Finite Group Databasebeta

← #13177465

Subgroups — #13177465

Order 1728.

1800 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723976C211yesmaximal, characteristic, complemented
2C3×#64145C211yesmaximal, characteristic, complemented
3C32×#937C211yesmaximal, characteristic
4#67⋊(C3⋊Dic3)C212yesmaximal, complemented
5#11723614C212yesmaximal, complemented
6D3×#3757C212yesmaximal, complemented
7#11723596C212yesmaximal, complemented
8#11723487C212yesmaximal, complemented
9#11723459C212yesmaximal, complemented
10#10596058—31nomaximal
11#10595176—34nomaximal
12C23×C32×D3C2211yescharacteristic
13C2×C33×D4C2211yescharacteristic
14C22×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
15C2×C3×#634C2211yescharacteristic, complemented
16C2×C32×#67C2212yes—
17C2×D3×C3⋊Dic3C2212yes—
18C2×D3×C3⋊Dic3C2212yes—
19#88864C2212yes—
20#88864C2211yescharacteristic, complemented
21#88857C2212yescomplemented
22C3×#3757C2212yes—
23C3×#3757C2212yescomplemented
24C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
25C32×#217C2211yescharacteristic
26C32×(C2×D3)⋊C4C2212yes—
27#88807C2212yescomplemented
28#88806C2212yescomplemented
29Dic3⋊(C3⋊Dic3)C2212yescomplemented
30(C2×D3)⋊(C3⋊Dic3)C2212yes—
31(C2×D3)⋊(C3⋊Dic3)C2212yes—
32Dic3×C3⋊Dic3C2212yescomplemented
33C32×#120—31no—
34C2×#3757—32no—
35#64145D311yescharacteristic, complemented
36#64144—31no—
37#64144—32no—
38#64144—34no—
39#64144—64no—
40D4×C3⋊Dic3—32no—
41#64088—32no—
42#64087—32no—
43#64086—32no—
44C3×#937D314yes—
45C3×#936—34no—
46#67⋊Dic3—38no—
47#63969—38no—
48D3×#217—38no—
49#63956—38no—
50#63887—38no—
51#63877—38no—
52C23×C33C2311yescharacteristic
53C23×C33C2311yescharacteristic
54C22×C32×D3—24no—
55C22×C32×D3—41no—
56C22×C32×D3C2312yes—
57C33×D4—41no—
58C2×C33×C4C2311yescharacteristic
59C2×C3⋊(C3⋊Dic3)—41no—
60C2×C3⋊(C3⋊Dic3)C2312yes—
61C2×C3⋊(C3⋊Dic3)C2312yes—
62C2×C3×C3⋊Dic3C2312yes—
63C2×C3×C3⋊Dic3C2312yes—
64C3×#634—41no—
65C32×#67—24no—
66C2×C32×Dic3C2312yes—
67D3×C3⋊Dic3—24no—
68D3×C3⋊Dic3—42no—
69#5679—91no—
70#5587—94no—
71C24×C32—31no—
72C23×C3×D3C2×D314yes—
73C2×C32×D4—24no—
74C2×C32×D4—32no—
75C2×C32×D4C2×D314yes—
76C2×C32×D4C2×D311yescharacteristic
77C22×C3⋊Dic3—31no—
78C22×C3⋊Dic3—34no—
79C22×C3⋊Dic3—32no—
80C22×C3⋊Dic3—32no—
81C22×C3⋊Dic3—64no—
82C2×#634C2×D311yescharacteristic, complemented
83C2×C3×#67C2×D318yes—
84C2×C3×Dic6—34no—
85C2×D3×Dic3—38no—
86C2×D3×Dic3—38no—
87C32×C22⋊C4—31no—
88C32×C22⋊C4—32no—
89#3757—32no—
90#3757—31no—
91#3757—38no—
92#3757—34no—
93#3757—32no—
94#3757—32no—
95#3757—68no—
96#3757—64no—
97#3757C2×D312yes—
98#3757C2×D312yescomplemented
99C4⋊(C3⋊Dic3)—32no—
100#3750—32no—
101#3750—38no—
102#3750—32no—
103#3750—32no—
104#3750—68no—
105C4×C3⋊Dic3—32no—
106C4×C3⋊Dic3C2×D311yescharacteristic, complemented
107C3×#217—38no—
108C3×#217—38no—
109C3×#217C2×D314yes—
110C3×(C2×D3)⋊C4C2×D318yes—
111C3×C4×Dic3—34no—
112#3724—38no—
113Dic3⋊Dic3—38no—
114Dic3⋊Dic3—38no—
115(C2×D3)⋊Dic3—38no—
116(C2×D3)⋊Dic3—38no—
117Dic32—38no—
118C22×C33—22no—
119C22×C33—22no—
120C22×C33—21no—
121C22×C33—21no—
122C22×C33C2411yescharacteristic
123C2×C32×D3—22no—
124C2×C32×D3—22no—
125C2×C32×D3—44no—
126C2×C32×D3#4114yes—
127C33×C4—21no—
128C3⋊(C3⋊Dic3)—22no—
129C3⋊(C3⋊Dic3)—22no—
130C3×C3⋊Dic3—22no—
131C3×C3⋊Dic3—22no—
132C32×Dic3—22no—
133C3×#124—91no—
134C3×#120—34no—
135C2×#217—98no—
136#946—92no—
137#937C3⋊D311yescharacteristic
138#936—34no—
139#935—98no—
140#935—94no—
141#935—91no—
142#935—184no—
143D4×Dic3—98no—
144#883—92no—
145#67⋊C4—92no—
146D3×C22⋊C4—92no—
147#879—98no—
148#879—92no—
149#878—98no—
150#877—98no—
151#872—92no—
152C23×C32—24no—
153C23×C32—24no—
154C23×C32—32no—
155C23×C32—64no—
156C23×C32—121no—
157C23×C32C22×D314yes—
158C23×C32C22×D314yes—
159C23×C32C22×D311yescharacteristic
160C23×C32C22×D311yescharacteristic
161C22×C3×D3—216no—
162C22×C3×D3—44no—
163C22×C3×D3C22×D318yes—
164C32×D4—41no—
165C32×D4—44no—
166C32×D4—48no—
167C32×D4—64no—
168C2×C32×C4—24no—
169C2×C32×C4—32no—
170C2×C32×C4C22×D314yes—
171C2×C32×C4C22×D311yescharacteristic
172C2×C3⋊Dic3—32no—
173C2×C3⋊Dic3—32no—
174C2×C3⋊Dic3—38no—
175C2×C3⋊Dic3—38no—
176C2×C3⋊Dic3—68no—
177C2×C3⋊Dic3—68no—
178C2×C3⋊Dic3—64no—
179C2×C3⋊Dic3—122no—
180C2×C3⋊Dic3—121no—
181C2×C3⋊Dic3—124no—
182C2×C3⋊Dic3—128no—
183C2×C3⋊Dic3C22×D312yes—
184C2×C3⋊Dic3C22×D312yes—
185#634—41no—
186C3×#67—216no—
187C2×C3×Dic3—38no—
188C2×C3×Dic3—38no—
189C2×C3×Dic3C22×D318yes—
190C3×Dic6—124no—
191D3×Dic3—616no—
192D3×Dic3—128no—
193#551—271noSylow
194C2×C33—41no—
195C2×C33—41no—
196C2×C33C2×#4112yes—
197C2×C33H(5,𝔽2)11yescharacteristic
198C32×D3—24no—
199C24×C3—34no—
200C23×D3C2×C3⋊D311yescharacteristic
201C2×C3×Q8—91no—
202C2×C3×D4—24no—
203C2×C3×D4—38no—
204C2×C3×D4D3214yes—
205C2×C3×D4C2×C3⋊D311yescharacteristic
206C2×#67C2×C3⋊D312yes—
207C22×Dic3—98no—
208C22×Dic3—98no—
209C22×Dic3—94no—
210C22×Dic3—91no—
211C22×Dic3—184no—
212C2×C4×D3—92no—
213C2×C4×D3—92no—
214C2×Dic6—34no—
215C3×C22⋊C4—34no—
216C3×C22⋊C4—38no—
217C3×C22⋊C4—92no—
218C3×C22⋊C4—92no—
219C3×C42—91no—
220#217—38no—
221#217—38no—
222#217—98no—
223#217—98no—
224#217—98no—
225#217—94no—
226#217—92no—
227#217—91no—
228#217—188no—
229#217—184no—
230#217C2×C3⋊D311yescharacteristic
231(C2×D3)⋊C4—92no—
232(C2×D3)⋊C4—92no—
233(C2×D3)⋊C4C2×C3⋊D312yes—
234C4⋊Dic3—98no—
235C4⋊Dic3—92no—
236Dic3⋊C4—98no—
237Dic3⋊C4—98no—
238Dic3⋊C4—98no—
239Dic3⋊C4—92no—
240Dic3⋊C4—92no—
241Dic3⋊C4—92no—
242Dic3⋊C4—188no—
243C4×Dic3—34no—
244C4×Dic3—98no—
245C4×Dic3—92no—
246C22×C32—28no—
247C22×C32—24no—
248C22×C32—21no—
249C22×C32—21no—
250C22×C32—22no—
251C22×C32—22no—
252C22×C32—24no—
253C22×C32—28no—
254C22×C32—28no—
255C22×C32—24no—
256C22×C32—34no—
257C22×C32—48no—
258C22×C32—44no—
259C22×C32—48no—
260C22×C32—62no—
261C22×C32—62no—
262C22×C32—124no—
263C22×C32C23×D314yes—
264C22×C32C23×D311yescharacteristic
265C2×C3×D3—28no—
266C2×C3×D3—28no—
267C2×C3×D3—416no—
268C2×C3×D3#237116yes—
269C32×C4—21no—
270C32×C4—24no—
271C32×C4—44no—
272C32×C4—62no—
273C3⋊Dic3—22no—
274C3⋊Dic3—22no—
275C3⋊Dic3—62no—
276C3⋊Dic3—62no—
277C3⋊Dic3—68no—
278C3⋊Dic3—68no—
279C3⋊Dic3—128no—
280C3⋊Dic3—128no—
281C3×Dic3—28no—
282C3×Dic3—68no—
283C3×Dic3—68no—
284#126—272no—
285#124—91no—
286#123—272no—
287#123—271no—
288C22⋊Q8—272no—
289#120—31no—
290C4×D4—272no—
291#117—272no—
292C2×C22⋊C4—272no—
293C33#55111yescharacteristic, complemented, Sylow
294C23×C3—24no—
295C23×C3—24no—
296C23×C3—38no—
297C23×C3—616no—
298C23×C3—124no—
299C23×C3C2×D3214yes—
300C23×C3C2×D3214yes—
301C23×C3C22×C3⋊D311yescharacteristic
302C23×C3C22×C3⋊D311yescharacteristic
303C22×D3—24no—
304C22×D3—41no—
305C22×D3C22×C3⋊D312yes—
306C3×Q8—361no—
307C3×D4—44no—
308C3×D4—41no—
309C3×D4—48no—
310C3×D4—616no—
311C2×C3×C4—24no—
312C2×C3×C4—38no—
313C2×C3×C4—92no—
314C2×C3×C4—92no—
315C2×C3×C4C2×D3214yes—
316C2×C3×C4C22×C3⋊D311yescharacteristic
317#67—24no—
318C2×Dic3—38no—
319C2×Dic3—38no—
320C2×Dic3—98no—
321C2×Dic3—98no—
322C2×Dic3—92no—
323C2×Dic3—92no—
324C2×Dic3—188no—
325C2×Dic3—188no—
326C2×Dic3—1816no—
327C2×Dic3—368no—
328C2×Dic3—364no—
329C2×Dic3—361no—
330C2×Dic3—368no—
331C2×Dic3C22×C3⋊D312yes—
332C4×D3—184no—
333C4×D3—362no—
334Dic6—124no—
335C2×C32—28no—
336C2×C32—24no—
337C2×C32—48no—
338C2×C32—41no—
339C2×C32—41no—
340C2×C32—44no—
341C2×C32—44no—
342C2×C32—64no—
343C2×C32—84no—
344C2×C32C2×#23718yes—
345C2×C32#100414yes—
346C2×C32#100411yescharacteristic
347C2×C32D3×#4112yes—
348C3×D3—216no—
349C24—31no—
350C2×Q8—91no—
351C2×D4—32no—
352C2×D4D3×C3⋊D311yescharacteristic
353C22×C4—272no—
354C22×C4—272no—
355C22×C4—271no—
356C4⋊C4—272no—
357C4⋊C4—272no—
358C4⋊C4—272no—
359C4⋊C4—272no—
360C22⋊C4—31no—
361C22⋊C4—32no—
362C22⋊C4—92no—
363C22⋊C4—92no—
364C22⋊C4—272no—
365C22⋊C4—272no—
366C22⋊C4—272no—
367C22⋊C4—271no—
368C42—91no—
369C42—272no—
370C22×C3—28no—
371C22×C3—28no—
372C22×C3—28no—
373C22×C3—24no—
374C22×C3—24no—
375C22×C3—22no—
376C22×C3—22no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—24no—
380C22×C3—316no—
381C22×C3—48no—
382C22×C3—44no—
383C22×C3—48no—
384C22×C3—68no—
385C22×C3—68no—
386C22×C3—1216no—
387C22×C3C22×D3214yes—
388C22×C3C23×C3⋊D311yescharacteristic
389C2×D3—22no—
390C2×D3—22no—
391C2×D3—44no—
392C2×D3#383014yes—
393C3×C4—24no—
394C3×C4—21no—
395C3×C4—44no—
396C3×C4—68no—
397C3×C4—182no—
398C3×C4—182no—
399Dic3—22no—
400Dic3—68no—
401Dic3—68no—
402Dic3—188no—
403Dic3—188no—
404Dic3—182no—
405Dic3—182no—
406Dic3—368no—
407Dic3—368no—
408C32—24no—
409C32#558714yescomplemented
410C32#567911yescharacteristic, complemented
411C23—32no—
412C23—64no—
413C23—121no—
414C23C2×D3×C3⋊D311yescharacteristic
415C23C2×D3×C3⋊D311yescharacteristic
416Q8—361no—
417D4—41no—
418D4—64no—
419C2×C4—32no—
420C2×C4—92no—
421C2×C4—92no—
422C2×C4—272no—
423C2×C4—272no—
424C2×C4—544no—
425C2×C4—1082no—
426C2×C4—1081no—
427C2×C4C2×D3×C3⋊D311yescharacteristic
428C2×C3—28no—
429C2×C3—24no—
430C2×C3—48no—
431C2×C3—44no—
432C2×C3—44no—
433C2×C3—41no—
434C2×C3—41no—
435C2×C3—616no—
436C2×C3—84no—
437C2×C3D3×#23718yes—
438C2×C3#6433014yes—
439C2×C3C2×#383012yes—
440C2×C3#6436111yescharacteristic
441D3—24no—
442C22—22no—
443C22—22no—
444C22—21no—
445C22—21no—
446C22—34no—
447C22—62no—
448C22—62no—
449C22—124no—
450C22C22×D3×C3⋊D311yescharacteristic
451C4—21no—
452C4—62no—
453C4—182no—
454C4—182no—
455C4—542no—
456C4—542no—
457C3—24no—
458C3#1059517614yescomplemented
459C3#1059605811yescharacteristic, complemented
460C2—41no—
461C2—41no—
462C2—64no—
463C2D3×#383012yes—
464C2#1172461111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
prime radicalC32×#937C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#551
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11723976C2semidirect
2. C3×#64145C2semidirect
4. #67⋊(C3⋊Dic3)C2semidirect
5. #11723614C2semidirect
6. D3×#3757C2semidirect
7. #11723596C2semidirect
8. #11723487C2semidirect
9. #11723459C2semidirect
14. C22×C3⋊(C3⋊Dic3)C22semidirect
15. C2×C3×#634C22semidirect
20. #88864C22semidirect
21. #88857C22semidirect
23. C3×#3757C22semidirect
24. C3×C4×C3⋊Dic3C22semidirect
27. #88807C22semidirect
28. #88806C22semidirect
29. Dic3⋊(C3⋊Dic3)C22semidirect
32. Dic3×C3⋊Dic3C22semidirect
35. #64145D3semidirect
82. C2×#634C2×D3semidirect
98. #3757C2×D3semidirect
106. C4×C3⋊Dic3C2×D3semidirect
293. C33#551semidirect
409. C32#5587semidirect
410. C32#5679semidirect
458. C3#10595176semidirect
459. C3#10596058semidirect

Maximal subgroups

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

subgroupindex
#117239762
C3×#641452
C32×#9372
#67⋊(C3⋊Dic3)2
#117236142
D3×#37572
#117235962
#117234872
#117234592
#105960583
#105951763