Finite Group Databasebeta

← C2×C3×#3218 #10679393

Subgroups — C2×C3×#3218 #10679393

Order 768.

2146 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×C3×#529C211yesmaximal, characteristic
3C3×#3218C2116yesmaximal, direct product factor
4C2×C3×#497C212yesmaximal, complemented
5C2×C3×#491C212yesmaximal, complemented
6C2×C3×#462C212yesmaximal, complemented
7C2×C3×#461C212yesmaximal, complemented
8C2×C3×#451C212yesmaximal, complemented
9C2×C3×#448C212yesmaximal
10C2×C3×#432C211yesmaximal, characteristic, complemented
11C2×#3218C311yesmaximal, characteristic, direct product factor, Sylow
12C23×C3×Q8C2212yes—
13C22×C3×Q16C2214yes—
14C22×C3×SD16C2214yes—
15C2×C3×C4⋊Q8—21no—
16C2×C3×C4⋊Q8—21no—
17C2×C3×C4⋊Q8—24no—
18C2×C3×C4⋊Q8C2212yes—
19C2×C3×C4⋊Q8C2212yes—
20C2×C3×C4⋊Q8C2218yescomplemented
21C3×#529—24no—
22C3×#529C2218yes—
23C2×C3×#126—21no—
24C2×C3×#125C2211yescharacteristic
25C2×C3×#124C2211yescharacteristic
26C2×C3×#123—21no—
27C2×C3×C22⋊Q8C2211yescharacteristic
28C2×C3×#121C2211yescharacteristic
29C2×C3×C4×Q8C2211yescharacteristic
30C2×C3×C4×D4—21no—
31C2×C3×C4×D4C2211yescharacteristic
32C2×C3×#117—21no—
33C22×C3×C4⋊C4—21no—
34C22×C3×C4⋊C4—21no—
35C22×C3×C42C2211yescharacteristic
36C3×#497C22116yescomplemented
37C3×#491C22116yescomplemented
38C3×#462C22116yescomplemented
39C3×#461C22116yescomplemented
40C3×#451C22116yescomplemented
41C3×#448C22116yes—
42C3×#432C2218yescomplemented
43C2×C3×C4⋊C8C2212yes—
44C2×C3×#103C2214yes—
45C2×C3×#103C2214yes—
46C2×C3×C22⋊C8C2212yes—
47C2×C3×#97C2212yes—
48C22×C4⋊Q8C2×C311yescharacteristic, complemented
49C2×#529C2×C311yescharacteristic
50#3218C2×C3116yesdirect product factor
51C2×#497C2×C312yescomplemented
52C2×#491C2×C312yescomplemented
53C2×#462C2×C312yescomplemented
54C2×#461C2×C312yescomplemented
55C2×#451C2×C312yescomplemented
56C2×#448C2×C312yes—
57C2×#432C2×C311yescharacteristic, complemented
58C22×C3×Q8—24no—
59C22×C3×Q8—22no—
60C22×C3×Q8—48no—
61C22×C3×Q8C2314yes—
62C22×C3×Q8C2311yescharacteristic
63C22×C3×Q8C2318yes—
64C22×C3×D4—21no—
65C22×C3×D4C2311yescharacteristic
66C23×C3×C4—22no—
67C23×C3×C4—21no—
68C23×C3×C4D411yescharacteristic
69C23×C3×C4D411yescharacteristic
70C23×C3×C4C2311yescharacteristic
71C2×C3×Q16—48no—
72C2×C3×Q16C23116yes—
73C2×C3×SD16—48no—
74C2×C3×SD16C23116yes—
75C22×C3×C8C2314yes—
76C3×C4⋊Q8—24no—
77C3×C4⋊Q8—24no—
78C3×C4⋊Q8—28no—
79C3×C4⋊Q8—28no—
80C3×C4⋊Q8C2318yes—
81C3×C4⋊Q8C2318yes—
82C3×#126—28no—
83C3×#125—22no—
84C3×#125C2314yes—
85C3×#124—22no—
86C3×#124C2314yes—
87C3×#123—28no—
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8C2314yes—
90C3×#121—22no—
91C3×#121C2314yes—
92C3×C4×Q8—22no—
93C3×C4×Q8C2314yes—
94C3×C4×D4—28no—
95C3×C4×D4—22no—
96C3×C4×D4C2314yes—
97C3×#117—28no—
98C2×C3×C4⋊C4—22no—
99C2×C3×C4⋊C4—22no—
100C2×C3×C4⋊C4—22no—
101C2×C3×C4⋊C4—22no—
102C2×C3×C4⋊C4—28no—
103C2×C3×C4⋊C4—28no—
104C2×C3×C4⋊C4—21no—
105C2×C3×C4⋊C4—21no—
106C2×C3×C4⋊C4—21no—
107C2×C3×C4⋊C4—21no—
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4—21no—
114C2×C3×C22⋊C4—21no—
115C2×C3×C42—21no—
116C2×C3×C42D411yescharacteristic
117C2×C3×C42D411yescharacteristic
118C2×C3×C42D414yes—
119C2×C3×C42C2311yescharacteristic
120C2×C3×C42C2311yescharacteristic
121C2×C3×C42C2314yes—
122C3×C4⋊C8C2318yes—
123C3×#103C23116yes—
124C3×#103C23116yes—
125C3×C22⋊C8C2318yes—
126C3×#97C2318yes—
127C23×Q8C22×C312yes—
128C22×Q16C22×C314yes—
129C22×SD16C22×C314yes—
130C2×C4⋊Q8—21no—
131C2×C4⋊Q8—21no—
132C2×C4⋊Q8—24no—
133C2×C4⋊Q8C22×C312yes—
134C2×C4⋊Q8C22×C312yes—
135C2×C4⋊Q8C22×C318yescomplemented
136#529—24no—
137#529C22×C318yes—
138C2×#126—21no—
139C2×#125C22×C311yescharacteristic
140C2×#124C22×C311yescharacteristic
141C2×#123—21no—
142C2×C22⋊Q8C22×C311yescharacteristic
143C2×#121C22×C311yescharacteristic
144C2×C4×Q8C22×C311yescharacteristic
145C2×C4×D4—21no—
146C2×C4×D4C22×C311yescharacteristic
147C2×#117—21no—
148C22×C4⋊C4—21no—
149C22×C4⋊C4—21no—
150C22×C42C22×C311yescharacteristic
151#497C22×C3116yescomplemented
152#491C22×C3116yescomplemented
153#462C22×C3116yescomplemented
154#461C22×C3116yescomplemented
155#451C22×C3116yescomplemented
156#448C22×C3116yes—
157#432C22×C318yescomplemented
158C2×C4⋊C8C22×C312yes—
159C2×#103C22×C314yes—
160C2×#103C22×C314yes—
161C2×C22⋊C8C22×C312yes—
162C2×#97C22×C312yes—
163C24×C3—21no—
164C24×C3C2×D411yescharacteristic
165C2×C3×Q8—21no—
166C2×C3×Q8—28no—
167C2×C3×Q8—24no—
168C2×C3×Q8—28no—
169C2×C3×Q8—24no—
170C2×C3×Q8—28no—
171C2×C3×Q8—416no—
172C2×C3×Q8—48no—
173C2×C3×Q8—416no—
174C2×C3×Q8C2418yes—
175C2×C3×Q8C2412yes—
176C2×C3×D4—24no—
177C2×C3×D4—21no—
178C2×C3×D4—24no—
179C2×C3×D4—44no—
180C2×C3×D4C2412yes—
181C22×C3×C4—24no—
182C22×C3×C4—24no—
183C22×C3×C4—24no—
184C22×C3×C4—28no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—24no—
189C22×C3×C4—24no—
190C22×C3×C4—48no—
191C22×C3×C4—44no—
192C22×C3×C4C2×D412yes—
193C22×C3×C4C2×D414yes—
194C22×C3×C4C2×D414yes—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2412yes—
201C22×C3×C4C2411yescharacteristic
202C3×Q16—416no—
203C3×SD16—416no—
204C2×C3×C8—28no—
205C2×C3×C8C2418yes—
206C3×C4⋊C4—28no—
207C3×C4⋊C4—28no—
208C3×C4⋊C4—28no—
209C3×C4⋊C4—28no—
210C3×C4⋊C4—24no—
211C3×C4⋊C4—24no—
212C3×C4⋊C4—24no—
213C3×C4⋊C4—24no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4C2412yes—
217C3×C4⋊C4C2412yes—
218C3×C22⋊C4—24no—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—24no—
221C3×C22⋊C4—24no—
222C3×C22⋊C4—24no—
223C3×C42—24no—
224C3×C42C2×D414yes—
225C3×C42C2×D412yes—
226C3×C42C2×D412yes—
227C3×C42C2×D414yes—
228C3×C42C2412yes—
229C3×C42C2412yes—
230C22×Q8—22no—
231C22×Q8—24no—
232C22×Q8—48no—
233C22×Q8C23×C311yescharacteristic
234C22×Q8C23×C314yes—
235C22×Q8C23×C318yes—
236C22×D4—21no—
237C22×D4C23×C311yescharacteristic
238C23×C4—22no—
239C23×C4—21no—
240C23×C4C3×D411yescharacteristic
241C23×C4C3×D411yescharacteristic
242C23×C4C23×C311yescharacteristic
243C2×Q16—48no—
244C2×Q16C23×C3116yes—
245C2×SD16—48no—
246C2×SD16C23×C3116yes—
247C22×C8C23×C314yes—
248C4⋊Q8—24no—
249C4⋊Q8—24no—
250C4⋊Q8—28no—
251C4⋊Q8—28no—
252C4⋊Q8C23×C318yes—
253C4⋊Q8C23×C318yes—
254#126—28no—
255#125—22no—
256#125C23×C314yes—
257#124—22no—
258#124C23×C314yes—
259#123—28no—
260C22⋊Q8—22no—
261C22⋊Q8C23×C314yes—
262#121—22no—
263#121C23×C314yes—
264C4×Q8—22no—
265C4×Q8C23×C314yes—
266C4×D4—28no—
267C4×D4—22no—
268C4×D4C23×C314yes—
269#117—28no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4C23×C311yescharacteristic
281C2×C4⋊C4C23×C311yescharacteristic
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C42—21no—
288C2×C42C3×D411yescharacteristic
289C2×C42C3×D411yescharacteristic
290C2×C42C3×D414yes—
291C2×C42C23×C311yescharacteristic
292C2×C42C23×C311yescharacteristic
293C2×C42C23×C314yes—
294C4⋊C8C23×C318yes—
295#103C23×C3116yes—
296#103C23×C3116yes—
297C22⋊C8C23×C318yes—
298#97C23×C318yes—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—22no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3—42no—
305C23×C3#13714yes—
306C23×C3C22×D412yes—
307C23×C3C22×D411yescharacteristic
308C3×Q8—28no—
309C3×Q8—24no—
310C3×Q8—28no—
311C3×Q8—416no—
312C3×Q8—42no—
313C3×Q8—48no—
314C3×D4—24no—
315C3×D4—48no—
316C3×D4—42no—
317C2×C3×C4—22no—
318C2×C3×C4—28no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—24no—
323C2×C3×C4—28no—
324C2×C3×C4—24no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—28no—
329C2×C3×C4—416no—
330C2×C3×C4—48no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—48no—
335C2×C3×C4#13714yes—
336C2×C3×C4C22×D412yes—
337C2×C3×C4C22×D412yes—
338C2×C3×C4C22×D412yes—
339C2×C3×C4C22×D412yes—
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C22×D412yes—
342C2×C3×C4H(5,𝔽2)14yes—
343C2×C3×C4C2511yescharacteristic
344C3×C8—28no—
345C24—21no—
346C24C2×C3×D411yescharacteristic
347C2×Q8—21no—
348C2×Q8—28no—
349C2×Q8—24no—
350C2×Q8—28no—
351C2×Q8—24no—
352C2×Q8—28no—
353C2×Q8—416no—
354C2×Q8—48no—
355C2×Q8—416no—
356C2×Q8C24×C318yes—
357C2×Q8C24×C312yes—
358C2×D4—24no—
359C2×D4—21no—
360C2×D4—24no—
361C2×D4—44no—
362C2×D4C24×C312yes—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—28no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—48no—
373C22×C4—44no—
374C22×C4C2×C3×D411yescharacteristic
375C22×C4C2×C3×D412yes—
376C22×C4C2×C3×D411yescharacteristic
377C22×C4C2×C3×D411yescharacteristic
378C22×C4C2×C3×D411yescharacteristic
379C22×C4C2×C3×D414yes—
380C22×C4C2×C3×D414yes—
381C22×C4C2×C3×D411yescharacteristic
382C22×C4C24×C312yes—
383C22×C4C24×C311yescharacteristic
384Q16—416no—
385SD16—416no—
386C2×C8—28no—
387C2×C8C24×C318yes—
388C4⋊C4—28no—
389C4⋊C4—28no—
390C4⋊C4—28no—
391C4⋊C4—28no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4C24×C312yes—
399C4⋊C4C24×C312yes—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—24no—
404C22⋊C4—24no—
405C42—24no—
406C42C2×C3×D414yes—
407C42C2×C3×D412yes—
408C42C2×C3×D412yes—
409C42C2×C3×D414yes—
410C42C24×C312yes—
411C42C24×C312yes—
412C22×C3—22no—
413C22×C3—28no—
414C22×C3—22no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3—84no—
419C22×C3#53512yes—
420C22×C3C2×#13714yes—
421C22×C3C2×#13712yes—
422C22×C3C2×#13712yes—
423C22×C3C23×D411yescharacteristic
424C3×C4—22no—
425C3×C4—22no—
426C3×C4—22no—
427C3×C4—22no—
428C3×C4—48no—
429C3×C4—42no—
430C3×C4—42no—
431C3×C4—42no—
432C3×C4#53512yes—
433C3×C4C2×#13714yes—
434C3×C4C2×H(5,𝔽2)12yes—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23C3×#13714yes—
442C23C22×C3×D412yes—
443C23C22×C3×D411yescharacteristic
444Q8—28no—
445Q8—24no—
446Q8—28no—
447Q8—416no—
448Q8—48no—
449Q8—42no—
450D4—24no—
451D4—48no—
452D4—42no—
453C2×C4—28no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—28no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—28no—
465C2×C4—416no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—48no—
469C2×C4—42no—
470C2×C4—48no—
471C2×C4C3×#13714yes—
472C2×C4C22×C3×D412yes—
473C2×C4C22×C3×D412yes—
474C2×C4C22×C3×D412yes—
475C2×C4C22×C3×D412yes—
476C2×C4C22×C3×D412yes—
477C2×C4C22×C3×D411yescharacteristic
478C2×C4C3×H(5,𝔽2)14yes—
479C2×C4C25×C311yescharacteristic
480C8—28no—
481C2×C3—24no—
482C2×C3—82no—
483C2×C3#321814yesdirect product factor
484C2×C3C2×#53511yescharacteristic
485C2×C3C22×#13711yescharacteristic
486C2×C3C22×#13711yescharacteristic
487C22—22no—
488C22—28no—
489C22—22no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—84no—
494C22C3×#53512yes—
495C22C2×C3×#13714yes—
496C22C2×C3×#13712yes—
497C22C2×C3×#13712yes—
498C22C23×C3×D411yescharacteristic
499C4—22no—
500C4—22no—
501C4—22no—
502C4—22no—
503C4—48no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4C3×#53512yes—
508C4C2×C3×#13714yes—
509C4C2×C3×H(5,𝔽2)12yes—
510C3C2×#321811yescharacteristic, direct product factor, Sylow
511C2—24no—
512C2—82no—
513C2C3×#321814yesdirect product factor
514C2C2×C3×#53511yescharacteristic
515C2C22×C3×#13711yescharacteristic
516C2C22×C3×#13711yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC2×C3×#121C22
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
1. C22×C3×C4⋊Q8C2semidirect
3. C3×#3218C2direct (N × H)
4. C2×C3×#497C2semidirect
5. C2×C3×#491C2semidirect
6. C2×C3×#462C2semidirect
7. C2×C3×#461C2semidirect
8. C2×C3×#451C2semidirect
10. C2×C3×#432C2semidirect
11. C2×#3218C3direct (N × H)
20. C2×C3×C4⋊Q8C22semidirect
36. C3×#497C22semidirect
37. C3×#491C22semidirect
38. C3×#462C22semidirect
39. C3×#461C22semidirect
40. C3×#451C22semidirect
42. C3×#432C22semidirect
48. C22×C4⋊Q8C2×C3semidirect
50. #3218C2×C3direct (N × H)
51. C2×#497C2×C3semidirect
52. C2×#491C2×C3semidirect
53. C2×#462C2×C3semidirect
54. C2×#461C2×C3semidirect
55. C2×#451C2×C3semidirect
57. C2×#432C2×C3semidirect
135. C2×C4⋊Q8C22×C3semidirect
151. #497C22×C3semidirect
152. #491C22×C3semidirect
153. #462C22×C3semidirect
154. #461C22×C3semidirect
155. #451C22×C3semidirect
157. #432C22×C3semidirect
483. C2×C3#3218direct (N × H)
510. C3C2×#3218direct (N × H)
513. C2C3×#3218direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊Q82
C2×C3×#5292
C3×#32182
C2×C3×#4972
C2×C3×#4912
C2×C3×#4622
C2×C3×#4612
C2×C3×#4512
C2×C3×#4482
C2×C3×#4322
C2×#32183