Finite Group Databasebeta

← C2×#11721248 #13151959

Subgroups — C2×#11721248 #13151959

Order 1728.

1226 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3⋊Dic9C211yesmaximal, characteristic
2C2×C9×#217C211yesmaximal, characteristic
3C2×C3×#3676C211yesmaximal, characteristic
4C2×#88442C212yesmaximal, complemented
5#11721248C218yesmaximal, direct product factor
6C2×Dic3×Dic9C212yesmaximal, complemented
7C2×#63957—31nomaximal
8C2×#63439—31nomaximal
9C24×C3×C9C2211yescharacteristic
10C22×C3⋊Dic9C418yescomplemented
11C22×C3⋊Dic9C2211yescharacteristic
12C22×C3⋊Dic9C2211yescharacteristic
13C22×C3⋊Dic9C2214yes—
14C22×C9×Dic3C2212yes—
15C22×C3×Dic9C2212yes—
16C9×#217C2214yes—
17C3×#3676C2214yes—
18#88442C2218yescomplemented
19Dic3×Dic9C2218yescomplemented
20C23×C3⋊Dic3—31no—
21C2×C3×#217—31no—
22C2×C3×#217D311yescharacteristic
23C2×#3724—32no—
24#63957—38no—
25C2×Dic32—32no—
26C23×Dic9—31no—
27C23×Dic9—61no—
28C2×C9×C22⋊C4—31no—
29C2×#3676D311yescharacteristic
30C2×Dic9⋊C4—32no—
31C2×C4×Dic9—32no—
32#63439—38no—
33C23×C3×C9—24no—
34C23×C3×C9C2×C414yes—
35C23×C3×C9C2312yes—
36C23×C3×C9C2311yescharacteristic
37C2×C3⋊Dic9C2×C414yes—
38C2×C3⋊Dic9C2×C414yes—
39C2×C3⋊Dic9C2×C4116yescomplemented
40C2×C3⋊Dic9C2312yes—
41C2×C3⋊Dic9C2312yes—
42C2×C9×Dic3—24no—
43C2×C9×Dic3C2314yes—
44C2×C3×Dic9—24no—
45C2×C3×Dic9C2314yes—
46C2×#877—91no—
47C2×#877—91no—
48C24×C32C2×D311yescharacteristic
49C22×C3⋊Dic3—38no—
50C22×C3⋊Dic3—31no—
51C22×C3⋊Dic3—34no—
52C22×C3⋊Dic3—31no—
53C22×C3×Dic3—32no—
54C22×C3×Dic3C2×D312yes—
55C24×C9—21no—
56C24×C9C2×D311yescharacteristic
57C3×#217—34no—
58C3×#217C2×D314yes—
59#3724—38no—
60Dic32—38no—
61C22×C4×C9—32no—
62C22×Dic9—38no—
63C22×Dic9—34no—
64C22×Dic9—31no—
65C22×Dic9—31no—
66C22×Dic9—68no—
67C22×Dic9—61no—
68C22×Dic9—61no—
69C22×Dic9—64no—
70C22×Dic9C2×D312yes—
71C9×C22⋊C4—34no—
72#3676C2×D314yes—
73Dic9⋊C4—38no—
74C4×Dic9—38no—
75C22×C3×C9—28no—
76C22×C3×C9—24no—
77C22×C3×C9C22×C414yes—
78C22×C3×C9C22×C412yes—
79C22×C3×C9#4114yes—
80C22×C3×C9C2411yescharacteristic
81C3⋊Dic9C22×C414yes—
82C3⋊Dic9C22×C414yes—
83C9×Dic3—24no—
84C3×Dic9—24no—
85C23×Dic3—91no—
86C23×Dic3—91no—
87C23×Dic3—181no—
88C2×C3×C22⋊C4—31no—
89C2×C3×C22⋊C4—91no—
90C2×#217—31no—
91C2×#217D911yescharacteristic
92C2×Dic3⋊C4—92no—
93C2×Dic3⋊C4—92no—
94C2×C4×Dic3—92no—
95C2×C4×Dic3—92no—
96#877—98no—
97#877—98no—
98C23×C32—24no—
99C23×C32C4×D314yes—
100C23×C32C22×D312yes—
101C23×C32C22×D311yescharacteristic
102C2×C3⋊Dic3—34no—
103C2×C3⋊Dic3—34no—
104C2×C3⋊Dic3—316no—
105C2×C3⋊Dic3—32no—
106C2×C3⋊Dic3—32no—
107C2×C3×Dic3—24no—
108C2×C3×Dic3—34no—
109C2×C3×Dic3—64no—
110C2×C3×Dic3C22×D314yes—
111C23×C9—28no—
112C23×C9—24no—
113C23×C9—24no—
114C23×C9—22no—
115C23×C9—21no—
116C23×C9C4×D314yes—
117C23×C9C22×D312yes—
118C23×C9C22×D311yescharacteristic
119C2×C4×C9—34no—
120C2×C4×C9—64no—
121C2×Dic9—24no—
122C2×Dic9—34no—
123C2×Dic9—34no—
124C2×Dic9—316no—
125C2×Dic9—32no—
126C2×Dic9—32no—
127C2×Dic9—64no—
128C2×Dic9—64no—
129C2×Dic9—616no—
130C2×Dic9—62no—
131C2×Dic9—62no—
132C2×Dic9C22×D314yes—
133C2×#117—271noSylow
134C2×C3×C9—24no—
135C2×C3×C9#11714yescomplemented
136C2×C3×C9C23×C411yescharacteristic
137C2×C3×C9C2×#4112yes—
138C24×C3—21no—
139C24×C3C2×D911yescharacteristic
140C24×C3D3211yescharacteristic
141C22×C3×C4—32no—
142C22×C3×C4—92no—
143C22×Dic3—32no—
144C22×Dic3—98no—
145C22×Dic3—94no—
146C22×Dic3—91no—
147C22×Dic3—91no—
148C22×Dic3—98no—
149C22×Dic3—94no—
150C22×Dic3—91no—
151C22×Dic3—91no—
152C22×Dic3—188no—
153C22×Dic3—184no—
154C22×Dic3—181no—
155C22×Dic3—181no—
156C22×Dic3C2×D912yes—
157C3×C22⋊C4—34no—
158C3×C22⋊C4—94no—
159#217—34no—
160#217C2×D914yes—
161Dic3⋊C4—98no—
162Dic3⋊C4—98no—
163C4×Dic3—98no—
164C4×Dic3—98no—
165C22×C32—28no—
166C22×C32—24no—
167C22×C32C2×C4×D314yes—
168C22×C32C2×C4×D312yes—
169C22×C32#23714yes—
170C22×C32C23×D311yescharacteristic
171C3⋊Dic3—34no—
172C3⋊Dic3—34no—
173C3×Dic3—24no—
174C3×Dic3—64no—
175C22×C9—28no—
176C22×C9—216no—
177C22×C9—28no—
178C22×C9—24no—
179C22×C9—21no—
180C22×C9—24no—
181C22×C9—24no—
182C22×C9—22no—
183C22×C9C2×C4×D314yes—
184C22×C9C2×C4×D312yes—
185C22×C9#23714yes—
186C22×C9C23×D311yescharacteristic
187C4×C9—64no—
188Dic9—24no—
189Dic9—34no—
190Dic9—34no—
191Dic9—64no—
192Dic9—64no—
193C23×C4—271no—
194#117—278no—
195C2×C4⋊C4—272no—
196C2×C22⋊C4—31no—
197C2×C22⋊C4—91no—
198C2×C42—272no—
199C3×C9C2×#11711yescharacteristic, complemented, Sylow
200C23×C3—28no—
201C23×C3—24no—
202C23×C3—24no—
203C23×C3—24no—
204C23×C3—22no—
205C23×C3—21no—
206C23×C3C4×D914yes—
207C23×C3C22×D912yes—
208C23×C3C22×D911yescharacteristic
209C23×C3#62414yes—
210C23×C3C2×D3212yes—
211C23×C3C2×D3211yescharacteristic
212C2×C3×C4—34no—
213C2×C3×C4—64no—
214C2×C3×C4—94no—
215C2×C3×C4—184no—
216C2×Dic3—24no—
217C2×Dic3—34no—
218C2×Dic3—64no—
219C2×Dic3—92no—
220C2×Dic3—92no—
221C2×Dic3—916no—
222C2×Dic3—94no—
223C2×Dic3—94no—
224C2×Dic3—92no—
225C2×Dic3—92no—
226C2×Dic3—916no—
227C2×Dic3—94no—
228C2×Dic3—94no—
229C2×Dic3—182no—
230C2×Dic3—182no—
231C2×Dic3—184no—
232C2×Dic3—184no—
233C2×Dic3—1816no—
234C2×Dic3C22×D914yes—
235C2×C32—24no—
236C2×C32#87714yes—
237C2×C32C22×C4×D311yescharacteristic
238C2×C32C2×#23712yes—
239C2×C9—28no—
240C2×C9—24no—
241C2×C9—22no—
242C2×C9—21no—
243C2×C9—24no—
244C2×C9#87714yescomplemented
245C2×C9C22×C4×D311yescharacteristic
246C2×C9C2×#23712yes—
247C24D3×D911yescharacteristic
248C22×C4—32no—
249C22×C4—92no—
250C22×C4—278no—
251C22×C4—274no—
252C22×C4—271no—
253C22×C4—271no—
254C4⋊C4—278no—
255C22⋊C4—34no—
256C22⋊C4—94no—
257C42—278no—
258C22×C3—216no—
259C22×C3—28no—
260C22×C3—24no—
261C22×C3—28no—
262C22×C3—28no—
263C22×C3—24no—
264C22×C3—21no—
265C22×C3—24no—
266C22×C3—22no—
267C22×C3—24no—
268C22×C3C2×C4×D914yes—
269C22×C3C2×C4×D912yes—
270C22×C3#369914yes—
271C22×C3C23×D911yescharacteristic
272C22×C3#380514yes—
273C22×C3C2×#62414yes—
274C22×C3C2×#62412yes—
275C22×C3C22×D3211yescharacteristic
276C3×C4—64no—
277C3×C4—184no—
278Dic3—24no—
279Dic3—64no—
280Dic3—94no—
281Dic3—94no—
282Dic3—94no—
283Dic3—94no—
284Dic3—184no—
285Dic3—184no—
286C32C2×#87711yescharacteristic
287C9—21no—
288C9C2×#87711yescharacteristic, complemented
289C23—24no—
290C23(C3⋊D9)⋊C414yes—
291C23C2×D3×D912yes—
292C23C2×D3×D911yescharacteristic
293C2×C4—34no—
294C2×C4—64no—
295C2×C4—94no—
296C2×C4—184no—
297C2×C4—272no—
298C2×C4—272no—
299C2×C4—2716no—
300C2×C4—274no—
301C2×C4—274no—
302C2×C3—28no—
303C2×C3—24no—
304C2×C3—24no—
305C2×C3—21no—
306C2×C3—24no—
307C2×C3—22no—
308C2×C3#6343914yescomplemented
309C2×C3C22×C4×D911yescharacteristic
310C2×C3C2×#369912yes—
311C2×C3#6395714yes—
312C2×C3C2×#380512yes—
313C2×C3C22×#62411yescharacteristic
314C22—28no—
315C22—24no—
316C22C2×(C3⋊D9)⋊C414yes—
317C22C2×(C3⋊D9)⋊C412yes—
318C22#8866314yes—
319C22C22×D3×D911yescharacteristic
320C4—64no—
321C4—184no—
322C4—274no—
323C4—274no—
324C3—21no—
325C3C2×#6343911yescharacteristic, complemented
326C3C2×#6395711yescharacteristic
327C2—24no—
328C2#1172124814yesdirect product factor
329C2C22×(C3⋊D9)⋊C411yescharacteristic
330C2C2×#8866312yes—
subgroupG/H
centerC23C2×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC24×C3×C9C22
socleC23×C32C22×D3
prime radicalC24×C32C2×D3
cosocleC22×C3×C9C24
hypercenterC24D3×D9
hypocenterC3×C9C2×#117
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
4. C2×#88442C2semidirect
5. #11721248C2direct (N × H)
6. C2×Dic3×Dic9C2semidirect
10. C22×C3⋊Dic9C4semidirect
18. #88442C22semidirect
19. Dic3×Dic9C22semidirect
39. C2×C3⋊Dic9C2×C4semidirect
135. C2×C3×C9#117semidirect
199. C3×C9C2×#117semidirect
244. C2×C9#877semidirect
288. C9C2×#877semidirect
308. C2×C3#63439semidirect
325. C3C2×#63439semidirect
328. C2#11721248direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3⋊Dic92
C2×C9×#2172
C2×C3×#36762
C2×#884422
#117212482
C2×Dic3×Dic92
C2×#639573
C2×#634393