Finite Group Databasebeta

← C3×#23914 #10642521

Subgroups — C3×#23914 #10642521

Order 768.

1706 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3485C211yesmaximal, characteristic
2C2×C3×#536C211yesmaximal, characteristic, complemented
3C2×C3×#455C212yesmaximal, complemented
4C3×#3004C212yesmaximal, complemented
5C3×#2005C214yesmaximal, complemented
6C3×#1897C212yesmaximal, complemented
7C3×#1885C212yesmaximal, complemented
8C3×#1785C211yesmaximal, characteristic, complemented
9#23914C311yesmaximal, characteristic, direct product factor, Sylow
10C2×C3×H(5,𝔽2)C2211yescharacteristic
11C22×C3×#41C2211yescharacteristic
12C23×C3×Q8C2211yescharacteristic, complemented
13C2×C3×#137C2214yes—
14C3×#547—24no—
15C3×#546—24no—
16C3×#536—28no—
17C3×#536C2212yes—
18C3×#534—22no—
19C2×C3×C4⋊Q8C2212yescomplemented
20C2×C3×#124C2212yes—
21C3×#525C2212yes—
22C2×C3×#123—22no—
23C2×C3×#123—22no—
24C2×C3×C22⋊Q8—22no—
25C3×#518C2211yescharacteristic
26C2×C3×#117C2211yescharacteristic, complemented
27C3×#455—28no—
28C3×#451—24no—
29C3×#448—24no—
30C2×C3×#104C2214yes—
31C3×#419C2214yescomplemented
32C2×C3×#101C2212yescomplemented
33C2×C3×#100C2212yes—
34C3×#407C2212yescomplemented
35C3×#339C2214yescomplemented
36#3485C2×C311yescharacteristic
37C2×#536C2×C311yescharacteristic, complemented
38C2×#455C2×C312yescomplemented
39#3004C2×C312yescomplemented
40#2005C2×C314yescomplemented
41#1897C2×C312yescomplemented
42#1885C2×C312yescomplemented
43#1785C2×C311yescharacteristic, complemented
44C3×H(5,𝔽2)—24no—
45C3×H(5,𝔽2)—42no—
46C2×C3×#41—22no—
47C2×C3×#41—24no—
48C2×C3×#41—24no—
49C2×C3×#41—42no—
50C2×C3×#41C2312yes—
51C22×C3×Q8—24no—
52C22×C3×Q8—21no—
53C22×C3×Q8—24no—
54C22×C3×Q8—42no—
55C22×C3×Q8C2312yes—
56C22×C3×Q8C2311yescharacteristic
57C22×C3×D4—22no—
58C22×C3×D4—22no—
59C22×C3×D4D412yes—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C23×C3×C4D412yes—
63C23×C3×C4C2311yescharacteristic
64C3×#137—28no—
65C3×#137—44no—
66C2×C3×Q16—24no—
67C2×C3×SD16—24no—
68C2×C3×M16C2314yes—
69C3×C4⋊Q8—24no—
70C3×C4⋊Q8—22no—
71C3×C4⋊Q8—24no—
72C3×#124—24no—
73C3×#124—22no—
74C3×#124D418yescomplemented
75C3×#123—216no—
76C3×#123—28no—
77C3×#123—24no—
78C3×#123—22no—
79C3×#123—24no—
80C3×C22⋊Q8—216no—
81C3×C22⋊Q8—24no—
82C3×C22⋊Q8D414yes—
83C3×#121—28no—
84C3×#121D414yes—
85C3×#120—24no—
86C3×#120—22no—
87C3×C4×D4—24no—
88C3×C4×D4—24no—
89C3×#117—24no—
90C3×#117C2312yes—
91C2×C3×C4⋊C4—24no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—24no—
94C2×C3×C22⋊C4—24no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4—24no—
98C2×C3×C42C2312yes—
99C3×#104—28no—
100C3×#103—28no—
101C3×#101—42no—
102C3×#100—24no—
103C3×C22⋊C8—24no—
104C2×H(5,𝔽2)C22×C311yescharacteristic
105C22×#41C22×C311yescharacteristic
106C23×Q8C22×C311yescharacteristic, complemented
107C2×#137C22×C314yes—
108#547—24no—
109#546—24no—
110#536—28no—
111#536C22×C312yes—
112#534—22no—
113C2×C4⋊Q8C22×C312yescomplemented
114C2×#124C22×C312yes—
115#525C22×C312yes—
116C2×#123—22no—
117C2×#123—22no—
118C2×C22⋊Q8—22no—
119#518C22×C311yescharacteristic
120C2×#117C22×C311yescharacteristic, complemented
121#455—28no—
122#451—24no—
123#448—24no—
124C2×#104C22×C314yes—
125#419C22×C314yescomplemented
126C2×#101C22×C312yescomplemented
127C2×#100C22×C312yes—
128#407C22×C312yescomplemented
129#339C22×C314yescomplemented
130C24×C3—22no—
131C24×C3C2×D412yes—
132C24×C3C2×D411yescharacteristic
133C3×#41—24no—
134C3×#41—24no—
135C3×#41—416no—
136C3×#41—48no—
137C2×C3×Q8—24no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—24no—
143C2×C3×Q8—24no—
144C2×C3×Q8—416no—
145C2×C3×Q8—42no—
146C2×C3×Q8—42no—
147C2×C3×Q8C2×D412yes—
148C2×C3×Q8C2×D414yes—
149C2×C3×D4—24no—
150C2×C3×D4—24no—
151C2×C3×D4—24no—
152C2×C3×D4—22no—
153C2×C3×D4—24no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—42no—
157C2×C3×D4—48no—
158C2×C3×D4—44no—
159C2×C3×D4—44no—
160C2×C3×D4C2×D412yes—
161C2×C3×D4C2×D414yes—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—24no—
166C22×C3×C4—24no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—24no—
170C22×C3×C4—24no—
171C22×C3×C4—22no—
172C22×C3×C4—42no—
173C22×C3×C4—42no—
174C22×C3×C4—42no—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2×D414yes—
178C22×C3×C4C2411yescharacteristic
179C3×Q16—48no—
180C3×SD16—48no—
181C3×M16—24no—
182C3×M16—24no—
183C2×C3×C8—24no—
184C3×C4⋊C4—24no—
185C3×C4⋊C4—28no—
186C3×C4⋊C4—216no—
187C3×C4⋊C4—24no—
188C3×C4⋊C4C2×D414yes—
189C3×C22⋊C4—24no—
190C3×C22⋊C4—24no—
191C3×C22⋊C4—28no—
192C3×C22⋊C4—216no—
193C3×C22⋊C4—24no—
194C3×C22⋊C4—22no—
195C3×C22⋊C4—24no—
196C3×C42—22no—
197C3×C42—22no—
198C3×C42C2×D414yes—
199H(5,𝔽2)—24no—
200H(5,𝔽2)—42no—
201C2×#41—22no—
202C2×#41—24no—
203C2×#41—24no—
204C2×#41—42no—
205C2×#41C23×C312yes—
206C22×Q8—24no—
207C22×Q8—21no—
208C22×Q8—24no—
209C22×Q8—42no—
210C22×Q8C23×C312yes—
211C22×Q8C23×C311yescharacteristic
212C22×D4—22no—
213C22×D4—22no—
214C22×D4C3×D412yes—
215C22×D4C23×C311yescharacteristic
216C23×C4—21no—
217C23×C4C3×D412yes—
218C23×C4C23×C311yescharacteristic
219#137—28no—
220#137—44no—
221C2×Q16—24no—
222C2×SD16—24no—
223C2×M16C23×C314yes—
224C4⋊Q8—24no—
225C4⋊Q8—22no—
226C4⋊Q8—24no—
227#124—24no—
228#124—22no—
229#124C3×D418yescomplemented
230#123—216no—
231#123—28no—
232#123—24no—
233#123—22no—
234#123—24no—
235C22⋊Q8—216no—
236C22⋊Q8—24no—
237C22⋊Q8C3×D414yes—
238#121—28no—
239#121C3×D414yes—
240#120—24no—
241#120—22no—
242C4×D4—24no—
243C4×D4—24no—
244#117—24no—
245#117C23×C312yes—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—24no—
253C2×C42C23×C312yes—
254#104—28no—
255#103—28no—
256#101—42no—
257#100—24no—
258C22⋊C8—24no—
259C23×C3—21no—
260C23×C3—22no—
261C23×C3—21no—
262C23×C3—22no—
263C23×C3—22no—
264C23×C3—42no—
265C23×C3—44no—
266C23×C3—44no—
267C23×C3—84no—
268C23×C3#12012yes—
269C23×C3#12014yes—
270C23×C3C22×D411yescharacteristic
271C3×Q8—28no—
272C3×Q8—24no—
273C3×Q8—48no—
274C3×Q8—44no—
275C3×Q8—44no—
276C3×D4—28no—
277C3×D4—24no—
278C3×D4—48no—
279C3×D4—48no—
280C3×D4—42no—
281C3×D4—42no—
282C3×D4—84no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—28no—
286C2×C3×C4—24no—
287C2×C3×C4—24no—
288C2×C3×C4—22no—
289C2×C3×C4—24no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—48no—
294C2×C3×C4—44no—
295C2×C3×C4—48no—
296C2×C3×C4—44no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—44no—
300C2×C3×C4—84no—
301C2×C3×C4#12012yes—
302C2×C3×C4C22×D412yes—
303C2×C3×C4C22×D412yes—
304C2×C3×C4H(5,𝔽2)12yes—
305C3×C8—44no—
306C24—22no—
307C24C2×C3×D412yes—
308C24C2×C3×D411yescharacteristic
309#41—24no—
310#41—24no—
311#41—416no—
312#41—48no—
313C2×Q8—24no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—24no—
319C2×Q8—24no—
320C2×Q8—416no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8C2×C3×D412yes—
324C2×Q8C2×C3×D414yes—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—22no—
329C2×D4—24no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—42no—
333C2×D4—48no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4C2×C3×D412yes—
337C2×D4C2×C3×D414yes—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4C2×C3×D412yes—
352C22×C4C2×C3×D411yescharacteristic
353C22×C4C2×C3×D414yes—
354C22×C4C24×C311yescharacteristic
355Q16—48no—
356SD16—48no—
357M16—24no—
358M16—24no—
359C2×C8—24no—
360C4⋊C4—24no—
361C4⋊C4—28no—
362C4⋊C4—216no—
363C4⋊C4—24no—
364C4⋊C4C2×C3×D414yes—
365C22⋊C4—24no—
366C22⋊C4—24no—
367C22⋊C4—28no—
368C22⋊C4—216no—
369C22⋊C4—24no—
370C22⋊C4—22no—
371C22⋊C4—24no—
372C42—22no—
373C42—22no—
374C42C2×C3×D414yes—
375C22×C3—22no—
376C22×C3—22no—
377C22×C3—24no—
378C22×C3—24no—
379C22×C3—44no—
380C22×C3—42no—
381C22×C3—42no—
382C22×C3—44no—
383C22×C3—42no—
384C22×C3—44no—
385C22×C3—82no—
386C22×C3—164no—
387C22×C3C2×#12011yescharacteristic
388C22×C3C2×#12011yescharacteristic
389C22×C3#53411yescharacteristic
390C3×C4—22no—
391C3×C4—44no—
392C3×C4—42no—
393C3×C4—42no—
394C3×C4—44no—
395C3×C4—44no—
396C3×C4D4214yes—
397C23—21no—
398C23—22no—
399C23—21no—
400C23—22no—
401C23—22no—
402C23—42no—
403C23—44no—
404C23—44no—
405C23—84no—
406C23C3×#12012yes—
407C23C3×#12014yes—
408C23C22×C3×D411yescharacteristic
409Q8—28no—
410Q8—24no—
411Q8—48no—
412Q8—44no—
413Q8—44no—
414D4—28no—
415D4—24no—
416D4—48no—
417D4—48no—
418D4—42no—
419D4—42no—
420D4—84no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—28no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—48no—
432C2×C4—48no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—84no—
439C2×C4C3×#12012yes—
440C2×C4C22×C3×D412yes—
441C2×C4C22×C3×D412yes—
442C2×C4C3×H(5,𝔽2)12yes—
443C8—44no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—42no—
447C2×C3—44no—
448C2×C3—82no—
449C2×C3#240311yescharacteristic
450C2×C3#302212yes—
451C22—22no—
452C22—22no—
453C22—24no—
454C22—24no—
455C22—44no—
456C22—42no—
457C22—42no—
458C22—44no—
459C22—42no—
460C22—44no—
461C22—82no—
462C22—164no—
463C22C2×C3×#12011yescharacteristic
464C22C2×C3×#12011yescharacteristic
465C22C3×#53411yescharacteristic
466C4—22no—
467C4—44no—
468C4—42no—
469C4—42no—
470C4—44no—
471C4—44no—
472C4C3×D4214yes—
473C3#2391411yescharacteristic, direct product factor, Sylow
474C2—21no—
475C2—21no—
476C2—42no—
477C2—44no—
478C2—82no—
479C2C3×#240311yescharacteristic
480C2C3×#302212yes—
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3485C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#536C2semidirect
3. C2×C3×#455C2semidirect
4. C3×#3004C2semidirect
5. C3×#2005C2semidirect
6. C3×#1897C2semidirect
7. C3×#1885C2semidirect
8. C3×#1785C2semidirect
9. #23914C3direct (N × H)
12. C23×C3×Q8C22semidirect
19. C2×C3×C4⋊Q8C22semidirect
26. C2×C3×#117C22semidirect
31. C3×#419C22semidirect
32. C2×C3×#101C22semidirect
34. C3×#407C22semidirect
35. C3×#339C22semidirect
37. C2×#536C2×C3semidirect
38. C2×#455C2×C3semidirect
39. #3004C2×C3semidirect
40. #2005C2×C3semidirect
41. #1897C2×C3semidirect
42. #1885C2×C3semidirect
43. #1785C2×C3semidirect
74. C3×#124D4semidirect
106. C23×Q8C22×C3semidirect
113. C2×C4⋊Q8C22×C3semidirect
120. C2×#117C22×C3semidirect
125. #419C22×C3semidirect
126. C2×#101C22×C3semidirect
128. #407C22×C3semidirect
129. #339C22×C3semidirect
229. #124C3×D4semidirect
473. C3#23914direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34852
C2×C3×#5362
C2×C3×#4552
C3×#30042
C3×#20052
C3×#18972
C3×#18852
C3×#17852
#239143