Finite Group Databasebeta

← C3×#2457 #84321

Subgroups — C3×#2457 #84321

Order 384.

596 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#123C211yesmaximal, characteristic, complemented
2C2×C3×C4×D4C211yesmaximal, characteristic, complemented
3C3×#400C211yesmaximal, characteristic, complemented
4C3×#396C211yesmaximal, characteristic, complemented
5C3×#394C211yesmaximal, characteristic, complemented
6C3×#392C211yesmaximal, characteristic
7C3×#388C211yesmaximal, characteristic, complemented
8C3×#388C211yesmaximal, characteristic, complemented
9C3×#387C211yesmaximal, characteristic, complemented
10C3×#386C211yesmaximal, characteristic, complemented
11C3×#386C211yesmaximal, characteristic, complemented
12C3×#386C211yesmaximal, characteristic, complemented
13C3×#381C211yesmaximal, characteristic, complemented
14C3×#380C211yesmaximal, characteristic, complemented
15C3×C4×C22⋊C4C211yesmaximal, characteristic, complemented
16#2457C311yesmaximal, characteristic, direct product factor, Sylow
17C22×C3×D4C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×C3×D4C2211yescharacteristic
20C23×C3×C4C2211yescharacteristic
21C23×C3×C4C2211yescharacteristic
22C23×C3×C4C2211yescharacteristic
23C3×#123—24no—
24C3×C4×D4—24no—
25C2×C3×C4⋊C4C2211yescharacteristic
26C2×C3×C4⋊C4C2211yescharacteristic
27C2×C3×C4⋊C4C2211yescharacteristic
28C2×C3×C4⋊C4C2211yescharacteristic
29C2×C3×C4⋊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×C42C2211yescharacteristic
43C2×C3×C42C2211yescharacteristic
44C3×#95C2211yescharacteristic
45C3×#95C2211yescharacteristic
46C3×#95C2211yescharacteristic
47C3×#95C2211yescharacteristic
48C3×#95C2211yescharacteristic
49C3×#95C2211yescharacteristic
50C3×#95C2211yescharacteristic
51C3×#95C2211yescharacteristic
52C3×#95C2211yescharacteristic
53C3×#95C2211yescharacteristic
54C2×#123C2×C311yescharacteristic, complemented
55C2×C4×D4C2×C311yescharacteristic, complemented
56#400C2×C311yescharacteristic, complemented
57#396C2×C311yescharacteristic, complemented
58#394C2×C311yescharacteristic, complemented
59#392C2×C311yescharacteristic
60#388C2×C311yescharacteristic, complemented
61#388C2×C311yescharacteristic, complemented
62#387C2×C311yescharacteristic, complemented
63#386C2×C311yescharacteristic, complemented
64#386C2×C311yescharacteristic, complemented
65#386C2×C311yescharacteristic, complemented
66#381C2×C311yescharacteristic, complemented
67#380C2×C311yescharacteristic, complemented
68C4×C22⋊C4C2×C311yescharacteristic, complemented
69C24×C3C2311yescharacteristic
70C24×C3C2311yescharacteristic
71C24×C3C2311yescharacteristic
72C2×C3×D4—22no—
73C2×C3×D4—22no—
74C2×C3×D4—22no—
75C2×C3×D4—22no—
76C2×C3×D4—22no—
77C2×C3×D4—22no—
78C2×C3×D4—22no—
79C2×C3×D4—41no—
80C2×C3×D4—41no—
81C22×C3×C4—22no—
82C22×C3×C4—22no—
83C22×C3×C4—22no—
84C22×C3×C4—22no—
85C22×C3×C4—22no—
86C22×C3×C4—22no—
87C22×C3×C4—41no—
88C22×C3×C4—41no—
89C22×C3×C4—41no—
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
100C22×C3×C4C2311yescharacteristic
101C22×C3×C4C2311yescharacteristic
102C3×C4⋊C4—22no—
103C3×C4⋊C4—22no—
104C3×C4⋊C4—22no—
105C3×C4⋊C4—22no—
106C3×C4⋊C4—41no—
107C3×C22⋊C4—22no—
108C3×C22⋊C4—22no—
109C3×C22⋊C4—22no—
110C3×C22⋊C4—22no—
111C3×C22⋊C4—22no—
112C3×C22⋊C4—22no—
113C3×C22⋊C4—41no—
114C3×C22⋊C4—41no—
115C3×C22⋊C4—41no—
116C3×C22⋊C4—41no—
117C3×C22⋊C4—41no—
118C3×C22⋊C4D414yescomplemented
119C3×C42—22no—
120C3×C42—41no—
121C22×D4C22×C311yescharacteristic
122C22×D4C22×C311yescharacteristic
123C22×D4C22×C311yescharacteristic
124C23×C4C22×C311yescharacteristic
125C23×C4C22×C311yescharacteristic
126C23×C4C22×C311yescharacteristic
127#123—24no—
128C4×D4—24no—
129C2×C4⋊C4C22×C311yescharacteristic
130C2×C4⋊C4C22×C311yescharacteristic
131C2×C4⋊C4C22×C311yescharacteristic
132C2×C4⋊C4C22×C311yescharacteristic
133C2×C4⋊C4C22×C311yescharacteristic
134C2×C22⋊C4C22×C311yescharacteristic
135C2×C22⋊C4C22×C311yescharacteristic
136C2×C22⋊C4C22×C311yescharacteristic
137C2×C22⋊C4C22×C311yescharacteristic
138C2×C22⋊C4C22×C311yescharacteristic
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×C42C22×C311yescharacteristic
147C2×C42C22×C311yescharacteristic
148#95C22×C311yescharacteristic
149#95C22×C311yescharacteristic
150#95C22×C311yescharacteristic
151#95C22×C311yescharacteristic
152#95C22×C311yescharacteristic
153#95C22×C311yescharacteristic
154#95C22×C311yescharacteristic
155#95C22×C311yescharacteristic
156#95C22×C311yescharacteristic
157#95C22×C311yescharacteristic
158C23×C3—21no—
159C23×C3—21no—
160C23×C3—21no—
161C23×C3—21no—
162C23×C3—21no—
163C23×C3—21no—
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×C3C2×D412yes—
177C23×C3#4112yes—
178C23×C3#4112yes—
179C23×C3C2411yescharacteristic
180C3×D4—28no—
181C3×D4—44no—
182C3×D4—44no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
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—44no—
216C2×C3×C4—44no—
217C2×C3×C4—44no—
218C2×C3×C4C2×D412yes—
219C2×C3×C4C2×D412yes—
220C2×C3×C4#4112yes—
221C2×C3×C4#4112yes—
222C24C23×C311yescharacteristic
223C24C23×C311yescharacteristic
224C24C23×C311yescharacteristic
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—41no—
233C2×D4—41no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4C23×C311yescharacteristic
244C22×C4C23×C311yescharacteristic
245C22×C4C23×C311yescharacteristic
246C22×C4C23×C311yescharacteristic
247C22×C4C23×C311yescharacteristic
248C22×C4C23×C311yescharacteristic
249C22×C4C23×C311yescharacteristic
250C22×C4C23×C311yescharacteristic
251C22×C4C23×C311yescharacteristic
252C22×C4C23×C311yescharacteristic
253C22×C4C23×C311yescharacteristic
254C22×C4C23×C311yescharacteristic
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—41no—
260C22⋊C4—22no—
261C22⋊C4—22no—
262C22⋊C4—22no—
263C22⋊C4—22no—
264C22⋊C4—22no—
265C22⋊C4—22no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4C3×D414yescomplemented
272C42—22no—
273C42—41no—
274C22×C3—22no—
275C22×C3—22no—
276C22×C3—22no—
277C22×C3—22no—
278C22×C3—22no—
279C22×C3—22no—
280C22×C3—22no—
281C22×C3—22no—
282C22×C3—22no—
283C22×C3—41no—
284C22×C3—41no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—41no—
295C22×C3C22×D411yescharacteristic
296C22×C3C2×#4111yescharacteristic
297C22×C3C2×#4111yescharacteristic
298C22×C3C2×#4111yescharacteristic
299C22×C3C2×#4111yescharacteristic
300C22×C3H(5,𝔽2)11yescharacteristic
301C22×C3H(5,𝔽2)11yescharacteristic
302C3×C4—22no—
303C3×C4—22no—
304C3×C4—22no—
305C3×C4—22no—
306C3×C4—41no—
307C3×C4—41no—
308C3×C4—41no—
309C3×C4—41no—
310C3×C4—41no—
311C3×C4—41no—
312C3×C4—41no—
313C3×C4—41no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23C2×C3×D412yes—
333C23C3×#4112yes—
334C23C3×#4112yes—
335C23C24×C311yescharacteristic
336D4—28no—
337D4—44no—
338D4—44no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4C2×C3×D412yes—
375C2×C4C2×C3×D412yes—
376C2×C4C3×#4112yes—
377C2×C4C3×#4112yes—
378C2×C3—42no—
379C2×C3—42no—
380C2×C3—42no—
381C2×C3#52511yescharacteristic
382C2×C3#52911yescharacteristic
383C2×C3#54011yescharacteristic
384C2×C3#54611yescharacteristic
385C2×C3#54611yescharacteristic
386C2×C3#55111yescharacteristic
387C2×C3#55311yescharacteristic
388C22—22no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22C22×C3×D411yescharacteristic
410C22C2×C3×#4111yescharacteristic
411C22C2×C3×#4111yescharacteristic
412C22C2×C3×#4111yescharacteristic
413C22C2×C3×#4111yescharacteristic
414C22C3×H(5,𝔽2)11yescharacteristic
415C22C3×H(5,𝔽2)11yescharacteristic
416C4—22no—
417C4—22no—
418C4—22no—
419C4—22no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—41no—
424C4—41no—
425C4—41no—
426C4—41no—
427C4—41no—
428C3#245711yescharacteristic, direct product factor, Sylow
429C2—42no—
430C2—42no—
431C2—42no—
432C2C3×#52511yescharacteristic
433C2C3×#52911yescharacteristic
434C2C3×#54011yescharacteristic
435C2C3×#54611yescharacteristic
436C2C3×#54611yescharacteristic
437C2C3×#55111yescharacteristic
438C2C3×#55311yescharacteristic
subgroupG/H
center, socleC23×C3C24
commutator, Frattini, cosocleC23C24×C3
prime radicalC3×#392C2
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×C4×D4C2semidirect
3. C3×#400C2semidirect
4. C3×#396C2semidirect
5. C3×#394C2semidirect
7. C3×#388C2semidirect
8. C3×#388C2semidirect
9. C3×#387C2semidirect
10. C3×#386C2semidirect
11. C3×#386C2semidirect
12. C3×#386C2semidirect
13. C3×#381C2semidirect
14. C3×#380C2semidirect
15. C3×C4×C22⋊C4C2semidirect
16. #2457C3direct (N × H)
54. C2×#123C2×C3semidirect
55. C2×C4×D4C2×C3semidirect
56. #400C2×C3semidirect
57. #396C2×C3semidirect
58. #394C2×C3semidirect
60. #388C2×C3semidirect
61. #388C2×C3semidirect
62. #387C2×C3semidirect
63. #386C2×C3semidirect
64. #386C2×C3semidirect
65. #386C2×C3semidirect
66. #381C2×C3semidirect
67. #380C2×C3semidirect
68. C4×C22⋊C4C2×C3semidirect
118. C3×C22⋊C4D4semidirect
271. C22⋊C4C3×D4semidirect
428. C3#2457direct (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×C4×D42
C3×#4002
C3×#3962
C3×#3942
C3×#3922
C3×#3882
C3×#3872
C3×#3862
C3×#3812
C3×#3802
C3×C4×C22⋊C42
#24573