Finite Group Databasebeta

← #13168274

Subgroups — #13168274

Order 1728.

1134 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6294C211yesmaximal, characteristic
2C3⋊D3×C4⋊Dic3C211yesmaximal, characteristic
3#11723447C212yesmaximal, complemented
4C3×#63900C211yesmaximal, characteristic
5C3×#63872C212yesmaximal, complemented
6#11723017C214yesmaximal, complemented
7#11722911C212yesmaximal
8#11722908C212yesmaximal
9#10594482—31nomaximal
10D3×#890—32nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×#6294C2211yescharacteristic
13C2×#6294C2211yescharacteristic
14C2×Dic3×C3⋊D3C2212yes—
15C2×C3×#624C2212yes—
16C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
17C3×C4×C3⋊Dic3C2212yes—
18C32×C4⋊Dic3C2212yes—
19C32×C4⋊Dic3C2211yescharacteristic
20#88813C2212yes—
21#88807C2212yes—
22#88806C2214yes—
23Dic3⋊(C3⋊Dic3)C2214yes—
24C3×#3724C2212yes—
25C3×Dic3⋊Dic3C2214yes—
26C3×Dic3⋊Dic3C2214yes—
27C4⋊C4×C3⋊D3—31no—
28#64099—32no—
29C3×D3×C4⋊C4—32no—
30C3×#890—32no—
31C3×#890—32no—
32#63900—32no—
33#63900D311yescharacteristic
34D3×C4⋊Dic3—32no—
35D3×C4⋊Dic3—34no—
36D3×C4⋊Dic3—61no—
37#63883—32no—
38C4×#624—31no—
39D3×Dic3⋊C4—34no—
40C4×D3×Dic3—32no—
41#63872D312yescomplemented
42#63869—32no—
43#63869—32no—
44#63869—62no—
45#63847—34no—
46#63846—34no—
47#63845—34no—
48C22×C3×C3⋊D3C2311yescharacteristic
49C2×C33×C4C2311yescharacteristic
50C2×C3⋊(C3⋊Dic3)C2312yes—
51C2×C3×C3⋊Dic3C2312yes—
52C2×C3×C3⋊Dic3C2312yes—
53C2×C3×C3⋊Dic3C2311yescharacteristic
54C3×C4×C3⋊D3Q814yes—
55C2×C32×Dic3C2314yes—
56C2×C32×Dic3C2312yes—
57#6294—22no—
58#6294—22no—
59Dic3×C3⋊D3—24no—
60C3×#624—24no—
61D3×#125—92no—
62C2×#890—91no—
63C2×C4×C3⋊D3—32no—
64C2×C4×C3⋊D3C2×D311yescharacteristic
65C2×C3×C4×D3—32no—
66C2×C3×C4×D3—34no—
67C2×C3×C4×D3—61no—
68C2×#624—31no—
69C2×#624—31no—
70C2×#624C2×D312yes—
71C2×D3×Dic3—32no—
72C2×D3×Dic3—32no—
73C2×D3×Dic3—34no—
74C2×D3×Dic3—62no—
75C32×C4⋊C4—31no—
76C32×C4⋊C4—32no—
77C4⋊(C3⋊Dic3)—32no—
78C4⋊(C3⋊Dic3)—31no—
79C4⋊(C3⋊Dic3)—34no—
80C4⋊(C3⋊Dic3)—62no—
81C4⋊(C3⋊Dic3)—61no—
82C4⋊(C3⋊Dic3)—121no—
83#3750—34no—
84#3750—32no—
85C4×C3⋊Dic3C2×D312yes—
86C3×C4⋊Dic3—22no—
87C3×C4⋊Dic3—21no—
88C3×C4⋊Dic3—34no—
89C3×C4⋊Dic3—34no—
90C3×C4⋊Dic3C2×D312yes—
91C3×C4⋊Dic3C2×D312yes—
92C3×C4⋊Dic3C2×D312yes—
93C3×Dic3⋊C4—34no—
94C3×Dic3⋊C4—34no—
95C3×Dic3⋊C4—34no—
96C3×C4×Dic3—32no—
97C3×C4×Dic3—32no—
98C3×C4×Dic3—62no—
99#3724—34no—
100#3724—34no—
101#3724—64no—
102#3724C2×D312yes—
103Dic3⋊Dic3—34no—
104Dic3⋊Dic3—34no—
105Dic3⋊Dic3—34no—
106Dic3⋊Dic3—64no—
107Dic3⋊Dic3—62no—
108Dic3⋊Dic3C2×D314yes—
109Dic3⋊Dic3C2×D314yes—
110Dic32—32no—
111Dic32—32no—
112Dic32—32no—
113C22×C33C2411yescharacteristic
114C2×C3×C3⋊D3C2×Q812yes—
115C2×C3×C3⋊D3#4112yes—
116C2×C3×C3⋊D3#4112yes—
117C33×C4C2×Q812yes—
118C3⋊(C3⋊Dic3)—22no—
119C3×C3⋊Dic3—22no—
120C3×C3⋊Dic3—22no—
121C3×C3⋊Dic3C2×Q812yes—
122C32×Dic3—24no—
123C32×Dic3—22no—
124C3×#125—92no—
125C2×C3×C4⋊C4—91no—
126C2×C4⋊Dic3—92no—
127C2×C4⋊Dic3—91no—
128C2×Dic3⋊C4—92no—
129C2×C4×Dic3—91no—
130D3×C4⋊C4—32no—
131D3×C4⋊C4—94no—
132D3×C4⋊C4—94no—
133D3×C4⋊C4—92no—
134D3×C4⋊C4—181no—
135#890—32no—
136#890—32no—
137#890—92no—
138#890—92no—
139#890—94no—
140#890—182no—
141#889—94no—
142#889—94no—
143C42×D3—92no—
144#870—92no—
145C22×C3⋊D3C22×D311yescharacteristic
146C22×C3×D3—32no—
147C22×C3×D3—61no—
148C2×C32×C4—22no—
149C2×C32×C4—21no—
150C2×C32×C4—32no—
151C2×C32×C4—34no—
152C2×C32×C4—41no—
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C22×D312yes—
155C2×C3⋊Dic3—32no—
156C2×C3⋊Dic3—34no—
157C2×C3⋊Dic3—64no—
158C2×C3⋊Dic3—62no—
159C2×C3⋊Dic3—122no—
160C2×C3⋊Dic3C22×D312yes—
161C2×C3⋊Dic3C22×D312yes—
162C2×C3⋊Dic3C22×D311yescharacteristic
163C4×C3⋊D3—64no—
164C4×C3⋊D3Dic614yes—
165C2×C3×Dic3—24no—
166C2×C3×Dic3—22no—
167C2×C3×Dic3—32no—
168C2×C3×Dic3—32no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—62no—
174C2×C3×Dic3—62no—
175C2×C3×Dic3C22×D314yes—
176C2×C3×Dic3C22×D314yes—
177C2×C3×Dic3C22×D314yes—
178C3×C4×D3—38no—
179C3×C4×D3—64no—
180C3×C4×D3—68no—
181#624—24no—
182#624—62no—
183#624—62no—
184D3×Dic3—68no—
185D3×Dic3—64no—
186D3×Dic3—64no—
187D3×Dic3—124no—
188C2×#125—271noSylow
189C2×C33C22×Q811yescharacteristic
190C2×C33C2×#4111yescharacteristic
191C2×C33C2×#4111yescharacteristic
192C3×C3⋊D3#12514yescomplemented
193C22×C3×C4—92no—
194C22×C3×C4—91no—
195C22×Dic3—91no—
196C22×Dic3—91no—
197C22×Dic3—92no—
198C2×C4×D3—32no—
199C2×C4×D3—34no—
200C2×C4×D3—61no—
201C2×C4×D3—92no—
202C2×C4×D3—92no—
203C2×C4×D3—94no—
204C2×C4×D3—182no—
205C3×C4⋊C4—32no—
206C3×C4⋊C4—32no—
207C3×C4⋊C4—32no—
208C3×C4⋊C4—62no—
209C3×C4⋊C4—61no—
210C3×C4⋊C4—94no—
211C3×C4⋊C4—94no—
212C3×C4⋊C4—92no—
213C3×C42—92no—
214C4⋊Dic3—34no—
215C4⋊Dic3—34no—
216C4⋊Dic3—94no—
217C4⋊Dic3—94no—
218C4⋊Dic3—92no—
219C4⋊Dic3—91no—
220C4⋊Dic3—92no—
221C4⋊Dic3—184no—
222C4⋊Dic3—182no—
223C4⋊Dic3—181no—
224C4⋊Dic3—361no—
225C4⋊Dic3D3212yes—
226C4⋊Dic3D3211yescharacteristic
227Dic3⋊C4—34no—
228Dic3⋊C4—34no—
229Dic3⋊C4—34no—
230Dic3⋊C4—94no—
231Dic3⋊C4—94no—
232Dic3⋊C4—94no—
233Dic3⋊C4—94no—
234Dic3⋊C4—94no—
235Dic3⋊C4—184no—
236Dic3⋊C4—182no—
237C4×Dic3—32no—
238C4×Dic3—32no—
239C4×Dic3—62no—
240C4×Dic3—92no—
241C4×Dic3—92no—
242C4×Dic3—92no—
243C22×C32—22no—
244C22×C32—21no—
245C22×C32—41no—
246C22×C32C23×D311yescharacteristic
247C22×C32C23×D312yes—
248C2×C3⋊D3C2×Dic612yes—
249C2×C3⋊D3#23712yes—
250C2×C3⋊D3#23912yes—
251C2×C3×D3—34no—
252C2×C3×D3—34no—
253C2×C3×D3—34no—
254C2×C3×D3—62no—
255C2×C3×D3—62no—
256C2×C3×D3—62no—
257C32×C4—24no—
258C32×C4—22no—
259C32×C4—42no—
260C32×C4—62no—
261C32×C4—64no—
262C32×C4C2×Dic612yes—
263C32×C4D3×Q814yes—
264C3⋊Dic3—22no—
265C3⋊Dic3—22no—
266C3⋊Dic3—62no—
267C3⋊Dic3—64no—
268C3⋊Dic3—122no—
269C3⋊Dic3—124no—
270C3⋊Dic3—242no—
271C3⋊Dic3C2×Dic612yes—
272C3×Dic3—24no—
273C3×Dic3—24no—
274C3×Dic3—24no—
275C3×Dic3—34no—
276C3×Dic3—42no—
277C3×Dic3—44no—
278C3×Dic3—62no—
279C3×Dic3—62no—
280C3×Dic3—62no—
281C3×Dic3—62no—
282C3×Dic3—62no—
283C3×Dic3—122no—
284C3×Dic3—122no—
285#125—92no—
286#125—272no—
287#125—274no—
288C2×C4⋊C4—91no—
289C2×C4⋊C4—271no—
290C2×C4⋊C4—272no—
291C2×C4⋊C4—272no—
292C2×C42—271no—
293C33C2×#12511yescharacteristic, complemented, Sylow
294C23×C3—91no—
295C22×D3—32no—
296C22×D3—61no—
297C2×C3×C4—22no—
298C2×C3×C4—21no—
299C2×C3×C4—34no—
300C2×C3×C4—34no—
301C2×C3×C4—34no—
302C2×C3×C4—41no—
303C2×C3×C4—64no—
304C2×C3×C4—62no—
305C2×C3×C4—91no—
306C2×C3×C4—94no—
307C2×C3×C4—92no—
308C2×C3×C4—92no—
309C2×C3×C4—184no—
310C2×C3×C4C2×D3212yes—
311C2×C3×C4C2×D3211yescharacteristic
312C2×Dic3—32no—
313C2×Dic3—32no—
314C2×Dic3—32no—
315C2×Dic3—32no—
316C2×Dic3—32no—
317C2×Dic3—61no—
318C2×Dic3—62no—
319C2×Dic3—62no—
320C2×Dic3—94no—
321C2×Dic3—92no—
322C2×Dic3—184no—
323C2×Dic3—182no—
324C2×Dic3—184no—
325C2×Dic3—182no—
326C2×Dic3—182no—
327C2×Dic3—362no—
328C2×Dic3C2×D3214yes—
329C2×Dic3C2×D3212yes—
330C4×D3—38no—
331C4×D3—64no—
332C4×D3—68no—
333C4×D3—184no—
334C4×D3—184no—
335C4×D3—188no—
336C4×D3—364no—
337C2×C32—22no—
338C2×C32—21no—
339C2×C32—22no—
340C2×C32—21no—
341C2×C32—22no—
342C2×C32—21no—
343C2×C32—41no—
344C2×C32—41no—
345C2×C32—41no—
346C2×C32C22×Dic611yescharacteristic
347C2×C32C2×#23711yescharacteristic
348C2×C32C2×D3×Q812yes—
349C2×C32C2×#23911yescharacteristic
350C2×C32D3×#4112yes—
351C2×C32D3×#4112yes—
352C3⋊D3#89014yescomplemented
353C3×D3—38no—
354C3×D3—64no—
355C22×C4—92no—
356C22×C4—91no—
357C22×C4—272no—
358C22×C4—271no—
359C22×C4—271no—
360C4⋊C4—32no—
361C4⋊C4—31no—
362C4⋊C4—92no—
363C4⋊C4—94no—
364C4⋊C4—94no—
365C4⋊C4—274no—
366C4⋊C4—272no—
367C4⋊C4—274no—
368C4⋊C4—271no—
369C42—92no—
370C42—272no—
371C22×C3—22no—
372C22×C3—21no—
373C22×C3—41no—
374C22×C3—92no—
375C22×C3—92no—
376C22×C3—92no—
377C22×C3C22×D3212yes—
378C22×C3C22×D3211yescharacteristic
379C2×D3—34no—
380C2×D3—34no—
381C2×D3—34no—
382C2×D3—62no—
383C2×D3—62no—
384C2×D3—62no—
385C3×C4—22no—
386C3×C4—24no—
387C3×C4—42no—
388C3×C4—64no—
389C3×C4—64no—
390C3×C4—64no—
391C3×C4—92no—
392C3×C4—122no—
393C3×C4—124no—
394C3×C4—182no—
395C3×C4—182no—
396C3×C4D3×Dic614yes—
397C3×C4#379712yes—
398Dic3—24no—
399Dic3—22no—
400Dic3—34no—
401Dic3—62no—
402Dic3—62no—
403Dic3—62no—
404Dic3—62no—
405Dic3—62no—
406Dic3—122no—
407Dic3—122no—
408Dic3—182no—
409Dic3—184no—
410Dic3—362no—
411Dic3—364no—
412Dic3—722no—
413C32—22no—
414C32—21no—
415C32—41no—
416C32C2×#89011yescharacteristic, complemented
417C32D3×#12512yescomplemented
418C23—91no—
419C2×C4—32no—
420C2×C4—34no—
421C2×C4—91no—
422C2×C4—94no—
423C2×C4—92no—
424C2×C4—92no—
425C2×C4—184no—
426C2×C4—272no—
427C2×C4—544no—
428C2×C4—542no—
429C2×C4—542no—
430C2×C4D3311yescharacteristic
431C2×C3—22no—
432C2×C3—21no—
433C2×C3—22no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—22no—
437C2×C3—41no—
438C2×C3—41no—
439C2×C3—41no—
440C2×C3—94no—
441C2×C3C2×D3×Dic612yes—
442C2×C3C2×#379711yescharacteristic
443C2×C3D3×#23712yes—
444C2×C3#6431211yescharacteristic
445C2×C3D3×#23912yes—
446C2×C3#6431911yescharacteristic
447D3—38no—
448D3—64no—
449C22—92no—
450C22—92no—
451C22—92no—
452C22C2×D3311yescharacteristic
453C4—64no—
454C4—62no—
455C4—92no—
456C4—182no—
457C4—182no—
458C4—542no—
459C4#8895712yes—
460C3—21no—
461C3—22no—
462C3—41no—
463C3D3×#89012yescomplemented
464C3#1059448211yescharacteristic, complemented
465C2—94no—
466C2#1172437411yescharacteristic
467C2#1172437711yescharacteristic
468C2C2×#8895711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
prime radicalC22×C3×C3⋊D3C23
hypercenterC2×C4D33
hypocenterC33C2×#125
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #11723447C2semidirect
5. C3×#63872C2semidirect
6. #11723017C2semidirect
41. #63872D3semidirect
192. C3×C3⋊D3#125semidirect
293. C33C2×#125semidirect
352. C3⋊D3#890semidirect
416. C32C2×#890semidirect
417. C32D3×#125semidirect
463. C3D3×#890semidirect
464. C3#10594482semidirect

Maximal subgroups

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

subgroupindex
C4×#62942
C3⋊D3×C4⋊Dic32
#117234472
C3×#639002
C3×#638722
#117230172
#117229112
#117229082
#105944823
D3×#8903