Finite Group Databasebeta

← C2×C3×#3101 #10679135

Subgroups — C2×C3×#3101 #10679135

Order 768.

2570 conjugacy classes of subgroups.

The subgroup lattice has 500 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×#3101C2116yesmaximal, direct product factor
4C2×C3×#483C212yesmaximal, complemented
5C2×C3×#476C212yesmaximal, complemented
6C2×C3×#461C212yesmaximal, complemented
7C2×C3×#450C212yesmaximal
8C2×C3×C4×SD16C214yesmaximal, complemented
9C2×C3×#431C211yesmaximal, characteristic, complemented
10C2×#3101C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×D4C2211yescharacteristic
12C24×C3×C4—21no—
13C22×C3×SD16C2214yes—
14C3×#533—24no—
15C3×#533C2218yescomplemented
16C2×C3×C4⋊Q8C2211yescharacteristic, complemented
17C2×C3×C4⋊Q8C2211yescharacteristic, complemented
18C2×C3×#125—21no—
19C2×C3×C22⋊Q8—21no—
20C2×C3×C22⋊Q8C2212yes—
21C2×C3×C4×Q8—21no—
22C2×C3×C4×Q8C2212yescomplemented
23C2×C3×C4×D4—21no—
24C2×C3×C4×D4—24no—
25C2×C3×C4×D4—21no—
26C2×C3×C4×D4C2212yes—
27C2×C3×C4×D4C2212yes—
28C2×C3×C4×D4C2218yes—
29C2×C3×#117—21no—
30C22×C3×C4⋊C4C2211yescharacteristic
31C22×C3×C22⋊C4—21no—
32C22×C3×C42C2211yescharacteristic
33C3×#483C22116yescomplemented
34C3×#476C22116yescomplemented
35C3×#461C22116yescomplemented
36C3×#450C22116yes—
37C3×C4×SD16C22132yescomplemented
38C3×#431C2218yescomplemented
39C2×C3×#106C2214yescomplemented
40C2×C3×C4⋊C8C2212yescomplemented
41C2×C3×#103C2214yescomplemented
42C2×C3×#102C2214yes—
43C2×C3×C22⋊C8C2212yes—
44C2×C3×C4×C8C2212yescomplemented
45C2×#533C2×C311yescharacteristic, complemented
46C22×C4×D4C2×C311yescharacteristic
47#3101C2×C3116yesdirect product factor
48C2×#483C2×C312yescomplemented
49C2×#476C2×C312yescomplemented
50C2×#461C2×C312yescomplemented
51C2×#450C2×C312yes—
52C2×C4×SD16C2×C314yescomplemented
53C2×#431C2×C311yescharacteristic, complemented
54C25×C3—21no—
55C22×C3×Q8—21no—
56C22×C3×Q8C2312yes—
57C22×C3×D4—28no—
58C22×C3×D4—21no—
59C22×C3×D4—22no—
60C22×C3×D4C2314yes—
61C22×C3×D4C2312yes—
62C23×C3×C4—21no—
63C23×C3×C4—22no—
64C23×C3×C4—28no—
65C23×C3×C4—22no—
66C23×C3×C4—48no—
67C23×C3×C4D411yescharacteristic
68C23×C3×C4D411yescharacteristic
69C23×C3×C4C2311yescharacteristic
70C2×C3×SD16—216no—
71C2×C3×SD16C23116yes—
72C22×C3×C8C2314yes—
73C3×C4⋊Q8—22no—
74C3×C4⋊Q8—22no—
75C3×C4⋊Q8C2314yescomplemented
76C3×C4⋊Q8C2314yescomplemented
77C3×#125—28no—
78C3×C22⋊Q8—28no—
79C3×C22⋊Q8—24no—
80C3×C22⋊Q8C2318yes—
81C3×C4×Q8—28no—
82C3×C4×Q8—24no—
83C3×C4×Q8C2318yescomplemented
84C3×C4×D4—24no—
85C3×C4×D4—28no—
86C3×C4×D4—28no—
87C3×C4×D4—24no—
88C3×C4×D4C2318yes—
89C3×C4×D4C2318yes—
90C3×#117—28no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—22no—
97C2×C3×C4⋊C4—22no—
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2314yes—
101C2×C3×C22⋊C4—22no—
102C2×C3×C22⋊C4—22no—
103C2×C3×C22⋊C4—22no—
104C2×C3×C22⋊C4—28no—
105C2×C3×C42—22no—
106C2×C3×C42D411yescharacteristic
107C2×C3×C42D414yes—
108C2×C3×C42D411yescharacteristic
109C2×C3×C42C2311yescharacteristic
110C2×C3×C42C2314yes—
111C2×C3×C42C2311yescharacteristic
112C3×#106C23116yescomplemented
113C3×C4⋊C8C2318yescomplemented
114C3×#103C23116yescomplemented
115C3×#102C23116yes—
116C3×C22⋊C8C2318yes—
117C3×C4×C8C2318yescomplemented
118C23×D4C22×C311yescharacteristic
119C24×C4—21no—
120C22×SD16C22×C314yes—
121#533—24no—
122#533C22×C318yescomplemented
123C2×C4⋊Q8C22×C311yescharacteristic, complemented
124C2×C4⋊Q8C22×C311yescharacteristic, complemented
125C2×#125—21no—
126C2×C22⋊Q8—21no—
127C2×C22⋊Q8C22×C312yes—
128C2×C4×Q8—21no—
129C2×C4×Q8C22×C312yescomplemented
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4—24no—
133C2×C4×D4C22×C318yes—
134C2×C4×D4C22×C312yes—
135C2×C4×D4C22×C312yes—
136C2×#117—21no—
137C22×C4⋊C4C22×C311yescharacteristic
138C22×C22⋊C4—21no—
139C22×C42C22×C311yescharacteristic
140#483C22×C3116yescomplemented
141#476C22×C3116yescomplemented
142#461C22×C3116yescomplemented
143#450C22×C3116yes—
144C4×SD16C22×C3132yescomplemented
145#431C22×C318yescomplemented
146C2×#106C22×C314yescomplemented
147C2×C4⋊C8C22×C312yescomplemented
148C2×#103C22×C314yescomplemented
149C2×#102C22×C314yes—
150C2×C22⋊C8C22×C312yes—
151C2×C4×C8C22×C312yescomplemented
152C24×C3—28no—
153C24×C3—24no—
154C24×C3—22no—
155C24×C3—44no—
156C24×C3—44no—
157C24×C3C2×D411yescharacteristic
158C2×C3×Q8—22no—
159C2×C3×Q8—24no—
160C2×C3×Q8—28no—
161C2×C3×Q8—44no—
162C2×C3×Q8C2414yes—
163C2×C3×D4—28no—
164C2×C3×D4—216no—
165C2×C3×D4—216no—
166C2×C3×D4—24no—
167C2×C3×D4—22no—
168C2×C3×D4—24no—
169C2×C3×D4#4118yes—
170C2×C3×D4C2414yes—
171C22×C3×C4—24no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—24no—
175C22×C3×C4—22no—
176C22×C3×C4—28no—
177C22×C3×C4—28no—
178C22×C3×C4—42no—
179C22×C3×C4—44no—
180C22×C3×C4—48no—
181C22×C3×C4—48no—
182C22×C3×C4—432no—
183C22×C3×C4SD1618yescomplemented
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D414yes—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D414yes—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2412yes—
193C22×C3×C4C2411yescharacteristic
194C3×SD16—232no—
195C2×C3×C8—28no—
196C2×C3×C8C2418yes—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—24no—
202C3×C4⋊C4—28no—
203C3×C4⋊C4—28no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4C2414yes—
206C3×C4⋊C4C2414yes—
207C3×C22⋊C4—28no—
208C3×C22⋊C4—28no—
209C3×C22⋊C4—28no—
210C3×C42—28no—
211C3×C42C2×D414yes—
212C3×C42C2×D412yes—
213C3×C42C2×D414yes—
214C3×C42C2×D412yes—
215C3×C42C2412yes—
216C3×C42C2412yes—
217C25—21no—
218C22×Q8—21no—
219C22×Q8C23×C312yes—
220C22×D4—28no—
221C22×D4—21no—
222C22×D4—22no—
223C22×D4C23×C314yes—
224C22×D4C23×C312yes—
225C23×C4—21no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—28no—
229C23×C4—48no—
230C23×C4C3×D411yescharacteristic
231C23×C4C3×D411yescharacteristic
232C23×C4C23×C311yescharacteristic
233C2×SD16—216no—
234C2×SD16C23×C3116yes—
235C22×C8C23×C314yes—
236C4⋊Q8—22no—
237C4⋊Q8—22no—
238C4⋊Q8C23×C314yescomplemented
239C4⋊Q8C23×C314yescomplemented
240#125—28no—
241C22⋊Q8—28no—
242C22⋊Q8—24no—
243C22⋊Q8C23×C318yes—
244C4×Q8—28no—
245C4×Q8—24no—
246C4×Q8C23×C318yescomplemented
247C4×D4—28no—
248C4×D4—24no—
249C4×D4—28no—
250C4×D4—24no—
251C4×D4C23×C318yes—
252C4×D4C23×C318yes—
253#117—28no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4C23×C312yes—
262C2×C4⋊C4C23×C314yes—
263C2×C4⋊C4C23×C312yes—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—28no—
268C2×C42—22no—
269C2×C42C3×D411yescharacteristic
270C2×C42C3×D411yescharacteristic
271C2×C42C3×D414yes—
272C2×C42C23×C311yescharacteristic
273C2×C42C23×C311yescharacteristic
274C2×C42C23×C314yes—
275#106C23×C3116yescomplemented
276C4⋊C8C23×C318yescomplemented
277#103C23×C3116yescomplemented
278#102C23×C3116yes—
279C22⋊C8C23×C318yes—
280C4×C8C23×C318yescomplemented
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—216no—
284C23×C3—28no—
285C23×C3—24no—
286C23×C3—432no—
287C23×C3—48no—
288C23×C3—48no—
289C23×C3—44no—
290C23×C3—44no—
291C23×C3C2×SD1614yes—
292C23×C3C22×D412yes—
293C23×C3C22×D411yescharacteristic
294C3×Q8—28no—
295C3×Q8—48no—
296C3×Q8—44no—
297C3×D4—216no—
298C3×D4—28no—
299C3×D4—24no—
300C3×D4C2×#4118yes—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—24no—
309C2×C3×C4—44no—
310C2×C3×C4—44no—
311C2×C3×C4—48no—
312C2×C3×C4—44no—
313C2×C3×C4—44no—
314C2×C3×C4—416no—
315C2×C3×C4—416no—
316C2×C3×C4C2×SD1614yes—
317C2×C3×C4C2×SD1614yes—
318C2×C3×C4C2×SD16116yescomplemented
319C2×C3×C4C22×D412yes—
320C2×C3×C4C22×D412yes—
321C2×C3×C4C22×D412yes—
322C2×C3×C4C22×D411yescharacteristic
323C2×C3×C4C22×D412yes—
324C2×C3×C4C22×D412yes—
325C2×C3×C4C2×#4114yes—
326C2×C3×C4C2511yescharacteristic
327C3×C8—28no—
328C24—28no—
329C24—24no—
330C24—22no—
331C24—44no—
332C24—44no—
333C24C2×C3×D411yescharacteristic
334C2×Q8—22no—
335C2×Q8—24no—
336C2×Q8—28no—
337C2×Q8—44no—
338C2×Q8C24×C314yes—
339C2×D4—28no—
340C2×D4—216no—
341C2×D4—216no—
342C2×D4—24no—
343C2×D4—22no—
344C2×D4—24no—
345C2×D4C3×#4118yes—
346C2×D4C24×C314yes—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—42no—
355C22×C4—44no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—432no—
359C22×C4C3×SD1618yescomplemented
360C22×C4C2×C3×D411yescharacteristic
361C22×C4C2×C3×D412yes—
362C22×C4C2×C3×D411yescharacteristic
363C22×C4C2×C3×D411yescharacteristic
364C22×C4C2×C3×D414yes—
365C22×C4C2×C3×D411yescharacteristic
366C22×C4C2×C3×D411yescharacteristic
367C22×C4C2×C3×D414yes—
368C22×C4C24×C312yes—
369C22×C4C24×C311yescharacteristic
370SD16—232no—
371C2×C8—28no—
372C2×C8C24×C318yes—
373C4⋊C4—24no—
374C4⋊C4—22no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—28no—
379C4⋊C4—28no—
380C4⋊C4—22no—
381C4⋊C4C24×C314yes—
382C4⋊C4C24×C314yes—
383C22⋊C4—28no—
384C22⋊C4—28no—
385C22⋊C4—28no—
386C42—28no—
387C42C2×C3×D414yes—
388C42C2×C3×D412yes—
389C42C2×C3×D414yes—
390C42C2×C3×D412yes—
391C42C24×C312yes—
392C42C24×C312yes—
393C22×C3—28no—
394C22×C3—22no—
395C22×C3—22no—
396C22×C3—28no—
397C22×C3—432no—
398C22×C3—416no—
399C22×C3—44no—
400C22×C3—44no—
401C22×C3#52512yes—
402C22×C3C22×SD1614yes—
403C22×C3C22×SD1612yes—
404C22×C3#57512yes—
405C22×C3C23×D411yescharacteristic
406C3×C4—22no—
407C3×C4—22no—
408C3×C4—22no—
409C3×C4—44no—
410C3×C4—44no—
411C3×C4—44no—
412C3×C4#52512yes—
413C3×C4C22×SD1614yes—
414C3×C4C22×SD1614yes—
415C3×C4C22×#4112yes—
416C23—22no—
417C23—22no—
418C23—216no—
419C23—24no—
420C23—28no—
421C23—48no—
422C23—432no—
423C23—48no—
424C23—44no—
425C23—44no—
426C23C2×C3×SD1614yes—
427C23C22×C3×D412yes—
428C23C22×C3×D411yescharacteristic
429Q8—28no—
430Q8—48no—
431Q8—44no—
432D4—216no—
433D4—24no—
434D4—28no—
435D4C2×C3×#4118yes—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—44no—
445C2×C4—48no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—416no—
450C2×C4—416no—
451C2×C4C2×C3×SD1614yes—
452C2×C4C2×C3×SD1614yes—
453C2×C4C2×C3×SD16116yescomplemented
454C2×C4C22×C3×D412yes—
455C2×C4C22×C3×D412yes—
456C2×C4C22×C3×D412yes—
457C2×C4C22×C3×D412yes—
458C2×C4C22×C3×D412yes—
459C2×C4C22×C3×D411yescharacteristic
460C2×C4C2×C3×#4114yes—
461C2×C4C25×C311yescharacteristic
462C8—28no—
463C2×C3—24no—
464C2×C3—48no—
465C2×C3#310114yesdirect product factor
466C2×C3C2×#52511yescharacteristic
467C2×C3C23×SD1611yescharacteristic
468C2×C3C2×#57511yescharacteristic
469C22—22no—
470C22—22no—
471C22—28no—
472C22—28no—
473C22—432no—
474C22—44no—
475C22—44no—
476C22—416no—
477C22C3×#52512yes—
478C22C22×C3×SD1614yes—
479C22C22×C3×SD1612yes—
480C22C3×#57512yes—
481C22C23×C3×D411yescharacteristic
482C4—22no—
483C4—22no—
484C4—22no—
485C4—44no—
486C4—44no—
487C4—44no—
488C4C3×#52512yes—
489C4C22×C3×SD1614yes—
490C4C22×C3×SD1614yes—
491C4C22×C3×#4112yes—
492C3C2×#310111yescharacteristic, direct product factor, Sylow
493C2—24no—
494C2—48no—
495C2C3×#310114yesdirect product factor
496C2C2×C3×#52511yescharacteristic
497C2C23×C3×SD1611yescharacteristic
498C2C2×C3×#57511yescharacteristic
subgroupG/H
centerC22×C3×C4C2×D4
commutator, Frattini, cosocleC2×C4C25×C3
socleC23×C3C22×D4
prime radicalC23×C3×D4C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3×C4 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#533C2semidirect
3. C3×#3101C2direct (N × H)
4. C2×C3×#483C2semidirect
5. C2×C3×#476C2semidirect
6. C2×C3×#461C2semidirect
8. C2×C3×C4×SD16C2semidirect
9. C2×C3×#431C2semidirect
10. C2×#3101C3direct (N × H)
15. C3×#533C22semidirect
16. C2×C3×C4⋊Q8C22semidirect
17. C2×C3×C4⋊Q8C22semidirect
22. C2×C3×C4×Q8C22semidirect
33. C3×#483C22semidirect
34. C3×#476C22semidirect
35. C3×#461C22semidirect
37. C3×C4×SD16C22semidirect
38. C3×#431C22semidirect
39. C2×C3×#106C22semidirect
40. C2×C3×C4⋊C8C22semidirect
41. C2×C3×#103C22semidirect
44. C2×C3×C4×C8C22semidirect
45. C2×#533C2×C3semidirect
47. #3101C2×C3direct (N × H)
48. C2×#483C2×C3semidirect
49. C2×#476C2×C3semidirect
50. C2×#461C2×C3semidirect
52. C2×C4×SD16C2×C3semidirect
53. C2×#431C2×C3semidirect
75. C3×C4⋊Q8C23semidirect
76. C3×C4⋊Q8C23semidirect
83. C3×C4×Q8C23semidirect
112. C3×#106C23semidirect
113. C3×C4⋊C8C23semidirect
114. C3×#103C23semidirect
117. C3×C4×C8C23semidirect
122. #533C22×C3semidirect
123. C2×C4⋊Q8C22×C3semidirect
124. C2×C4⋊Q8C22×C3semidirect
129. C2×C4×Q8C22×C3semidirect
140. #483C22×C3semidirect
141. #476C22×C3semidirect
142. #461C22×C3semidirect
144. C4×SD16C22×C3semidirect
145. #431C22×C3semidirect
146. C2×#106C22×C3semidirect
147. C2×C4⋊C8C22×C3semidirect
148. C2×#103C22×C3semidirect
151. C2×C4×C8C22×C3semidirect
183. C22×C3×C4SD16semidirect
238. C4⋊Q8C23×C3semidirect
239. C4⋊Q8C23×C3semidirect
246. C4×Q8C23×C3semidirect
275. #106C23×C3semidirect
276. C4⋊C8C23×C3semidirect
277. #103C23×C3semidirect
280. C4×C8C23×C3semidirect
318. C2×C3×C4C2×SD16semidirect
359. C22×C4C3×SD16semidirect
453. C2×C4C2×C3×SD16semidirect
465. C2×C3#3101direct (N × H)
492. C3C2×#3101direct (N × H)
495. C2C3×#3101direct (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×#31012
C2×C3×#4832
C2×C3×#4762
C2×C3×#4612
C2×C3×#4502
C2×C3×C4×SD162
C2×C3×#4312
C2×#31013