Finite Group Databasebeta

← #13138556

Subgroups — #13138556

Order 1728.

494 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 105 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×#88524C211yesmaximal, characteristic, complemented
2C2×C9×#204C211yesmaximal, characteristic, complemented
3#11721040C211yesmaximal, characteristic
4#11720924C211yesmaximal, characteristic, complemented
5C2×#88426C211yesmaximal, characteristic, complemented
6C2×#88424C211yesmaximal, characteristic, complemented
7C3×C8×Dic9C211yesmaximal, characteristic, complemented
8#10590877—31nomaximal
9#10588855—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×#6226C2211yescharacteristic
12C2×C9×D12C2211yescharacteristic
13C2×C9×Dic6C2211yescharacteristic
14C2×#6195—22no—
15C2×C3×C8×C9C2211yescharacteristic
16#88524—22no—
17C3×C4×Dic9C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19C9×#204—22no—
20Dic9⋊Dic3—22no—
21#88426—22no—
22#88424—22no—
23C2×#3744—31no—
24C2×C3×#204D311yescharacteristic
25#63911—31no—
26#63866—31no—
27C2×#3717—31no—
28C2×#3716—31no—
29C3×C8×Dic3—31no—
30C2×C9×SD16—31no—
31#63506—31no—
32C2×#3675—31no—
33#63502—31no—
34C2×#3672—31no—
35C2×#3664—31no—
36C2×#3664—61no—
37C8×Dic9D311yescharacteristic, complemented
38C22×C3⋊D9—21no—
39C22×D3×C9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3⋊Dic9—21no—
42C4⋊(C3⋊D9)—21no—
43C4⋊(C3⋊D9)—21no—
44#6226—21no—
45#6226—21no—
46C2×C9×Dic3—21no—
47C2×C3×Dic9D412yes—
48C9×D12—21no—
49C9×D12—21no—
50C9×Dic6—21no—
51C9×Dic6—21no—
52#6195—44no—
53C3×C8×C9D412yescomplemented
54C3×C9⋊C8D412yescomplemented
55#5175—91no—
56#4693—91no—
57C2×C4⋊(C3⋊D3)—31no—
58C2×#634—31no—
59C2×C3×D12C2×D311yescharacteristic
60C2×C3×Dic6C2×D311yescharacteristic
61C2×#626—62no—
62C2×C32×C8C2×D311yescharacteristic
63#3744—62no—
64C3×C4×Dic3—31no—
65C2×C3×C3⋊C8—31no—
66C3×#204—22no—
67Dic3⋊Dic3—62no—
68#3717—62no—
69#3716—62no—
70C2×Q8×C9—31no—
71C2×D4×C9—31no—
72C2×#611—62no—
73C2×D36—31no—
74C2×D36—61no—
75C2×Dic18—31no—
76C2×Dic18—61no—
77C9×SD16—62no—
78C2×C8×C9—21no—
79C2×C8×C9C2×D311yescharacteristic
80#3675—62no—
81#3672—62no—
82Dic9⋊C4—62no—
83C4×Dic9C2×D311yescharacteristic
84C2×C9⋊C8C2×D311yescharacteristic
85#3664—64no—
86#3664—62no—
87C22×C3×C9C2×D411yescharacteristic
88C2×C3⋊D9—41no—
89C2×C3⋊D9—42no—
90C2×D3×C9—41no—
91C2×D3×C9—42no—
92C3×C4×C9C2×D411yescharacteristic
93C3×C4×C9C2×D411yescharacteristic
94C3⋊Dic9—41no—
95C9×Dic3—41no—
96C3×Dic9SD1614yescomplemented
97C2×C3×SD16—31no—
98#944—91no—
99C2×#215—91no—
100#940—91no—
101C2×#214—91no—
102C2×#204—91no—
103C2×#204—91no—
104C2×#204—91no—
105C2×#204—91no—
106C2×#204—181no—
107C2×#204D911yescharacteristic, complemented
108#874—91no—
109#869—91no—
110C3×C4×C8—91no—
111C8×Dic3—31no—
112C22×C3⋊D3—61no—
113C22×C3×D3—21no—
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—61no—
116C4⋊(C3⋊D3)—61no—
117C4⋊(C3⋊D3)—61no—
118#634—61no—
119#634—61no—
120C2×C3×Dic3—21no—
121C2×C3×Dic3—32no—
122C3×D12—21no—
123C3×D12—21no—
124C3×Dic6—21no—
125C3×Dic6—21no—
126#626—124no—
127C23×C9—61no—
128C22×D9—61no—
129C22×D9—121no—
130C32×C8#6712yes—
131C3×C3⋊C8—32no—
132Q8×C9—61no—
133Q8×C9—61no—
134D4×C9—61no—
135D4×C9—61no—
136C2×C4×C9—21no—
137C2×C4×C9—61no—
138C2×C4×C9C22×D311yescharacteristic
139#611—124no—
140C2×Dic9—61no—
141C2×Dic9—121no—
142C2×Dic9D1212yes—
143D36—62no—
144D36—61no—
145D36—61no—
146D36—121no—
147Dic18—62no—
148Dic18—61no—
149Dic18—61no—
150Dic18—121no—
151C8×C9—22no—
152C8×C9D1212yescomplemented
153C9⋊C8D1212yescomplemented
154#492—271noSylow
155C2×C3×C9#12711yescharacteristic
156C2×C3×C9C2×SD1612yes—
157C3⋊D9—81no—
158D3×C9—81no—
159C2×C3×Q8—31no—
160C2×C3×D4—31no—
161C2×#67—182no—
162C2×D12—91no—
163C2×D12—91no—
164C2×D12—181no—
165C2×D12—182no—
166C2×D12C2×D911yescharacteristic
167C2×Dic6—91no—
168C2×Dic6—91no—
169C2×Dic6—181no—
170C2×Dic6C2×D911yescharacteristic
171C3×SD16—62no—
172C2×C3×C8—21no—
173C2×C3×C8—91no—
174C2×C3×C8C2×D911yescharacteristic
175C2×C3×C8D3211yescharacteristic
176C3×C42—91no—
177#215—182no—
178#214—182no—
179C4⋊Dic3—182no—
180Dic3⋊C4—182no—
181C4×Dic3—31no—
182C2×C3⋊C8—31no—
183#204—22no—
184#204—184no—
185#204—182no—
186#204—182no—
187#204—182no—
188#204—182no—
189C22×C32D3×D411yescharacteristic
190C2×C3⋊D3—121no—
191C2×C3⋊D3—122no—
192C2×C3×D3—42no—
193C2×C3×D3—41no—
194C32×C4D3×D411yescharacteristic
195C32×C4C2×#6711yescharacteristic
196C3⋊Dic3—121no—
197C3×Dic3—34no—
198C3×Dic3—41no—
199C22×C9—21no—
200C22×C9—121no—
201C22×C9—122no—
202C22×C9C2×D1211yescharacteristic
203C2×D9—122no—
204C2×D9—121no—
205C2×D9—122no—
206C2×D9—242no—
207C4×C9—21no—
208C4×C9—21no—
209C4×C9—121no—
210C4×C9C2×D1211yescharacteristic
211C4×C9C2×D1211yescharacteristic
212Dic9—122no—
213Dic9—121no—
214Dic9#20414yescomplemented
215C2×SD16—31no—
216C2×SD16—271no—
217C2×SD16—271no—
218C2×SD16—271no—
219C4⋊Q8—271no—
220#127—271no—
221C4×C8—91no—
222C3×C9#49211yescharacteristic, complemented, Sylow
223C23×C3—61no—
224C22×D3—21no—
225C22×D3—181no—
226C22×D3—181no—
227C22×D3—361no—
228C3×Q8—61no—
229C3×Q8—61no—
230C3×D4—61no—
231C3×D4—61no—
232C2×C3×C4—21no—
233C2×C3×C4—61no—
234C2×C3×C4—92no—
235C2×C3×C4C22×D911yescharacteristic
236C2×C3×C4C2×D3211yescharacteristic
237#67—364no—
238C2×Dic3—21no—
239C2×Dic3—32no—
240C2×Dic3—181no—
241C2×Dic3—181no—
242C2×Dic3—361no—
243D12—21no—
244D12—21no—
245D12—182no—
246D12—181no—
247D12—181no—
248D12—181no—
249D12—181no—
250D12—361no—
251D12—364no—
252Dic6—21no—
253Dic6—21no—
254Dic6—182no—
255Dic6—181no—
256Dic6—181no—
257Dic6—181no—
258Dic6—181no—
259Dic6—361no—
260C3×C8—22no—
261C3×C8—92no—
262C3×C8#61112yescomplemented
263C3×C8#62612yes—
264C3⋊C8—32no—
265C2×C32D3×SD1612yes—
266C2×C32#94011yescharacteristic
267C3⋊D3—241no—
268C3×D3—81no—
269C2×C9—22no—
270C2×C9—21no—
271C2×C9—241no—
272C2×C9#87411yescharacteristic
273C2×C9C2×#20412yes—
274D9—242no—
275D9—241no—
276C2×Q8—31no—
277C2×Q8—271no—
278C2×D4—31no—
279C2×D4—271no—
280C2×D4—542no—
281SD16—62no—
282SD16—542no—
283SD16—542no—
284SD16—542no—
285C2×C8—91no—
286C2×C8D3×D911yescharacteristic
287C4⋊C4—542no—
288C42—91no—
289C22×C3—21no—
290C22×C3—122no—
291C22×C3—121no—
292C22×C3D4×D911yescharacteristic
293C22×C3D3×D1211yescharacteristic
294C2×D3—41no—
295C2×D3—42no—
296C2×D3—362no—
297C2×D3—361no—
298C2×D3—362no—
299C2×D3—361no—
300C2×D3—362no—
301C2×D3—722no—
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—94no—
305C3×C4—121no—
306C3×C4D4×D911yescharacteristic
307C3×C4C2×#61111yescharacteristic
308C3×C4D3×D1211yescharacteristic
309C3×C4C2×#62611yescharacteristic
310Dic3—34no—
311Dic3—41no—
312Dic3—362no—
313Dic3—361no—
314Dic3—361no—
315C32#517511yescharacteristic
316C9—21no—
317C9#469311yescharacteristic, complemented
318C23—61no—
319C23—541no—
320Q8—61no—
321Q8—61no—
322Q8—541no—
323Q8—541no—
324D4—61no—
325D4—61no—
326D4—541no—
327D4—541no—
328D4—1084no—
329C2×C4—61no—
330C2×C4—92no—
331C2×C4—541no—
332C2×C4C2×D3×D911yescharacteristic
333C8—92no—
334C8#619512yescomplemented
335C2×C3—21no—
336C2×C3—22no—
337C2×C3—241no—
338C2×C3SD16×D912yes—
339C2×C3#6350211yescharacteristic
340C2×C3D3×#20412yes—
341C2×C3#6391111yescharacteristic
342D3—81no—
343D3—722no—
344D3—721no—
345D3—721no—
346C22—121no—
347C22—122no—
348C22—1081no—
349C22—1082no—
350C22D9×D1211yescharacteristic
351C4—94no—
352C4—121no—
353C4—1081no—
354C4D9×D1211yescharacteristic
355C4C2×#619511yescharacteristic
356C3—21no—
357C3#1058885511yescharacteristic, complemented
358C3#1059087711yescharacteristic
359C2—241no—
360C2—2161no—
361C2D9×#20412yes—
362C2#1172104011yescharacteristic
subgroupG/H
centerC22D9×D12
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radical#11721040C2
hypercenterC2×C8D3×D9
hypocenterC3×C9#492
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×#88524C2semidirect
2. C2×C9×#204C2semidirect
4. #11720924C2semidirect
5. C2×#88426C2semidirect
6. C2×#88424C2semidirect
7. C3×C8×Dic9C2semidirect
37. C8×Dic9D3semidirect
53. C3×C8×C9D4semidirect
54. C3×C9⋊C8D4semidirect
96. C3×Dic9SD16semidirect
107. C2×#204D9semidirect
152. C8×C9D12semidirect
153. C9⋊C8D12semidirect
214. Dic9#204semidirect
222. C3×C9#492semidirect
262. C3×C8#611semidirect
317. C9#4693semidirect
334. C8#6195semidirect
357. C3#10588855semidirect

Maximal subgroups

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

subgroupindex
C2×#885242
C2×C9×#2042
#117210402
#117209242
C2×#884262
C2×#884242
C3×C8×Dic92
#105908773
#105888553