Finite Group Databasebeta

← #13178108

Subgroups — #13178108

Order 1728.

6964 conjugacy classes of subgroups; only the 336 normal subgroups are shown.

normal subgroups only

This group has 6964 conjugacy classes of subgroups — too many to draw. Only its 336 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×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C2×#89039C213yesmaximal, complemented
3C3×#64162C211yesmaximal, characteristic, complemented
4C32×#953C211yesmaximal, characteristic, complemented
5#11723639C213yesmaximal, complemented
6#11723638C213yesmaximal, complemented
7#11723627C213yesmaximal, complemented
8C3⋊D3×#120—31nomaximal
9#10595527—34nomaximal
10C24×C33C2211yescharacteristic, complemented
11C23×C3⋊(C3⋊D3)—212no—
12C23×C3⋊(C3⋊D3)C2213yes—
13C23×C3⋊(C3⋊D3)C2213yescomplemented
14C22×D3×C3⋊D3C2211yescharacteristic, complemented
15C2×C3×#638C2213yescomplemented
16C2×C32×#67C2213yescomplemented
17#89039—212no—
18C2×#6292C2213yescomplemented
19C2×#6291C2213yescomplemented
20C2×#6289C2213yescomplemented
21C3×#3757C2213yescomplemented
22C32×#217C2213yescomplemented
23#88804C2216yescomplemented
24C24×C3⋊D3—31no—
25C24×C3⋊D3—34no—
26C24×C3⋊D3—64no—
27C2×D4×C3⋊D3—33no—
28C2×#3811—312no—
29C32×#120—31no—
30#64162D311yescharacteristic, complemented
31#64146—33no—
32#64091—33no—
33C22⋊C4×C3⋊D3—33no—
34C3×#953—34no—
35C3×#953D314yescomplemented
36#63981—312no—
37#63981—312no—
38#63974—312no—
39C23×C33—26no—
40C23×C33C2313yes—
41C22×C3⋊(C3⋊D3)—232no—
42C22×C3⋊(C3⋊D3)—26no—
43C22×C3⋊(C3⋊D3)—212no—
44C22×C3⋊(C3⋊D3)—212no—
45C22×C3⋊(C3⋊D3)D4112yescomplemented
46C22×C3⋊(C3⋊D3)C2313yes—
47C22×C3⋊(C3⋊D3)C2311yescharacteristic
48C22×C3×C3⋊D3C2311yescharacteristic, complemented
49C22×C32×D3C2311yescharacteristic, complemented
50C2×D3×C3⋊D3—26no—
51C3×#638—26no—
52C2×C3×C3⋊Dic3C2313yescomplemented
53C32×#67—26no—
54C2×C32×Dic3C2313yescomplemented
55#6292—26no—
56#6291—26no—
57#6289—26no—
58D3×#120—94no—
59D3×#120—91no—
60C24×C32—24no—
61C24×C32C2×D314yescomplemented
62C24×C32C2×D311yescharacteristic, complemented
63C23×C3⋊D3—33no—
64C23×C3⋊D3—312no—
65C23×C3⋊D3—33no—
66C23×C3⋊D3—312no—
67C23×C3⋊D3—31no—
68C23×C3⋊D3—612no—
69C23×C3⋊D3—648no—
70C23×C3⋊D3—612no—
71C23×C3⋊D3—612no—
72C23×C3⋊D3—696no—
73C22×D32—34no—
74C2×C32×D4—33no—
75C2×#638—33no—
76C2×#638C2×D313yescomplemented
77D4×C3⋊D3—612no—
78C2×C4⋊(C3⋊D3)—33no—
79C2×C4×C3⋊D3—33no—
80C2×C3×#67—312no—
81C2×C3×#67C2×D3112yescomplemented
82#3811—648no—
83C2×#626—312no—
84C2×#626—312no—
85C2×#624—312no—
86C32×C22⋊C4—33no—
87#3757C2×D313yescomplemented
88#3752—36no—
89C3×#217—312no—
90C3×#217C2×D3112yescomplemented
91#3722—324no—
92C22×C33—28no—
93C22×C33—26no—
94C22×C33C2×D416yescomplemented
95C22×C33C2411yescharacteristic
96C2×C3⋊(C3⋊D3)—224no—
97C2×C3⋊(C3⋊D3)—26no—
98C2×C3⋊(C3⋊D3)—224no—
99C2×C3⋊(C3⋊D3)C2×D416yes—
100C2×C3⋊(C3⋊D3)C2×D416yescomplemented
101C2×C3×C3⋊D3—23no—
102C2×C32×D3—23no—
103D3×C3⋊D3—44no—
104C3×C3⋊Dic3—23no—
105C32×Dic3—23no—
106C24×D3—94no—
107C24×D3—91no—
108C24×D3—184no—
109C2×D3×D4—912no—
110C2×D3×D4—93no—
111C3×#120—34no—
112C3×#120—91no—
113#953—34no—
114#953C3⋊D311yescharacteristic, complemented
115#937—912no—
116#937—93no—
117#882—912no—
118#882—93no—
119D3×C22⋊C4—912no—
120D3×C22⋊C4—93no—
121C23×C32—248no—
122C23×C32—224no—
123C23×C32—26no—
124C23×C32—212no—
125C23×C32—31no—
126C23×C32C22×D3112yes—
127C23×C32C22×D313yes—
128C22×C3⋊D3—31no—
129C22×C3⋊D3—312no—
130C22×C3⋊D3—33no—
131C22×C3⋊D3—34no—
132C22×C3⋊D3—348no—
133C22×C3⋊D3—312no—
134C22×C3⋊D3—648no—
135C22×C3⋊D3—64no—
136C22×C3⋊D3—612no—
137C22×C3⋊D3—66no—
138C22×C3⋊D3—612no—
139C22×C3⋊D3—612no—
140C22×C3⋊D3—648no—
141C22×C3⋊D3—648no—
142C22×C3⋊D3—696no—
143C22×C3⋊D3—696no—
144C22×C3⋊D3—632no—
145C22×C3⋊D3—6128no—
146C22×C3⋊D3—6256no—
147C22×C3⋊D3—648no—
148C22×C3⋊D3—624no—
149C22×C3⋊D3—66no—
150C22×C3⋊D3C22×D311yescharacteristic, complemented
151C22×C3×D3—34no—
152C22×C3×D3C22×D314yescomplemented
153C2×D32—624no—
154C32×D4—66no—
155C2×C32×C4—33no—
156#638—26no—
157#638—66no—
158C2×C3⋊Dic3C22×D313yescomplemented
159C4⋊(C3⋊D3)—66no—
160C4×C3⋊D3—66no—
161C3×#67—224no—
162C3×#67—624no—
163C2×C3×Dic3—312no—
164C2×C3×Dic3C22×D3112yescomplemented
165#626—624no—
166#626—624no—
167#624—624no—
168C2×#120—271noSylow
169C2×C33—26no—
170C2×C33C22×D413yes—
171C3⋊(C3⋊D3)—26no—
172C3⋊(C3⋊D3)#12014yescomplemented
173C3×C3⋊D3—41no—
174C32×D3—41no—
175C24×C3—24no—
176C24×C3D3214yescomplemented
177C24×C3C2×C3⋊D311yescharacteristic, complemented
178C23×D3—94no—
179C23×D3—912no—
180C23×D3—93no—
181C23×D3—912no—
182C23×D3—93no—
183C23×D3—91no—
184C23×D3—1812no—
185C23×D3—1812no—
186C23×D3—1848no—
187C23×D3—1812no—
188C23×D3—1896no—
189C2×C3×D4—312no—
190C2×C3×D4—93no—
191C2×#67—312no—
192C2×#67—912no—
193C2×#67—93no—
194C2×#67C2×C3⋊D313yescomplemented
195D3×D4—1812no—
196D3×D4—1848no—
197C2×D12—912no—
198C2×D12—93no—
199C2×C4×D3—912no—
200C2×C4×D3—93no—
201C3×C22⋊C4—312no—
202C3×C22⋊C4—93no—
203#217—312no—
204#217C2×C3⋊D313yescomplemented
205(C2×D3)⋊C4—924no—
206(C2×D3)⋊C4—96no—
207C22×C32—264no—
208C22×C32—232no—
209C22×C32—28no—
210C22×C32—248no—
211C22×C32—224no—
212C22×C32—26no—
213C22×C32—24no—
214C22×C32—224no—
215C22×C32—63no—
216C22×C32D3×D4124yescomplemented
217C22×C32D3×D416yescomplemented
218C22×C32C23×D314yes—
219C22×C32C23×D311yescharacteristic
220C2×C3⋊D3—23no—
221C2×C3⋊D3—36no—
222C2×C3⋊D3—36no—
223C2×C3⋊D3—324no—
224C2×C3⋊D3—324no—
225C2×C3⋊D3—624no—
226C2×C3⋊D3—624no—
227C2×C3⋊D3—624no—
228C2×C3⋊D3—696no—
229C2×C3⋊D3—6192no—
230C2×C3⋊D3—6192no—
231C2×C3⋊D3—696no—
232C2×C3⋊D3—624no—
233C2×C3⋊D3—648no—
234C2×C3⋊D3—624no—
235C2×C3⋊D3—66no—
236C2×C3⋊D3—124no—
237C2×C3×D3—212no—
238C2×C3×D3—612no—
239D32—1216no—
240C32×C4—63no—
241C3⋊Dic3—23no—
242C3×Dic3—212no—
243C3×Dic3—612no—
244C25—271no—
245C22×D4—273no—
246#120—31no—
247#120—91no—
248#120—273no—
249#120—273no—
250C2×C22⋊C4—273no—
251C33C2×#12011yescharacteristic, complemented, Sylow
252C23×C3—248no—
253C23×C3—224no—
254C23×C3—26no—
255C23×C3—212no—
256C23×C3—34no—
257C23×C3—91no—
258C23×C3C2×D32112yes—
259C23×C3C22×C3⋊D313yes—
260C22×D3—34no—
261C22×D3—94no—
262C22×D3—91no—
263C22×D3—912no—
264C22×D3—93no—
265C22×D3—948no—
266C22×D3—912no—
267C22×D3—184no—
268C22×D3—1812no—
269C22×D3—1848no—
270C22×D3—186no—
271C22×D3—1824no—
272C22×D3—1848no—
273C22×D3—1812no—
274C22×D3—1848no—
275C22×D3—1812no—
276C22×D3—1896no—
277C22×D3—1896no—
278C22×D3—18128no—
279C22×D3—1832no—
280C22×D3—18256no—
281C22×D3—1848no—
282C22×D3—1824no—
283C22×D3—186no—
284C22×D3C22×C3⋊D311yescharacteristic, complemented
285C3×D4—624no—
286C3×D4—186no—
287C2×C3×C4—312no—
288C2×C3×C4—93no—
289#67—26no—
290#67—624no—
291#67—186no—
292#67—1824no—
293C2×Dic3—312no—
294C2×Dic3C22×C3⋊D313yescomplemented
295D12—1824no—
296D12—186no—
297C4×D3—186no—
298C4×D3—1824no—
299C2×C32—248no—
300C2×C32—224no—
301C2×C32—26no—
302C2×C32—212no—
303C2×C32—121no—
304C2×C32C2×D3×D4112yes—
305C2×C32C2×D3×D413yes—
306C3⋊D3—34no—
307C3⋊D3—316no—
308C3⋊D3—41no—
309C3⋊D3—616no—
310C3⋊D3—648no—
311C3⋊D3—624no—
312C3⋊D3—66no—
313C3×D3—44no—
314C3×D3—124no—
315C24—271no—
316C24—273no—
317C24—273no—
318C24—5412no—
319C24D3×C3⋊D311yescharacteristic, complemented
320C2×D4—33no—
321C2×D4—93no—
322C2×D4—273no—
323C2×D4—273no—
324C2×D4—5412no—
325C22×C4—273no—
326C22⋊C4—33no—
327C22⋊C4—93no—
328C22⋊C4—276no—
329C22×C3—264no—
330C22×C3—232no—
331C22×C3—28no—
332C22×C3—248no—
333C22×C3—224no—
334C22×C3—26no—
335C22×C3—24no—
336C22×C3—224no—
337C22×C3—612no—
338C22×C3—183no—
339C22×C3#3811124yescomplemented
340C22×C3D4×C3⋊D316yescomplemented
341C22×C3C22×D3214yes—
342C22×C3C23×C3⋊D311yescharacteristic
343C2×D3—23no—
344C2×D3—612no—
345C2×D3—924no—
346C2×D3—96no—
347C2×D3—924no—
348C2×D3—96no—
349C2×D3—1824no—
350C2×D3—1824no—
351C2×D3—1896no—
352C2×D3—1824no—
353C2×D3—18192no—
354C2×D3—18192no—
355C2×D3—1896no—
356C2×D3—1824no—
357C2×D3—1848no—
358C2×D3—1824no—
359C2×D3—186no—
360C2×D3—364no—
361C2×D3—3616no—
362C3×C4—612no—
363C3×C4—183no—
364Dic3—23no—
365Dic3—612no—
366C32—24no—
367C32D3×#12014yescomplemented
368C32D3×#12011yescharacteristic, complemented
369C23—26no—
370C23—31no—
371C23—91no—
372C23—271no—
373C23—273no—
374C23—2712no—
375C23—5412no—
376C23—5412no—
377C23—5432no—
378C23—546no—
379C23—546no—
380C23C2×D3×C3⋊D313yes—
381D4—66no—
382D4—186no—
383D4—546no—
384D4—546no—
385C2×C4—33no—
386C2×C4—93no—
387C2×C4—546no—
388C2×C3—248no—
389C2×C3—224no—
390C2×C3—26no—
391C2×C3—212no—
392C2×C3—124no—
393C2×C3—361no—
394C2×C3C2×#3811112yes—
395C2×C3C2×D4×C3⋊D313yes—
396D3—41no—
397D3—916no—
398D3—94no—
399D3—124no—
400D3—1816no—
401D3—1848no—
402D3—1824no—
403D3—186no—
404C22—28no—
405C22—26no—
406C22—63no—
407C22—183no—
408C22—276no—
409C22—276no—
410C22—5424no—
411C22—5424no—
412C22—546no—
413C22—1084no—
414C22#8903916yescomplemented
415C22C22×D3×C3⋊D311yescharacteristic
416C4—63no—
417C4—183no—
418C3—24no—
419C3#1059552714yescomplemented
420C3C3⋊D3×#12011yescharacteristic, complemented
421C2—26no—
422C2—121no—
423C2—274no—
424C2—361no—
425C2—546no—
426C2C2×#8903913yes—
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×C3⋊(C3⋊D3)C2semidirect
2. C2×#89039C2semidirect
3. C3×#64162C2semidirect
4. C32×#953C2semidirect
5. #11723639C2semidirect
6. #11723638C2semidirect
7. #11723627C2semidirect
10. C24×C33C22semidirect
13. C23×C3⋊(C3⋊D3)C22semidirect
14. C22×D3×C3⋊D3C22semidirect
15. C2×C3×#638C22semidirect
16. C2×C32×#67C22semidirect
18. C2×#6292C22semidirect
19. C2×#6291C22semidirect
20. C2×#6289C22semidirect
21. C3×#3757C22semidirect
22. C32×#217C22semidirect
23. #88804C22semidirect
30. #64162D3semidirect
35. C3×#953D3semidirect
45. C22×C3⋊(C3⋊D3)D4semidirect
48. C22×C3×C3⋊D3C23semidirect
49. C22×C32×D3C23semidirect
52. C2×C3×C3⋊Dic3C23semidirect
54. C2×C32×Dic3C23semidirect
61. C24×C32C2×D3semidirect
62. C24×C32C2×D3semidirect
76. C2×#638C2×D3semidirect
81. C2×C3×#67C2×D3semidirect
87. #3757C2×D3semidirect
90. C3×#217C2×D3semidirect
94. C22×C33C2×D4semidirect
100. C2×C3⋊(C3⋊D3)C2×D4semidirect
114. #953C3⋊D3semidirect
150. C22×C3⋊D3C22×D3semidirect
152. C22×C3×D3C22×D3semidirect
158. C2×C3⋊Dic3C22×D3semidirect
164. C2×C3×Dic3C22×D3semidirect
172. C3⋊(C3⋊D3)#120semidirect
176. C24×C3D32semidirect
177. C24×C3C2×C3⋊D3semidirect
194. C2×#67C2×C3⋊D3semidirect
204. #217C2×C3⋊D3semidirect
216. C22×C32D3×D4semidirect
217. C22×C32D3×D4semidirect
251. C33C2×#120semidirect
284. C22×D3C22×C3⋊D3semidirect
294. C2×Dic3C22×C3⋊D3semidirect
319. C24D3×C3⋊D3semidirect
339. C22×C3#3811semidirect
340. C22×C3D4×C3⋊D3semidirect
367. C32D3×#120semidirect
368. C32D3×#120semidirect
414. C22#89039semidirect
419. C3#10595527semidirect
420. C3C3⋊D3×#120semidirect

Maximal subgroups

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

subgroupindex
C24×C3⋊(C3⋊D3)2
C2×#890392
C3×#641622
C32×#9532
#117236392
#117236382
#117236272
C3⋊D3×#1203
#105955273