Finite Group Databasebeta

← #13146967

Subgroups — #13146967

Order 1728.

431 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 431 conjugacy classes of subgroups — too many to draw. Only its 83 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×C4×C9×A4C211yesmaximal, characteristic
2C2×C9⋊SL(2,ℤ/4)C211yesmaximal, characteristic
3C2×C9⋊SL(2,ℤ/4)C211yesmaximal, characteristic
4#10593645—31nomaximal
5((C22⋊C9)⋊C4)⋊C4—32nomaximal
6#10588890—31nomaximal
7C22×C9×A4C2211yescharacteristic
8C4×C9×A4—21no—
9C9⋊SL(2,ℤ/4)—21no—
10C9⋊SL(2,ℤ/4)C412yescomplemented
11#88534—41nomaximal
12C2×C3×C4×A4D311yescharacteristic
13C2×C3⋊SL(2,ℤ/4)—31no—
14C2×C3⋊SL(2,ℤ/4)—31no—
15C23×C4×C9D311yescharacteristic
16C2×C4×C22⋊C9D312yes—
17C2×(C22⋊C9)⋊C4—32no—
18C2×(C22⋊C9)⋊C4—32no—
19C2×#3676—31no—
20C2×#3676—31no—
21C2×Dic9⋊C4—31no—
22C2×Dic9⋊C4—31no—
23C9⋊#95—31no—
24C9⋊#95—31no—
25C2×C9×A4C2×C411yescharacteristic
26C2×C9×A4D411yescharacteristic
27C2×C9×A4Q811yescharacteristic
28C2×C3×C4×C9—41no—
29C2×C3⋊Dic9—41no—
30C2×C3⋊Dic9—41no—
31SL(2,ℤ/4)⋊C4—93no—
32#5210—91no—
33C22×C3×A4C2×D311yescharacteristic
34C3×C4×A4—21no—
35C3⋊SL(2,ℤ/4)—32no—
36C3⋊SL(2,ℤ/4)—61no—
37C24×C9C2×D311yescharacteristic
38C22×C22⋊C9C2×D312yes—
39#3750—121no—
40C22×C4×C9—21no—
41C22×C4×C9—31no—
42C22×C4×C9—31no—
43C22×C4×C9—61no—
44C22×Dic9—31no—
45C22×Dic9—31no—
46C22×Dic9—31no—
47C22×Dic9—31no—
48C4×C22⋊C9—22no—
49(C22⋊C9)⋊C4—34no—
50(C22⋊C9)⋊C4—62no—
51#3676—32no—
52#3676—32no—
53#3676—61no—
54#3676—61no—
55Dic9⋊C4—62no—
56Dic9⋊C4—61no—
57Dic9⋊C4—61no—
58Dic9⋊C4—122no—
59C22×C3×C9—41no—
60C9×A4C4⋊C411yescharacteristic, complemented
61C3×C4×C9—81no—
62C3⋊Dic9—42no—
63C3⋊Dic9—81no—
64C23×C3×C4C3⋊D311yescharacteristic
65C2×C4×A4D913yes—
66C2×SL(2,ℤ/4)—93no—
67C2×SL(2,ℤ/4)—93no—
68C2×#217—91no—
69C2×#217—91no—
70C2×Dic3⋊C4—91no—
71C2×Dic3⋊C4—91no—
72C3⋊#95—91no—
73C3⋊#95—91no—
74C2×C3×A4Dic611yescharacteristic
75C2×C3×A4C4×D311yescharacteristic
76C2×C3×A4#6711yescharacteristic
77C2×C32×C4—41no—
78C2×C3⋊Dic3—121no—
79C2×C3⋊Dic3—121no—
80C23×C9—31no—
81C23×C9—31no—
82C23×C9—31no—
83C23×C9—31no—
84C23×C9Dic611yescharacteristic
85C23×C9C4×D311yescharacteristic
86C23×C9#6711yescharacteristic
87C2×C22⋊C9Dic612yes—
88C2×C22⋊C9C4×D312yes—
89C2×C22⋊C9#6712yes—
90C2×C4×C9—32no—
91C2×C4×C9—31no—
92C2×C4×C9—42no—
93C2×C4×C9—61no—
94C2×C4×C9—61no—
95C2×C4×C9—61no—
96C2×C4×C9S411yescharacteristic
97C2×Dic9—32no—
98C2×Dic9—32no—
99C2×Dic9—61no—
100C2×Dic9—61no—
101C2×Dic9—61no—
102C2×Dic9—61no—
103C2×Dic9—61no—
104C2×Dic9—61no—
105C2×Dic9—61no—
106C2×Dic9—61no—
107C2×Dic9—61no—
108C2×Dic9—61no—
109C2×Dic9—122no—
110C2×Dic9—122no—
111#385—271noSylow
112C2×C3×C9—41no—
113C2×C3×C9—41no—
114C2×C3×C9—41no—
115C24×C3C2×C3⋊D311yescharacteristic
116C22×A4C2×D913yes—
117C22×C3×C4—21no—
118C22×C3×C4—31no—
119C22×C3×C4—31no—
120C22×C3×C4—61no—
121C22×Dic3—91no—
122C22×Dic3—91no—
123C22×Dic3—91no—
124C22×Dic3—91no—
125C4×A4—23no—
126SL(2,ℤ/4)—96no—
127SL(2,ℤ/4)—183no—
128#217—92no—
129#217—92no—
130#217—181no—
131#217—181no—
132Dic3⋊C4—182no—
133Dic3⋊C4—181no—
134Dic3⋊C4—181no—
135Dic3⋊C4—363no—
136C22×C32—41no—
137C3×A4Dic3⋊C411yescharacteristic
138C32×C4—81no—
139C3⋊Dic3—122no—
140C3⋊Dic3—241no—
141C22×C9—31no—
142C22×C9—31no—
143C22×C9—31no—
144C22×C9—31no—
145C22×C9—31no—
146C22×C9—31no—
147C22×C9—31no—
148C22×C9—31no—
149C22×C9—31no—
150C22×C9—42no—
151C22×C9—61no—
152C22×C9Dic3⋊C411yescharacteristic, complemented
153C22×C9C2×S411yescharacteristic
154C22⋊C9Dic3⋊C412yescomplemented
155C4×C9—21no—
156C4×C9—61no—
157C4×C9—82no—
158Dic9—62no—
159Dic9—62no—
160Dic9—124no—
161Dic9—121no—
162Dic9—121no—
163Dic9—242no—
164C23×C4C3⋊D911yescharacteristic
165C2×C4⋊C4—271no—
166C2×C4⋊C4—271no—
167C2×C22⋊C4—271no—
168C2×C22⋊C4—271no—
169#95—271no—
170#95—271no—
171C3×C9—41noSylow
172C23×C3—31no—
173C23×C3—31no—
174C23×C3—31no—
175C23×C3—31no—
176C23×C3#63411yescharacteristic
177C23×C3C4×C3⋊D311yescharacteristic
178C23×C3#63811yescharacteristic
179C2×A4Dic1813yes—
180C2×A4C4×D913yes—
181C2×A4#61113yes—
182C2×C3×C4—32no—
183C2×C3×C4—31no—
184C2×C3×C4—43no—
185C2×C3×C4—61no—
186C2×C3×C4—61no—
187C2×C3×C4—61no—
188C2×C3×C4C3⋊S411yescharacteristic
189C2×Dic3—92no—
190C2×Dic3—92no—
191C2×Dic3—181no—
192C2×Dic3—181no—
193C2×Dic3—181no—
194C2×Dic3—181no—
195C2×Dic3—181no—
196C2×Dic3—181no—
197C2×Dic3—181no—
198C2×Dic3—181no—
199C2×Dic3—181no—
200C2×Dic3—181no—
201C2×Dic3—363no—
202C2×Dic3—363no—
203C2×C32—41no—
204C2×C32—41no—
205C2×C32—41no—
206C2×C9—31no—
207C2×C9—31no—
208C2×C9—31no—
209C2×C9—31no—
210C2×C9—42no—
211C2×C9—42no—
212C2×C9—42no—
213C2×C9A4⋊Q811yescharacteristic
214C2×C9C4×S411yescharacteristic
215C2×C9GL(2,ℤ/4)11yescharacteristic
216C24C2×C3⋊D911yescharacteristic
217C22×C4—21no—
218C22×C4—31no—
219C22×C4—31no—
220C22×C4—61no—
221C22×C4—271no—
222C22×C4—271no—
223C22×C4—271no—
224C22×C4—271no—
225C4⋊C4—542no—
226C4⋊C4—541no—
227C4⋊C4—541no—
228C22⋊C4—272no—
229C22⋊C4—272no—
230C22⋊C4—541no—
231C22⋊C4—541no—
232C22×C3—31no—
233C22×C3—31no—
234C22×C3—31no—
235C22×C3—31no—
236C22×C3—31no—
237C22×C3—31no—
238C22×C3—31no—
239C22×C3—31no—
240C22×C3—31no—
241C22×C3—43no—
242C22×C3—61no—
243C22×C3#375011yescharacteristic
244C22×C3C2×C3⋊S411yescharacteristic
245A4Dic9⋊C413yescomplemented
246C3×C4—21no—
247C3×C4—61no—
248C3×C4—83no—
249Dic3—182no—
250Dic3—182no—
251Dic3—366no—
252Dic3—361no—
253Dic3—361no—
254Dic3—723no—
255C32—41no—
256C9—42no—
257C9SL(2,ℤ/4)⋊C411yescharacteristic, complemented
258C23—31no—
259C23—31no—
260C23—31no—
261C23—31no—
262C23#622611yescharacteristic
263C23C4×C3⋊D911yescharacteristic
264C23#623311yescharacteristic
265C2×C4—32no—
266C2×C4—31no—
267C2×C4—61no—
268C2×C4—61no—
269C2×C4—61no—
270C2×C4—272no—
271C2×C4—272no—
272C2×C4—541no—
273C2×C4—541no—
274C2×C4—541no—
275C2×C4—541no—
276C2×C4—541no—
277C2×C4—541no—
278C2×C4—541no—
279C2×C4—541no—
280C2×C4—541no—
281C2×C4—541no—
282C2×C4C9⋊S411yescharacteristic
283C2×C3—31no—
284C2×C3—31no—
285C2×C3—31no—
286C2×C3—31no—
287C2×C3—43no—
288C2×C3—43no—
289C2×C3—43no—
290C2×C3#6425911yescharacteristic
291C2×C3C4×C3⋊S411yescharacteristic
292C2×C3#6426911yescharacteristic
293C22—31no—
294C22—31no—
295C22—31no—
296C22—31no—
297C22—31no—
298C22—31no—
299C22—31no—
300C22—31no—
301C22—31no—
302C22—61no—
303C22#8853411yescharacteristic, complemented
304C22C2×C9⋊S411yescharacteristic
305C4—21no—
306C4—61no—
307C4—542no—
308C4—542no—
309C4—1081no—
310C4—1081no—
311C3—43no—
312C3#1059364511yescharacteristic
313C2—31no—
314C2—31no—
315C2—31no—
316C2—31no—
317C2#1172236611yescharacteristic
318C2C4×C9⋊S411yescharacteristic
319C2#1172237611yescharacteristic
subgroupG/H
centerC22C2×C9⋊S4
commutatorC2×C9×A4C2×C4
FrattiniC22×C3C2×C3⋊S4
FittingC23×C4×C9D3
socleC24×C3C2×C3⋊D3
prime radicalC22×C3×A4C2×D3
cosocleC22×C9×A4C22
hypercenterC2×C4C9⋊S4
hypocenterC9×A4C4⋊C4
LCSG ▷ C2×C9×A4 ▷ C9×A4
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
10. C9⋊SL(2,ℤ/4)C4semidirect
60. C9×A4C4⋊C4semidirect
152. C22×C9Dic3⋊C4semidirect
154. C22⋊C9Dic3⋊C4semidirect
245. A4Dic9⋊C4semidirect
257. C9SL(2,ℤ/4)⋊C4semidirect
303. C22#88534semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C9×A42
C2×C9⋊SL(2,ℤ/4)2
#105936453
((C22⋊C9)⋊C4)⋊C43
#105888903
#885344