Finite Group Databasebeta

← #13168647

Subgroups — #13168647

Order 1728.

1398 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3⋊D3×Dic6C211yesmaximal, characteristic
2#11723673C211yesmaximal, characteristic
3#11723406C212yesmaximal, complemented
4C3×#63849C211yesmaximal, characteristic
5C3×#63843C212yesmaximal, complemented
6#11722942C214yesmaximal, complemented
7#11722921C212yesmaximal
8#11722919C212yesmaximal
9#10595339—31nomaximal
10#10594357—32nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×#6293C2212yes—
15C2×Dic3×C3⋊D3C2212yes—
16C3⋊D3×Dic6C418yescomplemented
17C3×C4×C3⋊Dic3C2212yes—
18C32×C4⋊Dic3C2212yes—
19#88815C2212yes—
20#88814C2211yescharacteristic
21#88814C2211yescharacteristic
22#88806C2214yes—
23Dic3×C3⋊Dic3C2214yes—
24C3×Dic3⋊Dic3C2214yes—
25C3×#3722C2212yes—
26C3×Dic32C2214yes—
27C3×Dic32C2212yes—
28C2×Q8×C3⋊D3—31no—
29C2×D3×Dic6—32no—
30C2×D3×Dic6—61no—
31#64096—32no—
32C3×Q8×Dic3—32no—
33C3×#892—32no—
34C3×#887—32no—
35C4×#627—34no—
36#63878—34no—
37#63877—34no—
38#63874—31no—
39#63874—32no—
40#63855—32no—
41#63849—32no—
42#63849D311yescharacteristic
43#63844—32no—
44#63844—32no—
45#63844—34no—
46#63844—62no—
47#63843D312yescomplemented
48#63837—34no—
49C22×C3×C3⋊D3C2311yescharacteristic
50C2×C33×C4C2311yescharacteristic
51C2×C3⋊(C3⋊Dic3)C2312yes—
52#6308C2×C414yes—
53C2×C3×C3⋊Dic3C2312yes—
54C2×C3×C3⋊Dic3C2312yes—
55C2×C3×C3⋊Dic3C2311yescharacteristic
56C3×C4×C3⋊D3C2×C414yes—
57C2×C32×Dic3C2314yes—
58C2×C32×Dic3C2312yes—
59C32×Dic6C2×C414yes—
60#6293C2×C418yes—
61Dic3×C3⋊D3C2×C418yes—
62#5572—92no—
63#5516—91no—
64C2×C32×Q8—31no—
65Q8×C3⋊D3—38no—
66C2×C4×C3⋊D3—32no—
67C2×C4×C3⋊D3C2×D311yescharacteristic
68C2×#634—32no—
69C2×#634—31no—
70C2×#634—32no—
71C2×#634—61no—
72C2×#634—62no—
73C2×#634—121no—
74C2×C3×C4×D3—32no—
75C2×C3×C4×D3—61no—
76C2×C3×Dic6—21no—
77C2×C3×Dic6C2×D312yes—
78C2×#627—34no—
79C2×#627—62no—
80C2×D3×Dic3—34no—
81C2×D3×Dic3—62no—
82D3×Dic6—316no—
83D3×Dic6—68no—
84C32×C4⋊C4—32no—
85#3750—34no—
86C4×C3⋊Dic3—34no—
87C4×C3⋊Dic3C2×D312yes—
88C3×(C2×D3)⋊C4—34no—
89C3×C4⋊Dic3—22no—
90C3×C4⋊Dic3—34no—
91C3×C4⋊Dic3C2×D312yes—
92C3×C4⋊Dic3C2×D312yes—
93C3×Dic3⋊C4—34no—
94C3×C4×Dic3—32no—
95C3×C4×Dic3—32no—
96C3×C4×Dic3—34no—
97C3×C4×Dic3—34no—
98C3×C4×Dic3—34no—
99C3×C4×Dic3—62no—
100#3724—32no—
101#3724—34no—
102#3724—34no—
103#3724—64no—
104Dic3⋊Dic3—32no—
105Dic3⋊Dic3—32no—
106Dic3⋊Dic3C2×D314yes—
107#3722—31no—
108#3722—31no—
109#3722C2×D312yes—
110(C2×D3)⋊Dic3—32no—
111(C2×D3)⋊Dic3—32no—
112Dic32—34no—
113Dic32—34no—
114Dic32—64no—
115Dic32C2×D314yes—
116Dic32C2×D312yes—
117C22×C33C2411yescharacteristic
118C2×C3×C3⋊D3—22no—
119C2×C3×C3⋊D3C22×C412yes—
120C33×C4C22×C412yes—
121C3⋊(C3⋊Dic3)C22×C414yes—
122C3×C3⋊Dic3—22no—
123C3×C3⋊Dic3—22no—
124C3×C3⋊Dic3C22×C412yes—
125C32×Dic3—24no—
126C32×Dic3C22×C414yes—
127C2×D3×Q8—92no—
128C2×D3×Q8—181no—
129C22×Dic6—91no—
130C3×C4×Q8—92no—
131C3×#117—91no—
132Q8×Dic3—32no—
133#926—91no—
134#892—32no—
135#892—94no—
136#887—32no—
137#887—94no—
138#887—92no—
139#887—92no—
140#887—182no—
141#877—92no—
142#877—92no—
143#872—94no—
144#872—92no—
145C4×Dic6—94no—
146C4×Dic6—94no—
147C4×Dic6—92no—
148C22×C3⋊D3C22×D311yescharacteristic
149C22×C3×D3—32no—
150C22×C3×D3—61no—
151C32×Q8—34no—
152C2×C32×C4—22no—
153C2×C32×C4—21no—
154C2×C32×C4—32no—
155C2×C32×C4—34no—
156C2×C32×C4—41no—
157C2×C32×C4C22×D311yescharacteristic
158C2×C32×C4C22×D312yes—
159C2×C3⋊Dic3—32no—
160C2×C3⋊Dic3—34no—
161C2×C3⋊Dic3—64no—
162C2×C3⋊Dic3—62no—
163C2×C3⋊Dic3—122no—
164C2×C3⋊Dic3C22×D312yes—
165C2×C3⋊Dic3C22×D312yes—
166C2×C3⋊Dic3C22×D311yescharacteristic
167C4×C3⋊D3—38no—
168C4×C3⋊D3C4×D314yes—
169#634—38no—
170#634—34no—
171#634—38no—
172#634—64no—
173#634—68no—
174#634—124no—
175C2×C3×Dic3—22no—
176C2×C3×Dic3—24no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—32no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3—62no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3—62no—
185C2×C3×Dic3C22×D314yes—
186C2×C3×Dic3C22×D314yes—
187C2×C3×Dic3C22×D314yes—
188C3×C4×D3—38no—
189C3×C4×D3—64no—
190C3×Dic6—24no—
191C3×Dic6C4×D318yes—
192#627—316no—
193#627—68no—
194D3×Dic3—316no—
195D3×Dic3—68no—
196#519—271noSylow
197C2×C33C23×C411yescharacteristic
198C2×C332−1+412yes—
199C3×C3⋊D3—22no—
200C2×C3×Q8—32no—
201C2×C3×Q8—61no—
202C22×C3×C4—91no—
203C22×Dic3—92no—
204D3×Q8—916no—
205D3×Q8—188no—
206C2×C4×D3—32no—
207C2×C4×D3—61no—
208C2×C4×D3—94no—
209C2×C4×D3—182no—
210C2×Dic6—94no—
211C2×Dic6—92no—
212C2×Dic6—91no—
213C2×Dic6—92no—
214C2×Dic6—98no—
215C2×Dic6—182no—
216C2×Dic6—182no—
217C2×Dic6—181no—
218C2×Dic6—361no—
219C2×Dic6D3211yescharacteristic
220C3×C4⋊C4—32no—
221C3×C4⋊C4—32no—
222C3×C4⋊C4—62no—
223C3×C4⋊C4—94no—
224C3×C22⋊C4—92no—
225C3×C42—92no—
226C3×C42—94no—
227C3×C42—92no—
228#217—91no—
229#217—91no—
230(C2×D3)⋊C4—34no—
231(C2×D3)⋊C4—92no—
232(C2×D3)⋊C4—92no—
233C4⋊Dic3—34no—
234C4⋊Dic3—92no—
235C4⋊Dic3D3212yes—
236Dic3⋊C4—34no—
237Dic3⋊C4—92no—
238Dic3⋊C4—92no—
239Dic3⋊C4—94no—
240Dic3⋊C4—94no—
241Dic3⋊C4—94no—
242Dic3⋊C4—184no—
243C4×Dic3—32no—
244C4×Dic3—32no—
245C4×Dic3—34no—
246C4×Dic3—34no—
247C4×Dic3—34no—
248C4×Dic3—62no—
249C4×Dic3—94no—
250C4×Dic3—94no—
251C4×Dic3—94no—
252C4×Dic3—184no—
253C22×C32—22no—
254C22×C32—21no—
255C22×C32—41no—
256C22×C32C23×D312yes—
257C22×C32C23×D311yescharacteristic
258C2×C3⋊D3—22no—
259C2×C3⋊D3C2×C4×D312yes—
260C2×C3×D3—34no—
261C2×C3×D3—62no—
262C2×C3×D3—64no—
263C2×C3×D3—62no—
264C2×C3×D3—62no—
265C32×C4—24no—
266C32×C4—22no—
267C32×C4—34no—
268C32×C4—42no—
269C32×C4—64no—
270C32×C4C2×C4×D312yes—
271C32×C4C2×C4×D314yes—
272C3⋊Dic3—22no—
273C3⋊Dic3—22no—
274C3⋊Dic3—38no—
275C3⋊Dic3—34no—
276C3⋊Dic3—64no—
277C3⋊Dic3—68no—
278C3⋊Dic3—124no—
279C3⋊Dic3C2×C4×D312yes—
280C3×Dic3—24no—
281C3×Dic3—24no—
282C3×Dic3—24no—
283C3×Dic3—34no—
284C3×Dic3—44no—
285C3×Dic3—62no—
286C3×Dic3—64no—
287C3×Dic3—62no—
288C3×Dic3—62no—
289C3×Dic3—62no—
290C3×Dic3—62no—
291C3×Dic3—122no—
292C3×Dic3C2×C4×D318yes—
293C22×Q8—271no—
294C4×Q8—92no—
295C4×Q8—274no—
296C4×Q8—272no—
297#117—91no—
298#117—272no—
299#117—271no—
300#117—272no—
301C33#51911yescharacteristic, complemented, Sylow
302C23×C3—91no—
303C22×D3—32no—
304C22×D3—61no—
305C3×Q8—38no—
306C3×Q8—64no—
307C2×C3×C4—22no—
308C2×C3×C4—21no—
309C2×C3×C4—34no—
310C2×C3×C4—34no—
311C2×C3×C4—34no—
312C2×C3×C4—41no—
313C2×C3×C4—62no—
314C2×C3×C4—64no—
315C2×C3×C4—91no—
316C2×C3×C4—94no—
317C2×C3×C4—92no—
318C2×C3×C4—92no—
319C2×C3×C4C2×D3212yes—
320C2×C3×C4C2×D3211yescharacteristic
321C2×Dic3—32no—
322C2×Dic3—32no—
323C2×Dic3—32no—
324C2×Dic3—32no—
325C2×Dic3—32no—
326C2×Dic3—62no—
327C2×Dic3—61no—
328C2×Dic3—62no—
329C2×Dic3—94no—
330C2×Dic3—92no—
331C2×Dic3—98no—
332C2×Dic3—184no—
333C2×Dic3—182no—
334C2×Dic3—362no—
335C2×Dic3C2×D3214yes—
336C2×Dic3C2×D3212yes—
337C4×D3—38no—
338C4×D3—64no—
339C4×D3—916no—
340C4×D3—188no—
341Dic6—916no—
342Dic6—98no—
343Dic6—94no—
344Dic6—98no—
345Dic6—188no—
346Dic6—188no—
347Dic6—184no—
348Dic6—364no—
349Dic6#62414yes—
350C2×C32—22no—
351C2×C32—22no—
352C2×C32—22no—
353C2×C32—22no—
354C2×C32—21no—
355C2×C32—41no—
356C2×C32—42no—
357C2×C32C22×C4×D311yescharacteristic
358C2×C32C22×C4×D312yes—
359C2×C32#100712yes—
360C2×C32#101012yes—
361C2×C32#101012yes—
362C3⋊D3—22no—
363C3×D3—64no—
364C3×D3—64no—
365C2×Q8—31no—
366C2×Q8—272no—
367C2×Q8—278no—
368C2×Q8—271no—
369C22×C4—91no—
370C22×C4—272no—
371C4⋊C4—32no—
372C4⋊C4—94no—
373C4⋊C4—274no—
374C4⋊C4—272no—
375C22⋊C4—92no—
376C22⋊C4—271no—
377C22⋊C4—271no—
378C42—92no—
379C42—94no—
380C42—92no—
381C42—274no—
382C22×C3—22no—
383C22×C3—21no—
384C22×C3—41no—
385C22×C3—92no—
386C22×C3—182no—
387C22×C3C22×D3212yes—
388C22×C3C22×D3211yescharacteristic
389C2×D3—34no—
390C2×D3—62no—
391C2×D3—64no—
392C2×D3—62no—
393C2×D3—62no—
394C3×C4—22no—
395C3×C4—24no—
396C3×C4—38no—
397C3×C4—42no—
398C3×C4—64no—
399C3×C4—64no—
400C3×C4—64no—
401C3×C4—92no—
402C3×C4—124no—
403C3×C4—182no—
404C3×C4—182no—
405C3×C4C4×D3214yes—
406C3×C4C2×#62412yes—
407Dic3—24no—
408Dic3—34no—
409Dic3—62no—
410Dic3—64no—
411Dic3—62no—
412Dic3—62no—
413Dic3—62no—
414Dic3—62no—
415Dic3—98no—
416Dic3—94no—
417Dic3—122no—
418Dic3—184no—
419Dic3—188no—
420Dic3—364no—
421Dic3C2×#62414yes—
422C32—21no—
423C32—22no—
424C32—41no—
425C32#551611yescharacteristic, complemented
426C32#557212yescomplemented
427C23—91no—
428Q8—34no—
429Q8—278no—
430Q8—274no—
431C2×C4—32no—
432C2×C4—34no—
433C2×C4—91no—
434C2×C4—94no—
435C2×C4—92no—
436C2×C4—92no—
437C2×C4—278no—
438C2×C4—272no—
439C2×C4D3311yescharacteristic
440C2×C3—22no—
441C2×C3—22no—
442C2×C3—22no—
443C2×C3—21no—
444C2×C3—22no—
445C2×C3—42no—
446C2×C3—41no—
447C2×C3—182no—
448C2×C3C2×C4×D3212yes—
449C2×C3#6430912yes—
450C2×C3#6431512yes—
451C2×C3#6431512yes—
452C2×C3C22×#62411yescharacteristic
453D3—64no—
454D3—64no—
455C22—92no—
456C22—182no—
457C22C2×D3311yescharacteristic
458C4—34no—
459C4—64no—
460C4—92no—
461C4—182no—
462C4—182no—
463C4—274no—
464C4D3×#62412yes—
465C3—21no—
466C3—22no—
467C3—41no—
468C3#1059435712yescomplemented
469C3#1059533911yescharacteristic, complemented
470C2—182no—
471C2#1172439412yes—
472C2C2×D3×#62411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutatorC2×C33C23×C4
FittingC2×C33×C4C23
socle, cosocleC22×C33C24
prime radicalC22×C3×C3⋊D3C23
hypercenterC2×C4D33
hypocenterC33#519
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #11723406C2semidirect
5. C3×#63843C2semidirect
6. #11722942C2semidirect
16. C3⋊D3×Dic6C4semidirect
47. #63843D3semidirect
301. C33#519semidirect
425. C32#5516semidirect
426. C32#5572semidirect
468. C3#10594357semidirect
469. C3#10595339semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×Dic62
#117236732
#117234062
C3×#638492
C3×#638432
#117229422
#117229212
#117229192
#105953393
#105943573