Finite Group Databasebeta

← C2×#11720975 #13150165

Subgroups — C2×#11720975 #13150165

Order 1728.

1098 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Dic3×D9C211yesmaximal, characteristic
2C2×C4×C3⋊Dic9C211yesmaximal, characteristic, complemented
3C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic
4C2×C9×Dic3⋊C4C211yesmaximal, characteristic, complemented
5C2×(C2×D9)⋊Dic3C211yesmaximal, characteristic
6C2×Dic3⋊Dic9C211yesmaximal, characteristic, complemented
7C2×Dic3×Dic9C211yesmaximal, characteristic, complemented
8#11720975C218yesmaximal, direct product factor
9C2×#63878—31nomaximal
10C2×#63454—31nomaximal
11C23×C3×D9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C22×C3⋊Dic9C2211yescharacteristic
15C22×C9×Dic3C2211yescharacteristic
16C22×C9×Dic3C2211yescharacteristic
17C22×C3×Dic9C2211yescharacteristic
18C2×Dic3×D9C418yescomplemented
19C2×Dic3×D9C2214yes—
20C4×C3⋊Dic9C2214yescomplemented
21C3×(C2×D9)⋊C4C2214yes—
22C9×Dic3⋊C4C2214yescomplemented
23(C2×D9)⋊Dic3C2214yes—
24Dic3⋊Dic9C2214yescomplemented
25Dic3×Dic9C2214yescomplemented
26C22×D3×Dic3—31no—
27C2×C4×C3⋊Dic3—31no—
28C2×C3×(C2×D3)⋊C4—31no—
29C2×C3×Dic3⋊C4D311yescharacteristic
30C2×Dic3⋊Dic3—31no—
31C2×(C2×D3)⋊Dic3—31no—
32C2×Dic32—31no—
33#63878—38no—
34C22×C4×D9—31no—
35C2×C9×C4⋊C4—31no—
36C2×(C2×D9)⋊C4—31no—
37C2×(C2×D9)⋊C4D311yescharacteristic
38C2×C4⋊Dic9—31no—
39C2×C4×Dic9—31no—
40C2×C4×Dic9—31no—
41C2×C4×Dic9—61no—
42#63454—38no—
43C23×C3×C9C2311yescharacteristic
44C22×C3×D9—22no—
45C22×C3×D9—22no—
46C22×C3×D9C2×C414yes—
47C22×C3×D9C2312yes—
48C2×C3×C4×C9—22no—
49C2×C3×C4×C9C2312yes—
50C2×C3⋊Dic9—22no—
51C2×C3⋊Dic9C2×C414yes—
52C2×C3⋊Dic9C2312yes—
53C2×C3⋊Dic9C2312yes—
54C2×C9×Dic3—22no—
55C2×C9×Dic3C2×C414yes—
56C2×C9×Dic3C2312yes—
57C2×C9×Dic3C2312yes—
58C2×C3×Dic9—22no—
59C2×C3×Dic9C2312yes—
60Dic3×D9C2×C4116yescomplemented
61C2×#892—91no—
62C2×#877—91no—
63C23×C3×D3—31no—
64C22×C32×C4C2×D311yescharacteristic
65C22×C3⋊Dic3—31no—
66C22×C3⋊Dic3—31no—
67C22×C3×Dic3—31no—
68C22×C3×Dic3C2×D311yescharacteristic
69C22×C3×Dic3C2×D311yescharacteristic
70C2×D3×Dic3—38no—
71C2×D3×Dic3—34no—
72C23×D9C2×D311yescharacteristic
73C4×C3⋊Dic3—34no—
74C3×(C2×D3)⋊C4—34no—
75C3×Dic3⋊C4C2×D314yes—
76Dic3⋊Dic3—34no—
77(C2×D3)⋊Dic3—34no—
78Dic32—34no—
79C22×C4×C9—21no—
80C22×C4×C9—31no—
81C22×C4×C9—31no—
82C22×C4×C9C2×D311yescharacteristic
83C22×Dic9—31no—
84C22×Dic9—31no—
85C22×Dic9—61no—
86C22×Dic9—61no—
87C22×Dic9C2×D311yescharacteristic
88C2×C4×D9—38no—
89C2×C4×D9—34no—
90C9×C4⋊C4—34no—
91(C2×D9)⋊C4—34no—
92(C2×D9)⋊C4C2×D314yes—
93C4⋊Dic9—34no—
94C4×Dic9—34no—
95C4×Dic9—34no—
96C4×Dic9—64no—
97C22×C3×C9C22×C412yes—
98C22×C3×C9#4112yes—
99C22×C3×C9#4112yes—
100C22×C3×C9C2411yescharacteristic
101C2×C3×D9—28no—
102C2×C3×D9—22no—
103C2×C3×D9—22no—
104C2×C3×D9C22×C414yes—
105C3×C4×C9—22no—
106C3⋊Dic9—22no—
107C3⋊Dic9C22×C414yes—
108C9×Dic3—22no—
109C9×Dic3C22×C414yes—
110C3×Dic9—22no—
111C23×Dic3—91no—
112C22×C4×D3—91no—
113C2×C3×C4⋊C4—31no—
114C2×C3×C22⋊C4—91no—
115C2×#217—91no—
116C2×(C2×D3)⋊C4—31no—
117C2×(C2×D3)⋊C4—91no—
118C2×C4⋊Dic3—91no—
119C2×Dic3⋊C4—91no—
120C2×Dic3⋊C4D911yescharacteristic, complemented
121C2×C4×Dic3—91no—
122C2×C4×Dic3—91no—
123C2×C4×Dic3—91no—
124C2×C4×Dic3—91no—
125C2×C4×Dic3—181no—
126#892—98no—
127#877—98no—
128C23×C32C22×D311yescharacteristic
129C22×C3×D3—34no—
130C22×C3×D3—32no—
131C22×C3×D3—62no—
132C22×C3×D3—62no—
133C2×C32×C4—22no—
134C2×C32×C4C22×D312yes—
135C2×C3⋊Dic3—34no—
136C2×C3⋊Dic3—32no—
137C2×C3⋊Dic3—32no—
138C2×C3⋊Dic3—62no—
139C2×C3×Dic3—22no—
140C2×C3×Dic3—32no—
141C2×C3×Dic3—62no—
142C2×C3×Dic3C4×D314yes—
143C2×C3×Dic3C22×D312yes—
144C2×C3×Dic3C22×D312yes—
145D3×Dic3—316no—
146C23×C9—21no—
147C23×C9C22×D311yescharacteristic
148C22×D9—22no—
149C22×D9—22no—
150C22×D9C4×D314yes—
151C22×D9C22×D312yes—
152C2×C4×C9—24no—
153C2×C4×C9—22no—
154C2×C4×C9—22no—
155C2×C4×C9—34no—
156C2×C4×C9—32no—
157C2×C4×C9—32no—
158C2×C4×C9—62no—
159C2×C4×C9C22×D312yes—
160C2×Dic9—22no—
161C2×Dic9—34no—
162C2×Dic9—32no—
163C2×Dic9—32no—
164C2×Dic9—62no—
165C2×Dic9—62no—
166C2×Dic9—64no—
167C2×Dic9—64no—
168C2×Dic9—62no—
169C2×Dic9C22×D312yes—
170C4×D9—316no—
171C2×#117—271noSylow
172C2×C3×C9#11714yescomplemented
173C2×C3×C9C23×C411yescharacteristic
174C2×C3×C9C2×#4111yescharacteristic
175C2×C3×C9C2×#4111yescharacteristic
176C3×D9—24no—
177C24×C3—91no—
178C23×D3—31no—
179C22×C3×C4—21no—
180C22×C3×C4—31no—
181C22×C3×C4—31no—
182C22×C3×C4—91no—
183C22×C3×C4C2×D911yescharacteristic
184C22×C3×C4D3211yescharacteristic
185C22×Dic3—31no—
186C22×Dic3—91no—
187C22×Dic3—91no—
188C22×Dic3—98no—
189C22×Dic3—94no—
190C22×Dic3—91no—
191C22×Dic3—91no—
192C22×Dic3—181no—
193C22×Dic3—181no—
194C22×Dic3C2×D911yescharacteristic
195C22×Dic3C2×D911yescharacteristic
196C2×C4×D3—94no—
197C2×C4×D3—98no—
198C3×C4⋊C4—34no—
199C3×C22⋊C4—94no—
200#217—94no—
201(C2×D3)⋊C4—34no—
202(C2×D3)⋊C4—94no—
203C4⋊Dic3—94no—
204Dic3⋊C4—94no—
205Dic3⋊C4C2×D914yescomplemented
206C4×Dic3—94no—
207C4×Dic3—94no—
208C4×Dic3—94no—
209C4×Dic3—94no—
210C4×Dic3—184no—
211C22×C32C2×C4×D312yes—
212C22×C32#23712yes—
213C22×C32#23912yes—
214C22×C32C23×D311yescharacteristic
215C2×C3×D3—34no—
216C2×C3×D3—68no—
217C2×C3×D3—62no—
218C2×C3×D3—62no—
219C32×C4—22no—
220C3⋊Dic3—34no—
221C3⋊Dic3—62no—
222C3×Dic3—22no—
223C3×Dic3—62no—
224C3×Dic3C2×C4×D314yes—
225C22×C9—21no—
226C22×C9—22no—
227C22×C9—22no—
228C22×C9—22no—
229C22×C9C2×C4×D312yes—
230C22×C9#23712yes—
231C22×C9#23712yes—
232C22×C9C23×D311yescharacteristic
233C2×D9—28no—
234C2×D9—22no—
235C2×D9—22no—
236C2×D9C2×C4×D314yes—
237C4×C9—24no—
238C4×C9—22no—
239C4×C9—34no—
240C4×C9—62no—
241Dic9—22no—
242Dic9—34no—
243Dic9—64no—
244Dic9—64no—
245Dic9—62no—
246C23×C4—271no—
247#117—278no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—271no—
250C2×C22⋊C4—91no—
251C2×C22⋊C4—271no—
252C2×C42—271no—
253C2×C42—271no—
254C3×C9C2×#11711yescharacteristic, complemented, Sylow
255C23×C3—21no—
256C23×C3—94no—
257C23×C3—92no—
258C23×C3—182no—
259C23×C3—182no—
260C23×C3C22×D911yescharacteristic
261C23×C3C2×D3211yescharacteristic
262C22×D3—32no—
263C22×D3—34no—
264C22×D3—62no—
265C22×D3—62no—
266C2×C3×C4—24no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—32no—
271C2×C3×C4—34no—
272C2×C3×C4—32no—
273C2×C3×C4—62no—
274C2×C3×C4—92no—
275C2×C3×C4—182no—
276C2×C3×C4C22×D912yes—
277C2×C3×C4C2×D3212yes—
278C2×Dic3—22no—
279C2×Dic3—32no—
280C2×Dic3—62no—
281C2×Dic3—92no—
282C2×Dic3—94no—
283C2×Dic3—92no—
284C2×Dic3—94no—
285C2×Dic3—916no—
286C2×Dic3—92no—
287C2×Dic3—92no—
288C2×Dic3—182no—
289C2×Dic3—182no—
290C2×Dic3—184no—
291C2×Dic3—184no—
292C2×Dic3—182no—
293C2×Dic3—182no—
294C2×Dic3C4×D914yes—
295C2×Dic3C22×D912yes—
296C2×Dic3C22×D912yes—
297C4×D3—916no—
298C2×C32#89214yes—
299C2×C32C22×C4×D311yescharacteristic
300C2×C32C2×#23711yescharacteristic
301C2×C32C2×#23911yescharacteristic
302C3×D3—64no—
303C2×C9—21no—
304C2×C9—21no—
305C2×C9—21no—
306C2×C9—24no—
307C2×C9#87714yescomplemented
308C2×C9C22×C4×D311yescharacteristic
309C2×C9C2×#23711yescharacteristic
310C2×C9C2×#23711yescharacteristic
311D9—24no—
312C24—91no—
313C22×C4—31no—
314C22×C4—31no—
315C22×C4—91no—
316C22×C4—274no—
317C22×C4—278no—
318C22×C4—271no—
319C22×C4—271no—
320C22×C4D3×D911yescharacteristic
321C4⋊C4—34no—
322C4⋊C4—274no—
323C22⋊C4—94no—
324C22⋊C4—274no—
325C42—274no—
326C42—274no—
327C22×C3—22no—
328C22×C3—21no—
329C22×C3—22no—
330C22×C3—22no—
331C22×C3—94no—
332C22×C3—182no—
333C22×C3—188no—
334C22×C3—182no—
335C22×C3C2×C4×D912yes—
336C22×C3#369912yes—
337C22×C3#370112yes—
338C22×C3C23×D911yescharacteristic
339C22×C3#379612yes—
340C22×C3C4×D3212yes—
341C22×C3#380512yes—
342C22×C3C22×D3211yescharacteristic
343C2×D3—34no—
344C2×D3—62no—
345C2×D3—62no—
346C2×D3—68no—
347C3×C4—24no—
348C3×C4—22no—
349C3×C4—22no—
350C3×C4—34no—
351C3×C4—62no—
352C3×C4—182no—
353Dic3—22no—
354Dic3—62no—
355Dic3—94no—
356Dic3—94no—
357Dic3—184no—
358Dic3—184no—
359Dic3—182no—
360Dic3—182no—
361Dic3C2×C4×D914yes—
362C32C2×#89211yescharacteristic
363C9—21no—
364C9C2×#87711yescharacteristic, complemented
365C23—92no—
366C23—94no—
367C23—182no—
368C23—182no—
369C23C2×D3×D911yescharacteristic
370C2×C4—22no—
371C2×C4—32no—
372C2×C4—34no—
373C2×C4—32no—
374C2×C4—62no—
375C2×C4—92no—
376C2×C4—182no—
377C2×C4—2716no—
378C2×C4—272no—
379C2×C4—274no—
380C2×C4—272no—
381C2×C4—542no—
382C2×C4C2×D3×D912yes—
383C2×C3—24no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—184no—
388C2×C3#6345414yescomplemented
389C2×C3C22×C4×D911yescharacteristic
390C2×C3C2×#369911yescharacteristic
391C2×C3C2×#370111yescharacteristic
392C2×C3#6387814yes—
393C2×C3C2×#379611yescharacteristic
394C2×C3C2×C4×D3211yescharacteristic
395C2×C3C2×#380511yescharacteristic
396D3—64no—
397C22—94no—
398C22—188no—
399C22—182no—
400C22—182no—
401C22#8863412yes—
402C22C4×D3×D912yes—
403C22#8866312yes—
404C22C22×D3×D911yescharacteristic
405C4—22no—
406C4—34no—
407C4—62no—
408C4—182no—
409C4—274no—
410C4—542no—
411C3—21no—
412C3C2×#6345411yescharacteristic, complemented
413C3C2×#6387811yescharacteristic
414C2—184no—
415C2#1172097514yesdirect product factor
416C2C2×#8863411yescharacteristic
417C2C2×C4×D3×D911yescharacteristic
418C2C2×#8866311yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC23×C3×D9C22
cosocleC22×C3×C9C24
hypercenterC22×C4D3×D9
hypocenterC3×C9C2×#117
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C3⋊Dic9C2semidirect
4. C2×C9×Dic3⋊C4C2semidirect
6. C2×Dic3⋊Dic9C2semidirect
7. C2×Dic3×Dic9C2semidirect
8. #11720975C2direct (N × H)
18. C2×Dic3×D9C4semidirect
20. C4×C3⋊Dic9C22semidirect
22. C9×Dic3⋊C4C22semidirect
24. Dic3⋊Dic9C22semidirect
25. Dic3×Dic9C22semidirect
60. Dic3×D9C2×C4semidirect
120. C2×Dic3⋊C4D9semidirect
172. C2×C3×C9#117semidirect
205. Dic3⋊C4C2×D9semidirect
254. C3×C9C2×#117semidirect
307. C2×C9#877semidirect
364. C9C2×#877semidirect
388. C2×C3#63454semidirect
412. C3C2×#63454semidirect
415. C2#11720975direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Dic3×D92
C2×C4×C3⋊Dic92
C2×C3×(C2×D9)⋊C42
C2×C9×Dic3⋊C42
C2×(C2×D9)⋊Dic32
C2×Dic3⋊Dic92
C2×Dic3×Dic92
#117209752
C2×#638783
C2×#634543