Finite Group Databasebeta

← #13162050

Subgroups — #13162050

Order 1728.

1666 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C3⋊Dic3C211yesmaximal, characteristic
2C2×C3×#3757C212yesmaximal
3C2×#88813C212yesmaximal, complemented
4#11722082C212yesmaximal, complemented
5#10591705—31nomaximal
6Dic3×#217—32nomaximal
7C24×C33C2211yescharacteristic
8C22×C3×C3⋊Dic3—24no—
9C22×C3×C3⋊Dic3C414yescomplemented
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C3×C3⋊Dic3C2214yes—
12C22×C3×C3⋊Dic3C2211yescharacteristic
13C3×#3757C418yescomplemented
14#88813—24no—
15C23×C3⋊Dic3D311yescharacteristic
16C23×C3×Dic3—32no—
17C23×C3×Dic3—61no—
18C2×#3757D312yes—
19C2×C3×#217—32no—
20C2×C3×#217—32no—
21C2×C3×#217—62no—
22C2×Dic32—34no—
23C2×Dic32—32no—
24#63579—34no—
25#63579—32no—
26C23×C33—22no—
27C23×C33—22no—
28C23×C33C2×C414yes—
29C23×C33C2×C412yes—
30C23×C33C2311yescharacteristic
31C2×C3×C3⋊Dic3—22no—
32C2×C3×C3⋊Dic3—24no—
33C2×C3×C3⋊Dic3—24no—
34C2×C3×C3⋊Dic3—28no—
35C2×C3×C3⋊Dic3C2×C412yes—
36C2×C3×C3⋊Dic3C2×C412yes—
37C2×C3×C3⋊Dic3C2×C418yescomplemented
38C2×C3×C3⋊Dic3D414yes—
39C4×#217—91no—
40Dic3×C22⋊C4—92no—
41C24×C32—21no—
42C24×C32—22no—
43C24×C32—41no—
44C24×C32C2×D311yescharacteristic
45C24×C32C2×D312yes—
46C22×C3⋊Dic3—24no—
47C22×C3⋊Dic3Dic314yescomplemented
48C22×C3⋊Dic3C2×D311yescharacteristic
49C22×C3⋊Dic3C2×D314yes—
50C22×C3⋊Dic3C2×D311yescharacteristic
51C22×C3×Dic3—32no—
52C22×C3×Dic3—38no—
53C22×C3×Dic3—34no—
54C22×C3×Dic3—34no—
55C22×C3×Dic3—32no—
56C22×C3×Dic3—68no—
57C22×C3×Dic3—64no—
58C22×C3×Dic3—61no—
59C22×C3×Dic3—61no—
60C22×C3×Dic3—68no—
61C22×C3×Dic3—64no—
62#3757Dic318yescomplemented
63C3×#217—38no—
64C3×#217—38no—
65C3×#217—68no—
66Dic32—68no—
67Dic32—64no—
68C22×C33—22no—
69C22×C33—22no—
70C22×C33—28no—
71C22×C33C4214yescomplemented
72C22×C33C22×C411yescharacteristic
73C22×C33C22×C412yes—
74C22×C33C2×D412yes—
75C22×C33#4112yes—
76C3×C3⋊Dic3—22no—
77C3×C3⋊Dic3—28no—
78C3×C3⋊Dic3C22⋊C414yescomplemented
79C23×C3×C4—91no—
80C23×Dic3—32no—
81C23×Dic3—61no—
82C2×C3×C22⋊C4—92no—
83C2×#217—32no—
84C2×#217—32no—
85C2×#217—62no—
86C2×C4×Dic3—94no—
87C2×C4×Dic3—92no—
88C3⋊#95—94no—
89C3⋊#95—92no—
90C23×C32—22no—
91C23×C32—22no—
92C23×C32—24no—
93C23×C32—24no—
94C23×C32—24no—
95C23×C32—24no—
96C23×C32—24no—
97C23×C32—24no—
98C23×C32—24no—
99C23×C32—22no—
100C23×C32—24no—
101C23×C32—22no—
102C23×C32—21no—
103C23×C32—44no—
104C23×C32—42no—
105C23×C32—41no—
106C23×C32—44no—
107C23×C32—44no—
108C23×C32—44no—
109C23×C32—44no—
110C23×C32C4×D314yes—
111C23×C32C4×D314yes—
112C23×C32C4×D314yes—
113C23×C32C2×Dic314yes—
114C23×C32C2×Dic312yes—
115C23×C32C22×D311yescharacteristic
116C23×C32C22×D312yes—
117C2×C3⋊Dic3—22no—
118C2×C3⋊Dic3—28no—
119C2×C3⋊Dic3—24no—
120C2×C3⋊Dic3—24no—
121C2×C3⋊Dic3C2×Dic312yes—
122C2×C3⋊Dic3C2×Dic312yes—
123C2×C3⋊Dic3C2×Dic318yescomplemented
124C2×C3⋊Dic3#6714yes—
125C2×C3×Dic3—34no—
126C2×C3×Dic3—38no—
127C2×C3×Dic3—38no—
128C2×C3×Dic3—38no—
129C2×C3×Dic3—34no—
130C2×C3×Dic3—62no—
131C2×C3×Dic3—62no—
132C2×C3×Dic3—64no—
133C2×C3×Dic3—68no—
134C2×C3×Dic3—616no—
135C2×C3×Dic3—616no—
136C2×C3×Dic3—64no—
137C2×C3×Dic3—64no—
138C2×C3×Dic3—64no—
139C2×C3×Dic3—64no—
140C2×C3×Dic3—64no—
141C2×C3×Dic3—64no—
142C2×C3×Dic3—124no—
143C2×C3×Dic3—124no—
144C4×C22⋊C4—271noSylow
145C2×C33—24no—
146C2×C33C2×C4211yescharacteristic
147C2×C33C2×C22⋊C411yescharacteristic
148C2×C33#11711yescharacteristic
149C2×C33C4×D414yes—
150C24×C3—21no—
151C24×C3—22no—
152C24×C3—41no—
153C24×C3D3211yescharacteristic
154C24×C3D3212yes—
155C22×C3×C4—91no—
156C22×C3×C4—94no—
157C22×C3×C4—91no—
158C22×C3×C4—94no—
159C22×C3×C4—184no—
160C22×Dic3—32no—
161C22×Dic3—38no—
162C22×Dic3—32no—
163C22×Dic3—34no—
164C22×Dic3—34no—
165C22×Dic3—64no—
166C22×Dic3—61no—
167C22×Dic3—61no—
168C22×Dic3—64no—
169C22×Dic3—68no—
170C22×Dic3—68no—
171C3×C22⋊C4—98no—
172#217—38no—
173#217—38no—
174#217—68no—
175C4×Dic3—188no—
176C4×Dic3—184no—
177C22×C32—22no—
178C22×C32—28no—
179C22×C32—22no—
180C22×C32—24no—
181C22×C32—24no—
182C22×C32—216no—
183C22×C32—24no—
184C22×C32—24no—
185C22×C32—28no—
186C22×C32—28no—
187C22×C32—24no—
188C22×C32—28no—
189C22×C32—22no—
190C22×C32—22no—
191C22×C32—22no—
192C22×C32—24no—
193C22×C32—22no—
194C22×C32—22no—
195C22×C32—21no—
196C22×C32—24no—
197C22×C32—22no—
198C22×C32—44no—
199C22×C32—41no—
200C22×C32—42no—
201C22×C32—42no—
202C22×C32—42no—
203C22×C32—48no—
204C22×C32—44no—
205C22×C32—44no—
206C22×C32—48no—
207C22×C32—44no—
208C22×C32—44no—
209C22×C32—416no—
210C22×C32C4×Dic314yescomplemented
211C22×C32C4×Dic318yescomplemented
212C22×C32C2×C4×D312yes—
213C22×C32C2×C4×D312yes—
214C22×C32C2×C4×D312yes—
215C22×C32#23512yes—
216C22×C32D3×D414yes—
217C22×C32#23714yes—
218C22×C32C22×Dic311yescharacteristic
219C22×C32C22×Dic312yes—
220C22×C32C2×#6712yes—
221C3⋊Dic3—22no—
222C3⋊Dic3—28no—
223C3⋊Dic3#21714yescomplemented
224C3×Dic3—38no—
225C3×Dic3—64no—
226C3×Dic3—64no—
227C3×Dic3—64no—
228C3×Dic3—68no—
229C3×Dic3—68no—
230C3×Dic3—128no—
231C23×C4—91no—
232C2×C22⋊C4—92no—
233C2×C42—272no—
234#95—272no—
235C33C4×C22⋊C411yescharacteristic, complemented, Sylow
236C23×C3—24no—
237C23×C3—24no—
238C23×C3—24no—
239C23×C3—22no—
240C23×C3—24no—
241C23×C3—22no—
242C23×C3—24no—
243C23×C3—22no—
244C23×C3—24no—
245C23×C3—24no—
246C23×C3—24no—
247C23×C3—22no—
248C23×C3—21no—
249C23×C3—44no—
250C23×C3—44no—
251C23×C3—44no—
252C23×C3—44no—
253C23×C3—44no—
254C23×C3—42no—
255C23×C3—41no—
256C23×C3D3×Dic314yes—
257C23×C3D3×Dic314yes—
258C23×C3D3×Dic314yes—
259C23×C3#62414yes—
260C23×C3#62412yes—
261C23×C3C2×D3212yes—
262C23×C3C2×D3211yescharacteristic
263C2×C3×C4—92no—
264C2×C3×C4—94no—
265C2×C3×C4—98no—
266C2×C3×C4—92no—
267C2×C3×C4—188no—
268C2×C3×C4—184no—
269C2×C3×C4—184no—
270C2×C3×C4—182no—
271C2×Dic3—34no—
272C2×Dic3—38no—
273C2×Dic3—34no—
274C2×Dic3—38no—
275C2×Dic3—38no—
276C2×Dic3—62no—
277C2×Dic3—62no—
278C2×Dic3—64no—
279C2×Dic3—68no—
280C2×Dic3—616no—
281C2×Dic3—616no—
282C2×Dic3—64no—
283C2×Dic3—64no—
284C2×Dic3—64no—
285C2×Dic3—64no—
286C2×Dic3—64no—
287C2×Dic3—64no—
288C2×Dic3—124no—
289C2×Dic3—124no—
290C2×C32—24no—
291C2×C32—28no—
292C2×C32—28no—
293C2×C32—24no—
294C2×C32—22no—
295C2×C32—21no—
296C2×C32—22no—
297C2×C32—21no—
298C2×C32—22no—
299C2×C32—24no—
300C2×C32—24no—
301C2×C32—21no—
302C2×C32—41no—
303C2×C32—41no—
304C2×C32—41no—
305C2×C32—44no—
306C2×C32—48no—
307C2×C32—48no—
308C2×C32#87712yes—
309C2×C32D3×C22⋊C412yes—
310C2×C32#67⋊C414yes—
311C2×C32C2×C4×Dic312yes—
312C2×C32C2×C4×Dic311yescharacteristic
313C2×C32#92611yescharacteristic
314C2×C32C4×#6714yes—
315C2×C32D4×Dic314yes—
316C2×C32C2×#21711yescharacteristic
317C24#111311yescharacteristic
318C22×C4—91no—
319C22×C4—94no—
320C22×C4—91no—
321C22×C4—94no—
322C22×C4—184no—
323C22⋊C4—98no—
324C42—544no—
325C22×C3—24no—
326C22×C3—24no—
327C22×C3—216no—
328C22×C3—28no—
329C22×C3—24no—
330C22×C3—22no—
331C22×C3—24no—
332C22×C3—28no—
333C22×C3—28no—
334C22×C3—22no—
335C22×C3—21no—
336C22×C3—22no—
337C22×C3—22no—
338C22×C3—22no—
339C22×C3—22no—
340C22×C3—24no—
341C22×C3—22no—
342C22×C3—22no—
343C22×C3—24no—
344C22×C3—24no—
345C22×C3—28no—
346C22×C3—44no—
347C22×C3—44no—
348C22×C3—48no—
349C22×C3—48no—
350C22×C3—44no—
351C22×C3—44no—
352C22×C3—416no—
353C22×C3—41no—
354C22×C3—42no—
355C22×C3—42no—
356C22×C3—42no—
357C22×C3—44no—
358C22×C3Dic3218yescomplemented
359C22×C3Dic3214yescomplemented
360C22×C3C2×D3×Dic312yes—
361C22×C3C2×D3×Dic312yes—
362C22×C3C2×D3×Dic312yes—
363C22×C3#380414yes—
364C22×C3#380512yes—
365C22×C3C2×#62412yes—
366C22×C3C2×#62411yescharacteristic
367C22×C3D3×#6714yes—
368C22×C3#381112yes—
369C3×C4—94no—
370C3×C4—188no—
371C3×C4—182no—
372Dic3—38no—
373Dic3—64no—
374Dic3—64no—
375Dic3—64no—
376Dic3—68no—
377Dic3—68no—
378Dic3—128no—
379C32—22no—
380C32—21no—
381C32—41no—
382C32Dic3×C22⋊C412yescomplemented
383C32C4×#21711yescharacteristic, complemented
384C23—22no—
385C23—22no—
386C23#629414yes—
387C23#629412yes—
388C23C2×#111311yescharacteristic
389C2×C4—92no—
390C2×C4—94no—
391C2×C4—98no—
392C2×C4—92no—
393C2×C4—188no—
394C2×C4—184no—
395C2×C4—184no—
396C2×C4—182no—
397C2×C3—28no—
398C2×C3—28no—
399C2×C3—24no—
400C2×C3—21no—
401C2×C3—22no—
402C2×C3—21no—
403C2×C3—22no—
404C2×C3—21no—
405C2×C3—22no—
406C2×C3—24no—
407C2×C3—24no—
408C2×C3—24no—
409C2×C3—48no—
410C2×C3—48no—
411C2×C3—41no—
412C2×C3—41no—
413C2×C3—41no—
414C2×C3—44no—
415C2×C3#6395214yes—
416C2×C3C2×Dic3212yes—
417C2×C3C2×Dic3211yescharacteristic
418C2×C3#6395512yes—
419C2×C3#6395711yescharacteristic
420C2×C3D3×#21712yes—
421C2×C3Dic3×#6714yes—
422C2×C3#67⋊Dic314yes—
423C2×C3#6397411yescharacteristic
424C22—22no—
425C22—22no—
426C22—28no—
427C22#8881314yescomplemented
428C22C2×#629411yescharacteristic
429C22C2×#629412yes—
430C22#8904612yes—
431C22#8904912yes—
432C4—94no—
433C4—182no—
434C4—188no—
435C3—21no—
436C3—22no—
437C3—41no—
438C3Dic3×#21712yescomplemented
439C3#1059170511yescharacteristic, complemented
440C2—24no—
441C2C2×#8881311yescharacteristic
442C2#1172371511yescharacteristic
443C2#1172372311yescharacteristic
444C2#1172372614yes—
subgroupG/H
center, FrattiniC23C2×#1113
commutatorC2×C33C2×C42
Fitting, prime radicalC24×C33C22
socle, cosocleC23×C33C23
hypercenterC24#1113
hypocenterC33C4×C22⋊C4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
3. C2×#88813C2semidirect
4. #11722082C2semidirect
9. C22×C3×C3⋊Dic3C4semidirect
13. C3×#3757C4semidirect
37. C2×C3×C3⋊Dic3C2×C4semidirect
47. C22×C3⋊Dic3Dic3semidirect
62. #3757Dic3semidirect
71. C22×C33C42semidirect
78. C3×C3⋊Dic3C22⋊C4semidirect
123. C2×C3⋊Dic3C2×Dic3semidirect
210. C22×C32C4×Dic3semidirect
211. C22×C32C4×Dic3semidirect
223. C3⋊Dic3#217semidirect
235. C33C4×C22⋊C4semidirect
358. C22×C3Dic32semidirect
359. C22×C3Dic32semidirect
382. C32Dic3×C22⋊C4semidirect
383. C32C4×#217semidirect
427. C22#88813semidirect
438. C3Dic3×#217semidirect
439. C3#10591705semidirect

Maximal subgroups

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

subgroupindex
C23×C3×C3⋊Dic32
C2×C3×#37572
C2×#888132
#117220822
#105917053
Dic3×#2173