Finite Group Databasebeta

← #13177546

Subgroups — #13177546

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×C3×D4×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×#89018C211yesmaximal, characteristic, complemented
3#11723981C211yesmaximal, characteristic, complemented
4C32×#936C211yesmaximal, characteristic, complemented
5#11723636C214yesmaximal, complemented
6#11723623C214yesmaximal, complemented
7#11723517C211yesmaximal, characteristic, complemented
8#11723437C211yesmaximal, characteristic, complemented
9#11723436C211yesmaximal, characteristic, complemented
10#10596145—31nomaximal
11#10595235—34nomaximal
12C23×C3×C3⋊D3C2212yes—
13C2×C33×D4C2211yescharacteristic
14C2×#6312C2214yescomplemented
15C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
16C2×C3×#638C2212yes—
17C3×D4×C3⋊D3—24no—
18C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
19C2×C3×C4×C3⋊D3C2211yescharacteristic
20C2×C32×Dic6C2211yescharacteristic, complemented
21C2×#6292C2212yes—
22C2×#6292C2214yescomplemented
23C2×#6289C2212yescomplemented
24#89018—24no—
25C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
26C32×#217C2214yescomplemented
27C32×C4×Dic3C2211yescharacteristic, complemented
28#88807C2212yescomplemented
29#88803C2212yes—
30#88803C2214yescomplemented
31C2×#3832—31no—
32C2×D4×C3⋊D3D311yescharacteristic, complemented
33C2×C3×D3×D4—34no—
34C2×#3799—34no—
35C32×#124—31no—
36#64156—31no—
37#64149—31no—
38#64149—34no—
39#64149—64no—
40#64093—34no—
41#64091—34no—
42#64083—31no—
43#64081—31no—
44C3×#936D314yescomplemented
45#63977—316no—
46#63958—316no—
47#63911—34no—
48#63863—34no—
49#63855—34no—
50C23×C33C2312yes—
51C22×C3⋊(C3⋊D3)C2312yes—
52C22×C3×C3⋊D3—24no—
53C22×C3×C3⋊D3—44no—
54C22×C3×C3⋊D3C2312yes—
55C22×C3×C3⋊D3C2311yescharacteristic
56C33×D4—22no—
57C2×C33×C4C2311yescharacteristic
58#6312—44no—
59C2×C3⋊(C3⋊Dic3)C2312yes—
60C4⋊(C3⋊(C3⋊D3))—22no—
61C3×#638—24no—
62C2×C3×C3⋊Dic3C2311yescharacteristic
63C3×C4⋊(C3⋊D3)—22no—
64C3×C4×C3⋊D3D414yescomplemented
65C2×C32×Dic3C2312yes—
66C2×C32×Dic3C2312yes—
67C32×Dic6—22no—
68#6292—24no—
69#6292—44no—
70#6289—24no—
71#5805—91no—
72#5674—94no—
73C23×C3⋊D3C2×D312yes—
74C23×C3×D3—38no—
75C2×C32×Q8—31no—
76C2×C32×D4—24no—
77C2×C32×D4C2×D314yes—
78C2×C32×D4C2×D311yescharacteristic
79C2×#638—34no—
80C2×#638—316no—
81C2×#638—616no—
82C2×#638C2×D312yes—
83#3832—64no—
84D4×C3⋊D3—24no—
85C2×C4⋊(C3⋊D3)—32no—
86C2×C4⋊(C3⋊D3)—31no—
87C2×C4⋊(C3⋊D3)—34no—
88C2×C4⋊(C3⋊D3)—34no—
89C2×C4⋊(C3⋊D3)—64no—
90C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
91C2×C4×C3⋊D3—32no—
92C2×C4×C3⋊D3C2×D311yescharacteristic
93C2×C3×#67—38no—
94C3×D3×D4—616no—
95C2×C3×D12—34no—
96C2×C3×C4×D3—34no—
97C2×C3×Dic6C2×D314yescomplemented
98C2×#626—38no—
99C2×#626—316no—
100C2×#624—38no—
101#3799—616no—
102C32×C22⋊C4—34no—
103C32×C42—31no—
104#3752—32no—
105#3752—34no—
106#3750—32no—
107C4×C3⋊Dic3—31no—
108C4×C3⋊Dic3—34no—
109C4×C3⋊Dic3—64no—
110C3×#217C2×D3116yescomplemented
111C3×C4×Dic3C2×D314yescomplemented
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×C3×C3⋊D3—22no—
121C2×C3×C3⋊D3—22no—
122C2×C3×C3⋊D3—24no—
123C2×C3×C3⋊D3—48no—
124C2×C3×C3⋊D3C2×D412yes—
125C33×C4C2×D412yes—
126C3⋊(C3⋊Dic3)—22no—
127C3×C3⋊Dic3C2×D412yes—
128C32×Dic3—22no—
129C32×Dic3—22no—
130C22×C3×D4—91no—
131C2×#239—94no—
132C2×D3×D4—34no—
133C2×#235—91no—
134C3×#124—34no—
135#953—94no—
136#947—94no—
137#940—94no—
138#940—91no—
139#940—91no—
140#940—184no—
141#938—94no—
142#936—91no—
143#936C3⋊D311yescharacteristic, complemented
144#926—91no—
145#884—916no—
146#882—916no—
147#874—94no—
148#872—94no—
149C23×C32—28no—
150C23×C32C22×D318yes—
151C23×C32C22×D312yes—
152C22×C3⋊D3—24no—
153C22×C3⋊D3—32no—
154C22×C3⋊D3—38no—
155C22×C3⋊D3—44no—
156C22×C3⋊D3—68no—
157C22×C3⋊D3C22×D312yes—
158C22×C3⋊D3C22×D311yescharacteristic
159C22×C3×D3—34no—
160C22×C3×D3—38no—
161C22×C3×D3—616no—
162C22×C3×D3—1216no—
163C32×Q8—62no—
164C32×D4—22no—
165C32×D4—28no—
166C32×D4—48no—
167C2×C32×C4—24no—
168C2×C32×C4—32no—
169C2×C32×C4—32no—
170C2×C32×C4C22×D314yes—
171C2×C32×C4C22×D311yescharacteristic
172#638—24no—
173#638—124no—
174#638—1216no—
175#638—1232no—
176C2×C3⋊Dic3—32no—
177C2×C3⋊Dic3—38no—
178C2×C3⋊Dic3—68no—
179C2×C3⋊Dic3C22×D311yescharacteristic
180C4⋊(C3⋊D3)—22no—
181C4⋊(C3⋊D3)—64no—
182C4⋊(C3⋊D3)—62no—
183C4⋊(C3⋊D3)—68no—
184C4⋊(C3⋊D3)—128no—
185C4⋊(C3⋊D3)—124no—
186C4×C3⋊D3—64no—
187C4×C3⋊D3#6714yescomplemented
188C3×#67—616no—
189C2×C3×Dic3—34no—
190C2×C3×Dic3C22×D318yes—
191C2×C3×Dic3C22×D318yes—
192C3×D12—68no—
193C3×C4×D3—316no—
194C3×Dic6—28no—
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—
203C3×C3⋊D3—22no—
204C3×C3⋊D3—42no—
205C24×C3—92no—
206C23×D3—38no—
207C2×C3×Q8—34no—
208C2×C3×D4—24no—
209C2×C3×D4—92no—
210C2×C3×D4—91no—
211C2×C3×D4—184no—
212C2×C3×D4D3214yes—
213C2×C3×D4C2×C3⋊D311yescharacteristic
214C22×C3×C4—91no—
215C2×#67—38no—
216C2×#67—916no—
217C2×#67—94no—
218C2×#67—92no—
219C2×#67—94no—
220C2×#67—1816no—
221#239—1816no—
222D3×D4—616no—
223#235—184no—
224C2×D12—34no—
225C2×D12—98no—
226C2×D12—94no—
227C2×D12—91no—
228C2×D12—916no—
229C2×D12—184no—
230C2×C4×D3—34no—
231C2×C4×D3—98no—
232C2×C4×D3—92no—
233C2×Dic6C2×C3⋊D311yescharacteristic, complemented
234C3×C22⋊C4—316no—
235C3×C42—34no—
236#217—92no—
237#217—94no—
238#217C2×C3⋊D314yescomplemented
239(C2×D3)⋊C4—98no—
240(C2×D3)⋊C4—916no—
241C4⋊Dic3—92no—
242Dic3⋊C4—98no—
243C4×Dic3—94no—
244C4×Dic3—91no—
245C4×Dic3—184no—
246C4×Dic3C2×C3⋊D311yescharacteristic, complemented
247C22×C32—24no—
248C22×C32—22no—
249C22×C32—24no—
250C22×C32—28no—
251C22×C32—216no—
252C22×C32—416no—
253C22×C32—48no—
254C22×C32C23×D314yes—
255C22×C32C23×D311yescharacteristic
256C2×C3⋊D3—22no—
257C2×C3⋊D3—24no—
258C2×C3⋊D3—22no—
259C2×C3⋊D3—48no—
260C2×C3⋊D3—64no—
261C2×C3⋊D3—62no—
262C2×C3⋊D3—68no—
263C2×C3⋊D3—616no—
264C2×C3⋊D3—128no—
265C2×C3⋊D3—1216no—
266C2×C3⋊D3C2×#6712yes—
267C2×C3×D3—38no—
268C2×C3×D3—68no—
269C2×C3×D3—68no—
270C2×C3×D3—616no—
271C2×C3×D3—1232no—
272C32×C4—28no—
273C32×C4—62no—
274C32×C4—62no—
275C32×C4D3×D418yes—
276C32×C4C2×#6712yes—
277C3⋊Dic3—616no—
278C3⋊Dic3—62no—
279C3⋊Dic3—68no—
280C3⋊Dic3C2×#6712yes—
281C3×Dic3—28no—
282C3×Dic3—28no—
283C3×Dic3—38no—
284C2×#41—271no—
285C22×D4—91no—
286#127—271no—
287#127—271no—
288#124—31no—
289#124—271no—
290#121—274no—
291#120—274no—
292#117—271no—
293C33#53511yescharacteristic, complemented, Sylow
294C23×C3—28no—
295C23×C3—91no—
296C23×C3—92no—
297C23×C3—184no—
298C23×C3—364no—
299C23×C3C2×D3218yes—
300C23×C3C22×C3⋊D312yes—
301C22×D3—34no—
302C22×D3—38no—
303C22×D3—616no—
304C22×D3—98no—
305C22×D3—92no—
306C22×D3—1216no—
307C22×D3—188no—
308C3×Q8—68no—
309C3×D4—28no—
310C3×D4—22no—
311C3×D4—48no—
312C3×D4—184no—
313C3×D4—182no—
314C2×C3×C4—24no—
315C2×C3×C4—38no—
316C2×C3×C4—38no—
317C2×C3×C4—91no—
318C2×C3×C4—94no—
319C2×C3×C4C2×D3214yes—
320C2×C3×C4C22×C3⋊D311yescharacteristic
321#67—616no—
322#67—184no—
323#67—364no—
324#67—3616no—
325#67—364no—
326#67—3632no—
327C2×Dic3—34no—
328C2×Dic3—98no—
329C2×Dic3—92no—
330C2×Dic3—188no—
331C2×Dic3C22×C3⋊D312yes—
332C2×Dic3C22×C3⋊D312yes—
333D12—68no—
334D12—1816no—
335D12—188no—
336D12—182no—
337D12—368no—
338D12—3616no—
339C4×D3—316no—
340C4×D3—184no—
341C4×D3—1816no—
342Dic6—22no—
343C2×C32—28no—
344C2×C32—24no—
345C2×C32—42no—
346C2×C32—48no—
347C2×C32—88no—
348C2×C32C2×D3×D414yes—
349C2×C32#100412yes—
350C2×C32#100918yes—
351C2×C32C22×#6711yescharacteristic
352C3⋊D3—22no—
353C3⋊D3—42no—
354C3⋊D3—122no—
355C3⋊D3—128no—
356C3⋊D3—248no—
357C3×D3—68no—
358C3×D3—128no—
359C24—92no—
360#41—544no—
361C2×Q8—31no—
362C2×D4—92no—
363C2×D4—91no—
364C2×D4—184no—
365C2×D4—272no—
366C2×D4—274no—
367C2×D4—274no—
368C2×D4—271no—
369C2×D4D3×C3⋊D311yescharacteristic
370C22×C4—91no—
371C22×C4—272no—
372C4⋊C4—272no—
373C22⋊C4—34no—
374C22⋊C4—272no—
375C22⋊C4—274no—
376C42—31no—
377C42—271no—
378C22×C3—216no—
379C22×C3—28no—
380C22×C3—22no—
381C22×C3—24no—
382C22×C3—24no—
383C22×C3—416no—
384C22×C3—48no—
385C22×C3—92no—
386C22×C3—182no—
387C22×C3—182no—
388C22×C3—184no—
389C22×C3—368no—
390C22×C3C22×D3214yes—
391C22×C3C23×C3⋊D311yescharacteristic
392C2×D3—38no—
393C2×D3—68no—
394C2×D3—616no—
395C2×D3—68no—
396C2×D3—1232no—
397C2×D3—1816no—
398C2×D3—188no—
399C2×D3—182no—
400C2×D3—184no—
401C2×D3—368no—
402C2×D3—3616no—
403C3×C4—28no—
404C3×C4—68no—
405C3×C4—68no—
406C3×C4—92no—
407C3×C4D3×#6718yes—
408C3×C4D4×C3⋊D312yes—
409Dic3—22no—
410Dic3—22no—
411Dic3—38no—
412Dic3—1816no—
413Dic3—188no—
414Dic3—182no—
415C32—24no—
416C32#567414yescomplemented
417C32#580511yescharacteristic, complemented
418C23—91no—
419C23—92no—
420C23—184no—
421C23—272no—
422C23—364no—
423C23C2×D3×C3⋊D312yes—
424Q8—62no—
425D4—22no—
426D4—184no—
427D4—182no—
428D4—544no—
429D4—542no—
430D4—1084no—
431D4—1084no—
432C2×C4—32no—
433C2×C4—32no—
434C2×C4—91no—
435C2×C4—94no—
436C2×C4—272no—
437C2×C4—544no—
438C2×C4C2×D3×C3⋊D311yescharacteristic
439C2×C3—28no—
440C2×C3—24no—
441C2×C3—42no—
442C2×C3—48no—
443C2×C3—88no—
444C2×C3—182no—
445C2×C3—362no—
446C2×C3#6431418yes—
447C2×C3C2×D3×#6714yes—
448C2×C3C2×D4×C3⋊D311yescharacteristic
449C2×C3#6436612yes—
450D3—68no—
451D3—128no—
452D3—362no—
453D3—368no—
454D3—728no—
455C22—24no—
456C22—22no—
457C22—92no—
458C22—182no—
459C22—184no—
460C22—182no—
461C22—368no—
462C22—544no—
463C22—542no—
464C22C22×D3×C3⋊D311yescharacteristic
465C4—62no—
466C4—62no—
467C4—92no—
468C4—542no—
469C4C3⋊D3×#6712yes—
470C3—24no—
471C3#1059523514yescomplemented
472C3#1059614511yescharacteristic, complemented
473C2—42no—
474C2—182no—
475C2—362no—
476C2—1082no—
477C2#1172458912yes—
478C2C2×C3⋊D3×#6711yescharacteristic
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×C3×D4×C3⋊D3C2semidirect
2. C2×#89018C2semidirect
3. #11723981C2semidirect
4. C32×#936C2semidirect
5. #11723636C2semidirect
6. #11723623C2semidirect
7. #11723517C2semidirect
8. #11723437C2semidirect
9. #11723436C2semidirect
14. C2×#6312C22semidirect
15. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
20. C2×C32×Dic6C22semidirect
22. C2×#6292C22semidirect
23. C2×#6289C22semidirect
25. C4×C3⋊(C3⋊Dic3)C22semidirect
26. C32×#217C22semidirect
27. C32×C4×Dic3C22semidirect
28. #88807C22semidirect
30. #88803C22semidirect
32. C2×D4×C3⋊D3D3semidirect
44. C3×#936D3semidirect
64. C3×C4×C3⋊D3D4semidirect
97. C2×C3×Dic6C2×D3semidirect
110. C3×#217C2×D3semidirect
111. C3×C4×Dic3C2×D3semidirect
143. #936C3⋊D3semidirect
187. C4×C3⋊D3#67semidirect
233. C2×Dic6C2×C3⋊D3semidirect
238. #217C2×C3⋊D3semidirect
246. C4×Dic3C2×C3⋊D3semidirect
293. C33#535semidirect
416. C32#5674semidirect
417. C32#5805semidirect
471. C3#10595235semidirect
472. C3#10596145semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×C3⋊D32
C2×#890182
#117239812
C32×#9362
#117236362
#117236232
#117235172
#117234372
#117234362
#105961453
#105952353