Finite Group Databasebeta

← #13175409

Subgroups — #13175409

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×D3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C2×#89021C211yesmaximal, characteristic, complemented
3#11723916C211yesmaximal, characteristic, complemented
4C3×#64083C211yesmaximal, characteristic, complemented
5C32×#872C211yesmaximal, characteristic, complemented
6#11723541C214yesmaximal, complemented
7#11723517C211yesmaximal, characteristic, complemented
8#11723482C214yesmaximal, complemented
9#11723439C211yesmaximal, characteristic, complemented
10#10595996—31nomaximal
11#10594000—34nomaximal
12C22×D3×C3⋊D3C2212yes—
13C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
14C2×#6308C2211yescharacteristic, complemented
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
17C2×C3×C4⋊(C3⋊D3)C2214yescomplemented
18C2×C32×C4×D3C2211yescharacteristic
19C2×#6292C2212yes—
20C2×#6292C2214yescomplemented
21C2×#6290C2212yes—
22C2×Dic3×C3⋊D3C2212yescomplemented
23D3×C4⋊(C3⋊D3)—24no—
24#89021—24no—
25C33×C42C2211yescharacteristic, complemented
26#88859C2214yescomplemented
27C32×(C2×D3)⋊C4C2212yes—
28C32×Dic3⋊C4C2212yescomplemented
29C32×C4×Dic3C2211yescharacteristic, complemented
30#88803C2214yescomplemented
31C2×#3828—31no—
32C22×C4⋊(C3⋊D3)—31no—
33C2×D3×D12—34no—
34C2×#3795—34no—
35C32×#117—31no—
36#64104—34no—
37#64091—34no—
38#64084—31no—
39#64084—31no—
40#64084—34no—
41#64084—64no—
42#64083—31no—
43#64083D311yescharacteristic, complemented
44C3×#874—34no—
45C3×#872D314yescomplemented
46#63922—316no—
47#63911—34no—
48#63886—316no—
49#63860—34no—
50C22×C3⋊(C3⋊D3)C2312yes—
51C22×C3×C3⋊D3C2312yes—
52C22×C3×C3⋊D3C2312yes—
53C22×C32×D3C2311yescharacteristic
54C2×D3×C3⋊D3—24no—
55C2×D3×C3⋊D3—44no—
56C2×C33×C4C2312yes—
57C2×C33×C4C2311yescharacteristic
58C2×C3⋊(C3⋊Dic3)C2312yes—
59C4⋊(C3⋊(C3⋊D3))—22no—
60#6308—22no—
61C3×C4⋊(C3⋊D3)—22no—
62C3×C4⋊(C3⋊D3)—22no—
63C3×C4⋊(C3⋊D3)—44no—
64C2×C32×Dic3C2312yes—
65C2×C32×Dic3C2311yescharacteristic
66C32×C4×D3D414yescomplemented
67#6292—24no—
68#6292—44no—
69#6290—24no—
70Dic3×C3⋊D3—24no—
71#5715—91no—
72#5525—94no—
73C23×C3⋊D3—32no—
74C22×D32—38no—
75C22×C32×C4—31no—
76C2×#638—32no—
77#3828—64no—
78C2×C4⋊(C3⋊D3)—32no—
79C2×C4⋊(C3⋊D3)—31no—
80C2×C4⋊(C3⋊D3)—34no—
81C2×C4⋊(C3⋊D3)—34no—
82C2×C4⋊(C3⋊D3)—64no—
83C2×C4⋊(C3⋊D3)—64no—
84C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
85C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
86C2×C4⋊(C3⋊D3)C2×D314yescomplemented
87C2×C4×C3⋊D3—32no—
88C2×#634—31no—
89C2×#634—34no—
90C2×#634—64no—
91C2×C3×D12—34no—
92C2×C3×D12—34no—
93C2×C3×D12—316no—
94C2×C3×C4×D3C2×D314yes—
95C2×#626—38no—
96C2×#626—316no—
97C2×#625—38no—
98C2×D3×Dic3—38no—
99D3×D12—616no—
100#3795—616no—
101C32×C4⋊C4—32no—
102C32×C22⋊C4—32no—
103C32×C42—24no—
104C32×C42—31no—
105C32×C42C2×D314yescomplemented
106C32×C42C2×D311yescharacteristic, complemented
107#3752—34no—
108#3752—34no—
109#3752—316no—
110#3752—616no—
111C3×(C2×D3)⋊C4C2×D318yes—
112C3×Dic3⋊C4C2×D318yescomplemented
113C3×C4×Dic3C2×D314yescomplemented
114(C2×D3)⋊Dic3—316no—
115C22×C33C2411yescharacteristic
116C2×C3⋊(C3⋊D3)—22no—
117C2×C3⋊(C3⋊D3)—24no—
118C2×C3×C3⋊D3—22no—
119C2×C3×C3⋊D3—22no—
120C2×C3×C3⋊D3—24no—
121C2×C3×C3⋊D3—24no—
122C2×C32×D3—22no—
123C2×C32×D3C2×D412yes—
124D3×C3⋊D3—48no—
125C33×C4—22no—
126C33×C4C2×D412yes—
127C3⋊(C3⋊Dic3)—22no—
128C32×Dic3—22no—
129C32×Dic3C2×D412yes—
130C2×#237—91no—
131C2×D3×D4—91no—
132C2×#235—94no—
133C22×D12—94no—
134C3×#127—91no—
135C3×#117—34no—
136#940—91no—
137#939—94no—
138#937—94no—
139#936—91no—
140#895—916no—
141#882—916no—
142#875—94no—
143#875—94no—
144#875—91no—
145#875—184no—
146#874—34no—
147#874—94no—
148#872C3⋊D311yescharacteristic, complemented
149C23×C32—31no—
150C22×C3⋊D3—32no—
151C22×C3⋊D3—38no—
152C22×C3⋊D3—68no—
153C22×C3⋊D3—64no—
154C22×C3⋊D3—124no—
155C22×C3⋊D3C22×D312yes—
156C22×C3⋊D3C22×D312yes—
157C22×C3×D3—38no—
158C22×C3×D3—38no—
159C22×C3×D3C22×D314yes—
160C2×D32—616no—
161C2×D32—1216no—
162C2×C32×C4—24no—
163C2×C32×C4—28no—
164C2×C32×C4—31no—
165C2×C32×C4—34no—
166C2×C32×C4—32no—
167C2×C32×C4C22×D318yes—
168C2×C32×C4C22×D314yes—
169C2×C32×C4C22×D312yes—
170C2×C32×C4C22×D311yescharacteristic
171#638—64no—
172C2×C3⋊Dic3—32no—
173C2×C3⋊Dic3—38no—
174C2×C3⋊Dic3—68no—
175C4⋊(C3⋊D3)—22no—
176C4⋊(C3⋊D3)—22no—
177C4⋊(C3⋊D3)—44no—
178C4⋊(C3⋊D3)—64no—
179C4⋊(C3⋊D3)—62no—
180C4⋊(C3⋊D3)—68no—
181C4⋊(C3⋊D3)—128no—
182C4⋊(C3⋊D3)—124no—
183C4×C3⋊D3—64no—
184#634—62no—
185#634—68no—
186#634—128no—
187C2×C3×Dic3C22×D318yes—
188C2×C3×Dic3C22×D314yes—
189C3×D12—68no—
190C3×D12—68no—
191C3×D12—1216no—
192C3×C4×D3D12116yescomplemented
193#626—616no—
194#626—1216no—
195#625—616no—
196D3×Dic3—616no—
197#535—271noSylow
198C2×C33C22×D411yescharacteristic
199C2×C33H(5,𝔽2)12yes—
200C3⋊(C3⋊D3)—42no—
201C3×C3⋊D3—42no—
202C3×C3⋊D3—42no—
203C32×D3—22no—
204C23×D3—98no—
205C23×D3—92no—
206C2×C3×D4—91no—
207C2×C3×D4—91no—
208C2×C3×D4—94no—
209C22×C3×C4—34no—
210C2×#67—98no—
211C2×#67—92no—
212C2×#67—92no—
213C2×#67—94no—
214C22×Dic3—92no—
215#237—184no—
216D3×D4—184no—
217#235—1816no—
218C2×D12—34no—
219C2×D12—34no—
220C2×D12—316no—
221C2×D12—98no—
222C2×D12—94no—
223C2×D12—91no—
224C2×D12—916no—
225C2×D12—184no—
226C2×D12—1816no—
227C2×C4×D3—98no—
228C2×C4×D3C2×C3⋊D311yescharacteristic
229C2×Dic6—94no—
230C2×Dic6—91no—
231C2×Dic6—184no—
232C3×C4⋊C4—38no—
233C3×C22⋊C4—38no—
234C3×C42—24no—
235C3×C42—34no—
236C3×C42D3214yescomplemented
237C3×C42C2×C3⋊D311yescharacteristic, complemented
238#217—94no—
239(C2×D3)⋊C4—916no—
240(C2×D3)⋊C4—916no—
241(C2×D3)⋊C4—94no—
242(C2×D3)⋊C4—1816no—
243(C2×D3)⋊C4C2×C3⋊D312yes—
244Dic3⋊C4C2×C3⋊D312yescomplemented
245C4×Dic3C2×C3⋊D311yescharacteristic, complemented
246C22×C32—24no—
247C22×C32—32no—
248C22×C32—62no—
249C22×C32C23×D314yes—
250C22×C32C23×D311yescharacteristic
251C2×C3⋊D3—22no—
252C2×C3⋊D3—22no—
253C2×C3⋊D3—24no—
254C2×C3⋊D3—24no—
255C2×C3⋊D3—62no—
256C2×C3⋊D3—64no—
257C2×C3⋊D3—68no—
258C2×C3⋊D3—616no—
259C2×C3⋊D3—128no—
260C2×C3⋊D3—1216no—
261C2×C3⋊D3—128no—
262C2×C3×D3—28no—
263C2×C3×D3—68no—
264C2×C3×D3—68no—
265C2×C3×D3—616no—
266C2×C3×D3—616no—
267C2×C3×D3C2×D1218yes—
268D32—1232no—
269C32×C4—22no—
270C32×C4—28no—
271C32×C4—28no—
272C32×C4—32no—
273C32×C4—48no—
274C32×C4—62no—
275C32×C4C2×D1218yes—
276C32×C4D3×D412yes—
277C3⋊Dic3—62no—
278C3⋊Dic3—68no—
279C3⋊Dic3—128no—
280C3×Dic3—28no—
281C3×Dic3C2×D1218yes—
282C2×#41—271no—
283C22×D4—271no—
284#127—91no—
285#127—271no—
286#124—271no—
287#124—271no—
288#121—274no—
289#120—274no—
290#117—31no—
291C33#53511yescharacteristic, complemented, Sylow
292C23×C3—34no—
293C23×C3—92no—
294C23×C3—92no—
295C22×D3—38no—
296C22×D3—38no—
297C22×D3—98no—
298C22×D3—92no—
299C22×D3—188no—
300C22×D3—1816no—
301C22×D3—184no—
302C22×D3—364no—
303C22×D3—3616no—
304C22×D3C22×C3⋊D311yescharacteristic
305C3×D4—182no—
306C3×D4—182no—
307C3×D4—364no—
308C2×C3×C4—24no—
309C2×C3×C4—28no—
310C2×C3×C4—34no—
311C2×C3×C4—316no—
312C2×C3×C4—38no—
313C2×C3×C4C2×D3218yes—
314C2×C3×C4C2×D3214yes—
315C2×C3×C4C22×C3⋊D312yes—
316C2×C3×C4C22×C3⋊D311yescharacteristic
317#67—1816no—
318#67—184no—
319#67—184no—
320#67—364no—
321C2×Dic3—98no—
322C2×Dic3—92no—
323C2×Dic3—188no—
324C2×Dic3—184no—
325C2×Dic3C22×C3⋊D312yes—
326C2×Dic3C22×C3⋊D311yescharacteristic
327D12—68no—
328D12—68no—
329D12—1216no—
330D12—1816no—
331D12—188no—
332D12—182no—
333D12—368no—
334D12—3616no—
335C4×D3—1816no—
336C4×D3C4⋊(C3⋊D3)14yescomplemented
337Dic6—188no—
338Dic6—182no—
339Dic6—368no—
340C2×C32—24no—
341C2×C32—28no—
342C2×C32—62no—
343C2×C32C22×D1214yes—
344C2×C32C2×D3×D411yescharacteristic
345C2×C32#100412yes—
346C2×C32#100918yes—
347C3⋊D3—42no—
348C3⋊D3—42no—
349C3⋊D3—122no—
350C3⋊D3—128no—
351C3⋊D3—248no—
352C3×D3—28no—
353C3×D3—128no—
354C3×D3—128no—
355C24—272no—
356#41—544no—
357C2×Q8—271no—
358C2×D4—91no—
359C2×D4—91no—
360C2×D4—94no—
361C2×D4—272no—
362C2×D4—274no—
363C2×D4—272no—
364C2×D4—271no—
365C2×D4—544no—
366C22×C4—31no—
367C22×C4—272no—
368C4⋊C4—32no—
369C22⋊C4—32no—
370C22⋊C4—274no—
371C22⋊C4—274no—
372C42—31no—
373C42D3×C3⋊D311yescharacteristic, complemented
374C22×C3—24no—
375C22×C3—38no—
376C22×C3—68no—
377C22×C3—182no—
378C22×C3—182no—
379C22×C3—184no—
380C22×C3—184no—
381C22×C3C22×D3214yes—
382C22×C3C23×C3⋊D311yescharacteristic
383C2×D3—22no—
384C2×D3—68no—
385C2×D3—68no—
386C2×D3—616no—
387C2×D3—616no—
388C2×D3—188no—
389C2×D3—1816no—
390C2×D3—182no—
391C2×D3—184no—
392C2×D3—368no—
393C2×D3—3616no—
394C2×D3—368no—
395C2×D3—3632no—
396C2×D3C2×C4⋊(C3⋊D3)12yes—
397C3×C4—28no—
398C3×C4—22no—
399C3×C4—28no—
400C3×C4—38no—
401C3×C4—48no—
402C3×C4—68no—
403C3×C4D3×D1218yes—
404C3×C4C2×C4⋊(C3⋊D3)12yes—
405Dic3—22no—
406Dic3—188no—
407Dic3—182no—
408Dic3—368no—
409Dic3C2×C4⋊(C3⋊D3)12yes—
410C32—24no—
411C32#552514yescomplemented
412C32#571511yescharacteristic, complemented
413C23—31no—
414C23—92no—
415C23—92no—
416C23—272no—
417C23—544no—
418C23—1084no—
419Q8—542no—
420D4—182no—
421D4—182no—
422D4—364no—
423D4—544no—
424D4—544no—
425D4—542no—
426D4—1084no—
427C2×C4—31no—
428C2×C4—34no—
429C2×C4—32no—
430C2×C4—272no—
431C2×C4—544no—
432C2×C4C2×D3×C3⋊D312yes—
433C2×C4C2×D3×C3⋊D311yescharacteristic
434C2×C3—24no—
435C2×C3—28no—
436C2×C3—68no—
437C2×C3—362no—
438C2×C3—362no—
439C2×C3C2×D3×D1214yes—
440C2×C3#6430718yes—
441C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
442C2×C3#6436612yes—
443D3—22no—
444D3—128no—
445D3—128no—
446D3—362no—
447D3—368no—
448D3—728no—
449C22—32no—
450C22—62no—
451C22—182no—
452C22—182no—
453C22—184no—
454C22—184no—
455C22—542no—
456C22—544no—
457C22—1088no—
458C22C22×D3×C3⋊D311yescharacteristic
459C4—22no—
460C4—32no—
461C4—62no—
462C4—542no—
463C4D3×C4⋊(C3⋊D3)12yes—
464C3—24no—
465C3#1059400014yescomplemented
466C3#1059599611yescharacteristic, complemented
467C2—62no—
468C2—362no—
469C2—362no—
470C2—1082no—
471C2C2×D3×C4⋊(C3⋊D3)11yescharacteristic
472C2#1172458012yes—
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
hypercenterC42D3×C3⋊D3
hypocenterC33#535
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×D3×C4⋊(C3⋊D3)C2semidirect
2. C2×#89021C2semidirect
3. #11723916C2semidirect
4. C3×#64083C2semidirect
5. C32×#872C2semidirect
6. #11723541C2semidirect
7. #11723517C2semidirect
8. #11723482C2semidirect
9. #11723439C2semidirect
14. C2×#6308C22semidirect
16. C2×C3×C4⋊(C3⋊D3)C22semidirect
17. C2×C3×C4⋊(C3⋊D3)C22semidirect
20. C2×#6292C22semidirect
22. C2×Dic3×C3⋊D3C22semidirect
25. C33×C42C22semidirect
26. #88859C22semidirect
28. C32×Dic3⋊C4C22semidirect
29. C32×C4×Dic3C22semidirect
30. #88803C22semidirect
43. #64083D3semidirect
45. C3×#872D3semidirect
66. C32×C4×D3D4semidirect
85. C2×C4⋊(C3⋊D3)C2×D3semidirect
86. C2×C4⋊(C3⋊D3)C2×D3semidirect
105. C32×C42C2×D3semidirect
106. C32×C42C2×D3semidirect
112. C3×Dic3⋊C4C2×D3semidirect
113. C3×C4×Dic3C2×D3semidirect
148. #872C3⋊D3semidirect
192. C3×C4×D3D12semidirect
236. C3×C42D32semidirect
237. C3×C42C2×C3⋊D3semidirect
244. Dic3⋊C4C2×C3⋊D3semidirect
245. C4×Dic3C2×C3⋊D3semidirect
291. C33#535semidirect
336. C4×D3C4⋊(C3⋊D3)semidirect
373. C42D3×C3⋊D3semidirect
411. C32#5525semidirect
412. C32#5715semidirect
465. C3#10594000semidirect
466. C3#10595996semidirect

Maximal subgroups

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

subgroupindex
C2×D3×C4⋊(C3⋊D3)2
C2×#890212
#117239162
C3×#640832
C32×#8722
#117235412
#117235172
#117234822
#117234392
#105959963
#105940003