Finite Group Databasebeta

← D3×#64162 #13178105

Subgroups — D3×#64162 #13178105

Order 1728.

4492 conjugacy classes of subgroups; only the 476 normal subgroups are shown.

normal subgroups only

This group has 4492 conjugacy classes of subgroups — too many to draw. Only its 476 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C32×D3C211yesmaximal, characteristic, complemented
2C2×D3×#638C213yesmaximal, complemented
3#11723994C211yesmaximal, characteristic, complemented
4C3×#64162C211yesmaximal, characteristic, complemented
5#11723630C213yesmaximal, complemented
6#11723629C213yesmaximal, complemented
7D3×#3757C213yesmaximal, complemented
8C2×#64162—31nomaximal
9D3×#953—34nomaximal
10C24×C33C2211yescharacteristic, complemented
11C23×C32×D3—212no—
12C23×C32×D3C2213yes—
13C23×C32×D3C2213yescomplemented
14C22×D3×C3⋊D3C2211yescharacteristic, complemented
15C2×#6312C2213yescomplemented
16C2×C3×#638C2213yescomplemented
17D3×#638—212no—
18C2×#6291C2213yescomplemented
19C2×#6290C2213yescomplemented
20C2×D3×C3⋊Dic3C2213yescomplemented
21#88864C2213yescomplemented
22C3×#3757C2213yescomplemented
23(C2×D3)⋊(C3⋊Dic3)C2216yescomplemented
24C25×C32—31no—
25C24×C3×D3D314yescomplemented
26C22×#638—33no—
27C2×D3×#67—312no—
28#64162—33no—
29#64162—33no—
30#64162—31no—
31#64162—34no—
32#64162—64no—
33#64162D311yescharacteristic, direct product factor
34C2×#3757—33no—
35C3×#953—34no—
36#63977—312no—
37#63976—312no—
38D3×#217—312no—
39C23×C33—26no—
40C23×C33C2313yes—
41C22×C3⋊(C3⋊D3)C2311yescharacteristic, complemented
42C22×C3×C3⋊D3C2311yescharacteristic, complemented
43C22×C32×D3—232no—
44C22×C32×D3—26no—
45C22×C32×D3—212no—
46C22×C32×D3—212no—
47C22×C32×D3D4112yescomplemented
48C22×C32×D3C2313yes—
49C22×C32×D3C2311yescharacteristic
50C2×D3×C3⋊D3—26no—
51#6312—26no—
52C2×C3⋊(C3⋊Dic3)C2313yescomplemented
53C3×#638—26no—
54C2×C3×C3⋊Dic3C2313yescomplemented
55#6291—26no—
56#6290—26no—
57D3×C3⋊Dic3—26no—
58C2×#953—94no—
59D3×#120—91no—
60C24×C32—24no—
61C24×C32—33no—
62C24×C32—33no—
63C24×C32—612no—
64C24×C32C2×D314yescomplemented
65C24×C32C2×D311yescharacteristic, complemented
66C23×C3⋊D3—31no—
67C23×C3×D3—248no—
68C23×C3×D3C2×D3112yes—
69C23×C3×D3C2×D3112yescomplemented
70C22×D32—34no—
71C2×#638—33no—
72C2×#638—312no—
73C2×#638—33no—
74C2×#638—33no—
75C2×#638—612no—
76C2×#638—612no—
77C2×#638C2×D313yescomplemented
78C22×C3⋊Dic3—33no—
79C2×C3×#67—312no—
80D3×#67—648no—
81C2×#626—312no—
82C2×#625—312no—
83C2×D3×Dic3—312no—
84#3757—33no—
85#3757—312no—
86#3757—36no—
87#3757—612no—
88#3757C2×D313yescomplemented
89C3×#217—312no—
90(C2×D3)⋊Dic3—324no—
91C22×C33—28no—
92C22×C33—26no—
93C22×C33C2×D416yescomplemented
94C22×C33C2411yescharacteristic
95C2×C3⋊(C3⋊D3)—23no—
96C2×C3×C3⋊D3—23no—
97C2×C32×D3—224no—
98C2×C32×D3—26no—
99C2×C32×D3—224no—
100C2×C32×D3C2×D416yes—
101C2×C32×D3C2×D416yescomplemented
102D3×C3⋊D3—44no—
103C3⋊(C3⋊Dic3)—23no—
104C3×C3⋊Dic3—23no—
105C25×C3—34no—
106C24×D3C3⋊D311yescharacteristic, complemented
107C22×#67—912no—
108C2×D3×D4—93no—
109C3×#120—91no—
110#953—34no—
111#953—912no—
112#953—912no—
113#953—94no—
114#953—91no—
115#953—184no—
116C2×#217—912no—
117#937—93no—
118#882—93no—
119D3×C22⋊C4—93no—
120C23×C32—26no—
121C23×C32—212no—
122C23×C32—224no—
123C23×C32—31no—
124C23×C32—312no—
125C23×C32—33no—
126C23×C32—424no—
127C23×C32—612no—
128C23×C32—612no—
129C23×C32—632no—
130C23×C32—66no—
131C23×C32C22×D3112yes—
132C23×C32C22×D313yes—
133C22×C3⋊D3—31no—
134C22×C3⋊D3—34no—
135C22×C3⋊D3—64no—
136C22×C3⋊D3—66no—
137C22×C3⋊D3C22×D311yescharacteristic, complemented
138C22×C3×D3—2128no—
139C22×C3×D3—224no—
140C22×C3×D3—248no—
141C22×C3×D3—248no—
142C22×C3×D3—34no—
143C22×C3×D3#67148yescomplemented
144C22×C3×D3C22×D3112yes—
145C22×C3×D3C22×D314yes—
146C2×D32—624no—
147#638—26no—
148#638—66no—
149#638—66no—
150#638—66no—
151#638—624no—
152#638—1224no—
153C2×C3⋊Dic3—33no—
154C2×C3⋊Dic3—312no—
155C2×C3⋊Dic3—612no—
156C2×C3⋊Dic3—66no—
157C2×C3⋊Dic3C22×D313yescomplemented
158C3×#67—624no—
159C2×C3×Dic3—312no—
160#626—624no—
161#625—624no—
162D3×Dic3—624no—
163C2×#120—271noSylow
164C2×C33—26no—
165C2×C33C22×D413yes—
166C3⋊(C3⋊D3)—41no—
167C3×C3⋊D3—41no—
168C32×D3—26no—
169C32×D3#12014yescomplemented
170C24×C3—24no—
171C24×C3—312no—
172C24×C3—312no—
173C24×C3—648no—
174C24×C3D3214yescomplemented
175C24×C3C2×C3⋊D311yescharacteristic, complemented
176C23×D3—212no—
177C23×D3—94no—
178C23×D3—91no—
179C23×D3C2×C3⋊D313yes—
180C23×D3C2×C3⋊D313yescomplemented
181C2×C3×D4—93no—
182C2×#67—312no—
183C2×#67—912no—
184C2×#67—912no—
185C2×#67—912no—
186C2×#67—93no—
187C2×#67—93no—
188C2×#67—1812no—
189C2×#67—1848no—
190C22×Dic3—912no—
191D3×D4—1812no—
192C2×D12—93no—
193C2×C4×D3—93no—
194C3×C22⋊C4—93no—
195#217—312no—
196#217—924no—
197#217—912no—
198#217—93no—
199#217—1812no—
200(C2×D3)⋊C4—96no—
201C22×C32—28no—
202C22×C32—26no—
203C22×C32—24no—
204C22×C32—224no—
205C22×C32—232no—
206C22×C32—224no—
207C22×C32—36no—
208C22×C32—36no—
209C22×C32—424no—
210C22×C32—432no—
211C22×C32—624no—
212C22×C32—624no—
213C22×C32—66no—
214C22×C32D3×D416yescomplemented
215C22×C32C2×#67124yescomplemented
216C22×C32C23×D314yes—
217C22×C32C23×D311yescharacteristic
218C2×C3⋊D3—23no—
219C2×C3⋊D3—63no—
220C2×C3⋊D3—612no—
221C2×C3⋊D3—1212no—
222C2×C3⋊D3—124no—
223C2×C3×D3—296no—
224C2×C3×D3—224no—
225C2×C3×D3—296no—
226C2×C3×D3—612no—
227C2×C3×D3C2×#67124yes—
228C2×C3×D3C2×#67124yescomplemented
229D32—1216no—
230C3⋊Dic3—23no—
231C3⋊Dic3—63no—
232C3⋊Dic3—612no—
233C3⋊Dic3—1212no—
234C3×Dic3—612no—
235C25—31no—
236C22×D4—273no—
237#120—91no—
238#120—273no—
239#120—273no—
240#120—271no—
241C2×C22⋊C4—273no—
242C33C2×#12011yescharacteristic, complemented, Sylow
243C23×C3—224no—
244C23×C3—26no—
245C23×C3—212no—
246C23×C3—34no—
247C23×C3—312no—
248C23×C3—348no—
249C23×C3—424no—
250C23×C3—648no—
251C23×C3—648no—
252C23×C3—6128no—
253C23×C3—624no—
254C23×C3—91no—
255C23×C3C2×D32112yes—
256C23×C3C22×C3⋊D313yes—
257C22×D3—232no—
258C22×D3—26no—
259C22×D3—212no—
260C22×D3—212no—
261C22×D3—34no—
262C22×D3—94no—
263C22×D3—91no—
264C22×D3—184no—
265C22×D3—1824no—
266C22×D3—186no—
267C22×D3#638112yescomplemented
268C22×D3C22×C3⋊D313yes—
269C22×D3C22×C3⋊D311yescharacteristic
270C3×D4—186no—
271C2×C3×C4—93no—
272#67—624no—
273#67—1824no—
274#67—1824no—
275#67—1824no—
276#67—186no—
277#67—186no—
278#67—3624no—
279C2×Dic3—312no—
280C2×Dic3—912no—
281C2×Dic3—93no—
282C2×Dic3—1812no—
283C2×Dic3—1824no—
284D12—186no—
285C4×D3—186no—
286C2×C32—26no—
287C2×C32—212no—
288C2×C32—224no—
289C2×C32—34no—
290C2×C32—424no—
291C2×C32—66no—
292C2×C32C2×D3×D413yes—
293C2×C32C22×#67112yes—
294C3⋊D3—41no—
295C3⋊D3—121no—
296C3⋊D3—124no—
297C3⋊D3—244no—
298C3×D3—224no—
299C3×D3—124no—
300C3×D3#953116yescomplemented
301C24—33no—
302C24—33no—
303C24—612no—
304C24—271no—
305C24D3×C3⋊D311yescharacteristic, complemented
306C2×D4—93no—
307C2×D4—273no—
308C2×D4—273no—
309C2×D4—273no—
310C2×D4—5412no—
311C22×C4—273no—
312C22⋊C4—93no—
313C22⋊C4—276no—
314C22⋊C4—273no—
315C22×C3—28no—
316C22×C3—232no—
317C22×C3—224no—
318C22×C3—26no—
319C22×C3—24no—
320C22×C3—224no—
321C22×C3—324no—
322C22×C3—324no—
323C22×C3—424no—
324C22×C3—432no—
325C22×C3—696no—
326C22×C3—696no—
327C22×C3—624no—
328C22×C3—183no—
329C22×C3D3×#67124yescomplemented
330C22×C3C2×#63816yescomplemented
331C22×C3C22×D3214yes—
332C22×C3C23×C3⋊D311yescharacteristic
333C2×D3—224no—
334C2×D3—26no—
335C2×D3—224no—
336C2×D3—612no—
337C2×D3—1812no—
338C2×D3—183no—
339C2×D3—3612no—
340C2×D3—3616no—
341C2×D3—364no—
342C2×D3C2×#63816yes—
343C2×D3C2×#63816yescomplemented
344C3×C4—183no—
345Dic3—612no—
346Dic3—1812no—
347Dic3—183no—
348Dic3—3612no—
349C32—24no—
350C32D3×#12011yescharacteristic, complemented
351C32C2×#95314yescomplemented
352C23—26no—
353C23—31no—
354C23—33no—
355C23—312no—
356C23—612no—
357C23—612no—
358C23—632no—
359C23—66no—
360C23—91no—
361C23—271no—
362C23—546no—
363C23C2×D3×C3⋊D313yes—
364D4—186no—
365D4—546no—
366D4—546no—
367D4—546no—
368C2×C4—93no—
369C2×C4—273no—
370C2×C4—546no—
371C2×C3—224no—
372C2×C3—26no—
373C2×C3—212no—
374C2×C3—316no—
375C2×C3—424no—
376C2×C3—624no—
377C2×C3—361no—
378C2×C3C2×D3×#67112yes—
379C2×C3C22×#63813yes—
380D3—26no—
381D3—124no—
382D3—364no—
383D3—361no—
384D3—724no—
385D3#6416214yesdirect product factor
386C22—28no—
387C22—26no—
388C22—36no—
389C22—36no—
390C22—624no—
391C22—624no—
392C22—66no—
393C22—183no—
394C22—543no—
395C22—1084no—
396C22D3×#63816yescomplemented
397C22C22×D3×C3⋊D311yescharacteristic
398C4—183no—
399C4—543no—
400C3—24no—
401C3D3×#95314yescomplemented
402C3C2×#6416211yescharacteristic, complemented
403C2—26no—
404C2—34no—
405C2—66no—
406C2—361no—
407C2—1081no—
408C2C2×D3×#63813yes—
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC24×C33C22
hypercenterC24D3×C3⋊D3
hypocenterC33C2×#120
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C24×C32×D3C2semidirect
2. C2×D3×#638C2semidirect
3. #11723994C2semidirect
4. C3×#64162C2semidirect
5. #11723630C2semidirect
6. #11723629C2semidirect
7. D3×#3757C2semidirect
10. C24×C33C22semidirect
13. C23×C32×D3C22semidirect
14. C22×D3×C3⋊D3C22semidirect
15. C2×#6312C22semidirect
16. C2×C3×#638C22semidirect
18. C2×#6291C22semidirect
19. C2×#6290C22semidirect
20. C2×D3×C3⋊Dic3C22semidirect
21. #88864C22semidirect
22. C3×#3757C22semidirect
23. (C2×D3)⋊(C3⋊Dic3)C22semidirect
25. C24×C3×D3D3semidirect
33. #64162D3direct (N × H)
41. C22×C3⋊(C3⋊D3)C23semidirect
42. C22×C3×C3⋊D3C23semidirect
47. C22×C32×D3D4semidirect
52. C2×C3⋊(C3⋊Dic3)C23semidirect
54. C2×C3×C3⋊Dic3C23semidirect
64. C24×C32C2×D3semidirect
65. C24×C32C2×D3semidirect
69. C23×C3×D3C2×D3semidirect
77. C2×#638C2×D3semidirect
88. #3757C2×D3semidirect
93. C22×C33C2×D4semidirect
101. C2×C32×D3C2×D4semidirect
106. C24×D3C3⋊D3semidirect
137. C22×C3⋊D3C22×D3semidirect
143. C22×C3×D3#67semidirect
157. C2×C3⋊Dic3C22×D3semidirect
169. C32×D3#120semidirect
174. C24×C3D32semidirect
175. C24×C3C2×C3⋊D3semidirect
180. C23×D3C2×C3⋊D3semidirect
214. C22×C32D3×D4semidirect
215. C22×C32C2×#67semidirect
228. C2×C3×D3C2×#67semidirect
242. C33C2×#120semidirect
267. C22×D3#638semidirect
300. C3×D3#953semidirect
305. C24D3×C3⋊D3semidirect
329. C22×C3D3×#67semidirect
330. C22×C3C2×#638semidirect
343. C2×D3C2×#638semidirect
350. C32D3×#120semidirect
351. C32C2×#953semidirect
385. D3#64162direct (N × H)
396. C22D3×#638semidirect
401. C3D3×#953semidirect
402. C3C2×#64162semidirect

Maximal subgroups

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

subgroupindex
C24×C32×D32
C2×D3×#6382
#117239942
C3×#641622
#117236302
#117236292
D3×#37572
C2×#641623
D3×#9533