Finite Group Databasebeta

← #13177543

Subgroups — #13177543

Order 1728.

2308 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×#89014C211yesmaximal, characteristic, complemented
3#11723977C211yesmaximal, characteristic, complemented
4C32×#940C211yesmaximal, characteristic, complemented
5#11723634C214yesmaximal, complemented
6#11723615C214yesmaximal, complemented
7#11723529C211yesmaximal, characteristic, complemented
8#11723439C211yesmaximal, characteristic, complemented
9#11723437C211yesmaximal, characteristic, complemented
10#10596186—31nomaximal
11#10595233—34nomaximal
12C23×C3×C3⋊D3C2212yes—
13C2×C33×D4C2211yescharacteristic
14C2×#6308C2211yescharacteristic, complemented
15C2×C3×#638C2212yes—
16C3×D4×C3⋊D3—24no—
17C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
18C2×C3×C4×C3⋊D3C2211yescharacteristic
19C2×C32×#67C2214yescomplemented
20C2×C32×D12C2211yescharacteristic, complemented
21C2×#6290C2212yes—
22C2×#6290C2214yescomplemented
23C2×D3×C3⋊Dic3C2212yescomplemented
24#89014—24no—
25#88864C2214yescomplemented
26C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
27C32×C4×Dic3C2211yescharacteristic, complemented
28#88807C2212yescomplemented
29#88803C2212yes—
30#88803C2214yescomplemented
31C2×#3830—31no—
32C2×D4×C3⋊D3D311yescharacteristic, complemented
33C2×C3×D3×D4—34no—
34C2×#3795—34no—
35C32×#127—31no—
36#64149—31no—
37#64148—34no—
38#64146—34no—
39#64145—31no—
40#64145—34no—
41#64145—64no—
42#64084—31no—
43#64081—31no—
44C3×#940D314yescomplemented
45#63976—316no—
46#63970—316no—
47#63914—34no—
48#63860—34no—
49#63855—34no—
50C23×C33C2312yes—
51C22×C3×C3⋊D3—24no—
52C22×C3×C3⋊D3—44no—
53C22×C3×C3⋊D3C2312yes—
54C22×C3×C3⋊D3C2311yescharacteristic
55C22×C32×D3C2312yes—
56C33×D4—22no—
57C2×C33×C4C2311yescharacteristic
58C2×C3⋊(C3⋊Dic3)C2312yes—
59C2×C3⋊(C3⋊Dic3)C2312yes—
60#6308—22no—
61C3×#638—24no—
62C2×C3×C3⋊Dic3C2311yescharacteristic
63C3×C4⋊(C3⋊D3)—22no—
64C3×C4×C3⋊D3D414yescomplemented
65C32×#67—44no—
66C2×C32×Dic3C2312yes—
67C32×D12—22no—
68#6290—24no—
69#6290—44no—
70D3×C3⋊Dic3—24no—
71#5805—91no—
72#5715—94no—
73C23×C3⋊D3C2×D312yes—
74C23×C3×D3—38no—
75C2×C32×D4—24no—
76C2×C32×D4—34no—
77C2×C32×D4—31no—
78C2×C32×D4C2×D314yes—
79C2×C32×D4C2×D311yescharacteristic
80C2×#638—32no—
81C2×#638—34no—
82C2×#638C2×D312yes—
83C22×C3⋊Dic3—32no—
84#3830—64no—
85D4×C3⋊D3—24no—
86C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
87C2×C4×C3⋊D3C2×D311yescharacteristic
88C2×#634—31no—
89C2×#634—34no—
90C2×#634—64no—
91C2×C3×#67—38no—
92C2×C3×#67C2×D3116yescomplemented
93C3×D3×D4—616no—
94C2×C3×D12—34no—
95C2×C3×D12C2×D314yescomplemented
96C2×C3×C4×D3—34no—
97C2×#625—38no—
98C2×#625—316no—
99C2×D3×Dic3—38no—
100#3795—616no—
101C32×C42—31no—
102#3757—34no—
103#3757—316no—
104#3757—616no—
105#3752—32no—
106#3752—34no—
107#3750—32no—
108C4×C3⋊Dic3—31no—
109C4×C3⋊Dic3—34no—
110C4×C3⋊Dic3—64no—
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—22no—
120C2×C3×C3⋊D3—24no—
121C2×C3×C3⋊D3—48no—
122C2×C3×C3⋊D3C2×D412yes—
123C2×C32×D3—22no—
124C2×C32×D3—24no—
125C33×C4C2×D412yes—
126C3⋊(C3⋊Dic3)—22no—
127C3⋊(C3⋊Dic3)—22no—
128C3×C3⋊Dic3C2×D412yes—
129C32×Dic3—22no—
130C22×C3×D4—91no—
131C2×#237—94no—
132C2×D3×D4—34no—
133C2×#235—91no—
134C3×#127—34no—
135#953—94no—
136#940—94no—
137#940—91no—
138#940C3⋊D311yescharacteristic, complemented
139#939—916no—
140#938—94no—
141#937—916no—
142#936—94no—
143#936—91no—
144#936—91no—
145#936—184no—
146#926—91no—
147#875—94no—
148#872—94no—
149C23×C32—28no—
150C23×C32—32no—
151C23×C32C22×D318yes—
152C23×C32C22×D312yes—
153C22×C3⋊D3—24no—
154C22×C3⋊D3—44no—
155C22×C3⋊D3C22×D312yes—
156C22×C3⋊D3C22×D311yescharacteristic
157C22×C3×D3—34no—
158C22×C3×D3—38no—
159C22×C3×D3—616no—
160C22×C3×D3—1216no—
161C22×C3×D3C22×D318yes—
162C32×D4—22no—
163C32×D4—28no—
164C32×D4—48no—
165C32×D4—62no—
166C32×D4—124no—
167C2×C32×C4—24no—
168C2×C32×C4—32no—
169C2×C32×C4C22×D314yes—
170C2×C32×C4C22×D311yescharacteristic
171#638—24no—
172#638—64no—
173#638—124no—
174C2×C3⋊Dic3—32no—
175C2×C3⋊Dic3—32no—
176C2×C3⋊Dic3—38no—
177C2×C3⋊Dic3—38no—
178C2×C3⋊Dic3—68no—
179C2×C3⋊Dic3—68no—
180C2×C3⋊Dic3—64no—
181C2×C3⋊Dic3C22×D311yescharacteristic
182C4⋊(C3⋊D3)—22no—
183C4×C3⋊D3#6714yescomplemented
184#634—62no—
185#634—68no—
186#634—128no—
187C3×#67—416no—
188C3×#67—616no—
189C2×C3×Dic3—34no—
190C2×C3×Dic3C22×D318yes—
191C3×D12—28no—
192C3×D12—68no—
193C3×C4×D3—316no—
194#625—616no—
195#625—1216no—
196D3×Dic3—616no—
197#535—271noSylow
198C2×C33—42no—
199C2×C33C22×D411yescharacteristic
200C2×C33H(5,𝔽2)12yes—
201C3×C3⋊D3—22no—
202C3×C3⋊D3—42no—
203C32×D3—42no—
204C24×C3—92no—
205C23×D3—38no—
206C2×C3×D4—24no—
207C2×C3×D4—316no—
208C2×C3×D4—34no—
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—98no—
217C2×#67—92no—
218C2×#67—916no—
219C2×#67—94no—
220C2×#67C2×C3⋊D314yescomplemented
221C22×Dic3—98no—
222#237—1816no—
223D3×D4—616no—
224#235—184no—
225C2×D12—34no—
226C2×D12C2×C3⋊D311yescharacteristic, complemented
227C2×C4×D3—34no—
228C2×C4×D3—92no—
229C2×Dic6—94no—
230C2×Dic6—91no—
231C2×Dic6—184no—
232C3×C42—34no—
233#217—916no—
234#217—94no—
235#217—92no—
236#217—94no—
237#217—1816no—
238(C2×D3)⋊C4—98no—
239(C2×D3)⋊C4—916no—
240C4⋊Dic3—92no—
241Dic3⋊C4—98no—
242C4×Dic3—94no—
243C4×Dic3—91no—
244C4×Dic3—184no—
245C4×Dic3C2×C3⋊D311yescharacteristic, complemented
246C22×C32—24no—
247C22×C32—22no—
248C22×C32—24no—
249C22×C32—28no—
250C22×C32—216no—
251C22×C32—48no—
252C22×C32—416no—
253C22×C32—64no—
254C22×C32—62no—
255C22×C32C23×D314yes—
256C22×C32C23×D311yescharacteristic
257C2×C3⋊D3—22no—
258C2×C3⋊D3—24no—
259C2×C3⋊D3—22no—
260C2×C3⋊D3—48no—
261C2×C3⋊D3C2×#6712yes—
262C2×C3×D3—28no—
263C2×C3×D3—216no—
264C2×C3×D3—38no—
265C2×C3×D3—68no—
266C2×C3×D3—68no—
267C2×C3×D3—616no—
268C2×C3×D3—1232no—
269C32×C4—28no—
270C32×C4—62no—
271C32×C4D3×D418yes—
272C32×C4C2×#6712yes—
273C3⋊Dic3—62no—
274C3⋊Dic3—62no—
275C3⋊Dic3—68no—
276C3⋊Dic3—68no—
277C3⋊Dic3—128no—
278C3⋊Dic3—128no—
279C3⋊Dic3C2×#6712yes—
280C3×Dic3—28no—
281C3×Dic3—38no—
282C2×#41—271no—
283C22×D4—91no—
284#127—31no—
285#127—271no—
286#124—271no—
287#124—271no—
288#121—274no—
289#120—274no—
290#117—271no—
291C33#53511yescharacteristic, complemented, Sylow
292C23×C3—28no—
293C23×C3—38no—
294C23×C3—91no—
295C23×C3—92no—
296C23×C3—184no—
297C23×C3—364no—
298C23×C3C2×D3218yes—
299C23×C3C22×C3⋊D312yes—
300C22×D3—34no—
301C22×D3—38no—
302C22×D3—616no—
303C22×D3—1216no—
304C22×D3C22×C3⋊D312yes—
305C3×D4—28no—
306C3×D4—22no—
307C3×D4—48no—
308C3×D4—68no—
309C3×D4—1216no—
310C3×D4—184no—
311C3×D4—182no—
312C2×C3×C4—24no—
313C2×C3×C4—38no—
314C2×C3×C4—91no—
315C2×C3×C4—94no—
316C2×C3×C4C2×D3214yes—
317C2×C3×C4C22×C3⋊D311yescharacteristic
318#67—44no—
319#67—616no—
320#67—184no—
321#67—1816no—
322#67—364no—
323#67—3616no—
324C2×Dic3—34no—
325C2×Dic3—98no—
326C2×Dic3—98no—
327C2×Dic3—92no—
328C2×Dic3—92no—
329C2×Dic3—188no—
330C2×Dic3—188no—
331C2×Dic3—1816no—
332C2×Dic3C22×C3⋊D312yes—
333D12—22no—
334D12—68no—
335C4×D3—316no—
336C4×D3—184no—
337Dic6—188no—
338Dic6—182no—
339Dic6—368no—
340C2×C32—28no—
341C2×C32—24no—
342C2×C32—42no—
343C2×C32—48no—
344C2×C32—88no—
345C2×C32—122no—
346C2×C32C2×D3×D414yes—
347C2×C32#100418yes—
348C2×C32#100412yes—
349C2×C32C22×#6711yescharacteristic
350C3⋊D3—22no—
351C3⋊D3—42no—
352C3×D3—48no—
353C3×D3—68no—
354C3×D3—128no—
355C24—92no—
356#41—544no—
357C2×Q8—271no—
358C2×D4—34no—
359C2×D4—31no—
360C2×D4—92no—
361C2×D4—91no—
362C2×D4—184no—
363C2×D4—272no—
364C2×D4—274no—
365C2×D4D3×C3⋊D311yescharacteristic
366C22×C4—91no—
367C22×C4—272no—
368C4⋊C4—272no—
369C22⋊C4—272no—
370C22⋊C4—274no—
371C22⋊C4—274no—
372C42—31no—
373C42—271no—
374C22×C3—216no—
375C22×C3—28no—
376C22×C3—22no—
377C22×C3—24no—
378C22×C3—24no—
379C22×C3—48no—
380C22×C3—416no—
381C22×C3—616no—
382C22×C3—68no—
383C22×C3—92no—
384C22×C3—182no—
385C22×C3—182no—
386C22×C3—184no—
387C22×C3—368no—
388C22×C3C22×D3214yes—
389C22×C3C23×C3⋊D311yescharacteristic
390C2×D3—22no—
391C2×D3—24no—
392C2×D3—38no—
393C2×D3—68no—
394C2×D3—616no—
395C2×D3—68no—
396C2×D3—1232no—
397C3×C4—28no—
398C3×C4—68no—
399C3×C4—92no—
400C3×C4D3×#6718yes—
401C3×C4D4×C3⋊D312yes—
402Dic3—22no—
403Dic3—38no—
404Dic3—188no—
405Dic3—188no—
406Dic3—182no—
407Dic3—182no—
408Dic3—368no—
409Dic3—368no—
410C32—24no—
411C32#571514yescomplemented
412C32#580511yescharacteristic, complemented
413C23—32no—
414C23—91no—
415C23—92no—
416C23—184no—
417C23—364no—
418C23C2×D3×C3⋊D312yes—
419Q8—542no—
420D4—22no—
421D4—62no—
422D4—124no—
423D4—184no—
424D4—182no—
425D4—544no—
426D4—1084no—
427C2×C4—32no—
428C2×C4—91no—
429C2×C4—94no—
430C2×C4—272no—
431C2×C4—272no—
432C2×C4—544no—
433C2×C4C2×D3×C3⋊D311yescharacteristic
434C2×C3—28no—
435C2×C3—24no—
436C2×C3—42no—
437C2×C3—48no—
438C2×C3—88no—
439C2×C3—128no—
440C2×C3—182no—
441C2×C3—362no—
442C2×C3#6431318yes—
443C2×C3C2×D3×#6714yes—
444C2×C3C2×D4×C3⋊D311yescharacteristic
445C2×C3#6436112yes—
446D3—42no—
447D3—68no—
448D3—128no—
449C22—24no—
450C22—22no—
451C22—64no—
452C22—62no—
453C22—92no—
454C22—182no—
455C22—184no—
456C22—182no—
457C22—368no—
458C22C22×D3×C3⋊D311yescharacteristic
459C4—62no—
460C4—92no—
461C4—542no—
462C4—542no—
463C4C3⋊D3×#6712yes—
464C3—24no—
465C3#1059523314yescomplemented
466C3#1059618611yescharacteristic, complemented
467C2—42no—
468C2—122no—
469C2—182no—
470C2—362no—
471C2#1172458712yes—
472C2C2×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×#89014C2semidirect
3. #11723977C2semidirect
4. C32×#940C2semidirect
5. #11723634C2semidirect
6. #11723615C2semidirect
7. #11723529C2semidirect
8. #11723439C2semidirect
9. #11723437C2semidirect
14. C2×#6308C22semidirect
19. C2×C32×#67C22semidirect
20. C2×C32×D12C22semidirect
22. C2×#6290C22semidirect
23. C2×D3×C3⋊Dic3C22semidirect
25. #88864C22semidirect
26. C4×C3⋊(C3⋊Dic3)C22semidirect
27. C32×C4×Dic3C22semidirect
28. #88807C22semidirect
30. #88803C22semidirect
32. C2×D4×C3⋊D3D3semidirect
44. C3×#940D3semidirect
64. C3×C4×C3⋊D3D4semidirect
92. C2×C3×#67C2×D3semidirect
95. C2×C3×D12C2×D3semidirect
111. C3×C4×Dic3C2×D3semidirect
138. #940C3⋊D3semidirect
183. C4×C3⋊D3#67semidirect
220. C2×#67C2×C3⋊D3semidirect
226. C2×D12C2×C3⋊D3semidirect
245. C4×Dic3C2×C3⋊D3semidirect
291. C33#535semidirect
411. C32#5715semidirect
412. C32#5805semidirect
465. C3#10595233semidirect
466. C3#10596186semidirect

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×#890142
#117239772
C32×#9402
#117236342
#117236152
#117235292
#117234392
#117234372
#105961863
#105952333