Finite Group Databasebeta

← C22×C3×#3799 #13181536

Subgroups — C22×C3×#3799 #13181536

Order 1728.

4242 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C22×C32×C4×D3C211yesmaximal, characteristic, complemented
3C22×C32×Dic6C211yesmaximal, characteristic, complemented
4C22×C3×#626C212yesmaximal, complemented
5C22×C3×#624C212yesmaximal, complemented
6C2×C3×#3799C2124yesmaximal, direct product factor
7C22×C3×#239—31nomaximal
8C22×C3×#235—31nomaximal
9C22×#3799C311yesmaximal, characteristic, direct product factor
10C23×C3×C3⋊D3C2212yes—
11C23×C32×D3C2211yescharacteristic
12C22×C33×C4C2211yescharacteristic
13C2×C3×C4⋊(C3⋊D3)C22112yescomplemented
14C22×C32×Dic3C2211yescharacteristic
15C22×C32×Dic3C2212yes—
16C2×C32×C4×D3C22112yescomplemented
17C2×C32×Dic6C22112yescomplemented
18C2×C3×#626C22124yescomplemented
19C2×C3×#624C22124yescomplemented
20C3×#3799C22164yesdirect product factor
21C22×C32×Q8—31no—
22C23×C32×C4—31no—
23C22×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
24C22×C3×#67—32no—
25C2×C3×#239—324no—
26C2×C3×#235—324no—
27C22×C3×D12—31no—
28C22×C3×D12—31no—
29C22×C3×D12—32no—
30C22×C3×D12—61no—
31C22×C3×C4×D3—21no—
32C22×C3×C4×D3—32no—
33C22×C3×C4×D3—32no—
34C22×C3×C4×D3D311yescharacteristic, complemented
35C22×C3×C4×D3C2×C311yescharacteristic, complemented
36C22×C3×Dic6—21no—
37C22×C3×Dic6D311yescharacteristic, complemented
38C22×C3×Dic6C2×C311yescharacteristic, complemented
39C22×#626C2×C312yescomplemented
40C22×#624C2×C312yescomplemented
41C2×#3799C2×C3124yesdirect product factor
42C23×C33C2311yescharacteristic
43C22×C3×C3⋊D3—28no—
44C22×C3×C3⋊D3C23112yes—
45C22×C32×D3—24no—
46C22×C32×D3C2316yes—
47C2×C33×C4C2316yes—
48C3×C4⋊(C3⋊D3)C23116yescomplemented
49C2×C32×Dic3C2316yes—
50C2×C32×Dic3C23112yes—
51C32×C4×D3C23116yescomplemented
52C32×Dic6C23116yescomplemented
53C3×#626C23132yescomplemented
54C3×#624C23132yescomplemented
55C22×C3×#41—91no—
56C22×#239—31no—
57C22×#235—31no—
58C24×C32—31no—
59C23×C3⋊D3C22×C312yes—
60C23×C3×D3—21no—
61C23×C3×D3—32no—
62C23×C3×D3—32no—
63C23×C3×D3—62no—
64C23×C3×D3C2×D311yescharacteristic
65C23×C3×D3C22×C311yescharacteristic
66C2×C32×Q8—312no—
67C22×C32×C4—21no—
68C22×C32×C4—21no—
69C22×C32×C4—21no—
70C22×C32×C4—31no—
71C22×C32×C4—32no—
72C22×C32×C4—312no—
73C22×C32×C4—41no—
74C22×C32×C4C2×D311yescharacteristic
75C22×C32×C4C2×D311yescharacteristic
76C22×C32×C4C22×C311yescharacteristic
77C2×C4⋊(C3⋊D3)C22×C3112yescomplemented
78C2×C3×#67—324no—
79C22×C3×Dic3—22no—
80C22×C3×Dic3—21no—
81C22×C3×Dic3C2×D312yes—
82C22×C3×Dic3C2×D311yescharacteristic
83C22×C3×Dic3C22×C311yescharacteristic
84C22×C3×Dic3C22×C312yes—
85C3×#239—364no—
86C3×#235—364no—
87C2×C3×D12—324no—
88C2×C3×D12—312no—
89C2×C3×D12—312no—
90C2×C3×D12—612no—
91C2×C3×C4×D3—212no—
92C2×C3×C4×D3—324no—
93C2×C3×C4×D3—324no—
94C2×C3×C4×D3C2×D3112yescomplemented
95C2×C3×C4×D3C22×C3112yescomplemented
96C2×C3×Dic6—212no—
97C2×C3×Dic6C2×D3112yescomplemented
98C2×C3×Dic6C22×C3112yescomplemented
99C2×#626C22×C3124yescomplemented
100C2×#624C22×C3124yescomplemented
101#3799C22×C3164yesdirect product factor
102C22×C33#4114yescomplemented
103C22×C33C2413yes—
104C2×C3×C3⋊D3—224no—
105C2×C3×C3⋊D3C2418yes—
106C2×C32×D3—212no—
107C2×C32×D3C2414yes—
108C33×C4C2414yes—
109C32×Dic3C2414yes—
110C32×Dic3C2418yes—
111C2×C3×#41—924no—
112C22×C3×Q8—31no—
113C22×C3×Q8—31no—
114C22×C3×Q8—61no—
115C22×C3×D4—92no—
116C22×C3×D4—91no—
117C23×C3×C4—31no—
118C23×C3×C4—31no—
119C23×C3×C4—61no—
120C23×C3×C4—92no—
121C22×#67—32no—
122C2×#239—324no—
123C2×#235—324no—
124C22×D12—32no—
125C22×D12—31no—
126C22×D12—31no—
127C22×D12—61no—
128C22×C4×D3—32no—
129C22×C4×D3—32no—
130C22×C4×D3C3×D311yescharacteristic, complemented
131C22×Dic6C3×D311yescharacteristic, complemented
132C23×C32—21no—
133C23×C32—21no—
134C23×C32—21no—
135C23×C32—36no—
136C23×C32—41no—
137C23×C32—64no—
138C23×C32C22×D311yescharacteristic
139C23×C32C22×D311yescharacteristic
140C23×C32C23×C311yescharacteristic
141C22×C3⋊D3—28no—
142C22×C3⋊D3C23×C3112yes—
143C22×C3×D3—28no—
144C22×C3×D3—24no—
145C22×C3×D3—24no—
146C22×C3×D3—26no—
147C22×C3×D3—312no—
148C22×C3×D3—312no—
149C22×C3×D3—612no—
150C22×C3×D3—68no—
151C22×C3×D3—68no—
152C22×C3×D3—128no—
153C22×C3×D3C22×D316yes—
154C22×C3×D3C23×C316yes—
155C32×Q8—316no—
156C2×C32×C4—26no—
157C2×C32×C4—26no—
158C2×C32×C4—26no—
159C2×C32×C4—36no—
160C2×C32×C4—312no—
161C2×C32×C4—316no—
162C2×C32×C4—46no—
163C2×C32×C4C22×D316yes—
164C2×C32×C4C22×D316yes—
165C2×C32×C4C23×C316yes—
166C4⋊(C3⋊D3)C23×C3116yescomplemented
167C3×#67—332no—
168C2×C3×Dic3—212no—
169C2×C3×Dic3—26no—
170C2×C3×Dic3C22×D3112yes—
171C2×C3×Dic3C22×D316yes—
172C2×C3×Dic3C23×C316yes—
173C2×C3×Dic3C23×C3112yes—
174C3×D12—332no—
175C3×D12—316no—
176C3×D12—316no—
177C3×D12—616no—
178C3×C4×D3—216no—
179C3×C4×D3—332no—
180C3×C4×D3—332no—
181C3×C4×D3C22×D3116yescomplemented
182C3×C4×D3C23×C3116yescomplemented
183C3×Dic6—216no—
184C3×Dic6C22×D3116yescomplemented
185C3×Dic6C23×C3116yescomplemented
186#626C23×C3132yescomplemented
187#624C23×C3132yescomplemented
188C22×#41—91noSylow
189C2×C33C2×#4116yescomplemented
190C2×C33C2511yescharacteristic
191C3×C3⋊D3—28no—
192C32×D3—24no—
193C24×C3—31no—
194C24×C3—31no—
195C24×C3—61no—
196C24×C3—92no—
197C23×D3—32no—
198C23×D3—32no—
199C23×D3—62no—
200C23×D3C2×C3×D311yescharacteristic
201C3×#41—964no—
202C2×C3×Q8—312no—
203C2×C3×Q8—312no—
204C2×C3×Q8—612no—
205C2×C3×D4—924no—
206C2×C3×D4—912no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—31no—
211C22×C3×C4—32no—
212C22×C3×C4—31no—
213C22×C3×C4—32no—
214C22×C3×C4—312no—
215C22×C3×C4—312no—
216C22×C3×C4—41no—
217C22×C3×C4—612no—
218C22×C3×C4—61no—
219C22×C3×C4—62no—
220C22×C3×C4—924no—
221C22×C3×C4D3211yescharacteristic
222C22×C3×C4C2×C3×D311yescharacteristic
223C22×C3×C4C2×C3×D311yescharacteristic
224C2×#67—324no—
225C22×Dic3C2×C3×D312yes—
226C22×Dic3C2×C3×D311yescharacteristic
227#239—364no—
228#235—364no—
229C2×D12—324no—
230C2×D12—312no—
231C2×D12—312no—
232C2×D12—612no—
233C2×C4×D3—324no—
234C2×C4×D3—324no—
235C2×C4×D3C2×C3×D3112yescomplemented
236C2×Dic6C2×C3×D3112yescomplemented
237C22×C32—23no—
238C22×C32—23no—
239C22×C32—23no—
240C22×C32—24no—
241C22×C32—24no—
242C22×C32—24no—
243C22×C32—34no—
244C22×C32—44no—
245C22×C32—43no—
246C22×C32—612no—
247C22×C32#23514yescomplemented
248C22×C32#23914yescomplemented
249C22×C32C3×#4114yescomplemented
250C22×C32C23×D313yes—
251C22×C32C23×D313yes—
252C22×C32C24×C313yes—
253C2×C3⋊D3—224no—
254C2×C3⋊D3C24×C318yes—
255C2×C3×D3—224no—
256C2×C3×D3—212no—
257C2×C3×D3—212no—
258C2×C3×D3—24no—
259C2×C3×D3—38no—
260C2×C3×D3—38no—
261C2×C3×D3—68no—
262C2×C3×D3—624no—
263C2×C3×D3—624no—
264C2×C3×D3—1224no—
265C2×C3×D3C23×D314yes—
266C2×C3×D3C24×C314yes—
267C32×C4—24no—
268C32×C4—24no—
269C32×C4—24no—
270C32×C4—34no—
271C32×C4—38no—
272C32×C4—44no—
273C32×C4C23×D314yes—
274C32×C4C23×D314yes—
275C32×C4C24×C314yes—
276C3×Dic3—28no—
277C3×Dic3—24no—
278C3×Dic3C23×D318yes—
279C3×Dic3C23×D314yes—
280C3×Dic3C24×C314yes—
281C3×Dic3C24×C318yes—
282C2×#41—924no—
283C22×Q8—31no—
284C22×D4—92no—
285C22×D4—91no—
286C23×C4—31no—
287C23×C4—92no—
288C33C22×#4111yescharacteristic, complemented, Sylow
289C23×C3—21no—
290C23×C3—21no—
291C23×C3—21no—
292C23×C3—36no—
293C23×C3—36no—
294C23×C3—41no—
295C23×C3—66no—
296C23×C3—68no—
297C23×C3—64no—
298C23×C3—64no—
299C23×C3—912no—
300C23×C3—188no—
301C23×C3C2×D3211yescharacteristic
302C23×C3C22×C3×D311yescharacteristic
303C23×C3C22×C3×D311yescharacteristic
304C22×D3—24no—
305C22×D3—312no—
306C22×D3—312no—
307C22×D3—612no—
308C22×D3—68no—
309C22×D3—68no—
310C22×D3—128no—
311C22×D3C22×C3×D316yes—
312C3×Q8—316no—
313C3×Q8—316no—
314C3×Q8—616no—
315C3×D4—932no—
316C3×D4—916no—
317C2×C3×C4—26no—
318C2×C3×C4—26no—
319C2×C3×C4—26no—
320C2×C3×C4—36no—
321C2×C3×C4—312no—
322C2×C3×C4—36no—
323C2×C3×C4—312no—
324C2×C3×C4—316no—
325C2×C3×C4—316no—
326C2×C3×C4—46no—
327C2×C3×C4—616no—
328C2×C3×C4—66no—
329C2×C3×C4—612no—
330C2×C3×C4—932no—
331C2×C3×C4C2×D3216yes—
332C2×C3×C4C22×C3×D316yes—
333C2×C3×C4C22×C3×D316yes—
334#67—332no—
335C2×Dic3C22×C3×D3112yes—
336C2×Dic3C22×C3×D316yes—
337D12—332no—
338D12—316no—
339D12—316no—
340D12—616no—
341C4×D3—332no—
342C4×D3—332no—
343C4×D3C22×C3×D3116yescomplemented
344Dic6C22×C3×D3116yescomplemented
345C2×C32—26no—
346C2×C32—26no—
347C2×C32—26no—
348C2×C32—21no—
349C2×C32—21no—
350C2×C32—21no—
351C2×C32—46no—
352C2×C32—41no—
353C2×C32—64no—
354C2×C32C2×#23516yescomplemented
355C2×C32C2×#23916yescomplemented
356C2×C32C2×C3×#4116yescomplemented
357C2×C32C24×D311yescharacteristic
358C2×C32C24×D311yescharacteristic
359C2×C32C25×C311yescharacteristic
360C3⋊D3—28no—
361C3×D3—28no—
362C3×D3—24no—
363C3×D3—24no—
364C3×D3—68no—
365C3×D3—68no—
366C3×D3—128no—
367C24—31no—
368C24—92no—
369#41—964no—
370C2×Q8—312no—
371C2×D4—924no—
372C2×D4—912no—
373C22×C4—31no—
374C22×C4—32no—
375C22×C4—312no—
376C22×C4—924no—
377C22×C4C3×D3211yescharacteristic
378C22×C3—23no—
379C22×C3—23no—
380C22×C3—23no—
381C22×C3—24no—
382C22×C3—24no—
383C22×C3—24no—
384C22×C3—34no—
385C22×C3—34no—
386C22×C3—43no—
387C22×C3—44no—
388C22×C3—64no—
389C22×C3—624no—
390C22×C3—612no—
391C22×C3—612no—
392C22×C3—98no—
393C22×C3—1824no—
394C22×C3#379914yesdirect product factor
395C22×C3C3×#23514yescomplemented
396C22×C3C3×#23914yescomplemented
397C22×C3C22×D3213yes—
398C22×C3C23×C3×D313yes—
399C22×C3C23×C3×D313yes—
400C2×D3—212no—
401C2×D3—38no—
402C2×D3—38no—
403C2×D3—68no—
404C2×D3—624no—
405C2×D3—624no—
406C2×D3—1224no—
407C2×D3C23×C3×D314yes—
408C3×C4—24no—
409C3×C4—24no—
410C3×C4—24no—
411C3×C4—34no—
412C3×C4—38no—
413C3×C4—34no—
414C3×C4—38no—
415C3×C4—44no—
416C3×C4—64no—
417C3×C4—68no—
418C3×C4C22×D3214yes—
419C3×C4C23×C3×D314yes—
420C3×C4C23×C3×D314yes—
421Dic3C23×C3×D318yes—
422Dic3C23×C3×D314yes—
423C32—21no—
424C32—21no—
425C32—21no—
426C32—41no—
427C32C22×#23511yescharacteristic, complemented
428C32C22×#23911yescharacteristic, complemented
429C32C22×C3×#4111yescharacteristic, complemented
430C23—36no—
431C23—64no—
432C23—912no—
433C23—188no—
434C23C2×C3×D3211yescharacteristic
435Q8—316no—
436D4—932no—
437D4—916no—
438C2×C4—36no—
439C2×C4—312no—
440C2×C4—316no—
441C2×C4—932no—
442C2×C4C2×C3×D3216yes—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—26no—
447C2×C3—26no—
448C2×C3—26no—
449C2×C3—41no—
450C2×C3—46no—
451C2×C3—68no—
452C2×C3—64no—
453C2×C3—64no—
454C2×C3—188no—
455C2×C3C2×#379916yesdirect product factor
456C2×C3C2×C3×#23516yescomplemented
457C2×C3C2×C3×#23916yescomplemented
458C2×C3C23×D3211yescharacteristic
459C2×C3C24×C3×D311yescharacteristic
460C2×C3C24×C3×D311yescharacteristic
461D3—24no—
462D3—68no—
463D3—68no—
464D3—128no—
465C22—34no—
466C22—612no—
467C22—98no—
468C22—1824no—
469C22C3×#379914yesdirect product factor
470C22C22×C3×D3213yes—
471C4—34no—
472C4—38no—
473C4C22×C3×D3214yes—
474C3—21no—
475C3—21no—
476C3—21no—
477C3—41no—
478C3C22×#379911yescharacteristic, direct product factor
479C3C22×C3×#23511yescharacteristic, complemented
480C3C22×C3×#23911yescharacteristic, complemented
481C2—64no—
482C2—188no—
483C2C2×C3×#379916yesdirect product factor
484C2C23×C3×D3211yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC2×C32C25×C3
FrattiniC2C23×C3×D32
FittingC22×C33×C4C22
socleC23×C33C23
hypercenterC22×C3×C4D32
hypocenterC32C22×C3×#41
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4⋊(C3⋊D3)C2semidirect
2. C22×C32×C4×D3C2semidirect
3. C22×C32×Dic6C2semidirect
4. C22×C3×#626C2semidirect
5. C22×C3×#624C2semidirect
6. C2×C3×#3799C2direct (N × H)
9. C22×#3799C3direct (N × H)
13. C2×C3×C4⋊(C3⋊D3)C22semidirect
16. C2×C32×C4×D3C22semidirect
17. C2×C32×Dic6C22semidirect
18. C2×C3×#626C22semidirect
19. C2×C3×#624C22semidirect
20. C3×#3799C22direct (N × H)
23. C22×C4⋊(C3⋊D3)C2×C3semidirect
34. C22×C3×C4×D3D3semidirect
35. C22×C3×C4×D3C2×C3semidirect
37. C22×C3×Dic6D3semidirect
38. C22×C3×Dic6C2×C3semidirect
39. C22×#626C2×C3semidirect
40. C22×#624C2×C3semidirect
41. C2×#3799C2×C3direct (N × H)
48. C3×C4⋊(C3⋊D3)C23semidirect
51. C32×C4×D3C23semidirect
52. C32×Dic6C23semidirect
53. C3×#626C23semidirect
54. C3×#624C23semidirect
77. C2×C4⋊(C3⋊D3)C22×C3semidirect
94. C2×C3×C4×D3C2×D3semidirect
95. C2×C3×C4×D3C22×C3semidirect
97. C2×C3×Dic6C2×D3semidirect
98. C2×C3×Dic6C22×C3semidirect
99. C2×#626C22×C3semidirect
100. C2×#624C22×C3semidirect
101. #3799C22×C3direct (N × H)
102. C22×C33#41semidirect
130. C22×C4×D3C3×D3semidirect
131. C22×Dic6C3×D3semidirect
166. C4⋊(C3⋊D3)C23×C3semidirect
181. C3×C4×D3C22×D3semidirect
182. C3×C4×D3C23×C3semidirect
184. C3×Dic6C22×D3semidirect
185. C3×Dic6C23×C3semidirect
186. #626C23×C3semidirect
187. #624C23×C3semidirect
189. C2×C33C2×#41semidirect
235. C2×C4×D3C2×C3×D3semidirect
236. C2×Dic6C2×C3×D3semidirect
247. C22×C32#235semidirect
248. C22×C32#239semidirect
249. C22×C32C3×#41semidirect
288. C33C22×#41semidirect
343. C4×D3C22×C3×D3semidirect
344. Dic6C22×C3×D3semidirect
354. C2×C32C2×#235semidirect
355. C2×C32C2×#239semidirect
356. C2×C32C2×C3×#41semidirect
394. C22×C3#3799direct (N × H)
395. C22×C3C3×#235semidirect
396. C22×C3C3×#239semidirect
427. C32C22×#235semidirect
428. C32C22×#239semidirect
429. C32C22×C3×#41semidirect
455. C2×C3C2×#3799direct (N × H)
456. C2×C3C2×C3×#235semidirect
457. C2×C3C2×C3×#239semidirect
469. C22C3×#3799direct (N × H)
478. C3C22×#3799direct (N × H)
479. C3C22×C3×#235semidirect
480. C3C22×C3×#239semidirect
483. C2C2×C3×#3799direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊(C3⋊D3)2
C22×C32×C4×D32
C22×C32×Dic62
C22×C3×#6262
C22×C3×#6242
C2×C3×#37992
C22×C3×#2393
C22×C3×#2353
C22×#37993