Finite Group Databasebeta

← #13135188

Subgroups — #13135188

Order 1728.

412 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 51 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
1#11721837C211yesmaximal, characteristic
2C3×#63393C211yesmaximal, characteristic, complemented
3C9×#833C211yesmaximal, characteristic, complemented
4#10589174—31nomaximal
5#10588153—31nomaximal
6C23×C3⋊D9C411yescharacteristic, complemented
7C2×C3×D4×C9C2211yescharacteristic
8C2×#6233—21no—
9#88551C411yescharacteristic, complemented
10#88536—21no—
11C3×#3676—21no—
12C9×#208—21no—
13#64146—31no—
14C3×#834—31no—
15C3×#833D311yescharacteristic
16#63499—31no—
17#63499—61no—
18C9×#100—31no—
19#63393D311yescharacteristic, complemented
20C23×C3×C9C2×C411yescharacteristic
21C23×C3×C9D411yescharacteristic
22C22×C3⋊D9—21no—
23C22×C3⋊D9—42no—
24C22×C3⋊D9—41no—
25C3×D4×C9—41no—
26C2×C3×C4×C9D411yescharacteristic
27#6233—42no—
28C2×C3⋊Dic9—21no—
29C2×C3×Dic9—41no—
30C9×C3⋊C8—41no—
31#4475—91no—
32#4471—91no—
33C23×C3⋊D3—31no—
34C2×C32×D4C2×D311yescharacteristic
35C2×#638—61no—
36C23×D9—31no—
37C23×D9—61no—
38#3757—31no—
39#3752—61no—
40C3×#217—61no—
41C3×#208—21no—
42C2×D4×C9—21no—
43C2×D4×C9C2×D311yescharacteristic
44C2×#611—62no—
45C2×#611—61no—
46C9×M16—61no—
47#3676—21no—
48#3676—31no—
49#3676—61no—
50(C2×D9)⋊C4—62no—
51(C2×D9)⋊C4—61no—
52C22×C3×C9—21no—
53C22×C3×C9—21no—
54C22×C3×C9—41no—
55C22×C3×C9—41no—
56C22×C3×C9C22⋊C411yescharacteristic
57C2×C3⋊D9—22no—
58C2×C3⋊D9—41no—
59C2×C3⋊D9—42no—
60C2×C3⋊D9—41no—
61C2×C3⋊D9—42no—
62C3×C4×C9—21no—
63C3⋊Dic9—41no—
64C3×Dic9—81no—
65#937—91no—
66#937—91no—
67#937—181no—
68C3×#100—31no—
69C3×#99—91no—
70#834—31no—
71#833D911yescharacteristic, complemented
72C23×C32C4×D311yescharacteristic
73C23×C32D1211yescharacteristic
74C22×C3⋊D3—61no—
75C22×C3⋊D3—122no—
76C22×C3⋊D3—121no—
77C32×D4—41no—
78C2×C32×C4#6711yescharacteristic
79#638—122no—
80C2×C3⋊Dic3—61no—
81C2×C3×Dic3—121no—
82C23×C9—21no—
83C23×C9—21no—
84C23×C9C4×D311yescharacteristic
85C23×C9#6711yescharacteristic
86C22×D9—62no—
87C22×D9—61no—
88C22×D9—124no—
89C22×D9—122no—
90C22×D9—122no—
91C22×D9—121no—
92C3×C3⋊C8—41no—
93D4×C9—41no—
94D4×C9—42no—
95C2×C4×C9—21no—
96C2×C4×C9D1211yescharacteristic
97#611—122no—
98#611—124no—
99C2×Dic9—41no—
100C2×Dic9—62no—
101C2×Dic9—61no—
102C8×C9—121no—
103AΣL(1,𝔽16)—271noSylow
104C2×C3×C9—21no—
105C2×C3×C9—41no—
106C2×C3×C9—41no—
107C2×C3×C9#9911yescharacteristic
108C3⋊D9—42no—
109C23×D3—91no—
110C23×D3—91no—
111C23×D3—181no—
112C2×C3×D4—21no—
113C2×C3×D4C2×D911yescharacteristic
114C2×C3×D4D3211yescharacteristic
115C2×#67—182no—
116C2×#67—181no—
117C2×#67—181no—
118C3×M16—61no—
119C3×C22⋊C4—181no—
120#217—61no—
121#217—91no—
122#217—91no—
123#217—181no—
124(C2×D3)⋊C4—182no—
125(C2×D3)⋊C4—181no—
126(C2×D3)⋊C4—181no—
127#208—21no—
128C22×C32—21no—
129C22×C32—21no—
130C22×C32—41no—
131C22×C32—41no—
132C22×C32(C2×D3)⋊C411yescharacteristic
133C2×C3⋊D3—62no—
134C2×C3⋊D3—121no—
135C2×C3⋊D3—122no—
136C2×C3⋊D3—121no—
137C2×C3⋊D3—122no—
138C32×C4—21no—
139C3⋊Dic3—121no—
140C3×Dic3—241no—
141C22×C9—22no—
142C22×C9—21no—
143C22×C9—21no—
144C22×C9—21no—
145C22×C9—41no—
146C22×C9—41no—
147C22×C9—41no—
148C22×C9—42no—
149C22×C9—42no—
150C22×C9(C2×D3)⋊C411yescharacteristic
151C2×D9—64no—
152C2×D9—62no—
153C2×D9—122no—
154C2×D9—121no—
155C2×D9—122no—
156C2×D9—124no—
157C2×D9—122no—
158C2×D9—124no—
159C2×D9—121no—
160C2×D9—122no—
161C4×C9—21no—
162C4×C9—41no—
163Dic9—81no—
164Dic9—122no—
165Dic9—121no—
166#120—271no—
167#100—31no—
168#99—91no—
169C3×C9AΣL(1,𝔽16)11yescharacteristic, complemented, Sylow
170C23×C3—21no—
171C23×C3—21no—
172C23×C3C4×D911yescharacteristic
173C23×C3D3611yescharacteristic
174C23×C3#62411yescharacteristic
175C23×C3#62611yescharacteristic
176C22×D3—182no—
177C22×D3—181no—
178C22×D3—181no—
179C22×D3—364no—
180C22×D3—362no—
181C22×D3—362no—
182C22×D3—362no—
183C22×D3—361no—
184C22×D3—361no—
185C3×D4—41no—
186C3×D4—41no—
187C3×D4—42no—
188C2×C3×C4—21no—
189C2×C3×C4—361no—
190C2×C3×C4#61111yescharacteristic
191C2×C3×C4#62611yescharacteristic
192#67—362no—
193#67—362no—
194#67—364no—
195C2×Dic3—121no—
196C2×Dic3—182no—
197C2×Dic3—181no—
198C2×Dic3—181no—
199C3×C8—121no—
200C3⋊C8—41no—
201C2×C32—21no—
202C2×C32—41no—
203C2×C32—41no—
204C2×C32#80611yescharacteristic
205C3⋊D3—122no—
206C2×C9—22no—
207C2×C9—21no—
208C2×C9—21no—
209C2×C9—42no—
210C2×C9—41no—
211C2×C9—41no—
212C2×C9—81no—
213C2×C9#80611yescharacteristic
214D9—124no—
215D9—122no—
216C24—271no—
217C2×D4—541no—
218C2×D4D3×D911yescharacteristic
219M16—61no—
220C22⋊C4—181no—
221C22⋊C4—271no—
222C22⋊C4—541no—
223C22×C3—22no—
224C22×C3—21no—
225C22×C3—21no—
226C22×C3—21no—
227C22×C3—21no—
228C22×C3—21no—
229C22×C3—41no—
230C22×C3—41no—
231C22×C3—41no—
232C22×C3—41no—
233C22×C3—41no—
234C22×C3—42no—
235C22×C3—42no—
236C22×C3(C2×D9)⋊C411yescharacteristic
237C22×C3#372211yescharacteristic
238C2×D3—182no—
239C2×D3—184no—
240C2×D3—182no—
241C2×D3—361no—
242C2×D3—362no—
243C2×D3—361no—
244C2×D3—362no—
245C2×D3—362no—
246C2×D3—364no—
247C2×D3—364no—
248C2×D3—362no—
249C2×D3—361no—
250C2×D3—361no—
251C2×D3—362no—
252C2×D3—362no—
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—41no—
256C3×C4—721no—
257Dic3—241no—
258Dic3—361no—
259Dic3—362no—
260Dic3—361no—
261C32#447511yescharacteristic
262C9—21no—
263C9#447111yescharacteristic, complemented
264C23—541no—
265C23—1081no—
266C23—1082no—
267C23(C3⋊D9)⋊C411yescharacteristic
268C23#619211yescharacteristic
269D4—41no—
270D4—1082no—
271C2×C4—361no—
272C2×C4—541no—
273C2×C4#619511yescharacteristic
274C8—121no—
275C2×C3—22no—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3—21no—
279C2×C3—42no—
280C2×C3—41no—
281C2×C3—41no—
282C2×C3—41no—
283C2×C3—41no—
284C2×C3—81no—
285C2×C3#6336511yescharacteristic
286C2×C3#6358111yescharacteristic
287D3—364no—
288D3—362no—
289D3—362no—
290C22—21no—
291C22—21no—
292C22—41no—
293C22—41no—
294C22—542no—
295C22—1081no—
296C22—1082no—
297C22—1082no—
298C22—1081no—
299C22#8844511yescharacteristic
300C4—21no—
301C4—721no—
302C4—1081no—
303C3—21no—
304C3#1058815311yescharacteristic, complemented
305C3#1058917411yescharacteristic
306C2—21no—
307C2—41no—
308C2—41no—
309C2—1082no—
310C2#1172033011yescharacteristic
subgroupG/H
centerC2#11720330
commutatorC23×C3×C9C2×C4
FrattiniC2×C3×D4D32
Fitting, cosocleC2×C3×D4×C9C22
socleC2×C32#806
prime radical#11721837C2
hypercenterC2×D4D3×D9
hypocenterC3×C9AΣL(1,𝔽16)
LCSG ▷ C23×C3×C9 ▷ C22×C3×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C3×#63393C2semidirect
3. C9×#833C2semidirect
6. C23×C3⋊D9C4semidirect
9. #88551C4semidirect
19. #63393D3semidirect
71. #833D9semidirect
169. C3×C9AΣL(1,𝔽16)semidirect
263. C9#4471semidirect
304. C3#10588153semidirect

Maximal subgroups

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

subgroupindex
#117218372
C3×#633932
C9×#8332
#105891743
#105881533