Finite Group Databasebeta

← C2×#11723106 #13173406

Subgroups — C2×#11723106 #13173406

Order 1728.

1240 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3⋊#157C212yesmaximal
2C22×C3×C4×C3⋊D3C211yesmaximal, characteristic
3#11723106C214yesmaximal, direct product factor
4C2×#63781—31nomaximal
5C23×C3×C3⋊D3C2211yescharacteristic
6C22×C3⋊#157—24no—
7C22×C3⋊#157—24no—
8C22×C3⋊#157C418yescomplemented
9C22×C3⋊#157C2214yes—
10C22×C33×C4C411yescharacteristic, complemented
11C22×C3×C3⋊Dic3C411yescharacteristic, complemented
12C2×C3×C4×C3⋊D3—24no—
13C2×C3×C4×C3⋊D3C412yescomplemented
14C2×C3×C4×C3⋊D3C2212yes—
15C23×#157—32no—
16C22×C4×C3⋊D3D311yescharacteristic
17C22×C3×C4×D3—62no—
18#63781—34no—
19C23×C33C2×C411yescharacteristic
20C22×C3×C3⋊D3C2×C411yescharacteristic
21C22×C3×C3⋊D3C2×C414yes—
22C22×C3×C3⋊D3D412yes—
23C22×C3×C3⋊D3D414yes—
24C22×C3×C3⋊D3Q812yes—
25C22×C3×C3⋊D3C2311yescharacteristic
26C2×C3⋊#157—216no—
27C2×C3⋊#157—24no—
28C2×C3⋊#157—24no—
29C2×C3⋊#157C2×C418yescomplemented
30C2×C33×C4—22no—
31C2×C33×C4C2×C412yescomplemented
32C2×C3×C3⋊Dic3—22no—
33C2×C3×C3⋊Dic3C2×C412yescomplemented
34C3×C4×C3⋊D3—24no—
35C3×C4×C3⋊D3—24no—
36C2×C3⋊#95—91nomaximal
37C23×C3⋊D3C2×D311yescharacteristic
38C23×C3×D3—62no—
39C22×#157—38no—
40C22×#157—34no—
41C22×#157—64no—
42C22×#157—64no—
43C22×C32×C4—22no—
44C22×C32×C4—42no—
45C22×C32×C4Dic311yescharacteristic, complemented
46C22×C3⋊Dic3Dic311yescharacteristic, complemented
47C2×C4×C3⋊D3—24no—
48C2×C4×C3⋊D3Dic312yescomplemented
49C2×C4×C3⋊D3C2×D312yes—
50C22×C3×Dic3—62no—
51C2×C3×C4×D3—64no—
52C2×C3×C4×D3—64no—
53C2×C3×C4×D3—616no—
54C22×C33C4212yes—
55C22×C33C22⋊C412yes—
56C22×C33C4⋊C412yes—
57C22×C33C22×C411yescharacteristic
58C2×C3×C3⋊D3C4212yes—
59C2×C3×C3⋊D3C22⋊C412yes—
60C2×C3×C3⋊D3C22⋊C418yes—
61C2×C3×C3⋊D3C4⋊C412yes—
62C2×C3×C3⋊D3C4⋊C418yes—
63C2×C3×C3⋊D3C22×C412yes—
64C2×C3×C3⋊D3C2×D411yescharacteristic
65C2×C3×C3⋊D3C2×D412yes—
66C2×C3×C3⋊D3C2×Q811yescharacteristic
67C3⋊#157—28no—
68C33×C4—22no—
69C3×C3⋊Dic3—22no—
70C23×C3×C4—91no—
71C23×Dic3—92no—
72C22×C4×D3—62no—
73C3⋊#95—94no—
74C23×C32—22no—
75C23×C32—42no—
76C23×C32C2×Dic311yescharacteristic
77C22×C3⋊D3Dic612yes—
78C22×C3⋊D3C4×D314yes—
79C22×C3⋊D3D1212yes—
80C22×C3⋊D3C2×Dic311yescharacteristic
81C22×C3⋊D3#6714yes—
82C22×C3⋊D3C22×D311yescharacteristic
83C22×C3×D3—68no—
84C22×C3×D3—62no—
85C22×C3×D3—64no—
86C22×C3×D3—68no—
87C22×C3×D3—64no—
88C22×C3×D3—62no—
89C2×#157—38no—
90C2×#157—616no—
91C2×#157—64no—
92C2×#157—64no—
93C2×C32×C4—22no—
94C2×C32×C4—28no—
95C2×C32×C4—24no—
96C2×C32×C4—48no—
97C2×C32×C4—44no—
98C2×C32×C4C2×Dic312yescomplemented
99C2×C3⋊Dic3—22no—
100C2×C3⋊Dic3C2×Dic312yescomplemented
101C4×C3⋊D3—24no—
102C4×C3⋊D3—24no—
103C2×C3×Dic3—64no—
104C2×C3×Dic3—68no—
105C3×C4×D3—616no—
106C3×C4×D3—616no—
107C2×#95—271noSylow
108C2×C33#9514yescomplemented
109C2×C33C2×C4211yescharacteristic
110C2×C33C2×C22⋊C411yescharacteristic
111C2×C33C2×C4⋊C411yescharacteristic
112C3×C3⋊D3#9514yescomplemented
113C3×C3⋊D3C2×C22⋊C412yes—
114C3×C3⋊D3C2×C4⋊C412yes—
115C24×C3—91no—
116C23×D3—62no—
117C22×C3×C4—22no—
118C22×C3×C4—42no—
119C22×C3×C4—91no—
120C22×C3×C4—92no—
121C22×C3×C4—92no—
122C22×C3×C4—184no—
123C22×C3×C4#15711yescharacteristic, complemented
124C22×Dic3—62no—
125C22×Dic3—98no—
126C22×Dic3—94no—
127C22×Dic3—184no—
128C22×Dic3—184no—
129C2×C4×D3—64no—
130C2×C4×D3—64no—
131C2×C4×D3—616no—
132C22×C32—22no—
133C22×C32—24no—
134C22×C32—24no—
135C22×C32—24no—
136C22×C32—44no—
137C22×C32—44no—
138C22×C32—44no—
139C22×C32—42no—
140C22×C32C4×Dic312yes—
141C22×C32C4⋊Dic312yes—
142C22×C32#21712yes—
143C22×C32C22×Dic311yescharacteristic
144C2×C3⋊D3C4×Dic312yes—
145C2×C3⋊D3Dic3⋊C418yes—
146C2×C3⋊D3C4⋊Dic312yes—
147C2×C3⋊D3(C2×D3)⋊C418yes—
148C2×C3⋊D3#21712yes—
149C2×C3⋊D3C2×Dic611yescharacteristic
150C2×C3⋊D3C2×C4×D312yes—
151C2×C3⋊D3C2×D1211yescharacteristic
152C2×C3⋊D3C2×#6712yes—
153C2×C3×D3—64no—
154C2×C3×D3—64no—
155C2×C3×D3—616no—
156C2×C3×D3—616no—
157C2×C3×D3—64no—
158C2×C3×D3—64no—
159C2×C3×D3—64no—
160C2×C3×D3—62no—
161C2×C3×D3—62no—
162#157—68no—
163C32×C4—22no—
164C32×C4—28no—
165C32×C4—48no—
166C3⋊Dic3—22no—
167C3×Dic3—68no—
168C23×C4—91no—
169C23×C4—272no—
170#95—274no—
171C33C2×#9511yescharacteristic, complemented, Sylow
172C23×C3—22no—
173C23×C3—42no—
174C23×C3—91no—
175C23×C3—91no—
176C23×C3—94no—
177C23×C3—92no—
178C23×C3—92no—
179C23×C3—94no—
180C23×C3C2×#15711yescharacteristic
181C22×D3—64no—
182C22×D3—68no—
183C22×D3—64no—
184C22×D3—62no—
185C22×D3—68no—
186C22×D3—62no—
187C2×C3×C4—28no—
188C2×C3×C4—22no—
189C2×C3×C4—24no—
190C2×C3×C4—48no—
191C2×C3×C4—44no—
192C2×C3×C4—92no—
193C2×C3×C4—184no—
194C2×C3×C4—184no—
195C2×C3×C4—182no—
196C2×C3×C4C2×#15712yescomplemented
197C2×Dic3—64no—
198C2×Dic3—68no—
199C2×Dic3—98no—
200C2×Dic3—184no—
201C2×Dic3—1816no—
202C2×Dic3—184no—
203C4×D3—616no—
204C4×D3—616no—
205C2×C32—22no—
206C2×C32—22no—
207C2×C32—28no—
208C2×C32—22no—
209C2×C32—48no—
210C2×C32—42no—
211C2×C32—42no—
212C2×C32—42no—
213C2×C32C3⋊#9514yescomplemented
214C2×C32C2×C4×Dic311yescharacteristic
215C2×C32C2×C4⋊Dic311yescharacteristic
216C2×C32C2×#21711yescharacteristic
217C3⋊D3C3⋊#9514yescomplemented
218C3⋊D3C2×Dic3⋊C412yes—
219C3⋊D3C2×(C2×D3)⋊C412yes—
220C3×D3—68no—
221C3×D3—64no—
222C3×D3—64no—
223C24—91no—
224C22×C4—91no—
225C22×C4—92no—
226C22×C4—92no—
227C22×C4—184no—
228C22×C4—274no—
229C22×C4—278no—
230C22×C4—544no—
231C22×C4—544no—
232C22×C4C3⋊#15711yescharacteristic, complemented
233C22×C3—22no—
234C22×C3—24no—
235C22×C3—24no—
236C22×C3—24no—
237C22×C3—42no—
238C22×C3—44no—
239C22×C3—44no—
240C22×C3—44no—
241C22×C3—92no—
242C22×C3—92no—
243C22×C3—91no—
244C22×C3—91no—
245C22×C3—92no—
246C22×C3—98no—
247C22×C3—98no—
248C22×C3—92no—
249C22×C3—92no—
250C22×C3C4×#15712yes—
251C22×C3C4⋊#15712yes—
252C22×C3C22⋊#15712yes—
253C22×C3C22×#15711yescharacteristic
254C2×D3—64no—
255C2×D3—64no—
256C2×D3—64no—
257C2×D3—616no—
258C2×D3—64no—
259C2×D3—616no—
260C2×D3—62no—
261C2×D3—64no—
262C2×D3—62no—
263C3×C4—28no—
264C3×C4—22no—
265C3×C4—48no—
266C3×C4—182no—
267Dic3—68no—
268Dic3—188no—
269C32—22no—
270C32—42no—
271C32C2×C3⋊#9511yescharacteristic, complemented
272C23—91no—
273C23—91no—
274C23—94no—
275C23—92no—
276C23—92no—
277C23—94no—
278C23C2×C3⋊#15711yescharacteristic
279C2×C4—22no—
280C2×C4—92no—
281C2×C4—184no—
282C2×C4—184no—
283C2×C4—182no—
284C2×C4—278no—
285C2×C4—5416no—
286C2×C4—544no—
287C2×C4—544no—
288C2×C4C2×C3⋊#15712yescomplemented
289C2×C3—28no—
290C2×C3—22no—
291C2×C3—22no—
292C2×C3—22no—
293C2×C3—48no—
294C2×C3—42no—
295C2×C3—42no—
296C2×C3—42no—
297C2×C3—92no—
298C2×C3—92no—
299C2×C3—94no—
300C2×C3#6378114yescomplemented
301C2×C3C2×C4×#15711yescharacteristic
302C2×C3C2×C4⋊#15711yescharacteristic
303C2×C3C2×C22⋊#15711yescharacteristic
304D3—68no—
305D3—64no—
306D3—64no—
307C22—98no—
308C22—98no—
309C22—92no—
310C22—92no—
311C22—91no—
312C22—91no—
313C22—92no—
314C22—92no—
315C22—92no—
316C22C4×C3⋊#15712yes—
317C22C4⋊(C3⋊#157)12yes—
318C22#8899412yes—
319C22C22×C3⋊#15711yescharacteristic
320C4—22no—
321C4—182no—
322C4—548no—
323C3—22no—
324C3—42no—
325C3C2×#6378111yescharacteristic, complemented
326C2—92no—
327C2—94no—
328C2—92no—
329C2#1172310614yesdirect product factor
330C2C2×C4×C3⋊#15711yescharacteristic
331C2C2×C4⋊(C3⋊#157)11yescharacteristic
332C2C2×#8899411yescharacteristic
subgroupG/H
centerC23C2×C3⋊#157
commutatorC2×C33C2×C42
FrattiniC22C22×C3⋊#157
FittingC22×C33×C4C4
socleC23×C33C2×C4
prime radicalC23×C3×C3⋊D3C22
cosocleC22×C3×C3⋊D3C23
hypercenterC22×C4C3⋊#157
hypocenterC33C2×#95
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. #11723106C2direct (N × H)
8. C22×C3⋊#157C4semidirect
10. C22×C33×C4C4semidirect
11. C22×C3×C3⋊Dic3C4semidirect
13. C2×C3×C4×C3⋊D3C4semidirect
29. C2×C3⋊#157C2×C4semidirect
31. C2×C33×C4C2×C4semidirect
33. C2×C3×C3⋊Dic3C2×C4semidirect
45. C22×C32×C4Dic3semidirect
46. C22×C3⋊Dic3Dic3semidirect
48. C2×C4×C3⋊D3Dic3semidirect
98. C2×C32×C4C2×Dic3semidirect
100. C2×C3⋊Dic3C2×Dic3semidirect
108. C2×C33#95semidirect
112. C3×C3⋊D3#95semidirect
123. C22×C3×C4#157semidirect
171. C33C2×#95semidirect
196. C2×C3×C4C2×#157semidirect
213. C2×C32C3⋊#95semidirect
217. C3⋊D3C3⋊#95semidirect
232. C22×C4C3⋊#157semidirect
271. C32C2×C3⋊#95semidirect
288. C2×C4C2×C3⋊#157semidirect
300. C2×C3#63781semidirect
325. C3C2×#63781semidirect
329. C2#11723106direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3⋊#1572
C22×C3×C4×C3⋊D32
#117231062
C2×#637813
C2×C3⋊#959