Finite Group Databasebeta

← #13169323

Subgroups — #13169323

Order 1728.

1044 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723678C211yesmaximal, characteristic
2#11723443C212yesmaximal
3#11723406C211yesmaximal, characteristic
4C3×C4×#627C211yesmaximal, characteristic
5C3×#63844C212yesmaximal
6#11723018C212yesmaximal
7#11723011C214yesmaximal
8Dic3×#627C212yesmaximal
9#10594595—31nomaximal
10#10593860—32nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×#634C2212yes—
13C2×C3×#627C2211yescharacteristic
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C32×C4⋊Dic3C2211yescharacteristic
16C32×C4×Dic3C2212yes—
17#88815C2212yes—
18#88815C2212yes—
19#88806C2212yes—
20Dic3⋊(C3⋊Dic3)C2214yes—
21Dic3⋊(C3⋊Dic3)C2214yes—
22Dic3×C3⋊Dic3C2212yes—
23C3×#3724C2214yes—
24C3×#3724C2211yescharacteristic
25C3×Dic3⋊Dic3C2212yes—
26C3×Dic32C2214yes—
27#64096—31no—
28#64078—32no—
29C3×#887—32no—
30C3×C4×Dic6—32no—
31C3×C4×Dic6—32no—
32C4×#627—32no—
33C4×#627D311yescharacteristic
34#63873—31no—
35#63870—32no—
36#63869—34no—
37#63867—34no—
38#63866—32no—
39#63866—32no—
40#63866—32no—
41#63866—62no—
42#63845—34no—
43#63845—34no—
44#63844—32no—
45#63844—61no—
46#63844D312yes—
47Dic3×Dic6—34no—
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)C2312yes—
50#6308—22no—
51C2×C3×C3⋊Dic3C2311yescharacteristic
52C2×C3×C3⋊Dic3C2314yes—
53C2×C3×C3⋊Dic3C2311yescharacteristic
54C3×#634—24no—
55C2×C32×Dic3C2312yes—
56C2×C32×Dic3C2312yes—
57C2×C32×Dic3C2312yes—
58C3×#627Q814yescomplemented
59#5570—91no—
60#5567—92no—
61C2×#634—32no—
62C2×#634—31no—
63C2×#634—61no—
64C2×#634—62no—
65C2×#634—121no—
66C2×#634C2×D312yes—
67C2×C3×Dic6—32no—
68C2×C3×Dic6—32no—
69C2×C3×Dic6—32no—
70C2×C3×Dic6—62no—
71C2×#627C2×D311yescharacteristic
72C32×C4⋊C4—31no—
73C32×C42—32no—
74C4⋊(C3⋊Dic3)—34no—
75C4⋊(C3⋊Dic3)—34no—
76#3750—32no—
77C4×C3⋊Dic3—32no—
78C4×C3⋊Dic3C2×D311yescharacteristic
79C3×C4⋊Dic3—21no—
80C3×C4⋊Dic3—32no—
81C3×C4⋊Dic3C2×D312yes—
82C3×Dic3⋊C4—34no—
83C3×Dic3⋊C4—32no—
84C3×Dic3⋊C4—32no—
85C3×Dic3⋊C4—34no—
86C3×C4×Dic3—22no—
87C3×C4×Dic3—32no—
88C3×C4×Dic3—34no—
89C3×C4×Dic3—34no—
90C3×C4×Dic3—61no—
91C3×C4×Dic3C2×D312yes—
92C3×C4×Dic3C2×D312yes—
93#3724—32no—
94#3724—32no—
95#3724—34no—
96#3724—62no—
97#3724C2×D314yes—
98#3724C2×D311yescharacteristic
99Dic3⋊Dic3—34no—
100Dic3⋊Dic3—34no—
101Dic3⋊Dic3—34no—
102Dic3⋊Dic3—34no—
103Dic3⋊Dic3—34no—
104Dic3⋊Dic3—34no—
105Dic3⋊Dic3—64no—
106Dic3⋊Dic3—64no—
107Dic3⋊Dic3C2×D312yes—
108Dic32—34no—
109Dic32—62no—
110Dic32C2×D314yes—
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—91no—
121C3×C4×Q8—92no—
122Q8×Dic3—92no—
123#944—92no—
124#944—92no—
125#890—94no—
126#890—92no—
127#889—94no—
128#889—94no—
129#887—32no—
130#887—92no—
131#887—181no—
132#870—94no—
133#869—92no—
134#869—92no—
135#869—91no—
136#869—182no—
137C4×Dic6—32no—
138C4×Dic6—32no—
139C4×Dic6—94no—
140C4×Dic6—91no—
141C2×C32×C4—22no—
142C2×C32×C4—21no—
143C2×C32×C4—32no—
144C2×C32×C4—32no—
145C2×C32×C4—32no—
146C2×C32×C4—41no—
147C2×C32×C4C22×D312yes—
148C2×C32×C4C22×D311yescharacteristic
149C2×C3⋊Dic3—34no—
150C2×C3⋊Dic3—32no—
151C2×C3⋊Dic3—62no—
152C2×C3⋊Dic3—64no—
153C2×C3⋊Dic3—122no—
154C2×C3⋊Dic3C22×D314yes—
155C2×C3⋊Dic3C22×D311yescharacteristic
156C2×C3⋊Dic3C22×D311yescharacteristic
157#634—24no—
158#634—62no—
159#634—64no—
160#634—64no—
161#634—124no—
162#634—124no—
163C2×C3×Dic3—22no—
164C2×C3×Dic3—22no—
165C2×C3×Dic3—22no—
166C2×C3×Dic3—32no—
167C2×C3×Dic3—34no—
168C2×C3×Dic3—32no—
169C2×C3×Dic3—34no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—64no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3C22×D314yes—
174C2×C3×Dic3C22×D312yes—
175C2×C3×Dic3C22×D312yes—
176C2×C3×Dic3C22×D312yes—
177C2×C3×Dic3C22×D312yes—
178C3×Dic6—38no—
179C3×Dic6—64no—
180C3×Dic6—64no—
181C3×Dic6—124no—
182#627Dic614yescomplemented
183#557—271noSylow
184C2×C33C22×Q811yescharacteristic
185C2×C33C2×#4111yescharacteristic
186C2×C332−1+411yescharacteristic
187C2×C3×Q8—91no—
188C2×C3×Q8—92no—
189C2×Dic6—32no—
190C2×Dic6—32no—
191C2×Dic6—32no—
192C2×Dic6—62no—
193C2×Dic6—91no—
194C2×Dic6—92no—
195C2×Dic6—181no—
196C2×Dic6—182no—
197C2×Dic6—361no—
198C3×C4⋊C4—32no—
199C3×C4⋊C4—61no—
200C3×C4⋊C4—94no—
201C3×C4⋊C4—91no—
202C3×C4⋊C4—92no—
203C3×C42—32no—
204C3×C42—32no—
205C3×C42—62no—
206C3×C42—94no—
207C3×C42—91no—
208C4⋊Dic3—32no—
209C4⋊Dic3—92no—
210C4⋊Dic3—92no—
211C4⋊Dic3—94no—
212C4⋊Dic3—94no—
213C4⋊Dic3—94no—
214C4⋊Dic3—94no—
215C4⋊Dic3—184no—
216C4⋊Dic3—184no—
217C4⋊Dic3D3211yescharacteristic
218Dic3⋊C4—32no—
219Dic3⋊C4—32no—
220Dic3⋊C4—34no—
221Dic3⋊C4—34no—
222Dic3⋊C4—94no—
223Dic3⋊C4—94no—
224Dic3⋊C4—94no—
225Dic3⋊C4—94no—
226Dic3⋊C4—92no—
227Dic3⋊C4—94no—
228Dic3⋊C4—182no—
229C4×Dic3—32no—
230C4×Dic3—34no—
231C4×Dic3—34no—
232C4×Dic3—61no—
233C4×Dic3—94no—
234C4×Dic3—92no—
235C4×Dic3—182no—
236C4×Dic3D3212yes—
237C22×C32—22no—
238C22×C32—21no—
239C22×C32—41no—
240C22×C32C23×D311yescharacteristic
241C22×C32C23×D312yes—
242C32×C4—22no—
243C32×C4—24no—
244C32×C4—34no—
245C32×C4—42no—
246C32×C4—62no—
247C32×C4—62no—
248C32×C4#23514yes—
249C32×C4#23912yes—
250C3⋊Dic3—24no—
251C3⋊Dic3—21no—
252C3⋊Dic3—62no—
253C3⋊Dic3—64no—
254C3⋊Dic3—64no—
255C3⋊Dic3—124no—
256C3⋊Dic3—124no—
257C3⋊Dic3C2×Dic612yes—
258C3×Dic3—24no—
259C3×Dic3—22no—
260C3×Dic3—22no—
261C3×Dic3—24no—
262C3×Dic3—24no—
263C3×Dic3—34no—
264C3×Dic3—42no—
265C3×Dic3—62no—
266C3×Dic3—64no—
267C3×Dic3—64no—
268C3×Dic3—62no—
269C3×Dic3—124no—
270C3×Dic3—121no—
271C3×Dic3C2×Dic614yes—
272C3×Dic3D3×Q814yes—
273C4⋊Q8—272no—
274C4⋊Q8—271no—
275#125—274no—
276#125—272no—
277C4×Q8—91no—
278C4×Q8—92no—
279C4×Q8—272no—
280C4×Q8—271no—
281C33#55711yescharacteristic, complemented, Sylow
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×D3211yescharacteristic
299C2×C3×C4C2×D3212yes—
300C2×Dic3—32no—
301C2×Dic3—32no—
302C2×Dic3—34no—
303C2×Dic3—34no—
304C2×Dic3—61no—
305C2×Dic3—61no—
306C2×Dic3—64no—
307C2×Dic3—94no—
308C2×Dic3—92no—
309C2×Dic3—182no—
310C2×Dic3—184no—
311C2×Dic3—362no—
312C2×Dic3C2×D3212yes—
313C2×Dic3C2×D3212yes—
314C2×Dic3C2×D3212yes—
315Dic6—38no—
316Dic6—64no—
317Dic6—64no—
318Dic6—124no—
319Dic6—184no—
320Dic6—182no—
321Dic6—184no—
322Dic6—364no—
323Dic6—364no—
324C2×C32—22no—
325C2×C32—22no—
326C2×C32—21no—
327C2×C32—22no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—41no—
331C2×C32—41no—
332C2×C32—41no—
333C2×C32C22×Dic611yescharacteristic
334C2×C32C2×#23512yes—
335C2×C32C2×D3×Q812yes—
336C2×C32C2×#23911yescharacteristic
337C2×C32#101011yescharacteristic
338C2×C32#101012yes—
339C2×Q8—91no—
340C2×Q8—92no—
341C2×Q8—271no—
342C4⋊C4—31no—
343C4⋊C4—91no—
344C4⋊C4—94no—
345C4⋊C4—92no—
346C4⋊C4—272no—
347C4⋊C4—274no—
348C4⋊C4—274no—
349C4⋊C4—272no—
350C4⋊C4—272no—
351C42—32no—
352C42—94no—
353C42—91no—
354C42—272no—
355C22×C3—21no—
356C22×C3—22no—
357C22×C3—41no—
358C22×C3C22×D3212yes—
359C22×C3C22×D3211yescharacteristic
360C3×C4—24no—
361C3×C4—22no—
362C3×C4—34no—
363C3×C4—34no—
364C3×C4—42no—
365C3×C4—64no—
366C3×C4—64no—
367C3×C4—62no—
368C3×C4—64no—
369C3×C4—62no—
370C3×C4—92no—
371C3×C4—122no—
372C3×C4—184no—
373C3×C4—181no—
374C3×C4#379812yes—
375C3×C4#379914yes—
376Dic3—22no—
377Dic3—22no—
378Dic3—34no—
379Dic3—62no—
380Dic3—62no—
381Dic3—64no—
382Dic3—64no—
383Dic3—124no—
384Dic3—121no—
385Dic3—184no—
386Dic3—182no—
387Dic3—184no—
388Dic3—364no—
389Dic3—364no—
390Dic3D3×Dic614yes—
391C32—21no—
392C32—22no—
393C32—41no—
394C32#556712yescomplemented
395C32#557011yescharacteristic, complemented
396Q8—94no—
397Q8—184no—
398Q8—542no—
399C2×C4—32no—
400C2×C4—32no—
401C2×C4—32no—
402C2×C4—91no—
403C2×C4—91no—
404C2×C4—94no—
405C2×C4—272no—
406C2×C4D3311yescharacteristic
407C2×C3—21no—
408C2×C3—22no—
409C2×C3—21no—
410C2×C3—22no—
411C2×C3—22no—
412C2×C3—21no—
413C2×C3—41no—
414C2×C3—41no—
415C2×C3—41no—
416C2×C3C2×D3×Dic612yes—
417C2×C3#6429812yes—
418C2×C3C2×#379811yescharacteristic
419C2×C3C2×#379912yes—
420C2×C3#6430911yescharacteristic
421C2×C3D32×Q811yescharacteristic
422C22C2×D3311yescharacteristic
423C4—34no—
424C4—62no—
425C4—62no—
426C4—92no—
427C4—181no—
428C4—184no—
429C4—542no—
430C4#8896512yes—
431C3—22no—
432C3—21no—
433C3—41no—
434C3#1059386012yescomplemented
435C3#1059459511yescharacteristic, complemented
436C2D32×Dic611yescharacteristic
437C2#1172439211yescharacteristic
438C2C2×#8896511yescharacteristic
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
182. #627Dic6semidirect
281. C33#557semidirect
394. C32#5567semidirect
395. C32#5570semidirect
434. C3#10593860semidirect
435. C3#10594595semidirect

Maximal subgroups

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

subgroupindex
#117236782
#117234432
#117234062
C3×C4×#6272
C3×#638442
#117230182
#117230112
Dic3×#6272
#105945953
#105938603