Finite Group Databasebeta

← #13139313

Subgroups — #13139313

Order 1728.

436 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 101 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#11721819C211yesmaximal, characteristic
2C2×C9×C4⋊Dic3C211yesmaximal, characteristic, complemented
3C3×#63389C211yesmaximal, characteristic, complemented
4#11720267C212yesmaximal, complemented
5#11720262C212yesmaximal, complemented
6#10591133—31nomaximal
7#10588650—31nomaximal
8C22×C3×C4×C9C2211yescharacteristic
9C2×#6233—21no—
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C22×C9×Dic3—21no—
12#88536—21no—
13C4⋊(C3⋊Dic9)C2211yescharacteristic
14C9×C4⋊Dic3—21no—
15C9×C4⋊Dic3C2212yes—
16C2×C3×C9⋊C8C2212yes—
17#64139—31no—
18C2×C3×C4⋊Dic3D311yescharacteristic
19C3×#830—31no—
20#63575—32no—
21#63564—32no—
22C2×C9×C4⋊C4—31no—
23#63492—31no—
24#63492—61no—
25#63389D311yescharacteristic, complemented
26#63369—32no—
27#63366—32no—
28C23×C3×C9D411yescharacteristic
29C22×C3⋊D9—21no—
30C2×C3×C4×C9—22no—
31C2×C3×C4×C9D411yescharacteristic
32C2×C3×C4×C9C2311yescharacteristic
33#6233—42no—
34C2×C3⋊Dic9—21no—
35C4⋊(C3⋊D9)—22no—
36C2×C9×Dic3—22no—
37C2×C9×Dic3—42no—
38C3×C9⋊C8—22no—
39#4967—91no—
40#4736—91no—
41C22×C32×C4C2×D311yescharacteristic
42C2×#638—61no—
43C2×C4⋊(C3⋊D3)—31no—
44C22×C3×Dic3—21no—
45#3752—61no—
46C4⋊(C3⋊Dic3)—31no—
47C3×C4⋊Dic3—21no—
48C3×C4⋊Dic3C2×D312yes—
49C2×C3×C3⋊C8—32no—
50C22×C4×C9—21no—
51C22×C4×C9—61no—
52C22×C4×C9C2×D311yescharacteristic
53C2×#611—62no—
54C2×#611—61no—
55C2×D36—31no—
56C2×D36—61no—
57C9×C4⋊C4—32no—
58C9×C4⋊C4—61no—
59(C2×D9)⋊C4—62no—
60(C2×D9)⋊C4—61no—
61C4⋊Dic9—31no—
62C4⋊Dic9—61no—
63C2×C9⋊C8C2×D312yes—
64C22×C3×C9—21no—
65C22×C3×C9—21no—
66C22×C3×C9D812yes—
67C22×C3×C9C2×D411yescharacteristic
68C2×C3⋊D9—41no—
69C2×C3⋊D9—41no—
70C2×C3⋊D9—41no—
71C3×C4×C9—21no—
72C3×C4×C9#4112yes—
73C3⋊Dic9—41no—
74C9×Dic3—42no—
75C2×C3×C4⋊C4—31no—
76#930—91no—
77#930—91no—
78#930—181no—
79C2×C4⋊Dic3D911yescharacteristic, complemented
80C3×C22⋊C8—91no—
81#830—31no—
82#821—92no—
83#818—92no—
84#809—92no—
85#807—92no—
86C23×C32#6711yescharacteristic
87C22×C3⋊D3—61no—
88C2×C32×C4—22no—
89C2×C32×C4#6711yescharacteristic
90C2×C32×C4C22×D311yescharacteristic
91#638—122no—
92C2×C3⋊Dic3—61no—
93C4⋊(C3⋊D3)—62no—
94C2×C3×Dic3—22no—
95C2×C3×Dic3—42no—
96C23×C9—21no—
97C23×C9D1211yescharacteristic
98C22×D9—62no—
99C22×D9—61no—
100C3×C3⋊C8—62no—
101C2×C4×C9—22no—
102C2×C4×C9—21no—
103C2×C4×C9—21no—
104C2×C4×C9—42no—
105C2×C4×C9—62no—
106C2×C4×C9—122no—
107C2×C4×C9D1211yescharacteristic
108C2×C4×C9C22×D311yescharacteristic
109#611—124no—
110#611—122no—
111C2×Dic9—62no—
112C2×Dic9—61no—
113D36—62no—
114D36—122no—
115C9⋊C8—22no—
116#480—271noSylow
117C2×C3×C9—22no—
118C2×C3×C9#12311yescharacteristic
119C2×C3×C9C2×D811yescharacteristic
120C2×C3×C9#13711yescharacteristic
121C3⋊D9—81no—
122C22×C3×C4—21no—
123C22×C3×C4—61no—
124C22×C3×C4C2×D911yescharacteristic
125C22×C3×C4D3211yescharacteristic
126C2×#67—182no—
127C2×#67—181no—
128C2×#67—181no—
129C22×Dic3—21no—
130C2×D12—91no—
131C2×D12—91no—
132C2×D12—181no—
133C2×C3×C8—92no—
134C3×C4⋊C4—32no—
135C3×C4⋊C4—61no—
136(C2×D3)⋊C4—182no—
137(C2×D3)⋊C4—181no—
138(C2×D3)⋊C4—181no—
139C4⋊Dic3—21no—
140C4⋊Dic3—91no—
141C4⋊Dic3—91no—
142C4⋊Dic3—181no—
143C4⋊Dic3C2×D912yes—
144C2×C3⋊C8—32no—
145C22×C32—21no—
146C22×C32—21no—
147C22×C32#21312yes—
148C22×C32C2×#6711yescharacteristic
149C2×C3⋊D3—121no—
150C2×C3⋊D3—121no—
151C2×C3⋊D3—121no—
152C32×C4—21no—
153C32×C4#23512yes—
154C3⋊Dic3—121no—
155C3×Dic3—42no—
156C22×C9—21no—
157C22×C9—21no—
158C22×C9—22no—
159C22×C9—21no—
160C22×C9—41no—
161C22×C9—41no—
162C22×C9D2412yes—
163C22×C9C2×D1211yescharacteristic
164C2×D9—122no—
165C2×D9—122no—
166C2×D9—122no—
167C2×D9—121no—
168C2×D9—121no—
169C2×D9—121no—
170C4×C9—21no—
171C4×C9—22no—
172C4×C9—41no—
173C4×C9—122no—
174C4×C9#23712yes—
175Dic9—122no—
176Dic9—121no—
177#121—271no—
178C2×C4⋊C4—31no—
179#107—272no—
180#102—272no—
181C22⋊C8—91no—
182C3×C9#48011yescharacteristic, complemented, Sylow
183C23×C3—21no—
184C23×C3#61111yescharacteristic
185C23×C3#62611yescharacteristic
186C22×D3—182no—
187C22×D3—181no—
188C22×D3—181no—
189C2×C3×C4—22no—
190C2×C3×C4—22no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—42no—
194C2×C3×C4—62no—
195C2×C3×C4—122no—
196C2×C3×C4#61111yescharacteristic
197C2×C3×C4C22×D911yescharacteristic
198C2×C3×C4#62611yescharacteristic
199C2×C3×C4C2×D3211yescharacteristic
200#67—364no—
201#67—362no—
202#67—362no—
203C2×Dic3—22no—
204C2×Dic3—42no—
205C2×Dic3—182no—
206C2×Dic3—181no—
207C2×Dic3—181no—
208D12—182no—
209D12—182no—
210D12—362no—
211C3×C8—182no—
212C3⋊C8—62no—
213C2×C32—22no—
214C2×C32#92911yescharacteristic
215C2×C32C2×#21311yescharacteristic
216C2×C32#94211yescharacteristic
217C3⋊D3—241no—
218C2×C9—22no—
219C2×C9—21no—
220C2×C9—21no—
221C2×C9—21no—
222C2×C9—42no—
223C2×C9#88611yescharacteristic
224C2×C9C2×D2411yescharacteristic
225C2×C9#90911yescharacteristic
226D9—242no—
227D9—241no—
228C2×D4—271no—
229C2×D4—541no—
230C22×C4—61no—
231C22×C4D3×D911yescharacteristic
232C2×C8—92no—
233C4⋊C4—32no—
234C4⋊C4—61no—
235C4⋊C4—271no—
236C22⋊C4—541no—
237C22×C3—21no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—21no—
242C22×C3—22no—
243C22×C3—41no—
244C22×C3—41no—
245C22×C3#367312yes—
246C22×C3C2×#61111yescharacteristic
247C22×C3#371412yes—
248C22×C3C2×#62611yescharacteristic
249C2×D3—362no—
250C2×D3—362no—
251C2×D3—362no—
252C2×D3—361no—
253C2×D3—361no—
254C2×D3—361no—
255C2×D3—361no—
256C2×D3—361no—
257C2×D3—361no—
258C3×C4—21no—
259C3×C4—21no—
260C3×C4—22no—
261C3×C4—41no—
262C3×C4—122no—
263C3×C4#369712yes—
264C3×C4#380412yes—
265Dic3—42no—
266Dic3—362no—
267Dic3—361no—
268Dic3—361no—
269C32#496711yescharacteristic
270C9—21no—
271C9#473611yescharacteristic, complemented
272C23—541no—
273C23#619511yescharacteristic
274D4—542no—
275D4—1082no—
276C2×C4—22no—
277C2×C4—62no—
278C2×C4—122no—
279C2×C4—541no—
280C2×C4#619511yescharacteristic
281C2×C4C2×D3×D911yescharacteristic
282C8—182no—
283C2×C3—22no—
284C2×C3—22no—
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—42no—
289C2×C3#6349111yescharacteristic
290C2×C3C2×#367311yescharacteristic
291C2×C3#6350511yescharacteristic
292C2×C3C2×#371411yescharacteristic
293C2×C3#6383311yescharacteristic
294C2×C3#6396611yescharacteristic
295D3—722no—
296D3—721no—
297D3—721no—
298C22—21no—
299C22—21no—
300C22—1081no—
301C22—1081no—
302C22—1081no—
303C22#8842212yes—
304C22C2×#619511yescharacteristic
305C4—21no—
306C4—122no—
307C4—1081no—
308C4#8865812yes—
309C3—21no—
310C3#1058865011yescharacteristic, complemented
311C3#1059113311yescharacteristic
312C2—22no—
313C2—2161no—
314C2C2×#8842211yescharacteristic
315C2#1172089511yescharacteristic
316C2#1172125111yescharacteristic
subgroupG/H
centerC22C2×#6195
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC22×C3×C4×C9C22
socleC22×C32C2×#67
prime radical#11721819C2
hypercenterC22×C4D3×D9
hypocenterC3×C9#480
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C9×C4⋊Dic3C2semidirect
3. C3×#63389C2semidirect
4. #11720267C2semidirect
5. #11720262C2semidirect
25. #63389D3semidirect
79. C2×C4⋊Dic3D9semidirect
182. C3×C9#480semidirect
271. C9#4736semidirect
310. C3#10588650semidirect

Maximal subgroups

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

subgroupindex
#117218192
C2×C9×C4⋊Dic32
C3×#633892
#117202672
#117202622
#105911333
#105886503