Finite Group Databasebeta

← #13177418

Subgroups — #13177418

Order 1728.

2436 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×D3×D4C211yesmaximal, characteristic, complemented
2C2×#89022C211yesmaximal, characteristic, complemented
3#11723981C211yesmaximal, characteristic, complemented
4C3×#64145C211yesmaximal, characteristic, complemented
5#11723630C214yesmaximal, complemented
6#11723599C214yesmaximal, complemented
7#11723524C211yesmaximal, characteristic, complemented
8#11723434C211yesmaximal, characteristic, complemented
9#11723423C211yesmaximal, characteristic, complemented
10#10596276—31nomaximal
11#10595235—34nomaximal
12C23×C32×D3C2212yes—
13C2×C33×D4C2211yescharacteristic
14C2×#6312C2214yescomplemented
15C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
16C2×C3×#634C2211yescharacteristic, complemented
17C2×C32×#67C2212yes—
18C32×D3×D4—24no—
19C2×C32×D12C2211yescharacteristic
20C2×C32×C4×D3C2211yescharacteristic
21C2×#6291C2212yes—
22C2×#6291C2214yescomplemented
23C2×#6289C2212yescomplemented
24#89022—24no—
25C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
26C3×#3757C2214yescomplemented
27C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
28Dic3⋊(C3⋊Dic3)C2212yescomplemented
29(C2×D3)⋊(C3⋊Dic3)C2212yes—
30(C2×D3)⋊(C3⋊Dic3)C2214yescomplemented
31C22×C32×D4—31no—
32C2×#3828—31no—
33C2×C3×D3×D4D314yescomplemented
34C2×#3799—34no—
35#64162—34no—
36#64149—31no—
37#64149—34no—
38#64149—31no—
39#64149—64no—
40#64147—34no—
41#64145—31no—
42#64145D311yescharacteristic, complemented
43#64135—31no—
44C3×#936—34no—
45#63977—316no—
46#63958—316no—
47#63911—34no—
48#63863—34no—
49#63855—34no—
50C23×C33C2312yes—
51C22×C3⋊(C3⋊D3)C2312yes—
52C22×C32×D3—24no—
53C22×C32×D3—44no—
54C22×C32×D3C2312yes—
55C22×C32×D3C2311yescharacteristic
56C33×D4—22no—
57C2×C33×C4C2311yescharacteristic
58#6312—44no—
59C2×C3⋊(C3⋊Dic3)C2312yes—
60C4⋊(C3⋊(C3⋊D3))—22no—
61C2×C3×C3⋊Dic3C2312yes—
62C2×C3×C3⋊Dic3C2312yes—
63C3×#634—22no—
64C32×#67—24no—
65C2×C32×Dic3C2311yescharacteristic
66C32×D12—22no—
67C32×C4×D3D414yescomplemented
68#6291—24no—
69#6291—44no—
70#6289—24no—
71#5805—94no—
72#5674—91no—
73C24×C32—32no—
74C23×C3×D3C2×D318yes—
75C2×C32×D4—24no—
76C2×C32×D4—32no—
77C2×C32×D4—31no—
78C2×C32×D4—64no—
79C2×C32×D4C2×D314yes—
80C2×C32×D4C2×D311yescharacteristic
81C22×C32×C4—31no—
82C2×#638—34no—
83C2×#638—316no—
84C2×#638—32no—
85C2×#638—34no—
86C2×#638—616no—
87#3828—64no—
88C2×C4⋊(C3⋊D3)—31no—
89C2×C4⋊(C3⋊D3)—34no—
90C2×C4⋊(C3⋊D3)—64no—
91C2×C4×C3⋊D3—32no—
92C2×#634C2×D311yescharacteristic, complemented
93C2×C3×#67C2×D318yes—
94C3×D3×D4—216no—
95C2×C3×D12C2×D314yes—
96C2×C3×C4×D3C2×D314yes—
97C2×C3×Dic6—34no—
98C2×#626—38no—
99C2×#626—316no—
100C2×#624—38no—
101#3799—616no—
102#3757—32no—
103#3757—34no—
104#3757C2×D314yescomplemented
105C4⋊(C3⋊Dic3)—32no—
106C4×C3⋊Dic3—31no—
107C4×C3⋊Dic3—34no—
108C4×C3⋊Dic3—64no—
109C4×C3⋊Dic3C2×D311yescharacteristic, complemented
110C3×#217—316no—
111C3×C4×Dic3—34no—
112Dic3⋊Dic3—38no—
113(C2×D3)⋊Dic3—38no—
114(C2×D3)⋊Dic3—316no—
115C22×C33—22no—
116C22×C33—24no—
117C22×C33C2411yescharacteristic
118C2×C3⋊(C3⋊D3)—22no—
119C2×C3⋊(C3⋊D3)—24no—
120C2×C32×D3—22no—
121C2×C32×D3—22no—
122C2×C32×D3—24no—
123C2×C32×D3—48no—
124C2×C32×D3C2×D412yes—
125C33×C4C2×D412yes—
126C3⋊(C3⋊Dic3)—22no—
127C3×C3⋊Dic3—22no—
128C3×C3⋊Dic3—22no—
129C32×Dic3C2×D412yes—
130C22×C3×D4—34no—
131C2×#239—91no—
132C2×D3×D4C3⋊D311yescharacteristic, complemented
133C2×#235—94no—
134C3×#124—91no—
135#953—916no—
136#947—91no—
137#940—94no—
138#940—94no—
139#940—91no—
140#940—184no—
141#938—916no—
142#936—34no—
143#936—94no—
144#926—94no—
145#884—94no—
146#882—94no—
147#874—91no—
148#872—91no—
149C23×C32—28no—
150C23×C32—31no—
151C23×C32—32no—
152C23×C32—64no—
153C23×C32—124no—
154C23×C32C22×D318yes—
155C23×C32C22×D312yes—
156C22×C3⋊D3—32no—
157C22×C3⋊D3—38no—
158C22×C3⋊D3—68no—
159C22×C3×D3—216no—
160C22×C3×D3—416no—
161C22×C3×D3C22×D318yes—
162C22×C3×D3C22×D314yes—
163C32×D4—22no—
164C32×D4—28no—
165C32×D4—48no—
166C32×D4—64no—
167C32×D4—62no—
168C2×C32×C4—24no—
169C2×C32×C4—31no—
170C2×C32×C4—34no—
171C2×C32×C4C22×D314yes—
172C2×C32×C4C22×D311yescharacteristic
173#638—64no—
174#638—124no—
175#638—124no—
176#638—1216no—
177#638—1232no—
178C2×C3⋊Dic3—32no—
179C2×C3⋊Dic3—38no—
180C2×C3⋊Dic3—68no—
181C2×C3⋊Dic3C22×D312yes—
182C2×C3⋊Dic3C22×D312yes—
183C4⋊(C3⋊D3)—62no—
184C4⋊(C3⋊D3)—68no—
185C4⋊(C3⋊D3)—128no—
186C4×C3⋊D3—64no—
187#634—22no—
188C3×#67—216no—
189C2×C3×Dic3—38no—
190C2×C3×Dic3—38no—
191C2×C3×Dic3C22×D314yes—
192C3×D12—28no—
193C3×C4×D3#67116yescomplemented
194C3×Dic6—68no—
195#626—616no—
196#626—1216no—
197#624—616no—
198#535—271noSylow
199C2×C33—42no—
200C2×C33C22×D411yescharacteristic
201C2×C33H(5,𝔽2)12yes—
202C3⋊(C3⋊D3)—42no—
203C32×D3—22no—
204C32×D3—42no—
205C24×C3—38no—
206C23×D3C2×C3⋊D312yes—
207C2×C3×Q8—91no—
208C2×C3×D4—24no—
209C2×C3×D4—38no—
210C2×C3×D4—34no—
211C2×C3×D4—616no—
212C2×C3×D4D3214yes—
213C2×C3×D4C2×C3⋊D311yescharacteristic
214C22×C3×C4—34no—
215C2×#67—98no—
216C2×#67—916no—
217C2×#67—916no—
218C2×#67—94no—
219C2×#67—1816no—
220C2×#67C2×C3⋊D312yes—
221#239—184no—
222D3×D4—24no—
223#235—1816no—
224C2×D12—92no—
225C2×D12—94no—
226C2×D12—91no—
227C2×D12—94no—
228C2×D12—184no—
229C2×D12C2×C3⋊D311yescharacteristic
230C2×C4×D3—98no—
231C2×C4×D3—92no—
232C2×C4×D3C2×C3⋊D311yescharacteristic
233C2×Dic6—34no—
234C3×C22⋊C4—94no—
235C3×C42—91no—
236#217—316no—
237#217—98no—
238#217—916no—
239(C2×D3)⋊C4—92no—
240(C2×D3)⋊C4—94no—
241C4⋊Dic3—98no—
242Dic3⋊C4—92no—
243C4×Dic3—34no—
244C4×Dic3—94no—
245C4×Dic3—91no—
246C4×Dic3—184no—
247C22×C32—22no—
248C22×C32—24no—
249C22×C32—24no—
250C22×C32—216no—
251C22×C32—28no—
252C22×C32—32no—
253C22×C32—416no—
254C22×C32—48no—
255C22×C32—62no—
256C22×C32—62no—
257C22×C32—64no—
258C22×C32—128no—
259C22×C32C23×D314yes—
260C22×C32C23×D311yescharacteristic
261C2×C3⋊D3—62no—
262C2×C3⋊D3—64no—
263C2×C3⋊D3—616no—
264C2×C3⋊D3—68no—
265C2×C3⋊D3—128no—
266C2×C3⋊D3—1216no—
267C2×C3×D3—28no—
268C2×C3×D3—216no—
269C2×C3×D3—28no—
270C2×C3×D3—432no—
271C2×C3×D3C2×#6718yes—
272C32×C4—28no—
273C32×C4—32no—
274C32×C4D3×D412yes—
275C32×C4C2×#6718yes—
276C3⋊Dic3—22no—
277C3⋊Dic3—22no—
278C3⋊Dic3—616no—
279C3⋊Dic3—68no—
280C3⋊Dic3—62no—
281C3×Dic3—68no—
282C3×Dic3—68no—
283C3×Dic3C2×#6718yes—
284C2×#41—271no—
285C22×D4—31no—
286#127—271no—
287#127—271no—
288#124—91no—
289#124—271no—
290#121—274no—
291#120—274no—
292#117—271no—
293C33#53511yescharacteristic, complemented, Sylow
294C23×C3—28no—
295C23×C3—34no—
296C23×C3—38no—
297C23×C3—616no—
298C23×C3—1216no—
299C23×C3C2×D3218yes—
300C23×C3C22×C3⋊D312yes—
301C22×D3—24no—
302C22×D3—44no—
303C22×D3—98no—
304C22×D3—92no—
305C22×D3—188no—
306C22×D3C22×C3⋊D312yes—
307C22×D3C22×C3⋊D311yescharacteristic
308C3×Q8—182no—
309C3×D4—28no—
310C3×D4—22no—
311C3×D4—48no—
312C3×D4—616no—
313C3×D4—68no—
314C2×C3×C4—24no—
315C2×C3×C4—34no—
316C2×C3×C4—316no—
317C2×C3×C4—92no—
318C2×C3×C4—92no—
319C2×C3×C4C2×D3214yes—
320C2×C3×C4C22×C3⋊D311yescharacteristic
321#67—24no—
322#67—1816no—
323#67—3616no—
324#67—3616no—
325#67—364no—
326#67—3632no—
327C2×Dic3—38no—
328C2×Dic3—38no—
329C2×Dic3—98no—
330C2×Dic3—92no—
331C2×Dic3—188no—
332C2×Dic3C22×C3⋊D311yescharacteristic
333D12—22no—
334D12—184no—
335D12—188no—
336D12—182no—
337D12—368no—
338D12—364no—
339C4×D3—1816no—
340C4×D3—184no—
341C4×D3#63814yescomplemented
342Dic6—68no—
343C2×C32—28no—
344C2×C32—24no—
345C2×C32—42no—
346C2×C32—48no—
347C2×C32—62no—
348C2×C32—88no—
349C2×C32—122no—
350C2×C32C2×D3×D411yescharacteristic
351C2×C32#100418yes—
352C2×C32#100912yes—
353C2×C32C22×#6714yes—
354C3⋊D3—122no—
355C3⋊D3—128no—
356C3⋊D3—248no—
357C3×D3—28no—
358C3×D3—48no—
359C24—32no—
360#41—544no—
361C2×Q8—91no—
362C2×D4—32no—
363C2×D4—31no—
364C2×D4—64no—
365C2×D4—272no—
366C2×D4—274no—
367C2×D4—274no—
368C2×D4—271no—
369C2×D4D3×C3⋊D311yescharacteristic
370C22×C4—31no—
371C22×C4—272no—
372C4⋊C4—272no—
373C22⋊C4—94no—
374C22⋊C4—272no—
375C22⋊C4—274no—
376C42—91no—
377C42—271no—
378C22×C3—216no—
379C22×C3—28no—
380C22×C3—24no—
381C22×C3—22no—
382C22×C3—24no—
383C22×C3—38no—
384C22×C3—416no—
385C22×C3—48no—
386C22×C3—68no—
387C22×C3—616no—
388C22×C3—68no—
389C22×C3—1232no—
390C22×C3C22×D3214yes—
391C22×C3C23×C3⋊D311yescharacteristic
392C2×D3—22no—
393C2×D3—24no—
394C2×D3—22no—
395C2×D3—48no—
396C2×D3—1816no—
397C2×D3—188no—
398C2×D3—184no—
399C2×D3—182no—
400C2×D3—368no—
401C2×D3—3616no—
402C2×D3C2×#63812yes—
403C3×C4—28no—
404C3×C4—38no—
405C3×C4—182no—
406C3×C4—182no—
407C3×C4D3×#6718yes—
408C3×C4C2×#63812yes—
409Dic3—68no—
410Dic3—68no—
411Dic3—1816no—
412Dic3—188no—
413Dic3—182no—
414Dic3C2×#63812yes—
415C32—24no—
416C32#567411yescharacteristic, complemented
417C32#580514yescomplemented
418C23—31no—
419C23—32no—
420C23—64no—
421C23—124no—
422C23—272no—
423C23C2×D3×C3⋊D312yes—
424Q8—182no—
425D4—22no—
426D4—64no—
427D4—62no—
428D4—544no—
429D4—542no—
430D4—1084no—
431D4—1084no—
432C2×C4—31no—
433C2×C4—34no—
434C2×C4—92no—
435C2×C4—92no—
436C2×C4—272no—
437C2×C4—544no—
438C2×C4C2×D3×C3⋊D311yescharacteristic
439C2×C3—28no—
440C2×C3—24no—
441C2×C3—48no—
442C2×C3—42no—
443C2×C3—68no—
444C2×C3—88no—
445C2×C3—128no—
446C2×C3#6431418yes—
447C2×C3C2×D3×#6714yes—
448C2×C3#6436112yes—
449C2×C3C22×#63811yescharacteristic
450D3—22no—
451D3—42no—
452D3—368no—
453D3—362no—
454D3—728no—
455C22—24no—
456C22—22no—
457C22—32no—
458C22—62no—
459C22—64no—
460C22—62no—
461C22—128no—
462C22—544no—
463C22—542no—
464C22C22×D3×C3⋊D311yescharacteristic
465C4—32no—
466C4—182no—
467C4—182no—
468C4—542no—
469C4D3×#63812yes—
470C3—24no—
471C3#1059523514yescomplemented
472C3#1059627611yescharacteristic, complemented
473C2—42no—
474C2—62no—
475C2—122no—
476C2—1082no—
477C2#1172458812yes—
478C2C2×D3×#63811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#535
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C32×D3×D4C2semidirect
2. C2×#89022C2semidirect
3. #11723981C2semidirect
4. C3×#64145C2semidirect
5. #11723630C2semidirect
6. #11723599C2semidirect
7. #11723524C2semidirect
8. #11723434C2semidirect
9. #11723423C2semidirect
14. C2×#6312C22semidirect
15. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
16. C2×C3×#634C22semidirect
22. C2×#6291C22semidirect
23. C2×#6289C22semidirect
25. C4×C3⋊(C3⋊Dic3)C22semidirect
26. C3×#3757C22semidirect
27. C3×C4×C3⋊Dic3C22semidirect
28. Dic3⋊(C3⋊Dic3)C22semidirect
30. (C2×D3)⋊(C3⋊Dic3)C22semidirect
33. C2×C3×D3×D4D3semidirect
42. #64145D3semidirect
67. C32×C4×D3D4semidirect
92. C2×#634C2×D3semidirect
104. #3757C2×D3semidirect
109. C4×C3⋊Dic3C2×D3semidirect
132. C2×D3×D4C3⋊D3semidirect
193. C3×C4×D3#67semidirect
293. C33#535semidirect
341. C4×D3#638semidirect
416. C32#5674semidirect
417. C32#5805semidirect
471. C3#10595235semidirect
472. C3#10596276semidirect

Maximal subgroups

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

subgroupindex
C2×C32×D3×D42
C2×#890222
#117239812
C3×#641452
#117236302
#117235992
#117235242
#117234342
#117234232
#105962763
#105952353