Finite Group Databasebeta

← C3×#21876 #10640483

Subgroups — C3×#21876 #10640483

Order 768.

1254 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×#538C211yesmaximal, characteristic
2C3×#2315C211yesmaximal, characteristic, complemented
3C3×#2312C211yesmaximal, characteristic, complemented
4C3×#1996C212yesmaximal, complemented
5C3×#1964C212yesmaximal, complemented
6C3×#1898C214yesmaximal, complemented
7C3×#1799C214yesmaximal, complemented
8#21876C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#538—24no—
10C3×#538—24no—
11C2×C3×#126C412yescomplemented
12C2×C3×#124C412yescomplemented
13C2×C3×#123C412yescomplemented
14C2×C3×C22⋊Q8C2211yescharacteristic
15C2×C3×#121C412yescomplemented
16C2×C3×#121C2211yescharacteristic
17C2×C3×#120C2212yes—
18C2×C3×C4×D4C2212yes—
19C22×C3×C22⋊C4C2212yes—
20C22×C3×C22⋊C4C2211yescharacteristic
21C2×C3×#99C2218yes—
22C3×#388—24no—
23C3×#385—24no—
24C3×#380C2212yes—
25C3×#380C2212yes—
26C3×#380C2212yes—
27C3×#379C2212yes—
28C3×C4×C22⋊C4C2212yes—
29C3×#342C2218yes—
30C2×#538C2×C311yescharacteristic
31#2315C2×C311yescharacteristic, complemented
32#2312C2×C311yescharacteristic, complemented
33#1996C2×C312yescomplemented
34#1964C2×C312yescomplemented
35#1898C2×C314yescomplemented
36#1799C2×C314yescomplemented
37C25×C3C2311yescharacteristic
38C22×C3×Q8C2×C411yescharacteristic, complemented
39C22×C3×D4C2×C412yes—
40C22×C3×D4C2×C412yes—
41C22×C3×D4C2×C411yescharacteristic, complemented
42C22×C3×D4C2312yes—
43C23×C3×C4—22no—
44C23×C3×C4C2×C412yescomplemented
45C23×C3×C4C2311yescharacteristic
46C23×C3×C4C2311yescharacteristic
47C3×#126—28no—
48C3×#124—28no—
49C3×#123—28no—
50C3×C22⋊Q8—22no—
51C3×C22⋊Q8—22no—
52C3×#121—22no—
53C3×#121—22no—
54C3×#121—28no—
55C3×#120—24no—
56C3×#120—24no—
57C3×C4×D4—24no—
58C3×C4×D4—24no—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4C2×C412yescomplemented
63C2×C3×C4⋊C4C2×C412yescomplemented
64C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
65C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4—44no—
70C2×C3×C22⋊C4C2×C412yes—
71C2×C3×C22⋊C4C2×C412yes—
72C2×C3×C22⋊C4C2×C412yes—
73C2×C3×C22⋊C4C2×C412yes—
74C2×C3×C22⋊C4C2×C412yescomplemented
75C2×C3×C22⋊C4C2×C412yescomplemented
76C2×C3×C22⋊C4D414yes—
77C2×C3×C22⋊C4D414yes—
78C2×C3×C22⋊C4C2314yes—
79C2×C3×C22⋊C4C2314yes—
80C2×C3×C22⋊C4C2312yes—
81C2×C3×C42—22no—
82C2×C3×C42C2×C412yescomplemented
83C3×#99—48no—
84C3×#95—22no—
85C3×#95—22no—
86C3×#95—22no—
87C3×#95—22no—
88#538—24no—
89#538—24no—
90C2×#126C3×C412yescomplemented
91C2×#124C3×C412yescomplemented
92C2×#123C3×C412yescomplemented
93C2×C22⋊Q8C22×C311yescharacteristic
94C2×#121C3×C412yescomplemented
95C2×#121C22×C311yescharacteristic
96C2×#120C22×C312yes—
97C2×C4×D4C22×C312yes—
98C22×C22⋊C4C22×C312yes—
99C22×C22⋊C4C22×C311yescharacteristic
100C2×#99C22×C318yes—
101#388—24no—
102#385—24no—
103#380C22×C312yes—
104#380C22×C312yes—
105#380C22×C312yes—
106#379C22×C312yes—
107C4×C22⋊C4C22×C312yes—
108#342C22×C318yes—
109C24×C3—22no—
110C24×C3—22no—
111C24×C3—81no—
112C24×C3—81no—
113C24×C3C22×C412yes—
114C24×C3C22×C412yes—
115C24×C3C2×D412yes—
116C24×C3C2×D412yes—
117C24×C3C2411yescharacteristic
118C2×C3×Q8—22no—
119C2×C3×Q8—81no—
120C2×C3×D4—22no—
121C2×C3×D4—22no—
122C2×C3×D4—24no—
123C2×C3×D4—24no—
124C2×C3×D4—24no—
125C2×C3×D4—24no—
126C2×C3×D4—22no—
127C2×C3×D4—82no—
128C2×C3×D4—82no—
129C2×C3×D4—81no—
130C22×C3×C4—24no—
131C22×C3×C4—24no—
132C22×C3×C4—24no—
133C22×C3×C4—44no—
134C22×C3×C4—81no—
135C22×C3×C4—81no—
136C22×C3×C4—82no—
137C22×C3×C4—82no—
138C22×C3×C4—82no—
139C22×C3×C4C22×C412yes—
140C22×C3×C4C22×C412yes—
141C22×C3×C4C22×C412yes—
142C22×C3×C4C22×C412yes—
143C22×C3×C4C22×C411yescharacteristic
144C22×C3×C4C22×C411yescharacteristic
145C22×C3×C4C2×D412yes—
146C22×C3×C4C2×D412yes—
147C22×C3×C4C2×D412yes—
148C22×C3×C4C2×D412yes—
149C3×C4⋊C4—24no—
150C3×C4⋊C4—24no—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—82no—
154C3×C4⋊C4—82no—
155C3×C4⋊C4—82no—
156C3×C22⋊C4—24no—
157C3×C22⋊C4—24no—
158C3×C22⋊C4—22no—
159C3×C22⋊C4—22no—
160C3×C22⋊C4—24no—
161C3×C22⋊C4—24no—
162C3×C22⋊C4—24no—
163C3×C22⋊C4—28no—
164C3×C22⋊C4—24no—
165C3×C22⋊C4—44no—
166C3×C22⋊C4—82no—
167C3×C22⋊C4—82no—
168C3×C22⋊C4C22⋊C418yes—
169C3×C22⋊C4C22⋊C418yes—
170C3×C42—24no—
171C3×C42—82no—
172C25C23×C311yescharacteristic
173C22×Q8C2×C3×C411yescharacteristic, complemented
174C22×D4C2×C3×C412yes—
175C22×D4C2×C3×C412yes—
176C22×D4C2×C3×C411yescharacteristic, complemented
177C22×D4C23×C312yes—
178C23×C4—22no—
179C23×C4C2×C3×C412yescomplemented
180C23×C4C23×C311yescharacteristic
181C23×C4C23×C311yescharacteristic
182#126—28no—
183#124—28no—
184#123—28no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187#121—22no—
188#121—22no—
189#121—28no—
190#120—24no—
191#120—24no—
192C4×D4—24no—
193C4×D4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4C2×C3×C412yescomplemented
198C2×C4⋊C4C2×C3×C412yescomplemented
199C2×C4⋊C4C2×C3×C411yescharacteristic, complemented
200C2×C4⋊C4C2×C3×C411yescharacteristic, complemented
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4C2×C3×C412yes—
206C2×C22⋊C4C2×C3×C412yes—
207C2×C22⋊C4C2×C3×C412yes—
208C2×C22⋊C4C2×C3×C412yes—
209C2×C22⋊C4C2×C3×C412yescomplemented
210C2×C22⋊C4C2×C3×C412yescomplemented
211C2×C22⋊C4C3×D414yes—
212C2×C22⋊C4C3×D414yes—
213C2×C22⋊C4C23×C314yes—
214C2×C22⋊C4C23×C314yes—
215C2×C22⋊C4C23×C312yes—
216C2×C42—22no—
217C2×C42C2×C3×C412yescomplemented
218#99—48no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223C23×C3—22no—
224C23×C3—22no—
225C23×C3—24no—
226C23×C3—24no—
227C23×C3—22no—
228C23×C3—24no—
229C23×C3—42no—
230C23×C3—42no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3—42no—
234C23×C3—42no—
235C23×C3—42no—
236C23×C3—42no—
237C23×C3—42no—
238C23×C3—42no—
239C23×C3—84no—
240C23×C3—84no—
241C23×C3C2×C22⋊C414yes—
242C23×C3C23×C411yescharacteristic
243C23×C3C22×D411yescharacteristic
244C23×C3C22×D411yescharacteristic
245C23×C3H(5,𝔽2)12yes—
246C23×C32−1+412yes—
247C3×Q8—82no—
248C3×D4—42no—
249C3×D4—42no—
250C3×D4—44no—
251C3×D4—82no—
252C3×D4—84no—
253C3×D4—84no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—24no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—44no—
268C2×C3×C4—44no—
269C2×C3×C4—44no—
270C2×C3×C4—44no—
271C2×C3×C4—44no—
272C2×C3×C4—44no—
273C2×C3×C4—84no—
274C2×C3×C4—88no—
275C2×C3×C4C2×C22⋊C414yes—
276C2×C3×C4C2×C22⋊C412yes—
277C2×C3×C4C2×C22⋊C412yes—
278C24—22no—
279C24—22no—
280C24—81no—
281C24—81no—
282C24C22×C3×C412yes—
283C24C22×C3×C412yes—
284C24C2×C3×D412yes—
285C24C2×C3×D412yes—
286C24C24×C311yescharacteristic
287C2×Q8—22no—
288C2×Q8—81no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—22no—
296C2×D4—82no—
297C2×D4—82no—
298C2×D4—81no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—44no—
303C22×C4—81no—
304C22×C4—81no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4C22×C3×C412yes—
309C22×C4C22×C3×C412yes—
310C22×C4C22×C3×C412yes—
311C22×C4C22×C3×C412yes—
312C22×C4C22×C3×C411yescharacteristic
313C22×C4C22×C3×C411yescharacteristic
314C22×C4C2×C3×D412yes—
315C22×C4C2×C3×D412yes—
316C22×C4C2×C3×D412yes—
317C22×C4C2×C3×D412yes—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C22⋊C4—24no—
326C22⋊C4—24no—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C22⋊C4—28no—
333C22⋊C4—24no—
334C22⋊C4—44no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C22⋊C4C3×C22⋊C418yes—
338C22⋊C4C3×C22⋊C418yes—
339C42—24no—
340C42—82no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—24no—
348C22×C3—24no—
349C22×C3—24no—
350C22×C3—42no—
351C22×C3—42no—
352C22×C3—44no—
353C22×C3—44no—
354C22×C3—42no—
355C22×C3—42no—
356C22×C3—42no—
357C22×C3—84no—
358C22×C3—84no—
359C22×C3—84no—
360C22×C3C22×C22⋊C411yescharacteristic
361C22×C3#52012yes—
362C22×C3#53511yescharacteristic
363C22×C3#53611yescharacteristic
364C22×C3#53712yes—
365C3×C4—24no—
366C3×C4—22no—
367C3×C4—22no—
368C3×C4—42no—
369C3×C4—42no—
370C3×C4—42no—
371C3×C4—84no—
372C3×C4—84no—
373C23—22no—
374C23—22no—
375C23—24no—
376C23—24no—
377C23—22no—
378C23—24no—
379C23—42no—
380C23—42no—
381C23—41no—
382C23—41no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—84no—
390C23—84no—
391C23C2×C3×C22⋊C414yes—
392C23C23×C3×C411yescharacteristic
393C23C22×C3×D411yescharacteristic
394C23C22×C3×D411yescharacteristic
395C23C3×H(5,𝔽2)12yes—
396C23C3×2−1+412yes—
397Q8—82no—
398D4—42no—
399D4—42no—
400D4—44no—
401D4—82no—
402D4—84no—
403D4—84no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—84no—
424C2×C4—88no—
425C2×C4C2×C3×C22⋊C414yes—
426C2×C4C2×C3×C22⋊C412yes—
427C2×C4C2×C3×C22⋊C412yes—
428C2×C3—21no—
429C2×C3—22no—
430C2×C3—21no—
431C2×C3—22no—
432C2×C3—44no—
433C2×C3—82no—
434C2×C3#231111yescharacteristic
435C2×C3#288311yescharacteristic
436C2×C3#288311yescharacteristic
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—24no—
444C22—24no—
445C22—24no—
446C22—42no—
447C22—42no—
448C22—44no—
449C22—44no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—84no—
454C22—84no—
455C22—84no—
456C22C22×C3×C22⋊C411yescharacteristic
457C22C3×#52012yes—
458C22C3×#53511yescharacteristic
459C22C3×#53611yescharacteristic
460C22C3×#53712yes—
461C4—24no—
462C4—22no—
463C4—22no—
464C4—42no—
465C4—42no—
466C4—42no—
467C4—84no—
468C4—84no—
469C3#2187611yescharacteristic, direct product factor, Sylow
470C2—21no—
471C2—22no—
472C2—21no—
473C2—22no—
474C2—44no—
475C2—82no—
476C2C3×#231111yescharacteristic
477C2C3×#288311yescharacteristic
478C2C3×#288311yescharacteristic
subgroupG/H
center, socleC22×C3C22×C22⋊C4
commutatorC23C23×C3×C4
Frattini, cosocleC24C24×C3
prime radicalC2×C3×#538C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2315C2semidirect
3. C3×#2312C2semidirect
4. C3×#1996C2semidirect
5. C3×#1964C2semidirect
6. C3×#1898C2semidirect
7. C3×#1799C2semidirect
8. #21876C3direct (N × H)
11. C2×C3×#126C4semidirect
12. C2×C3×#124C4semidirect
13. C2×C3×#123C4semidirect
15. C2×C3×#121C4semidirect
31. #2315C2×C3semidirect
32. #2312C2×C3semidirect
33. #1996C2×C3semidirect
34. #1964C2×C3semidirect
35. #1898C2×C3semidirect
36. #1799C2×C3semidirect
38. C22×C3×Q8C2×C4semidirect
41. C22×C3×D4C2×C4semidirect
44. C23×C3×C4C2×C4semidirect
62. C2×C3×C4⋊C4C2×C4semidirect
63. C2×C3×C4⋊C4C2×C4semidirect
64. C2×C3×C4⋊C4C2×C4semidirect
65. C2×C3×C4⋊C4C2×C4semidirect
74. C2×C3×C22⋊C4C2×C4semidirect
75. C2×C3×C22⋊C4C2×C4semidirect
82. C2×C3×C42C2×C4semidirect
90. C2×#126C3×C4semidirect
91. C2×#124C3×C4semidirect
92. C2×#123C3×C4semidirect
94. C2×#121C3×C4semidirect
173. C22×Q8C2×C3×C4semidirect
176. C22×D4C2×C3×C4semidirect
179. C23×C4C2×C3×C4semidirect
197. C2×C4⋊C4C2×C3×C4semidirect
198. C2×C4⋊C4C2×C3×C4semidirect
199. C2×C4⋊C4C2×C3×C4semidirect
200. C2×C4⋊C4C2×C3×C4semidirect
209. C2×C22⋊C4C2×C3×C4semidirect
210. C2×C22⋊C4C2×C3×C4semidirect
217. C2×C42C2×C3×C4semidirect
469. C3#21876direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5382
C3×#23152
C3×#23122
C3×#19962
C3×#19642
C3×#18982
C3×#17992
#218763