Finite Group Databasebeta

← #13168259

Subgroups — #13168259

Order 1728.

1154 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723689C211yesmaximal, characteristic
2C4×Dic3×C3⋊D3C211yesmaximal, characteristic
3#11723450C212yesmaximal, complemented
4C3×#63900C211yesmaximal, characteristic
5C3×#63869C212yesmaximal, complemented
6#11723018C212yesmaximal, complemented
7#11723015C212yesmaximal, complemented
8#11722910C212yesmaximal
9#11722908C212yesmaximal
10D3×#890—32nomaximal
11#10593948—31nomaximal
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×#6294C2212yes—
14C2×Dic3×C3⋊D3C2211yescharacteristic
15C2×Dic3×C3⋊D3C2211yescharacteristic
16C2×C3×#624C2212yes—
17C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
18C3×C4⋊(C3⋊Dic3)C2212yes—
19C32×C4⋊Dic3C2212yes—
20C32×C4×Dic3C2211yescharacteristic
21#88815C2212yes—
22#88807C2214yes—
23Dic3⋊(C3⋊Dic3)C2214yes—
24Dic3×C3⋊Dic3C2211yescharacteristic
25Dic3×C3⋊Dic3C2211yescharacteristic
26C3×#3724C2212yes—
27C3×#3724C2214yes—
28C3×Dic3⋊Dic3C2214yes—
29#64099—32no—
30C42×C3⋊D3—31no—
31C3×D3×C4⋊C4—32no—
32C3×#890—32no—
33C3×#870—32no—
34#63900—31no—
35#63900D311yescharacteristic
36D3×C4⋊Dic3—32no—
37D3×C4⋊Dic3—34no—
38#63883—32no—
39#63883—32no—
40D3×Dic3⋊C4—34no—
41C4×D3×Dic3—32no—
42C4×D3×Dic3—61no—
43#63872—32no—
44#63872—32no—
45#63872—62no—
46#63870—32no—
47#63870—32no—
48#63869D312yescomplemented
49#63846—34no—
50#63845—34no—
51C22×C3×C3⋊D3C2311yescharacteristic
52C2×C33×C4C2311yescharacteristic
53C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
54C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
55C2×C3×C3⋊Dic3C2314yes—
56C2×C3×C3⋊Dic3C2311yescharacteristic
57C3×C4×C3⋊D3Q814yes—
58C2×C32×Dic3C2314yes—
59C2×C32×Dic3C2311yescharacteristic
60C2×C32×Dic3C2311yescharacteristic
61#6294—24no—
62Dic3×C3⋊D3—22no—
63Dic3×C3⋊D3—22no—
64C3×#624—24no—
65D3×#125—92no—
66C2×#890—91no—
67C2×C4×C3⋊D3—31no—
68C2×C4×C3⋊D3—31no—
69C2×C4×C3⋊D3C2×D311yescharacteristic
70C2×C3×C4×D3—32no—
71C2×C3×C4×D3—34no—
72C2×C3×C4×D3—61no—
73C2×#624—32no—
74C2×#624C2×D312yes—
75C2×D3×Dic3—34no—
76C2×D3×Dic3—32no—
77C2×D3×Dic3—32no—
78C2×D3×Dic3—61no—
79C2×D3×Dic3—61no—
80C32×C4⋊C4—32no—
81C32×C42—31no—
82C4⋊(C3⋊Dic3)—34no—
83C4⋊(C3⋊Dic3)C2×D312yes—
84#3750—34no—
85C4×C3⋊Dic3—31no—
86C4×C3⋊Dic3—32no—
87C4×C3⋊Dic3—31no—
88C4×C3⋊Dic3—31no—
89C4×C3⋊Dic3—62no—
90C4×C3⋊Dic3—61no—
91C4×C3⋊Dic3—121no—
92C3×C4⋊Dic3—22no—
93C3×C4⋊Dic3—32no—
94C3×C4⋊Dic3—32no—
95C3×C4⋊Dic3—34no—
96C3×C4⋊Dic3—62no—
97C3×C4⋊Dic3C2×D312yes—
98C3×C4⋊Dic3C2×D312yes—
99C3×Dic3⋊C4—34no—
100C3×Dic3⋊C4—34no—
101C3×Dic3⋊C4—34no—
102C3×Dic3⋊C4—34no—
103C3×C4×Dic3—21no—
104C3×C4×Dic3C2×D312yes—
105#3724—32no—
106#3724C2×D314yes—
107#3724C2×D312yes—
108Dic3⋊Dic3—34no—
109Dic3⋊Dic3—34no—
110Dic3⋊Dic3—34no—
111Dic3⋊Dic3—34no—
112Dic3⋊Dic3—34no—
113Dic3⋊Dic3—64no—
114Dic3⋊Dic3—64no—
115Dic3⋊Dic3C2×D314yes—
116Dic32—32no—
117Dic32—32no—
118Dic32—61no—
119Dic32—61no—
120C22×C33C2411yescharacteristic
121C2×C3×C3⋊D3C2×Q812yes—
122C2×C3×C3⋊D3#4112yes—
123C2×C3×C3⋊D3#4112yes—
124C33×C4C2×Q812yes—
125C3⋊(C3⋊Dic3)—21no—
126C3⋊(C3⋊Dic3)—21no—
127C3×C3⋊Dic3—24no—
128C3×C3⋊Dic3C2×Q812yes—
129C32×Dic3—21no—
130C32×Dic3—21no—
131C32×Dic3—24no—
132C3×#125—92no—
133C2×C3×C4⋊C4—91no—
134C2×C4⋊Dic3—91no—
135C2×C4⋊Dic3—92no—
136C2×Dic3⋊C4—92no—
137C2×C4×Dic3—91no—
138D3×C4⋊C4—32no—
139D3×C4⋊C4—94no—
140D3×C4⋊C4—94no—
141D3×C4⋊C4—92no—
142#890—32no—
143#890—92no—
144#890—92no—
145#890—92no—
146#890—92no—
147#890—92no—
148#890—182no—
149#889—94no—
150#889—94no—
151C42×D3—92no—
152C42×D3—181no—
153#870—32no—
154C22×C3⋊D3C22×D311yescharacteristic
155C22×C3×D3—32no—
156C22×C3×D3—61no—
157C2×C32×C4—21no—
158C2×C32×C4—22no—
159C2×C32×C4—31no—
160C2×C32×C4—31no—
161C2×C32×C4—34no—
162C2×C32×C4—41no—
163C2×C32×C4C22×D311yescharacteristic
164C2×C32×C4C22×D312yes—
165C2×C3⋊Dic3—32no—
166C2×C3⋊Dic3—32no—
167C2×C3⋊Dic3—31no—
168C2×C3⋊Dic3—31no—
169C2×C3⋊Dic3—61no—
170C2×C3⋊Dic3—61no—
171C2×C3⋊Dic3—62no—
172C2×C3⋊Dic3—62no—
173C2×C3⋊Dic3—121no—
174C2×C3⋊Dic3—121no—
175C2×C3⋊Dic3C22×D314yes—
176C2×C3⋊Dic3C22×D311yescharacteristic
177C4×C3⋊D3—62no—
178C4×C3⋊D3—62no—
179C4×C3⋊D3Dic614yes—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—24no—
183C2×C3×Dic3—32no—
184C2×C3×Dic3—34no—
185C2×C3×Dic3—34no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—64no—
188C2×C3×Dic3C22×D312yes—
189C2×C3×Dic3C22×D312yes—
190C2×C3×Dic3C22×D314yes—
191C2×C3×Dic3C22×D314yes—
192C3×C4×D3—38no—
193C3×C4×D3—64no—
194C3×C4×D3—68no—
195#624—24no—
196#624—64no—
197D3×Dic3—64no—
198D3×Dic3—64no—
199D3×Dic3—64no—
200D3×Dic3—64no—
201D3×Dic3—68no—
202C2×#125—271noSylow
203C2×C33C22×Q811yescharacteristic
204C2×C33C2×#4111yescharacteristic
205C2×C33C2×#4111yescharacteristic
206C3×C3⋊D3#12514yescomplemented
207C22×C3×C4—92no—
208C22×C3×C4—91no—
209C22×Dic3—92no—
210C22×Dic3—91no—
211C22×Dic3—91no—
212C2×C4×D3—32no—
213C2×C4×D3—34no—
214C2×C4×D3—61no—
215C2×C4×D3—94no—
216C2×C4×D3—92no—
217C2×C4×D3—92no—
218C2×C4×D3—181no—
219C2×C4×D3—181no—
220C3×C4⋊C4—32no—
221C3×C4⋊C4—32no—
222C3×C4⋊C4—62no—
223C3×C4⋊C4—94no—
224C3×C4⋊C4—94no—
225C3×C4⋊C4—92no—
226C3×C4⋊C4—92no—
227C3×C42—32no—
228C3×C42—61no—
229C4⋊Dic3—32no—
230C4⋊Dic3—32no—
231C4⋊Dic3—34no—
232C4⋊Dic3—62no—
233C4⋊Dic3—92no—
234C4⋊Dic3—94no—
235C4⋊Dic3—94no—
236C4⋊Dic3—94no—
237C4⋊Dic3—184no—
238C4⋊Dic3D3212yes—
239Dic3⋊C4—34no—
240Dic3⋊C4—34no—
241Dic3⋊C4—34no—
242Dic3⋊C4—34no—
243Dic3⋊C4—94no—
244Dic3⋊C4—94no—
245Dic3⋊C4—94no—
246Dic3⋊C4—94no—
247Dic3⋊C4—184no—
248C4×Dic3—92no—
249C4×Dic3—92no—
250C4×Dic3—92no—
251C4×Dic3—91no—
252C4×Dic3—91no—
253C4×Dic3—91no—
254C4×Dic3—181no—
255C4×Dic3—181no—
256C4×Dic3—181no—
257C4×Dic3—182no—
258C4×Dic3—361no—
259C4×Dic3D3211yescharacteristic
260C22×C32—22no—
261C22×C32—21no—
262C22×C32—41no—
263C22×C32C23×D311yescharacteristic
264C22×C32C23×D312yes—
265C2×C3⋊D3C2×Dic612yes—
266C2×C3⋊D3#23712yes—
267C2×C3⋊D3#23912yes—
268C2×C3×D3—34no—
269C2×C3×D3—34no—
270C2×C3×D3—34no—
271C2×C3×D3—62no—
272C2×C3×D3—62no—
273C2×C3×D3—62no—
274C32×C4—24no—
275C32×C4—22no—
276C32×C4—42no—
277C32×C4—61no—
278C32×C4—61no—
279C32×C4—64no—
280C32×C4C2×Dic612yes—
281C32×C4D3×Q814yes—
282C3⋊Dic3—24no—
283C3⋊Dic3—61no—
284C3⋊Dic3—61no—
285C3⋊Dic3—62no—
286C3⋊Dic3—62no—
287C3⋊Dic3—62no—
288C3⋊Dic3—62no—
289C3⋊Dic3—122no—
290C3⋊Dic3—122no—
291C3⋊Dic3—122no—
292C3⋊Dic3—122no—
293C3⋊Dic3C2×Dic612yes—
294C3×Dic3—22no—
295C3×Dic3—22no—
296C3×Dic3—24no—
297C3×Dic3—22no—
298C3×Dic3—22no—
299C3×Dic3—24no—
300C3×Dic3—34no—
301C3×Dic3—44no—
302C3×Dic3—62no—
303C3×Dic3—64no—
304C3×Dic3—64no—
305C3×Dic3—124no—
306#125—92no—
307#125—272no—
308#125—272no—
309#125—272no—
310C2×C4⋊C4—91no—
311C2×C4⋊C4—272no—
312C2×C4⋊C4—272no—
313C2×C4⋊C4—271no—
314C2×C42—271no—
315C33C2×#12511yescharacteristic, complemented, Sylow
316C23×C3—91no—
317C22×D3—32no—
318C22×D3—61no—
319C2×C3×C4—22no—
320C2×C3×C4—21no—
321C2×C3×C4—32no—
322C2×C3×C4—32no—
323C2×C3×C4—34no—
324C2×C3×C4—34no—
325C2×C3×C4—41no—
326C2×C3×C4—61no—
327C2×C3×C4—61no—
328C2×C3×C4—64no—
329C2×C3×C4—91no—
330C2×C3×C4—94no—
331C2×C3×C4—94no—
332C2×C3×C4—184no—
333C2×C3×C4C2×D3212yes—
334C2×C3×C4C2×D3211yescharacteristic
335C2×Dic3—32no—
336C2×Dic3—34no—
337C2×Dic3—34no—
338C2×Dic3—61no—
339C2×Dic3—64no—
340C2×Dic3—91no—
341C2×Dic3—92no—
342C2×Dic3—92no—
343C2×Dic3—91no—
344C2×Dic3—181no—
345C2×Dic3—181no—
346C2×Dic3—182no—
347C2×Dic3—182no—
348C2×Dic3—182no—
349C2×Dic3—182no—
350C2×Dic3—184no—
351C2×Dic3—361no—
352C2×Dic3—361no—
353C2×Dic3C2×D3211yescharacteristic
354C2×Dic3C2×D3211yescharacteristic
355C2×Dic3C2×D3214yes—
356C4×D3—38no—
357C4×D3—64no—
358C4×D3—68no—
359C4×D3—188no—
360C4×D3—184no—
361C4×D3—184no—
362C4×D3—184no—
363C4×D3—184no—
364C2×C32—22no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—21no—
368C2×C32—22no—
369C2×C32—22no—
370C2×C32—41no—
371C2×C32—41no—
372C2×C32—41no—
373C2×C32C22×Dic611yescharacteristic
374C2×C32C2×#23711yescharacteristic
375C2×C32C2×D3×Q812yes—
376C2×C32C2×#23911yescharacteristic
377C2×C32D3×#4112yes—
378C2×C32D3×#4112yes—
379C3⋊D3#89014yescomplemented
380C3×D3—38no—
381C3×D3—64no—
382C22×C4—92no—
383C22×C4—91no—
384C22×C4—271no—
385C22×C4—271no—
386C22×C4—272no—
387C4⋊C4—32no—
388C4⋊C4—92no—
389C4⋊C4—94no—
390C4⋊C4—94no—
391C4⋊C4—92no—
392C4⋊C4—274no—
393C4⋊C4—274no—
394C4⋊C4—272no—
395C42—31no—
396C42—271no—
397C42—271no—
398C42—271no—
399C22×C3—22no—
400C22×C3—21no—
401C22×C3—41no—
402C22×C3—92no—
403C22×C3—92no—
404C22×C3—92no—
405C22×C3C22×D3212yes—
406C22×C3C22×D3211yescharacteristic
407C2×D3—34no—
408C2×D3—34no—
409C2×D3—34no—
410C2×D3—62no—
411C2×D3—62no—
412C2×D3—62no—
413C3×C4—22no—
414C3×C4—24no—
415C3×C4—42no—
416C3×C4—62no—
417C3×C4—62no—
418C3×C4—64no—
419C3×C4—64no—
420C3×C4—62no—
421C3×C4—62no—
422C3×C4—92no—
423C3×C4—124no—
424C3×C4—184no—
425C3×C4D3×Dic614yes—
426C3×C4#379712yes—
427Dic3—21no—
428Dic3—21no—
429Dic3—24no—
430Dic3—34no—
431Dic3—62no—
432Dic3—64no—
433Dic3—64no—
434Dic3—124no—
435Dic3—182no—
436Dic3—182no—
437Dic3—181no—
438Dic3—181no—
439Dic3—182no—
440Dic3—182no—
441Dic3—362no—
442Dic3—362no—
443Dic3—362no—
444Dic3—362no—
445C32—22no—
446C32—21no—
447C32—41no—
448C32C2×#89011yescharacteristic, complemented
449C32D3×#12512yescomplemented
450C23—91no—
451C2×C4—31no—
452C2×C4—31no—
453C2×C4—34no—
454C2×C4—91no—
455C2×C4—94no—
456C2×C4—94no—
457C2×C4—184no—
458C2×C4—271no—
459C2×C4—271no—
460C2×C4—542no—
461C2×C4—542no—
462C2×C4—544no—
463C2×C4D3311yescharacteristic
464C2×C3—21no—
465C2×C3—22no—
466C2×C3—22no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—22no—
470C2×C3—41no—
471C2×C3—41no—
472C2×C3—41no—
473C2×C3—94no—
474C2×C3C2×D3×Dic612yes—
475C2×C3C2×#379711yescharacteristic
476C2×C3#6430611yescharacteristic
477C2×C3#6430611yescharacteristic
478C2×C3D3×#23712yes—
479C2×C3D3×#23912yes—
480D3—38no—
481D3—64no—
482C22—92no—
483C22—92no—
484C22—92no—
485C22C2×D3311yescharacteristic
486C4—64no—
487C4—61no—
488C4—61no—
489C4—92no—
490C4—184no—
491C4—541no—
492C4—541no—
493C4#8895712yes—
494C3—21no—
495C3—22no—
496C3—41no—
497C3#1059394811yescharacteristic, complemented
498C3D3×#89012yescomplemented
499C2—94no—
500C2#1172437311yescharacteristic
501C2#1172439111yescharacteristic
502C2C2×#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. #11723450C2semidirect
5. C3×#63869C2semidirect
6. #11723018C2semidirect
7. #11723015C2semidirect
48. #63869D3semidirect
206. C3×C3⋊D3#125semidirect
315. C33C2×#125semidirect
379. C3⋊D3#890semidirect
448. C32C2×#890semidirect
449. C32D3×#125semidirect
497. C3#10593948semidirect
498. C3D3×#890semidirect

Maximal subgroups

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

subgroupindex
#117236892
C4×Dic3×C3⋊D32
#117234502
C3×#639002
C3×#638692
#117230182
#117230152
#117229102
#117229082
D3×#8903
#105939483