Finite Group Databasebeta

← #13173061

Subgroups — #13173061

Order 1728.

1070 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 1070 conjugacy classes of subgroups — too many to draw. Only its 209 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×C32×C4×A4C211yesmaximal, characteristic
2C2×C3⋊(C3⋊SL(2,ℤ/4))C211yesmaximal, characteristic
3C2×C3⋊(C3⋊SL(2,ℤ/4))C211yesmaximal, characteristic
4#10593645—312nomaximal
5#10592909—31nomaximal
6C22×C32×A4C2211yescharacteristic
7C32×C4×A4—21no—
8C3⋊(C3⋊SL(2,ℤ/4))—21no—
9C3⋊(C3⋊SL(2,ℤ/4))C412yescomplemented
10#88857—41nomaximal
11C23×C32×C4D311yescharacteristic
12C2×C3×C4×A4D3112yes—
13C2×C3⋊SL(2,ℤ/4)—312no—
14C2×C3⋊SL(2,ℤ/4)—312no—
15C2×#3757—31no—
16C2×#3757—31no—
17C2×#3750—31no—
18C2×#3750—31no—
19#63658—31no—
20#63658—31no—
21C2×C32×A4C2×C411yescharacteristic
22C2×C32×A4D411yescharacteristic
23C2×C32×A4Q811yescharacteristic
24C2×C33×C4—41no—
25C2×C3⋊(C3⋊Dic3)—41no—
26C2×C3⋊(C3⋊Dic3)—41no—
27SL(2,ℤ/4)⋊C4—99no—
28#5210—94no—
29C24×C32C2×D311yescharacteristic
30C22×C3×A4C2×D3112yes—
31C22×C32×C4—21no—
32C22×C32×C4—31no—
33C22×C32×C4—31no—
34C22×C32×C4—61no—
35C22×C3⋊Dic3—31no—
36C22×C3⋊Dic3—31no—
37C22×C3⋊Dic3—31no—
38C22×C3⋊Dic3—31no—
39C3×C4×A4—212no—
40C3⋊SL(2,ℤ/4)—324no—
41C3⋊SL(2,ℤ/4)—612no—
42#3757—32no—
43#3757—32no—
44#3757—61no—
45#3757—61no—
46#3750—62no—
47#3750—61no—
48#3750—61no—
49#3750—1212no—
50C22×C33—41no—
51C32×A4C4⋊C411yescharacteristic, complemented
52C33×C4—81no—
53C3⋊(C3⋊Dic3)—42no—
54C3⋊(C3⋊Dic3)—81no—
55C23×C3×C4C3⋊D314yes—
56C2×C4×A4C3⋊D319yes—
57C2×SL(2,ℤ/4)—99no—
58C2×SL(2,ℤ/4)—99no—
59C2×#217—94no—
60C2×#217—94no—
61C2×Dic3⋊C4—94no—
62C2×Dic3⋊C4—94no—
63C3⋊#95—94no—
64C3⋊#95—94no—
65C23×C32—31no—
66C23×C32—31no—
67C23×C32—31no—
68C23×C32—31no—
69C23×C32Dic611yescharacteristic
70C23×C32C4×D311yescharacteristic
71C23×C32#6711yescharacteristic
72C2×C3×A4Dic6112yes—
73C2×C3×A4C4×D3112yes—
74C2×C3×A4#67112yes—
75C2×C32×C4—32no—
76C2×C32×C4—31no—
77C2×C32×C4—412no—
78C2×C32×C4—61no—
79C2×C32×C4—61no—
80C2×C32×C4—61no—
81C2×C32×C4S411yescharacteristic
82C2×C3⋊Dic3—32no—
83C2×C3⋊Dic3—32no—
84C2×C3⋊Dic3—61no—
85C2×C3⋊Dic3—61no—
86C2×C3⋊Dic3—61no—
87C2×C3⋊Dic3—61no—
88C2×C3⋊Dic3—61no—
89C2×C3⋊Dic3—61no—
90C2×C3⋊Dic3—61no—
91C2×C3⋊Dic3—61no—
92C2×C3⋊Dic3—61no—
93C2×C3⋊Dic3—61no—
94C2×C3⋊Dic3—1212no—
95C2×C3⋊Dic3—1212no—
96#385—271noSylow
97C2×C33—41no—
98C2×C33—41no—
99C2×C33—41no—
100C24×C3C2×C3⋊D314yes—
101C22×A4C2×C3⋊D319yes—
102C22×C3×C4—24no—
103C22×C3×C4—34no—
104C22×C3×C4—34no—
105C22×C3×C4—64no—
106C22×Dic3—94no—
107C22×Dic3—94no—
108C22×Dic3—94no—
109C22×Dic3—94no—
110C4×A4—29no—
111SL(2,ℤ/4)—918no—
112SL(2,ℤ/4)—189no—
113#217—98no—
114#217—98no—
115#217—184no—
116#217—184no—
117Dic3⋊C4—188no—
118Dic3⋊C4—184no—
119Dic3⋊C4—184no—
120Dic3⋊C4—369no—
121C22×C32—31no—
122C22×C32—31no—
123C22×C32—31no—
124C22×C32—31no—
125C22×C32—31no—
126C22×C32—31no—
127C22×C32—31no—
128C22×C32—31no—
129C22×C32—31no—
130C22×C32—412no—
131C22×C32—61no—
132C22×C32Dic3⋊C411yescharacteristic, complemented
133C22×C32C2×S411yescharacteristic
134C3×A4Dic3⋊C4112yescomplemented
135C32×C4—21no—
136C32×C4—61no—
137C32×C4—812no—
138C3⋊Dic3—62no—
139C3⋊Dic3—62no—
140C3⋊Dic3—1224no—
141C3⋊Dic3—121no—
142C3⋊Dic3—121no—
143C3⋊Dic3—2412no—
144C23×C4C3⋊(C3⋊D3)11yescharacteristic
145C2×C4⋊C4—271no—
146C2×C4⋊C4—271no—
147C2×C22⋊C4—271no—
148C2×C22⋊C4—271no—
149#95—271no—
150#95—271no—
151C33—41noSylow
152C23×C3—34no—
153C23×C3—34no—
154C23×C3—34no—
155C23×C3—34no—
156C23×C3#63414yes—
157C23×C3C4×C3⋊D314yes—
158C23×C3#63814yes—
159C2×A4#63419yes—
160C2×A4C4×C3⋊D319yes—
161C2×A4#63819yes—
162C2×C3×C4—38no—
163C2×C3×C4—34no—
164C2×C3×C4—49no—
165C2×C3×C4—64no—
166C2×C3×C4—64no—
167C2×C3×C4—64no—
168C2×C3×C4C3⋊S414yes—
169C2×Dic3—98no—
170C2×Dic3—98no—
171C2×Dic3—184no—
172C2×Dic3—184no—
173C2×Dic3—184no—
174C2×Dic3—184no—
175C2×Dic3—184no—
176C2×Dic3—184no—
177C2×Dic3—184no—
178C2×Dic3—184no—
179C2×Dic3—184no—
180C2×Dic3—184no—
181C2×Dic3—369no—
182C2×Dic3—369no—
183C2×C32—31no—
184C2×C32—31no—
185C2×C32—31no—
186C2×C32—31no—
187C2×C32—412no—
188C2×C32—412no—
189C2×C32—412no—
190C2×C32A4⋊Q811yescharacteristic
191C2×C32C4×S411yescharacteristic
192C2×C32GL(2,ℤ/4)11yescharacteristic
193C24C2×C3⋊(C3⋊D3)11yescharacteristic
194C22×C4—21no—
195C22×C4—31no—
196C22×C4—31no—
197C22×C4—61no—
198C22×C4—271no—
199C22×C4—271no—
200C22×C4—271no—
201C22×C4—271no—
202C4⋊C4—542no—
203C4⋊C4—541no—
204C4⋊C4—541no—
205C22⋊C4—272no—
206C22⋊C4—272no—
207C22⋊C4—541no—
208C22⋊C4—541no—
209C22×C3—34no—
210C22×C3—34no—
211C22×C3—34no—
212C22×C3—34no—
213C22×C3—34no—
214C22×C3—34no—
215C22×C3—34no—
216C22×C3—34no—
217C22×C3—34no—
218C22×C3—49no—
219C22×C3—64no—
220C22×C3#375014yescomplemented
221C22×C3C2×C3⋊S414yes—
222A4#375019yescomplemented
223C3×C4—24no—
224C3×C4—64no—
225C3×C4—89no—
226Dic3—188no—
227Dic3—188no—
228Dic3—3618no—
229Dic3—364no—
230Dic3—364no—
231Dic3—729no—
232C32—412no—
233C32SL(2,ℤ/4)⋊C411yescharacteristic, complemented
234C23—31no—
235C23—31no—
236C23—31no—
237C23—31no—
238C23#630811yescharacteristic
239C23C4×C3⋊(C3⋊D3)11yescharacteristic
240C23#631211yescharacteristic
241C2×C4—32no—
242C2×C4—31no—
243C2×C4—61no—
244C2×C4—61no—
245C2×C4—61no—
246C2×C4—272no—
247C2×C4—272no—
248C2×C4—541no—
249C2×C4—541no—
250C2×C4—541no—
251C2×C4—541no—
252C2×C4—541no—
253C2×C4—541no—
254C2×C4—541no—
255C2×C4—541no—
256C2×C4—541no—
257C2×C4—541no—
258C2×C4C3⋊(C3⋊S4)11yescharacteristic
259C2×C3—34no—
260C2×C3—34no—
261C2×C3—34no—
262C2×C3—34no—
263C2×C3—49no—
264C2×C3—49no—
265C2×C3—49no—
266C2×C3#6425914yes—
267C2×C3C4×C3⋊S414yes—
268C2×C3#6426914yes—
269C22—31no—
270C22—31no—
271C22—31no—
272C22—31no—
273C22—31no—
274C22—31no—
275C22—31no—
276C22—31no—
277C22—31no—
278C22—61no—
279C22#8885711yescharacteristic, complemented
280C22C2×C3⋊(C3⋊S4)11yescharacteristic
281C4—21no—
282C4—61no—
283C4—542no—
284C4—542no—
285C4—1081no—
286C4—1081no—
287C3—49no—
288C3#1059364514yescomplemented
289C2—31no—
290C2—31no—
291C2—31no—
292C2—31no—
293C2#1172446311yescharacteristic
294C2C4×C3⋊(C3⋊S4)11yescharacteristic
295C2#1172447311yescharacteristic
subgroupG/H
center, FrattiniC22C2×C3⋊(C3⋊S4)
commutatorC2×C32×A4C2×C4
FittingC23×C32×C4D3
socleC24×C32C2×D3
prime radical, cosocleC22×C32×A4C22
hypercenterC2×C4C3⋊(C3⋊S4)
hypocenterC32×A4C4⋊C4
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
9. C3⋊(C3⋊SL(2,ℤ/4))C4semidirect
51. C32×A4C4⋊C4semidirect
132. C22×C32Dic3⋊C4semidirect
134. C3×A4Dic3⋊C4semidirect
220. C22×C3#3750semidirect
222. A4#3750semidirect
233. C32SL(2,ℤ/4)⋊C4semidirect
279. C22#88857semidirect
288. C3#10593645semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×A42
C2×C3⋊(C3⋊SL(2,ℤ/4))2
#105936453
#105929093
#888574