Finite Group Databasebeta

← C2×#11721169 #13151099

Subgroups — C2×#11721169 #13151099

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×C9×#237C211yesmaximal, characteristic, complemented
2C2×#88641C211yesmaximal, characteristic, complemented
3C2×#88543C211yesmaximal, characteristic, complemented
4C2×C3×#3673C211yesmaximal, characteristic, complemented
5#11721169C218yesmaximal, direct product factor
6C2×#88428C211yesmaximal, characteristic, complemented
7C2×#88425C211yesmaximal, characteristic, complemented
8C2×D3×C9⋊C8C211yesmaximal, characteristic, complemented
9C2×#63933—31nomaximal
10C2×#63513—31nomaximal
11C2×C3×D4×C9C2211yescharacteristic
12C2×#6226C2211yescharacteristic
13C2×C9×#67—21no—
14C22×C9×Dic3—21no—
15C9×#237—22no—
16C9×#237C2214yescomplemented
17C2×C4×D3×C9C2211yescharacteristic
18C2×C3×D36C2211yescharacteristic
19C2×C9×Dic6C2211yescharacteristic
20C2×#6194—21no—
21C2×Dic3×D9—21no—
22#88641—22no—
23#88641C2214yescomplemented
24#88543C2214yescomplemented
25C2×C3⋊(C9⋊C8)C2211yescharacteristic
26C3×#3673C2214yescomplemented
27C2×C3×C9⋊C8C2211yescharacteristic
28#88428C2214yescomplemented
29#88425C2214yescomplemented
30D3×C9⋊C8C2214yescomplemented
31C2×C3×#237D311yescharacteristic
32C2×#3795—31no—
33C2×#3754—31no—
34C2×C3×#213—31no—
35#63933—38no—
36C2×#3719—31no—
37C2×#3715—31no—
38C2×D3×C3⋊C8—31no—
39C2×C9×#41—31no—
40C2×#3697—31no—
41#63513—38no—
42C2×#3675—31no—
43C2×#3674—31no—
44C2×#3673D311yescharacteristic, complemented
45C2×#3672—31no—
46C2×#3672—61no—
47C22×C9⋊C8—31no—
48C23×C3×C9—21no—
49C22×D3×C9D411yescharacteristic
50C22×C3×D9—21no—
51C3×D4×C9—21no—
52C3×D4×C9C2312yes—
53C2×C3×C4×C9C2311yescharacteristic
54C2×C3⋊Dic9—21no—
55#6226—21no—
56#6226C2312yes—
57C9×#67—24no—
58C2×C9×Dic3—21no—
59C2×C9×Dic3—24no—
60C2×C9×Dic3D411yescharacteristic
61C4×D3×C9D412yes—
62C4×D3×C9C2312yes—
63C3×D36—21no—
64C3×D36C2312yes—
65C9×Dic6—21no—
66C9×Dic6C2312yes—
67#6194—24no—
68Dic3×D9—24no—
69C3⋊(C9⋊C8)C2312yes—
70C3×C9⋊C8C2312yes—
71C2×#950—91no—
72C2×#912—91no—
73C2×C32×D4C2×D311yescharacteristic
74C2×#634—31no—
75C2×C3×#67—21no—
76C22×C3×Dic3—21no—
77C3×#237—22no—
78C3×#237C2×D314yes—
79C2×C3×D12—31no—
80C2×C3×C4×D3C2×D311yescharacteristic
81C2×C3×Dic6C2×D311yescharacteristic
82C2×#625—61no—
83C2×D3×Dic3—61no—
84#3795—34no—
85#3795—62no—
86#3754—34no—
87C2×C3⋊(C3⋊C8)—31no—
88C3×#213—34no—
89C2×C3×C3⋊C8—31no—
90#3719—34no—
91#3715—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×D36C2×D311yescharacteristic
105C2×C4×D9—61no—
106C2×Dic18—31no—
107C2×Dic18—61no—
108#3675—34no—
109#3674—34no—
110#3673C2×D314yescomplemented
111#3672—34no—
112#3672—64no—
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×D3×C9—22no—
121C2×D3×C9C2×D412yes—
122C2×C3×D9—22no—
123C2×C3×D9—42no—
124C3×C4×C9C2×D411yescharacteristic
125C3×C4×C9C2411yescharacteristic
126C3⋊Dic9—22no—
127C9×Dic3—22no—
128C9×Dic3C2×D412yes—
129C2×C3×#41—31no—
130C2×#237—91no—
131C2×#237D911yescharacteristic, complemented
132C2×#235—91no—
133C2×C3×D8—91no—
134#950—98no—
135C2×#216—91no—
136C2×#215—91no—
137C2×#214—91no—
138C2×#214—91no—
139C2×#214—91no—
140C2×#214—181no—
141C2×#213—31no—
142C22×C3⋊C8—91no—
143#912—98no—
144C2×Dic12—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
152C2×C3⋊Dic3—61no—
153#634—32no—
154#634—61no—
155C3×#67—24no—
156C2×C3×Dic3—21no—
157C2×C3×Dic3—24no—
158C2×C3×Dic3#6711yescharacteristic
159C3×D12—32no—
160C3×D12—61no—
161C3×C4×D3#6712yes—
162C3×C4×D3C22×D312yes—
163C3×Dic6—21no—
164C3×Dic6C22×D312yes—
165#625—64no—
166D3×Dic3—64no—
167C23×C9—21no—
168C23×C9—31no—
169C23×C9—41no—
170C22×D9—21no—
171C3⋊(C3⋊C8)—32no—
172C3×C3⋊C8—32no—
173Q8×C9—32no—
174Q8×C9—61no—
175D4×C9—21no—
176D4×C9—22no—
177D4×C9—41no—
178D4×C9—64no—
179D4×C9C22×D312yes—
180C2×C4×C9—21no—
181C2×C4×C9—31no—
182C2×C4×C9—32no—
183C2×C4×C9—32no—
184C2×C4×C9—64no—
185C2×C4×C9—61no—
186C2×C4×C9C22×D311yescharacteristic
187#611—64no—
188C2×Dic9—61no—
189C2×Dic9—121no—
190D36—21no—
191D36C22×D312yes—
192C4×D9—64no—
193Dic18—32no—
194Dic18—62no—
195Dic18—61no—
196Dic18—121no—
197C9⋊C8—32no—
198C9⋊C8—62no—
199C9⋊C8C22×D312yes—
200C2×#135—271noSylow
201C2×C3×C9—42no—
202C2×C3×C9#13512yescomplemented
203C2×C3×C9C22×D411yescharacteristic
204D3×C9—22no—
205C3×D9—42no—
206C3×#41—34no—
207C3×#41—62no—
208C2×C3×Q8—31no—
209C2×C3×D4—21no—
210C2×C3×D4—61no—
211C2×C3×D4—91no—
212C2×C3×D4C2×D911yescharacteristic
213C2×C3×D4D3211yescharacteristic
214C22×C3×C4—31no—
215C22×C3×C4—61no—
216C2×#67—21no—
217C2×#67—181no—
218C2×#67—181no—
219C22×Dic3—21no—
220C22×Dic3—181no—
221#237—22no—
222#237—94no—
223#237—182no—
224#237C2×D914yescomplemented
225#235—94no—
226#235—182no—
227C2×D12—31no—
228C2×C4×D3—181no—
229C2×C4×D3C2×D911yescharacteristic
230C2×Dic6—91no—
231C2×Dic6—91no—
232C2×Dic6—181no—
233C2×Dic6C2×D911yescharacteristic
234C3×D8—94no—
235C2×C3×C8—91no—
236#216—94no—
237#215—94no—
238#214—94no—
239#214—94no—
240#214—94no—
241#214—184no—
242#213—34no—
243C2×C3⋊C8—31no—
244C2×C3⋊C8—94no—
245C2×C3⋊C8—91no—
246C2×C3⋊C8—91no—
247C2×C3⋊C8—181no—
248Dic12—94no—
249D3×C8—94no—
250C22×C32—22no—
251C22×C32—42no—
252C22×C32C2×#6711yescharacteristic
253C2×C3×D3—22no—
254C2×C3×D3—62no—
255C2×C3×D3—122no—
256C2×C3×D3C2×#6712yes—
257C32×C4C2×#6711yescharacteristic
258C32×C4C23×D311yescharacteristic
259C3⋊Dic3—62no—
260C3×Dic3—22no—
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—22no—
271C2×D9—42no—
272C4×C9—21no—
273C4×C9—21no—
274C4×C9—32no—
275C4×C9—62no—
276C4×C9D3×D411yescharacteristic
277C4×C9C23×D311yescharacteristic
278Dic9—62no—
279Dic9—122no—
280C2×#41—31no—
281C2×#41—271no—
282#135—278no—
283C2×Q16—271no—
284C2×SD16—271no—
285C2×SD16—271no—
286C2×D8—91no—
287C22×C8—271no—
288C3×C9C2×#13511yescharacteristic, complemented, Sylow
289C23×C3—21no—
290C23×C3—21no—
291C23×C3—31no—
292C23×C3—41no—
293C23×C3—181no—
294C22×D3—61no—
295C22×D3#61111yescharacteristic
296C3×Q8—32no—
297C3×Q8—61no—
298C3×D4—21no—
299C3×D4—21no—
300C3×D4—22no—
301C3×D4—41no—
302C3×D4—64no—
303C3×D4—92no—
304C3×D4—181no—
305C3×D4C22×D912yes—
306C3×D4C2×D3212yes—
307C2×C3×C4—21no—
308C2×C3×C4—31no—
309C2×C3×C4—32no—
310C2×C3×C4—32no—
311C2×C3×C4—64no—
312C2×C3×C4—61no—
313C2×C3×C4C22×D911yescharacteristic
314C2×C3×C4C2×D3211yescharacteristic
315#67—24no—
316#67—184no—
317#67—184no—
318C2×Dic3—21no—
319C2×Dic3—24no—
320C2×Dic3—181no—
321C2×Dic3—181no—
322C2×Dic3—184no—
323C2×Dic3—361no—
324C2×Dic3#61111yescharacteristic
325D12—32no—
326D12—61no—
327C4×D3—184no—
328C4×D3#61112yes—
329C4×D3C22×D912yes—
330Dic6—21no—
331Dic6—92no—
332Dic6—92no—
333Dic6—182no—
334Dic6—181no—
335Dic6—181no—
336Dic6—361no—
337Dic6C22×D912yes—
338C3×C8—92no—
339C3⋊C8—32no—
340C3⋊C8—92no—
341C3⋊C8—92no—
342C3⋊C8—182no—
343C2×C32—42no—
344C2×C32#95012yes—
345C2×C32C22×#6711yescharacteristic
346C3×D3—22no—
347C3×D3—122no—
348C2×C9—21no—
349C2×C9—22no—
350C2×C9—42no—
351C2×C9—62no—
352C2×C9—82no—
353C2×C9#91212yescomplemented
354C2×C9C2×D3×D411yescharacteristic
355D9—42no—
356#41—34no—
357#41—62no—
358#41—274no—
359#41—542no—
360C2×Q8—31no—
361C2×Q8—271no—
362C2×D4—61no—
363C2×D4—91no—
364C2×D4—541no—
365C2×D4D3×D911yescharacteristic
366C22×C4—31no—
367C22×C4—61no—
368C22×C4—541no—
369Q16—274no—
370SD16—274no—
371SD16—274no—
372D8—94no—
373C2×C8—91no—
374C2×C8—274no—
375C2×C8—271no—
376C22×C3—22no—
377C22×C3—22no—
378C22×C3—21no—
379C22×C3—32no—
380C22×C3—42no—
381C22×C3—42no—
382C22×C3—42no—
383C22×C3—62no—
384C22×C3—82no—
385C22×C3—182no—
386C22×C3—362no—
387C22×C3C2×#61111yescharacteristic
388C22×C3D3×#6711yescharacteristic
389C2×D3—22no—
390C2×D3—62no—
391C2×D3—122no—
392C2×D3C2×#61112yes—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—32no—
396C3×C4—62no—
397C3×C4C2×#61111yescharacteristic
398C3×C4C23×D911yescharacteristic
399C3×C4D3×#6711yescharacteristic
400C3×C4C22×D3211yescharacteristic
401Dic3—22no—
402Dic3—182no—
403Dic3—182no—
404Dic3—362no—
405Dic3C2×#61112yes—
406C32C2×#95011yescharacteristic
407C9—21no—
408C9C2×#91211yescharacteristic, complemented
409C23—21no—
410C23—31no—
411C23—181no—
412Q8—32no—
413Q8—61no—
414Q8—272no—
415Q8—541no—
416D4—21no—
417D4—64no—
418D4—92no—
419D4—181no—
420D4—544no—
421D4C2×D3×D912yes—
422C2×C4—31no—
423C2×C4—32no—
424C2×C4—32no—
425C2×C4—64no—
426C2×C4—61no—
427C2×C4—544no—
428C2×C4—541no—
429C2×C4C2×D3×D911yescharacteristic
430C8—92no—
431C8—272no—
432C2×C3—21no—
433C2×C3—22no—
434C2×C3—42no—
435C2×C3—42no—
436C2×C3—62no—
437C2×C3—82no—
438C2×C3—362no—
439C2×C3#6351312yescomplemented
440C2×C3C22×#61111yescharacteristic
441C2×C3#6393312yes—
442C2×C3C2×D3×#6711yescharacteristic
443D3—22no—
444D3—122no—
445C22—22no—
446C22—32no—
447C22—42no—
448C22—62no—
449C22—182no—
450C22—362no—
451C22D3×#61111yescharacteristic
452C4—32no—
453C4—62no—
454C4—542no—
455C4D3×#61111yescharacteristic
456C4C22×D3×D911yescharacteristic
457C3—21no—
458C3C2×#6351311yescharacteristic, complemented
459C3C2×#6393311yescharacteristic
460C2—42no—
461C2—62no—
462C2—362no—
463C2#1172116912yesdirect product factor
464C2C2×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×#88641C2semidirect
3. C2×#88543C2semidirect
4. C2×C3×#3673C2semidirect
5. #11721169C2direct (N × H)
6. C2×#88428C2semidirect
7. C2×#88425C2semidirect
8. C2×D3×C9⋊C8C2semidirect
16. C9×#237C22semidirect
23. #88641C22semidirect
24. #88543C22semidirect
26. C3×#3673C22semidirect
28. #88428C22semidirect
29. #88425C22semidirect
30. D3×C9⋊C8C22semidirect
44. C2×#3673D3semidirect
110. #3673C2×D3semidirect
131. C2×#237D9semidirect
202. C2×C3×C9#135semidirect
224. #237C2×D9semidirect
288. C3×C9C2×#135semidirect
353. C2×C9#912semidirect
408. C9C2×#912semidirect
439. C2×C3#63513semidirect
458. C3C2×#63513semidirect
463. C2#11721169direct (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×#886412
C2×#885432
C2×C3×#36732
#117211692
C2×#884282
C2×#884252
C2×D3×C9⋊C82
C2×#639333
C2×#635133