Finite Group Databasebeta

← #157×SL(2,ℤ/4) #13180191

Subgroups — #157×SL(2,ℤ/4) #13180191

Order 1728.

400 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 38 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×A4×#157C211yesmaximal, characteristic
2C2×A4⋊#157C211yesmaximal, characteristic
3C3⋊D3×SL(2,ℤ/4)C211yesmaximal, characteristic
4C22⋊C4×#157—31nomaximal
5C2×A4×C3⋊D3C2211yescharacteristic
6A4×#157C412yescomplemented
7A4⋊#157C412yescomplemented
8C3⋊(C3⋊SL(2,ℤ/4))C411yescharacteristic, complemented
9C32×SL(2,ℤ/4)C411yescharacteristic, complemented
10Dic3×#157—41nomaximal
11C23×#157D311yescharacteristic
12C2×C22⋊#157—31no—
13C2×C4×#157—31no—
14C2×C4×#157—31no—
15D3×SL(2,ℤ/4)—62no—
16C22⋊C4×C3⋊D3—31no—
17#63781—31no—
18#63781—31no—
19C2×C32×A4C2×C411yescharacteristic
20C2×C3⋊#157—41no—
21C2×C3×#157—41no—
22A4×C3⋊D3C2×C411yescharacteristic
23A4×C3⋊D3C2×C411yescharacteristic
24Dic3×C3⋊D3—41no—
25C4×SL(2,ℤ/4)—91nomaximal
26C23×C3⋊D3C2×D311yescharacteristic
27C22×#157—31no—
28C22×#157—31no—
29C22×#157—32no—
30C22×#157—31no—
31C22×#157—31no—
32C22×#157Dic312yescomplemented
33C2×D3×A4—62no—
34C2×C4×C3⋊D3—31no—
35C2×C4×C3⋊D3—31no—
36C22⋊#157—32no—
37C22⋊#157—32no—
38C4×#157—62no—
39C4×#157—61no—
40C4×#157—61no—
41C3⋊SL(2,ℤ/4)—62no—
42C3⋊SL(2,ℤ/4)—122no—
43C3×SL(2,ℤ/4)—22no—
44C32×C22⋊C4—31no—
45#3757—31no—
46#3752—31no—
47#3752—31no—
48C2×C3×C3⋊D3—41no—
49C32×A4C4211yescharacteristic, complemented
50C3⋊#157—42no—
51C3×#157—42no—
52C3⋊(C3⋊Dic3)—41no—
53C32×Dic3—41no—
54C2×C4×A4—91no—
55C2×SL(2,ℤ/4)—91no—
56C2×SL(2,ℤ/4)—91no—
57D3×C22⋊C4—182no—
58C23×C32C4×D311yescharacteristic
59C22×C3⋊D3—31no—
60C22×C3⋊D3—31no—
61C22×C3⋊D3—31no—
62C22×C3⋊D3—31no—
63C22×C3⋊D3C4×D311yescharacteristic
64C22×C3⋊D3C2×Dic311yescharacteristic
65C2×C3×A4—22no—
66C2×C3×A4—42no—
67C2×#157—31no—
68C2×#157—32no—
69C2×#157—32no—
70C2×#157—32no—
71C2×#157—32no—
72C2×#157—32no—
73C2×#157—61no—
74C2×#157—61no—
75C2×#157—61no—
76C2×#157—61no—
77C2×#157—61no—
78C2×#157S411yescharacteristic
79D3×A4—62no—
80D3×A4—62no—
81C2×C32×C4—31no—
82C2×C32×C4—31no—
83C2×C3⋊Dic3—31no—
84C2×C3⋊Dic3—31no—
85C4×C3⋊D3—61no—
86C4×C3⋊D3—61no—
87C4×C3⋊D3—61no—
88C4×C3⋊D3—61no—
89D3×Dic3—242no—
90C4×C22⋊C4—271noSylow
91C2×C33—41no—
92C3×C3⋊D3—41no—
93C3×C3⋊D3—41no—
94C23×D3—62no—
95C22×A4—91no—
96C2×C4×D3—182no—
97C2×C4×D3—182no—
98C4×A4—92no—
99SL(2,ℤ/4)—92no—
100SL(2,ℤ/4)—91no—
101SL(2,ℤ/4)—362no—
102SL(2,ℤ/4)#15711yescharacteristic, direct product factor
103C3×C22⋊C4—62no—
104#217—182no—
105(C2×D3)⋊C4—182no—
106(C2×D3)⋊C4—182no—
107C4×Dic3—361no—
108C22×C32—31no—
109C22×C32—31no—
110C22×C32C4×Dic311yescharacteristic, complemented
111C2×C3⋊D3—31no—
112C2×C3⋊D3—31no—
113C2×C3⋊D3—31no—
114C2×C3⋊D3—31no—
115C2×C3⋊D3—31no—
116C2×C3⋊D3—31no—
117C2×C3⋊D3—31no—
118C2×C3⋊D3—61no—
119C2×C3⋊D3C2×S411yescharacteristic
120C2×C3×D3—242no—
121C3×A4—22no—
122C3×A4—42no—
123#157—32no—
124#157—62no—
125#157—62no—
126#157SL(2,ℤ/4)12yesdirect product factor
127C32×C4—61no—
128C32×C4—61no—
129C3⋊Dic3—61no—
130C3⋊Dic3—61no—
131C3⋊Dic3—242no—
132C3⋊Dic3—482no—
133C3×Dic3—82no—
134C23×C4—91no—
135C2×C22⋊C4—271no—
136C2×C22⋊C4—271no—
137C2×C42—271no—
138C2×C42—271no—
139#95—271no—
140#95—271no—
141C33—41noSylow
142C23×C3—22no—
143C22×D3—62no—
144C22×D3—62no—
145C22×D3—182no—
146C22×D3—182no—
147C22×D3—182no—
148C22×D3—182no—
149C2×A4—42no—
150C2×A4—91no—
151C2×A4—91no—
152C2×A4C2×#15711yescharacteristic
153C2×C3×C4—62no—
154C2×C3×C4—62no—
155C2×C3×C4—361no—
156C2×Dic3—182no—
157C2×Dic3—182no—
158C2×Dic3—361no—
159C2×Dic3—361no—
160C4×D3—362no—
161C4×D3—362no—
162C4×D3—362no—
163C4×D3—362no—
164C2×C32—31no—
165C2×C32—31no—
166C2×C32—82no—
167C2×C32—162no—
168C2×C32C4×S411yescharacteristic
169C3⋊D3—31no—
170C3⋊D3—31no—
171C3⋊D3C4×S411yescharacteristic
172C3⋊D3C2×SL(2,ℤ/4)11yescharacteristic
173C3×D3—242no—
174C3×D3—242no—
175C24—91no—
176C22×C4—92no—
177C22×C4—271no—
178C22×C4—271no—
179C22×C4—271no—
180C22×C4—271no—
181C22×C4—272no—
182C22×C4—271no—
183C22×C4—271no—
184C22⋊C4—31no—
185C22⋊C4—271no—
186C22⋊C4—271no—
187C22⋊C4—272no—
188C22⋊C4—272no—
189C22⋊C4—271no—
190C42—541no—
191C42—542no—
192C42—541no—
193C22×C3—22no—
194C22×C3—62no—
195C22×C3—62no—
196C22×C3—361no—
197C2×D3—62no—
198C2×D3—182no—
199C2×D3—182no—
200C2×D3—182no—
201C2×D3—182no—
202C2×D3—182no—
203C2×D3—182no—
204C2×D3—182no—
205C2×D3—362no—
206A4—42no—
207A4C4×#15711yescharacteristic, complemented
208C3×C4—122no—
209C3×C4—122no—
210C3×C4—362no—
211Dic3—41no—
212Dic3—362no—
213Dic3—361no—
214Dic3—362no—
215Dic3—362no—
216Dic3—1442no—
217C32—82no—
218C32—162no—
219C32C4×SL(2,ℤ/4)11yescharacteristic, complemented
220C23—91no—
221C23—91no—
222C23—271no—
223C23—271no—
224C23—271no—
225C23—271no—
226C23D3×#15711yescharacteristic
227C2×C4—31no—
228C2×C4—31no—
229C2×C4—91no—
230C2×C4—272no—
231C2×C4—272no—
232C2×C4—271no—
233C2×C4—272no—
234C2×C4—271no—
235C2×C4—271no—
236C2×C4—272no—
237C2×C4—272no—
238C2×C4—541no—
239C2×C4—541no—
240C2×C4—541no—
241C2×C4—541no—
242C2×C4—541no—
243C2×C4—541no—
244C2×C4—541no—
245C2×C4—541no—
246C2×C4—541no—
247C2×C3—22no—
248C2×C3—41no—
249C2×C3—62no—
250C2×C3—62no—
251C2×C3—162no—
252C2×C3—361no—
253C2×C3—361no—
254D3—62no—
255D3—62no—
256D3—182no—
257D3—182no—
258C22—31no—
259C22—31no—
260C22—91no—
261C22—271no—
262C22—271no—
263C22—271no—
264C22—271no—
265C22—271no—
266C22—271no—
267C22—271no—
268C22—541no—
269C22Dic3×#15711yescharacteristic, complemented
270C4—61no—
271C4—61no—
272C4—92no—
273C4—272no—
274C4—541no—
275C4—542no—
276C4—542no—
277C4—541no—
278C3—22no—
279C3—41no—
280C3—162no—
281C2—31no—
282C2—31no—
283C2—91no—
284C2—91no—
285C2—271no—
286C2—271no—
287C2S4×#15711yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2S4×#157
commutator, hypocenterC32×A4C42
Fitting, socleC23×C32C4×D3
prime radical, cosocleC2×A4×C3⋊D3C22
LCSG ▷ C32×A4
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
6. A4×#157C4semidirect
7. A4⋊#157C4semidirect
8. C3⋊(C3⋊SL(2,ℤ/4))C4semidirect
9. C32×SL(2,ℤ/4)C4semidirect
32. C22×#157Dic3semidirect
49. C32×A4C42semidirect
102. SL(2,ℤ/4)#157direct (N × H)
110. C22×C32C4×Dic3semidirect
126. #157SL(2,ℤ/4)direct (N × H)
207. A4C4×#157semidirect
219. C32C4×SL(2,ℤ/4)semidirect
269. C22Dic3×#157semidirect

Maximal subgroups

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

subgroupindex
C2×A4×#1572
C2×A4⋊#1572
C3⋊D3×SL(2,ℤ/4)2
C22⋊C4×#1573
Dic3×#1574
C4×SL(2,ℤ/4)9