Finite Group Databasebeta

← C3×#34967 #10653574

Subgroups — C3×#34967 #10653574

Order 768.

2164 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2522C214yesmaximal, complemented
2C3×#2516C214yesmaximal, complemented
3C3×#2464C214yesmaximal, complemented
4C3×#2457C214yesmaximal, complemented
5C3×#2402C211yesmaximal, characteristic, complemented
6C3×#2397C211yesmaximal, characteristic, complemented
7C3×#2392C212yesmaximal, complemented
8C3×C4×#123C214yesmaximal, complemented
9C3×C4×#121C214yesmaximal, complemented
10C3×#2298C211yesmaximal, characteristic, complemented
11C3×C42×D4C212yesmaximal, complemented
12#34967C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#123C2214yes—
14C2×C3×#121C2214yes—
15C2×C3×C4×D4C2212yes—
16C2×C3×C4×D4C2214yes—
17C2×C3×C4×D4C2214yes—
18C2×C3×C4×D4C2214yes—
19C2×C3×#117C2211yescharacteristic
20C22×C3×C42C2214yes—
21C3×#400C2214yes—
22C3×#399C2214yes—
23C3×#398C2212yes—
24C3×#398C2214yes—
25C3×#397C2212yes—
26C3×#396C2214yes—
27C3×#394C2212yes—
28C3×#394C2214yes—
29C3×#392C2212yes—
30C3×#392C2214yes—
31C3×#390C2214yes—
32C3×#389C2214yes—
33C3×#388C2214yes—
34C3×#388C2214yes—
35C3×#388C2214yes—
36C3×#388C2214yes—
37C3×#388C2214yes—
38C3×#387C2214yes—
39C3×#387C2214yes—
40C3×#387C2214yes—
41C3×#386C2214yes—
42C3×#386C2214yes—
43C3×#386C2214yes—
44C3×#385C2214yes—
45C3×#381C2214yes—
46C3×#380C2214yes—
47C3×C4×C4⋊C4C2212yes—
48C3×C4×C4⋊C4C2214yes—
49C3×C4×C4⋊C4C2214yes—
50C3×C4×C4⋊C4C2212yes—
51C3×C4×C22⋊C4C2214yes—
52C3×C4×C22⋊C4C2214yes—
53C3×C4×C22⋊C4C2214yes—
54C3×C4×C22⋊C4C2214yes—
55C3×C4×C22⋊C4C2212yes—
56C3×#376C2211yescharacteristic
57C3×C43C2211yescharacteristic
58#2522C2×C314yescomplemented
59#2516C2×C314yescomplemented
60#2464C2×C314yescomplemented
61#2457C2×C314yescomplemented
62#2402C2×C311yescharacteristic, complemented
63#2397C2×C311yescharacteristic, complemented
64#2392C2×C312yescomplemented
65C4×#123C2×C314yescomplemented
66C4×#121C2×C314yescomplemented
67#2298C2×C311yescharacteristic, complemented
68C42×D4C2×C312yescomplemented
69C22×C3×D4C2312yes—
70C22×C3×D4C2314yes—
71C22×C3×D4C2314yes—
72C23×C3×C4C2314yes—
73C23×C3×C4C2311yescharacteristic
74C23×C3×C4C2314yes—
75C23×C3×C4C2314yes—
76C3×#123—216no—
77C3×#121—216no—
78C3×C4×D4—216no—
79C3×C4×D4—28no—
80C3×C4×D4—48no—
81C3×C4×D4—48no—
82C3×#117—24no—
83C2×C3×C4⋊C4C2314yes—
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C4⋊C4C2314yes—
86C2×C3×C4⋊C4C2314yes—
87C2×C3×C4⋊C4C2314yes—
88C2×C3×C4⋊C4C2312yes—
89C2×C3×C4⋊C4C2314yes—
90C2×C3×C4⋊C4C2314yes—
91C2×C3×C22⋊C4C2314yes—
92C2×C3×C22⋊C4C2314yes—
93C2×C3×C22⋊C4C2314yes—
94C2×C3×C22⋊C4C2314yes—
95C2×C3×C22⋊C4C2314yes—
96C2×C3×C22⋊C4C2312yes—
97C2×C3×C22⋊C4C2314yes—
98C2×C3×C22⋊C4C2314yes—
99C2×C3×C22⋊C4C2314yes—
100C2×C3×C22⋊C4C2314yes—
101C2×C3×C22⋊C4C2314yes—
102C2×C3×C42—48no—
103C2×C3×C42C2314yes—
104C2×C3×C42C2312yes—
105C2×C3×C42C2312yes—
106C2×C3×C42C2311yescharacteristic
107C2×C3×C42C2314yes—
108C2×C3×C42C2314yes—
109C2×C3×C42C2314yes—
110C2×C3×C42C2311yescharacteristic
111C3×#95C2314yes—
112C3×#95C2314yes—
113C3×#95C2314yes—
114C3×#95C2312yes—
115C3×#95C2312yes—
116C3×#95C2314yes—
117C3×#95C2314yes—
118C3×#95C2314yes—
119C3×#95C2314yes—
120C3×#95C2314yes—
121C3×#95C2314yes—
122C2×#123C22×C314yes—
123C2×#121C22×C314yes—
124C2×C4×D4C22×C312yes—
125C2×C4×D4C22×C314yes—
126C2×C4×D4C22×C314yes—
127C2×C4×D4C22×C314yes—
128C2×#117C22×C311yescharacteristic
129C22×C42C22×C314yes—
130#400C22×C314yes—
131#399C22×C314yes—
132#398C22×C312yes—
133#398C22×C314yes—
134#397C22×C312yes—
135#396C22×C314yes—
136#394C22×C312yes—
137#394C22×C314yes—
138#392C22×C312yes—
139#392C22×C314yes—
140#390C22×C314yes—
141#389C22×C314yes—
142#388C22×C314yes—
143#388C22×C314yes—
144#388C22×C314yes—
145#388C22×C314yes—
146#388C22×C314yes—
147#387C22×C314yes—
148#387C22×C314yes—
149#387C22×C314yes—
150#386C22×C314yes—
151#386C22×C314yes—
152#386C22×C314yes—
153#385C22×C314yes—
154#381C22×C314yes—
155#380C22×C314yes—
156C4×C4⋊C4C22×C312yes—
157C4×C4⋊C4C22×C314yes—
158C4×C4⋊C4C22×C314yes—
159C4×C4⋊C4C22×C312yes—
160C4×C22⋊C4C22×C314yes—
161C4×C22⋊C4C22×C314yes—
162C4×C22⋊C4C22×C314yes—
163C4×C22⋊C4C22×C314yes—
164C4×C22⋊C4C22×C312yes—
165#376C22×C311yescharacteristic
166C43C22×C311yescharacteristic
167C24×C3C2411yescharacteristic
168C24×C3C2414yes—
169C2×C3×D4—28no—
170C2×C3×D4—28no—
171C2×C3×D4—28no—
172C2×C3×D4—28no—
173C2×C3×D4—24no—
174C2×C3×D4—28no—
175C2×C3×D4—44no—
176C2×C3×D4—44no—
177C22×C3×C4—28no—
178C22×C3×C4—22no—
179C22×C3×C4—28no—
180C22×C3×C4—28no—
181C22×C3×C4—28no—
182C22×C3×C4—28no—
183C22×C3×C4—28no—
184C22×C3×C4—28no—
185C22×C3×C4—24no—
186C22×C3×C4—44no—
187C22×C3×C4—44no—
188C22×C3×C4C2414yes—
189C22×C3×C4C2412yes—
190C22×C3×C4C2414yes—
191C22×C3×C4C2414yes—
192C22×C3×C4C2411yescharacteristic
193C22×C3×C4C2414yes—
194C22×C3×C4C2412yes—
195C22×C3×C4C2414yes—
196C22×C3×C4C2411yescharacteristic
197C3×C4⋊C4—24no—
198C3×C4⋊C4—28no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—28no—
201C3×C4⋊C4—28no—
202C3×C4⋊C4—28no—
203C3×C4⋊C4—44no—
204C3×C4⋊C4—44no—
205C3×C22⋊C4—28no—
206C3×C22⋊C4—24no—
207C3×C22⋊C4—28no—
208C3×C22⋊C4—28no—
209C3×C22⋊C4—28no—
210C3×C22⋊C4—28no—
211C3×C22⋊C4—44no—
212C3×C22⋊C4—44no—
213C3×C22⋊C4—44no—
214C3×C22⋊C4—44no—
215C3×C22⋊C4#41116yes—
216C3×C42—22no—
217C3×C42—28no—
218C3×C42—24no—
219C3×C42—28no—
220C3×C42—24no—
221C3×C42—44no—
222C3×C42—44no—
223C3×C42#4114yes—
224C22×D4C23×C312yes—
225C22×D4C23×C314yes—
226C22×D4C23×C314yes—
227C23×C4C23×C311yescharacteristic
228C23×C4C23×C314yes—
229C23×C4C23×C314yes—
230C23×C4C23×C314yes—
231#123—216no—
232#121—216no—
233C4×D4—216no—
234C4×D4—28no—
235C4×D4—48no—
236C4×D4—48no—
237#117—24no—
238C2×C4⋊C4C23×C312yes—
239C2×C4⋊C4C23×C314yes—
240C2×C4⋊C4C23×C312yes—
241C2×C4⋊C4C23×C314yes—
242C2×C4⋊C4C23×C314yes—
243C2×C4⋊C4C23×C314yes—
244C2×C4⋊C4C23×C314yes—
245C2×C4⋊C4C23×C314yes—
246C2×C22⋊C4C23×C312yes—
247C2×C22⋊C4C23×C314yes—
248C2×C22⋊C4C23×C314yes—
249C2×C22⋊C4C23×C314yes—
250C2×C22⋊C4C23×C314yes—
251C2×C22⋊C4C23×C314yes—
252C2×C22⋊C4C23×C314yes—
253C2×C22⋊C4C23×C314yes—
254C2×C22⋊C4C23×C314yes—
255C2×C22⋊C4C23×C314yes—
256C2×C22⋊C4C23×C314yes—
257C2×C42—48no—
258C2×C42C23×C311yescharacteristic
259C2×C42C23×C312yes—
260C2×C42C23×C314yes—
261C2×C42C23×C314yes—
262C2×C42C23×C312yes—
263C2×C42C23×C314yes—
264C2×C42C23×C314yes—
265C2×C42C23×C311yescharacteristic
266#95C23×C314yes—
267#95C23×C314yes—
268#95C23×C314yes—
269#95C23×C312yes—
270#95C23×C312yes—
271#95C23×C314yes—
272#95C23×C314yes—
273#95C23×C314yes—
274#95C23×C314yes—
275#95C23×C314yes—
276#95C23×C314yes—
277C23×C3—24no—
278C23×C3—24no—
279C23×C3—24no—
280C23×C3—24no—
281C23×C3—24no—
282C23×C3—24no—
283C23×C3—21no—
284C23×C3—24no—
285C23×C3—22no—
286C23×C3C2×#4118yes—
287C23×C3C2511yescharacteristic
288C3×D4—216no—
289C3×D4—416no—
290C3×D4—416no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—24no—
294C2×C3×C4—24no—
295C2×C3×C4—24no—
296C2×C3×C4—22no—
297C2×C3×C4—24no—
298C2×C3×C4—24no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—24no—
302C2×C3×C4—21no—
303C2×C3×C4—24no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—22no—
309C2×C3×C4—416no—
310C2×C3×C4—416no—
311C2×C3×C4—416no—
312C2×C3×C4—44no—
313C2×C3×C4C2×#4118yes—
314C2×C3×C4C2×#4114yes—
315C2×C3×C4C2×#4118yes—
316C2×C3×C4C2×#4112yes—
317C2×C3×C4H(5,𝔽2)14yes—
318C24C24×C311yescharacteristic
319C24C24×C314yes—
320C2×D4—28no—
321C2×D4—28no—
322C2×D4—28no—
323C2×D4—28no—
324C2×D4—24no—
325C2×D4—28no—
326C2×D4—44no—
327C2×D4—44no—
328C22×C4—28no—
329C22×C4—22no—
330C22×C4—28no—
331C22×C4—28no—
332C22×C4—28no—
333C22×C4—28no—
334C22×C4—28no—
335C22×C4—28no—
336C22×C4—24no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4C24×C314yes—
340C22×C4C24×C312yes—
341C22×C4C24×C314yes—
342C22×C4C24×C314yes—
343C22×C4C24×C311yescharacteristic
344C22×C4C24×C314yes—
345C22×C4C24×C312yes—
346C22×C4C24×C314yes—
347C22×C4C24×C311yescharacteristic
348C4⋊C4—24no—
349C4⋊C4—28no—
350C4⋊C4—24no—
351C4⋊C4—28no—
352C4⋊C4—28no—
353C4⋊C4—28no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C22⋊C4—28no—
357C22⋊C4—24no—
358C22⋊C4—28no—
359C22⋊C4—28no—
360C22⋊C4—28no—
361C22⋊C4—28no—
362C22⋊C4—44no—
363C22⋊C4—44no—
364C22⋊C4—44no—
365C22⋊C4—44no—
366C22⋊C4C3×#41116yes—
367C42—22no—
368C42—28no—
369C42—24no—
370C42—28no—
371C42—24no—
372C42—44no—
373C42—44no—
374C42C3×#4114yes—
375C22×C3—28no—
376C22×C3—28no—
377C22×C3—28no—
378C22×C3—44no—
379C22×C3—44no—
380C22×C3—44no—
381C22×C3—44no—
382C22×C3—42no—
383C22×C3—41no—
384C22×C3—44no—
385C22×C3C22×#4111yescharacteristic
386C22×C3C22×#4114yes—
387C22×C3C2×H(5,𝔽2)12yes—
388C3×C4—24no—
389C3×C4—28no—
390C3×C4—28no—
391C3×C4—42no—
392C3×C4—44no—
393C3×C4—41no—
394C3×C4—44no—
395C3×C4—44no—
396C3×C4#54014yes—
397C23—24no—
398C23—24no—
399C23—24no—
400C23—24no—
401C23—24no—
402C23—24no—
403C23—21no—
404C23—24no—
405C23—22no—
406C23C2×C3×#4118yes—
407C23C25×C311yescharacteristic
408D4—216no—
409D4—416no—
410D4—416no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—21no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—416no—
430C2×C4—416no—
431C2×C4—416no—
432C2×C4—44no—
433C2×C4C2×C3×#4118yes—
434C2×C4C2×C3×#4114yes—
435C2×C4C2×C3×#4118yes—
436C2×C4C2×C3×#4112yes—
437C2×C4C3×H(5,𝔽2)14yes—
438C2×C3—48no—
439C2×C3—81no—
440C2×C3#344412yes—
441C2×C3C2×#54011yescharacteristic
442C2×C3#348114yes—
443C22—28no—
444C22—28no—
445C22—28no—
446C22—44no—
447C22—44no—
448C22—44no—
449C22—44no—
450C22—42no—
451C22—41no—
452C22—44no—
453C22C22×C3×#4111yescharacteristic
454C22C22×C3×#4114yes—
455C22C2×C3×H(5,𝔽2)12yes—
456C4—24no—
457C4—28no—
458C4—28no—
459C4—42no—
460C4—44no—
461C4—41no—
462C4—44no—
463C4—44no—
464C4C3×#54014yes—
465C3#3496711yescharacteristic, direct product factor, Sylow
466C2—48no—
467C2—81no—
468C2C3×#344412yes—
469C2C2×C3×#54011yescharacteristic
470C2C3×#348114yes—
subgroupG/H
centerC22×C3×C4C24
commutator, Frattini, cosocleC23C25×C3
socleC23×C3C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2522C2semidirect
2. C3×#2516C2semidirect
3. C3×#2464C2semidirect
4. C3×#2457C2semidirect
5. C3×#2402C2semidirect
6. C3×#2397C2semidirect
7. C3×#2392C2semidirect
8. C3×C4×#123C2semidirect
9. C3×C4×#121C2semidirect
10. C3×#2298C2semidirect
11. C3×C42×D4C2semidirect
12. #34967C3direct (N × H)
58. #2522C2×C3semidirect
59. #2516C2×C3semidirect
60. #2464C2×C3semidirect
61. #2457C2×C3semidirect
62. #2402C2×C3semidirect
63. #2397C2×C3semidirect
64. #2392C2×C3semidirect
65. C4×#123C2×C3semidirect
66. C4×#121C2×C3semidirect
67. #2298C2×C3semidirect
68. C42×D4C2×C3semidirect
465. C3#34967direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#25222
C3×#25162
C3×#24642
C3×#24572
C3×#24022
C3×#23972
C3×#23922
C3×C4×#1232
C3×C4×#1212
C3×#22982
C3×C42×D42
#349673