Finite Group Databasebeta

← #13165988

Subgroups — #13165988

Order 1728.

782 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 103 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×C3⋊Dic3C211yesmaximal, characteristic
2C2×C3⋊(C3⋊SL(2,ℤ/4))C211yesmaximal, characteristic
3C2×C32×SL(2,ℤ/4)C211yesmaximal, characteristic
4#10593238—34nomaximal
5#10592645—31nomaximal
6C22×C32×A4C2211yescharacteristic
7A4×C3⋊Dic3—21no—
8C3⋊(C3⋊SL(2,ℤ/4))C412yescomplemented
9C32×SL(2,ℤ/4)—21no—
10#88806—41nomaximal
11C23×C3⋊Dic3D311yescharacteristic
12C2×Dic3×A4—34no—
13C2×C3⋊SL(2,ℤ/4)—34no—
14C2×C3⋊SL(2,ℤ/4)—64no—
15C2×C3×SL(2,ℤ/4)D314yes—
16C2×C32×C22⋊C4—31no—
17C2×#3757—31no—
18C2×#3750—31no—
19C2×#3750—31no—
20#63658—31no—
21#63658—31no—
22C2×C32×A4C2×C411yescharacteristic
23C2×C32×A4D411yescharacteristic
24C2×C32×A4Q811yescharacteristic
25C2×C3⋊(C3⋊Dic3)—41no—
26C2×C3×C3⋊Dic3—41no—
27C2×C32×Dic3—41no—
28SL(2,ℤ/4)⋊C4—91no—
29#4946—94no—
30C24×C32C2×D311yescharacteristic
31C22×C3×A4—24no—
32C22×C3×A4C2×D314yes—
33C22×C32×C4—31no—
34C22×C32×C4—31no—
35C22×C3⋊Dic3—21no—
36C22×C3⋊Dic3—31no—
37C22×C3⋊Dic3—31no—
38C22×C3⋊Dic3—31no—
39C22×C3⋊Dic3—31no—
40C22×C3⋊Dic3—61no—
41Dic3×A4—64no—
42C3⋊SL(2,ℤ/4)—38no—
43C3⋊SL(2,ℤ/4)—68no—
44C3×SL(2,ℤ/4)—24no—
45C32×C22⋊C4—61no—
46C32×C22⋊C4—61no—
47#3757—32no—
48#3757—32no—
49#3750—62no—
50#3750—61no—
51#3750—61no—
52#3724—124no—
53C22×C33—41no—
54C32×A4C4⋊C411yescharacteristic, complemented
55C3⋊(C3⋊Dic3)—42no—
56C3×C3⋊Dic3—81no—
57C32×Dic3—81no—
58C23×Dic3—34no—
59C2×C4×A4—91no—
60C2×SL(2,ℤ/4)—91no—
61C2×SL(2,ℤ/4)—184no—
62C2×SL(2,ℤ/4)C3⋊D311yescharacteristic
63C2×C3×C22⋊C4—34no—
64C2×#217—94no—
65C2×Dic3⋊C4—94no—
66C2×Dic3⋊C4—94no—
67C3⋊#95—94no—
68C3⋊#95—94no—
69C23×C32—31no—
70C23×C32—31no—
71C23×C32—31no—
72C23×C32—31no—
73C23×C32Dic611yescharacteristic
74C23×C32C4×D311yescharacteristic
75C23×C32#6711yescharacteristic
76C2×C3×A4—24no—
77C2×C3×A4—24no—
78C2×C3×A4—24no—
79C2×C3×A4Dic614yes—
80C2×C3×A4C4×D314yes—
81C2×C3×A4#6714yes—
82C2×C32×C4—61no—
83C2×C32×C4—61no—
84C2×C32×C4—61no—
85C2×C32×C4—61no—
86C2×C32×C4—61no—
87C2×C32×C4—61no—
88C2×C3⋊Dic3—32no—
89C2×C3⋊Dic3—32no—
90C2×C3⋊Dic3—32no—
91C2×C3⋊Dic3—31no—
92C2×C3⋊Dic3—61no—
93C2×C3⋊Dic3—61no—
94C2×C3⋊Dic3—61no—
95C2×C3⋊Dic3—61no—
96C2×C3⋊Dic3—61no—
97C2×C3⋊Dic3—61no—
98C2×C3⋊Dic3—61no—
99C2×C3⋊Dic3—124no—
100C2×C3⋊Dic3—244no—
101C2×C3⋊Dic3S411yescharacteristic
102C2×C3×Dic3—44no—
103C2×C3×Dic3—124no—
104#385—271noSylow
105C2×C33—41no—
106C2×C33—41no—
107C2×C33—41no—
108C24×C3D3214yes—
109C22×A4—24no—
110C22×A4C2×C3⋊D311yescharacteristic
111C22×C3×C4—34no—
112C22×C3×C4—34no—
113C22×Dic3—64no—
114C22×Dic3—94no—
115C22×Dic3—94no—
116C22×Dic3—94no—
117C22×Dic3—94no—
118C22×Dic3—184no—
119C4×A4—181no—
120SL(2,ℤ/4)—21no—
121SL(2,ℤ/4)—92no—
122SL(2,ℤ/4)—188no—
123C3×C22⋊C4—64no—
124C3×C22⋊C4—64no—
125#217—98no—
126#217—98no—
127Dic3⋊C4—184no—
128Dic3⋊C4—188no—
129Dic3⋊C4—184no—
130Dic3⋊C4—361no—
131C22×C32—31no—
132C22×C32—31no—
133C22×C32—31no—
134C22×C32—31no—
135C22×C32—31no—
136C22×C32—31no—
137C22×C32—31no—
138C22×C32—31no—
139C22×C32—31no—
140C22×C32—44no—
141C22×C32—61no—
142C22×C32—84no—
143C22×C32Dic3⋊C411yescharacteristic, complemented
144C22×C32C2×S411yescharacteristic
145C3×A4—24no—
146C3×A4Dic3⋊C414yescomplemented
147C32×C4—121no—
148C32×C4—121no—
149C3⋊Dic3—21no—
150C3⋊Dic3—62no—
151C3⋊Dic3—62no—
152C3⋊Dic3—61no—
153C3⋊Dic3—128no—
154C3⋊Dic3—248no—
155C3×Dic3—84no—
156C3×Dic3—244no—
157C23×C4—91no—
158C2×C4⋊C4—271no—
159C2×C4⋊C4—271no—
160C2×C22⋊C4—31no—
161C2×C22⋊C4—271no—
162#95—271no—
163#95—271no—
164C33—41noSylow
165C23×C3—34no—
166C23×C3—34no—
167C23×C3—34no—
168C23×C3—34no—
169C23×C3#62414yes—
170C23×C3#62514yes—
171C23×C3#62714yes—
172C2×A4—24no—
173C2×A4—24no—
174C2×A4—24no—
175C2×A4#63411yescharacteristic
176C2×A4C4×C3⋊D311yescharacteristic
177C2×A4#63811yescharacteristic
178C2×C3×C4—64no—
179C2×C3×C4—64no—
180C2×C3×C4—64no—
181C2×C3×C4—64no—
182C2×C3×C4—64no—
183C2×C3×C4—64no—
184C2×C3×C4—361no—
185C2×Dic3—34no—
186C2×Dic3—41no—
187C2×Dic3—98no—
188C2×Dic3—98no—
189C2×Dic3—98no—
190C2×Dic3—94no—
191C2×Dic3—184no—
192C2×Dic3—184no—
193C2×Dic3—184no—
194C2×Dic3—184no—
195C2×Dic3—184no—
196C2×Dic3—184no—
197C2×Dic3—184no—
198C2×Dic3—361no—
199C2×Dic3—724no—
200C2×C32—31no—
201C2×C32—31no—
202C2×C32—31no—
203C2×C32—31no—
204C2×C32—44no—
205C2×C32—44no—
206C2×C32—44no—
207C2×C32—84no—
208C2×C32—84no—
209C2×C32—84no—
210C2×C32A4⋊Q811yescharacteristic
211C2×C32C4×S411yescharacteristic
212C2×C32GL(2,ℤ/4)11yescharacteristic
213C24D3×C3⋊D311yescharacteristic
214C22×C4—31no—
215C22×C4—31no—
216C22×C4—181no—
217C22×C4—271no—
218C22×C4—271no—
219C22×C4—271no—
220C22×C4—271no—
221C22×C4—541no—
222C4⋊C4—541no—
223C4⋊C4—542no—
224C4⋊C4—541no—
225C22⋊C4—61no—
226C22⋊C4—61no—
227C22⋊C4—272no—
228C22⋊C4—272no—
229C22×C3—34no—
230C22×C3—34no—
231C22×C3—34no—
232C22×C3—34no—
233C22×C3—34no—
234C22×C3—34no—
235C22×C3—34no—
236C22×C3—34no—
237C22×C3—34no—
238C22×C3—41no—
239C22×C3—64no—
240C22×C3—84no—
241C22×C3#372414yescomplemented
242C22×C3D3×S414yes—
243A4—24no—
244A4#375011yescharacteristic, complemented
245C3×C4—124no—
246C3×C4—124no—
247C3×C4—721no—
248Dic3—64no—
249Dic3—81no—
250Dic3—188no—
251Dic3—188no—
252Dic3—184no—
253Dic3—362no—
254Dic3—728no—
255C32—44no—
256C32—84no—
257C32SL(2,ℤ/4)⋊C411yescharacteristic, complemented
258C23—31no—
259C23—31no—
260C23—31no—
261C23—31no—
262C23#628911yescharacteristic
263C23#629011yescharacteristic
264C23#629311yescharacteristic
265C2×C4—61no—
266C2×C4—61no—
267C2×C4—61no—
268C2×C4—61no—
269C2×C4—61no—
270C2×C4—61no—
271C2×C4—91no—
272C2×C4—272no—
273C2×C4—272no—
274C2×C4—272no—
275C2×C4—271no—
276C2×C4—541no—
277C2×C4—541no—
278C2×C4—541no—
279C2×C4—541no—
280C2×C4—541no—
281C2×C4—541no—
282C2×C4—541no—
283C2×C3—34no—
284C2×C3—34no—
285C2×C3—34no—
286C2×C3—34no—
287C2×C3—41no—
288C2×C3—41no—
289C2×C3—41no—
290C2×C3—84no—
291C2×C3—84no—
292C2×C3—84no—
293C2×C3#6420414yes—
294C2×C3(C3⋊S4)⋊C414yes—
295C2×C3#6420914yes—
296C22—31no—
297C22—31no—
298C22—31no—
299C22—31no—
300C22—31no—
301C22—31no—
302C22—31no—
303C22—31no—
304C22—31no—
305C22—61no—
306C22#8880611yescharacteristic, complemented
307C22C3⋊D3×S411yescharacteristic
308C4—121no—
309C4—121no—
310C4—181no—
311C4—542no—
312C4—542no—
313C4—541no—
314C3—41no—
315C3—84no—
316C3#1059323814yescomplemented
317C2—31no—
318C2—31no—
319C2—31no—
320C2—31no—
321C2#1172418411yescharacteristic
322C2#1172418611yescharacteristic
323C2#1172418911yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22C3⋊D3×S4
commutatorC2×C32×A4C2×C4
Fitting, socleC24×C32C2×D3
prime radical, cosocleC22×C32×A4C22
hypocenterC32×A4C4⋊C4
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
8. C3⋊(C3⋊SL(2,ℤ/4))C4semidirect
54. C32×A4C4⋊C4semidirect
143. C22×C32Dic3⋊C4semidirect
146. C3×A4Dic3⋊C4semidirect
241. C22×C3#3724semidirect
244. A4#3750semidirect
257. C32SL(2,ℤ/4)⋊C4semidirect
306. C22#88806semidirect
316. C3#10593238semidirect

Maximal subgroups

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

subgroupindex
C2×A4×C3⋊Dic32
C2×C3⋊(C3⋊SL(2,ℤ/4))2
C2×C32×SL(2,ℤ/4)2
#105932383
#105926453
#888064