Finite Group Databasebeta

← #13137807

Subgroups — #13137807

Order 1728.

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

normal subgroups only

This group has 386 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×#63449C211yesmaximal, characteristic, complemented
2#11720953C211yesmaximal, characteristic
3C2×#88427C211yesmaximal, characteristic, complemented
4C2×#88416C211yesmaximal, characteristic
5#11720528C211yesmaximal, characteristic
6C9×#810C211yesmaximal, characteristic, complemented
7Dic9⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10590536—31nomaximal
9#10588478—31nomaximal
10C2×C4⋊(C3⋊D9)C2211yescharacteristic
11C2×C9×Dic6C2211yescharacteristic
12C2×C3×Dic18C2211yescharacteristic
13C3×C9×C4⋊C4C2211yescharacteristic
14C2×C3⋊(C9⋊C8)C2211yescharacteristic
15C3×Dic9⋊C4—21no—
16C3×C4×Dic9—21no—
17C3×C4×Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19#88445—21no—
20#88445—21no—
21#88427—22no—
22#88416—22no—
23C3×#887—31no—
24#63864—31no—
25C2×#3718—31no—
26C2×#3717—31no—
27#63636—31no—
28C3×#810D311yescharacteristic
29Dic3⋊(C3⋊C8)—31no—
30#63509—31no—
31C2×#3674—31no—
32C2×#3664—31no—
33#63449D311yescharacteristic, complemented
34C9×#103—31no—
35Dic9⋊C8—31no—
36#63369—31no—
37#63369—61no—
38C22×C3⋊D9—21no—
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41C4⋊(C3⋊D9)—21no—
42C4⋊(C3⋊D9)—21no—
43C2×C9×Dic3—21no—
44C2×C3×Dic9—21no—
45C2×C3×Dic9D411yescharacteristic
46C2×C3×Dic9D411yescharacteristic
47C9×Dic6—21no—
48C9×Dic6—21no—
49C3×Dic18—21no—
50C3×Dic18D412yes—
51C3⋊(C9⋊C8)—21no—
52C9×C3⋊C8—21no—
53#5028—91no—
54#4798—91no—
55C2×C4⋊(C3⋊D3)—31no—
56C2×C3×Dic6—31no—
57C2×C3×Dic6C2×D311yescharacteristic
58C32×C4⋊C4C2×D311yescharacteristic
59C2×C3⋊(C3⋊C8)—31no—
60C3×Dic3⋊C4—61no—
61C3×C4×Dic3—31no—
62C3×C4×Dic3—61no—
63C2×C3×C3⋊C8C2×D311yescharacteristic
64#3722—61no—
65#3722—61no—
66#3718—62no—
67#3717—62no—
68C2×Q8×C9—31no—
69C2×D36—31no—
70C2×D36—61no—
71C2×Dic18C2×D311yescharacteristic
72C2×C8×C9—31no—
73C9×C4⋊C4—21no—
74C9×C4⋊C4C2×D311yescharacteristic
75#3674—62no—
76(C2×D9)⋊C4—61no—
77(C2×D9)⋊C4—61no—
78Dic9⋊C4—21no—
79C4×Dic9—21no—
80C4×Dic9C2×D311yescharacteristic
81C2×C9⋊C8—31no—
82C2×C9⋊C8—61no—
83#3664—62no—
84C22×C3×C9C2×D411yescharacteristic
85C2×C3⋊D9—41no—
86C2×C3⋊D9—41no—
87C2×C3⋊D9—41no—
88C3×C4×C9—41no—
89C3×C4×C9C2×D411yescharacteristic
90C3×C4×C9#4111yescharacteristic
91C9×Dic3—41no—
92C3×Dic9—22no—
93C3×Dic9—21no—
94C3×Dic9—21no—
95C3×C4×Q8—91no—
96#947—91no—
97C2×#216—91no—
98C2×#216—91no—
99C2×#215—91no—
100C2×#204—91no—
101#887—31no—
102#875—91no—
103C3×#103—31no—
104Dic3⋊C8—91no—
105#810D911yescharacteristic, complemented
106#809—91no—
107#809—91no—
108#809—181no—
109C4⋊(C3⋊C8)—91no—
110C22×C3⋊D3—61no—
111C2×C32×C4—21no—
112C2×C32×C4C22×D311yescharacteristic
113C4⋊(C3⋊D3)—61no—
114C4⋊(C3⋊D3)—61no—
115C2×C3×Dic3—21no—
116C2×C3×Dic3—31no—
117C2×C3×Dic3—31no—
118C2×C3×Dic3—61no—
119C3×Dic6—21no—
120C3×Dic6—21no—
121C3×Dic6—32no—
122C3×Dic6—61no—
123C22×D9—61no—
124C22×D9—121no—
125C3⋊(C3⋊C8)—61no—
126C3×C3⋊C8—21no—
127Q8×C9—61no—
128Q8×C9—61no—
129C2×C4×C9—21no—
130C2×C4×C9—21no—
131C2×C4×C9—41no—
132C2×C4×C9—61no—
133C2×C4×C9C22×D311yescharacteristic
134C2×Dic9—21no—
135C2×Dic9#6711yescharacteristic
136C2×Dic9#6711yescharacteristic
137D36—62no—
138D36—61no—
139D36—61no—
140D36—121no—
141Dic18—21no—
142Dic18D1212yes—
143C8×C9—61no—
144C9⋊C8—61no—
145C9⋊C8—121no—
146#464—271noSylow
147C2×C3×C9#12111yescharacteristic
148C2×C3×C9#13511yescharacteristic
149C2×C3×C9#13711yescharacteristic
150C3⋊D9—81no—
151C2×C3×Q8—31no—
152C2×C3×Q8—91no—
153C2×D12—91no—
154C2×D12—91no—
155C2×D12—181no—
156C2×Dic6—31no—
157C2×Dic6C2×D911yescharacteristic
158C2×C3×C8—31no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—181no—
161C3×C4⋊C4C2×D911yescharacteristic
162C3×C4⋊C4D3211yescharacteristic
163C3×C42—91no—
164C3×C42—181no—
165#216—182no—
166#216—182no—
167#215—182no—
168(C2×D3)⋊C4—181no—
169(C2×D3)⋊C4—181no—
170(C2×D3)⋊C4—181no—
171(C2×D3)⋊C4—181no—
172Dic3⋊C4—61no—
173C4×Dic3—31no—
174C4×Dic3—61no—
175C2×C3⋊C8—91no—
176C2×C3⋊C8—91no—
177C2×C3⋊C8—181no—
178C2×C3⋊C8C2×D911yescharacteristic
179#204—182no—
180C22×C32D3×D411yescharacteristic
181C2×C3⋊D3—121no—
182C2×C3⋊D3—121no—
183C2×C3⋊D3—121no—
184C32×C4—41no—
185C32×C4#23511yescharacteristic
186C32×C4D3×D411yescharacteristic
187C3×Dic3—41no—
188C3×Dic3—62no—
189C3×Dic3—61no—
190C3×Dic3—61no—
191C22×C9—21no—
192C22×C9C2×#6711yescharacteristic
193C2×D9—122no—
194C2×D9—121no—
195C2×D9—121no—
196C2×D9—121no—
197C2×D9—241no—
198C2×D9—241no—
199C4×C9—21no—
200C4×C9—21no—
201C4×C9—41no—
202C4×C9—81no—
203C4×C9—121no—
204C4×C9C2×D1211yescharacteristic
205C4×C9#23511yescharacteristic
206Dic9—22no—
207Dic9—21no—
208Dic9—21no—
209C2×Q16—271no—
210C2×SD16—271no—
211#124—271no—
212C4×Q8—91no—
213C4⋊C8—271no—
214#103—31no—
215#102—271no—
216C3×C9#46411yescharacteristic, complemented, Sylow
217C22×D3—181no—
218C22×D3—181no—
219C22×D3—361no—
220C3×Q8—61no—
221C3×Q8—61no—
222C3×Q8—92no—
223C3×Q8—181no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—41no—
228C2×C3×C4—61no—
229C2×C3×C4—91no—
230C2×C3×C4—91no—
231C2×C3×C4—181no—
232C2×C3×C4C22×D911yescharacteristic
233C2×C3×C4C2×D3211yescharacteristic
234C2×Dic3—21no—
235C2×Dic3—31no—
236C2×Dic3—31no—
237C2×Dic3—61no—
238D12—182no—
239D12—181no—
240D12—181no—
241D12—181no—
242D12—181no—
243D12—361no—
244Dic6—21no—
245Dic6—21no—
246Dic6—32no—
247Dic6—61no—
248C3×C8—61no—
249C3⋊C8—21no—
250C3⋊C8—181no—
251C3⋊C8—181no—
252C3⋊C8—361no—
253C2×C32#88411yescharacteristic
254C2×C32#91611yescharacteristic
255C2×C32#91811yescharacteristic
256C3⋊D3—241no—
257C2×C9—21no—
258C2×C9—21no—
259C2×C9—21no—
260C2×C9#93011yescharacteristic
261C2×C9#94211yescharacteristic
262C2×C9#95011yescharacteristic
263D9—242no—
264D9—241no—
265C2×Q8—31no—
266C2×Q8—91no—
267C2×D4—271no—
268Q16—542no—
269SD16—542no—
270C2×C8—31no—
271C2×C8—271no—
272C4⋊C4—181no—
273C4⋊C4D3×D911yescharacteristic
274C22⋊C4—541no—
275C22⋊C4—541no—
276C42—91no—
277C42—181no—
278C22×C3—21no—
279C22×C3D4×D911yescharacteristic
280C22×C3D3×#6711yescharacteristic
281C2×D3—362no—
282C2×D3—361no—
283C2×D3—361no—
284C2×D3—361no—
285C2×D3—361no—
286C2×D3—361no—
287C2×D3—361no—
288C2×D3—721no—
289C2×D3—721no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—41no—
293C3×C4—41no—
294C3×C4—81no—
295C3×C4—121no—
296C3×C4—182no—
297C3×C4—181no—
298C3×C4—181no—
299C3×C4#369711yescharacteristic
300C3×C4D4×D911yescharacteristic
301C3×C4#379811yescharacteristic
302C3×C4D3×D1211yescharacteristic
303Dic3—41no—
304Dic3—62no—
305Dic3—61no—
306Dic3—61no—
307C32#479811yescharacteristic
308C9—21no—
309C9#502811yescharacteristic, complemented
310C23—541no—
311Q8—61no—
312Q8—61no—
313Q8—92no—
314Q8—181no—
315D4—541no—
316D4—541no—
317C2×C4—21no—
318C2×C4—61no—
319C2×C4—91no—
320C2×C4—91no—
321C2×C4—181no—
322C2×C4C2×D3×D911yescharacteristic
323C8—61no—
324C8—541no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3#6344611yescharacteristic
329C2×C3#6347811yescharacteristic
330C2×C3#6348011yescharacteristic
331C2×C3#6390611yescharacteristic
332C2×C3#6393511yescharacteristic
333C2×C3#6394811yescharacteristic
334D3—722no—
335D3—721no—
336D3—721no—
337C22—1081no—
338C22—1081no—
339C22—1081no—
340C22D9×#6711yescharacteristic
341C4—41no—
342C4—121no—
343C4—181no—
344C4—182no—
345C4—181no—
346C4#8864011yescharacteristic
347C4D9×D1211yescharacteristic
348C3—21no—
349C3#1058847811yescharacteristic, complemented
350C3#1059053611yescharacteristic
351C2—2161no—
352C2#1172103311yescharacteristic
353C2#1172118511yescharacteristic
354C2#1172122211yescharacteristic
subgroupG/H
centerC22D9×#67
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC2×C4⋊(C3⋊D9)C22
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#464
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×#63449C2semidirect
3. C2×#88427C2semidirect
6. C9×#810C2semidirect
7. Dic9⋊(C3⋊C8)C2semidirect
33. #63449D3semidirect
105. #810D9semidirect
216. C3×C9#464semidirect
309. C9#5028semidirect
349. C3#10588478semidirect

Maximal subgroups

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

subgroupindex
C3×#634492
#117209532
C2×#884272
C2×#884162
#117205282
C9×#8102
Dic9⋊(C3⋊C8)2
#105905363
#105884783