Finite Group Databasebeta

← #13169338

Subgroups — #13169338

Order 1728.

1036 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723678C211yesmaximal, characteristic
2Dic3×#634C212yesmaximal
3Dic6×C3⋊Dic3C211yesmaximal, characteristic
4C3×C4×#627C211yesmaximal, characteristic
5C3×#63866C212yesmaximal
6#11723015C212yesmaximal
7#11723013C214yesmaximal
8#11722941C212yesmaximal
9#10595338—31nomaximal
10#10593861—32nomaximal
11C2×C3×#634C2212yes—
12C2×C32×Dic6C2211yescharacteristic
13C2×C3×#627C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C32×C4×Dic3C2212yes—
17#88815C2212yes—
18#88815C2212yes—
19#88807C2214yes—
20Dic3⋊(C3⋊Dic3)C2212yes—
21Dic3×C3⋊Dic3C2214yes—
22Dic3×C3⋊Dic3C2212yes—
23C3×#3724C2211yescharacteristic
24C3×Dic3⋊Dic3C2212yes—
25C3×Dic3⋊Dic3C2214yes—
26C3×Dic3⋊Dic3C2214yes—
27Q8×C3⋊Dic3—31no—
28C4×#634—32no—
29C3×#944—32no—
30C3×#869—32no—
31C3×C4×Dic6—32no—
32C4×#627D311yescharacteristic
33#63873—31no—
34#63872—34no—
35#63870—32no—
36#63866—32no—
37#63866D312yes—
38#63846—34no—
39#63846—34no—
40#63845—34no—
41#63843—34no—
42Dic3×Dic6—32no—
43Dic3×Dic6—32no—
44Dic3×Dic6—32no—
45Dic3×Dic6—61no—
46Dic3×Dic6—62no—
47#63838—32no—
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)C2312yes—
50C2×C3×C3⋊Dic3C2311yescharacteristic
51C2×C3×C3⋊Dic3C2314yes—
52C2×C3×C3⋊Dic3C2311yescharacteristic
53C3×#634—24no—
54C2×C32×Dic3C2312yes—
55C2×C32×Dic3C2312yes—
56C2×C32×Dic3C2312yes—
57C32×Dic6—22no—
58C3×#627Q814yescomplemented
59#5721—92no—
60#5570—91no—
61C2×C32×Q8—31no—
62C2×#634C2×D312yes—
63C2×C3×Dic6—21no—
64C2×C3×Dic6—32no—
65C2×C3×Dic6—32no—
66C2×C3×Dic6—32no—
67C2×C3×Dic6—62no—
68C2×C3×Dic6C2×D312yes—
69C2×#627C2×D311yescharacteristic
70C32×C42—32no—
71C4⋊(C3⋊Dic3)—32no—
72C4⋊(C3⋊Dic3)—31no—
73C4⋊(C3⋊Dic3)—32no—
74C4⋊(C3⋊Dic3)—62no—
75C4⋊(C3⋊Dic3)—61no—
76C4⋊(C3⋊Dic3)—121no—
77#3750—34no—
78C4×C3⋊Dic3—34no—
79C4×C3⋊Dic3—32no—
80C4×C3⋊Dic3C2×D311yescharacteristic
81C3×C4⋊Dic3—34no—
82C3×C4⋊Dic3—34no—
83C3×C4⋊Dic3—32no—
84C3×Dic3⋊C4—34no—
85C3×Dic3⋊C4—34no—
86C3×Dic3⋊C4—32no—
87C3×Dic3⋊C4—32no—
88C3×C4×Dic3—22no—
89C3×C4×Dic3—32no—
90C3×C4×Dic3—61no—
91C3×C4×Dic3C2×D312yes—
92C3×C4×Dic3C2×D312yes—
93#3724—32no—
94#3724—32no—
95#3724C2×D311yescharacteristic
96Dic3⋊Dic3—34no—
97Dic3⋊Dic3—34no—
98Dic3⋊Dic3—34no—
99Dic3⋊Dic3—34no—
100Dic3⋊Dic3—34no—
101Dic3⋊Dic3—62no—
102Dic3⋊Dic3—64no—
103Dic3⋊Dic3C2×D314yes—
104Dic3⋊Dic3C2×D312yes—
105Dic3⋊Dic3C2×D314yes—
106Dic32—34no—
107Dic32—34no—
108Dic32—34no—
109Dic32—64no—
110Dic32—62no—
111C22×C33C2411yescharacteristic
112C33×C4#4112yes—
113C3⋊(C3⋊Dic3)—22no—
114C3×C3⋊Dic3—24no—
115C3×C3⋊Dic3—21no—
116C3×C3⋊Dic3C2×Q812yes—
117C32×Dic3—22no—
118C32×Dic3—22no—
119C32×Dic3C2×Q814yes—
120C3×C4⋊Q8—92no—
121C3×C4×Q8—91no—
122Q8×Dic3—92no—
123Q8×Dic3—92no—
124Q8×Dic3—92no—
125Q8×Dic3—181no—
126#944—32no—
127#944—92no—
128#890—94no—
129#890—92no—
130#890—94no—
131#889—94no—
132#889—94no—
133#887—94no—
134#869—32no—
135#869—91no—
136C4×Dic6—32no—
137C4×Dic6—92no—
138C4×Dic6—92no—
139C4×Dic6—91no—
140C4×Dic6—182no—
141C32×Q8—62no—
142C2×C32×C4—22no—
143C2×C32×C4—21no—
144C2×C32×C4—32no—
145C2×C32×C4—32no—
146C2×C32×C4—32no—
147C2×C32×C4—41no—
148C2×C32×C4C22×D312yes—
149C2×C32×C4C22×D311yescharacteristic
150C2×C3⋊Dic3—34no—
151C2×C3⋊Dic3—32no—
152C2×C3⋊Dic3—62no—
153C2×C3⋊Dic3—64no—
154C2×C3⋊Dic3—122no—
155C2×C3⋊Dic3C22×D314yes—
156C2×C3⋊Dic3C22×D311yescharacteristic
157C2×C3⋊Dic3C22×D311yescharacteristic
158#634—24no—
159C2×C3×Dic3—22no—
160C2×C3×Dic3—22no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—32no—
163C2×C3×Dic3—34no—
164C2×C3×Dic3—34no—
165C2×C3×Dic3—32no—
166C2×C3×Dic3—61no—
167C2×C3×Dic3—64no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3C22×D314yes—
170C2×C3×Dic3C22×D312yes—
171C2×C3×Dic3C22×D312yes—
172C2×C3×Dic3C22×D312yes—
173C2×C3×Dic3C22×D312yes—
174C3×Dic6—24no—
175C3×Dic6—24no—
176C3×Dic6—38no—
177C3×Dic6—68no—
178C3×Dic6—64no—
179C3×Dic6—64no—
180#627Dic614yescomplemented
181#557—271noSylow
182C2×C33C22×Q811yescharacteristic
183C2×C33C2×#4111yescharacteristic
184C2×C332−1+411yescharacteristic
185C2×C3×Q8—32no—
186C2×C3×Q8—61no—
187C2×C3×Q8—91no—
188C2×C3×Q8—92no—
189C2×Dic6—32no—
190C2×Dic6—32no—
191C2×Dic6—32no—
192C2×Dic6—62no—
193C2×Dic6D3211yescharacteristic
194C3×C4⋊C4—94no—
195C3×C4⋊C4—94no—
196C3×C4⋊C4—91no—
197C3×C4⋊C4—92no—
198C3×C42—32no—
199C3×C42—32no—
200C3×C42—62no—
201C3×C42—91no—
202C4⋊Dic3—32no—
203C4⋊Dic3—34no—
204C4⋊Dic3—34no—
205C4⋊Dic3—92no—
206C4⋊Dic3—92no—
207C4⋊Dic3—94no—
208C4⋊Dic3—91no—
209C4⋊Dic3—92no—
210C4⋊Dic3—94no—
211C4⋊Dic3—182no—
212C4⋊Dic3—182no—
213C4⋊Dic3—181no—
214C4⋊Dic3—361no—
215Dic3⋊C4—32no—
216Dic3⋊C4—32no—
217Dic3⋊C4—34no—
218Dic3⋊C4—34no—
219Dic3⋊C4—94no—
220Dic3⋊C4—94no—
221Dic3⋊C4—94no—
222Dic3⋊C4—94no—
223Dic3⋊C4—92no—
224Dic3⋊C4—184no—
225C4×Dic3—32no—
226C4×Dic3—61no—
227C4×Dic3—94no—
228C4×Dic3—94no—
229C4×Dic3—94no—
230C4×Dic3—92no—
231C4×Dic3—94no—
232C4×Dic3—184no—
233C4×Dic3—182no—
234C4×Dic3D3212yes—
235C22×C32—22no—
236C22×C32—21no—
237C22×C32—41no—
238C22×C32C23×D311yescharacteristic
239C22×C32C23×D312yes—
240C32×C4—24no—
241C32×C4—22no—
242C32×C4—34no—
243C32×C4—42no—
244C32×C4—62no—
245C32×C4—62no—
246C32×C4#23714yes—
247C32×C4#23912yes—
248C3⋊Dic3—24no—
249C3⋊Dic3—21no—
250C3⋊Dic3—64no—
251C3⋊Dic3—62no—
252C3⋊Dic3—124no—
253C3⋊Dic3—122no—
254C3⋊Dic3—242no—
255C3⋊Dic3C2×Dic612yes—
256C3×Dic3—24no—
257C3×Dic3—24no—
258C3×Dic3—22no—
259C3×Dic3—22no—
260C3×Dic3—24no—
261C3×Dic3—34no—
262C3×Dic3—42no—
263C3×Dic3—62no—
264C3×Dic3—68no—
265C3×Dic3—64no—
266C3×Dic3—64no—
267C3×Dic3—62no—
268C3×Dic3—121no—
269C3×Dic3C2×Dic614yes—
270C3×Dic3D3×Q814yes—
271C4⋊Q8—92no—
272C4⋊Q8—271no—
273#125—272no—
274#125—274no—
275C4×Q8—91no—
276C4×Q8—272no—
277C4×Q8—271no—
278C4×Q8—272no—
279C33#55711yescharacteristic, complemented, Sylow
280C3×Q8—64no—
281C3×Q8—64no—
282C3×Q8—94no—
283C3×Q8—184no—
284C2×C3×C4—21no—
285C2×C3×C4—22no—
286C2×C3×C4—32no—
287C2×C3×C4—32no—
288C2×C3×C4—34no—
289C2×C3×C4—32no—
290C2×C3×C4—32no—
291C2×C3×C4—41no—
292C2×C3×C4—62no—
293C2×C3×C4—62no—
294C2×C3×C4—62no—
295C2×C3×C4—91no—
296C2×C3×C4—94no—
297C2×C3×C4—91no—
298C2×C3×C4C2×D3212yes—
299C2×C3×C4C2×D3211yescharacteristic
300C2×Dic3—32no—
301C2×Dic3—32no—
302C2×Dic3—34no—
303C2×Dic3—34no—
304C2×Dic3—61no—
305C2×Dic3—64no—
306C2×Dic3—61no—
307C2×Dic3—92no—
308C2×Dic3—94no—
309C2×Dic3—184no—
310C2×Dic3—182no—
311C2×Dic3—362no—
312C2×Dic3C2×D3212yes—
313C2×Dic3C2×D3212yes—
314C2×Dic3C2×D3212yes—
315Dic6—22no—
316Dic6—38no—
317Dic6—68no—
318Dic6—64no—
319Dic6—64no—
320C2×C32—22no—
321C2×C32—22no—
322C2×C32—22no—
323C2×C32—21no—
324C2×C32—21no—
325C2×C32—21no—
326C2×C32—41no—
327C2×C32—41no—
328C2×C32—41no—
329C2×C32C22×Dic611yescharacteristic
330C2×C32C2×#23712yes—
331C2×C32C2×D3×Q812yes—
332C2×C32C2×#23911yescharacteristic
333C2×C32#100712yes—
334C2×C32#101011yescharacteristic
335C2×Q8—31no—
336C2×Q8—91no—
337C2×Q8—92no—
338C4⋊C4—91no—
339C4⋊C4—92no—
340C4⋊C4—94no—
341C4⋊C4—94no—
342C4⋊C4—274no—
343C4⋊C4—272no—
344C4⋊C4—272no—
345C4⋊C4—272no—
346C4⋊C4—271no—
347C42—32no—
348C42—91no—
349C42—274no—
350C42—272no—
351C22×C3—21no—
352C22×C3—22no—
353C22×C3—41no—
354C22×C3C22×D3212yes—
355C22×C3C22×D3211yescharacteristic
356C3×C4—24no—
357C3×C4—22no—
358C3×C4—34no—
359C3×C4—34no—
360C3×C4—42no—
361C3×C4—64no—
362C3×C4—64no—
363C3×C4—64no—
364C3×C4—62no—
365C3×C4—62no—
366C3×C4—92no—
367C3×C4—122no—
368C3×C4—184no—
369C3×C4—181no—
370C3×C4#379614yes—
371C3×C4#380512yes—
372Dic3—22no—
373Dic3—22no—
374Dic3—34no—
375Dic3—62no—
376Dic3—68no—
377Dic3—64no—
378Dic3—62no—
379Dic3—64no—
380Dic3—121no—
381Dic3—184no—
382Dic3—182no—
383Dic3—362no—
384Dic3—364no—
385Dic3—722no—
386Dic3D3×Dic614yes—
387C32—21no—
388C32—22no—
389C32—41no—
390C32#557011yescharacteristic, complemented
391C32#572112yescomplemented
392Q8—62no—
393Q8—94no—
394Q8—184no—
395C2×C4—32no—
396C2×C4—32no—
397C2×C4—32no—
398C2×C4—91no—
399C2×C4—91no—
400C2×C4—94no—
401C2×C4—272no—
402C2×C4D3311yescharacteristic
403C2×C3—21no—
404C2×C3—22no—
405C2×C3—21no—
406C2×C3—22no—
407C2×C3—22no—
408C2×C3—21no—
409C2×C3—41no—
410C2×C3—41no—
411C2×C3—41no—
412C2×C3C2×D3×Dic612yes—
413C2×C3C2×#379612yes—
414C2×C3#6429712yes—
415C2×C3#6431611yescharacteristic
416C2×C3D32×Q811yescharacteristic
417C2×C3C2×#380511yescharacteristic
418C22C2×D3311yescharacteristic
419C4—34no—
420C4—62no—
421C4—62no—
422C4—92no—
423C4—181no—
424C4—184no—
425C4—542no—
426C4#8896412yes—
427C3—22no—
428C3—21no—
429C3—41no—
430C3#1059386112yescomplemented
431C3#1059533811yescharacteristic, complemented
432C2D32×Dic611yescharacteristic
433C2#1172439511yescharacteristic
434C2C2×#8896411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, prime radical, cosocleC22×C33C24
FittingC2×C33×C4C23
hypercenterC2×C4D33
hypocenterC33#557
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
58. C3×#627Q8semidirect
180. #627Dic6semidirect
279. C33#557semidirect
390. C32#5570semidirect
391. C32#5721semidirect
430. C3#10593861semidirect
431. C3#10595338semidirect

Maximal subgroups

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

subgroupindex
#117236782
Dic3×#6342
Dic6×C3⋊Dic32
C3×C4×#6272
C3×#638662
#117230152
#117230132
#117229412
#105953383
#105938613