Finite Group Databasebeta

← C2×C3×#63458 #13152388

Subgroups — C2×C3×#63458 #13152388

Order 1728.

1058 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D9C211yesmaximal, characteristic
2C22×C3×Dic18C211yesmaximal, characteristic, complemented
3C2×C3×C9×C4⋊C4C211yesmaximal, characteristic, complemented
4C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic
5C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic
6C2×C3×Dic9⋊C4C211yesmaximal, characteristic, complemented
7C2×C3×Dic9⋊C4C211yesmaximal, characteristic, complemented
8C3×#63458C218yesmaximal, direct product factor
9C2×C3×#896—31nomaximal
10C2×#63458C311yesmaximal, characteristic, direct product factor
11C23×C3×D9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3×C4×C9C2211yescharacteristic
14C22×C3×C4×C9C2211yescharacteristic
15C22×C3×Dic9C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17C22×C3×Dic9C2211yescharacteristic
18C2×C3×C4×D9—24no—
19C2×C3×C4×D9C2214yes—
20C2×C3×Dic18—24no—
21C2×C3×Dic18C2214yescomplemented
22C3×C9×C4⋊C4C2214yescomplemented
23C3×(C2×D9)⋊C4C2214yes—
24C3×(C2×D9)⋊C4C2214yes—
25C3×Dic9⋊C4C2214yescomplemented
26C3×Dic9⋊C4C2214yescomplemented
27C22×C3×C4×D3—31no—
28C22×C3×Dic6—31no—
29C2×C32×C4⋊C4D311yescharacteristic
30C2×C3×(C2×D3)⋊C4—31no—
31C2×C3×(C2×D3)⋊C4—31no—
32C2×C3×Dic3⋊C4—31no—
33C2×C3×Dic3⋊C4—31no—
34C3×#896—38no—
35C22×C4×D9C2×C311yescharacteristic
36C22×Dic18C2×C311yescharacteristic, complemented
37C2×C9×C4⋊C4—21no—
38C2×C9×C4⋊C4C2×C311yescharacteristic, complemented
39C2×(C2×D9)⋊C4C2×C311yescharacteristic
40C2×(C2×D9)⋊C4C2×C311yescharacteristic
41C2×Dic9⋊C4C2×C311yescharacteristic, complemented
42C2×Dic9⋊C4C2×C311yescharacteristic, complemented
43#63458C2×C318yesdirect product factor
44C23×C3×C9C2311yescharacteristic
45C22×C3×D9—22no—
46C22×C3×D9—22no—
47C22×C3×D9Q814yescomplemented
48C22×C3×D9C2312yes—
49C2×C3×C4×C9—22no—
50C2×C3×C4×C9—22no—
51C2×C3×C4×C9—22no—
52C2×C3×C4×C9C2312yes—
53C2×C3×C4×C9C2312yes—
54C2×C3×C4×C9C2312yes—
55C2×C3×Dic9—22no—
56C2×C3×Dic9—22no—
57C2×C3×Dic9D414yes—
58C2×C3×Dic9C2312yes—
59C2×C3×Dic9C2312yes—
60C2×C3×Dic9C2312yes—
61C3×C4×D9—28no—
62C3×Dic18—28no—
63C2×C3×C22⋊Q8—91no—
64C2×#896—31no—
65C23×C3×D3—31no—
66C22×C32×C4C2×D311yescharacteristic
67C22×C32×C4C2×D311yescharacteristic
68C22×C32×C4C2×D311yescharacteristic
69C22×C3×Dic3—31no—
70C22×C3×Dic3—31no—
71C22×C3×Dic3—31no—
72C2×C3×C4×D3—34no—
73C2×C3×C4×D3—64no—
74C2×C3×Dic6—34no—
75C2×C3×Dic6—64no—
76C23×D9C22×C311yescharacteristic
77C32×C4⋊C4C2×D314yes—
78C3×(C2×D3)⋊C4—34no—
79C3×(C2×D3)⋊C4—34no—
80C3×Dic3⋊C4—34no—
81C3×Dic3⋊C4—34no—
82C22×C4×C9—21no—
83C22×C4×C9—21no—
84C22×C4×C9—21no—
85C22×C4×C9C22×C311yescharacteristic
86C22×C4×C9C22×C311yescharacteristic
87C22×C4×C9C22×C311yescharacteristic
88C22×Dic9C22×C311yescharacteristic
89C22×Dic9C22×C311yescharacteristic
90C22×Dic9C22×C311yescharacteristic
91C2×C4×D9—24no—
92C2×C4×D9C22×C314yes—
93C2×Dic18—24no—
94C2×Dic18C22×C314yescomplemented
95C9×C4⋊C4—24no—
96C9×C4⋊C4C22×C314yescomplemented
97(C2×D9)⋊C4C22×C314yes—
98(C2×D9)⋊C4C22×C314yes—
99Dic9⋊C4C22×C314yescomplemented
100Dic9⋊C4C22×C314yescomplemented
101C22×C3×C9C2×D412yes—
102C22×C3×C9C2×Q812yes—
103C22×C3×C9#4112yes—
104C22×C3×C9C2411yescharacteristic
105C2×C3×D9—28no—
106C2×C3×D9—22no—
107C2×C3×D9—22no—
108C2×C3×D9C2×Q814yescomplemented
109C3×C4×C9—22no—
110C3×C4×C9—22no—
111C3×C4×C9—22no—
112C3×Dic9—22no—
113C3×Dic9—22no—
114C3×Dic9C2×D414yes—
115C22×C3×Q8—91no—
116C23×C3×C4—91no—
117C22×C4×D3—31no—
118C22×Dic6—31no—
119C3×C22⋊Q8—98no—
120C2×C3×C4⋊C4—21no—
121C2×C3×C4⋊C4—91no—
122C2×C3×C4⋊C4—91no—
123C2×C3×C4⋊C4D911yescharacteristic, complemented
124C2×C3×C4⋊C4C3×D311yescharacteristic
125C2×C3×C22⋊C4—91no—
126C2×C3×C22⋊C4—91no—
127C2×(C2×D3)⋊C4—31no—
128C2×(C2×D3)⋊C4—31no—
129C2×Dic3⋊C4—31no—
130C2×Dic3⋊C4—31no—
131#896—38no—
132C23×C32C22×D311yescharacteristic
133C22×C3×D3—34no—
134C22×C3×D3—32no—
135C22×C3×D3—62no—
136C22×C3×D3—62no—
137C2×C32×C4—22no—
138C2×C32×C4—22no—
139C2×C32×C4—22no—
140C2×C32×C4C22×D312yes—
141C2×C32×C4C22×D312yes—
142C2×C32×C4C22×D312yes—
143C2×C3×Dic3—34no—
144C2×C3×Dic3—32no—
145C2×C3×Dic3—32no—
146C2×C3×Dic3—32no—
147C2×C3×Dic3—62no—
148C2×C3×Dic3—62no—
149C3×C4×D3—68no—
150C3×Dic6—68no—
151C23×C9—21no—
152C23×C9C23×C311yescharacteristic
153C22×D9—22no—
154C22×D9—22no—
155C22×D9C3×Q814yescomplemented
156C22×D9C23×C312yes—
157C2×C4×C9—24no—
158C2×C4×C9—22no—
159C2×C4×C9—22no—
160C2×C4×C9—22no—
161C2×C4×C9—22no—
162C2×C4×C9—22no—
163C2×C4×C9—22no—
164C2×C4×C9—42no—
165C2×C4×C9—42no—
166C2×C4×C9C23×C312yes—
167C2×C4×C9C23×C312yes—
168C2×C4×C9C23×C312yes—
169C2×Dic9—22no—
170C2×Dic9—22no—
171C2×Dic9C3×D414yes—
172C2×Dic9C23×C312yes—
173C2×Dic9C23×C312yes—
174C2×Dic9C23×C312yes—
175C4×D9—28no—
176Dic18—28no—
177C2×C22⋊Q8—91noSylow
178C2×C3×C9C22⋊Q814yescomplemented
179C2×C3×C9C22×D411yescharacteristic
180C2×C3×C9C22×Q811yescharacteristic
181C2×C3×C9C2×#4111yescharacteristic
182C3×D9—24no—
183C24×C3—91no—
184C23×D3—31no—
185C2×C3×Q8—94no—
186C2×C3×Q8—184no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—91no—
191C22×C3×C4—91no—
192C22×C3×C4—91no—
193C22×C3×C4—94no—
194C22×C3×C4—184no—
195C22×C3×C4C2×D911yescharacteristic
196C22×C3×C4C2×D911yescharacteristic
197C22×C3×C4C2×D911yescharacteristic
198C22×C3×C4C2×C3×D311yescharacteristic
199C22×C3×C4C2×C3×D311yescharacteristic
200C22×C3×C4C2×C3×D311yescharacteristic
201C22×Dic3—31no—
202C22×Dic3—31no—
203C22×Dic3—31no—
204C2×C4×D3—34no—
205C2×C4×D3—64no—
206C2×Dic6—34no—
207C2×Dic6—64no—
208C3×C4⋊C4—24no—
209C3×C4⋊C4—94no—
210C3×C4⋊C4—94no—
211C3×C4⋊C4C2×D914yescomplemented
212C3×C4⋊C4C2×C3×D314yes—
213C3×C22⋊C4—94no—
214C3×C22⋊C4—94no—
215(C2×D3)⋊C4—34no—
216(C2×D3)⋊C4—34no—
217Dic3⋊C4—34no—
218Dic3⋊C4—34no—
219C22×C32#23512yes—
220C22×C32D3×D412yes—
221C22×C32D3×Q812yes—
222C22×C32C23×D311yescharacteristic
223C2×C3×D3—34no—
224C2×C3×D3—68no—
225C2×C3×D3—62no—
226C2×C3×D3—62no—
227C32×C4—22no—
228C32×C4—22no—
229C32×C4—22no—
230C3×Dic3—34no—
231C3×Dic3—62no—
232C3×Dic3—62no—
233C22×C9—21no—
234C22×C9—22no—
235C22×C9—22no—
236C22×C9—22no—
237C22×C9C2×C3×D412yes—
238C22×C9C2×C3×Q812yes—
239C22×C9C3×#4112yes—
240C22×C9C24×C311yescharacteristic
241C2×D9—28no—
242C2×D9—22no—
243C2×D9—22no—
244C2×D9C2×C3×Q814yescomplemented
245C4×C9—24no—
246C4×C9—22no—
247C4×C9—22no—
248C4×C9—22no—
249C4×C9—42no—
250C4×C9—42no—
251Dic9—22no—
252Dic9—22no—
253Dic9C2×C3×D414yes—
254C22×Q8—91no—
255C23×C4—91no—
256C22⋊Q8—98no—
257C2×C4⋊C4—91no—
258C2×C4⋊C4—91no—
259C2×C4⋊C4C3×D911yescharacteristic, complemented
260C2×C22⋊C4—91no—
261C2×C22⋊C4—91no—
262C3×C9C2×C22⋊Q811yescharacteristic, complemented, Sylow
263C23×C3—21no—
264C23×C3—94no—
265C23×C3—92no—
266C23×C3—182no—
267C23×C3—182no—
268C23×C3C22×D911yescharacteristic
269C23×C3C22×C3×D311yescharacteristic
270C22×D3—32no—
271C22×D3—34no—
272C22×D3—62no—
273C22×D3—62no—
274C3×Q8—188no—
275C2×C3×C4—24no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—94no—
288C2×C3×C4—92no—
289C2×C3×C4—92no—
290C2×C3×C4—92no—
291C2×C3×C4—188no—
292C2×C3×C4—182no—
293C2×C3×C4—182no—
294C2×C3×C4C22×D912yes—
295C2×C3×C4C22×D912yes—
296C2×C3×C4C22×D912yes—
297C2×C3×C4C22×C3×D312yes—
298C2×C3×C4C22×C3×D312yes—
299C2×C3×C4C22×C3×D312yes—
300C2×Dic3—32no—
301C2×Dic3—34no—
302C2×Dic3—32no—
303C2×Dic3—32no—
304C2×Dic3—62no—
305C2×Dic3—62no—
306C4×D3—68no—
307Dic6—68no—
308C2×C32#89614yes—
309C2×C32C2×#23511yescharacteristic
310C2×C32C2×D3×D411yescharacteristic
311C2×C32C2×D3×Q811yescharacteristic
312C3×D3—64no—
313C2×C9—24no—
314C2×C9—21no—
315C2×C9—21no—
316C2×C9—21no—
317C2×C9C3×C22⋊Q814yescomplemented
318C2×C9C22×C3×D411yescharacteristic
319C2×C9C22×C3×Q811yescharacteristic
320C2×C9C2×C3×#4111yescharacteristic
321D9—24no—
322C24—91no—
323C2×Q8—94no—
324C2×Q8—184no—
325C22×C4—91no—
326C22×C4—91no—
327C22×C4—91no—
328C22×C4—94no—
329C22×C4—184no—
330C22×C4C2×C3×D911yescharacteristic
331C22×C4C2×C3×D911yescharacteristic
332C22×C4C2×C3×D911yescharacteristic
333C4⋊C4—94no—
334C4⋊C4—94no—
335C4⋊C4C2×C3×D914yescomplemented
336C22⋊C4—94no—
337C22⋊C4—94no—
338C22×C3—21no—
339C22×C3—22no—
340C22×C3—22no—
341C22×C3—22no—
342C22×C3—94no—
343C22×C3—182no—
344C22×C3—182no—
345C22×C3—188no—
346C22×C3#369712yes—
347C22×C3D4×D912yes—
348C22×C3Q8×D912yes—
349C22×C3C23×D911yescharacteristic
350C22×C3C3×#23512yes—
351C22×C3C3×D3×D412yes—
352C22×C3C3×D3×Q812yes—
353C22×C3C23×C3×D311yescharacteristic
354C2×D3—34no—
355C2×D3—68no—
356C2×D3—62no—
357C2×D3—62no—
358C3×C4—24no—
359C3×C4—22no—
360C3×C4—22no—
361C3×C4—22no—
362C3×C4—22no—
363C3×C4—22no—
364C3×C4—22no—
365C3×C4—42no—
366C3×C4—42no—
367C3×C4—94no—
368C3×C4—182no—
369C3×C4—182no—
370Dic3—34no—
371Dic3—62no—
372Dic3—62no—
373C32C2×#89611yescharacteristic
374C9—21no—
375C9C2×C3×C22⋊Q811yescharacteristic, complemented
376C23—92no—
377C23—94no—
378C23—182no—
379C23—182no—
380C23C22×C3×D911yescharacteristic
381Q8—188no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—92no—
386C2×C4—94no—
387C2×C4—92no—
388C2×C4—92no—
389C2×C4—188no—
390C2×C4—182no—
391C2×C4—182no—
392C2×C4C22×C3×D912yes—
393C2×C4C22×C3×D912yes—
394C2×C4C22×C3×D912yes—
395C2×C3—21no—
396C2×C3—21no—
397C2×C3—21no—
398C2×C3—24no—
399C2×C3—184no—
400C2×C3#6345814yesdirect product factor
401C2×C3C2×#369711yescharacteristic
402C2×C3C2×D4×D911yescharacteristic
403C2×C3C2×Q8×D911yescharacteristic
404C2×C3C3×#89614yes—
405C2×C3C2×C3×#23511yescharacteristic
406C2×C3C2×C3×D3×D411yescharacteristic
407C2×C3C2×C3×D3×Q811yescharacteristic
408D3—64no—
409C22—94no—
410C22—182no—
411C22—182no—
412C22—188no—
413C22C3×#369712yes—
414C22C3×D4×D912yes—
415C22C3×Q8×D912yes—
416C22C23×C3×D911yescharacteristic
417C4—22no—
418C4—22no—
419C4—22no—
420C4—94no—
421C4—182no—
422C4—182no—
423C3—21no—
424C3C2×#6345811yescharacteristic, direct product factor
425C3C2×C3×#89611yescharacteristic
426C2—184no—
427C2C3×#6345814yesdirect product factor
428C2C2×C3×#369711yescharacteristic
429C2C2×C3×D4×D911yescharacteristic
430C2C2×C3×Q8×D911yescharacteristic
subgroupG/H
centerC23×C3C22×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC2×C3×C9×C4⋊C4C2
socleC23×C32C22×D3
prime radicalC23×C3×D9C22
hypercenterC2×C3×C4⋊C4D9
hypocenterC9C2×C3×C22⋊Q8
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×C3×Dic18C2semidirect
3. C2×C3×C9×C4⋊C4C2semidirect
6. C2×C3×Dic9⋊C4C2semidirect
7. C2×C3×Dic9⋊C4C2semidirect
8. C3×#63458C2direct (N × H)
10. C2×#63458C3direct (N × H)
21. C2×C3×Dic18C22semidirect
22. C3×C9×C4⋊C4C22semidirect
25. C3×Dic9⋊C4C22semidirect
26. C3×Dic9⋊C4C22semidirect
36. C22×Dic18C2×C3semidirect
38. C2×C9×C4⋊C4C2×C3semidirect
41. C2×Dic9⋊C4C2×C3semidirect
42. C2×Dic9⋊C4C2×C3semidirect
43. #63458C2×C3direct (N × H)
47. C22×C3×D9Q8semidirect
94. C2×Dic18C22×C3semidirect
96. C9×C4⋊C4C22×C3semidirect
99. Dic9⋊C4C22×C3semidirect
100. Dic9⋊C4C22×C3semidirect
108. C2×C3×D9C2×Q8semidirect
123. C2×C3×C4⋊C4D9semidirect
155. C22×D9C3×Q8semidirect
178. C2×C3×C9C22⋊Q8semidirect
211. C3×C4⋊C4C2×D9semidirect
244. C2×D9C2×C3×Q8semidirect
259. C2×C4⋊C4C3×D9semidirect
262. C3×C9C2×C22⋊Q8semidirect
317. C2×C9C3×C22⋊Q8semidirect
335. C4⋊C4C2×C3×D9semidirect
375. C9C2×C3×C22⋊Q8semidirect
400. C2×C3#63458direct (N × H)
424. C3C2×#63458direct (N × H)
427. C2C3×#63458direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×D92
C22×C3×Dic182
C2×C3×C9×C4⋊C42
C2×C3×(C2×D9)⋊C42
C2×C3×Dic9⋊C42
C3×#634582
C2×C3×#8963
C2×#634583