Finite Group Databasebeta

← #13178679

Subgroups — #13178679

Order 1728.

1430 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1430 conjugacy classes of subgroups — too many to draw. Only its 167 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
1C3×#64146C212yesmaximal, complemented
2C3×#64145C211yesmaximal, characteristic, complemented
3#11723727C214yesmaximal, complemented
4#11723718C214yesmaximal, complemented
5#11723672C212yesmaximal, complemented
6#11723670C212yesmaximal, complemented
7#10595262—32nomaximal
8#10595256—31nomaximal
9C23×C3×C3⋊D3C2212yescomplemented
10C2×C33×D4C2211yescharacteristic
11C2×C3×#638C2214yes—
12C2×C3×#634C2211yescharacteristic, complemented
13C2×#6296C2212yescomplemented
14C2×#6295C2212yes—
15C2×#6295C2212yes—
16C3×#3757C2214yescomplemented
17C3×#3757C2212yescomplemented
18C3×#3752C2214yescomplemented
19C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
20#88814C2214yescomplemented
21#88814C2214yescomplemented
22#88813C2212yescomplemented
23#64146D312yescomplemented
24#64145D311yescharacteristic, complemented
25C3×#937—32no—
26C3×#937—32no—
27C3×#937—62no—
28C3×#936—32no—
29C3×#936—61no—
30#63981—34no—
31#63977—34no—
32#63976—34no—
33#63959—34no—
34#63959—34no—
35#63953—34no—
36#63915—32no—
37#63913—32no—
38#63907—34no—
39#63852—34no—
40C23×C33C2312yes—
41C22×C3×C3⋊D3—44no—
42C22×C3×C3⋊D3—42no—
43C22×C3×C3⋊D3C2314yes—
44C33×D4—41no—
45C2×C33×C4C2311yescharacteristic
46C3×#638—44no—
47C2×C3×C3⋊Dic3C2312yes—
48C2×C3×C3⋊Dic3C2314yes—
49C2×C3×C3⋊Dic3C2312yes—
50C3×#634—41no—
51#6296—42no—
52#6295—42no—
53#6295—42no—
54#5683—91no—
55#5594—92no—
56C23×C3⋊D3C2×D312yescomplemented
57C23×C3×D3—32no—
58C23×C3×D3—32no—
59C23×C3×D3—62no—
60C2×C32×D4—21no—
61C2×C32×D4—22no—
62C2×C32×D4—41no—
63C2×C32×D4C2×D312yes—
64C2×C32×D4C2×D311yescharacteristic
65C2×#638C2×D314yes—
66C2×#634C2×D311yescharacteristic, complemented
67C2×C3×#67—34no—
68C2×C3×#67—34no—
69C2×C3×#67—64no—
70C2×C3×Dic6—32no—
71C2×C3×Dic6—61no—
72C2×#627—34no—
73C2×#627—32no—
74C2×#626—34no—
75C2×#626—34no—
76C2×#625—32no—
77C2×#625—32no—
78#3757C2×D314yescomplemented
79#3757C2×D312yescomplemented
80#3752C2×D314yescomplemented
81C4×C3⋊Dic3C2×D311yescharacteristic, complemented
82C3×#217—34no—
83C3×#217—34no—
84C3×#217—32no—
85C3×#217—32no—
86C3×#217—64no—
87C3×#217—62no—
88C3×(C2×D3)⋊C4—34no—
89C3×(C2×D3)⋊C4—34no—
90C3×(C2×D3)⋊C4—64no—
91C3×C4×Dic3—32no—
92C3×C4×Dic3—61no—
93#3722—34no—
94#3722—34no—
95(C2×D3)⋊Dic3—34no—
96(C2×D3)⋊Dic3—34no—
97(C2×D3)⋊Dic3—34no—
98(C2×D3)⋊Dic3—34no—
99Dic32—34no—
100Dic32—32no—
101C22×C33—22no—
102C22×C33—24no—
103C22×C33C2411yescharacteristic
104C2×C3×C3⋊D3—24no—
105C2×C3×C3⋊D3—24no—
106C2×C3×C3⋊D3—24no—
107C2×C3×C3⋊D3—48no—
108C33×C4—21no—
109C3×C3⋊Dic3—22no—
110C3×C3⋊Dic3—22no—
111C3×C3⋊Dic3—24no—
112C3×#124—91no—
113C3×#120—92no—
114#953—94no—
115#947—92no—
116#937—32no—
117#937—32no—
118#937—62no—
119#937—94no—
120#936—32no—
121#936—61no—
122#936—94no—
123#885—94no—
124#885—94no—
125#885—94no—
126#885—94no—
127#882—94no—
128#875—92no—
129C23×C32—22no—
130C23×C32—22no—
131C23×C32—22no—
132C23×C32—42no—
133C23×C32C22×D312yes—
134C23×C32C22×D312yes—
135C23×C32C22×D312yes—
136C22×C3⋊D3—44no—
137C22×C3⋊D3—42no—
138C22×C3⋊D3C22×D314yes—
139C22×C3×D3—34no—
140C22×C3×D3—34no—
141C22×C3×D3—64no—
142C22×C3×D3—122no—
143C22×C3×D3—122no—
144C22×C3×D3—124no—
145C22×C3×D3—124no—
146C22×C3×D3—128no—
147C22×C3×D3—124no—
148C32×D4—41no—
149C32×D4—42no—
150C32×D4—42no—
151C32×D4—44no—
152C32×D4—44no—
153C2×C32×C4—21no—
154C2×C32×C4—22no—
155C2×C32×C4—41no—
156C2×C32×C4C22×D312yes—
157C2×C32×C4C22×D311yescharacteristic
158#638—44no—
159C2×C3⋊Dic3C22×D314yes—
160C2×C3⋊Dic3C22×D312yes—
161C2×C3⋊Dic3C22×D312yes—
162#634—41no—
163C3×#67—124no—
164C3×#67—124no—
165C3×#67—128no—
166C2×C3×Dic3—34no—
167C2×C3×Dic3—34no—
168C2×C3×Dic3—34no—
169C2×C3×Dic3—34no—
170C2×C3×Dic3—62no—
171C2×C3×Dic3—64no—
172C2×C3×Dic3—62no—
173C3×Dic6—122no—
174C3×Dic6—122no—
175#627—124no—
176#627—122no—
177#626—124no—
178#626—124no—
179#625—122no—
180#625—122no—
181H(3,𝔽4)—271noSylow
182C2×C33—42no—
183C2×C33H(5,𝔽2)11yescharacteristic
184C2×C33H(5,𝔽2)12yes—
185C3×C3⋊D3—44no—
186C24×C3—92no—
187C23×D3—32no—
188C23×D3—32no—
189C23×D3—62no—
190C2×C3×Q8—91no—
191C2×C3×D4—21no—
192C2×C3×D4—22no—
193C2×C3×D4—41no—
194C2×C3×D4—94no—
195C2×C3×D4D3212yes—
196C2×C3×D4D3211yescharacteristic
197C2×#67—34no—
198C2×#67—34no—
199C2×#67—64no—
200C2×#67—94no—
201C2×#67—94no—
202C2×D12—92no—
203C2×D12—92no—
204C2×Dic6—32no—
205C2×Dic6—61no—
206C2×Dic6—92no—
207C2×Dic6—94no—
208C3×C22⋊C4—94no—
209C3×C22⋊C4—92no—
210C3×C22⋊C4—94no—
211C3×C42—91no—
212#217—32no—
213#217—34no—
214#217—34no—
215#217—32no—
216#217—64no—
217#217—62no—
218#217—94no—
219#217—94no—
220(C2×D3)⋊C4—34no—
221(C2×D3)⋊C4—34no—
222(C2×D3)⋊C4—64no—
223(C2×D3)⋊C4—94no—
224(C2×D3)⋊C4—94no—
225(C2×D3)⋊C4—94no—
226(C2×D3)⋊C4—94no—
227C4×Dic3—32no—
228C4×Dic3—61no—
229C4×Dic3—94no—
230C4×Dic3—92no—
231C22×C32—22no—
232C22×C32—24no—
233C22×C32—24no—
234C22×C32—24no—
235C22×C32—22no—
236C22×C32—22no—
237C22×C32—24no—
238C22×C32—21no—
239C22×C32—22no—
240C22×C32—41no—
241C22×C32—42no—
242C22×C32—42no—
243C22×C32—44no—
244C22×C32—44no—
245C22×C32—44no—
246C22×C32—44no—
247C22×C32—84no—
248C22×C32C23×D312yes—
249C22×C32C23×D311yescharacteristic
250C2×C3⋊D3—24no—
251C2×C3⋊D3—24no—
252C2×C3⋊D3—24no—
253C2×C3⋊D3—48no—
254C2×C3×D3—68no—
255C2×C3×D3—64no—
256C2×C3×D3—64no—
257C2×C3×D3—64no—
258C2×C3×D3—64no—
259C2×C3×D3—64no—
260C2×C3×D3—64no—
261C2×C3×D3—124no—
262C2×C3×D3—124no—
263C2×C3×D3—1216no—
264C2×C3×D3—128no—
265C2×C3×D3—128no—
266C32×C4—21no—
267C32×C4—22no—
268C32×C4—22no—
269C32×C4—42no—
270C32×C4—42no—
271C3⋊Dic3—22no—
272C3⋊Dic3—24no—
273C3⋊Dic3—22no—
274C3×Dic3—64no—
275C3×Dic3—64no—
276C3×Dic3—64no—
277C3×Dic3—64no—
278C3×Dic3—122no—
279C3×Dic3—122no—
280C3×Dic3—124no—
281#124—91no—
282#124—272no—
283#124—274no—
284#124—272no—
285#120—92no—
286#120—274no—
287C33H(3,𝔽4)11yescharacteristic, complemented, Sylow
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—42no—
292C23×C3—94no—
293C23×C3—364no—
294C23×C3—362no—
295C23×C3C2×D3212yes—
296C23×C3C2×D3212yes—
297C23×C3C2×D3212yes—
298C22×D3—34no—
299C22×D3—34no—
300C22×D3—64no—
301C22×D3—124no—
302C22×D3—124no—
303C22×D3—122no—
304C22×D3—122no—
305C22×D3—128no—
306C22×D3—124no—
307C3×Q8—361no—
308C3×D4—42no—
309C3×D4—41no—
310C3×D4—42no—
311C3×D4—44no—
312C3×D4—44no—
313C3×D4—364no—
314C2×C3×C4—21no—
315C2×C3×C4—22no—
316C2×C3×C4—41no—
317C2×C3×C4—94no—
318C2×C3×C4—92no—
319C2×C3×C4—92no—
320C2×C3×C4C2×D3212yes—
321C2×C3×C4C2×D3211yescharacteristic
322#67—124no—
323#67—124no—
324#67—128no—
325#67—364no—
326#67—364no—
327C2×Dic3—34no—
328C2×Dic3—34no—
329C2×Dic3—34no—
330C2×Dic3—34no—
331C2×Dic3—62no—
332C2×Dic3—64no—
333C2×Dic3—62no—
334D12—362no—
335D12—362no—
336Dic6—122no—
337Dic6—122no—
338Dic6—362no—
339Dic6—364no—
340C2×C32—22no—
341C2×C32—24no—
342C2×C32—21no—
343C2×C32—22no—
344C2×C32—41no—
345C2×C32—42no—
346C2×C32—42no—
347C2×C32—44no—
348C2×C32—42no—
349C2×C32—42no—
350C2×C32—82no—
351C2×C32—82no—
352C2×C32—84no—
353C2×C32#100414yes—
354C2×C32#100411yescharacteristic
355C2×C32#100412yes—
356C2×C32#100912yes—
357C3⋊D3—44no—
358C3×D3—128no—
359C3×D3—124no—
360C3×D3—124no—
361C24—92no—
362C2×Q8—91no—
363C2×Q8—272no—
364C2×D4—94no—
365C2×D4—272no—
366C2×D4—272no—
367C2×D4#111311yescharacteristic
368C22⋊C4—92no—
369C22⋊C4—94no—
370C22⋊C4—94no—
371C22⋊C4—274no—
372C22⋊C4—274no—
373C42—91no—
374C42—272no—
375C22×C3—24no—
376C22×C3—24no—
377C22×C3—24no—
378C22×C3—24no—
379C22×C3—22no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—21no—
383C22×C3—22no—
384C22×C3—44no—
385C22×C3—44no—
386C22×C3—42no—
387C22×C3—42no—
388C22×C3—44no—
389C22×C3—44no—
390C22×C3—41no—
391C22×C3—84no—
392C22×C3—184no—
393C22×C3—184no—
394C22×C3—184no—
395C22×C3—368no—
396C22×C3C22×D3211yescharacteristic
397C22×C3C22×D3212yes—
398C2×D3—68no—
399C2×D3—64no—
400C2×D3—64no—
401C2×D3—64no—
402C2×D3—64no—
403C2×D3—64no—
404C2×D3—64no—
405C2×D3—124no—
406C2×D3—124no—
407C2×D3—1216no—
408C2×D3—128no—
409C2×D3—128no—
410C3×C4—22no—
411C3×C4—22no—
412C3×C4—21no—
413C3×C4—42no—
414C3×C4—42no—
415C3×C4—182no—
416C3×C4—184no—
417C3×C4—182no—
418Dic3—64no—
419Dic3—64no—
420Dic3—64no—
421Dic3—64no—
422Dic3—122no—
423Dic3—124no—
424Dic3—122no—
425C32—21no—
426C32—22no—
427C32—41no—
428C32#559412yescomplemented
429C32#568311yescharacteristic, complemented
430C23—94no—
431C23—364no—
432C23—362no—
433C23C2×#111312yes—
434Q8—361no—
435Q8—1082no—
436D4—41no—
437D4—364no—
438D4—1082no—
439D4—1082no—
440C2×C4—92no—
441C2×C4—94no—
442C2×C4—92no—
443C2×C4C2×#111311yescharacteristic
444C2×C3—24no—
445C2×C3—22no—
446C2×C3—21no—
447C2×C3—22no—
448C2×C3—44no—
449C2×C3—42no—
450C2×C3—42no—
451C2×C3—42no—
452C2×C3—42no—
453C2×C3—41no—
454C2×C3—82no—
455C2×C3—82no—
456C2×C3—84no—
457C2×C3—364no—
458C2×C3#6431312yes—
459C2×C3#6431414yes—
460C2×C3#6433011yescharacteristic
461C2×C3#6433012yes—
462D3—128no—
463D3—124no—
464D3—124no—
465C22—24no—
466C22—22no—
467C22—184no—
468C22—184no—
469C22—184no—
470C22—368no—
471C22C22×#111311yescharacteristic
472C4—21no—
473C4—184no—
474C4—182no—
475C4—182no—
476C3—22no—
477C3—21no—
478C3—41no—
479C3#1059525611yescharacteristic, complemented
480C3#1059526212yescomplemented
481C2—42no—
482C2—364no—
483C2#1172462312yes—
484C2#1172463311yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4#1113
hypocenterC33H(3,𝔽4)
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C3×#64146C2semidirect
2. C3×#64145C2semidirect
3. #11723727C2semidirect
4. #11723718C2semidirect
5. #11723672C2semidirect
6. #11723670C2semidirect
9. C23×C3×C3⋊D3C22semidirect
12. C2×C3×#634C22semidirect
13. C2×#6296C22semidirect
16. C3×#3757C22semidirect
17. C3×#3757C22semidirect
18. C3×#3752C22semidirect
19. C3×C4×C3⋊Dic3C22semidirect
20. #88814C22semidirect
21. #88814C22semidirect
22. #88813C22semidirect
23. #64146D3semidirect
24. #64145D3semidirect
56. C23×C3⋊D3C2×D3semidirect
66. C2×#634C2×D3semidirect
78. #3757C2×D3semidirect
79. #3757C2×D3semidirect
80. #3752C2×D3semidirect
81. C4×C3⋊Dic3C2×D3semidirect
287. C33H(3,𝔽4)semidirect
428. C32#5594semidirect
429. C32#5683semidirect
479. C3#10595256semidirect
480. C3#10595262semidirect

Maximal subgroups

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

subgroupindex
C3×#641462
C3×#641452
#117237272
#117237182
#117236722
#117236702
#105952623
#105952563