Finite Group Databasebeta

← #13144450

Subgroups — #13144450

Order 1728.

670 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×#121C211yesmaximal, characteristic, complemented
2C2×#88544C211yesmaximal, characteristic, complemented
3#11721819C211yesmaximal, characteristic, complemented
4C22×C3⋊(C9⋊C8)C211yesmaximal, characteristic, complemented
5#11720576C211yesmaximal, characteristic, complemented
6#11720528C211yesmaximal, characteristic, complemented
7#11720525C211yesmaximal, characteristic, complemented
8#10592737—31nomaximal
9#10588721—33nomaximal
10C2×C3×D4×C9—21no—
11C2×C3×D4×C9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C2×#6233—21no—
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C3×C9×C4⋊C4C2211yescharacteristic
16C3×C9×C22⋊C4—21no—
17#88544—22no—
18#88536—21no—
19C4⋊(C3⋊Dic9)C2211yescharacteristic
20C2×C3⋊(C9⋊C8)—22no—
21C2×C3⋊(C9⋊C8)C2211yescharacteristic
22C2×C3⋊(C9⋊C8)C2211yescharacteristic
23C32×#121D311yescharacteristic
24C2×#3753—31no—
25#64139—31no—
26C22×C3⋊(C3⋊C8)—31no—
27#63659—31no—
28#63636—31no—
29#63634—31no—
30C9×#121D313yescomplemented
31C2×#3673—33no—
32#63492—33no—
33C22×C9⋊C8—33no—
34#63392—33no—
35#63369—33no—
36#63366—33no—
37C23×C3×C9—21no—
38C23×C3×C9D411yescharacteristic
39C22×C3⋊D9—21no—
40C3×D4×C9—21no—
41C3×D4×C9—21no—
42C3×D4×C9—42no—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9—21no—
46C2×C3×C4×C9D411yescharacteristic
47C2×C3×C4×C9C2311yescharacteristic
48#6233—42no—
49C2×C3⋊Dic9—21no—
50C4⋊(C3⋊D9)—21no—
51C4⋊(C3⋊D9)—21no—
52C3⋊(C9⋊C8)—21no—
53C3⋊(C9⋊C8)D412yescomplemented
54#5038—93no—
55#5038—91no—
56C2×C32×D4—21no—
57C2×C32×D4C2×D311yescharacteristic
58C22×C32×C4C2×D311yescharacteristic
59C2×#638—61no—
60C2×C4⋊(C3⋊D3)—31no—
61C32×C4⋊C4C2×D311yescharacteristic
62C32×C22⋊C4—21no—
63#3753—62no—
64#3752—61no—
65C4⋊(C3⋊Dic3)—31no—
66C2×C3⋊(C3⋊C8)—31no—
67C2×C3⋊(C3⋊C8)—31no—
68C2×C3⋊(C3⋊C8)—62no—
69C2×D4×C9—23no—
70C2×D4×C9C2×D313yes—
71C22×C4×C9C2×D313yes—
72C2×#611—63no—
73C2×D36—33no—
74C9×C4⋊C4C2×D313yes—
75C9×C22⋊C4—23no—
76#3673—66no—
77(C2×D9)⋊C4—63no—
78C4⋊Dic9—33no—
79C2×C9⋊C8—33no—
80C2×C9⋊C8—33no—
81C2×C9⋊C8—66no—
82C22×C3×C9—21no—
83C22×C3×C9—21no—
84C22×C3×C9—41no—
85C22×C3×C9—41no—
86C22×C3×C9—41no—
87C22×C3×C9D812yescomplemented
88C22×C3×C9C2×D411yescharacteristic
89C2×C3⋊D9—41no—
90C2×C3⋊D9—41no—
91C2×C3⋊D9—41no—
92C3×C4×C9—21no—
93C3×C4×C9—41no—
94C3×C4×C9C2×D411yescharacteristic
95C3×C4×C9#4111yescharacteristic
96C3⋊Dic9—41no—
97C3×#121D913yescomplemented
98C3×#121C3⋊D311yescharacteristic
99C2×#213—93no—
100C2×#213—91no—
101#930—93no—
102#930—91no—
103C22×C3⋊C8—93no—
104C22×C3⋊C8—91no—
105#832—93no—
106#832—91no—
107#809—93no—
108#809—91no—
109#807—93no—
110#807—91no—
111C23×C32—21no—
112C23×C32#6711yescharacteristic
113C22×C3⋊D3—61no—
114C32×D4—21no—
115C32×D4—21no—
116C32×D4—42no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4#6711yescharacteristic
121C2×C32×C4C22×D311yescharacteristic
122#638—122no—
123C2×C3⋊Dic3—61no—
124C4⋊(C3⋊D3)—61no—
125C4⋊(C3⋊D3)—61no—
126C23×C9—23no—
127C23×C9#6713yes—
128C22×D9—63no—
129C3⋊(C3⋊C8)—32no—
130C3⋊(C3⋊C8)—61no—
131D4×C9—23no—
132D4×C9—23no—
133D4×C9—46no—
134C2×C4×C9—23no—
135C2×C4×C9—23no—
136C2×C4×C9—23no—
137C2×C4×C9#6713yes—
138C2×C4×C9C22×D313yes—
139#611—126no—
140C2×Dic9—63no—
141D36—63no—
142D36—63no—
143C9⋊C8—36no—
144C9⋊C8—63no—
145#466—271noSylow
146C2×C3×C9—22no—
147C2×C3×C9—81no—
148C2×C3×C9#12111yescharacteristic
149C2×C3×C9C2×D811yescharacteristic
150C2×C3×C9#13511yescharacteristic
151C3⋊D9—81no—
152C2×C3×D4—23no—
153C2×C3×D4—21no—
154C2×C3×D4C2×D913yes—
155C2×C3×D4C2×C3⋊D311yescharacteristic
156C22×C3×C4C2×D913yes—
157C22×C3×C4C2×C3⋊D311yescharacteristic
158C2×#67—183no—
159C2×#67—181no—
160C2×D12—93no—
161C2×D12—91no—
162C3×C4⋊C4C2×D913yes—
163C3×C4⋊C4C2×C3⋊D311yescharacteristic
164C3×C22⋊C4—23no—
165C3×C22⋊C4—21no—
166#213—182no—
167#213—186no—
168(C2×D3)⋊C4—183no—
169(C2×D3)⋊C4—181no—
170C4⋊Dic3—93no—
171C4⋊Dic3—91no—
172C2×C3⋊C8—93no—
173C2×C3⋊C8—91no—
174C2×C3⋊C8—93no—
175C2×C3⋊C8—91no—
176C2×C3⋊C8—182no—
177C2×C3⋊C8—186no—
178C22×C32—21no—
179C22×C32—21no—
180C22×C32—41no—
181C22×C32—41no—
182C22×C32—41no—
183C22×C32#21312yes—
184C22×C32C2×#6711yescharacteristic
185C2×C3⋊D3—121no—
186C2×C3⋊D3—121no—
187C2×C3⋊D3—121no—
188C32×C4—21no—
189C32×C4—41no—
190C32×C4D3×D411yescharacteristic
191C32×C4#23711yescharacteristic
192C3⋊Dic3—121no—
193C22×C9—23no—
194C22×C9—23no—
195C22×C9—43no—
196C22×C9—43no—
197C22×C9—43no—
198C22×C9#21316yescomplemented
199C22×C9C2×#6713yes—
200C2×D9—123no—
201C2×D9—123no—
202C2×D9—123no—
203C4×C9—23no—
204C4×C9—43no—
205C4×C9D3×D413yes—
206C4×C9#23713yes—
207Dic9—123no—
208C2×D8—271no—
209C22×C8—271no—
210#121—271no—
211#121C3⋊D911yescharacteristic, complemented
212#107—271no—
213#102—271no—
214#102—271no—
215C3×C9#46611yescharacteristic, complemented, Sylow
216C23×C3—23no—
217C23×C3—21no—
218C23×C3#61113yes—
219C23×C3#63811yescharacteristic
220C22×D3—183no—
221C22×D3—181no—
222C3×D4—23no—
223C3×D4—23no—
224C3×D4—21no—
225C3×D4—21no—
226C3×D4—46no—
227C3×D4—42no—
228C2×C3×C4—23no—
229C2×C3×C4—23no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—23no—
233C2×C3×C4—21no—
234C2×C3×C4#61113yes—
235C2×C3×C4C22×D913yes—
236C2×C3×C4#63811yescharacteristic
237C2×C3×C4C22×C3⋊D311yescharacteristic
238#67—366no—
239#67—362no—
240C2×Dic3—183no—
241C2×Dic3—181no—
242D12—183no—
243D12—183no—
244D12—181no—
245D12—181no—
246C3⋊C8—96no—
247C3⋊C8—92no—
248C3⋊C8—183no—
249C3⋊C8—181no—
250C2×C32—22no—
251C2×C32—81no—
252C2×C32C2×#21311yescharacteristic
253C2×C32#93911yescharacteristic
254C2×C32#95011yescharacteristic
255C3⋊D3—241no—
256C2×C9—26no—
257C2×C9—83no—
258C2×C9C2×#21313yes—
259C2×C9#93913yes—
260C2×C9#95013yes—
261D9—243no—
262C2×D4—21no—
263C2×D4—271no—
264C2×D4—541no—
265C2×D4C2×C3⋊D911yescharacteristic
266C22×C4C2×C3⋊D911yescharacteristic
267D8—542no—
268C2×C8—271no—
269C2×C8—271no—
270C2×C8—542no—
271C4⋊C4—271no—
272C4⋊C4C2×C3⋊D911yescharacteristic
273C22⋊C4—21no—
274C22⋊C4—541no—
275C22×C3—23no—
276C22×C3—23no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—43no—
280C22×C3—43no—
281C22×C3—43no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3—41no—
285C22×C3#367316yescomplemented
286C22×C3C2×#61113yes—
287C22×C3#375312yes—
288C22×C3C2×#63811yescharacteristic
289C2×D3—363no—
290C2×D3—363no—
291C2×D3—363no—
292C2×D3—361no—
293C2×D3—361no—
294C2×D3—361no—
295C3×C4—23no—
296C3×C4—21no—
297C3×C4—43no—
298C3×C4—41no—
299C3×C4D4×D913yes—
300C3×C4#369913yes—
301C3×C4D4×C3⋊D311yescharacteristic
302C3×C4#383011yescharacteristic
303Dic3—363no—
304Dic3—361no—
305C32#503811yescharacteristic
306C9#503813yescomplemented
307C23—21no—
308C23—541no—
309C23#623311yescharacteristic
310D4—21no—
311D4—21no—
312D4—42no—
313D4—541no—
314D4—541no—
315D4—1082no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—541no—
320C2×C4#623311yescharacteristic
321C2×C4C22×C3⋊D911yescharacteristic
322C8—272no—
323C8—541no—
324C2×C3—26no—
325C2×C3—22no—
326C2×C3—83no—
327C2×C3—81no—
328C2×C3C2×#367313yes—
329C2×C3#6350113yes—
330C2×C3#6351313yes—
331C2×C3C2×#375311yescharacteristic
332C2×C3#6414811yescharacteristic
333C2×C3#6415911yescharacteristic
334D3—723no—
335D3—721no—
336C22—21no—
337C22—21no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—1081no—
342C22—1081no—
343C22—1081no—
344C22#8854412yescomplemented
345C22C2×#623311yescharacteristic
346C4—21no—
347C4—41no—
348C4—1081no—
349C4D4×C3⋊D911yescharacteristic
350C4#8874211yescharacteristic
351C3#1058872113yescomplemented
352C3#1059273711yescharacteristic
353C2—22no—
354C2—81no—
355C2—2161no—
356C2C2×#8854411yescharacteristic
357C2#1172183911yescharacteristic
358C2#1172186811yescharacteristic
subgroupG/H
centerC22C2×#6233
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#121C2
socleC22×C32C2×#67
hypercenter#121C3⋊D9
hypocenterC3×C9#466
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C3×C9×#121C2semidirect
2. C2×#88544C2semidirect
3. #11721819C2semidirect
4. C22×C3⋊(C9⋊C8)C2semidirect
5. #11720576C2semidirect
6. #11720528C2semidirect
7. #11720525C2semidirect
30. C9×#121D3semidirect
53. C3⋊(C9⋊C8)D4semidirect
87. C22×C3×C9D8semidirect
97. C3×#121D9semidirect
198. C22×C9#213semidirect
211. #121C3⋊D9semidirect
215. C3×C9#466semidirect
285. C22×C3#3673semidirect
306. C9#5038semidirect
344. C22#88544semidirect
351. C3#10588721semidirect

Maximal subgroups

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

subgroupindex
C3×C9×#1212
C2×#885442
#117218192
C22×C3⋊(C9⋊C8)2
#117205762
#117205282
#117205252
#105927373
#105887213