Finite Group Databasebeta

← C2×C3×#3095 #10679117

Subgroups — C2×C3×#3095 #10679117

Order 768.

2326 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C22⋊Q8C211yesmaximal, characteristic
2C22×C3×#117C211yesmaximal, characteristic
3C3×#3095C2116yesmaximal, direct product factor
4C2×C3×#484C212yesmaximal, complemented
5C2×C3×#485C214yesmaximal, complemented
6C2×C3×#483C212yesmaximal, complemented
7C2×C3×#428C212yesmaximal, complemented
8C2×C3×#419C212yesmaximal, complemented
9C2×C3×#407C211yesmaximal, characteristic
10C2×#3095C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×Q8C2211yescharacteristic
12C24×C3×C4C2211yescharacteristic
13C22×C3×M16C2212yes—
14C2×C3×C22⋊Q8—21no—
15C2×C3×C22⋊Q8—24no—
16C2×C3×C22⋊Q8—21no—
17C2×C3×C22⋊Q8C2212yes—
18C2×C3×C22⋊Q8C2212yes—
19C2×C3×C22⋊Q8C2218yes—
20C2×C3×#117—21no—
21C2×C3×#117—24no—
22C2×C3×#117—21no—
23C2×C3×#117C2212yes—
24C2×C3×#117C2212yes—
25C2×C3×#117C2218yes—
26C22×C3×C4⋊C4—21no—
27C22×C3×C4⋊C4C2211yescharacteristic
28C22×C3×C4⋊C4C2211yescharacteristic
29C22×C3×C4⋊C4C2211yescharacteristic
30C22×C3×C22⋊C4—21no—
31C22×C3×C22⋊C4—21no—
32C22×C3×C42—21no—
33C3×#485C22132yescomplemented
34C3×#484C22116yescomplemented
35C3×#483C22116yescomplemented
36C3×#428C22116yescomplemented
37C2×C3×#107C2214yescomplemented
38C2×C3×#106C2214yescomplemented
39C3×#419C22116yescomplemented
40C2×C3×#103C2214yescomplemented
41C2×C3×#103C2214yescomplemented
42C3×#407C2218yes—
43C2×C3×C22⋊C8C2212yes—
44C2×C3×C22⋊C8C2212yes—
45C22×C22⋊Q8C2×C311yescharacteristic
46C22×#117C2×C311yescharacteristic
47#3095C2×C3116yesdirect product factor
48C2×#484C2×C312yescomplemented
49C2×#485C2×C314yescomplemented
50C2×#483C2×C312yescomplemented
51C2×#428C2×C312yescomplemented
52C2×#419C2×C312yescomplemented
53C2×#407C2×C311yescharacteristic
54C25×C3D411yescharacteristic
55C22×C3×Q8—22no—
56C22×C3×Q8—21no—
57C22×C3×Q8—28no—
58C22×C3×Q8C2312yes—
59C22×C3×Q8C2314yes—
60C23×C3×C4—28no—
61C23×C3×C4—21no—
62C23×C3×C4—21no—
63C23×C3×C4—21no—
64C23×C3×C4—21no—
65C23×C3×C4D411yescharacteristic
66C23×C3×C4D411yescharacteristic
67C23×C3×C4D414yes—
68C23×C3×C4D411yescharacteristic
69C23×C3×C4C2311yescharacteristic
70C23×C3×C4C2311yescharacteristic
71C23×C3×C4C2314yes—
72C23×C3×C4C2311yescharacteristic
73C2×C3×M16—28no—
74C2×C3×M16C2318yes—
75C22×C3×C8C2314yes—
76C3×C22⋊Q8—28no—
77C3×C22⋊Q8—24no—
78C3×C22⋊Q8—24no—
79C3×C22⋊Q8—28no—
80C3×C22⋊Q8C2318yes—
81C3×C22⋊Q8C2318yes—
82C3×#117—28no—
83C3×#117—24no—
84C3×#117—28no—
85C3×#117—24no—
86C3×#117C2318yes—
87C3×#117C2318yes—
88C2×C3×C4⋊C4—28no—
89C2×C3×C4⋊C4—22no—
90C2×C3×C4⋊C4—22no—
91C2×C3×C4⋊C4—22no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—22no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4C2312yes—
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2312yes—
100C2×C3×C4⋊C4C2314yes—
101C2×C3×C4⋊C4C2314yes—
102C2×C3×C4⋊C4C2314yes—
103C2×C3×C22⋊C4—28no—
104C2×C3×C22⋊C4—22no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4—28no—
107C2×C3×C22⋊C4—22no—
108C2×C3×C22⋊C4—22no—
109C2×C3×C42—28no—
110C2×C3×C42—22no—
111C2×C3×C42—22no—
112C3×#107C23116yescomplemented
113C3×#106C23116yescomplemented
114C3×#103C23116yescomplemented
115C3×#103C23116yescomplemented
116C3×C22⋊C8C2318yes—
117C3×C22⋊C8C2318yes—
118C23×Q8C22×C311yescharacteristic
119C24×C4C22×C311yescharacteristic
120C22×M16C22×C312yes—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8—24no—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8C22×C312yes—
125C2×C22⋊Q8C22×C312yes—
126C2×C22⋊Q8C22×C318yes—
127C2×#117—21no—
128C2×#117—24no—
129C2×#117—21no—
130C2×#117C22×C312yes—
131C2×#117C22×C318yes—
132C2×#117C22×C312yes—
133C22×C4⋊C4—21no—
134C22×C4⋊C4C22×C311yescharacteristic
135C22×C4⋊C4C22×C311yescharacteristic
136C22×C4⋊C4C22×C311yescharacteristic
137C22×C22⋊C4—21no—
138C22×C22⋊C4—21no—
139C22×C42—21no—
140#485C22×C3132yescomplemented
141#484C22×C3116yescomplemented
142#483C22×C3116yescomplemented
143#428C22×C3116yescomplemented
144C2×#107C22×C314yescomplemented
145C2×#106C22×C314yescomplemented
146#419C22×C3116yescomplemented
147C2×#103C22×C314yescomplemented
148C2×#103C22×C314yescomplemented
149#407C22×C318yes—
150C2×C22⋊C8C22×C312yes—
151C2×C22⋊C8C22×C312yes—
152C24×C3—24no—
153C24×C3—24no—
154C24×C3—42no—
155C24×C3C2×D411yescharacteristic
156C24×C3C2×D414yes—
157C24×C3C2×D411yescharacteristic
158C24×C3C2×D411yescharacteristic
159C2×C3×Q8—24no—
160C2×C3×Q8—22no—
161C2×C3×Q8—28no—
162C2×C3×Q8—216no—
163C2×C3×Q8—48no—
164C2×C3×Q8—44no—
165C2×C3×Q8C2414yes—
166C22×C3×C4—24no—
167C22×C3×C4—24no—
168C22×C3×C4—216no—
169C22×C3×C4—24no—
170C22×C3×C4—24no—
171C22×C3×C4—24no—
172C22×C3×C4—216no—
173C22×C3×C4—24no—
174C22×C3×C4—24no—
175C22×C3×C4—24no—
176C22×C3×C4—24no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—44no—
182C22×C3×C4—44no—
183C22×C3×C4—44no—
184C22×C3×C4—44no—
185C22×C3×C4C2×D414yes—
186C22×C3×C4C2×D414yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×D414yes—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4#4118yes—
195C22×C3×C4C2412yes—
196C22×C3×C4C2412yes—
197C22×C3×C4C2412yes—
198C22×C3×C4C2411yescharacteristic
199C3×M16—216no—
200C2×C3×C8—28no—
201C2×C3×C8C2418yes—
202C3×C4⋊C4—28no—
203C3×C4⋊C4—28no—
204C3×C4⋊C4—24no—
205C3×C4⋊C4—24no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—24no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4C2414yes—
211C3×C4⋊C4C2414yes—
212C3×C4⋊C4C2414yes—
213C3×C22⋊C4—28no—
214C3×C22⋊C4—28no—
215C3×C22⋊C4—28no—
216C3×C22⋊C4—28no—
217C3×C42—28no—
218C3×C42—28no—
219C25C3×D411yescharacteristic
220C22×Q8—21no—
221C22×Q8—22no—
222C22×Q8—28no—
223C22×Q8C23×C312yes—
224C22×Q8C23×C314yes—
225C23×C4—28no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4C3×D411yescharacteristic
231C23×C4C3×D414yes—
232C23×C4C3×D411yescharacteristic
233C23×C4C3×D411yescharacteristic
234C23×C4C23×C311yescharacteristic
235C23×C4C23×C311yescharacteristic
236C23×C4C23×C314yes—
237C23×C4C23×C311yescharacteristic
238C2×M16—28no—
239C2×M16C23×C318yes—
240C22×C8C23×C314yes—
241C22⋊Q8—28no—
242C22⋊Q8—24no—
243C22⋊Q8—24no—
244C22⋊Q8—28no—
245C22⋊Q8C23×C318yes—
246C22⋊Q8C23×C318yes—
247#117—28no—
248#117—24no—
249#117—28no—
250#117—24no—
251#117C23×C318yes—
252#117C23×C318yes—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4C23×C312yes—
263C2×C4⋊C4C23×C312yes—
264C2×C4⋊C4C23×C314yes—
265C2×C4⋊C4C23×C314yes—
266C2×C4⋊C4C23×C312yes—
267C2×C4⋊C4C23×C314yes—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—22no—
274C2×C42—22no—
275C2×C42—28no—
276C2×C42—22no—
277#107C23×C3116yescomplemented
278#106C23×C3116yescomplemented
279#103C23×C3116yescomplemented
280#103C23×C3116yescomplemented
281C22⋊C8C23×C318yes—
282C22⋊C8C23×C318yes—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—28no—
288C23×C3—22no—
289C23×C3—28no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—44no—
293C23×C3—416no—
294C23×C3#12314yes—
295C23×C3#13714yes—
296C23×C3C22×D412yes—
297C23×C3C22×D412yes—
298C23×C3C22×D412yes—
299C23×C3C22×D411yescharacteristic
300C3×Q8—28no—
301C3×Q8—48no—
302C3×Q8—44no—
303C2×C3×C4—28no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—28no—
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—24no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—48no—
317C2×C3×C4—48no—
318C2×C3×C4—48no—
319C2×C3×C4—44no—
320C2×C3×C4—48no—
321C2×C3×C4—44no—
322C2×C3×C4—44no—
323C2×C3×C4—44no—
324C2×C3×C4#12318yes—
325C2×C3×C4#12314yes—
326C2×C3×C4C22×D412yes—
327C2×C3×C4C22×D412yes—
328C2×C3×C4C22×D412yes—
329C2×C3×C4C22×D411yescharacteristic
330C2×C3×C4C2×#4114yes—
331C2×C3×C4C2×#4118yes—
332C2×C3×C4C2511yescharacteristic
333C3×C8—28no—
334C24—24no—
335C24—24no—
336C24—42no—
337C24C2×C3×D411yescharacteristic
338C24C2×C3×D414yes—
339C24C2×C3×D411yescharacteristic
340C24C2×C3×D411yescharacteristic
341C2×Q8—24no—
342C2×Q8—22no—
343C2×Q8—28no—
344C2×Q8—216no—
345C2×Q8—48no—
346C2×Q8—44no—
347C2×Q8C24×C314yes—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—216no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—216no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4C2×C3×D411yescharacteristic
368C22×C4C2×C3×D414yes—
369C22×C4C2×C3×D411yescharacteristic
370C22×C4C2×C3×D414yes—
371C22×C4C2×C3×D412yes—
372C22×C4C2×C3×D412yes—
373C22×C4C2×C3×D414yes—
374C22×C4C2×C3×D411yescharacteristic
375C22×C4C2×C3×D412yes—
376C22×C4C3×#4118yes—
377C22×C4C24×C312yes—
378C22×C4C24×C312yes—
379C22×C4C24×C311yescharacteristic
380C22×C4C24×C312yes—
381M16—216no—
382C2×C8—28no—
383C2×C8C24×C318yes—
384C4⋊C4—28no—
385C4⋊C4—28no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—22no—
392C4⋊C4C24×C314yes—
393C4⋊C4C24×C314yes—
394C4⋊C4C24×C314yes—
395C22⋊C4—28no—
396C22⋊C4—28no—
397C22⋊C4—28no—
398C22⋊C4—28no—
399C42—28no—
400C42—28no—
401C22×C3—28no—
402C22×C3—28no—
403C22×C3—22no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—22no—
410C22×C3—416no—
411C22×C3—44no—
412C22×C3C2×#12314yes—
413C22×C3C2×#12312yes—
414C22×C3C2×#13714yes—
415C22×C3C2×#13712yes—
416C22×C3#57512yes—
417C22×C3C23×D411yescharacteristic
418C3×C4—22no—
419C3×C4—22no—
420C3×C4—44no—
421C3×C4—44no—
422C3×C4—44no—
423C3×C4—44no—
424C3×C4C2×#12314yes—
425C3×C4C2×#12312yes—
426C3×C4C22×#4112yes—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—28no—
432C23—22no—
433C23—22no—
434C23—28no—
435C23—22no—
436C23—44no—
437C23—416no—
438C23C3×#12314yes—
439C23C3×#13714yes—
440C23C22×C3×D412yes—
441C23C22×C3×D412yes—
442C23C22×C3×D412yes—
443C23C22×C3×D411yescharacteristic
444Q8—28no—
445Q8—48no—
446Q8—44no—
447C2×C4—28no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—28no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—28no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—48no—
461C2×C4—48no—
462C2×C4—48no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—48no—
467C2×C4—44no—
468C2×C4C3×#12318yes—
469C2×C4C3×#12314yes—
470C2×C4C22×C3×D412yes—
471C2×C4C22×C3×D412yes—
472C2×C4C22×C3×D412yes—
473C2×C4C22×C3×D411yescharacteristic
474C2×C4C2×C3×#4114yes—
475C2×C4C2×C3×#4118yes—
476C2×C4C25×C311yescharacteristic
477C8—28no—
478C2×C3—24no—
479C2×C3—24no—
480C2×C3—42no—
481C2×C3#309514yesdirect product factor
482C2×C3C22×#12311yescharacteristic
483C2×C3C22×#13711yescharacteristic
484C2×C3C2×#57511yescharacteristic
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—28no—
490C22—28no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—416no—
495C22—44no—
496C22C2×C3×#12314yes—
497C22C2×C3×#12312yes—
498C22C2×C3×#13714yes—
499C22C2×C3×#13712yes—
500C22C3×#57512yes—
501C22C23×C3×D411yescharacteristic
502C4—22no—
503C4—22no—
504C4—44no—
505C4—44no—
506C4—44no—
507C4—44no—
508C4C2×C3×#12314yes—
509C4C2×C3×#12312yes—
510C4C22×C3×#4112yes—
511C3C2×#309511yescharacteristic, direct product factor, Sylow
512C2—24no—
513C2—24no—
514C2—42no—
515C2C3×#309514yesdirect product factor
516C2C22×C3×#12311yescharacteristic
517C2C22×C3×#13711yescharacteristic
518C2C2×C3×#57511yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC25×C3D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#3095C2direct (N × H)
4. C2×C3×#484C2semidirect
5. C2×C3×#485C2semidirect
6. C2×C3×#483C2semidirect
7. C2×C3×#428C2semidirect
8. C2×C3×#419C2semidirect
10. C2×#3095C3direct (N × H)
33. C3×#485C22semidirect
34. C3×#484C22semidirect
35. C3×#483C22semidirect
36. C3×#428C22semidirect
37. C2×C3×#107C22semidirect
38. C2×C3×#106C22semidirect
39. C3×#419C22semidirect
40. C2×C3×#103C22semidirect
41. C2×C3×#103C22semidirect
47. #3095C2×C3direct (N × H)
48. C2×#484C2×C3semidirect
49. C2×#485C2×C3semidirect
50. C2×#483C2×C3semidirect
51. C2×#428C2×C3semidirect
52. C2×#419C2×C3semidirect
112. C3×#107C23semidirect
113. C3×#106C23semidirect
114. C3×#103C23semidirect
115. C3×#103C23semidirect
140. #485C22×C3semidirect
141. #484C22×C3semidirect
142. #483C22×C3semidirect
143. #428C22×C3semidirect
144. C2×#107C22×C3semidirect
145. C2×#106C22×C3semidirect
146. #419C22×C3semidirect
147. C2×#103C22×C3semidirect
148. C2×#103C22×C3semidirect
277. #107C23×C3semidirect
278. #106C23×C3semidirect
279. #103C23×C3semidirect
280. #103C23×C3semidirect
481. C2×C3#3095direct (N × H)
511. C3C2×#3095direct (N × H)
515. C2C3×#3095direct (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×#1172
C3×#30952
C2×C3×#4842
C2×C3×#4852
C2×C3×#4832
C2×C3×#4282
C2×C3×#4192
C2×C3×#4072
C2×#30953