Finite Group Databasebeta

← #13137592

Subgroups — #13137592

Order 1728.

422 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C3×#63459C211yesmaximal, characteristic, complemented
2#11721004C211yesmaximal, characteristic
3C2×#88416C211yesmaximal, characteristic, complemented
4C2×D9×C3⋊C8C211yesmaximal, characteristic, complemented
5#11720528C211yesmaximal, characteristic, complemented
6C9×#808C211yesmaximal, characteristic, complemented
7#11720268C211yesmaximal, characteristic, complemented
8#10590577—31nomaximal
9#10588544—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C3×C4×D9C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13C2×#6192—21no—
14C3×C9×C4⋊C4C2211yescharacteristic
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16C3×(C2×D9)⋊C4—21no—
17C9×C4⋊Dic3C2211yescharacteristic
18C3×C4⋊Dic9—21no—
19C2×C9×C3⋊C8C2211yescharacteristic
20#88445—21no—
21#88416—22no—
22D9×C3⋊C8—22no—
23C3×#897—31no—
24#63909—31no—
25C2×#3717—31no—
26C2×D3×C3⋊C8—31no—
27#63636—31no—
28C3×#808D311yescharacteristic
29#63565—31no—
30C2×#3664—31no—
31C2×C8×D9—31no—
32#63459D311yescharacteristic, complemented
33#63457—31no—
34C9×#106—31no—
35#63369—31no—
36#63369—61no—
37#63368—31no—
38C22×C3⋊D9—21no—
39C22×C3×D9D411yescharacteristic
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C4⋊(C3⋊D9)—21no—
43C4⋊(C3⋊D9)—21no—
44C2×C9×Dic3—21no—
45C2×C3×Dic9—21no—
46C2×C3×Dic9D411yescharacteristic
47C3×C4×D9—21no—
48C3×C4×D9—21no—
49C3×Dic18—21no—
50C3×Dic18—21no—
51#6192—42no—
52C3⋊(C9⋊C8)—21no—
53C9×C3⋊C8D412yescomplemented
54#5048—91no—
55#4864—91no—
56C2×C4⋊(C3⋊D3)—31no—
57C2×C3×C4×D3—31no—
58C2×C3×Dic6—31no—
59C2×#626—61no—
60C32×C4⋊C4C2×D311yescharacteristic
61C2×C3⋊(C3⋊C8)—31no—
62C3×(C2×D3)⋊C4—61no—
63C3×C4⋊Dic3—61no—
64C3×C4⋊Dic3C2×D311yescharacteristic
65C2×C3×C3⋊C8C2×D311yescharacteristic
66#3722—61no—
67#3717—62no—
68D3×C3⋊C8—62no—
69C2×D36—31no—
70C2×D36—61no—
71C2×D36—61no—
72C2×C4×D9C2×D311yescharacteristic
73C2×Dic18C2×D311yescharacteristic
74C2×C8×C9—31no—
75C9×C4⋊C4—21no—
76C9×C4⋊C4—31no—
77C9×C4⋊C4C2×D311yescharacteristic
78(C2×D9)⋊C4—21no—
79(C2×D9)⋊C4—61no—
80C4⋊Dic9—21no—
81C2×C9⋊C8—31no—
82C2×C9⋊C8—61no—
83#3664—62no—
84C8×D9—62no—
85C22×C3×C9C2×D411yescharacteristic
86C2×C3⋊D9—41no—
87C2×C3⋊D9—41no—
88C2×C3⋊D9—41no—
89C2×C3×D9—21no—
90C2×C3×D9—21no—
91C2×C3×D9SD1612yescomplemented
92C3×C4×C9—41no—
93C3×C4×C9C2×D411yescharacteristic
94C3×C4×C9#4111yescharacteristic
95C9×Dic3—41no—
96C3×Dic9—21no—
97C3×Dic9—41no—
98C3×C22⋊Q8—91no—
99C2×#215—91no—
100#930—91no—
101C22×C3⋊C8—91no—
102C2×#204—91no—
103C2×D3×C8—91no—
104#897—31no—
105#895—91no—
106C3×#106—31no—
107#835—91no—
108#810—91no—
109#809—91no—
110#809—91no—
111#809—181no—
112#808D911yescharacteristic, complemented
113C22×C3⋊D3—61no—
114C22×C3×D3—31no—
115C2×C32×C4—21no—
116C2×C32×C4C22×D311yescharacteristic
117C4⋊(C3⋊D3)—61no—
118C4⋊(C3⋊D3)—61no—
119C2×C3×Dic3—21no—
120C2×C3×Dic3—31no—
121C2×C3×Dic3—61no—
122C3×C4×D3—61no—
123C3×C4×D3—61no—
124C3×Dic6—61no—
125C3×Dic6—61no—
126#626—122no—
127C22×D9—61no—
128C22×D9—121no—
129C22×D9#6711yescharacteristic
130C3⋊(C3⋊C8)—61no—
131C3×C3⋊C8D1212yes—
132C2×C4×C9—21no—
133C2×C4×C9—21no—
134C2×C4×C9—41no—
135C2×C4×C9—61no—
136C2×C4×C9C22×D311yescharacteristic
137C2×Dic9—21no—
138C2×Dic9#6711yescharacteristic
139D36—62no—
140D36—61no—
141D36—61no—
142D36—121no—
143D36—122no—
144C4×D9—21no—
145C4×D9—21no—
146Dic18—21no—
147Dic18—21no—
148C8×C9—32no—
149C9⋊C8—61no—
150C9⋊C8—121no—
151#465—271noSylow
152C2×C3×C9#12111yescharacteristic
153C2×C3×C9C2×SD1611yescharacteristic
154C2×C3×C9#13511yescharacteristic
155C3⋊D9—81no—
156C3×D9—22no—
157C2×C3×Q8—91no—
158C22×C3×C4—91no—
159C2×#67—181no—
160C2×D12—91no—
161C2×D12—91no—
162C2×D12—181no—
163C2×D12—181no—
164C2×C4×D3—31no—
165C2×Dic6—31no—
166C2×C3×C8—31no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—31no—
169C3×C4⋊C4—181no—
170C3×C4⋊C4C2×D911yescharacteristic
171C3×C4⋊C4D3211yescharacteristic
172C3×C22⋊C4—181no—
173#215—182no—
174(C2×D3)⋊C4—61no—
175(C2×D3)⋊C4—181no—
176(C2×D3)⋊C4—181no—
177C4⋊Dic3—61no—
178C4⋊Dic3C2×D911yescharacteristic
179C2×C3⋊C8—91no—
180C2×C3⋊C8—91no—
181C2×C3⋊C8—181no—
182C2×C3⋊C8—182no—
183C2×C3⋊C8C2×D911yescharacteristic
184#204—182no—
185D3×C8—182no—
186C22×C32D3×D411yescharacteristic
187C2×C3⋊D3—121no—
188C2×C3⋊D3—121no—
189C2×C3⋊D3—121no—
190C2×C3×D3—32no—
191C2×C3×D3—61no—
192C2×C3×D3—61no—
193C32×C4—41no—
194C32×C4C2×D1211yescharacteristic
195C32×C4#23911yescharacteristic
196C3×Dic3—41no—
197C3×Dic3—61no—
198C3×Dic3—121no—
199C22×C9—21no—
200C22×C9C2×#6711yescharacteristic
201C2×D9—21no—
202C2×D9—21no—
203C2×D9—122no—
204C2×D9—121no—
205C2×D9—121no—
206C2×D9—121no—
207C2×D9—241no—
208C2×D9—241no—
209C2×D9#21512yescomplemented
210C4×C9—21no—
211C4×C9—21no—
212C4×C9—41no—
213C4×C9—81no—
214C4×C9—121no—
215C4×C9D3×D411yescharacteristic
216C4×C9#23711yescharacteristic
217Dic9—21no—
218Dic9—41no—
219C2×SD16—271no—
220C22×C8—271no—
221C22⋊Q8—91no—
222#121—271no—
223#106—31no—
224#103—271no—
225#102—271no—
226C3×C9#46511yescharacteristic, complemented, Sylow
227C23×C3—91no—
228C22×D3—31no—
229C22×D3—181no—
230C22×D3—181no—
231C22×D3—361no—
232C3×Q8—181no—
233C3×Q8—181no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—41no—
238C2×C3×C4—61no—
239C2×C3×C4—91no—
240C2×C3×C4—181no—
241C2×C3×C4—181no—
242C2×C3×C4—181no—
243C2×C3×C4C22×D911yescharacteristic
244C2×C3×C4C2×D3211yescharacteristic
245#67—362no—
246C2×Dic3—21no—
247C2×Dic3—31no—
248C2×Dic3—61no—
249D12—182no—
250D12—181no—
251D12—181no—
252D12—181no—
253D12—181no—
254D12—361no—
255D12—362no—
256C4×D3—61no—
257C4×D3—61no—
258Dic6—61no—
259Dic6—61no—
260C3×C8—32no—
261C3⋊C8—181no—
262C3⋊C8—181no—
263C3⋊C8—361no—
264C3⋊C8D3612yescomplemented
265C2×C32#89511yescharacteristic
266C2×C32D3×SD1611yescharacteristic
267C2×C32#91611yescharacteristic
268C3⋊D3—241no—
269C3×D3—62no—
270C2×C9—21no—
271C2×C9—21no—
272C2×C9—21no—
273C2×C9#93911yescharacteristic
274C2×C9C2×#21511yescharacteristic
275C2×C9#95011yescharacteristic
276D9—22no—
277D9—242no—
278D9—241no—
279C2×Q8—91no—
280C2×D4—271no—
281C2×D4—541no—
282C22×C4—91no—
283SD16—542no—
284C2×C8—31no—
285C2×C8—271no—
286C2×C8—542no—
287C4⋊C4—31no—
288C4⋊C4—181no—
289C4⋊C4D3×D911yescharacteristic
290C22⋊C4—181no—
291C22⋊C4—541no—
292C22×C3—21no—
293C22×C3—92no—
294C22×C3—181no—
295C22×C3—181no—
296C22×C3D4×D911yescharacteristic
297C22×C3D3×#6711yescharacteristic
298C2×D3—32no—
299C2×D3—61no—
300C2×D3—61no—
301C2×D3—362no—
302C2×D3—361no—
303C2×D3—361no—
304C2×D3—361no—
305C2×D3—361no—
306C2×D3—361no—
307C2×D3—361no—
308C2×D3—721no—
309C2×D3—721no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—41no—
313C3×C4—41no—
314C3×C4—81no—
315C3×C4—121no—
316C3×C4—181no—
317C3×C4—361no—
318C3×C4C2×D3611yescharacteristic
319C3×C4#370111yescharacteristic
320C3×C4#379611yescharacteristic
321C3×C4D3×D1211yescharacteristic
322Dic3—41no—
323Dic3—61no—
324Dic3—121no—
325C32#486411yescharacteristic
326C9—21no—
327C9#504811yescharacteristic, complemented
328C23—91no—
329C23—541no—
330Q8—181no—
331Q8—181no—
332D4—541no—
333D4—541no—
334D4—1082no—
335C2×C4—21no—
336C2×C4—61no—
337C2×C4—91no—
338C2×C4—181no—
339C2×C4—181no—
340C2×C4—181no—
341C2×C4C2×D3×D911yescharacteristic
342C8—32no—
343C8—541no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—182no—
348C2×C3#6345711yescharacteristic
349C2×C3SD16×D911yescharacteristic
350C2×C3#6347811yescharacteristic
351C2×C3#6388611yescharacteristic
352C2×C3#6393511yescharacteristic
353C2×C3D3×#21511yescharacteristic
354D3—62no—
355D3—722no—
356D3—721no—
357D3—721no—
358C22—92no—
359C22—181no—
360C22—181no—
361C22—1081no—
362C22—1081no—
363C22—1081no—
364C22D9×#6711yescharacteristic
365C4—41no—
366C4—121no—
367C4—181no—
368C4—361no—
369C4#8863411yescharacteristic
370C4D3×D3611yescharacteristic
371C3—21no—
372C3#1058854411yescharacteristic, complemented
373C3#1059057711yescharacteristic
374C2—182no—
375C2—2161no—
376C2#1172098311yescharacteristic
377C2#1172118511yescharacteristic
378C2D9×#21511yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radical#11721004C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#465
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#63459C2semidirect
3. C2×#88416C2semidirect
4. C2×D9×C3⋊C8C2semidirect
5. #11720528C2semidirect
6. C9×#808C2semidirect
7. #11720268C2semidirect
32. #63459D3semidirect
53. C9×C3⋊C8D4semidirect
91. C2×C3×D9SD16semidirect
112. #808D9semidirect
209. C2×D9#215semidirect
226. C3×C9#465semidirect
264. C3⋊C8D36semidirect
327. C9#5048semidirect
372. C3#10588544semidirect

Maximal subgroups

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

subgroupindex
C3×#634592
#117210042
C2×#884162
C2×D9×C3⋊C82
#117205282
C9×#8082
#117202682
#105905773
#105885443