Finite Group Databasebeta

← C2×#11721171 #13151101

Subgroups — C2×#11721171 #13151101

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×C9×#237C211yesmaximal, characteristic, complemented
2C2×#88642C211yesmaximal, characteristic, complemented
3C2×#88544C211yesmaximal, characteristic, complemented
4C2×C3×#3672C211yesmaximal, characteristic, complemented
5#11721171C218yesmaximal, direct product factor
6C2×#88427C211yesmaximal, characteristic, complemented
7C2×#88426C211yesmaximal, characteristic, complemented
8C2×D3×C9⋊C8C211yesmaximal, characteristic, complemented
9C2×#63935—31nomaximal
10C2×#63513—31nomaximal
11C2×C3×D4×C9C2211yescharacteristic
12C2×C4⋊(C3⋊D9)C2211yescharacteristic
13C2×C9×#67—21no—
14C22×C9×Dic3—21no—
15C9×#237—22no—
16C9×#237C2214yescomplemented
17C2×C4×D3×C9C2211yescharacteristic
18C2×C9×Dic6C2211yescharacteristic
19C2×C3×Dic18C2211yescharacteristic
20C2×#6195—21no—
21C2×(C3⋊D9)⋊C4—21no—
22#88642—22no—
23#88642C2214yescomplemented
24#88544C2214yescomplemented
25C2×C3⋊(C9⋊C8)C2211yescharacteristic
26C3×#3672C2214yescomplemented
27C2×C3×C9⋊C8C2211yescharacteristic
28#88427C2214yescomplemented
29#88426C2214yescomplemented
30D3×C9⋊C8C2214yescomplemented
31C2×C3×#237D311yescharacteristic
32C2×#3799—31no—
33C2×#3753—31no—
34C2×C3×#214—31no—
35#63935—38no—
36C2×#3718—31no—
37C2×#3717—31no—
38C2×D3×C3⋊C8—31no—
39C2×C9×#41—31no—
40C2×#3697—31no—
41#63513—38no—
42C2×#3675—31no—
43C2×#3674—31no—
44C2×#3673—31no—
45C2×#3673—61no—
46C2×#3672D311yescharacteristic, complemented
47C22×C9⋊C8—31no—
48C23×C3×C9—21no—
49C22×C3⋊D9—21no—
50C22×D3×C9D411yescharacteristic
51C3×D4×C9—21no—
52C3×D4×C9C2312yes—
53C2×C3×C4×C9C2311yescharacteristic
54C4⋊(C3⋊D9)—21no—
55C4⋊(C3⋊D9)C2312yes—
56C9×#67—24no—
57C2×C9×Dic3—21no—
58C2×C9×Dic3—24no—
59C2×C9×Dic3D411yescharacteristic
60C2×C3×Dic9—21no—
61C4×D3×C9D412yes—
62C4×D3×C9C2312yes—
63C9×Dic6—21no—
64C9×Dic6C2312yes—
65C3×Dic18—21no—
66C3×Dic18C2312yes—
67#6195—24no—
68(C3⋊D9)⋊C4—24no—
69C3⋊(C9⋊C8)C2312yes—
70C3×C9⋊C8C2312yes—
71C2×#950—91no—
72C2×#916—91no—
73C2×C32×D4C2×D311yescharacteristic
74C2×C4⋊(C3⋊D3)—31no—
75C2×C3×#67—21no—
76C22×C3×Dic3—21no—
77C3×#237—22no—
78C3×#237C2×D314yes—
79C2×C3×C4×D3C2×D311yescharacteristic
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×#214—34no—
89C2×C3×C3⋊C8—31no—
90#3718—34no—
91#3717—34no—
92D3×C3⋊C8—34no—
93C9×#41—34no—
94C9×#41—62no—
95C2×Q8×C9—31no—
96C2×D4×C9—21no—
97C2×D4×C9—61no—
98C2×D4×C9C2×D311yescharacteristic
99C22×C4×C9—31no—
100C22×C4×C9—61no—
101C2×#611—61no—
102#3697—34no—
103#3697—62no—
104C2×D36—31no—
105C2×D36—61no—
106C2×C4×D9—61no—
107C2×Dic18C2×D311yescharacteristic
108#3675—34no—
109#3674—34no—
110#3673—34no—
111#3673—64no—
112#3672C2×D314yescomplemented
113C2×C9⋊C8—31no—
114C2×C9⋊C8—34no—
115C2×C9⋊C8—61no—
116C2×C9⋊C8C2×D311yescharacteristic
117C22×C3×C9—22no—
118C22×C3×C9—42no—
119C22×C3×C9C2×D411yescharacteristic
120C2×C3⋊D9—22no—
121C2×C3⋊D9—42no—
122C2×D3×C9—22no—
123C2×D3×C9C2×D412yes—
124C3×C4×C9C2×D411yescharacteristic
125C3×C4×C9C2411yescharacteristic
126C9×Dic3—22no—
127C9×Dic3C2×D412yes—
128C3×Dic9—22no—
129C2×C3×#41—31no—
130C2×#239—91no—
131C2×#237D911yescharacteristic, complemented
132C2×#235—91no—
133C2×C3×SD16—91no—
134#950—98no—
135C2×#216—91no—
136C2×#216—91no—
137C2×#215—91no—
138C2×#214—31no—
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#6711yescharacteristic
149C32×D4—21no—
150C32×D4C22×D312yes—
151C2×C32×C4C22×D311yescharacteristic
152C4⋊(C3⋊D3)—32no—
153C4⋊(C3⋊D3)—61no—
154C3×#67—24no—
155C2×C3×Dic3—21no—
156C2×C3×Dic3—24no—
157C2×C3×Dic3—61no—
158C2×C3×Dic3#6711yescharacteristic
159C3×C4×D3#6712yes—
160C3×C4×D3C22×D312yes—
161C3×Dic6—21no—
162C3×Dic6—32no—
163C3×Dic6—61no—
164C3×Dic6C22×D312yes—
165#626—64no—
166#624—64no—
167C23×C9—21no—
168C23×C9—31no—
169C23×C9—41no—
170C22×D9—61no—
171C22×D9—121no—
172C3⋊(C3⋊C8)—32no—
173C3×C3⋊C8—32no—
174Q8×C9—32no—
175Q8×C9—61no—
176D4×C9—21no—
177D4×C9—22no—
178D4×C9—41no—
179D4×C9—64no—
180D4×C9C22×D312yes—
181C2×C4×C9—21no—
182C2×C4×C9—31no—
183C2×C4×C9—32no—
184C2×C4×C9—32no—
185C2×C4×C9—64no—
186C2×C4×C9—61no—
187C2×C4×C9C22×D311yescharacteristic
188#611—64no—
189C2×Dic9—21no—
190D36—32no—
191D36—62no—
192D36—61no—
193D36—121no—
194C4×D9—64no—
195Dic18—21no—
196Dic18C22×D312yes—
197C9⋊C8—32no—
198C9⋊C8—62no—
199C9⋊C8C22×D312yes—
200C2×#135—271noSylow
201C2×C3×C9—42no—
202C2×C3×C9#13512yescomplemented
203C2×C3×C9C22×D411yescharacteristic
204C3⋊D9—42no—
205D3×C9—22no—
206C3×#41—34no—
207C3×#41—62no—
208C2×C3×Q8—31no—
209C2×C3×Q8—91no—
210C2×C3×D4—21no—
211C2×C3×D4—61no—
212C2×C3×D4C2×D911yescharacteristic
213C2×C3×D4D3211yescharacteristic
214C22×C3×C4—31no—
215C22×C3×C4—61no—
216C2×#67—21no—
217C2×#67—181no—
218C22×Dic3—21no—
219#239—94no—
220#239—182no—
221#237—22no—
222#237C2×D914yescomplemented
223#235—94no—
224#235—182no—
225C2×D12—91no—
226C2×D12—91no—
227C2×D12—181no—
228C2×D12—181no—
229C2×C4×D3—181no—
230C2×C4×D3—181no—
231C2×C4×D3C2×D911yescharacteristic
232C2×Dic6—31no—
233C2×Dic6C2×D911yescharacteristic
234C3×SD16—94no—
235C2×C3×C8—91no—
236#216—94no—
237#216—94no—
238#215—94no—
239#214—34no—
240#213—94no—
241#213—94no—
242#213—184no—
243C2×C3⋊C8—31no—
244C2×C3⋊C8—94no—
245C2×C3⋊C8—91no—
246C2×C3⋊C8—91no—
247C2×C3⋊C8—181no—
248#204—94no—
249D3×C8—94no—
250C22×C32—22no—
251C22×C32—42no—
252C22×C32C2×#6711yescharacteristic
253C2×C3⋊D3—62no—
254C2×C3⋊D3—122no—
255C2×C3×D3—22no—
256C2×C3×D3C2×#6712yes—
257C32×C4C2×#6711yescharacteristic
258C32×C4C23×D311yescharacteristic
259C3×Dic3—22no—
260C3×Dic3—62no—
261C3×Dic3C2×#6712yes—
262C22×C9—22no—
263C22×C9—21no—
264C22×C9—32no—
265C22×C9—42no—
266C22×C9—42no—
267C22×C9—62no—
268C22×C9—82no—
269C22×C9D3×D411yescharacteristic
270C2×D9—62no—
271C2×D9—122no—
272C2×D9—122no—
273C2×D9—242no—
274C4×C9—21no—
275C4×C9—21no—
276C4×C9—32no—
277C4×C9—62no—
278C4×C9D3×D411yescharacteristic
279C4×C9C23×D311yescharacteristic
280Dic9—22no—
281C2×#41—31no—
282C2×#41—271no—
283#135—278no—
284C2×Q16—271no—
285C2×SD16—91no—
286C2×SD16—271no—
287C2×D8—271no—
288C22×C8—271no—
289C3×C9C2×#13511yescharacteristic, complemented, Sylow
290C23×C3—21no—
291C23×C3—21no—
292C23×C3—31no—
293C23×C3—41no—
294C22×D3—181no—
295C22×D3—181no—
296C22×D3—361no—
297C22×D3#61111yescharacteristic
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—64no—
307C3×D4C22×D912yes—
308C3×D4C2×D3212yes—
309C2×C3×C4—21no—
310C2×C3×C4—31no—
311C2×C3×C4—32no—
312C2×C3×C4—32no—
313C2×C3×C4—64no—
314C2×C3×C4—61no—
315C2×C3×C4—181no—
316C2×C3×C4C22×D911yescharacteristic
317C2×C3×C4C2×D3211yescharacteristic
318#67—24no—
319#67—184no—
320C2×Dic3—21no—
321C2×Dic3—24no—
322C2×Dic3—61no—
323C2×Dic3#61111yescharacteristic
324D12—92no—
325D12—92no—
326D12—182no—
327D12—184no—
328D12—181no—
329D12—181no—
330D12—361no—
331C4×D3—184no—
332C4×D3—184no—
333C4×D3#61112yes—
334C4×D3C22×D912yes—
335Dic6—21no—
336Dic6—32no—
337Dic6—61no—
338Dic6C22×D912yes—
339C3×C8—92no—
340C3⋊C8—32no—
341C3⋊C8—92no—
342C3⋊C8—92no—
343C3⋊C8—182no—
344C2×C32—42no—
345C2×C32#95012yes—
346C2×C32C22×#6711yescharacteristic
347C3⋊D3—122no—
348C3×D3—22no—
349C2×C9—21no—
350C2×C9—22no—
351C2×C9—42no—
352C2×C9—62no—
353C2×C9—82no—
354C2×C9#91612yescomplemented
355C2×C9C2×D3×D411yescharacteristic
356D9—122no—
357D9—242no—
358#41—34no—
359#41—62no—
360#41—274no—
361#41—542no—
362C2×Q8—31no—
363C2×Q8—91no—
364C2×D4—61no—
365C2×D4—271no—
366C2×D4—541no—
367C2×D4D3×D911yescharacteristic
368C22×C4—31no—
369C22×C4—61no—
370C22×C4—541no—
371Q16—274no—
372SD16—94no—
373SD16—274no—
374D8—274no—
375C2×C8—91no—
376C2×C8—274no—
377C2×C8—271no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—21no—
381C22×C3—32no—
382C22×C3—42no—
383C22×C3—42no—
384C22×C3—42no—
385C22×C3—62no—
386C22×C3—82no—
387C22×C3C2×#61111yescharacteristic
388C22×C3D3×#6711yescharacteristic
389C2×D3—22no—
390C2×D3—182no—
391C2×D3—182no—
392C2×D3—362no—
393C2×D3—362no—
394C2×D3—362no—
395C2×D3—722no—
396C2×D3C2×#61112yes—
397C3×C4—21no—
398C3×C4—21no—
399C3×C4—32no—
400C3×C4—62no—
401C3×C4—182no—
402C3×C4C2×#61111yescharacteristic
403C3×C4C23×D911yescharacteristic
404C3×C4D3×#6711yescharacteristic
405C3×C4C22×D3211yescharacteristic
406Dic3—22no—
407Dic3—62no—
408Dic3C2×#61112yes—
409C32C2×#95011yescharacteristic
410C9—21no—
411C9C2×#91611yescharacteristic, complemented
412C23—21no—
413C23—31no—
414C23—541no—
415Q8—32no—
416Q8—61no—
417Q8—92no—
418Q8—181no—
419D4—21no—
420D4—64no—
421D4—272no—
422D4—544no—
423D4—541no—
424D4C2×D3×D912yes—
425C2×C4—31no—
426C2×C4—32no—
427C2×C4—32no—
428C2×C4—64no—
429C2×C4—61no—
430C2×C4—181no—
431C2×C4—544no—
432C2×C4C2×D3×D911yescharacteristic
433C8—92no—
434C8—272no—
435C2×C3—21no—
436C2×C3—22no—
437C2×C3—42no—
438C2×C3—42no—
439C2×C3—62no—
440C2×C3—82no—
441C2×C3#6351312yescomplemented
442C2×C3C22×#61111yescharacteristic
443C2×C3#6393512yes—
444C2×C3C2×D3×#6711yescharacteristic
445D3—22no—
446D3—362no—
447D3—362no—
448D3—722no—
449C22—22no—
450C22—32no—
451C22—42no—
452C22—62no—
453C22—542no—
454C22—1082no—
455C22D3×#61111yescharacteristic
456C4—32no—
457C4—62no—
458C4—182no—
459C4D3×#61111yescharacteristic
460C4C22×D3×D911yescharacteristic
461C3—21no—
462C3C2×#6351311yescharacteristic, complemented
463C3C2×#6393511yescharacteristic
464C2—42no—
465C2—62no—
466C2—1082no—
467C2#1172117112yesdirect product factor
468C2C2×D3×#61111yescharacteristic
subgroupG/H
centerC22D3×#611
commutator, cosocleC3×C4×C9C24
FrattiniC3×C4C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C2×#67
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×C9×#237C2semidirect
2. C2×#88642C2semidirect
3. C2×#88544C2semidirect
4. C2×C3×#3672C2semidirect
5. #11721171C2direct (N × H)
6. C2×#88427C2semidirect
7. C2×#88426C2semidirect
8. C2×D3×C9⋊C8C2semidirect
16. C9×#237C22semidirect
23. #88642C22semidirect
24. #88544C22semidirect
26. C3×#3672C22semidirect
28. #88427C22semidirect
29. #88426C22semidirect
30. D3×C9⋊C8C22semidirect
46. C2×#3672D3semidirect
112. #3672C2×D3semidirect
131. C2×#237D9semidirect
202. C2×C3×C9#135semidirect
222. #237C2×D9semidirect
289. C3×C9C2×#135semidirect
354. C2×C9#916semidirect
411. C9C2×#916semidirect
441. C2×C3#63513semidirect
462. C3C2×#63513semidirect
467. C2#11721171direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C9×#2372
C2×#886422
C2×#885442
C2×C3×#36722
#117211712
C2×#884272
C2×#884262
C2×D3×C9⋊C82
C2×#639353
C2×#635133