Finite Group Databasebeta

← C2×#11721185 #13151126

Subgroups — C2×#11721185 #13151126

Order 1728.

874 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3699C211yesmaximal, characteristic, complemented
2C2×#88635C211yesmaximal, characteristic, complemented
3C2×#88544C211yesmaximal, characteristic, complemented
4C2×C9×#214C211yesmaximal, characteristic, complemented
5#11721185C218yesmaximal, direct product factor
6C2×#88427C211yesmaximal, characteristic, complemented
7C2×#88416C211yesmaximal, characteristic, complemented
8C2×D9×C3⋊C8C211yesmaximal, characteristic, complemented
9C2×#63935—31nomaximal
10C2×#63478—31nomaximal
11C2×C3×D4×C9C2211yescharacteristic
12C2×C4⋊(C3⋊D9)C2211yescharacteristic
13C2×C3×#611—21no—
14C22×C3×Dic9—21no—
15C3×#3699—22no—
16C3×#3699C2214yescomplemented
17C2×C3×C4×D9C2211yescharacteristic
18C2×C9×Dic6C2211yescharacteristic
19C2×C3×Dic18C2211yescharacteristic
20C2×#6192—21no—
21C2×(C3⋊D9)⋊C4—21no—
22#88635—22no—
23#88635C2214yescomplemented
24#88544C2214yescomplemented
25C2×C3⋊(C9⋊C8)C2211yescharacteristic
26C9×#214C2214yescomplemented
27C2×C9×C3⋊C8C2211yescharacteristic
28#88427C2214yescomplemented
29#88416C2214yescomplemented
30D9×C3⋊C8C2214yescomplemented
31C2×C3×#237—31no—
32C2×#3799—31no—
33C2×#3753—31no—
34C2×C3×#214D311yescharacteristic
35#63935—38no—
36C2×#3718—31no—
37C2×#3717—31no—
38C2×D3×C3⋊C8—31no—
39C2×#3701—31no—
40C2×#3699D311yescharacteristic, complemented
41C2×C9×SD16—31no—
42C2×#3674—31no—
43C2×#3673—31no—
44C2×#3673—61no—
45#63478—38no—
46C2×#3664—31no—
47C2×C8×D9—31no—
48C23×C3×C9—21no—
49C22×C3⋊D9—21no—
50C22×C3×D9D411yescharacteristic
51C3×D4×C9—21no—
52C3×D4×C9C2312yes—
53C2×C3×C4×C9C2311yescharacteristic
54C4⋊(C3⋊D9)—21no—
55C4⋊(C3⋊D9)C2312yes—
56C3×#611—24no—
57C2×C9×Dic3—21no—
58C2×C3×Dic9—21no—
59C2×C3×Dic9—24no—
60C2×C3×Dic9D411yescharacteristic
61C3×C4×D9D412yes—
62C3×C4×D9C2312yes—
63C9×Dic6—21no—
64C9×Dic6C2312yes—
65C3×Dic18—21no—
66C3×Dic18C2312yes—
67#6192—24no—
68(C3⋊D9)⋊C4—24no—
69C3⋊(C9⋊C8)C2312yes—
70C9×C3⋊C8C2312yes—
71C2×#950—91no—
72C2×#916—91no—
73C2×C32×D4C2×D311yescharacteristic
74C2×C4⋊(C3⋊D3)—31no—
75C2×C3×#67—61no—
76C22×C3×Dic3—61no—
77C3×#237—34no—
78C3×#237—62no—
79C2×C3×C4×D3—31no—
80C2×C3×Dic6—31no—
81C2×C3×Dic6C2×D311yescharacteristic
82C2×#626—61no—
83C2×#624—61no—
84#3799—34no—
85#3799—62no—
86#3753—34no—
87C2×C3⋊(C3⋊C8)—31no—
88C3×#214C2×D314yes—
89C2×C3×C3⋊C8C2×D311yescharacteristic
90#3718—34no—
91#3717—34no—
92D3×C3⋊C8—34no—
93C2×Q8×C9—31no—
94C2×D4×C9—21no—
95C2×D4×C9C2×D311yescharacteristic
96C2×#611—21no—
97C22×Dic9—21no—
98#3701—34no—
99#3701—62no—
100#3699—22no—
101#3699C2×D314yescomplemented
102C2×D36—31no—
103C2×D36—61no—
104C2×D36—61no—
105C2×C4×D9—61no—
106C2×C4×D9C2×D311yescharacteristic
107C2×Dic18C2×D311yescharacteristic
108C9×SD16—34no—
109C2×C8×C9—31no—
110#3674—34no—
111#3673—34no—
112#3673—64no—
113C2×C9⋊C8—31no—
114C2×C9⋊C8—61no—
115#3664—34no—
116C8×D9—34no—
117C22×C3×C9—22no—
118C22×C3×C9—42no—
119C22×C3×C9C2×D411yescharacteristic
120C2×C3⋊D9—22no—
121C2×C3⋊D9—42no—
122C2×C3×D9—22no—
123C2×C3×D9C2×D412yes—
124C3×C4×C9C2×D411yescharacteristic
125C3×C4×C9C2411yescharacteristic
126C9×Dic3—22no—
127C3×Dic9—22no—
128C3×Dic9C2×D412yes—
129C2×C3×#41—91no—
130C2×#239—91no—
131C2×#237—31no—
132C2×#235—91no—
133C2×C3×SD16—31no—
134#950—98no—
135C2×#216—91no—
136C2×#216—91no—
137C2×#215—91no—
138C2×#214D911yescharacteristic, complemented
139C2×#213—91no—
140C2×#213—91no—
141C2×#213—181no—
142C22×C3⋊C8—91no—
143#916—98no—
144C2×#204—91no—
145C2×D3×C8—91no—
146C23×C32—21no—
147C22×C3⋊D3—61no—
148C22×C3×D3—31no—
149C32×D4—21no—
150C32×D4C22×D312yes—
151C2×C32×C4C22×D311yescharacteristic
152C4⋊(C3⋊D3)—32no—
153C4⋊(C3⋊D3)—61no—
154C3×#67—64no—
155C2×C3×Dic3—21no—
156C2×C3×Dic3—31no—
157C2×C3×Dic3—64no—
158C2×C3×Dic3—61no—
159C3×C4×D3—32no—
160C3×C4×D3—32no—
161C3×Dic6—21no—
162C3×Dic6—32no—
163C3×Dic6—61no—
164C3×Dic6C22×D312yes—
165#626—64no—
166#624—64no—
167C23×C9—21no—
168C23×C9—41no—
169C22×D9—61no—
170C22×D9—121no—
171C22×D9#6711yescharacteristic
172C3⋊(C3⋊C8)—32no—
173C3×C3⋊C8C22×D312yes—
174Q8×C9—32no—
175Q8×C9—61no—
176D4×C9—21no—
177D4×C9—22no—
178D4×C9—41no—
179D4×C9C22×D312yes—
180C2×C4×C9—21no—
181C2×C4×C9—61no—
182C2×C4×C9C22×D311yescharacteristic
183#611—24no—
184C2×Dic9—21no—
185C2×Dic9—24no—
186C2×Dic9#6711yescharacteristic
187D36—32no—
188D36—62no—
189D36—64no—
190D36—61no—
191D36—121no—
192C4×D9—64no—
193C4×D9#6712yes—
194C4×D9C22×D312yes—
195Dic18—21no—
196Dic18C22×D312yes—
197C8×C9—32no—
198C9⋊C8—32no—
199C9⋊C8—62no—
200C2×#135—271noSylow
201C2×C3×C9—42no—
202C2×C3×C9#13512yescomplemented
203C2×C3×C9C22×D411yescharacteristic
204C3⋊D9—42no—
205C3×D9—22no—
206C3×#41—94no—
207C3×#41—182no—
208C2×C3×Q8—31no—
209C2×C3×Q8—91no—
210C2×C3×D4—21no—
211C2×C3×D4—181no—
212C2×C3×D4C2×D911yescharacteristic
213C2×C3×D4D3211yescharacteristic
214C22×C3×C4—91no—
215C22×C3×C4—181no—
216C2×#67—61no—
217C2×#67—181no—
218C22×Dic3—61no—
219#239—94no—
220#239—182no—
221#237—34no—
222#237—62no—
223#235—94no—
224#235—182no—
225C2×D12—91no—
226C2×D12—91no—
227C2×D12—181no—
228C2×D12—181no—
229C2×C4×D3—31no—
230C2×C4×D3—181no—
231C2×C4×D3—181no—
232C2×Dic6—31no—
233C2×Dic6C2×D911yescharacteristic
234C3×SD16—34no—
235C2×C3×C8—31no—
236#216—94no—
237#216—94no—
238#215—94no—
239#214C2×D914yescomplemented
240#213—94no—
241#213—94no—
242#213—184no—
243C2×C3⋊C8—91no—
244C2×C3⋊C8—91no—
245C2×C3⋊C8—94no—
246C2×C3⋊C8—181no—
247C2×C3⋊C8C2×D911yescharacteristic
248#204—94no—
249D3×C8—94no—
250C22×C32—22no—
251C22×C32—42no—
252C22×C32D3×D411yescharacteristic
253C2×C3⋊D3—62no—
254C2×C3⋊D3—122no—
255C2×C3×D3—32no—
256C2×C3×D3—62no—
257C32×C4D3×D411yescharacteristic
258C32×C4C23×D311yescharacteristic
259C3×Dic3—22no—
260C3×Dic3—32no—
261C3×Dic3—62no—
262C22×C9—22no—
263C22×C9—21no—
264C22×C9—42no—
265C22×C9—42no—
266C22×C9—82no—
267C22×C9C2×#6711yescharacteristic
268C2×D9—22no—
269C2×D9—62no—
270C2×D9—122no—
271C2×D9—122no—
272C2×D9—242no—
273C2×D9C2×#6712yes—
274C4×C9—21no—
275C4×C9—21no—
276C4×C9—62no—
277C4×C9C2×#6711yescharacteristic
278C4×C9C23×D311yescharacteristic
279Dic9—22no—
280Dic9C2×#6712yes—
281C2×#41—91no—
282C2×#41—271no—
283#135—278no—
284C2×Q16—271no—
285C2×SD16—31no—
286C2×SD16—271no—
287C2×D8—271no—
288C22×C8—271no—
289C3×C9C2×#13511yescharacteristic, complemented, Sylow
290C23×C3—21no—
291C23×C3—21no—
292C23×C3—41no—
293C23×C3—91no—
294C22×D3—31no—
295C22×D3—181no—
296C22×D3—181no—
297C22×D3—361no—
298C3×Q8—32no—
299C3×Q8—61no—
300C3×Q8—92no—
301C3×Q8—181no—
302C3×D4—21no—
303C3×D4—21no—
304C3×D4—22no—
305C3×D4—41no—
306C3×D4—184no—
307C3×D4C22×D912yes—
308C3×D4C2×D3212yes—
309C2×C3×C4—21no—
310C2×C3×C4—61no—
311C2×C3×C4—91no—
312C2×C3×C4—92no—
313C2×C3×C4—92no—
314C2×C3×C4—184no—
315C2×C3×C4—181no—
316C2×C3×C4C22×D911yescharacteristic
317C2×C3×C4C2×D3211yescharacteristic
318#67—64no—
319#67—184no—
320C2×Dic3—21no—
321C2×Dic3—31no—
322C2×Dic3—64no—
323C2×Dic3—61no—
324D12—92no—
325D12—92no—
326D12—182no—
327D12—184no—
328D12—181no—
329D12—181no—
330D12—361no—
331C4×D3—32no—
332C4×D3—32no—
333C4×D3—184no—
334C4×D3—184no—
335Dic6—21no—
336Dic6—32no—
337Dic6—61no—
338Dic6C22×D912yes—
339C3×C8—32no—
340C3⋊C8—92no—
341C3⋊C8—92no—
342C3⋊C8—182no—
343C3⋊C8C22×D912yes—
344C2×C32—42no—
345C2×C32#91612yes—
346C2×C32C2×D3×D411yescharacteristic
347C3⋊D3—122no—
348C3×D3—62no—
349C2×C9—21no—
350C2×C9—22no—
351C2×C9—42no—
352C2×C9—82no—
353C2×C9#95012yescomplemented
354C2×C9C22×#6711yescharacteristic
355D9—22no—
356D9—122no—
357D9—242no—
358#41—94no—
359#41—182no—
360#41—274no—
361#41—542no—
362C2×Q8—31no—
363C2×Q8—91no—
364C2×D4—181no—
365C2×D4—271no—
366C2×D4—541no—
367C2×D4D3×D911yescharacteristic
368C22×C4—91no—
369C22×C4—181no—
370C22×C4—541no—
371Q16—274no—
372SD16—34no—
373SD16—274no—
374D8—274no—
375C2×C8—31no—
376C2×C8—274no—
377C2×C8—271no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—21no—
381C22×C3—42no—
382C22×C3—42no—
383C22×C3—42no—
384C22×C3—82no—
385C22×C3—92no—
386C22×C3—182no—
387C22×C3D4×D911yescharacteristic
388C22×C3D3×#6711yescharacteristic
389C2×D3—32no—
390C2×D3—62no—
391C2×D3—182no—
392C2×D3—182no—
393C2×D3—362no—
394C2×D3—362no—
395C2×D3—362no—
396C2×D3—722no—
397C3×C4—21no—
398C3×C4—21no—
399C3×C4—62no—
400C3×C4—92no—
401C3×C4—182no—
402C3×C4D4×D911yescharacteristic
403C3×C4C23×D911yescharacteristic
404C3×C4D3×#6711yescharacteristic
405C3×C4C22×D3211yescharacteristic
406Dic3—22no—
407Dic3—32no—
408Dic3—62no—
409C32C2×#91611yescharacteristic
410C9—21no—
411C9C2×#95011yescharacteristic, complemented
412C23—21no—
413C23—91no—
414C23—541no—
415Q8—32no—
416Q8—61no—
417Q8—92no—
418Q8—181no—
419D4—21no—
420D4—184no—
421D4—272no—
422D4—544no—
423D4—541no—
424D4C2×D3×D912yes—
425C2×C4—61no—
426C2×C4—91no—
427C2×C4—92no—
428C2×C4—92no—
429C2×C4—184no—
430C2×C4—181no—
431C2×C4—544no—
432C2×C4C2×D3×D911yescharacteristic
433C8—32no—
434C8—272no—
435C2×C3—21no—
436C2×C3—22no—
437C2×C3—42no—
438C2×C3—42no—
439C2×C3—82no—
440C2×C3—182no—
441C2×C3#6347812yescomplemented
442C2×C3C2×D4×D911yescharacteristic
443C2×C3#6393512yes—
444C2×C3C2×D3×#6711yescharacteristic
445D3—62no—
446D3—362no—
447D3—362no—
448D3—722no—
449C22—22no—
450C22—42no—
451C22—92no—
452C22—182no—
453C22—542no—
454C22—1082no—
455C22D9×#6711yescharacteristic
456C4—62no—
457C4—92no—
458C4—182no—
459C4D9×#6711yescharacteristic
460C4C22×D3×D911yescharacteristic
461C3—21no—
462C3C2×#6347811yescharacteristic, complemented
463C3C2×#6393511yescharacteristic
464C2—42no—
465C2—182no—
466C2—1082no—
467C2#1172118512yesdirect product factor
468C2C2×D9×#6711yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC3×C4×C9C24
FrattiniC3×C4C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
hypercenterC2×D4D3×D9
hypocenterC3×C9C2×#135
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3699C2semidirect
2. C2×#88635C2semidirect
3. C2×#88544C2semidirect
4. C2×C9×#214C2semidirect
5. #11721185C2direct (N × H)
6. C2×#88427C2semidirect
7. C2×#88416C2semidirect
8. C2×D9×C3⋊C8C2semidirect
16. C3×#3699C22semidirect
23. #88635C22semidirect
24. #88544C22semidirect
26. C9×#214C22semidirect
28. #88427C22semidirect
29. #88416C22semidirect
30. D9×C3⋊C8C22semidirect
40. C2×#3699D3semidirect
101. #3699C2×D3semidirect
138. C2×#214D9semidirect
202. C2×C3×C9#135semidirect
239. #214C2×D9semidirect
289. C3×C9C2×#135semidirect
353. C2×C9#950semidirect
411. C9C2×#950semidirect
441. C2×C3#63478semidirect
462. C3C2×#63478semidirect
467. C2#11721185direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#36992
C2×#886352
C2×#885442
C2×C9×#2142
#117211852
C2×#884272
C2×#884162
C2×D9×C3⋊C82
C2×#639353
C2×#634783