Finite Group Databasebeta

← #13137805

Subgroups — #13137805

Order 1728.

422 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 85 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×#63455C211yesmaximal, characteristic, complemented
2#11720953C211yesmaximal, characteristic, complemented
3C2×#88425C211yesmaximal, characteristic, complemented
4C2×#88417C211yesmaximal, characteristic
5#11720528C211yesmaximal, characteristic, complemented
6C9×#810C211yesmaximal, characteristic, complemented
7Dic9⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10590534—31nomaximal
9#10588499—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C3×D36C2211yescharacteristic
12C2×C3×C4×D9—21no—
13C2×C9×Dic6C2211yescharacteristic
14C3×C9×C4⋊C4C2211yescharacteristic
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16C3×(C2×D9)⋊C4—21no—
17C3×C4×Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19#88445—21no—
20#88445—21no—
21#88425—22no—
22#88417—22no—
23C3×#893—31no—
24#63864—31no—
25C2×#3715—31no—
26C2×#3714—31no—
27#63636—31no—
28C3×#810D311yescharacteristic
29Dic3⋊(C3⋊C8)—31no—
30#63509—31no—
31C2×#3675—31no—
32C2×D72—31no—
33#63455D311yescharacteristic, complemented
34C9×#103—31no—
35Dic9⋊C8—31no—
36#63369—31no—
37#63369—61no—
38C22×C3⋊D9—21no—
39C22×C3×D9—21no—
40C2×C3×C4×C9—21no—
41C2×C3×C4×C9C2311yescharacteristic
42C4⋊(C3⋊D9)—21no—
43C4⋊(C3⋊D9)—21no—
44C2×C9×Dic3—21no—
45C2×C3×Dic9D411yescharacteristic
46C2×C3×Dic9D411yescharacteristic
47C3×D36—21no—
48C3×D36D412yes—
49C3×C4×D9—42no—
50C9×Dic6—21no—
51C9×Dic6—21no—
52C3⋊(C9⋊C8)—21no—
53C9×C3⋊C8—21no—
54#5016—91no—
55#4819—91no—
56C2×C4⋊(C3⋊D3)—31no—
57C2×C3×D12—31no—
58C2×C3×C4×D3—61no—
59C2×C3×Dic6C2×D311yescharacteristic
60C32×C4⋊C4C2×D311yescharacteristic
61C2×C3⋊(C3⋊C8)—31no—
62C3×(C2×D3)⋊C4—61no—
63C3×C4×Dic3—31no—
64C2×C3×C3⋊C8C2×D311yescharacteristic
65#3722—61no—
66#3722—61no—
67#3715—62no—
68#3714—62no—
69C2×Q8×C9—31no—
70C2×D36—31no—
71C2×D36—61no—
72C2×D36C2×D311yescharacteristic
73C2×C4×D9—21no—
74C2×C8×C9—31no—
75C9×C4⋊C4—21no—
76C9×C4⋊C4C2×D311yescharacteristic
77#3675—62no—
78(C2×D9)⋊C4—21no—
79(C2×D9)⋊C4—61no—
80(C2×D9)⋊C4—61no—
81C4×Dic9C2×D311yescharacteristic
82C2×C9⋊C8—31no—
83C2×C9⋊C8—61no—
84D72—62no—
85C22×C3×C9C2×D411yescharacteristic
86C2×C3⋊D9—41no—
87C2×C3⋊D9—41no—
88C2×C3⋊D9—41no—
89C2×C3×D9—22no—
90C2×C3×D9—41no—
91C2×C3×D9—41no—
92C3×C4×C9—41no—
93C3×C4×C9C2×D411yescharacteristic
94C3×C4×C9#4111yescharacteristic
95C9×Dic3—41no—
96C3×Dic9—21no—
97C3×Dic9—21no—
98C3×C4×D4—91no—
99#947—91no—
100C2×#215—91no—
101C2×#214—91no—
102C2×#213—91no—
103C2×D24—91no—
104#893—31no—
105#875—91no—
106C3×#103—31no—
107Dic3⋊C8—91no—
108#810D911yescharacteristic, complemented
109#809—91no—
110#809—91no—
111#809—181no—
112C4⋊(C3⋊C8)—91no—
113C22×C3⋊D3—61no—
114C22×C3×D3—61no—
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—31no—
122C3×D12—32no—
123C3×D12—61no—
124C3×C4×D3—122no—
125C3×Dic6—21no—
126C3×Dic6—21no—
127C22×D9—21no—
128C22×D9—61no—
129C22×D9—121no—
130C3⋊(C3⋊C8)—61no—
131C3×C3⋊C8—21no—
132Q8×C9—61no—
133Q8×C9—61no—
134C2×C4×C9—21no—
135C2×C4×C9—21no—
136C2×C4×C9—41no—
137C2×C4×C9—61no—
138C2×C4×C9C22×D311yescharacteristic
139C2×Dic9#6711yescharacteristic
140C2×Dic9#6711yescharacteristic
141D36—21no—
142D36—62no—
143D36—61no—
144D36—61no—
145D36—121no—
146D36D1212yes—
147C4×D9—42no—
148C8×C9—61no—
149C9⋊C8—61no—
150C9⋊C8—121no—
151#463—271noSylow
152C2×C3×C9#12111yescharacteristic
153C2×C3×C9#13511yescharacteristic
154C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
155C3⋊D9—81no—
156C3×D9—42no—
157C2×C3×Q8—31no—
158C2×C3×D4—91no—
159C22×C3×C4—181no—
160C2×D12—31no—
161C2×D12—91no—
162C2×D12—91no—
163C2×D12—181no—
164C2×C4×D3—61no—
165C2×Dic6C2×D911yescharacteristic
166C2×C3×C8—31no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4C2×D911yescharacteristic
169C3×C4⋊C4D3211yescharacteristic
170C3×C22⋊C4—181no—
171C3×C42—91no—
172#215—182no—
173#214—182no—
174#213—182no—
175(C2×D3)⋊C4—61no—
176(C2×D3)⋊C4—181no—
177(C2×D3)⋊C4—181no—
178(C2×D3)⋊C4—181no—
179(C2×D3)⋊C4—181no—
180C4×Dic3—31no—
181C2×C3⋊C8—91no—
182C2×C3⋊C8—91no—
183C2×C3⋊C8—181no—
184C2×C3⋊C8C2×D911yescharacteristic
185D24—182no—
186C22×C32D3×D411yescharacteristic
187C2×C3⋊D3—121no—
188C2×C3⋊D3—121no—
189C2×C3⋊D3—121no—
190C2×C3×D3—62no—
191C2×C3×D3—121no—
192C2×C3×D3—121no—
193C32×C4—41no—
194C32×C4#23511yescharacteristic
195C32×C4D3×D411yescharacteristic
196C3×Dic3—41no—
197C3×Dic3—61no—
198C3×Dic3—61no—
199C22×C9—21no—
200C22×C9C2×#6711yescharacteristic
201C2×D9—22no—
202C2×D9—41no—
203C2×D9—41no—
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—81no—
214C4×C9—121no—
215C4×C9C2×D1211yescharacteristic
216C4×C9#23511yescharacteristic
217Dic9—21no—
218Dic9—21no—
219C2×SD16—271no—
220C2×D8—271no—
221#124—271no—
222C4×D4—91no—
223C4⋊C8—271no—
224#103—31no—
225#102—271no—
226C3×C9#46311yescharacteristic, complemented, Sylow
227C23×C3—181no—
228C22×D3—61no—
229C22×D3—181no—
230C22×D3—181no—
231C22×D3—361no—
232C3×Q8—61no—
233C3×Q8—61no—
234C3×D4—92no—
235C3×D4—181no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—41no—
240C2×C3×C4—61no—
241C2×C3×C4—91no—
242C2×C3×C4—91no—
243C2×C3×C4—362no—
244C2×C3×C4C22×D911yescharacteristic
245C2×C3×C4C2×D3211yescharacteristic
246C2×Dic3—21no—
247C2×Dic3—31no—
248C2×Dic3—31no—
249D12—32no—
250D12—61no—
251D12—182no—
252D12—181no—
253D12—181no—
254D12—181no—
255D12—181no—
256D12—361no—
257C4×D3—122no—
258Dic6—21no—
259Dic6—21no—
260C3×C8—61no—
261C3⋊C8—21no—
262C3⋊C8—181no—
263C3⋊C8—181no—
264C3⋊C8—361no—
265C2×C32#88411yescharacteristic
266C2×C32#91411yescharacteristic
267C2×C32#91911yescharacteristic
268C3⋊D3—241no—
269C3×D3—122no—
270C2×C9—21no—
271C2×C9—21no—
272C2×C9—21no—
273C2×C9#93011yescharacteristic
274C2×C9#93211yescharacteristic
275C2×C9#95011yescharacteristic
276D9—42no—
277D9—242no—
278D9—241no—
279C2×Q8—31no—
280C2×D4—91no—
281C2×D4—271no—
282C22×C4—181no—
283SD16—542no—
284D8—542no—
285C2×C8—31no—
286C2×C8—271no—
287C4⋊C4D3×D911yescharacteristic
288C22⋊C4—181no—
289C22⋊C4—541no—
290C22⋊C4—541no—
291C42—91no—
292C22×C3—21no—
293C22×C3—182no—
294C22×C3—361no—
295C22×C3—361no—
296C22×C3D4×D911yescharacteristic
297C22×C3D3×#6711yescharacteristic
298C2×D3—62no—
299C2×D3—121no—
300C2×D3—121no—
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—181no—
318C3×C4#369711yescharacteristic
319C3×C4D4×D911yescharacteristic
320C3×C4#379811yescharacteristic
321C3×C4D3×D1211yescharacteristic
322Dic3—41no—
323Dic3—61no—
324Dic3—61no—
325C32#481911yescharacteristic
326C9—21no—
327C9#501611yescharacteristic, complemented
328C23—181no—
329C23—541no—
330Q8—61no—
331Q8—61no—
332D4—92no—
333D4—181no—
334D4—541no—
335D4—541no—
336C2×C4—21no—
337C2×C4—61no—
338C2×C4—91no—
339C2×C4—91no—
340C2×C4—362no—
341C2×C4C2×D3×D911yescharacteristic
342C8—61no—
343C8—541no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—362no—
348C2×C3#6344611yescharacteristic
349C2×C3#6347611yescharacteristic
350C2×C3#6348111yescharacteristic
351C2×C3#6390611yescharacteristic
352C2×C3#6393411yescharacteristic
353C2×C3#6394911yescharacteristic
354D3—122no—
355D3—722no—
356D3—721no—
357D3—721no—
358C22—182no—
359C22—361no—
360C22—361no—
361C22—1081no—
362C22—1081no—
363C22—1081no—
364C22D9×#6711yescharacteristic
365C4—41no—
366C4—121no—
367C4—181no—
368C4—181no—
369C4#8864011yescharacteristic
370C4D9×D1211yescharacteristic
371C3—21no—
372C3#1058849911yescharacteristic, complemented
373C3#1059053411yescharacteristic
374C2—362no—
375C2—2161no—
376C2#1172103311yescharacteristic
377C2#1172118411yescharacteristic
378C2#1172122311yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC2×#88417C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#463
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×#63455C2semidirect
2. #11720953C2semidirect
3. C2×#88425C2semidirect
5. #11720528C2semidirect
6. C9×#810C2semidirect
7. Dic9⋊(C3⋊C8)C2semidirect
33. #63455D3semidirect
108. #810D9semidirect
226. C3×C9#463semidirect
327. C9#5016semidirect
372. C3#10588499semidirect

Maximal subgroups

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

subgroupindex
C3×#634552
#117209532
C2×#884252
C2×#884172
#117205282
C9×#8102
Dic9⋊(C3⋊C8)2
#105905343
#105884993