Finite Group Databasebeta

← C2×#11723715 #13178782

Subgroups — C2×#11723715 #13178782

Order 1728.

1954 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C3⋊Dic3C211yesmaximal, characteristic
2C2×C3×#3757C212yesmaximal
3C2×#88815C212yesmaximal, complemented
4#11723715C218yesmaximal, direct product factor
5C2×#88813C212yesmaximal, complemented
6C2×#63957—31nomaximal
7C2×#63955—32nomaximal
8C24×C33C2211yescharacteristic
9C22×C3×C3⋊Dic3C418yescomplemented
10C22×C3×C3⋊Dic3C2214yes—
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C22×C3×C3⋊Dic3C2211yescharacteristic
13C22×C3×C3⋊Dic3C2214yes—
14C3×#3757C2218yes—
15#88815C2218yescomplemented
16#88813C2218yescomplemented
17C23×C3⋊Dic3D311yescharacteristic
18C23×C3×Dic3—32no—
19C23×C3×Dic3—61no—
20C2×#3757D312yes—
21C2×C3×#217—32no—
22C2×C3×#217—32no—
23C2×C3×#217—62no—
24C2×#3724—32no—
25C2×Dic3⋊Dic3—34no—
26#63957—38no—
27#63955—316no—
28C2×Dic32—34no—
29C2×Dic32—32no—
30C23×C33—24no—
31C23×C33C2×C414yes—
32C23×C33C2312yes—
33C23×C33C2311yescharacteristic
34C2×C3×C3⋊Dic3—28no—
35C2×C3×C3⋊Dic3C2×C414yes—
36C2×C3×C3⋊Dic3C2×C414yes—
37C2×C3×C3⋊Dic3C2×C4116yescomplemented
38C2×C3×C3⋊Dic3C2318yes—
39C2×C3×C3⋊Dic3C2312yes—
40C2×C3×C3⋊Dic3C2312yes—
41C2×#926—91no—
42C2×#877—92no—
43C24×C32—21no—
44C24×C32—22no—
45C24×C32—41no—
46C24×C32C2×D312yes—
47C24×C32C2×D311yescharacteristic
48C22×C3⋊Dic3Dic318yescomplemented
49C22×C3⋊Dic3C2×D314yes—
50C22×C3⋊Dic3C2×D311yescharacteristic
51C22×C3⋊Dic3C2×D311yescharacteristic
52C22×C3⋊Dic3C2×D314yes—
53C22×C3×Dic3—316no—
54C22×C3×Dic3—32no—
55C22×C3×Dic3—38no—
56C22×C3×Dic3—32no—
57C22×C3×Dic3—34no—
58C22×C3×Dic3—34no—
59C22×C3×Dic3—68no—
60C22×C3×Dic3—61no—
61C22×C3×Dic3—61no—
62C22×C3×Dic3—64no—
63C22×C3×Dic3—64no—
64#3757C2×D318yes—
65C3×#217—38no—
66C3×#217—38no—
67C3×#217—68no—
68#3724—38no—
69Dic3⋊Dic3—316no—
70Dic32—316no—
71Dic32—38no—
72C22×C33—28no—
73C22×C33—24no—
74C22×C33C22×C414yes—
75C22×C33C22×C412yes—
76C22×C33#4114yes—
77C22×C33C2411yescharacteristic
78C3×C3⋊Dic3—28no—
79C3×C3⋊Dic3C22×C414yes—
80C3×C3⋊Dic3C22×C414yes—
81C23×C3×C4—91no—
82C23×Dic3—32no—
83C23×Dic3—61no—
84C2×C3×C22⋊C4—92no—
85C2×#217—32no—
86C2×#217—32no—
87C2×#217—62no—
88#926—98no—
89C2×C4⋊Dic3—92no—
90C2×Dic3⋊C4—94no—
91C2×C4×Dic3—94no—
92C2×C4×Dic3—92no—
93#877—916no—
94C23×C32—24no—
95C23×C32—28no—
96C23×C32—28no—
97C23×C32—28no—
98C23×C32—24no—
99C23×C32—22no—
100C23×C32—22no—
101C23×C32—21no—
102C23×C32—24no—
103C23×C32—44no—
104C23×C32—42no—
105C23×C32—41no—
106C23×C32—48no—
107C23×C32—48no—
108C23×C32C4×D318yes—
109C23×C32C2×Dic314yes—
110C23×C32C22×D314yes—
111C23×C32C22×D312yes—
112C23×C32C22×D312yes—
113C23×C32C22×D311yescharacteristic
114C2×C3⋊Dic3—28no—
115C2×C3⋊Dic3C2×Dic314yes—
116C2×C3⋊Dic3C2×Dic314yes—
117C2×C3⋊Dic3C2×Dic3116yescomplemented
118C2×C3⋊Dic3C22×D318yes—
119C2×C3⋊Dic3C22×D312yes—
120C2×C3⋊Dic3C22×D312yes—
121C2×C3×Dic3—38no—
122C2×C3×Dic3—38no—
123C2×C3×Dic3—332no—
124C2×C3×Dic3—34no—
125C2×C3×Dic3—34no—
126C2×C3×Dic3—38no—
127C2×C3×Dic3—38no—
128C2×C3×Dic3—64no—
129C2×C3×Dic3—64no—
130C2×C3×Dic3—616no—
131C2×C3×Dic3—62no—
132C2×C3×Dic3—62no—
133C2×C3×Dic3—68no—
134C2×C3×Dic3—68no—
135C2×C3×Dic3—68no—
136C2×C3×Dic3—128no—
137C2×#117—271noSylow
138C2×C33—24no—
139C2×C33#11714yescomplemented
140C2×C33C23×C411yescharacteristic
141C2×C33C2×#4112yes—
142C24×C3—21no—
143C24×C3—22no—
144C24×C3—41no—
145C24×C3D3212yes—
146C24×C3D3211yescharacteristic
147C22×C3×C4—98no—
148C22×C3×C4—91no—
149C22×C3×C4—91no—
150C22×C3×C4—94no—
151C22×C3×C4—94no—
152C22×Dic3—316no—
153C22×Dic3—38no—
154C22×Dic3—32no—
155C22×Dic3—32no—
156C22×Dic3—34no—
157C22×Dic3—34no—
158C22×Dic3—68no—
159C22×Dic3—64no—
160C22×Dic3—61no—
161C22×Dic3—61no—
162C22×Dic3—64no—
163C3×C22⋊C4—98no—
164#217—38no—
165#217—38no—
166#217—68no—
167C4⋊Dic3—98no—
168Dic3⋊C4—916no—
169C4×Dic3—916no—
170C4×Dic3—98no—
171C22×C32—28no—
172C22×C32—24no—
173C22×C32—28no—
174C22×C32—216no—
175C22×C32—216no—
176C22×C32—28no—
177C22×C32—22no—
178C22×C32—21no—
179C22×C32—24no—
180C22×C32—28no—
181C22×C32—28no—
182C22×C32—24no—
183C22×C32—24no—
184C22×C32—22no—
185C22×C32—42no—
186C22×C32—44no—
187C22×C32—44no—
188C22×C32—41no—
189C22×C32—48no—
190C22×C32—416no—
191C22×C32—416no—
192C22×C32—48no—
193C22×C32C2×C4×D318yes—
194C22×C32C2×C4×D314yes—
195C22×C32#23514yes—
196C22×C32#23718yes—
197C22×C32C22×Dic314yes—
198C22×C32C22×Dic312yes—
199C22×C32C23×D312yes—
200C22×C32C23×D311yescharacteristic
201C3⋊Dic3—28no—
202C3⋊Dic3C22×Dic314yes—
203C3⋊Dic3C22×Dic314yes—
204C3×Dic3—38no—
205C3×Dic3—38no—
206C3×Dic3—64no—
207C3×Dic3—64no—
208C3×Dic3—68no—
209C3×Dic3—68no—
210C3×Dic3—128no—
211C23×C4—91no—
212#117—278no—
213C2×C4⋊C4—272no—
214C2×C22⋊C4—92no—
215C2×C42—272no—
216C33C2×#11711yescharacteristic, complemented, Sylow
217C23×C3—28no—
218C23×C3—28no—
219C23×C3—24no—
220C23×C3—24no—
221C23×C3—28no—
222C23×C3—22no—
223C23×C3—24no—
224C23×C3—21no—
225C23×C3—22no—
226C23×C3—48no—
227C23×C3—48no—
228C23×C3—44no—
229C23×C3—42no—
230C23×C3—41no—
231C23×C3D3×Dic318yes—
232C23×C3#62414yes—
233C23×C3C2×D3214yes—
234C23×C3C2×D3212yes—
235C23×C3C2×D3212yes—
236C23×C3C2×D3211yescharacteristic
237C2×C3×C4—92no—
238C2×C3×C4—92no—
239C2×C3×C4—916no—
240C2×C3×C4—94no—
241C2×C3×C4—94no—
242C2×C3×C4—98no—
243C2×C3×C4—188no—
244C2×Dic3—34no—
245C2×Dic3—34no—
246C2×Dic3—38no—
247C2×Dic3—38no—
248C2×Dic3—332no—
249C2×Dic3—38no—
250C2×Dic3—38no—
251C2×Dic3—64no—
252C2×Dic3—64no—
253C2×Dic3—616no—
254C2×Dic3—62no—
255C2×Dic3—62no—
256C2×Dic3—68no—
257C2×Dic3—68no—
258C2×Dic3—68no—
259C2×Dic3—128no—
260C2×C32—24no—
261C2×C32—28no—
262C2×C32—28no—
263C2×C32—22no—
264C2×C32—24no—
265C2×C32—22no—
266C2×C32—21no—
267C2×C32—28no—
268C2×C32—24no—
269C2×C32—44no—
270C2×C32—41no—
271C2×C32—42no—
272C2×C32—48no—
273C2×C32—48no—
274C2×C32#87718yescomplemented
275C2×C32#92614yescomplemented
276C2×C32C22×C4×D312yes—
277C2×C32C2×#23512yes—
278C2×C32C2×#23714yes—
279C2×C32C23×Dic311yescharacteristic
280C24#111311yescharacteristic
281C22×C4—98no—
282C22×C4—94no—
283C22×C4—91no—
284C22×C4—91no—
285C22×C4—94no—
286C4⋊C4—278no—
287C22⋊C4—98no—
288C42—278no—
289C22×C3—28no—
290C22×C3—216no—
291C22×C3—28no—
292C22×C3—24no—
293C22×C3—216no—
294C22×C3—28no—
295C22×C3—21no—
296C22×C3—22no—
297C22×C3—24no—
298C22×C3—22no—
299C22×C3—24no—
300C22×C3—28no—
301C22×C3—28no—
302C22×C3—24no—
303C22×C3—48no—
304C22×C3—416no—
305C22×C3—48no—
306C22×C3—416no—
307C22×C3—41no—
308C22×C3—44no—
309C22×C3—42no—
310C22×C3—44no—
311C22×C3C2×D3×Dic318yes—
312C22×C3C2×D3×Dic314yes—
313C22×C3#380418yes—
314C22×C3#380514yes—
315C22×C3C2×#62414yes—
316C22×C3C2×#62412yes—
317C22×C3C22×D3212yes—
318C22×C3C22×D3211yescharacteristic
319C3×C4—94no—
320C3×C4—94no—
321C3×C4—188no—
322Dic3—38no—
323Dic3—38no—
324Dic3—64no—
325Dic3—64no—
326Dic3—68no—
327Dic3—68no—
328Dic3—128no—
329C32—22no—
330C32—21no—
331C32—41no—
332C32C2×#87712yescomplemented
333C32C2×#92611yescharacteristic, complemented
334C23—24no—
335C23#629414yes—
336C23C2×#111312yes—
337C23C2×#111311yescharacteristic
338C2×C4—92no—
339C2×C4—92no—
340C2×C4—916no—
341C2×C4—94no—
342C2×C4—94no—
343C2×C4—98no—
344C2×C4—188no—
345C2×C3—28no—
346C2×C3—28no—
347C2×C3—24no—
348C2×C3—21no—
349C2×C3—22no—
350C2×C3—24no—
351C2×C3—22no—
352C2×C3—24no—
353C2×C3—28no—
354C2×C3—48no—
355C2×C3—48no—
356C2×C3—41no—
357C2×C3—42no—
358C2×C3—44no—
359C2×C3#6395518yescomplemented
360C2×C3#6395714yescomplemented
361C2×C3C22×D3×Dic312yes—
362C2×C3C2×#380414yes—
363C2×C3C2×#380512yes—
364C2×C3C22×#62411yescharacteristic
365C22—24no—
366C22—28no—
367C22C2×#629414yes—
368C22C2×#629412yes—
369C22#8904614yes—
370C22C22×#111311yescharacteristic
371C4—94no—
372C4—94no—
373C4—188no—
374C3—21no—
375C3—22no—
376C3—41no—
377C3C2×#6395512yescomplemented
378C3C2×#6395711yescharacteristic, complemented
379C2—24no—
380C2#1172371514yesdirect product factor
381C2C22×#629411yescharacteristic
382C2C2×#8904612yes—
subgroupG/H
centerC23C2×#1113
commutatorC2×C33C23×C4
FrattiniC22C22×#1113
Fitting, prime radicalC24×C33C22
socleC23×C33C23
cosocleC22×C33C24
hypercenterC24#1113
hypocenterC33C2×#117
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
3. C2×#88815C2semidirect
4. #11723715C2direct (N × H)
5. C2×#88813C2semidirect
9. C22×C3×C3⋊Dic3C4semidirect
15. #88815C22semidirect
16. #88813C22semidirect
37. C2×C3×C3⋊Dic3C2×C4semidirect
48. C22×C3⋊Dic3Dic3semidirect
117. C2×C3⋊Dic3C2×Dic3semidirect
139. C2×C33#117semidirect
216. C33C2×#117semidirect
274. C2×C32#877semidirect
275. C2×C32#926semidirect
332. C32C2×#877semidirect
333. C32C2×#926semidirect
359. C2×C3#63955semidirect
360. C2×C3#63957semidirect
377. C3C2×#63955semidirect
378. C3C2×#63957semidirect
380. C2#11723715direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C3⋊Dic32
C2×C3×#37572
C2×#888152
#117237152
C2×#888132
C2×#639573
C2×#639553