Finite Group Databasebeta

← #13168305

Subgroups — #13168305

Order 1728.

1518 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89014C211yesmaximal, characteristic
2#11723673C211yesmaximal, characteristic, complemented
3#11723406C212yesmaximal, complemented
4C3×#63900C211yesmaximal, characteristic, complemented
5C3×#63898C212yesmaximal, complemented
6#11722930C214yesmaximal, complemented
7#11722917C212yesmaximal, complemented
8D3×#3724C212yesmaximal, complemented
9#10595219—31nomaximal
10#10594359—32nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×#6290C2212yes—
15C2×D3×C3⋊Dic3C2212yes—
16#89014C418yescomplemented
17C2×C3×#624C2212yes—
18C2×C3×D3×Dic3C2214yes—
19C3×C4×C3⋊Dic3C2212yes—
20C32×C4⋊Dic3C2212yes—
21#88815C2211yescharacteristic
22#88815C2211yescharacteristic
23#88814C2212yes—
24#88806C2214yes—
25Dic3×C3⋊Dic3C2214yes—
26C3×#3724C2212yes—
27C3×(C2×D3)⋊Dic3C2214yes—
28C2×#3830—31no—
29C2×#3795—32no—
30C2×#3795—61no—
31#64096—32no—
32C3×D4×Dic3—32no—
33C3×#893—32no—
34C3×D3×C4⋊C4—32no—
35C2×#3724—32no—
36#63957—32no—
37#63952—34no—
38C4×#625—34no—
39#63900D311yescharacteristic, complemented
40#63898D312yescomplemented
41#63879—34no—
42#63878—34no—
43D3×Dic3⋊C4—34no—
44#63874—31no—
45#63855—32no—
46#63844—32no—
47#63844—32no—
48#63844—62no—
49C22×C3×C3⋊D3C2311yescharacteristic
50C22×C32×D3C2312yes—
51C2×C33×C4C2311yescharacteristic
52C2×C3⋊(C3⋊Dic3)C2312yes—
53#6308C2×C414yescomplemented
54C2×C3×C3⋊Dic3C2312yes—
55C2×C3×C3⋊Dic3C2312yes—
56C2×C3×C3⋊Dic3C2311yescharacteristic
57C3×C4×C3⋊D3C2×C414yescomplemented
58C2×C32×Dic3C2314yes—
59C32×D12C2×C414yescomplemented
60#6290C2×C418yescomplemented
61D3×C3⋊Dic3C2×C418yescomplemented
62C3×#624—42no—
63C3×D3×Dic3—44no—
64#5546—92no—
65#5504—91no—
66C2×C32×D4—31no—
67C2×#638—32no—
68C22×C3⋊Dic3—32no—
69#3830—38no—
70C2×C4×C3⋊D3C2×D311yescharacteristic
71C2×#634—32no—
72C2×#634—31no—
73C2×#634—61no—
74C2×#634—62no—
75C2×#634—121no—
76C22×C3×Dic3—34no—
77C2×C3×D12—21no—
78C2×C3×D12C2×D312yes—
79C2×C3×C4×D3—32no—
80C2×C3×C4×D3—34no—
81C2×C3×C4×D3—34no—
82C2×C3×C4×D3—61no—
83C2×#625—34no—
84C2×#625—62no—
85C2×#624C2×D312yes—
86C2×D3×Dic3—34no—
87C2×D3×Dic3—62no—
88C2×D3×Dic3C2×D314yes—
89#3795—316no—
90#3795—68no—
91C32×C4⋊C4—32no—
92#3750—34no—
93C4×C3⋊Dic3—34no—
94C4×C3⋊Dic3C2×D312yes—
95C3×#217—34no—
96C3×(C2×D3)⋊C4—34no—
97C3×C4⋊Dic3—22no—
98C3×C4⋊Dic3C2×D312yes—
99C3×C4⋊Dic3C2×D312yes—
100C3×Dic3⋊C4—34no—
101C3×C4×Dic3—32no—
102C3×C4×Dic3—32no—
103C3×C4×Dic3—62no—
104#3724—31no—
105#3724—31no—
106#3724—34no—
107#3724—34no—
108#3724—64no—
109#3724C2×D312yes—
110Dic3⋊Dic3—32no—
111Dic3⋊Dic3—32no—
112#3722—32no—
113(C2×D3)⋊Dic3—32no—
114(C2×D3)⋊Dic3—32no—
115(C2×D3)⋊Dic3C2×D314yes—
116Dic32—34no—
117Dic32—34no—
118Dic32—64no—
119C22×C33C2411yescharacteristic
120C2×C3×C3⋊D3—21no—
121C2×C3×C3⋊D3—21no—
122C2×C3×C3⋊D3C22×C412yes—
123C2×C32×D3—22no—
124C2×C32×D3—22no—
125C2×C32×D3C22×C414yes—
126C33×C4C22×C412yes—
127C3⋊(C3⋊Dic3)C22×C414yes—
128C3×C3⋊Dic3—22no—
129C3×C3⋊Dic3—22no—
130C3×C3⋊Dic3C22×C412yes—
131C32×Dic3—24no—
132C2×#237—92no—
133C2×#237—181no—
134C2×#235—91no—
135C3×C4×D4—92no—
136C2×C3×C4⋊C4—91no—
137D4×Dic3—32no—
138C4×#67—94no—
139C4×#67—94no—
140#926—91no—
141C2×Dic3⋊C4—94no—
142#893—32no—
143#892—92no—
144D3×C4⋊C4—32no—
145D3×C4⋊C4—92no—
146#887—92no—
147#887—92no—
148#887—182no—
149#67⋊C4—94no—
150#877—94no—
151#872—92no—
152C4×Dic6—92no—
153C23×C32—32no—
154C22×C3⋊D3C22×D311yescharacteristic
155C22×C3×D3—22no—
156C22×C3×D3—32no—
157C22×C3×D3—61no—
158C22×C3×D3C22×D314yes—
159C32×D4—34no—
160C2×C32×C4—22no—
161C2×C32×C4—21no—
162C2×C32×C4—34no—
163C2×C32×C4—41no—
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D312yes—
166#638—38no—
167C2×C3⋊Dic3—32no—
168C2×C3⋊Dic3—34no—
169C2×C3⋊Dic3—38no—
170C2×C3⋊Dic3—64no—
171C2×C3⋊Dic3—62no—
172C2×C3⋊Dic3—122no—
173C2×C3⋊Dic3C22×D312yes—
174C2×C3⋊Dic3C22×D312yes—
175C2×C3⋊Dic3C22×D311yescharacteristic
176C4×C3⋊D3C4×D314yescomplemented
177#634—38no—
178#634—34no—
179#634—64no—
180#634—68no—
181#634—124no—
182C2×C3×Dic3—24no—
183C2×C3×Dic3—32no—
184C2×C3×Dic3—32no—
185C2×C3×Dic3—32no—
186C2×C3×Dic3—32no—
187C2×C3×Dic3—32no—
188C2×C3×Dic3—62no—
189C2×C3×Dic3—61no—
190C2×C3×Dic3—62no—
191C2×C3×Dic3—124no—
192C2×C3×Dic3C22×D314yes—
193C2×C3×Dic3C22×D314yes—
194C3×D12—24no—
195C3×D12C4×D318yescomplemented
196C3×C4×D3—38no—
197C3×C4×D3—64no—
198C3×C4×D3—124no—
199C3×C4×D3—124no—
200#625—316no—
201#625—68no—
202#624—42no—
203D3×Dic3—316no—
204D3×Dic3—44no—
205D3×Dic3—68no—
206#520—271noSylow
207C2×C33C23×C411yescharacteristic
208C2×C33H(5,𝔽2)11yescharacteristic
209C2×C332−1+411yescharacteristic
210C3×C3⋊D3—22no—
211C32×D3—24no—
212C2×C3×D4—32no—
213C2×C3×D4—61no—
214C22×C3×C4—92no—
215C22×C3×C4—94no—
216C22×C3×C4—91no—
217C2×#67—94no—
218C2×#67—92no—
219C2×#67—182no—
220C22×Dic3—34no—
221C22×Dic3—94no—
222C22×Dic3—182no—
223#237—916no—
224#237—188no—
225#235—98no—
226C2×D12D3211yescharacteristic
227C2×C4×D3—32no—
228C2×C4×D3—34no—
229C2×C4×D3—34no—
230C2×C4×D3—61no—
231C2×C4×D3—92no—
232C2×Dic6—92no—
233C2×Dic6—91no—
234C2×Dic6—182no—
235C2×Dic6—181no—
236C2×Dic6—361no—
237C3×C4⋊C4—32no—
238C3×C4⋊C4—32no—
239C3×C4⋊C4—62no—
240C3×C4⋊C4—92no—
241C3×C22⋊C4—94no—
242C3×C42—92no—
243#217—34no—
244#217—92no—
245(C2×D3)⋊C4—34no—
246(C2×D3)⋊C4—92no—
247(C2×D3)⋊C4—92no—
248C4⋊Dic3—91no—
249C4⋊Dic3—91no—
250C4⋊Dic3D3212yes—
251Dic3⋊C4—34no—
252Dic3⋊C4—92no—
253Dic3⋊C4—92no—
254Dic3⋊C4—94no—
255Dic3⋊C4—94no—
256Dic3⋊C4—94no—
257Dic3⋊C4—184no—
258C4×Dic3—32no—
259C4×Dic3—32no—
260C4×Dic3—62no—
261C4×Dic3—94no—
262C4×Dic3—94no—
263C4×Dic3—94no—
264C4×Dic3—184no—
265C22×C32—22no—
266C22×C32—21no—
267C22×C32—34no—
268C22×C32—41no—
269C22×C32—62no—
270C22×C32—62no—
271C22×C32C23×D311yescharacteristic
272C22×C32C23×D312yes—
273C2×C3⋊D3—21no—
274C2×C3⋊D3—21no—
275C2×C3⋊D3C2×C4×D312yes—
276C2×C3×D3—24no—
277C2×C3×D3—24no—
278C2×C3×D3—24no—
279C2×C3×D3—34no—
280C2×C3×D3—42no—
281C2×C3×D3—42no—
282C2×C3×D3—62no—
283C2×C3×D3—62no—
284C2×C3×D3—62no—
285C2×C3×D3—121no—
286C2×C3×D3—121no—
287C2×C3×D3C2×C4×D318yes—
288C32×C4—24no—
289C32×C4—22no—
290C32×C4—42no—
291C32×C4—64no—
292C32×C4C2×C4×D312yes—
293C32×C4C2×C4×D314yes—
294C3⋊Dic3—22no—
295C3⋊Dic3—22no—
296C3⋊Dic3—38no—
297C3⋊Dic3—34no—
298C3⋊Dic3—64no—
299C3⋊Dic3—68no—
300C3⋊Dic3—124no—
301C3⋊Dic3C2×C4×D312yes—
302C3×Dic3—24no—
303C3×Dic3—24no—
304C3×Dic3—34no—
305C3×Dic3—44no—
306C3×Dic3—62no—
307C3×Dic3—64no—
308C3×Dic3—62no—
309C3×Dic3—62no—
310C3×Dic3—62no—
311C3×Dic3—62no—
312C3×Dic3—122no—
313C2×#41—271no—
314C4×Q8—272no—
315C4×D4—92no—
316C4×D4—274no—
317#117—271no—
318#117—272no—
319C2×C4⋊C4—91no—
320C2×C4⋊C4—272no—
321C33#52011yescharacteristic, complemented, Sylow
322C23×C3—34no—
323C23×C3—62no—
324C23×C3—91no—
325C22×D3—32no—
326C22×D3—61no—
327C22×D3C2×D3212yes—
328C3×D4—38no—
329C3×D4—64no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—34no—
333C2×C3×C4—34no—
334C2×C3×C4—41no—
335C2×C3×C4—64no—
336C2×C3×C4—91no—
337C2×C3×C4—94no—
338C2×C3×C4—92no—
339C2×C3×C4—92no—
340C2×C3×C4—362no—
341C2×C3×C4—364no—
342C2×C3×C4C2×D3212yes—
343C2×C3×C4C2×D3211yescharacteristic
344#67—916no—
345#67—98no—
346#67—188no—
347C2×Dic3—32no—
348C2×Dic3—32no—
349C2×Dic3—32no—
350C2×Dic3—32no—
351C2×Dic3—32no—
352C2×Dic3—62no—
353C2×Dic3—61no—
354C2×Dic3—62no—
355C2×Dic3—916no—
356C2×Dic3—92no—
357C2×Dic3—94no—
358C2×Dic3—124no—
359C2×Dic3—188no—
360C2×Dic3—184no—
361C2×Dic3—182no—
362C2×Dic3—362no—
363C2×Dic3C2×D3214yes—
364D12#62414yescomplemented
365C4×D3—38no—
366C4×D3—64no—
367C4×D3—98no—
368C4×D3—124no—
369C4×D3—124no—
370Dic6—98no—
371Dic6—94no—
372Dic6—188no—
373Dic6—184no—
374Dic6—364no—
375C2×C32—22no—
376C2×C32—21no—
377C2×C32—22no—
378C2×C32—21no—
379C2×C32—22no—
380C2×C32—21no—
381C2×C32—41no—
382C2×C32—41no—
383C2×C32—41no—
384C2×C32—64no—
385C2×C32C22×C4×D311yescharacteristic
386C2×C32C22×C4×D312yes—
387C2×C32#100411yescharacteristic
388C2×C32#100412yes—
389C2×C32#100711yescharacteristic
390C2×C32#101012yes—
391C3⋊D3—22no—
392C3×D3—28no—
393C3×D3—44no—
394C3×D3—64no—
395C3×D3—122no—
396#41—278no—
397C2×Q8—271no—
398C2×D4—31no—
399C2×D4—272no—
400C22×C4—92no—
401C22×C4—94no—
402C22×C4—91no—
403C22×C4—272no—
404C4⋊C4—32no—
405C4⋊C4—92no—
406C4⋊C4—274no—
407C4⋊C4—271no—
408C4⋊C4—271no—
409C22⋊C4—94no—
410C22⋊C4—272no—
411C42—92no—
412C42—274no—
413C22×C3—22no—
414C22×C3—21no—
415C22×C3—38no—
416C22×C3—41no—
417C22×C3—64no—
418C22×C3—64no—
419C22×C3—64no—
420C22×C3—92no—
421C22×C3—122no—
422C22×C3—122no—
423C22×C3—181no—
424C22×C3—181no—
425C22×C3C22×D3212yes—
426C22×C3C22×D3211yescharacteristic
427C2×D3—22no—
428C2×D3—22no—
429C2×D3—34no—
430C2×D3—62no—
431C2×D3—62no—
432C2×D3—62no—
433C2×D3—121no—
434C2×D3—121no—
435C2×D3C2×#62414yes—
436C3×C4—22no—
437C3×C4—24no—
438C3×C4—42no—
439C3×C4—64no—
440C3×C4—64no—
441C3×C4—92no—
442C3×C4—124no—
443C3×C4—182no—
444C3×C4—182no—
445C3×C4C4×D3214yes—
446C3×C4C2×#62412yes—
447Dic3—24no—
448Dic3—34no—
449Dic3—62no—
450Dic3—64no—
451Dic3—62no—
452Dic3—62no—
453Dic3—62no—
454Dic3—62no—
455Dic3—98no—
456Dic3—94no—
457Dic3—122no—
458Dic3—184no—
459Dic3—188no—
460Dic3—364no—
461C32—22no—
462C32—21no—
463C32—41no—
464C32#550411yescharacteristic, complemented
465C32#554612yescomplemented
466C23—32no—
467C23—91no—
468Q8—274no—
469D4—34no—
470D4—278no—
471C2×C4—34no—
472C2×C4—91no—
473C2×C4—94no—
474C2×C4—92no—
475C2×C4—92no—
476C2×C4—278no—
477C2×C4—272no—
478C2×C4—362no—
479C2×C4—364no—
480C2×C4D3311yescharacteristic
481C2×C3—22no—
482C2×C3—21no—
483C2×C3—22no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—22no—
487C2×C3—41no—
488C2×C3—41no—
489C2×C3—41no—
490C2×C3—68no—
491C2×C3—124no—
492C2×C3—182no—
493C2×C3C2×C4×D3212yes—
494C2×C3#6430911yescharacteristic
495C2×C3#6431312yes—
496C2×C3#6431311yescharacteristic
497C2×C3#6431512yes—
498C2×C3C22×#62411yescharacteristic
499D3—24no—
500D3—64no—
501D3—122no—
502C22—34no—
503C22—62no—
504C22—62no—
505C22—92no—
506C22—181no—
507C22—181no—
508C22C2×D3311yescharacteristic
509C4—64no—
510C4—92no—
511C4—182no—
512C4—182no—
513C4—274no—
514C4D3×#62412yes—
515C3—21no—
516C3—22no—
517C3—41no—
518C3#1059435912yescomplemented
519C3#1059521911yescharacteristic, complemented
520C2—64no—
521C2—182no—
522C2#1172438211yescharacteristic
523C2#1172439411yescharacteristic
524C2C2×D3×#62411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutatorC2×C33C23×C4
FittingC2×C33×C4C23
socle, cosocleC22×C33C24
prime radicalC2×#89014C2
hypercenterC2×C4D33
hypocenterC33#520
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723673C2semidirect
3. #11723406C2semidirect
4. C3×#63900C2semidirect
5. C3×#63898C2semidirect
6. #11722930C2semidirect
7. #11722917C2semidirect
8. D3×#3724C2semidirect
16. #89014C4semidirect
39. #63900D3semidirect
40. #63898D3semidirect
53. #6308C2×C4semidirect
57. C3×C4×C3⋊D3C2×C4semidirect
59. C32×D12C2×C4semidirect
60. #6290C2×C4semidirect
61. D3×C3⋊Dic3C2×C4semidirect
176. C4×C3⋊D3C4×D3semidirect
195. C3×D12C4×D3semidirect
321. C33#520semidirect
364. D12#624semidirect
464. C32#5504semidirect
465. C32#5546semidirect
518. C3#10594359semidirect
519. C3#10595219semidirect

Maximal subgroups

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

subgroupindex
C2×#890142
#117236732
#117234062
C3×#639002
C3×#638982
#117229302
#117229172
D3×#37242
#105952193
#105943593