Finite Group Databasebeta

← C3×#34522 #10653129

Subgroups — C3×#34522 #10653129

Order 768.

2216 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#562C211yesmaximal, characteristic
2C3×#2382C211yesmaximal, characteristic, complemented
3C3×#2379C212yesmaximal, complemented
4C3×#2368C214yesmaximal, complemented
5C3×#2363C214yesmaximal, complemented
6C3×#2359C214yesmaximal, complemented
7C3×#2335C214yesmaximal, complemented
8C3×#2325C214yesmaximal, complemented
9C3×#2323C214yesmaximal, complemented
10C3×C4×#125C211yesmaximal, characteristic, complemented
11C3×C4×#124C212yesmaximal, complemented
12#34522C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#562C4116yescomplemented
14C2×C3×#125C2211yescharacteristic
15C2×C3×#124C2212yes—
16C2×C3×#123C2214yes—
17C2×C3×C22⋊Q8C2214yes—
18C2×C3×#121C2214yes—
19C2×C3×C4×Q8C2212yes—
20C2×C3×C4×D4C2214yes—
21C2×C3×C4×D4C2212yes—
22C2×C3×#117C2214yes—
23C3×#391C2212yes—
24C3×#391C2214yes—
25C3×#390C2212yes—
26C3×#390C2214yes—
27C3×#389C2214yes—
28C3×#389C2212yes—
29C3×#389C2214yes—
30C3×#388C2214yes—
31C3×#388C2214yes—
32C3×#388C2214yes—
33C3×#388C2214yes—
34C3×#388C2214yes—
35C3×#388C2214yes—
36C3×#387C2214yes—
37C3×#387C2214yes—
38C3×#387C2214yes—
39C3×#387C2214yes—
40C3×#387C2214yes—
41C3×#386C2214yes—
42C3×#386C2214yes—
43C3×#386C2214yes—
44C3×#385C2214yes—
45C3×#385C2214yes—
46C3×#385C2214yes—
47C3×#382C2211yescharacteristic
48C3×#382C2212yes—
49C3×#380C2214yes—
50C3×C4×C4⋊C4C2214yes—
51C3×C4×C4⋊C4C2214yes—
52C3×C4×C4⋊C4C2212yes—
53C3×C4×C22⋊C4C2214yes—
54C3×C4×C22⋊C4C2214yes—
55C3×C4×C22⋊C4C2214yes—
56C3×C4×C22⋊C4C2214yes—
57C2×C3×#95C2214yes—
58C3×C43C2211yescharacteristic
59C2×#562C2×C311yescharacteristic
60#2382C2×C311yescharacteristic, complemented
61#2379C2×C312yescomplemented
62#2368C2×C314yescomplemented
63#2363C2×C314yescomplemented
64#2359C2×C314yescomplemented
65#2335C2×C314yescomplemented
66#2325C2×C314yescomplemented
67#2323C2×C314yescomplemented
68C4×#125C2×C311yescharacteristic, complemented
69C4×#124C2×C312yescomplemented
70C22×C3×Q8C2312yes—
71C22×C3×D4C2314yes—
72C22×C3×D4C2312yes—
73C23×C3×C4C2314yes—
74C23×C3×C4C2314yes—
75C23×C3×C4C2314yes—
76C3×#125C2×C418yescomplemented
77C3×#124C2×C4116yescomplemented
78C3×#123C2×C4132yescomplemented
79C3×C22⋊Q8C2×C4132yescomplemented
80C3×#121C2×C4132yescomplemented
81C3×C4×Q8—44no—
82C3×C4×D4—44no—
83C3×C4×D4—48no—
84C3×#117—48no—
85C2×C3×C4⋊C4C2314yes—
86C2×C3×C4⋊C4C2314yes—
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C4⋊C4C2314yes—
89C2×C3×C4⋊C4C2312yes—
90C2×C3×C4⋊C4C2312yes—
91C2×C3×C4⋊C4C2314yes—
92C2×C3×C4⋊C4C2314yes—
93C2×C3×C4⋊C4C2314yes—
94C2×C3×C4⋊C4C2312yes—
95C2×C3×C22⋊C4C2314yes—
96C2×C3×C22⋊C4C2314yes—
97C2×C3×C22⋊C4C2314yes—
98C2×C3×C22⋊C4C2314yes—
99C2×C3×C22⋊C4C2314yes—
100C2×C3×C22⋊C4C2314yes—
101C2×C3×C22⋊C4C2314yes—
102C2×C3×C22⋊C4C2314yes—
103C2×C3×C22⋊C4C2314yes—
104C2×C3×C42C2311yescharacteristic
105C2×C3×C42C2312yes—
106C2×C3×C42C2314yes—
107C2×C3×C42C2311yescharacteristic
108C2×C3×C42C2312yes—
109C2×C3×C42C2314yes—
110C2×C3×C42C2314yes—
111C2×C3×C42C2314yes—
112C2×C3×C42C2311yescharacteristic
113C3×#95C2314yes—
114C3×#95C2314yes—
115C3×#95C2314yes—
116C3×#95C2314yes—
117C3×#95C2314yes—
118C3×#95C2314yes—
119C3×#95C2314yes—
120C3×#95C2314yes—
121C3×#95C2314yes—
122C3×#95C2314yes—
123C3×#95C2314yes—
124#562C3×C4116yescomplemented
125C2×#125C22×C311yescharacteristic
126C2×#124C22×C312yes—
127C2×#123C22×C314yes—
128C2×C22⋊Q8C22×C314yes—
129C2×#121C22×C314yes—
130C2×C4×Q8C22×C312yes—
131C2×C4×D4C22×C312yes—
132C2×C4×D4C22×C314yes—
133C2×#117C22×C314yes—
134#391C22×C314yes—
135#391C22×C312yes—
136#390C22×C314yes—
137#390C22×C312yes—
138#389C22×C312yes—
139#389C22×C314yes—
140#389C22×C314yes—
141#388C22×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#387C22×C314yes—
151#387C22×C314yes—
152#386C22×C314yes—
153#386C22×C314yes—
154#386C22×C314yes—
155#385C22×C314yes—
156#385C22×C314yes—
157#385C22×C314yes—
158#382C22×C311yescharacteristic
159#382C22×C312yes—
160#380C22×C314yes—
161C4×C4⋊C4C22×C312yes—
162C4×C4⋊C4C22×C314yes—
163C4×C4⋊C4C22×C314yes—
164C4×C22⋊C4C22×C314yes—
165C4×C22⋊C4C22×C314yes—
166C4×C22⋊C4C22×C314yes—
167C4×C22⋊C4C22×C314yes—
168C2×#95C22×C314yes—
169C43C22×C311yescharacteristic
170C24×C3C2414yes—
171C2×C3×Q8—42no—
172C2×C3×Q8—42no—
173C2×C3×Q8C22×C418yes—
174C2×C3×D4—42no—
175C2×C3×D4—44no—
176C2×C3×D4—44no—
177C2×C3×D4—42no—
178C2×C3×D4C22×C4116yes—
179C2×C3×D4C22×C418yes—
180C22×C3×C4—28no—
181C22×C3×C4—44no—
182C22×C3×C4—44no—
183C22×C3×C4—44no—
184C22×C3×C4—44no—
185C22×C3×C4—44no—
186C22×C3×C4—44no—
187C22×C3×C4—44no—
188C22×C3×C4C22×C4116yes—
189C22×C3×C4C2414yes—
190C22×C3×C4C2414yes—
191C22×C3×C4C2411yescharacteristic
192C22×C3×C4C2414yes—
193C22×C3×C4C2411yescharacteristic
194C22×C3×C4C2412yes—
195C22×C3×C4C2414yes—
196C22×C3×C4C2414yes—
197C22×C3×C4C2412yes—
198C22×C3×C4C2411yescharacteristic
199C3×C4⋊C4—28no—
200C3×C4⋊C4—42no—
201C3×C4⋊C4—44no—
202C3×C4⋊C4—44no—
203C3×C4⋊C4—42no—
204C3×C4⋊C4—44no—
205C3×C4⋊C4—42no—
206C3×C4⋊C4C22×C4116yes—
207C3×C4⋊C4C22×C4116yes—
208C3×C4⋊C4C22×C418yes—
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⋊C4C22×C4116yes—
216C3×C22⋊C4C22×C4116yes—
217C3×C22⋊C4C22×C4116yes—
218C3×C42—28no—
219C3×C42—24no—
220C3×C42—22no—
221C3×C42—42no—
222C3×C42—44no—
223C3×C42—41no—
224C3×C42—44no—
225C3×C42—44no—
226C3×C42C22×C414yes—
227C22×Q8C23×C312yes—
228C22×D4C23×C314yes—
229C22×D4C23×C312yes—
230C23×C4C23×C314yes—
231C23×C4C23×C314yes—
232C23×C4C23×C314yes—
233#125C2×C3×C418yescomplemented
234#124C2×C3×C4116yescomplemented
235#123C2×C3×C4132yescomplemented
236C22⋊Q8C2×C3×C4132yescomplemented
237#121C2×C3×C4132yescomplemented
238C4×Q8—44no—
239C4×D4—44no—
240C4×D4—48no—
241#117—48no—
242C2×C4⋊C4C23×C314yes—
243C2×C4⋊C4C23×C314yes—
244C2×C4⋊C4C23×C312yes—
245C2×C4⋊C4C23×C314yes—
246C2×C4⋊C4C23×C312yes—
247C2×C4⋊C4C23×C314yes—
248C2×C4⋊C4C23×C312yes—
249C2×C4⋊C4C23×C314yes—
250C2×C4⋊C4C23×C314yes—
251C2×C4⋊C4C23×C312yes—
252C2×C22⋊C4C23×C314yes—
253C2×C22⋊C4C23×C314yes—
254C2×C22⋊C4C23×C314yes—
255C2×C22⋊C4C23×C314yes—
256C2×C22⋊C4C23×C314yes—
257C2×C22⋊C4C23×C314yes—
258C2×C22⋊C4C23×C314yes—
259C2×C22⋊C4C23×C314yes—
260C2×C22⋊C4C23×C314yes—
261C2×C42C23×C314yes—
262C2×C42C23×C312yes—
263C2×C42C23×C311yescharacteristic
264C2×C42C23×C314yes—
265C2×C42C23×C314yes—
266C2×C42C23×C311yescharacteristic
267C2×C42C23×C312yes—
268C2×C42C23×C314yes—
269C2×C42C23×C311yescharacteristic
270#95C23×C314yes—
271#95C23×C314yes—
272#95C23×C314yes—
273#95C23×C314yes—
274#95C23×C314yes—
275#95C23×C314yes—
276#95C23×C314yes—
277#95C23×C314yes—
278#95C23×C314yes—
279#95C23×C314yes—
280#95C23×C314yes—
281C23×C3—24no—
282C23×C3—24no—
283C23×C3—24no—
284C23×C3—24no—
285C23×C3—24no—
286C23×C3—24no—
287C23×C3C23×C418yes—
288C23×C3C2511yescharacteristic
289C3×Q8—48no—
290C3×D4—416no—
291C3×D4—48no—
292C2×C3×C4—24no—
293C2×C3×C4—22no—
294C2×C3×C4—24no—
295C2×C3×C4—21no—
296C2×C3×C4—22no—
297C2×C3×C4—21no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—24no—
301C2×C3×C4—22no—
302C2×C3×C4—24no—
303C2×C3×C4—22no—
304C2×C3×C4—24no—
305C2×C3×C4—24no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—21no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—416no—
315C2×C3×C4—88no—
316C2×C3×C4—88no—
317C2×C3×C4C23×C412yes—
318C2×C3×C4C23×C414yes—
319C2×C3×C4C23×C418yes—
320C2×C3×C4C23×C418yes—
321C2×C3×C4H(5,𝔽2)14yes—
322C2×C3×C42−1+412yes—
323C24C24×C314yes—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8C22×C3×C418yes—
327C2×D4—42no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—42no—
331C2×D4C22×C3×C4116yes—
332C2×D4C22×C3×C418yes—
333C22×C4—28no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4C22×C3×C4116yes—
342C22×C4C24×C314yes—
343C22×C4C24×C314yes—
344C22×C4C24×C311yescharacteristic
345C22×C4C24×C314yes—
346C22×C4C24×C311yescharacteristic
347C22×C4C24×C312yes—
348C22×C4C24×C314yes—
349C22×C4C24×C314yes—
350C22×C4C24×C312yes—
351C22×C4C24×C311yescharacteristic
352C4⋊C4—28no—
353C4⋊C4—44no—
354C4⋊C4—42no—
355C4⋊C4—44no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—44no—
359C4⋊C4C22×C3×C4116yes—
360C4⋊C4C22×C3×C4116yes—
361C4⋊C4C22×C3×C418yes—
362C22⋊C4—28no—
363C22⋊C4—28no—
364C22⋊C4—44no—
365C22⋊C4—44no—
366C22⋊C4—44no—
367C22⋊C4—44no—
368C22⋊C4C22×C3×C4116yes—
369C22⋊C4C22×C3×C4116yes—
370C22⋊C4C22×C3×C4116yes—
371C42—28no—
372C42—24no—
373C42—22no—
374C42—44no—
375C42—44no—
376C42—42no—
377C42—44no—
378C42—41no—
379C42C22×C3×C414yes—
380C22×C3—28no—
381C22×C3—28no—
382C22×C3—28no—
383C22×C3—44no—
384C22×C3—44no—
385C22×C3—44no—
386C22×C3—44no—
387C22×C3C24×C411yescharacteristic
388C22×C3C2×H(5,𝔽2)12yes—
389C22×C3C2×2−1+411yescharacteristic
390C22×C3#58512yes—
391C22×C3#58511yescharacteristic
392C3×C4—28no—
393C3×C4—28no—
394C3×C4—24no—
395C3×C4—22no—
396C3×C4—24no—
397C3×C4—22no—
398C3×C4—44no—
399C3×C4—44no—
400C3×C4—41no—
401C3×C4—44no—
402C23—24no—
403C23—24no—
404C23—24no—
405C23—24no—
406C23—24no—
407C23—24no—
408C23C23×C3×C418yes—
409C23C25×C311yescharacteristic
410Q8—48no—
411D4—416no—
412D4—48no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—21no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—416no—
436C2×C4—88no—
437C2×C4—88no—
438C2×C4C23×C3×C412yes—
439C2×C4C23×C3×C414yes—
440C2×C4C23×C3×C418yes—
441C2×C4C23×C3×C418yes—
442C2×C4C3×H(5,𝔽2)14yes—
443C2×C4C3×2−1+412yes—
444C2×C3—48no—
445C2×C3C4×H(5,𝔽2)12yes—
446C2×C3C4×2−1+411yescharacteristic
447C2×C3#354512yes—
448C2×C3#354811yescharacteristic
449C2×C3#355311yescharacteristic
450C22—28no—
451C22—28no—
452C22—28no—
453C22—44no—
454C22—44no—
455C22—44no—
456C22—44no—
457C22C24×C3×C411yescharacteristic
458C22C2×C3×H(5,𝔽2)12yes—
459C22C2×C3×2−1+411yescharacteristic
460C22C3×#58512yes—
461C22C3×#58511yescharacteristic
462C4—28no—
463C4—28no—
464C4—24no—
465C4—22no—
466C4—24no—
467C4—22no—
468C4—44no—
469C4—44no—
470C4—41no—
471C4—44no—
472C3#3452211yescharacteristic, direct product factor, Sylow
473C2—48no—
474C2C3×C4×H(5,𝔽2)12yes—
475C2C3×C4×2−1+411yescharacteristic
476C2C3×#354512yes—
477C2C3×#354811yescharacteristic
478C2C3×#355311yescharacteristic
subgroupG/H
center, socleC23×C3C25
commutatorC22C24×C3×C4
Frattini, cosocleC23C25×C3
prime radicalC2×C3×#562C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2382C2semidirect
3. C3×#2379C2semidirect
4. C3×#2368C2semidirect
5. C3×#2363C2semidirect
6. C3×#2359C2semidirect
7. C3×#2335C2semidirect
8. C3×#2325C2semidirect
9. C3×#2323C2semidirect
10. C3×C4×#125C2semidirect
11. C3×C4×#124C2semidirect
12. #34522C3direct (N × H)
13. C3×#562C4semidirect
60. #2382C2×C3semidirect
61. #2379C2×C3semidirect
62. #2368C2×C3semidirect
63. #2363C2×C3semidirect
64. #2359C2×C3semidirect
65. #2335C2×C3semidirect
66. #2325C2×C3semidirect
67. #2323C2×C3semidirect
68. C4×#125C2×C3semidirect
69. C4×#124C2×C3semidirect
76. C3×#125C2×C4semidirect
77. C3×#124C2×C4semidirect
78. C3×#123C2×C4semidirect
79. C3×C22⋊Q8C2×C4semidirect
80. C3×#121C2×C4semidirect
124. #562C3×C4semidirect
233. #125C2×C3×C4semidirect
234. #124C2×C3×C4semidirect
235. #123C2×C3×C4semidirect
236. C22⋊Q8C2×C3×C4semidirect
237. #121C2×C3×C4semidirect
472. C3#34522direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5622
C3×#23822
C3×#23792
C3×#23682
C3×#23632
C3×#23592
C3×#23352
C3×#23252
C3×#23232
C3×C4×#1252
C3×C4×#1242
#345223