Finite Group Databasebeta

← C2×#11721241 #13151987

Subgroups — C2×#11721241 #13151987

Order 1728.

1514 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6233C211yesmaximal, characteristic
2C23×C3×Dic9C211yesmaximal, characteristic, complemented
3C22×(C3⋊D9)⋊C4C211yesmaximal, characteristic, complemented
4C2×C9×#217C211yesmaximal, characteristic, complemented
5C2×#88445C211yesmaximal, characteristic, complemented
6C2×#88442C211yesmaximal, characteristic, complemented
7C2×Dic3×Dic9C211yesmaximal, characteristic, complemented
8#11721241C218yesmaximal, direct product factor
9C2×#63952—31nomaximal
10C2×#611⋊C4—31nomaximal
11C24×C3×C9C2211yescharacteristic
12C23×C3⋊D9C2211yescharacteristic
13C2×#6233C418yescomplemented
14C2×#6233C2214yes—
15C22×C3⋊Dic9C2211yescharacteristic
16C22×C9×Dic3C2211yescharacteristic
17C22×C9×Dic3C2211yescharacteristic
18C22×C3×Dic9—24no—
19C22×C3×Dic9C2211yescharacteristic
20C22×C3×Dic9C2211yescharacteristic
21C22×C3×Dic9C2214yescomplemented
22C2×(C3⋊D9)⋊C4—24no—
23C2×(C3⋊D9)⋊C4C2214yescomplemented
24C9×#217C2214yescomplemented
25#88445C2214yescomplemented
26#88442C2214yescomplemented
27Dic3×Dic9C2214yescomplemented
28C22×#638—31no—
29C23×C3×Dic3—31no—
30C22×#624—31no—
31C2×C3×#217D311yescharacteristic
32C2×#3724—31no—
33C2×#3722—31no—
34C2×Dic32—31no—
35#63952—38no—
36C22×#611—31no—
37C22×#611—61no—
38C23×Dic9D311yescharacteristic, complemented
39C22×C4×D9—31no—
40C2×C9×C22⋊C4—31no—
41C2×(C2×D9)⋊C4—31no—
42C2×Dic9⋊C4—31no—
43C2×C4×Dic9—31no—
44#611⋊C4—38no—
45C23×C3×C9—22no—
46C23×C3×C9—22no—
47C23×C3×C9C2×C414yescomplemented
48C23×C3×C9C2312yes—
49C23×C3×C9C2311yescharacteristic
50C22×C3⋊D9—22no—
51C22×C3⋊D9—22no—
52C22×C3⋊D9C2×C414yescomplemented
53C22×C3⋊D9C2312yes—
54#6233C2×C4116yescomplemented
55C2×C3⋊Dic9C2×C414yescomplemented
56C2×C3⋊Dic9C2312yes—
57C2×C9×Dic3—22no—
58C2×C9×Dic3—22no—
59C2×C9×Dic3C2312yes—
60C2×C9×Dic3C2312yes—
61C2×C3×Dic9—22no—
62C2×C3×Dic9—28no—
63C2×C3×Dic9D414yescomplemented
64C2×C3×Dic9C2312yes—
65C2×C3×Dic9C2312yes—
66(C3⋊D9)⋊C4—28no—
67C2×C4×#67—91no—
68C2×#67⋊C4—91no—
69C24×C32C2×D311yescharacteristic
70C23×C3⋊D3—31no—
71C2×#638—38no—
72C2×#638—34no—
73C22×C3⋊Dic3—31no—
74C22×C3×Dic3—31no—
75C22×C3×Dic3—31no—
76C22×C3×Dic3—34no—
77C22×C3×Dic3—64no—
78C22×C3×Dic3C2×D311yescharacteristic
79C22×C3×Dic3C2×D311yescharacteristic
80C2×#624—34no—
81C2×#624—64no—
82C24×C9—21no—
83C24×C9C2×D311yescharacteristic
84C23×D9—31no—
85C23×D9—61no—
86C3×#217C2×D314yes—
87#3724—34no—
88#3722—34no—
89Dic32—34no—
90C22×C4×C9—31no—
91C22×C4×C9—31no—
92C2×#611—38no—
93C2×#611—34no—
94C2×#611—68no—
95C2×#611—64no—
96C22×Dic9—24no—
97C22×Dic9—31no—
98C22×Dic9—61no—
99C22×Dic9C2×D311yescharacteristic
100C22×Dic9C2×D311yescharacteristic
101C22×Dic9C2×D314yescomplemented
102C2×C4×D9—34no—
103C2×C4×D9—64no—
104C9×C22⋊C4—34no—
105(C2×D9)⋊C4—34no—
106Dic9⋊C4—34no—
107C4×Dic9—34no—
108C22×C3×C9—28no—
109C22×C3×C9—22no—
110C22×C3×C9—22no—
111C22×C3×C9C22×C412yes—
112C22×C3×C9C22×C414yescomplemented
113C22×C3×C9C2×D412yes—
114C22×C3×C9#4112yes—
115C22×C3×C9C2411yescharacteristic
116C2×C3⋊D9—28no—
117C2×C3⋊D9—22no—
118C2×C3⋊D9—22no—
119C2×C3⋊D9C22×C414yescomplemented
120C3⋊Dic9C22×C414yescomplemented
121C9×Dic3—22no—
122C9×Dic3—22no—
123C3×Dic9—22no—
124C3×Dic9C2×D414yescomplemented
125C23×C3×C4—91no—
126C22×#67—91no—
127C22×#67—91no—
128C22×#67—181no—
129C23×Dic3—31no—
130C22×C4×D3—91no—
131C22×C4×D3—91no—
132C2×C3×C22⋊C4—31no—
133C2×#217D911yescharacteristic, complemented
134C4×#67—98no—
135C2×(C2×D3)⋊C4—91no—
136C2×(C2×D3)⋊C4—91no—
137C2×Dic3⋊C4—91no—
138C2×Dic3⋊C4—91no—
139C2×C4×Dic3—91no—
140C2×C4×Dic3—91no—
141#67⋊C4—98no—
142C23×C32—22no—
143C23×C32—22no—
144C23×C32C4×D314yes—
145C23×C32C22×D312yes—
146C23×C32C22×D311yescharacteristic
147C22×C3⋊D3—34no—
148C22×C3⋊D3—32no—
149C22×C3⋊D3—62no—
150C22×C3⋊D3—62no—
151#638—316no—
152C2×C3⋊Dic3—34no—
153C2×C3⋊Dic3—32no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—34no—
157C2×C3×Dic3—32no—
158C2×C3×Dic3—32no—
159C2×C3×Dic3—68no—
160C2×C3×Dic3—62no—
161C2×C3×Dic3C22×D312yes—
162C2×C3×Dic3C22×D312yes—
163#624—68no—
164C23×C9—22no—
165C23×C9—22no—
166C23×C9—24no—
167C23×C9—21no—
168C23×C9—22no—
169C23×C9—42no—
170C23×C9—42no—
171C23×C9C4×D314yescomplemented
172C23×C9C22×D312yes—
173C23×C9C22×D311yescharacteristic
174C22×D9—34no—
175C22×D9—32no—
176C22×D9—64no—
177C22×D9—62no—
178C22×D9—62no—
179C22×D9—62no—
180C22×D9—122no—
181C22×D9—122no—
182C2×C4×C9—32no—
183C2×C4×C9—32no—
184C2×C4×C9—62no—
185C2×C4×C9—62no—
186#611—316no—
187#611—616no—
188C2×Dic9—22no—
189C2×Dic9—28no—
190C2×Dic9—34no—
191C2×Dic9—32no—
192C2×Dic9—64no—
193C2×Dic9—62no—
194C2×Dic9#6714yescomplemented
195C2×Dic9C22×D312yes—
196C2×Dic9C22×D312yes—
197C4×D9—68no—
198C2×C4×D4—271noSylow
199C2×C3×C9—24no—
200C2×C3×C9C4×D414yescomplemented
201C2×C3×C9C23×C411yescharacteristic
202C2×C3×C9C22×D411yescharacteristic
203C2×C3×C9C2×#4111yescharacteristic
204C3⋊D9—24no—
205C24×C3—21no—
206C24×C3C2×D911yescharacteristic
207C24×C3D3211yescharacteristic
208C23×D3—91no—
209C23×D3—91no—
210C23×D3—181no—
211C22×C3×C4—31no—
212C22×C3×C4—31no—
213C22×C3×C4—91no—
214C22×C3×C4—91no—
215C22×C3×C4—94no—
216C22×C3×C4—184no—
217C2×#67—98no—
218C2×#67—94no—
219C2×#67—98no—
220C2×#67—94no—
221C2×#67—188no—
222C2×#67—184no—
223C22×Dic3—31no—
224C22×Dic3—34no—
225C22×Dic3—31no—
226C22×Dic3—64no—
227C22×Dic3—91no—
228C22×Dic3—91no—
229C22×Dic3—181no—
230C22×Dic3C2×D911yescharacteristic
231C22×Dic3C2×D911yescharacteristic
232C2×C4×D3—94no—
233C2×C4×D3—94no—
234C2×C4×D3—184no—
235C2×C4×D3—184no—
236C3×C22⋊C4—34no—
237#217C2×D914yescomplemented
238(C2×D3)⋊C4—94no—
239(C2×D3)⋊C4—94no—
240Dic3⋊C4—94no—
241Dic3⋊C4—94no—
242C4×Dic3—94no—
243C4×Dic3—94no—
244C22×C32—28no—
245C22×C32—22no—
246C22×C32—22no—
247C22×C32C2×C4×D312yes—
248C22×C32C2×C4×D314yes—
249C22×C32D3×D412yes—
250C22×C32#23712yes—
251C22×C32C23×D311yescharacteristic
252C2×C3⋊D3—34no—
253C2×C3⋊D3—68no—
254C2×C3⋊D3—62no—
255C2×C3⋊D3—62no—
256C3⋊Dic3—34no—
257C3×Dic3—22no—
258C3×Dic3—22no—
259C3×Dic3—34no—
260C3×Dic3—62no—
261C22×C9—28no—
262C22×C9—22no—
263C22×C9—22no—
264C22×C9—21no—
265C22×C9—22no—
266C22×C9—22no—
267C22×C9—24no—
268C22×C9—22no—
269C22×C9—42no—
270C22×C9—42no—
271C22×C9—48no—
272C22×C9C2×C4×D312yes—
273C22×C9C2×C4×D314yescomplemented
274C22×C9#23512yes—
275C22×C9C2×#6712yes—
276C22×C9C23×D311yescharacteristic
277C2×D9—34no—
278C2×D9—64no—
279C2×D9—68no—
280C2×D9—62no—
281C2×D9—62no—
282C2×D9—122no—
283C2×D9—122no—
284C2×D9—128no—
285C4×C9—62no—
286C4×C9—62no—
287Dic9—22no—
288Dic9—34no—
289Dic9—64no—
290Dic9C2×#6714yescomplemented
291C22×D4—271no—
292C23×C4—91no—
293C23×C4—271no—
294C4×D4—278no—
295C2×C4⋊C4—271no—
296C2×C22⋊C4—31no—
297C2×C22⋊C4—271no—
298C2×C42—271no—
299C3×C9C2×C4×D411yescharacteristic, complemented, Sylow
300C23×C3—22no—
301C23×C3—22no—
302C23×C3—22no—
303C23×C3—22no—
304C23×C3—24no—
305C23×C3—22no—
306C23×C3—21no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3C4×D914yescomplemented
310C23×C3C22×D912yes—
311C23×C3C22×D911yescharacteristic
312C23×C3#62414yes—
313C23×C3C2×D3212yes—
314C23×C3C2×D3211yescharacteristic
315C22×D3—92no—
316C22×D3—94no—
317C22×D3—92no—
318C22×D3—94no—
319C22×D3—182no—
320C22×D3—184no—
321C22×D3—182no—
322C22×D3—182no—
323C22×D3—182no—
324C22×D3—182no—
325C22×D3—362no—
326C22×D3—362no—
327C2×C3×C4—32no—
328C2×C3×C4—32no—
329C2×C3×C4—62no—
330C2×C3×C4—62no—
331C2×C3×C4—94no—
332C2×C3×C4—92no—
333C2×C3×C4—92no—
334C2×C3×C4—188no—
335C2×C3×C4—182no—
336#67—916no—
337#67—916no—
338#67—1816no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—32no—
342C2×Dic3—34no—
343C2×Dic3—32no—
344C2×Dic3—68no—
345C2×Dic3—62no—
346C2×Dic3—92no—
347C2×Dic3—94no—
348C2×Dic3—92no—
349C2×Dic3—94no—
350C2×Dic3—182no—
351C2×Dic3—184no—
352C2×Dic3C22×D912yes—
353C2×Dic3C22×D912yes—
354C4×D3—188no—
355C4×D3—188no—
356C2×C32—24no—
357C2×C32#67⋊C414yes—
358C2×C32C22×C4×D311yescharacteristic
359C2×C32C2×D3×D411yescharacteristic
360C2×C32C2×#23711yescharacteristic
361C3⋊D3—64no—
362C2×C9—24no—
363C2×C9—21no—
364C2×C9—21no—
365C2×C9—21no—
366C2×C9—24no—
367C2×C9—44no—
368C2×C9C4×#6714yescomplemented
369C2×C9C22×C4×D311yescharacteristic
370C2×C9C2×#23511yescharacteristic
371C2×C9C22×#6711yescharacteristic
372D9—64no—
373D9—124no—
374C24—271no—
375C24D3×D911yescharacteristic
376C2×D4—278no—
377C2×D4—274no—
378C22×C4—31no—
379C22×C4—31no—
380C22×C4—91no—
381C22×C4—94no—
382C22×C4—91no—
383C22×C4—184no—
384C22×C4—274no—
385C22×C4—271no—
386C22×C4—544no—
387C4⋊C4—274no—
388C22⋊C4—34no—
389C22⋊C4—274no—
390C42—274no—
391C22×C3—22no—
392C22×C3—22no—
393C22×C3—28no—
394C22×C3—28no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—21no—
398C22×C3—22no—
399C22×C3—24no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—48no—
403C22×C3—42no—
404C22×C3—42no—
405C22×C3C2×C4×D912yes—
406C22×C3C2×C4×D914yescomplemented
407C22×C3D4×D912yes—
408C22×C3#369912yes—
409C22×C3C23×D911yescharacteristic
410C22×C3#380412yes—
411C22×C3C2×#62412yes—
412C22×C3C2×#62414yes—
413C22×C3D3×#6712yes—
414C22×C3C22×D3211yescharacteristic
415C2×D3—94no—
416C2×D3—94no—
417C2×D3—184no—
418C2×D3—182no—
419C2×D3—182no—
420C2×D3—188no—
421C2×D3—188no—
422C2×D3—182no—
423C2×D3—182no—
424C2×D3—368no—
425C2×D3—362no—
426C2×D3—362no—
427C3×C4—62no—
428C3×C4—62no—
429C3×C4—94no—
430C3×C4—182no—
431Dic3—22no—
432Dic3—22no—
433Dic3—34no—
434Dic3—62no—
435Dic3—94no—
436Dic3—94no—
437Dic3—184no—
438C32C2×#67⋊C411yescharacteristic
439C9—21no—
440C9C2×C4×#6711yescharacteristic, complemented
441C23—22no—
442C23—22no—
443C23—272no—
444C23—274no—
445C23—542no—
446C23—542no—
447C23(C3⋊D9)⋊C414yescomplemented
448C23C2×D3×D912yes—
449C23C2×D3×D911yescharacteristic
450D4—2716no—
451C2×C4—32no—
452C2×C4—32no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4—92no—
456C2×C4—94no—
457C2×C4—92no—
458C2×C4—188no—
459C2×C4—182no—
460C2×C4—272no—
461C2×C4—274no—
462C2×C4—548no—
463C2×C3—24no—
464C2×C3—24no—
465C2×C3—24no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—44no—
470C2×C3#611⋊C414yescomplemented
471C2×C3C22×C4×D911yescharacteristic
472C2×C3C2×D4×D911yescharacteristic
473C2×C3C2×#369911yescharacteristic
474C2×C3#6395214yes—
475C2×C3C2×#380411yescharacteristic
476C2×C3C22×#62411yescharacteristic
477C2×C3C2×D3×#6711yescharacteristic
478D3—184no—
479D3—184no—
480D3—364no—
481C22—28no—
482C22—22no—
483C22—22no—
484C22—274no—
485C22—548no—
486C22—542no—
487C22—542no—
488C22C2×(C3⋊D9)⋊C412yes—
489C22C2×(C3⋊D9)⋊C414yescomplemented
490C22#8866212yes—
491C22D9×#6712yes—
492C22C22×D3×D911yescharacteristic
493C4—62no—
494C4—62no—
495C4—94no—
496C4—182no—
497C4—274no—
498C3—21no—
499C3C2×#611⋊C411yescharacteristic, complemented
500C3C2×#6395211yescharacteristic
501C2—24no—
502C2—544no—
503C2#1172124114yesdirect product factor
504C2C22×(C3⋊D9)⋊C411yescharacteristic
505C2C2×#8866211yescharacteristic
506C2C2×D9×#6711yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC24×C3×C9C22
socleC23×C32C22×D3
prime radicalC22×#6233C2
cosocleC22×C3×C9C24
hypercenterC24D3×D9
hypocenterC3×C9C2×C4×D4
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C23×C3×Dic9C2semidirect
3. C22×(C3⋊D9)⋊C4C2semidirect
4. C2×C9×#217C2semidirect
5. C2×#88445C2semidirect
6. C2×#88442C2semidirect
7. C2×Dic3×Dic9C2semidirect
8. #11721241C2direct (N × H)
13. C2×#6233C4semidirect
21. C22×C3×Dic9C22semidirect
23. C2×(C3⋊D9)⋊C4C22semidirect
24. C9×#217C22semidirect
25. #88445C22semidirect
26. #88442C22semidirect
27. Dic3×Dic9C22semidirect
38. C23×Dic9D3semidirect
47. C23×C3×C9C2×C4semidirect
52. C22×C3⋊D9C2×C4semidirect
54. #6233C2×C4semidirect
55. C2×C3⋊Dic9C2×C4semidirect
63. C2×C3×Dic9D4semidirect
101. C22×Dic9C2×D3semidirect
112. C22×C3×C9C22×C4semidirect
119. C2×C3⋊D9C22×C4semidirect
120. C3⋊Dic9C22×C4semidirect
124. C3×Dic9C2×D4semidirect
133. C2×#217D9semidirect
171. C23×C9C4×D3semidirect
194. C2×Dic9#67semidirect
200. C2×C3×C9C4×D4semidirect
237. #217C2×D9semidirect
273. C22×C9C2×C4×D3semidirect
290. Dic9C2×#67semidirect
299. C3×C9C2×C4×D4semidirect
309. C23×C3C4×D9semidirect
368. C2×C9C4×#67semidirect
406. C22×C3C2×C4×D9semidirect
440. C9C2×C4×#67semidirect
447. C23(C3⋊D9)⋊C4semidirect
470. C2×C3#611⋊C4semidirect
489. C22C2×(C3⋊D9)⋊C4semidirect
499. C3C2×#611⋊C4semidirect
503. C2#11721241direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62332
C23×C3×Dic92
C22×(C3⋊D9)⋊C42
C2×C9×#2172
C2×#884452
C2×#884422
C2×Dic3×Dic92
#117212412
C2×#639523
C2×#611⋊C43