Finite Group Databasebeta

← C2×#11721165 #13150916

Subgroups — C2×#11721165 #13150916

Order 1728.

842 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C9×#237C211yesmaximal, characteristic
2C2×D3×Dic18C211yesmaximal, characteristic, complemented
3C2×#88543C211yesmaximal, characteristic, complemented
4C2×C3×#3672C211yesmaximal, characteristic, complemented
5#11721165C218yesmaximal, direct product factor
6C2×#88428C211yesmaximal, characteristic, complemented
7C2×#88427C211yesmaximal, characteristic, complemented
8C2×#88420C211yesmaximal, characteristic, complemented
9C2×#63929—31nomaximal
10C2×#63511—31nomaximal
11C2×C3×D4×C9C2211yescharacteristic
12C2×#6226C2211yescharacteristic
13C2×C9×#67—21no—
14C22×C9×Dic3—21no—
15C9×#237—22no—
16C9×#237C2214yes—
17C2×C4×D3×C9C2211yescharacteristic
18C2×C9×Dic6C2211yescharacteristic
19C2×C3×Dic18C2211yescharacteristic
20C2×D3×Dic9—21no—
21C2×#6189—21no—
22D3×Dic18—22no—
23D3×Dic18C2214yescomplemented
24#88543C2214yescomplemented
25C2×C3⋊(C9⋊C8)C2211yescharacteristic
26C3×#3672C2214yescomplemented
27C2×C3×C9⋊C8C2211yescharacteristic
28#88428C2214yescomplemented
29#88427C2214yescomplemented
30#88420C2214yescomplemented
31C2×C3×#237D311yescharacteristic
32C2×D3×Dic6—31no—
33C2×#3754—31no—
34C2×C3×#214—31no—
35#63929—38no—
36C2×#3719—31no—
37C2×#3718—31no—
38C2×#3711—31no—
39C2×C9×#41—31no—
40C22×Dic18—31no—
41#63511—38no—
42C2×#3674—31no—
43C2×#3674—31no—
44C2×#3672—31no—
45C2×#3672—61no—
46C2×#3672D311yescharacteristic, complemented
47C2×#3667—31no—
48C23×C3×C9—21no—
49C22×D3×C9D411yescharacteristic
50C3×D4×C9—21no—
51C3×D4×C9C2312yes—
52C2×C3×C4×C9C2311yescharacteristic
53C2×C3⋊Dic9—21no—
54#6226—21no—
55#6226C2312yes—
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—
67D3×Dic9—24no—
68#6189—24no—
69C3⋊(C9⋊C8)C2312yes—
70C3×C9⋊C8C2312yes—
71C2×#951—91no—
72C2×#915—91no—
73C2×C32×D4C2×D311yescharacteristic
74C2×#634—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×#627—61no—
83C2×D3×Dic3—61no—
84D3×Dic6—34no—
85D3×Dic6—62no—
86#3754—34no—
87C2×C3⋊(C3⋊C8)—31no—
88C3×#214—34no—
89C2×C3×C3⋊C8—31no—
90#3719—34no—
91#3718—34no—
92#3711—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—
101C22×Dic9—61no—
102C2×Dic18—31no—
103C2×Dic18—34no—
104C2×Dic18—62no—
105C2×Dic18—61no—
106C2×Dic18—61no—
107C2×Dic18C2×D311yescharacteristic
108#3674—34no—
109#3674—34no—
110#3672—34no—
111#3672—64no—
112#3672C2×D314yescomplemented
113#3667—34no—
114C2×C9⋊C8—31no—
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—
122C3×C4×C9C2×D411yescharacteristic
123C3×C4×C9C2411yescharacteristic
124C3⋊Dic9—22no—
125C9×Dic3—22no—
126C9×Dic3C2×D412yes—
127C3×Dic9—22no—
128C2×C3×#41—31no—
129C2×D3×Q8—91no—
130C2×#237D911yescharacteristic
131C22×Dic6—91no—
132C2×C3×SD16—91no—
133#951—98no—
134C2×#216—91no—
135C2×#216—91no—
136C2×#216—91no—
137C2×#214—31no—
138C2×#214—91no—
139C2×#214—91no—
140C2×#214—181no—
141C2×#208—91no—
142#915—98no—
143C2×Dic12—91no—
144C2×ΓL(1,𝔽25)—91no—
145C23×C32—21no—
146C22×C3×D3#6711yescharacteristic
147C32×D4—21no—
148C32×D4C22×D312yes—
149C2×C32×C4C22×D311yescharacteristic
150C2×C3⋊Dic3—61no—
151#634—32no—
152#634—61no—
153C3×#67—24no—
154C2×C3×Dic3—21no—
155C2×C3×Dic3—24no—
156C2×C3×Dic3—61no—
157C2×C3×Dic3#6711yescharacteristic
158C3×C4×D3#6712yes—
159C3×C4×D3C22×D312yes—
160C3×Dic6—21no—
161C3×Dic6—32no—
162C3×Dic6—61no—
163C3×Dic6C22×D312yes—
164#627—64no—
165D3×Dic3—64no—
166C23×C9—21no—
167C23×C9—31no—
168C23×C9—41no—
169C3⋊(C3⋊C8)—32no—
170C3×C3⋊C8—32no—
171Q8×C9—32no—
172Q8×C9—61no—
173D4×C9—21no—
174D4×C9—22no—
175D4×C9—41no—
176D4×C9—64no—
177D4×C9C22×D312yes—
178C2×C4×C9—21no—
179C2×C4×C9—31no—
180C2×C4×C9—32no—
181C2×C4×C9—32no—
182C2×C4×C9—64no—
183C2×C4×C9—61no—
184C2×C4×C9C22×D311yescharacteristic
185C2×Dic9—21no—
186C2×Dic9—64no—
187C2×Dic9—61no—
188C2×Dic9—121no—
189Dic18—21no—
190Dic18—32no—
191Dic18—62no—
192Dic18—64no—
193Dic18—61no—
194Dic18—121no—
195Dic18C22×D312yes—
196C9⋊C8—32no—
197C9⋊C8—62no—
198C9⋊C8C22×D312yes—
199C2×#137—271noSylow
200C2×C3×C9—42no—
201C2×C3×C9#13712yescomplemented
202C2×C3×C9C22×D411yescharacteristic
203D3×C9—22no—
204C3×#41—34no—
205C3×#41—62no—
206C2×C3×Q8—31no—
207C2×C3×Q8—91no—
208C2×C3×D4—21no—
209C2×C3×D4—61no—
210C2×C3×D4C2×D911yescharacteristic
211C2×C3×D4D3211yescharacteristic
212C22×C3×C4—31no—
213C22×C3×C4—61no—
214C2×#67—21no—
215C22×Dic3—21no—
216C22×Dic3—181no—
217D3×Q8—94no—
218D3×Q8—182no—
219#237—22no—
220#237C2×D914yes—
221C2×C4×D3—181no—
222C2×C4×D3C2×D911yescharacteristic
223C2×Dic6—31no—
224C2×Dic6—94no—
225C2×Dic6—91no—
226C2×Dic6—91no—
227C2×Dic6—181no—
228C2×Dic6—182no—
229C2×Dic6—181no—
230C2×Dic6—181no—
231C2×Dic6C2×D911yescharacteristic
232C3×SD16—94no—
233C2×C3×C8—91no—
234#216—94no—
235#216—94no—
236#216—94no—
237#214—34no—
238#214—94no—
239#214—94no—
240#214—184no—
241#208—94no—
242C2×C3⋊C8—31no—
243C2×C3⋊C8—91no—
244C2×C3⋊C8—91no—
245C2×C3⋊C8—181no—
246Dic12—94no—
247ΓL(1,𝔽25)—94no—
248C22×C32—22no—
249C22×C32—42no—
250C22×C32C2×#6711yescharacteristic
251C2×C3×D3—22no—
252C2×C3×D3C2×#6712yes—
253C32×C4C2×#6711yescharacteristic
254C32×C4C23×D311yescharacteristic
255C3⋊Dic3—62no—
256C3×Dic3—22no—
257C3×Dic3—62no—
258C3×Dic3C2×#6712yes—
259C22×C9—22no—
260C22×C9—21no—
261C22×C9—32no—
262C22×C9—42no—
263C22×C9—42no—
264C22×C9—62no—
265C22×C9—82no—
266C22×C9D3×D411yescharacteristic
267C4×C9—21no—
268C4×C9—21no—
269C4×C9—32no—
270C4×C9—62no—
271C4×C9D3×D411yescharacteristic
272C4×C9C23×D311yescharacteristic
273Dic9—22no—
274Dic9—62no—
275Dic9—122no—
276C2×#41—31no—
277C22×Q8—271no—
278#137—278no—
279C2×Q16—271no—
280C2×Q16—271no—
281C2×SD16—91no—
282C2×SD16—271no—
283C2×M16—271no—
284C3×C9C2×#13711yescharacteristic, complemented, Sylow
285C23×C3—21no—
286C23×C3—21no—
287C23×C3—31no—
288C23×C3—41no—
289C22×D3#61111yescharacteristic
290C3×Q8—32no—
291C3×Q8—61no—
292C3×Q8—92no—
293C3×Q8—181no—
294C3×D4—21no—
295C3×D4—21no—
296C3×D4—22no—
297C3×D4—41no—
298C3×D4—64no—
299C3×D4C22×D912yes—
300C3×D4C2×D3212yes—
301C2×C3×C4—21no—
302C2×C3×C4—31no—
303C2×C3×C4—32no—
304C2×C3×C4—32no—
305C2×C3×C4—64no—
306C2×C3×C4—61no—
307C2×C3×C4—181no—
308C2×C3×C4C22×D911yescharacteristic
309C2×C3×C4C2×D3211yescharacteristic
310#67—24no—
311C2×Dic3—21no—
312C2×Dic3—24no—
313C2×Dic3—61no—
314C2×Dic3—184no—
315C2×Dic3—181no—
316C2×Dic3—181no—
317C2×Dic3—361no—
318C2×Dic3#61111yescharacteristic
319C4×D3—184no—
320C4×D3#61112yes—
321C4×D3C22×D912yes—
322Dic6—21no—
323Dic6—32no—
324Dic6—61no—
325Dic6—92no—
326Dic6—92no—
327Dic6—182no—
328Dic6—184no—
329Dic6—181no—
330Dic6—181no—
331Dic6—184no—
332Dic6—361no—
333Dic6C22×D912yes—
334C3×C8—92no—
335C3⋊C8—32no—
336C3⋊C8—92no—
337C3⋊C8—92no—
338C3⋊C8—182no—
339C2×C32—42no—
340C2×C32#95112yes—
341C2×C32C22×#6711yescharacteristic
342C3×D3—22no—
343C2×C9—21no—
344C2×C9—22no—
345C2×C9—42no—
346C2×C9—62no—
347C2×C9—82no—
348C2×C9#91512yescomplemented
349C2×C9C2×D3×D411yescharacteristic
350#41—34no—
351#41—62no—
352C2×Q8—31no—
353C2×Q8—91no—
354C2×Q8—274no—
355C2×Q8—271no—
356C2×Q8—541no—
357C2×Q8—542no—
358C2×D4—61no—
359C2×D4D3×D911yescharacteristic
360C22×C4—31no—
361C22×C4—61no—
362C22×C4—541no—
363Q16—274no—
364Q16—274no—
365SD16—94no—
366SD16—274no—
367M16—274no—
368C2×C8—91no—
369C2×C8—271no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3—21no—
373C22×C3—32no—
374C22×C3—42no—
375C22×C3—42no—
376C22×C3—42no—
377C22×C3—62no—
378C22×C3—82no—
379C22×C3C2×#61111yescharacteristic
380C22×C3D3×#6711yescharacteristic
381C2×D3—22no—
382C2×D3C2×#61112yes—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—32no—
386C3×C4—62no—
387C3×C4—182no—
388C3×C4C2×#61111yescharacteristic
389C3×C4C23×D911yescharacteristic
390C3×C4D3×#6711yescharacteristic
391C3×C4C22×D3211yescharacteristic
392Dic3—22no—
393Dic3—62no—
394Dic3—182no—
395Dic3—182no—
396Dic3—362no—
397Dic3C2×#61112yes—
398C32C2×#95111yescharacteristic
399C9—21no—
400C9C2×#91511yescharacteristic, complemented
401C23—21no—
402C23—31no—
403Q8—32no—
404Q8—61no—
405Q8—92no—
406Q8—181no—
407Q8—272no—
408Q8—544no—
409Q8—541no—
410D4—21no—
411D4—64no—
412D4C2×D3×D912yes—
413C2×C4—31no—
414C2×C4—32no—
415C2×C4—32no—
416C2×C4—64no—
417C2×C4—61no—
418C2×C4—181no—
419C2×C4—544no—
420C2×C4—541no—
421C2×C4C2×D3×D911yescharacteristic
422C8—92no—
423C8—272no—
424C2×C3—21no—
425C2×C3—22no—
426C2×C3—42no—
427C2×C3—42no—
428C2×C3—62no—
429C2×C3—82no—
430C2×C3#6351112yescomplemented
431C2×C3C22×#61111yescharacteristic
432C2×C3#6392912yes—
433C2×C3C2×D3×#6711yescharacteristic
434D3—22no—
435C22—22no—
436C22—32no—
437C22—42no—
438C22—62no—
439C22D3×#61111yescharacteristic
440C4—32no—
441C4—62no—
442C4—182no—
443C4—542no—
444C4D3×#61111yescharacteristic
445C4C22×D3×D911yescharacteristic
446C3—21no—
447C3C2×#6351111yescharacteristic, complemented
448C3C2×#6392911yescharacteristic
449C2—42no—
450C2—62no—
451C2#1172116512yesdirect product factor
452C2C2×D3×#61111yescharacteristic
subgroupG/H
centerC22D3×#611
commutator, cosocleC3×C4×C9C24
FrattiniC3×C4C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C2×#67
prime radicalC2×C3×#237D3
hypercenterC2×D4D3×D9
hypocenterC3×C9C2×#137
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
2. C2×D3×Dic18C2semidirect
3. C2×#88543C2semidirect
4. C2×C3×#3672C2semidirect
5. #11721165C2direct (N × H)
6. C2×#88428C2semidirect
7. C2×#88427C2semidirect
8. C2×#88420C2semidirect
23. D3×Dic18C22semidirect
24. #88543C22semidirect
26. C3×#3672C22semidirect
28. #88428C22semidirect
29. #88427C22semidirect
30. #88420C22semidirect
46. C2×#3672D3semidirect
112. #3672C2×D3semidirect
201. C2×C3×C9#137semidirect
284. C3×C9C2×#137semidirect
348. C2×C9#915semidirect
400. C9C2×#915semidirect
430. C2×C3#63511semidirect
447. C3C2×#63511semidirect
451. C2#11721165direct (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×D3×Dic182
C2×#885432
C2×C3×#36722
#117211652
C2×#884282
C2×#884272
C2×#884202
C2×#639293
C2×#635113