Finite Group Databasebeta

← C3×#2678 #84542

Subgroups — C3×#2678 #84542

Order 384.

802 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#123C211yesmaximal, characteristic, complemented
2C2×C3×#121C211yesmaximal, characteristic, complemented
3C2×C3×#120C211yesmaximal, characteristic, complemented
4C2×C3×#120C211yesmaximal, characteristic, complemented
5C3×#397C211yesmaximal, characteristic, complemented
6C3×#396C211yesmaximal, characteristic, complemented
7C3×#394C211yesmaximal, characteristic, complemented
8C3×#394C211yesmaximal, characteristic, complemented
9C3×#392C211yesmaximal, characteristic, complemented
10C3×#392C211yesmaximal, characteristic, complemented
11C3×#390C211yesmaximal, characteristic, complemented
12C3×#390C211yesmaximal, characteristic, complemented
13C3×#388C211yesmaximal, characteristic, complemented
14C3×#386C211yesmaximal, characteristic, complemented
15C3×#379C211yesmaximal, characteristic, complemented
16#2678C311yesmaximal, characteristic, direct product factor, Sylow
17C25×C3C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×C3×D4C2211yescharacteristic
20C22×C3×D4C2211yescharacteristic
21C22×C3×D4C2211yescharacteristic
22C22×C3×D4C2211yescharacteristic
23C22×C3×D4C2211yescharacteristic
24C22×C3×D4C2211yescharacteristic, complemented
25C23×C3×C4C2211yescharacteristic, complemented
26C3×#123—24no—
27C3×#121—24no—
28C3×#120—24no—
29C3×#120—24no—
30C2×C3×C4⋊C4C2211yescharacteristic, complemented
31C2×C3×C4⋊C4C2211yescharacteristic, complemented
32C2×C3×C4⋊C4C2211yescharacteristic, complemented
33C2×C3×C4⋊C4C2211yescharacteristic, complemented
34C2×C3×C22⋊C4C2211yescharacteristic
35C2×C3×C22⋊C4C2211yescharacteristic
36C2×C3×C22⋊C4C2211yescharacteristic
37C2×C3×C22⋊C4C2211yescharacteristic
38C2×C3×C22⋊C4C2211yescharacteristic
39C2×C3×C22⋊C4C2211yescharacteristic
40C2×C3×C22⋊C4C2211yescharacteristic
41C2×C3×C22⋊C4C2211yescharacteristic
42C2×C3×C22⋊C4C2211yescharacteristic
43C2×C3×C22⋊C4C2211yescharacteristic
44C2×C3×C22⋊C4C2211yescharacteristic
45C2×C3×C22⋊C4C2211yescharacteristic
46C2×C3×C22⋊C4C2211yescharacteristic, complemented
47C2×C3×C22⋊C4C2211yescharacteristic, complemented
48C2×C3×C22⋊C4C2211yescharacteristic, complemented
49C2×C3×C22⋊C4C2211yescharacteristic, complemented
50C2×C3×C22⋊C4C2211yescharacteristic, complemented
51C2×C3×C42C2211yescharacteristic, complemented
52C3×#95C2211yescharacteristic, complemented
53C3×#95C2211yescharacteristic, complemented
54C3×#95C2211yescharacteristic, complemented
55C3×#95C2211yescharacteristic, complemented
56C2×#123C2×C311yescharacteristic, complemented
57C2×#121C2×C311yescharacteristic, complemented
58C2×#120C2×C311yescharacteristic, complemented
59C2×#120C2×C311yescharacteristic, complemented
60#397C2×C311yescharacteristic, complemented
61#396C2×C311yescharacteristic, complemented
62#394C2×C311yescharacteristic, complemented
63#394C2×C311yescharacteristic, complemented
64#392C2×C311yescharacteristic, complemented
65#392C2×C311yescharacteristic, complemented
66#390C2×C311yescharacteristic, complemented
67#390C2×C311yescharacteristic, complemented
68#388C2×C311yescharacteristic, complemented
69#386C2×C311yescharacteristic, complemented
70#379C2×C311yescharacteristic, complemented
71C24×C3—22no—
72C24×C3—22no—
73C24×C3—22no—
74C24×C3—41no—
75C24×C3—41no—
76C24×C3—41no—
77C24×C3—41no—
78C24×C3C2311yescharacteristic
79C24×C3C2311yescharacteristic
80C24×C3C2311yescharacteristic
81C24×C3C2311yescharacteristic
82C24×C3C2311yescharacteristic
83C2×C3×D4—22no—
84C2×C3×D4—22no—
85C2×C3×D4—22no—
86C2×C3×D4—22no—
87C2×C3×D4—22no—
88C2×C3×D4—22no—
89C2×C3×D4—22no—
90C2×C3×D4—22no—
91C2×C3×D4—22no—
92C2×C3×D4—22no—
93C2×C3×D4—41no—
94C2×C3×D4—41no—
95C2×C3×D4—41no—
96C2×C3×D4—41no—
97C2×C3×D4—41no—
98C2×C3×D4—41no—
99C2×C3×D4—41no—
100C2×C3×D4—41no—
101C2×C3×D4—41no—
102C2×C3×D4—41no—
103C2×C3×D4D414yescomplemented
104C22×C3×C4—22no—
105C22×C3×C4—22no—
106C22×C3×C4—41no—
107C22×C3×C4C2311yescharacteristic
108C22×C3×C4C2311yescharacteristic
109C22×C3×C4C2311yescharacteristic
110C22×C3×C4C2311yescharacteristic
111C22×C3×C4C2311yescharacteristic
112C22×C3×C4C2311yescharacteristic
113C22×C3×C4C2311yescharacteristic
114C22×C3×C4C2311yescharacteristic
115C22×C3×C4C2311yescharacteristic
116C22×C3×C4C2311yescharacteristic
117C3×C4⋊C4—22no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4—22no—
120C3×C4⋊C4—41no—
121C3×C22⋊C4—22no—
122C3×C22⋊C4—22no—
123C3×C22⋊C4—22no—
124C3×C22⋊C4—22no—
125C3×C22⋊C4—22no—
126C3×C22⋊C4—22no—
127C3×C22⋊C4—22no—
128C3×C22⋊C4—22no—
129C3×C22⋊C4—22no—
130C3×C22⋊C4—41no—
131C3×C22⋊C4—41no—
132C3×C22⋊C4—41no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4D414yescomplemented
138C3×C42—41no—
139C25C22×C311yescharacteristic
140C22×D4C22×C311yescharacteristic
141C22×D4C22×C311yescharacteristic
142C22×D4C22×C311yescharacteristic
143C22×D4C22×C311yescharacteristic
144C22×D4C22×C311yescharacteristic
145C22×D4C22×C311yescharacteristic
146C22×D4C22×C311yescharacteristic, complemented
147C23×C4C22×C311yescharacteristic, complemented
148#123—24no—
149#121—24no—
150#120—24no—
151#120—24no—
152C2×C4⋊C4C22×C311yescharacteristic, complemented
153C2×C4⋊C4C22×C311yescharacteristic, complemented
154C2×C4⋊C4C22×C311yescharacteristic, complemented
155C2×C4⋊C4C22×C311yescharacteristic, complemented
156C2×C22⋊C4C22×C311yescharacteristic
157C2×C22⋊C4C22×C311yescharacteristic
158C2×C22⋊C4C22×C311yescharacteristic
159C2×C22⋊C4C22×C311yescharacteristic
160C2×C22⋊C4C22×C311yescharacteristic
161C2×C22⋊C4C22×C311yescharacteristic
162C2×C22⋊C4C22×C311yescharacteristic
163C2×C22⋊C4C22×C311yescharacteristic
164C2×C22⋊C4C22×C311yescharacteristic
165C2×C22⋊C4C22×C311yescharacteristic
166C2×C22⋊C4C22×C311yescharacteristic
167C2×C22⋊C4C22×C311yescharacteristic
168C2×C22⋊C4C22×C311yescharacteristic, complemented
169C2×C22⋊C4C22×C311yescharacteristic, complemented
170C2×C22⋊C4C22×C311yescharacteristic, complemented
171C2×C22⋊C4C22×C311yescharacteristic, complemented
172C2×C22⋊C4C22×C311yescharacteristic, complemented
173C2×C42C22×C311yescharacteristic, complemented
174#95C22×C311yescharacteristic, complemented
175#95C22×C311yescharacteristic, complemented
176#95C22×C311yescharacteristic, complemented
177#95C22×C311yescharacteristic, complemented
178C23×C3—21no—
179C23×C3—21no—
180C23×C3—21no—
181C23×C3—21no—
182C23×C3—21no—
183C23×C3—21no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—21no—
189C23×C3—21no—
190C23×C3—21no—
191C23×C3—21no—
192C23×C3—21no—
193C23×C3—21no—
194C23×C3—21no—
195C23×C3—21no—
196C23×C3—21no—
197C23×C3—21no—
198C23×C3—21no—
199C23×C3—21no—
200C23×C3—21no—
201C23×C3—21no—
202C23×C3—21no—
203C23×C3—21no—
204C23×C3—21no—
205C23×C3—21no—
206C23×C3—21no—
207C23×C3—21no—
208C23×C3—21no—
209C23×C3—44no—
210C23×C3—44no—
211C23×C3—44no—
212C23×C3—44no—
213C23×C3—44no—
214C23×C3—44no—
215C23×C3—44no—
216C23×C3C2×D412yes—
217C23×C3C2×D412yes—
218C23×C3C2×D412yes—
219C23×C3#4112yes—
220C23×C3C2411yescharacteristic
221C3×D4—44no—
222C3×D4—44no—
223C3×D4—44no—
224C3×D4—44no—
225C3×D4—44no—
226C3×D4—44no—
227C3×D4—44no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—44no—
256C2×C3×C4C2×D412yes—
257C2×C3×C4C2×D412yes—
258C2×C3×C4C2×D412yes—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—41no—
263C24—41no—
264C24—41no—
265C24—41no—
266C24C23×C311yescharacteristic
267C24C23×C311yescharacteristic
268C24C23×C311yescharacteristic
269C24C23×C311yescharacteristic
270C24C23×C311yescharacteristic
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4C3×D414yescomplemented
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—41no—
295C22×C4C23×C311yescharacteristic
296C22×C4C23×C311yescharacteristic
297C22×C4C23×C311yescharacteristic
298C22×C4C23×C311yescharacteristic
299C22×C4C23×C311yescharacteristic
300C22×C4C23×C311yescharacteristic
301C22×C4C23×C311yescharacteristic
302C22×C4C23×C311yescharacteristic
303C22×C4C23×C311yescharacteristic
304C22×C4C23×C311yescharacteristic
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—41no—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4C3×D414yescomplemented
326C42—41no—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3—22no—
330C22×C3—22no—
331C22×C3—22no—
332C22×C3—22no—
333C22×C3—22no—
334C22×C3—22no—
335C22×C3—22no—
336C22×C3—22no—
337C22×C3—22no—
338C22×C3—22no—
339C22×C3—416no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3—41no—
358C22×C3—41no—
359C22×C3—41no—
360C22×C3—41no—
361C22×C3—41no—
362C22×C3—41no—
363C22×C3C22×D411yescharacteristic
364C22×C3C22×D411yescharacteristic
365C22×C3C2×#4111yescharacteristic
366C22×C3H(5,𝔽2)11yescharacteristic
367C22×C3H(5,𝔽2)11yescharacteristic
368C22×C3H(5,𝔽2)11yescharacteristic
369C22×C3H(5,𝔽2)11yescharacteristic
370C3×C4—22no—
371C3×C4—22no—
372C3×C4—22no—
373C3×C4—41no—
374C3×C4—41no—
375C3×C4—41no—
376C3×C4—41no—
377C3×C4—41no—
378C3×C4—41no—
379C3×C4—41no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23C2×C3×D412yes—
419C23C2×C3×D412yes—
420C23C2×C3×D412yes—
421C23C3×#4112yes—
422C23C24×C311yescharacteristic
423D4—44no—
424D4—44no—
425D4—44no—
426D4—44no—
427D4—44no—
428D4—44no—
429D4—44no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—44no—
458C2×C4C2×C3×D412yes—
459C2×C4C2×C3×D412yes—
460C2×C4C2×C3×D412yes—
461C2×C3—42no—
462C2×C3—42no—
463C2×C3—42no—
464C2×C3—42no—
465C2×C3—81no—
466C2×C3#53411yescharacteristic
467C2×C3#53511yescharacteristic
468C2×C3#53811yescharacteristic
469C2×C3D4211yescharacteristic
470C2×C3#54611yescharacteristic
471C2×C3#54611yescharacteristic
472C2×C3#56011yescharacteristic
473C22—22no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—416no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22C22×C3×D411yescharacteristic
510C22C22×C3×D411yescharacteristic
511C22C2×C3×#4111yescharacteristic
512C22C3×H(5,𝔽2)11yescharacteristic
513C22C3×H(5,𝔽2)11yescharacteristic
514C22C3×H(5,𝔽2)11yescharacteristic
515C22C3×H(5,𝔽2)11yescharacteristic
516C4—22no—
517C4—22no—
518C4—22no—
519C4—41no—
520C4—41no—
521C4—41no—
522C4—41no—
523C4—41no—
524C4—41no—
525C4—41no—
526C3#267811yescharacteristic, direct product factor, Sylow
527C2—42no—
528C2—42no—
529C2—42no—
530C2—42no—
531C2—81no—
532C2C3×#53411yescharacteristic
533C2C3×#53511yescharacteristic
534C2C3×#53811yescharacteristic
535C2C3×D4211yescharacteristic
536C2C3×#54611yescharacteristic
537C2C3×#54611yescharacteristic
538C2C3×#56011yescharacteristic
subgroupG/H
center, socleC23×C3C24
commutator, Frattini, cosocleC23C24×C3
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#123C2semidirect
2. C2×C3×#121C2semidirect
3. C2×C3×#120C2semidirect
4. C2×C3×#120C2semidirect
5. C3×#397C2semidirect
6. C3×#396C2semidirect
7. C3×#394C2semidirect
8. C3×#394C2semidirect
9. C3×#392C2semidirect
10. C3×#392C2semidirect
11. C3×#390C2semidirect
12. C3×#390C2semidirect
13. C3×#388C2semidirect
14. C3×#386C2semidirect
15. C3×#379C2semidirect
16. #2678C3direct (N × H)
24. C22×C3×D4C22semidirect
25. C23×C3×C4C22semidirect
30. C2×C3×C4⋊C4C22semidirect
31. C2×C3×C4⋊C4C22semidirect
32. C2×C3×C4⋊C4C22semidirect
33. C2×C3×C4⋊C4C22semidirect
46. C2×C3×C22⋊C4C22semidirect
47. C2×C3×C22⋊C4C22semidirect
48. C2×C3×C22⋊C4C22semidirect
49. C2×C3×C22⋊C4C22semidirect
50. C2×C3×C22⋊C4C22semidirect
51. C2×C3×C42C22semidirect
52. C3×#95C22semidirect
53. C3×#95C22semidirect
54. C3×#95C22semidirect
55. C3×#95C22semidirect
56. C2×#123C2×C3semidirect
57. C2×#121C2×C3semidirect
58. C2×#120C2×C3semidirect
59. C2×#120C2×C3semidirect
60. #397C2×C3semidirect
61. #396C2×C3semidirect
62. #394C2×C3semidirect
63. #394C2×C3semidirect
64. #392C2×C3semidirect
65. #392C2×C3semidirect
66. #390C2×C3semidirect
67. #390C2×C3semidirect
68. #388C2×C3semidirect
69. #386C2×C3semidirect
70. #379C2×C3semidirect
103. C2×C3×D4D4semidirect
137. C3×C22⋊C4D4semidirect
146. C22×D4C22×C3semidirect
147. C23×C4C22×C3semidirect
152. C2×C4⋊C4C22×C3semidirect
153. C2×C4⋊C4C22×C3semidirect
154. C2×C4⋊C4C22×C3semidirect
155. C2×C4⋊C4C22×C3semidirect
168. C2×C22⋊C4C22×C3semidirect
169. C2×C22⋊C4C22×C3semidirect
170. C2×C22⋊C4C22×C3semidirect
171. C2×C22⋊C4C22×C3semidirect
172. C2×C22⋊C4C22×C3semidirect
173. C2×C42C22×C3semidirect
174. #95C22×C3semidirect
175. #95C22×C3semidirect
176. #95C22×C3semidirect
177. #95C22×C3semidirect
291. C2×D4C3×D4semidirect
325. C22⋊C4C3×D4semidirect
526. C3#2678direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#1232
C2×C3×#1212
C2×C3×#1202
C3×#3972
C3×#3962
C3×#3942
C3×#3922
C3×#3902
C3×#3882
C3×#3862
C3×#3792
#26783