Finite Group Databasebeta

← C2×#11721177 #13151115

Subgroups — C2×#11721177 #13151115

Order 1728.

866 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3699C211yesmaximal, characteristic, complemented
2C2×#88638C211yesmaximal, characteristic, complemented
3C2×#88543C211yesmaximal, characteristic, complemented
4C2×C9×#213C211yesmaximal, characteristic, complemented
5#11721177C218yesmaximal, direct product factor
6C2×#88423C211yesmaximal, characteristic, complemented
7C2×#88418C211yesmaximal, characteristic, complemented
8C2×D9×C3⋊C8C211yesmaximal, characteristic, complemented
9C2×#63933—31nomaximal
10C2×#63474—31nomaximal
11C2×C3×D4×C9C2211yescharacteristic
12C2×#6226C2211yescharacteristic
13C2×C3×#611—21no—
14C22×C3×Dic9—21no—
15C3×#3699—22no—
16C3×#3699C2214yescomplemented
17C2×C9×D12C2211yescharacteristic
18C2×C3×C4×D9C2211yescharacteristic
19C2×C3×Dic18C2211yescharacteristic
20C2×#6194—21no—
21C2×D3×Dic9—21no—
22#88638—22no—
23#88638C2214yescomplemented
24#88543C2214yescomplemented
25C2×C3⋊(C9⋊C8)C2211yescharacteristic
26C9×#213C2214yescomplemented
27C2×C9×C3⋊C8C2211yescharacteristic
28#88423C2214yescomplemented
29#88418C2214yescomplemented
30D9×C3⋊C8C2214yescomplemented
31C2×C3×#237—31no—
32C2×#3795—31no—
33C2×#3754—31no—
34C2×C3×#213D311yescharacteristic
35#63933—38no—
36C2×#3719—31no—
37C2×#3715—31no—
38C2×D3×C3⋊C8—31no—
39C2×#3699—31no—
40C2×#3699D311yescharacteristic, complemented
41C2×C9×D8—31no—
42C2×#3672—31no—
43C2×#3672—31no—
44C2×#3672—61no—
45#63474—38no—
46C2×C8×D9—31no—
47C2×Dic36—31no—
48C23×C3×C9—21no—
49C22×D3×C9—21no—
50C22×C3×D9D411yescharacteristic
51C3×D4×C9—21no—
52C3×D4×C9C2312yes—
53C2×C3×C4×C9C2311yescharacteristic
54C2×C3⋊Dic9—21no—
55#6226—21no—
56#6226C2312yes—
57C3×#611—24no—
58C2×C3×Dic9—21no—
59C2×C3×Dic9—24no—
60C2×C3×Dic9D411yescharacteristic
61C9×D12—21no—
62C9×D12C2312yes—
63C3×C4×D9D412yes—
64C3×C4×D9C2312yes—
65C3×Dic18—21no—
66C3×Dic18C2312yes—
67#6194—24no—
68D3×Dic9—24no—
69C3⋊(C9⋊C8)C2312yes—
70C9×C3⋊C8C2312yes—
71C2×#950—91no—
72C2×#912—91no—
73C2×C32×D4C2×D311yescharacteristic
74C2×#634—31no—
75C2×C3×#67—61no—
76C22×C3×Dic3—61no—
77C3×#237—34no—
78C3×#237—62no—
79C2×C3×D12C2×D311yescharacteristic
80C2×C3×C4×D3—31no—
81C2×C3×Dic6—31no—
82C2×#625—61no—
83C2×D3×Dic3—61no—
84#3795—34no—
85#3795—62no—
86#3754—34no—
87C2×C3⋊(C3⋊C8)—31no—
88C3×#213C2×D314yes—
89C2×C3×C3⋊C8C2×D311yescharacteristic
90#3719—34no—
91#3715—34no—
92D3×C3⋊C8—34no—
93C2×D4×C9—21no—
94C2×D4×C9—31no—
95C2×D4×C9C2×D311yescharacteristic
96C2×#611—21no—
97C2×#611—61no—
98C22×Dic9—21no—
99C22×Dic9—61no—
100#3699—22no—
101#3699—34no—
102#3699—62no—
103#3699C2×D314yescomplemented
104C2×C4×D9C2×D311yescharacteristic
105C2×Dic18—31no—
106C2×Dic18—61no—
107C2×Dic18C2×D311yescharacteristic
108C9×D8—34no—
109C2×C8×C9—31no—
110#3672—34no—
111#3672—34no—
112#3672—64no—
113C2×C9⋊C8—31no—
114C2×C9⋊C8—61no—
115C8×D9—34no—
116Dic36—34no—
117C22×C3×C9—22no—
118C22×C3×C9—42no—
119C22×C3×C9C2×D411yescharacteristic
120C2×D3×C9—22no—
121C2×D3×C9—42no—
122C2×C3×D9—22no—
123C2×C3×D9C2×D412yes—
124C3×C4×C9C2×D411yescharacteristic
125C3×C4×C9C2411yescharacteristic
126C3⋊Dic9—22no—
127C3×Dic9—22no—
128C3×Dic9C2×D412yes—
129C2×C3×#41—91no—
130C2×#237—31no—
131C2×#237—91no—
132C2×#235—91no—
133C2×C3×D8—31no—
134#950—98no—
135C2×#216—91no—
136C2×#215—91no—
137C2×#214—91no—
138C2×#214—91no—
139C2×#214—91no—
140C2×#214—181no—
141C2×#213D911yescharacteristic, complemented
142C22×C3⋊C8—91no—
143#912—98no—
144C2×Dic12—91no—
145C2×D3×C8—91no—
146C23×C32—21no—
147C22×C3×D3—21no—
148C22×C3×D3—31no—
149C32×D4—21no—
150C32×D4C22×D312yes—
151C2×C32×C4C22×D311yescharacteristic
152C2×C3⋊Dic3—61no—
153#634—32no—
154#634—61no—
155C3×#67—64no—
156C2×C3×Dic3—31no—
157C2×C3×Dic3—64no—
158C2×C3×Dic3—61no—
159C3×D12—21no—
160C3×D12C22×D312yes—
161C3×C4×D3—32no—
162C3×C4×D3—32no—
163C3×Dic6—32no—
164C3×Dic6—61no—
165#625—64no—
166D3×Dic3—64no—
167C23×C9—21no—
168C23×C9—41no—
169C23×C9—61no—
170C22×D9#6711yescharacteristic
171C3⋊(C3⋊C8)—32no—
172C3×C3⋊C8C22×D312yes—
173D4×C9—21no—
174D4×C9—22no—
175D4×C9—32no—
176D4×C9—41no—
177D4×C9—61no—
178D4×C9C22×D312yes—
179C2×C4×C9—21no—
180C2×C4×C9C22×D311yescharacteristic
181#611—24no—
182#611—64no—
183C2×Dic9—21no—
184C2×Dic9—24no—
185C2×Dic9—61no—
186C2×Dic9—64no—
187C2×Dic9—121no—
188C2×Dic9#6711yescharacteristic
189C4×D9#6712yes—
190C4×D9C22×D312yes—
191Dic18—21no—
192Dic18—32no—
193Dic18—62no—
194Dic18—61no—
195Dic18—121no—
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
204D3×C9—42no—
205C3×D9—22no—
206C3×#41—94no—
207C3×#41—182no—
208C2×C3×Q8—91no—
209C2×C3×D4—21no—
210C2×C3×D4—31no—
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—
218C2×#67—181no—
219C22×Dic3—61no—
220C22×Dic3—181no—
221#237—34no—
222#237—62no—
223#237—94no—
224#237—182no—
225#235—94no—
226#235—182no—
227C2×D12C2×D911yescharacteristic
228C2×C4×D3—31no—
229C2×C4×D3—181no—
230C2×Dic6—31no—
231C2×Dic6—91no—
232C2×Dic6—91no—
233C2×Dic6—181no—
234C3×D8—34no—
235C2×C3×C8—31no—
236#216—94no—
237#215—94no—
238#214—94no—
239#214—94no—
240#214—94no—
241#214—184no—
242#213C2×D914yescomplemented
243C2×C3⋊C8—91no—
244C2×C3⋊C8—91no—
245C2×C3⋊C8—94no—
246C2×C3⋊C8—181no—
247C2×C3⋊C8C2×D911yescharacteristic
248Dic12—94no—
249D3×C8—94no—
250C22×C32—22no—
251C22×C32—42no—
252C22×C32D3×D411yescharacteristic
253C2×C3×D3—22no—
254C2×C3×D3—32no—
255C2×C3×D3—42no—
256C2×C3×D3—62no—
257C32×C4D3×D411yescharacteristic
258C32×C4C23×D311yescharacteristic
259C3⋊Dic3—62no—
260C3×Dic3—32no—
261C3×Dic3—62no—
262C22×C9—22no—
263C22×C9—21no—
264C22×C9—42no—
265C22×C9—42no—
266C22×C9—62no—
267C22×C9—82no—
268C22×C9—122no—
269C22×C9C2×#6711yescharacteristic
270C2×D9—22no—
271C2×D9C2×#6712yes—
272C4×C9—21no—
273C4×C9—21no—
274C4×C9C2×#6711yescharacteristic
275C4×C9C23×D311yescharacteristic
276Dic9—22no—
277Dic9—62no—
278Dic9—122no—
279Dic9C2×#6712yes—
280C2×#41—91no—
281C2×#41—271no—
282#135—278no—
283C2×Q16—271no—
284C2×SD16—271no—
285C2×SD16—271no—
286C2×D8—31no—
287C22×C8—271no—
288C3×C9C2×#13511yescharacteristic, complemented, Sylow
289C23×C3—21no—
290C23×C3—21no—
291C23×C3—41no—
292C23×C3—61no—
293C23×C3—91no—
294C22×D3—21no—
295C22×D3—31no—
296C3×Q8—92no—
297C3×Q8—181no—
298C3×D4—21no—
299C3×D4—21no—
300C3×D4—22no—
301C3×D4—32no—
302C3×D4—41no—
303C3×D4—61no—
304C3×D4—184no—
305C3×D4C22×D912yes—
306C3×D4C2×D3212yes—
307C2×C3×C4—21no—
308C2×C3×C4—91no—
309C2×C3×C4—92no—
310C2×C3×C4—92no—
311C2×C3×C4—184no—
312C2×C3×C4—181no—
313C2×C3×C4C22×D911yescharacteristic
314C2×C3×C4C2×D3211yescharacteristic
315#67—64no—
316#67—184no—
317#67—184no—
318C2×Dic3—31no—
319C2×Dic3—64no—
320C2×Dic3—61no—
321C2×Dic3—184no—
322C2×Dic3—181no—
323C2×Dic3—181no—
324C2×Dic3—361no—
325D12—21no—
326D12C22×D912yes—
327C4×D3—32no—
328C4×D3—32no—
329C4×D3—184no—
330Dic6—32no—
331Dic6—61no—
332Dic6—92no—
333Dic6—92no—
334Dic6—182no—
335Dic6—181no—
336Dic6—181no—
337Dic6—361no—
338C3×C8—32no—
339C3⋊C8—92no—
340C3⋊C8—92no—
341C3⋊C8—182no—
342C3⋊C8C22×D912yes—
343C2×C32—42no—
344C2×C32#91212yes—
345C2×C32C2×D3×D411yescharacteristic
346C3×D3—42no—
347C3×D3—62no—
348C2×C9—21no—
349C2×C9—22no—
350C2×C9—42no—
351C2×C9—82no—
352C2×C9—122no—
353C2×C9#95012yescomplemented
354C2×C9C22×#6711yescharacteristic
355D9—22no—
356#41—94no—
357#41—182no—
358#41—274no—
359#41—542no—
360C2×Q8—91no—
361C2×Q8—271no—
362C2×D4—31no—
363C2×D4—181no—
364C2×D4—541no—
365C2×D4D3×D911yescharacteristic
366C22×C4—91no—
367C22×C4—181no—
368C22×C4—541no—
369Q16—274no—
370SD16—274no—
371SD16—274no—
372D8—34no—
373C2×C8—31no—
374C2×C8—274no—
375C2×C8—271no—
376C22×C3—22no—
377C22×C3—22no—
378C22×C3—21no—
379C22×C3—42no—
380C22×C3—42no—
381C22×C3—42no—
382C22×C3—62no—
383C22×C3—82no—
384C22×C3—92no—
385C22×C3—122no—
386C22×C3—182no—
387C22×C3D4×D911yescharacteristic
388C22×C3D3×#6711yescharacteristic
389C2×D3—22no—
390C2×D3—32no—
391C2×D3—42no—
392C2×D3—62no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—92no—
396C3×C4—182no—
397C3×C4D4×D911yescharacteristic
398C3×C4C23×D911yescharacteristic
399C3×C4D3×#6711yescharacteristic
400C3×C4C22×D3211yescharacteristic
401Dic3—32no—
402Dic3—62no—
403Dic3—182no—
404Dic3—182no—
405Dic3—362no—
406C32C2×#91211yescharacteristic
407C9—21no—
408C9C2×#95011yescharacteristic, complemented
409C23—21no—
410C23—61no—
411C23—91no—
412Q8—92no—
413Q8—181no—
414Q8—272no—
415Q8—541no—
416D4—21no—
417D4—32no—
418D4—61no—
419D4—184no—
420D4—544no—
421D4C2×D3×D912yes—
422C2×C4—91no—
423C2×C4—92no—
424C2×C4—92no—
425C2×C4—184no—
426C2×C4—181no—
427C2×C4—544no—
428C2×C4—541no—
429C2×C4C2×D3×D911yescharacteristic
430C8—32no—
431C8—272no—
432C2×C3—21no—
433C2×C3—22no—
434C2×C3—42no—
435C2×C3—42no—
436C2×C3—82no—
437C2×C3—122no—
438C2×C3—182no—
439C2×C3#6347412yescomplemented
440C2×C3C2×D4×D911yescharacteristic
441C2×C3#6393312yes—
442C2×C3C2×D3×#6711yescharacteristic
443D3—42no—
444D3—62no—
445C22—22no—
446C22—42no—
447C22—62no—
448C22—92no—
449C22—122no—
450C22—182no—
451C22D9×#6711yescharacteristic
452C4—92no—
453C4—182no—
454C4—542no—
455C4D9×#6711yescharacteristic
456C4C22×D3×D911yescharacteristic
457C3—21no—
458C3C2×#6347411yescharacteristic, complemented
459C3C2×#6393311yescharacteristic
460C2—42no—
461C2—122no—
462C2—182no—
463C2#1172117712yesdirect product factor
464C2C2×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×#88638C2semidirect
3. C2×#88543C2semidirect
4. C2×C9×#213C2semidirect
5. #11721177C2direct (N × H)
6. C2×#88423C2semidirect
7. C2×#88418C2semidirect
8. C2×D9×C3⋊C8C2semidirect
16. C3×#3699C22semidirect
23. #88638C22semidirect
24. #88543C22semidirect
26. C9×#213C22semidirect
28. #88423C22semidirect
29. #88418C22semidirect
30. D9×C3⋊C8C22semidirect
40. C2×#3699D3semidirect
103. #3699C2×D3semidirect
141. C2×#213D9semidirect
202. C2×C3×C9#135semidirect
242. #213C2×D9semidirect
288. C3×C9C2×#135semidirect
353. C2×C9#950semidirect
408. C9C2×#950semidirect
439. C2×C3#63474semidirect
458. C3C2×#63474semidirect
463. C2#11721177direct (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×#886382
C2×#885432
C2×C9×#2132
#117211772
C2×#884232
C2×#884182
C2×D9×C3⋊C82
C2×#639333
C2×#634743