Finite Group Databasebeta

← #13144472

Subgroups — #13144472

Order 1728.

620 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#88548C211yesmaximal, characteristic, complemented
3#11721819C211yesmaximal, characteristic
4C22×C3⋊(C9⋊C8)C211yesmaximal, characteristic, complemented
5#11720582C211yesmaximal, characteristic, complemented
6#11720528C211yesmaximal, characteristic, complemented
7#11720526C211yesmaximal, characteristic, complemented
8#10592747—31nomaximal
9#10588731—33nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C22×C3×C4×C9C2211yescharacteristic
12C2×#6233—21no—
13C2×C4⋊(C3⋊D9)C2211yescharacteristic
14C3×C9×C4⋊C4—21no—
15C3×C9×C4⋊C4C2211yescharacteristic
16C3×C9×C22⋊C4—21no—
17#88548—22no—
18#88536—21no—
19C4⋊(C3⋊Dic9)C2211yescharacteristic
20C2×C3⋊(C9⋊C8)—22no—
21C2×C3⋊(C9⋊C8)C2211yescharacteristic
22C2×C3⋊(C9⋊C8)C2211yescharacteristic
23C32×C22⋊Q8D311yescharacteristic
24C2×#3755—31no—
25#64139—31no—
26C22×C3⋊(C3⋊C8)—31no—
27#63662—31no—
28#63636—31no—
29#63635—31no—
30C9×C22⋊Q8D313yescomplemented
31C2×#3675—33no—
32#63492—33no—
33C22×C9⋊C8—33no—
34#63395—33no—
35#63369—33no—
36#63367—33no—
37C23×C3×C9D411yescharacteristic
38C22×C3⋊D9—21no—
39C3×Q8×C9—21no—
40C3×Q8×C9—21no—
41C2×C3×C4×C9—21no—
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9D411yescharacteristic
46C2×C3×C4×C9C2311yescharacteristic
47#6233—42no—
48C2×C3⋊Dic9—21no—
49C4⋊(C3⋊D9)—21no—
50C4⋊(C3⋊D9)—21no—
51C3⋊(C9⋊C8)—21no—
52C3⋊(C9⋊C8)D412yescomplemented
53#5048—93no—
54#5048—91no—
55C2×C32×Q8C2×D311yescharacteristic
56C22×C32×C4C2×D311yescharacteristic
57C2×#638—61no—
58C2×C4⋊(C3⋊D3)—31no—
59C32×C4⋊C4—21no—
60C32×C4⋊C4C2×D311yescharacteristic
61C32×C22⋊C4—21no—
62#3755—62no—
63#3752—61no—
64C4⋊(C3⋊Dic3)—31no—
65C2×C3⋊(C3⋊C8)—31no—
66C2×C3⋊(C3⋊C8)—31no—
67C2×C3⋊(C3⋊C8)—62no—
68C2×Q8×C9C2×D313yes—
69C22×C4×C9C2×D313yes—
70C2×#611—63no—
71C2×D36—33no—
72C9×C4⋊C4—23no—
73C9×C4⋊C4C2×D313yes—
74C9×C22⋊C4—23no—
75#3675—66no—
76(C2×D9)⋊C4—63no—
77C4⋊Dic9—33no—
78C2×C9⋊C8—33no—
79C2×C9⋊C8—33no—
80C2×C9⋊C8—66no—
81C22×C3×C9—21no—
82C22×C3×C9—21no—
83C22×C3×C9SD1612yescomplemented
84C22×C3×C9C2×D411yescharacteristic
85C2×C3⋊D9—41no—
86C2×C3⋊D9—41no—
87C2×C3⋊D9—41no—
88C3×C4×C9—21no—
89C3×C4×C9—41no—
90C3×C4×C9—41no—
91C3×C4×C9C2×D411yescharacteristic
92C3×C4×C9#4111yescharacteristic
93C3⋊Dic9—41no—
94C3×C22⋊Q8D913yescomplemented
95C3×C22⋊Q8C3⋊D311yescharacteristic
96C2×#215—93no—
97C2×#215—91no—
98#930—93no—
99#930—91no—
100C22×C3⋊C8—93no—
101C22×C3⋊C8—91no—
102#835—93no—
103#835—91no—
104#809—93no—
105#809—91no—
106#808—93no—
107#808—91no—
108C23×C32#6711yescharacteristic
109C22×C3⋊D3—61no—
110C32×Q8—21no—
111C32×Q8—21no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4#6711yescharacteristic
117C2×C32×C4C22×D311yescharacteristic
118#638—122no—
119C2×C3⋊Dic3—61no—
120C4⋊(C3⋊D3)—61no—
121C4⋊(C3⋊D3)—61no—
122C23×C9#6713yes—
123C22×D9—63no—
124C3⋊(C3⋊C8)—32no—
125C3⋊(C3⋊C8)—61no—
126Q8×C9—23no—
127Q8×C9—23no—
128C2×C4×C9—23no—
129C2×C4×C9—23no—
130C2×C4×C9—23no—
131C2×C4×C9—23no—
132C2×C4×C9#6713yes—
133C2×C4×C9C22×D313yes—
134#611—126no—
135C2×Dic9—63no—
136D36—63no—
137D36—63no—
138C9⋊C8—36no—
139C9⋊C8—63no—
140#465—271noSylow
141C2×C3×C9—22no—
142C2×C3×C9#12111yescharacteristic
143C2×C3×C9C2×SD1611yescharacteristic
144C2×C3×C9#13511yescharacteristic
145C3⋊D9—81no—
146C2×C3×Q8C2×D913yes—
147C2×C3×Q8C2×C3⋊D311yescharacteristic
148C22×C3×C4C2×D913yes—
149C22×C3×C4C2×C3⋊D311yescharacteristic
150C2×#67—183no—
151C2×#67—181no—
152C2×D12—93no—
153C2×D12—91no—
154C3×C4⋊C4—23no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4C2×D913yes—
157C3×C4⋊C4C2×C3⋊D311yescharacteristic
158C3×C22⋊C4—23no—
159C3×C22⋊C4—21no—
160#215—182no—
161#215—186no—
162(C2×D3)⋊C4—183no—
163(C2×D3)⋊C4—181no—
164C4⋊Dic3—93no—
165C4⋊Dic3—91no—
166C2×C3⋊C8—93no—
167C2×C3⋊C8—91no—
168C2×C3⋊C8—93no—
169C2×C3⋊C8—91no—
170C2×C3⋊C8—182no—
171C2×C3⋊C8—186no—
172C22×C32—21no—
173C22×C32—21no—
174C22×C32#21512yes—
175C22×C32C2×#6711yescharacteristic
176C2×C3⋊D3—121no—
177C2×C3⋊D3—121no—
178C2×C3⋊D3—121no—
179C32×C4—21no—
180C32×C4—41no—
181C32×C4—41no—
182C32×C4D3×D411yescharacteristic
183C32×C4#23711yescharacteristic
184C3⋊Dic3—121no—
185C22×C9—23no—
186C22×C9—23no—
187C22×C9#21516yescomplemented
188C22×C9C2×#6713yes—
189C2×D9—123no—
190C2×D9—123no—
191C2×D9—123no—
192C4×C9—23no—
193C4×C9—43no—
194C4×C9—43no—
195C4×C9D3×D413yes—
196C4×C9#23713yes—
197Dic9—123no—
198C2×SD16—271no—
199C22×C8—271no—
200C22⋊Q8C3⋊D911yescharacteristic, complemented
201#121—271no—
202#106—271no—
203#103—271no—
204#102—271no—
205C3×C9#46511yescharacteristic, complemented, Sylow
206C23×C3#61113yes—
207C23×C3#63811yescharacteristic
208C22×D3—183no—
209C22×D3—181no—
210C3×Q8—23no—
211C3×Q8—23no—
212C3×Q8—21no—
213C3×Q8—21no—
214C2×C3×C4—23no—
215C2×C3×C4—23no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—23no—
219C2×C3×C4—23no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4#61113yes—
223C2×C3×C4C22×D913yes—
224C2×C3×C4#63811yescharacteristic
225C2×C3×C4C22×C3⋊D311yescharacteristic
226#67—366no—
227#67—362no—
228C2×Dic3—183no—
229C2×Dic3—181no—
230D12—183no—
231D12—183no—
232D12—181no—
233D12—181no—
234C3⋊C8—96no—
235C3⋊C8—92no—
236C3⋊C8—183no—
237C3⋊C8—181no—
238C2×C32—22no—
239C2×C32#93911yescharacteristic
240C2×C32C2×#21511yescharacteristic
241C2×C32#95011yescharacteristic
242C3⋊D3—241no—
243C2×C9—26no—
244C2×C9#93913yes—
245C2×C9C2×#21513yes—
246C2×C9#95013yes—
247D9—243no—
248C2×Q8C2×C3⋊D911yescharacteristic
249C2×D4—271no—
250C2×D4—541no—
251C22×C4C2×C3⋊D911yescharacteristic
252SD16—542no—
253C2×C8—271no—
254C2×C8—271no—
255C2×C8—542no—
256C4⋊C4—21no—
257C4⋊C4—271no—
258C4⋊C4C2×C3⋊D911yescharacteristic
259C22⋊C4—21no—
260C22⋊C4—541no—
261C22×C3—23no—
262C22×C3—23no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3#367516yescomplemented
266C22×C3C2×#61113yes—
267C22×C3#375512yes—
268C22×C3C2×#63811yescharacteristic
269C2×D3—363no—
270C2×D3—363no—
271C2×D3—363no—
272C2×D3—361no—
273C2×D3—361no—
274C2×D3—361no—
275C3×C4—23no—
276C3×C4—21no—
277C3×C4—43no—
278C3×C4—43no—
279C3×C4—41no—
280C3×C4—41no—
281C3×C4D4×D913yes—
282C3×C4#369913yes—
283C3×C4D4×C3⋊D311yescharacteristic
284C3×C4#383011yescharacteristic
285Dic3—363no—
286Dic3—361no—
287C32#504811yescharacteristic
288C9#504813yescomplemented
289C23—541no—
290C23#623311yescharacteristic
291Q8—21no—
292Q8—21no—
293D4—541no—
294D4—541no—
295D4—1082no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—541no—
301C2×C4#623311yescharacteristic
302C2×C4C22×C3⋊D911yescharacteristic
303C8—272no—
304C8—541no—
305C2×C3—26no—
306C2×C3—22no—
307C2×C3#6350113yes—
308C2×C3C2×#367513yes—
309C2×C3#6351313yes—
310C2×C3#6414811yescharacteristic
311C2×C3C2×#375511yescharacteristic
312C2×C3#6415911yescharacteristic
313D3—723no—
314D3—721no—
315C22—21no—
316C22—21no—
317C22—1081no—
318C22—1081no—
319C22—1081no—
320C22#8854812yescomplemented
321C22C2×#623311yescharacteristic
322C4—21no—
323C4—41no—
324C4—41no—
325C4—1081no—
326C4D4×C3⋊D911yescharacteristic
327C4#8874211yescharacteristic
328C3#1058873113yescomplemented
329C3#1059274711yescharacteristic
330C2—22no—
331C2—2161no—
332C2#1172183911yescharacteristic
333C2C2×#8854811yescharacteristic
334C2#1172186811yescharacteristic
subgroupG/H
centerC22C2×#6233
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×C22⋊Q8C2
socleC22×C32C2×#67
prime radical#11721819C2
hypercenterC22⋊Q8C3⋊D9
hypocenterC3×C9#465
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×C9×C22⋊Q8C2semidirect
2. C2×#88548C2semidirect
4. C22×C3⋊(C9⋊C8)C2semidirect
5. #11720582C2semidirect
6. #11720528C2semidirect
7. #11720526C2semidirect
30. C9×C22⋊Q8D3semidirect
52. C3⋊(C9⋊C8)D4semidirect
83. C22×C3×C9SD16semidirect
94. C3×C22⋊Q8D9semidirect
187. C22×C9#215semidirect
200. C22⋊Q8C3⋊D9semidirect
205. C3×C9#465semidirect
265. C22×C3#3675semidirect
288. C9#5048semidirect
320. C22#88548semidirect
328. C3#10588731semidirect

Maximal subgroups

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

subgroupindex
C3×C9×C22⋊Q82
C2×#885482
#117218192
C22×C3⋊(C9⋊C8)2
#117205822
#117205282
#117205262
#105927473
#105887313