Finite Group Databasebeta

← #13168713

Subgroups — #13168713

Order 1728.

1192 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4⋊#6296C211yesmaximal, characteristic, complemented
2D3×C4⋊(C3⋊Dic3)C212yesmaximal, complemented
3Dic6×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C3×C4×#625C211yesmaximal, characteristic
5C3×#63856C212yesmaximal, complemented
6#11723018C212yesmaximal, complemented
7#11722981C214yesmaximal, complemented
8#11722925C212yesmaximal
9#10595407—31nomaximal
10#10594993—32nomaximal
11C2×C32×C4×D3C2212yes—
12C2×C32×Dic6C2211yescharacteristic
13C2×#6296C2212yes—
14C2×D3×C3⋊Dic3C2214yes—
15C2×C3×#625C2211yescharacteristic
16C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2212yes—
18C3×C4×C3⋊Dic3C2211yescharacteristic
19#88815C2212yes—
20Dic3⋊(C3⋊Dic3)C2214yes—
21Dic3⋊(C3⋊Dic3)C2212yes—
22Dic3×C3⋊Dic3C2212yes—
23C3×#3724C2211yescharacteristic
24C3×#3724C2214yes—
25C3×(C2×D3)⋊Dic3C2214yes—
26C3×(C2×D3)⋊Dic3C2212yes—
27Q8×C3⋊Dic3—31no—
28C2×C4⋊(C3⋊Dic3)—32no—
29C3×#946—32no—
30C3×C4×#67—32no—
31C3×#924—32no—
32#63982—34no—
33#67⋊Dic3—34no—
34#63919—31no—
35#63918—32no—
36#63902—32no—
37C4×#625D311yescharacteristic
38#63893—34no—
39#63893—34no—
40D3×C4⋊Dic3—32no—
41D3×C4⋊Dic3—32no—
42D3×C4⋊Dic3—62no—
43#63870—32no—
44#63856D312yescomplemented
45#63845—34no—
46Dic3×Dic6—32no—
47Dic3×Dic6—61no—
48C22×C32×D3C2312yes—
49C2×C33×C4C2311yescharacteristic
50C2×C3⋊(C3⋊Dic3)C2312yes—
51C2×C3×C3⋊Dic3C2311yescharacteristic
52C2×C3×C3⋊Dic3C2314yes—
53C2×C3×C3⋊Dic3C2311yescharacteristic
54C2×C32×Dic3C2312yes—
55C2×C32×Dic3C2312yes—
56C32×C4×D3—24no—
57C32×Dic6—22no—
58#6296—42no—
59D3×C3⋊Dic3—44no—
60C3×#625Q814yescomplemented
61#5635—92no—
62#5543—91no—
63C2×C32×Q8—31no—
64C22×C32×C4—32no—
65C22×C3⋊Dic3—34no—
66C2×C3×#67—32no—
67C2×C3×C4×D3—22no—
68C2×C3×C4×D3C2×D312yes—
69C2×C3×C4×D3C2×D312yes—
70C2×C3×Dic6—21no—
71C2×C3×Dic6C2×D312yes—
72C2×#627—32no—
73C2×#627—34no—
74C2×#625C2×D311yescharacteristic
75C2×D3×Dic3—34no—
76C2×D3×Dic3—34no—
77C2×D3×Dic3—64no—
78C4⋊(C3⋊Dic3)—32no—
79C4⋊(C3⋊Dic3)—31no—
80C4⋊(C3⋊Dic3)—34no—
81C4⋊(C3⋊Dic3)—32no—
82C4⋊(C3⋊Dic3)—61no—
83C4⋊(C3⋊Dic3)—62no—
84C4⋊(C3⋊Dic3)—121no—
85C4⋊(C3⋊Dic3)C2×D312yes—
86C4×C3⋊Dic3—32no—
87C4×C3⋊Dic3C2×D311yescharacteristic
88C3×#217—34no—
89C3×#217—32no—
90C3×(C2×D3)⋊C4—32no—
91C3×(C2×D3)⋊C4—34no—
92C3×C4⋊Dic3—32no—
93C3×C4⋊Dic3—32no—
94C3×C4⋊Dic3—62no—
95C3×Dic3⋊C4—34no—
96C3×Dic3⋊C4—32no—
97C3×Dic3⋊C4—34no—
98C3×C4×Dic3—32no—
99C3×C4×Dic3—61no—
100#3724—32no—
101#3724C2×D314yes—
102#3724C2×D311yescharacteristic
103Dic3⋊Dic3—34no—
104Dic3⋊Dic3—34no—
105Dic3⋊Dic3—34no—
106Dic3⋊Dic3—34no—
107Dic3⋊Dic3—62no—
108Dic3⋊Dic3—64no—
109(C2×D3)⋊Dic3C2×D314yes—
110(C2×D3)⋊Dic3C2×D312yes—
111Dic32—34no—
112Dic32—62no—
113C22×C33C2411yescharacteristic
114C2×C32×D3—22no—
115C2×C32×D3—22no—
116C2×C32×D3C2×Q814yes—
117C33×C4#4112yes—
118C3⋊(C3⋊Dic3)—22no—
119C3×C3⋊Dic3—24no—
120C3×C3⋊Dic3—21no—
121C3×C3⋊Dic3C2×Q812yes—
122C32×Dic3—22no—
123C32×Dic3—22no—
124C3×C22⋊Q8—92no—
125C3×C4×D4—91no—
126#946—32no—
127Q8×Dic3—92no—
128Q8×Dic3—181no—
129D4×Dic3—92no—
130C4×#67—32no—
131C2×C4⋊Dic3—92no—
132C2×C4⋊Dic3—92no—
133C2×C4⋊Dic3—182no—
134#924—32no—
135#897—94no—
136D3×C4⋊C4—92no—
137#890—92no—
138#889—94no—
139#888—92no—
140#67⋊C4—94no—
141#878—94no—
142#878—94no—
143#869—91no—
144C4×Dic6—91no—
145C23×C32—32no—
146C22×C3×D3—22no—
147C22×C3×D3C22×D312yes—
148C22×C3×D3C22×D312yes—
149C32×Q8—62no—
150C2×C32×C4—21no—
151C2×C32×C4—22no—
152C2×C32×C4—32no—
153C2×C32×C4—32no—
154C2×C32×C4—41no—
155C2×C32×C4—64no—
156C2×C32×C4C22×D312yes—
157C2×C32×C4C22×D311yescharacteristic
158C2×C3⋊Dic3—34no—
159C2×C3⋊Dic3—32no—
160C2×C3⋊Dic3—62no—
161C2×C3⋊Dic3—64no—
162C2×C3⋊Dic3—122no—
163C2×C3⋊Dic3—124no—
164C2×C3⋊Dic3C22×D311yescharacteristic
165C2×C3⋊Dic3C22×D314yes—
166C2×C3⋊Dic3C22×D311yescharacteristic
167C3×#67—38no—
168C2×C3×Dic3—22no—
169C2×C3×Dic3—22no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—34no—
173C2×C3×Dic3—34no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—64no—
177C2×C3×Dic3C22×D314yes—
178C2×C3×Dic3C22×D312yes—
179C2×C3×Dic3C22×D312yes—
180C3×C4×D3—28no—
181C3×C4×D3—24no—
182C3×C4×D3—24no—
183C3×Dic6—24no—
184C3×Dic6—24no—
185#627—122no—
186#627—124no—
187#625Dic614yescomplemented
188D3×Dic3—124no—
189D3×Dic3—124no—
190D3×Dic3—128no—
191#554—271noSylow
192C2×C33C22×Q811yescharacteristic
193C2×C33C2×#4111yescharacteristic
194C2×C33H(5,𝔽2)11yescharacteristic
195C32×D3—24no—
196C2×C3×Q8—32no—
197C2×C3×Q8—61no—
198C2×C3×D4—91no—
199C22×C3×C4—32no—
200C22×C3×C4—32no—
201C22×C3×C4—62no—
202C2×#67—32no—
203C22×Dic3—94no—
204C22×Dic3—94no—
205C22×Dic3—184no—
206C2×C4×D3—94no—
207C2×C4×D3D3212yes—
208C2×Dic6—92no—
209C2×Dic6—94no—
210C2×Dic6D3211yescharacteristic
211C3×C4⋊C4—94no—
212C3×C4⋊C4—91no—
213C3×C4⋊C4—92no—
214C3×C22⋊C4—92no—
215C3×C22⋊C4—94no—
216C3×C42—91no—
217#217—32no—
218#217—34no—
219(C2×D3)⋊C4—34no—
220(C2×D3)⋊C4—32no—
221C4⋊Dic3—32no—
222C4⋊Dic3—32no—
223C4⋊Dic3—62no—
224C4⋊Dic3—92no—
225C4⋊Dic3—94no—
226C4⋊Dic3—94no—
227C4⋊Dic3—94no—
228C4⋊Dic3—91no—
229C4⋊Dic3—92no—
230C4⋊Dic3—184no—
231C4⋊Dic3—182no—
232C4⋊Dic3—182no—
233C4⋊Dic3—181no—
234C4⋊Dic3—361no—
235Dic3⋊C4—34no—
236Dic3⋊C4—34no—
237Dic3⋊C4—32no—
238Dic3⋊C4—94no—
239Dic3⋊C4—92no—
240Dic3⋊C4—94no—
241C4×Dic3—32no—
242C4×Dic3—61no—
243C4×Dic3—94no—
244C4×Dic3—92no—
245C4×Dic3—182no—
246C22×C32—21no—
247C22×C32—22no—
248C22×C32—34no—
249C22×C32—41no—
250C22×C32—62no—
251C22×C32—62no—
252C22×C32C23×D311yescharacteristic
253C22×C32C23×D312yes—
254C2×C3×D3—24no—
255C2×C3×D3—24no—
256C2×C3×D3—22no—
257C2×C3×D3—22no—
258C2×C3×D3—22no—
259C2×C3×D3—22no—
260C2×C3×D3—24no—
261C2×C3×D3C2×Dic614yes—
262C2×C3×D3D3×Q814yes—
263C32×C4—22no—
264C32×C4—24no—
265C32×C4—42no—
266C32×C4—62no—
267C32×C4—62no—
268C32×C4#23714yes—
269C32×C4#23712yes—
270C3⋊Dic3—21no—
271C3⋊Dic3—24no—
272C3⋊Dic3—62no—
273C3⋊Dic3—64no—
274C3⋊Dic3—124no—
275C3⋊Dic3—122no—
276C3⋊Dic3—242no—
277C3⋊Dic3C2×Dic612yes—
278C3×Dic3—24no—
279C3×Dic3—24no—
280C3×Dic3—24no—
281C3×Dic3—22no—
282C3×Dic3—22no—
283C3×Dic3—34no—
284C3×Dic3—62no—
285C3×Dic3—64no—
286C3×Dic3—64no—
287C3×Dic3—62no—
288C3×Dic3—124no—
289C3×Dic3—121no—
290C4⋊Q8—271no—
291#125—272no—
292C22⋊Q8—92no—
293C22⋊Q8—274no—
294C4×Q8—271no—
295C4×D4—91no—
296C4×D4—272no—
297C2×C4⋊C4—272no—
298C33#55411yescharacteristic, complemented, Sylow
299C23×C3—32no—
300C23×C3—32no—
301C23×C3—62no—
302C22×D3C2×D3212yes—
303C3×Q8—64no—
304C3×Q8—64no—
305C3×D4—94no—
306C2×C3×C4—21no—
307C2×C3×C4—22no—
308C2×C3×C4—34no—
309C2×C3×C4—32no—
310C2×C3×C4—32no—
311C2×C3×C4—41no—
312C2×C3×C4—62no—
313C2×C3×C4—62no—
314C2×C3×C4—68no—
315C2×C3×C4—64no—
316C2×C3×C4—64no—
317C2×C3×C4—91no—
318C2×C3×C4—91no—
319C2×C3×C4—94no—
320C2×C3×C4C2×D3212yes—
321C2×C3×C4C2×D3211yescharacteristic
322#67—38no—
323C2×Dic3—32no—
324C2×Dic3—32no—
325C2×Dic3—34no—
326C2×Dic3—34no—
327C2×Dic3—61no—
328C2×Dic3—61no—
329C2×Dic3—64no—
330C2×Dic3—94no—
331C2×Dic3—92no—
332C2×Dic3—182no—
333C2×Dic3—184no—
334C2×Dic3—362no—
335C2×Dic3—364no—
336C2×Dic3—364no—
337C2×Dic3—368no—
338C2×Dic3C2×D3212yes—
339C2×Dic3C2×D3212yes—
340C4×D3—24no—
341C4×D3—364no—
342Dic6—22no—
343Dic6—364no—
344Dic6—362no—
345C2×C32—22no—
346C2×C32—22no—
347C2×C32—21no—
348C2×C32—21no—
349C2×C32—21no—
350C2×C32—22no—
351C2×C32—41no—
352C2×C32—41no—
353C2×C32—41no—
354C2×C32—64no—
355C2×C32C22×Dic611yescharacteristic
356C2×C32C2×#23712yes—
357C2×C32C2×#23711yescharacteristic
358C2×C32C2×D3×Q812yes—
359C2×C32#100911yescharacteristic
360C2×C32#100912yes—
361C3×D3—28no—
362C3×D3—24no—
363C3×D3—24no—
364C2×Q8—31no—
365C2×Q8—272no—
366C2×D4—91no—
367C22×C4—32no—
368C22×C4—274no—
369C4⋊C4—91no—
370C4⋊C4—94no—
371C4⋊C4—92no—
372C4⋊C4—274no—
373C4⋊C4—272no—
374C4⋊C4—272no—
375C4⋊C4—271no—
376C22⋊C4—92no—
377C22⋊C4—94no—
378C42—91no—
379C42—272no—
380C22×C3—21no—
381C22×C3—22no—
382C22×C3—34no—
383C22×C3—34no—
384C22×C3—41no—
385C22×C3—64no—
386C22×C3—64no—
387C22×C3—64no—
388C22×C3—62no—
389C22×C3—62no—
390C22×C3—62no—
391C22×C3—62no—
392C22×C3C22×D3212yes—
393C22×C3C22×D3211yescharacteristic
394C2×D3—22no—
395C2×D3—22no—
396C2×D3D3×Dic614yes—
397C3×C4—24no—
398C3×C4—22no—
399C3×C4—42no—
400C3×C4—64no—
401C3×C4—64no—
402C3×C4—62no—
403C3×C4—62no—
404C3×C4—64no—
405C3×C4—92no—
406C3×C4—184no—
407C3×C4—181no—
408C3×C4#380512yes—
409C3×C4#380514yes—
410Dic3—22no—
411Dic3—22no—
412Dic3—34no—
413Dic3—62no—
414Dic3—62no—
415Dic3—64no—
416Dic3—64no—
417Dic3—124no—
418Dic3—121no—
419Dic3—184no—
420Dic3—182no—
421Dic3—364no—
422Dic3—362no—
423Dic3—722no—
424C32—21no—
425C32—22no—
426C32—41no—
427C32#554311yescharacteristic, complemented
428C32#563512yescomplemented
429C23—32no—
430Q8—62no—
431Q8—1082no—
432D4—94no—
433C2×C4—32no—
434C2×C4—32no—
435C2×C4—64no—
436C2×C4—91no—
437C2×C4—91no—
438C2×C4—94no—
439C2×C4—272no—
440C2×C4—1084no—
441C2×C4D3311yescharacteristic
442C2×C3—21no—
443C2×C3—22no—
444C2×C3—22no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—22no—
448C2×C3—41no—
449C2×C3—41no—
450C2×C3—41no—
451C2×C3—68no—
452C2×C3—64no—
453C2×C3—64no—
454C2×C3C2×D3×Dic612yes—
455C2×C3#6430812yes—
456C2×C3D32×Q811yescharacteristic
457C2×C3#6432011yescharacteristic
458C2×C3C2×#380511yescharacteristic
459C2×C3C2×#380512yes—
460D3—24no—
461C22—34no—
462C22—62no—
463C22—62no—
464C22C2×D3311yescharacteristic
465C4—62no—
466C4—62no—
467C4—92no—
468C4—184no—
469C4—181no—
470C4—542no—
471C4#8896312yes—
472C3—22no—
473C3—21no—
474C3—41no—
475C3#1059499312yescomplemented
476C3#1059540711yescharacteristic, complemented
477C2—64no—
478C2D32×Dic611yescharacteristic
479C2#1172439011yescharacteristic
480C2C2×#8896311yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
prime radicalC2×C3×#625C22
hypercenterC2×C4D33
hypocenterC33#554
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C4⋊#6296C2semidirect
2. D3×C4⋊(C3⋊Dic3)C2semidirect
3. Dic6×C3⋊Dic3C2semidirect
5. C3×#63856C2semidirect
6. #11723018C2semidirect
7. #11722981C2semidirect
44. #63856D3semidirect
60. C3×#625Q8semidirect
187. #625Dic6semidirect
298. C33#554semidirect
427. C32#5543semidirect
428. C32#5635semidirect
475. C3#10594993semidirect
476. C3#10595407semidirect

Maximal subgroups

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

subgroupindex
C4⋊#62962
D3×C4⋊(C3⋊Dic3)2
Dic6×C3⋊Dic32
C3×C4×#6252
C3×#638562
#117230182
#117229812
#117229252
#105954073
#105949933