Finite Group Databasebeta

← C2×Dic6×#157 #13180948

Subgroups — C2×Dic6×#157 #13180948

Order 1728.

984 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 984 conjugacy classes of subgroups — too many to draw. Only its 210 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3⋊D3×Dic6C211yesmaximal, characteristic
2C2×C4⋊(C3⋊#157)C211yesmaximal, characteristic
3C2×C3×C4×#157C211yesmaximal, characteristic
4C2×Dic3⋊#157C212yesmaximal
5C2×Dic3×#157C212yesmaximal
6Dic6×#157C218yesmaximal, direct product factor
7C2×Q8×#157—31nomaximal
8C22×C3⋊#157C2212yes—
9C22×C3×#157C2211yescharacteristic
10C22×C3×#157C2211yescharacteristic
11C2×#6308C411yescharacteristic, complemented
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×Dic6C411yescharacteristic, complemented
14C2×#6293C412yescomplemented
15C2×Dic3×C3⋊D3C2212yes—
16C3⋊D3×Dic6C414yescomplemented
17C3⋊D3×Dic6C2214yes—
18C4⋊(C3⋊#157)C2214yes—
19C3×C4×#157C2214yes—
20Dic3⋊#157C2218yes—
21Dic3×#157C2218yes—
22C2×Q8×C3⋊D3—31no—
23C2×D3×Dic6—62no—
24Q8×#157—38no—
25C2×C4⋊#157—31no—
26C2×C4⋊#157—32no—
27C2×C4×#157—32no—
28C2×C4×#157D311yescharacteristic
29C22×C3×C3⋊D3C2311yescharacteristic
30C2×C3⋊#157—24no—
31C2×C3⋊#157C2314yes—
32C2×C3×#157—22no—
33C2×C3×#157Q814yescomplemented
34C2×C3×#157C2312yes—
35C2×C3×#157C2312yes—
36C2×C33×C4C2×C411yescharacteristic
37C2×C3⋊(C3⋊Dic3)C2×C412yes—
38#6308C2×C414yescomplemented
39C2×C3×C3⋊Dic3C2×C411yescharacteristic
40C3×C4×C3⋊D3C2×C412yes—
41C3×C4×C3⋊D3C2312yes—
42C2×C32×Dic3C2×C412yes—
43C32×Dic6C2×C414yescomplemented
44#6293C2×C418yescomplemented
45Dic3×C3⋊D3C2×C414yes—
46Dic3×C3⋊D3C2314yes—
47C2×C4×Dic6—91nomaximal
48C22×#157—32no—
49C22×#157C2×D311yescharacteristic
50C22×#157C2×D311yescharacteristic
51C2×C32×Q8—31no—
52Q8×C3⋊D3—34no—
53Q8×C3⋊D3—34no—
54C2×C4×C3⋊D3—32no—
55C2×C4×C3⋊D3C2×D311yescharacteristic
56C2×#634—31no—
57C2×#634—32no—
58C2×#634—62no—
59C2×#634—122no—
60C2×C3×C4×D3—62no—
61C2×C3×Dic6—22no—
62C2×#627—64no—
63C2×D3×Dic3—64no—
64D3×Dic6—68no—
65D3×Dic6—68no—
66C4⋊#157—38no—
67C4⋊#157—34no—
68C4×#157—38no—
69C4×#157C2×D314yes—
70C22×C33C22×C411yescharacteristic
71C2×C3×C3⋊D3C22×C411yescharacteristic
72C2×C3×C3⋊D3C2×Q812yes—
73C2×C3×C3⋊D3#4112yes—
74C2×C3×C3⋊D3C2411yescharacteristic
75C3⋊#157—24no—
76C3×#157—22no—
77C3×#157C2×Q814yescomplemented
78C33×C4C22×C412yes—
79C3⋊(C3⋊Dic3)C22×C414yes—
80C3×C3⋊Dic3C22×C412yes—
81C32×Dic3C22×C414yes—
82C2×D3×Q8—182no—
83C22×Dic6—91no—
84C2×C3×C42—91no—
85C2×C4⋊Dic3—91no—
86C2×Dic3⋊C4—92no—
87C2×C4×Dic3—92no—
88C4×Dic6—98no—
89C22×C3⋊D3C22×D311yescharacteristic
90C22×C3×D3—62no—
91C2×#157—22no—
92C2×#157—34no—
93C2×#157—64no—
94C2×#157Dic614yesdirect product factor
95C2×#157C22×D312yes—
96C2×#157C22×D312yes—
97C32×Q8—34no—
98C2×C32×C4—22no—
99C2×C32×C4—32no—
100C2×C32×C4—42no—
101C2×C32×C4C4×D311yescharacteristic
102C2×C3⋊Dic3—32no—
103C2×C3⋊Dic3—64no—
104C2×C3⋊Dic3—124no—
105C2×C3⋊Dic3C4×D311yescharacteristic
106C4×C3⋊D3—34no—
107C4×C3⋊D3—34no—
108C4×C3⋊D3C4×D312yes—
109C4×C3⋊D3C22×D312yes—
110#634—34no—
111#634—38no—
112#634—68no—
113#634—128no—
114C2×C3×Dic3—24no—
115C2×C3×Dic3—62no—
116C3×C4×D3—64no—
117C3×C4×D3—64no—
118C3×Dic6—28no—
119#627—616no—
120D3×Dic3—68no—
121D3×Dic3—68no—
122C2×C4×Q8—271noSylow
123C2×C33C4×Q812yescomplemented
124C2×C33C23×C411yescharacteristic
125C3×C3⋊D3C4×Q812yescomplemented
126C3×C3⋊D3C22×Q811yescharacteristic
127C3×C3⋊D3C2×#4111yescharacteristic
128C2×C3×Q8—62no—
129C22×C3×C4—91no—
130C22×C3×C4—91no—
131C22×C3×C4—91no—
132C22×Dic3—92no—
133C22×Dic3—92no—
134D3×Q8—188no—
135D3×Q8—188no—
136C2×C4×D3—62no—
137C2×C4×D3—184no—
138C2×Dic6—91no—
139C2×Dic6—92no—
140C2×Dic6—94no—
141C2×Dic6—94no—
142C2×Dic6—184no—
143C2×Dic6—182no—
144C2×Dic6—362no—
145C2×Dic6#15711yescharacteristic, direct product factor
146C3×C42—94no—
147C4⋊Dic3—94no—
148Dic3⋊C4—98no—
149C4×Dic3—98no—
150C22×C32—22no—
151C22×C32—42no—
152C22×C32C2×C4×D311yescharacteristic
153C2×C3⋊D3C2×Dic612yes—
154C2×C3⋊D3C2×C4×D311yescharacteristic
155C2×C3⋊D3#23512yes—
156C2×C3⋊D3C23×D311yescharacteristic
157C2×C3×D3—62no—
158C2×C3×D3—64no—
159C2×C3×D3—64no—
160C2×C3×D3—62no—
161#157—22no—
162#157—64no—
163#157C2×Dic614yesdirect product factor
164C32×C4—24no—
165C32×C4—34no—
166C32×C4—44no—
167C32×C4C2×C4×D312yes—
168C3⋊Dic3—34no—
169C3⋊Dic3—68no—
170C3⋊Dic3—128no—
171C3⋊Dic3C2×C4×D312yes—
172C3×Dic3—28no—
173C3×Dic3—64no—
174C22×Q8—271no—
175C4×Q8—278no—
176C2×C4⋊C4—272no—
177C2×C4⋊C4—271no—
178C2×C42—91no—
179C2×C42—272no—
180C33C2×C4×Q811yescharacteristic, complemented, Sylow
181C23×C3—91no—
182C22×D3—62no—
183C3×Q8—68no—
184C2×C3×C4—22no—
185C2×C3×C4—42no—
186C2×C3×C4—64no—
187C2×C3×C4—94no—
188C2×C3×C4—92no—
189C2×C3×C4—91no—
190C2×C3×C4—92no—
191C2×C3×C4—92no—
192C2×C3×C4—92no—
193C2×C3×C4—182no—
194C2×C3×C4C2×#15711yescharacteristic
195C2×Dic3—62no—
196C2×Dic3—92no—
197C2×Dic3—94no—
198C2×Dic3—94no—
199C2×Dic3—94no—
200C2×Dic3—184no—
201C2×Dic3—184no—
202C2×Dic3—364no—
203C2×Dic3C2×#15712yes—
204C4×D3—64no—
205C4×D3—64no—
206C4×D3—188no—
207C4×D3—188no—
208Dic6—94no—
209Dic6—98no—
210Dic6—1816no—
211Dic6—188no—
212Dic6—368no—
213Dic6C2×#15714yesdirect product factor
214C2×C32—24no—
215C2×C32—22no—
216C2×C32—44no—
217C2×C32—42no—
218C2×C32C4×Dic612yescomplemented
219C2×C32C22×C4×D311yescharacteristic
220C3⋊D3C4×Dic612yescomplemented
221C3⋊D3C22×Dic611yescharacteristic
222C3⋊D3C2×#23511yescharacteristic
223C3×D3—64no—
224C3×D3—62no—
225C3×D3—62no—
226C2×Q8—31no—
227C2×Q8—272no—
228C2×Q8—274no—
229C2×Q8—274no—
230C2×Q8—271no—
231C22×C4—91no—
232C22×C4—91no—
233C22×C4—91no—
234C22×C4—272no—
235C22×C4—272no—
236C4⋊C4—278no—
237C4⋊C4—274no—
238C42—94no—
239C42—278no—
240C22×C3—22no—
241C22×C3—42no—
242C22×C3—92no—
243C22×C3—92no—
244C22×C3—91no—
245C22×C3—91no—
246C22×C3C22×#15711yescharacteristic
247C2×D3—62no—
248C2×D3—62no—
249C2×D3—64no—
250C2×D3—64no—
251C3×C4—24no—
252C3×C4—44no—
253C3×C4—68no—
254C3×C4—94no—
255C3×C4—92no—
256C3×C4—182no—
257C3×C4C22×#15712yes—
258Dic3—64no—
259Dic3—94no—
260Dic3—188no—
261Dic3—184no—
262Dic3—368no—
263Dic3C22×#15714yes—
264C32—22no—
265C32—42no—
266C32C2×C4×Dic611yescharacteristic, complemented
267C23—91no—
268Q8—34no—
269Q8—278no—
270Q8—274no—
271C2×C4—32no—
272C2×C4—92no—
273C2×C4—94no—
274C2×C4—91no—
275C2×C4—92no—
276C2×C4—92no—
277C2×C4—92no—
278C2×C4—182no—
279C2×C4—274no—
280C2×C4—274no—
281C2×C4—272no—
282C2×C4—274no—
283C2×C4—544no—
284C2×C4D3×#15711yescharacteristic
285C2×C3—22no—
286C2×C3—24no—
287C2×C3—42no—
288C2×C3—44no—
289C2×C3—91no—
290C2×C3—91no—
291C2×C3—92no—
292C2×C3Q8×#15712yescomplemented
293C2×C3C23×#15711yescharacteristic
294D3—64no—
295D3—62no—
296D3—62no—
297C22—91no—
298C22—91no—
299C22—92no—
300C22—92no—
301C22C2×D3×#15711yescharacteristic
302C4—34no—
303C4—94no—
304C4—92no—
305C4—182no—
306C4—274no—
307C4—544no—
308C4C2×D3×#15712yes—
309C3—22no—
310C3—42no—
311C3C2×Q8×#15711yescharacteristic, complemented
312C2—91no—
313C2—91no—
314C2—92no—
315C2Dic6×#15712yesdirect product factor
316C2C22×D3×#15711yescharacteristic
subgroupG/H
centerC22C2×D3×#157
commutatorC2×C33C23×C4
FrattiniC2C22×D3×#157
FittingC2×C33×C4C2×C4
socleC22×C33C22×C4
prime radicalC22×C3×C3⋊D3C23
cosocleC2×C3×C3⋊D3C24
hypercenterC2×C4D3×#157
hypocenterC33C2×C4×Q8
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
6. Dic6×#157C2direct (N × H)
11. C2×#6308C4semidirect
13. C2×C32×Dic6C4semidirect
14. C2×#6293C4semidirect
16. C3⋊D3×Dic6C4semidirect
33. C2×C3×#157Q8semidirect
38. #6308C2×C4semidirect
43. C32×Dic6C2×C4semidirect
44. #6293C2×C4semidirect
77. C3×#157C2×Q8semidirect
94. C2×#157Dic6direct (N × H)
123. C2×C33C4×Q8semidirect
125. C3×C3⋊D3C4×Q8semidirect
145. C2×Dic6#157direct (N × H)
163. #157C2×Dic6direct (N × H)
180. C33C2×C4×Q8semidirect
213. Dic6C2×#157direct (N × H)
218. C2×C32C4×Dic6semidirect
220. C3⋊D3C4×Dic6semidirect
266. C32C2×C4×Dic6semidirect
292. C2×C3Q8×#157semidirect
311. C3C2×Q8×#157semidirect
315. C2Dic6×#157direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×Dic62
C2×C4⋊(C3⋊#157)2
C2×C3×C4×#1572
C2×Dic3⋊#1572
C2×Dic3×#1572
Dic6×#1572
C2×Q8×#1573
C2×C4×Dic69