Finite Group Databasebeta

← #13166111

Subgroups — #13166111

Order 1728.

650 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 82 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)C212yesmaximal
3Dic3×SL(2,ℤ/4)—32nomaximal
4#10591705—31nomaximal
5C22×C32×A4C2211yescharacteristic
6A4×C3⋊Dic3C412yescomplemented
7C3×C3⋊SL(2,ℤ/4)C414yescomplemented
8#88813—41nomaximal
9C23×C3⋊Dic3D311yescharacteristic
10C2×Dic3×A4—32no—
11C2×Dic3×A4—61no—
12C2×C3⋊SL(2,ℤ/4)D312yes—
13C2×C3×SL(2,ℤ/4)—32no—
14C2×C3×SL(2,ℤ/4)—62no—
15C2×C3×#217—32no—
16C2×Dic32—31no—
17C2×Dic32—31no—
18#63579—32no—
19C2×C32×A4C2×C411yescharacteristic
20C2×C32×A4C2×C412yes—
21C2×C3×C3⋊Dic3—42no—
22C2×C3×C3⋊Dic3—41no—
23C4×SL(2,ℤ/4)—91no—
24Dic3×C22⋊C4—92no—
25C24×C32C2×D311yescharacteristic
26C22×C3×A4—22no—
27C22×C3×A4—21no—
28C22×C3×A4—41no—
29C22×C3×A4C2×D312yes—
30C22×C3⋊Dic3—31no—
31C22×C3⋊Dic3—31no—
32C22×C3⋊Dic3—32no—
33C22×C3⋊Dic3Dic312yescomplemented
34C22×C3×Dic3—32no—
35C22×C3×Dic3—32no—
36Dic3×A4—34no—
37Dic3×A4—62no—
38C3⋊SL(2,ℤ/4)Dic314yescomplemented
39C3×SL(2,ℤ/4)—34no—
40C3×SL(2,ℤ/4)—64no—
41C3×#217—34no—
42C3×#217—34no—
43Dic32—62no—
44Dic32—61no—
45Dic32—61no—
46Dic32—122no—
47C22×C33—41no—
48C32×A4C4211yescharacteristic, complemented
49C3×C3⋊Dic3—44no—
50C3×C3⋊Dic3—42no—
51C23×Dic3—32no—
52C23×Dic3—61no—
53C2×C4×A4—91no—
54C2×SL(2,ℤ/4)—32no—
55C2×SL(2,ℤ/4)—62no—
56C2×C3×C22⋊C4—92no—
57C2×#217—32no—
58C2×C4×Dic3—92no—
59C2×C4×Dic3—92no—
60C3⋊#95—92no—
61C3⋊#95—92no—
62C23×C32—31no—
63C23×C32—32no—
64C23×C32—31no—
65C23×C32C4×D312yes—
66C23×C32C2×Dic311yescharacteristic
67C2×C3×A4—22no—
68C2×C3×A4—22no—
69C2×C3×A4—22no—
70C2×C3×A4—22no—
71C2×C3×A4—21no—
72C2×C3×A4—41no—
73C2×C3×A4—42no—
74C2×C3×A4C4×D312yes—
75C2×C3×A4C4×D312yes—
76C2×C3×A4C2×Dic312yes—
77C2×C3⋊Dic3—31no—
78C2×C3⋊Dic3—32no—
79C2×C3⋊Dic3—32no—
80C2×C3⋊Dic3—32no—
81C2×C3⋊Dic3—42no—
82C2×C3⋊Dic3—61no—
83C2×C3⋊Dic3S411yescharacteristic
84C2×C3×Dic3—34no—
85C2×C3×Dic3—34no—
86C2×C3×Dic3—62no—
87C2×C3×Dic3—62no—
88C2×C3×Dic3—62no—
89C2×C3×Dic3—62no—
90C2×C3×Dic3—122no—
91C2×C3×Dic3—122no—
92C2×C3×Dic3—241no—
93C2×C3×Dic3—242no—
94C4×C22⋊C4—271noSylow
95C2×C33—42no—
96C2×C33—41no—
97C24×C3—21no—
98C24×C3D3212yes—
99C22×A4—22no—
100C22×A4—41no—
101C22×A4D3211yescharacteristic
102C22×C3×C4—92no—
103C22×C3×C4—92no—
104C22×Dic3—32no—
105C22×Dic3—32no—
106C22×Dic3—34no—
107C22×Dic3—62no—
108C22×Dic3—92no—
109C22×Dic3—92no—
110C22×Dic3—94no—
111C22×Dic3—181no—
112C22×Dic3—181no—
113C22×Dic3—182no—
114C4×A4—92no—
115SL(2,ℤ/4)—34no—
116SL(2,ℤ/4)—64no—
117C3×C22⋊C4—94no—
118C3×C22⋊C4—94no—
119#217—34no—
120#217—34no—
121C4×Dic3—182no—
122C4×Dic3—184no—
123C4×Dic3—182no—
124C4×Dic3—361no—
125C22×C32—32no—
126C22×C32—31no—
127C22×C32—32no—
128C22×C32—32no—
129C22×C32—31no—
130C22×C32—31no—
131C22×C32—42no—
132C22×C32—61no—
133C22×C32—81no—
134C22×C32—82no—
135C22×C32—161no—
136C22×C32C4×Dic311yescharacteristic, complemented
137C22×C32C2×S411yescharacteristic
138C3×A4—22no—
139C3×A4—21no—
140C3×A4—41no—
141C3×A4C4×Dic312yescomplemented
142C3⋊Dic3—32no—
143C3⋊Dic3—44no—
144C3⋊Dic3SL(2,ℤ/4)12yescomplemented
145C3×Dic3—64no—
146C3×Dic3—64no—
147C3×Dic3—124no—
148C3×Dic3—124no—
149C3×Dic3—242no—
150C3×Dic3—244no—
151C23×C4—91no—
152C2×C22⋊C4—92no—
153C2×C42—271no—
154C2×C42—271no—
155#95—272no—
156C33—41noSylow
157C23×C3—22no—
158C23×C3—21no—
159C23×C3—32no—
160C23×C3—32no—
161C23×C3—32no—
162C23×C3—32no—
163C23×C3—61no—
164C23×C3—61no—
165C23×C3—62no—
166C23×C3D3×Dic312yes—
167C23×C3D3×Dic312yes—
168C23×C3#62412yes—
169C2×A4—22no—
170C2×A4—22no—
171C2×A4—22no—
172C2×A4—42no—
173C2×A4—41no—
174C2×A4D3×Dic312yes—
175C2×A4#62411yescharacteristic
176C2×C3×C4—94no—
177C2×C3×C4—94no—
178C2×C3×C4—182no—
179C2×C3×C4—182no—
180C2×C3×C4—182no—
181C2×C3×C4—182no—
182C2×C3×C4—361no—
183C2×Dic3—32no—
184C2×Dic3—34no—
185C2×Dic3—34no—
186C2×Dic3—61no—
187C2×Dic3—62no—
188C2×Dic3—62no—
189C2×Dic3—62no—
190C2×Dic3—62no—
191C2×Dic3—94no—
192C2×Dic3—94no—
193C2×Dic3—92no—
194C2×Dic3—94no—
195C2×Dic3—122no—
196C2×Dic3—182no—
197C2×Dic3—182no—
198C2×Dic3—181no—
199C2×Dic3—182no—
200C2×Dic3—182no—
201C2×Dic3—182no—
202C2×Dic3—242no—
203C2×C32—32no—
204C2×C32—31no—
205C2×C32—31no—
206C2×C32—42no—
207C2×C32—42no—
208C2×C32—42no—
209C2×C32—81no—
210C2×C32—82no—
211C2×C32—82no—
212C2×C32—82no—
213C2×C32—82no—
214C2×C32—162no—
215C2×C32—161no—
216C2×C32C4×S412yes—
217C2×C32C2×SL(2,ℤ/4)11yescharacteristic
218C24#111311yescharacteristic
219C22×C4—92no—
220C22×C4—92no—
221C22×C4—92no—
222C22×C4—271no—
223C22×C4—271no—
224C22×C4—272no—
225C22⋊C4—94no—
226C22⋊C4—94no—
227C42—541no—
228C42—542no—
229C42—541no—
230C22×C3—21no—
231C22×C3—21no—
232C22×C3—32no—
233C22×C3—32no—
234C22×C3—32no—
235C22×C3—32no—
236C22×C3—32no—
237C22×C3—32no—
238C22×C3—32no—
239C22×C3—32no—
240C22×C3—32no—
241C22×C3—41no—
242C22×C3—61no—
243C22×C3—61no—
244C22×C3—62no—
245C22×C3—62no—
246C22×C3—62no—
247C22×C3—61no—
248C22×C3—62no—
249C22×C3—62no—
250C22×C3—82no—
251C22×C3—161no—
252C22×C3Dic3212yescomplemented
253C22×C3D3×S412yes—
254A4—22no—
255A4—41no—
256A4Dic3211yescharacteristic, complemented
257C3×C4—184no—
258C3×C4—184no—
259C3×C4—362no—
260Dic3—34no—
261Dic3—62no—
262Dic3—64no—
263Dic3—64no—
264Dic3—94no—
265Dic3—124no—
266Dic3—182no—
267Dic3—244no—
268C32—42no—
269C32—81no—
270C32—82no—
271C32—161no—
272C32C4×SL(2,ℤ/4)11yescharacteristic, complemented
273C23—31no—
274C23—32no—
275C23—31no—
276C23#629412yes—
277C23#629411yescharacteristic
278C2×C4—91no—
279C2×C4—94no—
280C2×C4—94no—
281C2×C4—182no—
282C2×C4—182no—
283C2×C4—182no—
284C2×C4—182no—
285C2×C4—272no—
286C2×C4—272no—
287C2×C4—271no—
288C2×C4—272no—
289C2×C4—541no—
290C2×C3—22no—
291C2×C3—21no—
292C2×C3—32no—
293C2×C3—32no—
294C2×C3—32no—
295C2×C3—32no—
296C2×C3—42no—
297C2×C3—41no—
298C2×C3—61no—
299C2×C3—62no—
300C2×C3—61no—
301C2×C3—82no—
302C2×C3—82no—
303C2×C3—82no—
304C2×C3—162no—
305C2×C3—161no—
306C2×C3Dic3×S412yes—
307C2×C3(C3⋊S4)⋊C412yes—
308C2×C3D3×SL(2,ℤ/4)12yes—
309C22—32no—
310C22—31no—
311C22—32no—
312C22—32no—
313C22—31no—
314C22—31no—
315C22—61no—
316C22#8881311yescharacteristic, complemented
317C22#8910111yescharacteristic
318C4—92no—
319C4—184no—
320C4—184no—
321C4—272no—
322C3—21no—
323C3—41no—
324C3—82no—
325C3—161no—
326C3Dic3×SL(2,ℤ/4)12yescomplemented
327C2—32no—
328C2—31no—
329C2—31no—
330C2#1172421312yes—
331C2#1172421511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22#89101
commutator, hypocenterC32×A4C42
Fitting, socleC24×C32C2×D3
prime radical, cosocleC22×C32×A4C22
LCSG ▷ C32×A4
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
6. A4×C3⋊Dic3C4semidirect
7. C3×C3⋊SL(2,ℤ/4)C4semidirect
33. C22×C3⋊Dic3Dic3semidirect
38. C3⋊SL(2,ℤ/4)Dic3semidirect
48. C32×A4C42semidirect
136. C22×C32C4×Dic3semidirect
141. C3×A4C4×Dic3semidirect
144. C3⋊Dic3SL(2,ℤ/4)semidirect
252. C22×C3Dic32semidirect
256. A4Dic32semidirect
272. C32C4×SL(2,ℤ/4)semidirect
316. C22#88813semidirect
326. C3Dic3×SL(2,ℤ/4)semidirect

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
Dic3×SL(2,ℤ/4)3
#105917053
#888134