Finite Group Databasebeta

← #13137699

Subgroups — #13137699

Order 1728.

422 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C9×#897C211yesmaximal, characteristic, complemented
2#11721038C211yesmaximal, characteristic
3C2×#88426C211yesmaximal, characteristic, complemented
4C2×D3×C9⋊C8C211yesmaximal, characteristic, complemented
5#11720528C211yesmaximal, characteristic, complemented
6C3×#63367C211yesmaximal, characteristic, complemented
7#11720297C211yesmaximal, characteristic, complemented
8#10590577—31nomaximal
9#10588731—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C4×D3×C9C2211yescharacteristic
12C2×C9×Dic6C2211yescharacteristic
13C2×#6195—21no—
14C3×C9×C4⋊C4C2211yescharacteristic
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16C9×(C2×D3)⋊C4—21no—
17C9×C4⋊Dic3—21no—
18C3×C4⋊Dic9C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20#88445—21no—
21#88426—22no—
22D3×C9⋊C8—22no—
23C3×#897D311yescharacteristic
24#63909—31no—
25C2×#3717—31no—
26C2×D3×C3⋊C8—31no—
27#63636—31no—
28C3×#808—31no—
29#63565—31no—
30C9×C22⋊Q8—31no—
31C2×#3675—31no—
32#63492—31no—
33C22×C9⋊C8—31no—
34#63395—31no—
35#63369—31no—
36#63369—61no—
37#63367D311yescharacteristic, complemented
38C22×C3⋊D9—21no—
39C22×D3×C9D411yescharacteristic
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C4⋊(C3⋊D9)—21no—
43C4⋊(C3⋊D9)—21no—
44C2×C9×Dic3—21no—
45C2×C9×Dic3D411yescharacteristic
46C2×C3×Dic9—21no—
47C4×D3×C9—21no—
48C4×D3×C9—21no—
49C9×Dic6—21no—
50C9×Dic6—21no—
51#6195—42no—
52C3⋊(C9⋊C8)—21no—
53C3×C9⋊C8D412yescomplemented
54#5048—91no—
55#4864—91no—
56C2×C4⋊(C3⋊D3)—31no—
57C2×C3×C4×D3C2×D311yescharacteristic
58C2×C3×Dic6C2×D311yescharacteristic
59C2×#626—61no—
60C32×C4⋊C4C2×D311yescharacteristic
61C2×C3⋊(C3⋊C8)—31no—
62C3×(C2×D3)⋊C4—21no—
63C3×C4⋊Dic3—21no—
64C3×C4⋊Dic3—31no—
65C2×C3×C3⋊C8—31no—
66#3722—61no—
67#3717—62no—
68D3×C3⋊C8—62no—
69C2×Q8×C9—31no—
70C22×C4×C9—31no—
71C2×#611—61no—
72C2×D36—31no—
73C2×D36—61no—
74C9×C4⋊C4—21no—
75C9×C4⋊C4—61no—
76C9×C4⋊C4C2×D311yescharacteristic
77C9×C22⋊C4—61no—
78#3675—62no—
79(C2×D9)⋊C4—61no—
80C4⋊Dic9C2×D311yescharacteristic
81C2×C9⋊C8—31no—
82C2×C9⋊C8—62no—
83C2×C9⋊C8—61no—
84C2×C9⋊C8C2×D311yescharacteristic
85C22×C3×C9C2×D411yescharacteristic
86C2×C3⋊D9—41no—
87C2×C3⋊D9—41no—
88C2×C3⋊D9—41no—
89C2×D3×C9—21no—
90C2×D3×C9—21no—
91C2×D3×C9SD1612yescomplemented
92C3×C4×C9—41no—
93C3×C4×C9C2×D411yescharacteristic
94C3×C4×C9#4111yescharacteristic
95C9×Dic3—21no—
96C9×Dic3—41no—
97C3×Dic9—41no—
98C3×C22⋊Q8—31no—
99C2×#215—91no—
100#930—91no—
101C22×C3⋊C8—91no—
102C2×#204—91no—
103C2×D3×C8—91no—
104#897D911yescharacteristic, complemented
105#895—91no—
106C3×#106—91no—
107#835—91no—
108#810—91no—
109#809—91no—
110#809—91no—
111#809—181no—
112#808—31no—
113C22×C3⋊D3—61no—
114C22×C3×D3#6711yescharacteristic
115C2×C32×C4—21no—
116C2×C32×C4C22×D311yescharacteristic
117C4⋊(C3⋊D3)—61no—
118C4⋊(C3⋊D3)—61no—
119C2×C3×Dic3—21no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3#6711yescharacteristic
122C3×C4×D3—21no—
123C3×C4×D3—21no—
124C3×Dic6—21no—
125C3×Dic6—21no—
126#626—122no—
127C23×C9—31no—
128C22×D9—61no—
129C22×D9—121no—
130C3⋊(C3⋊C8)—61no—
131C3×C3⋊C8—32no—
132Q8×C9—61no—
133Q8×C9—61no—
134C2×C4×C9—21no—
135C2×C4×C9—21no—
136C2×C4×C9—31no—
137C2×C4×C9—41no—
138C2×C4×C9—61no—
139C2×C4×C9—61no—
140C2×C4×C9—61no—
141C2×C4×C9C22×D311yescharacteristic
142#611—122no—
143C2×Dic9—21no—
144D36—62no—
145D36—61no—
146D36—61no—
147D36—121no—
148C9⋊C8—61no—
149C9⋊C8—121no—
150C9⋊C8D1212yescomplemented
151#465—271noSylow
152C2×C3×C9#12111yescharacteristic
153C2×C3×C9C2×SD1611yescharacteristic
154C2×C3×C9#13511yescharacteristic
155C3⋊D9—81no—
156D3×C9—22no—
157C2×C3×Q8—31no—
158C22×C3×C4—31no—
159C2×#67—181no—
160C2×D12—91no—
161C2×D12—91no—
162C2×D12—181no—
163C2×D12—181no—
164C2×C4×D3C2×D911yescharacteristic
165C2×Dic6C2×D911yescharacteristic
166C2×C3×C8—91no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—61no—
169C3×C4⋊C4—91no—
170C3×C4⋊C4C2×D911yescharacteristic
171C3×C4⋊C4D3211yescharacteristic
172C3×C22⋊C4—61no—
173#215—182no—
174(C2×D3)⋊C4—21no—
175(C2×D3)⋊C4—181no—
176(C2×D3)⋊C4—181no—
177C4⋊Dic3—21no—
178C4⋊Dic3—31no—
179C2×C3⋊C8—31no—
180C2×C3⋊C8—91no—
181C2×C3⋊C8—91no—
182C2×C3⋊C8—181no—
183C2×C3⋊C8—182no—
184#204—182no—
185D3×C8—182no—
186C22×C32C2×#6711yescharacteristic
187C2×C3⋊D3—121no—
188C2×C3⋊D3—121no—
189C2×C3⋊D3—121no—
190C2×C3×D3—21no—
191C2×C3×D3—21no—
192C2×C3×D3#21512yes—
193C32×C4—41no—
194C32×C4D3×D411yescharacteristic
195C32×C4#23711yescharacteristic
196C3×Dic3—21no—
197C3×Dic3—41no—
198C3×Dic3—121no—
199C22×C9—21no—
200C22×C9—32no—
201C22×C9—61no—
202C22×C9—61no—
203C22×C9D3×D411yescharacteristic
204C2×D9—122no—
205C2×D9—121no—
206C2×D9—121no—
207C2×D9—121no—
208C2×D9—241no—
209C2×D9—241no—
210C4×C9—21no—
211C4×C9—21no—
212C4×C9—41no—
213C4×C9—61no—
214C4×C9—81no—
215C4×C9—121no—
216C4×C9C2×D1211yescharacteristic
217C4×C9#23911yescharacteristic
218Dic9—41no—
219C2×SD16—271no—
220C22×C8—271no—
221C22⋊Q8—31no—
222#121—271no—
223#106—91no—
224#103—271no—
225#102—271no—
226C3×C9#46511yescharacteristic, complemented, Sylow
227C23×C3—31no—
228C22×D3—181no—
229C22×D3—181no—
230C22×D3—361no—
231C22×D3#61111yescharacteristic
232C3×Q8—61no—
233C3×Q8—61no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—31no—
238C2×C3×C4—41no—
239C2×C3×C4—61no—
240C2×C3×C4—61no—
241C2×C3×C4—61no—
242C2×C3×C4—181no—
243C2×C3×C4C22×D911yescharacteristic
244C2×C3×C4C2×D3211yescharacteristic
245#67—362no—
246C2×Dic3—21no—
247C2×Dic3—61no—
248C2×Dic3#61111yescharacteristic
249D12—182no—
250D12—181no—
251D12—181no—
252D12—181no—
253D12—181no—
254D12—361no—
255D12—362no—
256C4×D3—21no—
257C4×D3—21no—
258Dic6—21no—
259Dic6—21no—
260C3×C8—92no—
261C3⋊C8—32no—
262C3⋊C8—181no—
263C3⋊C8—181no—
264C3⋊C8—361no—
265C2×C32#93911yescharacteristic
266C2×C32C2×#21511yescharacteristic
267C2×C32#95011yescharacteristic
268C3⋊D3—241no—
269C3×D3—22no—
270C2×C9—21no—
271C2×C9—21no—
272C2×C9—21no—
273C2×C9—62no—
274C2×C9#89511yescharacteristic
275C2×C9D3×SD1611yescharacteristic
276C2×C9#91611yescharacteristic
277D9—242no—
278D9—241no—
279C2×Q8—31no—
280C2×D4—271no—
281C2×D4—541no—
282C22×C4—31no—
283SD16—542no—
284C2×C8—91no—
285C2×C8—271no—
286C2×C8—542no—
287C4⋊C4—61no—
288C4⋊C4—91no—
289C4⋊C4D3×D911yescharacteristic
290C22⋊C4—61no—
291C22⋊C4—541no—
292C22×C3—21no—
293C22×C3—32no—
294C22×C3—61no—
295C22×C3—61no—
296C22×C3C2×#61111yescharacteristic
297C22×C3D3×#6711yescharacteristic
298C2×D3—21no—
299C2×D3—21no—
300C2×D3—362no—
301C2×D3—361no—
302C2×D3—361no—
303C2×D3—361no—
304C2×D3—361no—
305C2×D3—361no—
306C2×D3—361no—
307C2×D3—721no—
308C2×D3—721no—
309C2×D3#367512yescomplemented
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—41no—
313C3×C4—41no—
314C3×C4—61no—
315C3×C4—81no—
316C3×C4—121no—
317C3×C4—361no—
318C3×C4D4×D911yescharacteristic
319C3×C4#369911yescharacteristic
320C3×C4#379611yescharacteristic
321C3×C4D3×D1211yescharacteristic
322Dic3—21no—
323Dic3—41no—
324Dic3—121no—
325C32#504811yescharacteristic
326C9—21no—
327C9#486411yescharacteristic, complemented
328C23—31no—
329C23—541no—
330Q8—61no—
331Q8—61no—
332D4—541no—
333D4—541no—
334D4—1082no—
335C2×C4—21no—
336C2×C4—31no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—181no—
341C2×C4C2×D3×D911yescharacteristic
342C8—92no—
343C8—541no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—62no—
348C2×C3#6350111yescharacteristic
349C2×C3C2×#367511yescharacteristic
350C2×C3#6351311yescharacteristic
351C2×C3#6388611yescharacteristic
352C2×C3#6393511yescharacteristic
353C2×C3D3×#21511yescharacteristic
354D3—22no—
355D3—722no—
356D3—721no—
357D3—721no—
358C22—32no—
359C22—61no—
360C22—61no—
361C22—1081no—
362C22—1081no—
363C22—1081no—
364C22D3×#61111yescharacteristic
365C4—41no—
366C4—61no—
367C4—121no—
368C4—361no—
369C4#8863911yescharacteristic
370C4D9×D1211yescharacteristic
371C3—21no—
372C3#1058873111yescharacteristic, complemented
373C3#1059057711yescharacteristic
374C2—62no—
375C2—2161no—
376C2#1172101711yescharacteristic
377C2#1172117111yescharacteristic
378C2D3×#367511yescharacteristic
subgroupG/H
centerC22D3×#611
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C2×#67
prime radical#11721038C2
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. C9×#897C2semidirect
3. C2×#88426C2semidirect
4. C2×D3×C9⋊C8C2semidirect
5. #11720528C2semidirect
6. C3×#63367C2semidirect
7. #11720297C2semidirect
37. #63367D3semidirect
53. C3×C9⋊C8D4semidirect
91. C2×D3×C9SD16semidirect
104. #897D9semidirect
150. C9⋊C8D12semidirect
226. C3×C9#465semidirect
309. C2×D3#3675semidirect
327. C9#4864semidirect
372. C3#10588731semidirect

Maximal subgroups

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

subgroupindex
C9×#8972
#117210382
C2×#884262
C2×D3×C9⋊C82
#117205282
C3×#633672
#117202972
#105905773
#105887313