711yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC2×C3×#532C2
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
2. C22×C3×C4×Q8C2semidirect
3. C3×#3112C2direct (N × H)
4. C2×C3×#480C2semidirect
5. C2×C3×#477C2semidirect
6. C2×C3×#460C2semidirect
7. C2×C3×#448C2semidirect
8. C2×C3×#440C2semidirect
9. C2×C3×#432C2semidirect
10. C2×#3112C3direct (N × H)
26. C2×C3×C4×Q8C22semidirect
33. C3×#480C22semidirect
34. C3×#477C22semidirect
35. C3×#460C22semidirect
36. C3×#448C22semidirect
37. C3×#440C22semidirect
38. C3×#432C22semidirect
46. C22×C4×Q8C2×C3semidirect
47. #3112C2×C3direct (N × H)
48. C2×#480C2×C3semidirect
49. C2×#477C2×C3semidirect
50. C2×#460C2×C3semidirect
51. C2×#448C2×C3semidirect
52. C2×#440C2×C3semidirect
53. C2×#432C2×C3semidirect
133. C2×C4×Q8C22×C3semidirect
140. #480C22×C3semidirect
141. #477C22×C3semidirect
142. #460C22×C3semidirect
143. #448C22×C3semidirect
144. #440C22×C3semidirect
145. #432C22×C3semidirect
163. C2×C3×Q8#41semidirect
294. C3×Q8C2×#41semidirect
338. C2×Q8C3×#41semidirect
427. Q8C2×C3×#41semidirect
462. C2×C3#3112direct (N × H)
488. C3C2×#3112direct (N × H)
491. C2C3×#3112direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C22×C3×C4×Q82
C3×#31122
C2×C3×#4802
C2×C3×#4772
C2×C3×#4602
C2×C3×#4482
C2×C3×#4402
C2×C3×#4322
C2×#31123