Finite Group Databasebeta

← C22×C3×D3×Dic6 #13181529

Subgroups — C22×C3×D3×Dic6 #13181529

Order 1728.

4002 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#634C211yesmaximal, characteristic, complemented
2C22×C32×C4×D3C211yesmaximal, characteristic
3C22×C32×Dic6C211yesmaximal, characteristic, complemented
4C22×C3×#627C212yesmaximal, complemented
5C22×C3×D3×Dic3C212yesmaximal
6C2×C3×D3×Dic6C2124yesmaximal, direct product factor
7C22×C3×D3×Q8—31nomaximal
8C23×C3×Dic6—31nomaximal
9C22×D3×Dic6C311yesmaximal, characteristic, direct product factor
10C23×C32×D3C2211yescharacteristic
11C22×C33×C4C2211yescharacteristic
12C22×C3×C3⋊Dic3C2212yes—
13C2×C3×#634C22112yescomplemented
14C22×C32×Dic3C2212yes—
15C22×C32×Dic3C2211yescharacteristic
16C2×C32×C4×D3C22112yes—
17C2×C32×Dic6C22112yescomplemented
18C2×C3×#627C22124yescomplemented
19C2×C3×D3×Dic3C22124yes—
20C3×D3×Dic6C22164yesdirect product factor
21C22×C32×Q8—31no—
22C23×C32×C4—31no—
23C22×#634C2×C311yescharacteristic, complemented
24C23×C3×Dic3—32no—
25C2×C3×D3×Q8—324no—
26C22×C3×C4×D3—21no—
27C22×C3×C4×D3—32no—
28C22×C3×C4×D3D311yescharacteristic
29C22×C3×C4×D3C2×C311yescharacteristic
30C22×C3×Dic6—21no—
31C22×C3×Dic6—31no—
32C22×C3×Dic6—31no—
33C22×C3×Dic6—32no—
34C22×C3×Dic6—32no—
35C22×C3×Dic6—324no—
36C22×C3×Dic6—61no—
37C22×C3×Dic6D311yescharacteristic, direct product factor
38C22×C3×Dic6C2×C311yescharacteristic, complemented
39C22×#627C2×C312yescomplemented
40C22×D3×Dic3C2×C312yes—
41C2×D3×Dic6C2×C3124yesdirect product factor
42C23×C33C2311yescharacteristic
43C22×C32×D3Q818yescomplemented
44C22×C32×D3C2316yes—
45C2×C33×C4C2316yes—
46C2×C3×C3⋊Dic3C23112yes—
47C3×#634C23116yescomplemented
48C2×C32×Dic3C23112yes—
49C2×C32×Dic3C2316yes—
50C32×C4×D3C23116yes—
51C32×Dic6C23116yescomplemented
52C3×#627C23132yescomplemented
53C3×D3×Dic3C23132yes—
54C23×C3×Q8—91no—
55C22×D3×Q8—31no—
56C23×Dic6—31no—
57C24×C32—31no—
58C23×C3×D3—21no—
59C23×C3×D3C2×D311yescharacteristic
60C23×C3×D3C22×C311yescharacteristic
61C2×C32×Q8—312no—
62C22×C32×C4—21no—
63C22×C32×C4—21no—
64C22×C32×C4—21no—
65C22×C32×C4—31no—
66C22×C32×C4—32no—
67C22×C32×C4—312no—
68C22×C32×C4—41no—
69C22×C32×C4C2×D311yescharacteristic
70C22×C32×C4C2×D311yescharacteristic
71C22×C32×C4C22×C311yescharacteristic
72C22×C3⋊Dic3C22×C312yes—
73C2×#634C22×C3112yescomplemented
74C22×C3×Dic3—22no—
75C22×C3×Dic3—21no—
76C22×C3×Dic3—32no—
77C22×C3×Dic3—32no—
78C22×C3×Dic3—324no—
79C22×C3×Dic3—62no—
80C22×C3×Dic3C2×D311yescharacteristic
81C22×C3×Dic3C2×D312yes—
82C22×C3×Dic3C22×C312yes—
83C22×C3×Dic3C22×C311yescharacteristic
84C3×D3×Q8—364no—
85C2×C3×C4×D3—212no—
86C2×C3×C4×D3—324no—
87C2×C3×C4×D3C2×D3112yes—
88C2×C3×C4×D3C22×C3112yes—
89C2×C3×Dic6—212no—
90C2×C3×Dic6—312no—
91C2×C3×Dic6—312no—
92C2×C3×Dic6—324no—
93C2×C3×Dic6—324no—
94C2×C3×Dic6—364no—
95C2×C3×Dic6—612no—
96C2×C3×Dic6C2×D3112yesdirect product factor
97C2×C3×Dic6C22×C3112yescomplemented
98C2×#627C22×C3124yescomplemented
99C2×D3×Dic3C22×C3124yes—
100D3×Dic6C22×C3164yesdirect product factor
101C22×C33C2×Q814yescomplemented
102C22×C33C2413yes—
103C2×C32×D3C2×Q8124yescomplemented
104C2×C32×D3C2414yes—
105C33×C4C2414yes—
106C3×C3⋊Dic3C2418yes—
107C32×Dic3C2418yes—
108C32×Dic3C2414yes—
109C22×C3×Q8—31no—
110C22×C3×Q8—31no—
111C22×C3×Q8—61no—
112C22×C3×Q8—92no—
113C22×C3×Q8—924no—
114C22×C3×Q8—91no—
115C23×C3×C4—31no—
116C23×C3×C4—31no—
117C23×C3×C4—61no—
118C23×C3×C4—92no—
119C23×Dic3—32no—
120C2×D3×Q8—324no—
121C22×C4×D3—32no—
122C22×C4×D3C3×D311yescharacteristic
123C22×Dic6—31no—
124C22×Dic6—31no—
125C22×Dic6—32no—
126C22×Dic6—32no—
127C22×Dic6—324no—
128C22×Dic6—61no—
129C22×Dic6C3×D311yescharacteristic, direct product factor
130C23×C32—21no—
131C23×C32—21no—
132C23×C32—21no—
133C23×C32—38no—
134C23×C32—36no—
135C23×C32—41no—
136C23×C32C22×D311yescharacteristic
137C23×C32C22×D311yescharacteristic
138C23×C32C23×C311yescharacteristic
139C22×C3×D3—28no—
140C22×C3×D3—26no—
141C22×C3×D3Dic618yesdirect product factor
142C22×C3×D3C3×Q818yescomplemented
143C22×C3×D3C22×D316yes—
144C22×C3×D3C23×C316yes—
145C32×Q8—316no—
146C2×C32×C4—26no—
147C2×C32×C4—26no—
148C2×C32×C4—26no—
149C2×C32×C4—36no—
150C2×C32×C4—312no—
151C2×C32×C4—316no—
152C2×C32×C4—46no—
153C2×C32×C4C22×D316yes—
154C2×C32×C4C22×D316yes—
155C2×C32×C4C23×C316yes—
156C2×C3⋊Dic3C23×C3112yes—
157#634C23×C3116yescomplemented
158C2×C3×Dic3—212no—
159C2×C3×Dic3—26no—
160C2×C3×Dic3—312no—
161C2×C3×Dic3—312no—
162C2×C3×Dic3—332no—
163C2×C3×Dic3—612no—
164C2×C3×Dic3C22×D316yes—
165C2×C3×Dic3C22×D3112yes—
166C2×C3×Dic3C23×C3112yes—
167C2×C3×Dic3C23×C316yes—
168C3×C4×D3—216no—
169C3×C4×D3—332no—
170C3×C4×D3C22×D3116yes—
171C3×C4×D3C23×C3116yes—
172C3×Dic6—216no—
173C3×Dic6—316no—
174C3×Dic6—316no—
175C3×Dic6—332no—
176C3×Dic6—332no—
177C3×Dic6—616no—
178C3×Dic6C22×D3116yesdirect product factor
179C3×Dic6C23×C3116yescomplemented
180#627C23×C3132yescomplemented
181D3×Dic3C23×C3132yes—
182C23×Q8—91noSylow
183C2×C33C22×Q816yescomplemented
184C2×C33C2511yescharacteristic
185C32×D3C22×Q818yescomplemented
186C24×C3—31no—
187C24×C3—31no—
188C24×C3—61no—
189C23×D3C2×C3×D311yescharacteristic
190C2×C3×Q8—312no—
191C2×C3×Q8—312no—
192C2×C3×Q8—612no—
193C2×C3×Q8—924no—
194C2×C3×Q8—964no—
195C2×C3×Q8—912no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—31no—
200C22×C3×C4—32no—
201C22×C3×C4—31no—
202C22×C3×C4—32no—
203C22×C3×C4—312no—
204C22×C3×C4—312no—
205C22×C3×C4—41no—
206C22×C3×C4—612no—
207C22×C3×C4—61no—
208C22×C3×C4—62no—
209C22×C3×C4—924no—
210C22×C3×C4—92no—
211C22×C3×C4D3211yescharacteristic
212C22×C3×C4C2×C3×D311yescharacteristic
213C22×C3×C4C2×C3×D311yescharacteristic
214C22×Dic3—32no—
215C22×Dic3—32no—
216C22×Dic3—324no—
217C22×Dic3—62no—
218C22×Dic3C2×C3×D311yescharacteristic
219C22×Dic3C2×C3×D312yes—
220D3×Q8—364no—
221C2×C4×D3—324no—
222C2×C4×D3C2×C3×D3112yes—
223C2×Dic6—312no—
224C2×Dic6—312no—
225C2×Dic6—324no—
226C2×Dic6—324no—
227C2×Dic6—364no—
228C2×Dic6—612no—
229C2×Dic6C2×C3×D3112yesdirect product factor
230C22×C32—23no—
231C22×C32—23no—
232C22×C32—23no—
233C22×C32—24no—
234C22×C32—24no—
235C22×C32—24no—
236C22×C32—324no—
237C22×C32—34no—
238C22×C32—44no—
239C22×C32—43no—
240C22×C32C2×Dic614yescomplemented
241C22×C32D3×Q814yescomplemented
242C22×C32C2×C3×Q814yescomplemented
243C22×C32C23×D313yes—
244C22×C32C23×D313yes—
245C22×C32C24×C313yes—
246C2×C3×D3—24no—
247C2×C3×D3—224no—
248C2×C3×D3C2×Dic6124yesdirect product factor
249C2×C3×D3C2×C3×Q8124yescomplemented
250C2×C3×D3C23×D314yes—
251C2×C3×D3C24×C314yes—
252C32×C4—24no—
253C32×C4—24no—
254C32×C4—24no—
255C32×C4—34no—
256C32×C4—38no—
257C32×C4—44no—
258C32×C4C23×D314yes—
259C32×C4C23×D314yes—
260C32×C4C24×C314yes—
261C3⋊Dic3C24×C318yes—
262C3×Dic3—28no—
263C3×Dic3—24no—
264C3×Dic3—38no—
265C3×Dic3—38no—
266C3×Dic3—68no—
267C3×Dic3C23×D314yes—
268C3×Dic3C23×D318yes—
269C3×Dic3C24×C318yes—
270C3×Dic3C24×C314yes—
271C22×Q8—31no—
272C22×Q8—92no—
273C22×Q8—924no—
274C22×Q8—91no—
275C23×C4—31no—
276C23×C4—92no—
277C33C23×Q811yescharacteristic, complemented, Sylow
278C23×C3—21no—
279C23×C3—21no—
280C23×C3—21no—
281C23×C3—36no—
282C23×C3—36no—
283C23×C3—38no—
284C23×C3—38no—
285C23×C3—41no—
286C23×C3—66no—
287C23×C3—68no—
288C23×C3C2×D3211yescharacteristic
289C23×C3C22×C3×D311yescharacteristic
290C23×C3C22×C3×D311yescharacteristic
291C22×D3C3×Dic618yesdirect product factor
292C22×D3C22×C3×D316yes—
293C3×Q8—316no—
294C3×Q8—316no—
295C3×Q8—616no—
296C3×Q8—932no—
297C3×Q8—916no—
298C2×C3×C4—26no—
299C2×C3×C4—26no—
300C2×C3×C4—26no—
301C2×C3×C4—36no—
302C2×C3×C4—312no—
303C2×C3×C4—36no—
304C2×C3×C4—312no—
305C2×C3×C4—316no—
306C2×C3×C4—316no—
307C2×C3×C4—46no—
308C2×C3×C4—616no—
309C2×C3×C4—66no—
310C2×C3×C4—612no—
311C2×C3×C4—932no—
312C2×C3×C4—912no—
313C2×C3×C4C2×D3216yes—
314C2×C3×C4C22×C3×D316yes—
315C2×C3×C4C22×C3×D316yes—
316C2×Dic3—312no—
317C2×Dic3—312no—
318C2×Dic3—332no—
319C2×Dic3—612no—
320C2×Dic3C22×C3×D316yes—
321C2×Dic3C22×C3×D3112yes—
322C4×D3—332no—
323C4×D3C22×C3×D3116yes—
324Dic6—316no—
325Dic6—316no—
326Dic6—332no—
327Dic6—332no—
328Dic6—616no—
329Dic6C22×C3×D3116yesdirect product factor
330C2×C32—26no—
331C2×C32—26no—
332C2×C32—26no—
333C2×C32—21no—
334C2×C32—21no—
335C2×C32—21no—
336C2×C32—38no—
337C2×C32—46no—
338C2×C32—41no—
339C2×C32C22×Dic616yescomplemented
340C2×C32C2×D3×Q816yescomplemented
341C2×C32C22×C3×Q816yescomplemented
342C2×C32C24×D311yescharacteristic
343C2×C32C24×D311yescharacteristic
344C2×C32C25×C311yescharacteristic
345C3×D3—28no—
346C3×D3C22×Dic618yesdirect product factor
347C3×D3C22×C3×Q818yescomplemented
348C24—31no—
349C2×Q8—312no—
350C2×Q8—924no—
351C2×Q8—964no—
352C2×Q8—912no—
353C22×C4—31no—
354C22×C4—32no—
355C22×C4—312no—
356C22×C4—924no—
357C22×C4—92no—
358C22×C4C3×D3211yescharacteristic
359C22×C3—23no—
360C22×C3—23no—
361C22×C3—23no—
362C22×C3—24no—
363C22×C3—24no—
364C22×C3—24no—
365C22×C3—324no—
366C22×C3—324no—
367C22×C3—34no—
368C22×C3—34no—
369C22×C3—43no—
370C22×C3—44no—
371C22×C3—64no—
372C22×C3—624no—
373C22×C3D3×Dic614yesdirect product factor
374C22×C3C2×C3×Dic614yescomplemented
375C22×C3C3×D3×Q814yescomplemented
376C22×C3C22×D3213yes—
377C22×C3C23×C3×D313yes—
378C22×C3C23×C3×D313yes—
379C2×D3C2×C3×Dic6124yesdirect product factor
380C2×D3C23×C3×D314yes—
381C3×C4—24no—
382C3×C4—24no—
383C3×C4—24no—
384C3×C4—34no—
385C3×C4—38no—
386C3×C4—34no—
387C3×C4—38no—
388C3×C4—44no—
389C3×C4—64no—
390C3×C4—68no—
391C3×C4—98no—
392C3×C4C22×D3214yes—
393C3×C4C23×C3×D314yes—
394C3×C4C23×C3×D314yes—
395Dic3—38no—
396Dic3—38no—
397Dic3—68no—
398Dic3C23×C3×D314yes—
399Dic3C23×C3×D318yes—
400C32—21no—
401C32—21no—
402C32—21no—
403C32—41no—
404C32C23×Dic611yescharacteristic, complemented
405C32C22×D3×Q811yescharacteristic, complemented
406C32C23×C3×Q811yescharacteristic, complemented
407C23—36no—
408C23—38no—
409C23C2×C3×D3211yescharacteristic
410Q8—316no—
411Q8—932no—
412Q8—916no—
413C2×C4—36no—
414C2×C4—312no—
415C2×C4—316no—
416C2×C4—932no—
417C2×C4—912no—
418C2×C4C2×C3×D3216yes—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—21no—
422C2×C3—26no—
423C2×C3—26no—
424C2×C3—26no—
425C2×C3—38no—
426C2×C3—38no—
427C2×C3—41no—
428C2×C3—46no—
429C2×C3—68no—
430C2×C3C2×D3×Dic616yesdirect product factor
431C2×C3C22×C3×Dic616yescomplemented
432C2×C3C2×C3×D3×Q816yescomplemented
433C2×C3C23×D3211yescharacteristic
434C2×C3C24×C3×D311yescharacteristic
435C2×C3C24×C3×D311yescharacteristic
436D3C22×C3×Dic618yesdirect product factor
437C22—34no—
438C22—324no—
439C22C3×D3×Dic614yesdirect product factor
440C22C22×C3×D3213yes—
441C4—34no—
442C4—38no—
443C4—98no—
444C4C22×C3×D3214yes—
445C3—21no—
446C3—21no—
447C3—21no—
448C3—41no—
449C3C22×D3×Dic611yescharacteristic, direct product factor
450C3C23×C3×Dic611yescharacteristic, complemented
451C3C22×C3×D3×Q811yescharacteristic, complemented
452C2—38no—
453C2C2×C3×D3×Dic616yesdirect product factor
454C2C23×C3×D3211yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC2×C32C25×C3
FrattiniC2C23×C3×D32
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC23×C32×D3C22
hypercenterC22×C3×C4D32
hypocenterC32C23×C3×Q8
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×#634C2semidirect
3. C22×C32×Dic6C2semidirect
4. C22×C3×#627C2semidirect
6. C2×C3×D3×Dic6C2direct (N × H)
9. C22×D3×Dic6C3direct (N × H)
13. C2×C3×#634C22semidirect
17. C2×C32×Dic6C22semidirect
18. C2×C3×#627C22semidirect
20. C3×D3×Dic6C22direct (N × H)
23. C22×#634C2×C3semidirect
37. C22×C3×Dic6D3direct (N × H)
38. C22×C3×Dic6C2×C3semidirect
39. C22×#627C2×C3semidirect
41. C2×D3×Dic6C2×C3direct (N × H)
43. C22×C32×D3Q8semidirect
47. C3×#634C23semidirect
51. C32×Dic6C23semidirect
52. C3×#627C23semidirect
73. C2×#634C22×C3semidirect
96. C2×C3×Dic6C2×D3direct (N × H)
97. C2×C3×Dic6C22×C3semidirect
98. C2×#627C22×C3semidirect
100. D3×Dic6C22×C3direct (N × H)
101. C22×C33C2×Q8semidirect
103. C2×C32×D3C2×Q8semidirect
129. C22×Dic6C3×D3direct (N × H)
141. C22×C3×D3Dic6direct (N × H)
142. C22×C3×D3C3×Q8semidirect
157. #634C23×C3semidirect
178. C3×Dic6C22×D3direct (N × H)
179. C3×Dic6C23×C3semidirect
180. #627C23×C3semidirect
183. C2×C33C22×Q8semidirect
185. C32×D3C22×Q8semidirect
229. C2×Dic6C2×C3×D3direct (N × H)
240. C22×C32C2×Dic6semidirect
241. C22×C32D3×Q8semidirect
242. C22×C32C2×C3×Q8semidirect
248. C2×C3×D3C2×Dic6direct (N × H)
249. C2×C3×D3C2×C3×Q8semidirect
277. C33C23×Q8semidirect
291. C22×D3C3×Dic6direct (N × H)
329. Dic6C22×C3×D3direct (N × H)
339. C2×C32C22×Dic6semidirect
340. C2×C32C2×D3×Q8semidirect
341. C2×C32C22×C3×Q8semidirect
346. C3×D3C22×Dic6direct (N × H)
347. C3×D3C22×C3×Q8semidirect
373. C22×C3D3×Dic6direct (N × H)
374. C22×C3C2×C3×Dic6semidirect
375. C22×C3C3×D3×Q8semidirect
379. C2×D3C2×C3×Dic6direct (N × H)
404. C32C23×Dic6semidirect
405. C32C22×D3×Q8semidirect
406. C32C23×C3×Q8semidirect
430. C2×C3C2×D3×Dic6direct (N × H)
431. C2×C3C22×C3×Dic6semidirect
432. C2×C3C2×C3×D3×Q8semidirect
436. D3C22×C3×Dic6direct (N × H)
439. C22C3×D3×Dic6direct (N × H)
449. C3C22×D3×Dic6direct (N × H)
450. C3C23×C3×Dic6semidirect
451. C3C22×C3×D3×Q8semidirect
453. C2C2×C3×D3×Dic6direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6342
C22×C32×C4×D32
C22×C32×Dic62
C22×C3×#6272
C22×C3×D3×Dic32
C2×C3×D3×Dic62
C22×C3×D3×Q83
C23×C3×Dic63
C22×D3×Dic63