Finite Group Databasebeta

← C3×#2685 #84549

Subgroups — C3×#2685 #84549

Order 384.

666 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#126C211yesmaximal, characteristic, complemented
2C2×C3×#120C211yesmaximal, characteristic
3C3×#399C211yesmaximal, characteristic, complemented
4C3×#396C211yesmaximal, characteristic, complemented
5C3×#394C211yesmaximal, characteristic, complemented
6C3×#394C211yesmaximal, characteristic, complemented
7C3×#394C211yesmaximal, characteristic, complemented
8C3×#394C211yesmaximal, characteristic, complemented
9C3×#390C211yesmaximal, characteristic, complemented
10C3×#390C211yesmaximal, characteristic, complemented
11C3×#388C211yesmaximal, characteristic, complemented
12C3×#388C211yesmaximal, characteristic, complemented
13C3×#388C211yesmaximal, characteristic, complemented
14C3×#379C211yesmaximal, characteristic, complemented
15C3×#379C211yesmaximal, characteristic, complemented
16#2685C311yesmaximal, characteristic, direct product factor, Sylow
17C25×C3C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×C3×D4C2211yescharacteristic
20C22×C3×D4C2211yescharacteristic
21C3×#126—24no—
22C3×#120—24no—
23C2×C3×C4⋊C4C2211yescharacteristic, complemented
24C2×C3×C4⋊C4C2211yescharacteristic, complemented
25C2×C3×C4⋊C4C2211yescharacteristic, complemented
26C2×C3×C4⋊C4C2211yescharacteristic, complemented
27C2×C3×C4⋊C4C2211yescharacteristic, complemented
28C2×C3×C4⋊C4C2211yescharacteristic, complemented
29C2×C3×C22⋊C4C2211yescharacteristic
30C2×C3×C22⋊C4C2211yescharacteristic
31C2×C3×C22⋊C4C2211yescharacteristic
32C2×C3×C22⋊C4C2211yescharacteristic
33C2×C3×C22⋊C4C2211yescharacteristic
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, complemented
45C2×C3×C22⋊C4C2211yescharacteristic, complemented
46C2×C3×C22⋊C4C2211yescharacteristic, complemented
47C2×C3×C22⋊C4C2211yescharacteristic, complemented
48C2×C3×C42C2211yescharacteristic, complemented
49C2×C3×C42C2211yescharacteristic, complemented
50C3×#95C2211yescharacteristic, complemented
51C3×#95C2211yescharacteristic, complemented
52C3×#95C2211yescharacteristic, complemented
53C3×#95C2211yescharacteristic, complemented
54C2×#126C2×C311yescharacteristic, complemented
55C2×#120C2×C311yescharacteristic
56#399C2×C311yescharacteristic, complemented
57#396C2×C311yescharacteristic, complemented
58#394C2×C311yescharacteristic, complemented
59#394C2×C311yescharacteristic, complemented
60#394C2×C311yescharacteristic, complemented
61#394C2×C311yescharacteristic, complemented
62#390C2×C311yescharacteristic, complemented
63#390C2×C311yescharacteristic, complemented
64#388C2×C311yescharacteristic, complemented
65#388C2×C311yescharacteristic, complemented
66#388C2×C311yescharacteristic, complemented
67#379C2×C311yescharacteristic, complemented
68#379C2×C311yescharacteristic, complemented
69C24×C3—22no—
70C24×C3—22no—
71C24×C3—22no—
72C24×C3—41no—
73C24×C3—41no—
74C24×C3—41no—
75C24×C3—41no—
76C24×C3C2311yescharacteristic
77C24×C3C2311yescharacteristic
78C24×C3C2311yescharacteristic
79C24×C3C2311yescharacteristic
80C2×C3×D4—22no—
81C2×C3×D4—22no—
82C2×C3×D4—22no—
83C2×C3×D4—22no—
84C2×C3×D4—22no—
85C2×C3×D4—41no—
86C2×C3×D4—41no—
87C2×C3×D4—41no—
88C2×C3×D4—41no—
89C22×C3×C4C2311yescharacteristic
90C22×C3×C4C2311yescharacteristic
91C22×C3×C4C2311yescharacteristic
92C22×C3×C4C2311yescharacteristic
93C22×C3×C4C2311yescharacteristic
94C22×C3×C4C2311yescharacteristic
95C22×C3×C4C2311yescharacteristic
96C22×C3×C4C2311yescharacteristic
97C22×C3×C4C2311yescharacteristic
98C22×C3×C4C2311yescharacteristic
99C22×C3×C4C2311yescharacteristic
100C3×C4⋊C4—22no—
101C3×C4⋊C4—22no—
102C3×C4⋊C4—22no—
103C3×C4⋊C4—41no—
104C3×C4⋊C4—41no—
105C3×C4⋊C4—41no—
106C3×C22⋊C4—22no—
107C3×C22⋊C4—22no—
108C3×C22⋊C4—22no—
109C3×C22⋊C4—22no—
110C3×C22⋊C4—22no—
111C3×C22⋊C4—22no—
112C3×C22⋊C4—41no—
113C3×C22⋊C4—41no—
114C3×C22⋊C4—41no—
115C3×C22⋊C4—41no—
116C3×C22⋊C4—41no—
117C3×C22⋊C4—41no—
118C3×C22⋊C4—41no—
119C3×C22⋊C4—41no—
120C3×C22⋊C4—41no—
121C3×C22⋊C4—41no—
122C3×C22⋊C4—41no—
123C3×C22⋊C4—41no—
124C3×C22⋊C4D414yes—
125C3×C42—22no—
126C3×C42—41no—
127C25C22×C311yescharacteristic
128C22×D4C22×C311yescharacteristic
129C22×D4C22×C311yescharacteristic
130C22×D4C22×C311yescharacteristic
131#126—24no—
132#120—24no—
133C2×C4⋊C4C22×C311yescharacteristic, complemented
134C2×C4⋊C4C22×C311yescharacteristic, complemented
135C2×C4⋊C4C22×C311yescharacteristic, complemented
136C2×C4⋊C4C22×C311yescharacteristic, complemented
137C2×C4⋊C4C22×C311yescharacteristic, complemented
138C2×C4⋊C4C22×C311yescharacteristic, complemented
139C2×C22⋊C4C22×C311yescharacteristic
140C2×C22⋊C4C22×C311yescharacteristic
141C2×C22⋊C4C22×C311yescharacteristic
142C2×C22⋊C4C22×C311yescharacteristic
143C2×C22⋊C4C22×C311yescharacteristic
144C2×C22⋊C4C22×C311yescharacteristic
145C2×C22⋊C4C22×C311yescharacteristic
146C2×C22⋊C4C22×C311yescharacteristic
147C2×C22⋊C4C22×C311yescharacteristic
148C2×C22⋊C4C22×C311yescharacteristic
149C2×C22⋊C4C22×C311yescharacteristic
150C2×C22⋊C4C22×C311yescharacteristic
151C2×C22⋊C4C22×C311yescharacteristic
152C2×C22⋊C4C22×C311yescharacteristic
153C2×C22⋊C4C22×C311yescharacteristic
154C2×C22⋊C4C22×C311yescharacteristic, complemented
155C2×C22⋊C4C22×C311yescharacteristic, complemented
156C2×C22⋊C4C22×C311yescharacteristic, complemented
157C2×C22⋊C4C22×C311yescharacteristic, complemented
158C2×C42C22×C311yescharacteristic, complemented
159C2×C42C22×C311yescharacteristic, complemented
160#95C22×C311yescharacteristic, complemented
161#95C22×C311yescharacteristic, complemented
162#95C22×C311yescharacteristic, complemented
163#95C22×C311yescharacteristic, complemented
164C23×C3—21no—
165C23×C3—21no—
166C23×C3—21no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—21no—
177C23×C3—21no—
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—44no—
190C23×C3—44no—
191C23×C3—44no—
192C23×C3—44no—
193C23×C3—44no—
194C23×C3—44no—
195C23×C3—44no—
196C23×C3C2×D412yes—
197C23×C3#4112yes—
198C23×C3#4112yes—
199C23×C3C2411yescharacteristic
200C3×D4—44no—
201C3×D4—44no—
202C3×D4—44no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
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×C4C2×D412yes—
235C2×C3×C4C2×D412yes—
236C24—22no—
237C24—22no—
238C24—22no—
239C24—41no—
240C24—41no—
241C24—41no—
242C24—41no—
243C24C23×C311yescharacteristic
244C24C23×C311yescharacteristic
245C24C23×C311yescharacteristic
246C24C23×C311yescharacteristic
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C22×C4C23×C311yescharacteristic
257C22×C4C23×C311yescharacteristic
258C22×C4C23×C311yescharacteristic
259C22×C4C23×C311yescharacteristic
260C22×C4C23×C311yescharacteristic
261C22×C4C23×C311yescharacteristic
262C22×C4C23×C311yescharacteristic
263C22×C4C23×C311yescharacteristic
264C22×C4C23×C311yescharacteristic
265C22×C4C23×C311yescharacteristic
266C22×C4C23×C311yescharacteristic
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C22⋊C4—22no—
274C22⋊C4—22no—
275C22⋊C4—22no—
276C22⋊C4—22no—
277C22⋊C4—22no—
278C22⋊C4—22no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4C3×D414yes—
292C42—22no—
293C42—41no—
294C22×C3—22no—
295C22×C3—22no—
296C22×C3—22no—
297C22×C3—22no—
298C22×C3—22no—
299C22×C3—22no—
300C22×C3—22no—
301C22×C3—22no—
302C22×C3—22no—
303C22×C3—416no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3—41no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3—41no—
322C22×C3—41no—
323C22×C3C22×D411yescharacteristic
324C22×C3C2×#4111yescharacteristic
325C22×C3C2×#4111yescharacteristic
326C22×C3H(5,𝔽2)11yescharacteristic
327C22×C3H(5,𝔽2)11yescharacteristic
328C22×C3H(5,𝔽2)11yescharacteristic
329C22×C3H(5,𝔽2)11yescharacteristic
330C3×C4—22no—
331C3×C4—22no—
332C3×C4—41no—
333C3×C4—41no—
334C3×C4—41no—
335C3×C4—41no—
336C3×C4—41no—
337C3×C4—41no—
338C3×C4—41no—
339C3×C4—41no—
340C3×C4—41no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—44no—
367C23—44no—
368C23—44no—
369C23—44no—
370C23—44no—
371C23—44no—
372C23—44no—
373C23C2×C3×D412yes—
374C23C3×#4112yes—
375C23C3×#4112yes—
376C23C24×C311yescharacteristic
377D4—44no—
378D4—44no—
379D4—44no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4C2×C3×D412yes—
412C2×C4C2×C3×D412yes—
413C2×C3—42no—
414C2×C3—42no—
415C2×C3—42no—
416C2×C3—81no—
417C2×C3#53511yescharacteristic
418C2×C3#53811yescharacteristic
419C2×C3#53811yescharacteristic
420C2×C3#54611yescharacteristic
421C2×C3#54611yescharacteristic
422C2×C3#55111yescharacteristic
423C2×C3#56011yescharacteristic
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—416no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22C22×C3×D411yescharacteristic
454C22C2×C3×#4111yescharacteristic
455C22C2×C3×#4111yescharacteristic
456C22C3×H(5,𝔽2)11yescharacteristic
457C22C3×H(5,𝔽2)11yescharacteristic
458C22C3×H(5,𝔽2)11yescharacteristic
459C22C3×H(5,𝔽2)11yescharacteristic
460C4—22no—
461C4—22no—
462C4—41no—
463C4—41no—
464C4—41no—
465C4—41no—
466C4—41no—
467C4—41no—
468C4—41no—
469C4—41no—
470C4—41no—
471C3#268511yescharacteristic, direct product factor, Sylow
472C2—42no—
473C2—42no—
474C2—42no—
475C2—81no—
476C2C3×#53511yescharacteristic
477C2C3×#53811yescharacteristic
478C2C3×#53811yescharacteristic
479C2C3×#54611yescharacteristic
480C2C3×#54611yescharacteristic
481C2C3×#55111yescharacteristic
482C2C3×#56011yescharacteristic
subgroupG/H
center, socleC23×C3C24
commutator, Frattini, cosocleC23C24×C3
prime radicalC2×C3×#120C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#126C2semidirect
3. C3×#399C2semidirect
4. C3×#396C2semidirect
5. C3×#394C2semidirect
6. C3×#394C2semidirect
7. C3×#394C2semidirect
8. C3×#394C2semidirect
9. C3×#390C2semidirect
10. C3×#390C2semidirect
11. C3×#388C2semidirect
12. C3×#388C2semidirect
13. C3×#388C2semidirect
14. C3×#379C2semidirect
15. C3×#379C2semidirect
16. #2685C3direct (N × H)
23. C2×C3×C4⋊C4C22semidirect
24. C2×C3×C4⋊C4C22semidirect
25. C2×C3×C4⋊C4C22semidirect
26. C2×C3×C4⋊C4C22semidirect
27. C2×C3×C4⋊C4C22semidirect
28. C2×C3×C4⋊C4C22semidirect
44. C2×C3×C22⋊C4C22semidirect
45. C2×C3×C22⋊C4C22semidirect
46. C2×C3×C22⋊C4C22semidirect
47. C2×C3×C22⋊C4C22semidirect
48. C2×C3×C42C22semidirect
49. C2×C3×C42C22semidirect
50. C3×#95C22semidirect
51. C3×#95C22semidirect
52. C3×#95C22semidirect
53. C3×#95C22semidirect
54. C2×#126C2×C3semidirect
56. #399C2×C3semidirect
57. #396C2×C3semidirect
58. #394C2×C3semidirect
59. #394C2×C3semidirect
60. #394C2×C3semidirect
61. #394C2×C3semidirect
62. #390C2×C3semidirect
63. #390C2×C3semidirect
64. #388C2×C3semidirect
65. #388C2×C3semidirect
66. #388C2×C3semidirect
67. #379C2×C3semidirect
68. #379C2×C3semidirect
133. C2×C4⋊C4C22×C3semidirect
134. C2×C4⋊C4C22×C3semidirect
135. C2×C4⋊C4C22×C3semidirect
136. C2×C4⋊C4C22×C3semidirect
137. C2×C4⋊C4C22×C3semidirect
138. C2×C4⋊C4C22×C3semidirect
154. C2×C22⋊C4C22×C3semidirect
155. C2×C22⋊C4C22×C3semidirect
156. C2×C22⋊C4C22×C3semidirect
157. C2×C22⋊C4C22×C3semidirect
158. C2×C42C22×C3semidirect
159. C2×C42C22×C3semidirect
160. #95C22×C3semidirect
161. #95C22×C3semidirect
162. #95C22×C3semidirect
163. #95C22×C3semidirect
471. C3#2685direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#1262
C2×C3×#1202
C3×#3992
C3×#3962
C3×#3942
C3×#3902
C3×#3882
C3×#3792
#26853