Finite Group Databasebeta

← #13138547

Subgroups — #13138547

Order 1728.

354 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 81 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
1#11721036C211yesmaximal, characteristic
2#11720907C211yesmaximal, characteristic, complemented
3#11720540C211yesmaximal, characteristic, complemented
4C9×#821C211yesmaximal, characteristic, complemented
5C3×(C2×D9)⋊C8C211yesmaximal, characteristic, complemented
6#11720295C211yesmaximal, characteristic, complemented
7#11720262C211yesmaximal, characteristic, complemented
8#10590837—31nomaximal
9#10588459—31nomaximal
10C2×C9×D12C2211yescharacteristic
11C2×C3×C4×D9C2211yescharacteristic
12C2×#6194—21no—
13C2×C3×C8×C9C2211yescharacteristic
14C4⋊(C3⋊Dic9)C2211yescharacteristic
15C4⋊(C3⋊Dic9)C2211yescharacteristic
16C9×C4⋊Dic3C2211yescharacteristic
17C2×C3×C9⋊C8C2211yescharacteristic
18(C2×D3)⋊Dic9—21no—
19(C2×D9)⋊Dic3—21no—
20Dic3×Dic9—21no—
21#63908—31no—
22#63841—31no—
23#63644—31no—
24C3×#821D311yescharacteristic
25C3×(C2×D3)⋊C8—31no—
26#63575—31no—
27#63563—31no—
28#63500—31no—
29#63454—31no—
30C9×#102—31no—
31#63392—31no—
32(C2×D9)⋊C8D311yescharacteristic, complemented
33#63377—31no—
34#63377—61no—
35#63366—31no—
36C22×D3×C9—21no—
37C22×C3×D9D411yescharacteristic
38C2×C3×C4×C9C2311yescharacteristic
39C2×C3⋊Dic9—21no—
40C2×C3⋊Dic9—21no—
41C2×C9×Dic3—21no—
42C2×C3×Dic9D411yescharacteristic
43C9×D12—21no—
44C9×D12—21no—
45C3×C4×D9—21no—
46C3×C4×D9—21no—
47#6194—42no—
48C3×C8×C9—21no—
49C3×C9⋊C8—21no—
50#4779—91no—
51#4737—91no—
52C2×C3×D12C2×D311yescharacteristic
53C2×C3×C4×D3—31no—
54C2×#625—61no—
55C2×C32×C8C2×D311yescharacteristic
56C4⋊(C3⋊Dic3)—31no—
57C4⋊(C3⋊Dic3)—31no—
58C3×C4⋊Dic3C2×D311yescharacteristic
59C2×C3×C3⋊C8—31no—
60(C2×D3)⋊Dic3—61no—
61(C2×D3)⋊Dic3—61no—
62Dic32—61no—
63C2×D4×C9—31no—
64C2×#611—61no—
65C2×C4×D9C2×D311yescharacteristic
66C2×C8×C9—21no—
67C2×C8×C9C2×D311yescharacteristic
68C9×C4⋊C4—31no—
69#3676—61no—
70(C2×D9)⋊C4—61no—
71C4⋊Dic9—31no—
72C4⋊Dic9—31no—
73C4⋊Dic9—61no—
74C4⋊Dic9—61no—
75C4×Dic9—61no—
76C2×C9⋊C8C2×D311yescharacteristic
77C22×C3×C9C2×D411yescharacteristic
78C2×D3×C9—41no—
79C2×D3×C9—41no—
80C2×D3×C9—41no—
81C2×C3×D9—21no—
82C2×C3×D9—21no—
83C2×C3×D9—21no—
84C3×C4×C9#4111yescharacteristic
85C3×C4×C9#4111yescharacteristic
86C3⋊Dic9—41no—
87C3⋊Dic9—41no—
88C9×Dic3—41no—
89C3×Dic9—21no—
90#938—91no—
91#930—91no—
92#926—91no—
93#892—91no—
94C3×#102—31no—
95C3×C22⋊C8—91no—
96#832—91no—
97#821—91no—
98#821D911yescharacteristic, complemented
99(C2×D3)⋊C8—31no—
100#818—91no—
101#817—91no—
102#817—91no—
103#817—181no—
104#807—91no—
105C22×C3×D3—21no—
106C22×C3×D3—31no—
107C2×C32×C4C22×D311yescharacteristic
108C2×C3⋊Dic3—61no—
109C2×C3⋊Dic3—61no—
110C2×C3×Dic3—21no—
111C2×C3×Dic3—31no—
112C3×D12—21no—
113C3×D12—21no—
114C3×C4×D3—61no—
115C3×C4×D3—61no—
116#625—122no—
117C23×C9—61no—
118C22×D9D1211yescharacteristic
119C32×C8—21no—
120C3×C3⋊C8—61no—
121D4×C9—61no—
122D4×C9—61no—
123C2×C4×C9—21no—
124C2×C4×C9—61no—
125C2×C4×C9C22×D311yescharacteristic
126#611—122no—
127C2×Dic9—61no—
128C2×Dic9—61no—
129C2×Dic9—121no—
130C2×Dic9—121no—
131C2×Dic9D1211yescharacteristic
132C4×D9—21no—
133C4×D9—21no—
134C8×C9—22no—
135C8×C9—21no—
136C9⋊C8—21no—
137#482—271noSylow
138C2×C3×C9#12311yescharacteristic
139C2×C3×C9#13511yescharacteristic
140C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
141D3×C9—81no—
142C3×D9—41no—
143C2×C3×D4—31no—
144C22×C3×C4—91no—
145C2×#67—181no—
146C2×#67—181no—
147C2×D12C2×D911yescharacteristic
148C2×C4×D3—31no—
149C2×C3×C8—21no—
150C2×C3×C8—91no—
151C2×C3×C8C2×D911yescharacteristic
152C2×C3×C8D3211yescharacteristic
153C3×C4⋊C4—31no—
154#217—181no—
155#217—181no—
156(C2×D3)⋊C4—181no—
157(C2×D3)⋊C4—181no—
158C4⋊Dic3—91no—
159C4⋊Dic3—91no—
160C4⋊Dic3—91no—
161C4⋊Dic3—91no—
162C4⋊Dic3—181no—
163C4⋊Dic3—181no—
164C4⋊Dic3C2×D911yescharacteristic
165C4×Dic3—181no—
166C4×Dic3—181no—
167C2×C3⋊C8—31no—
168C22×C32D3×D411yescharacteristic
169C2×C3×D3—41no—
170C2×C3×D3—41no—
171C2×C3×D3—41no—
172C2×C3×D3—61no—
173C2×C3×D3—61no—
174C2×C3×D3—61no—
175C32×C4#23511yescharacteristic
176C32×C4#23711yescharacteristic
177C3⋊Dic3—121no—
178C3⋊Dic3—121no—
179C3×Dic3—41no—
180C3×Dic3—61no—
181C22×C9—21no—
182C22×C9—121no—
183C22×C9—121no—
184C22×C9—121no—
185C22×C9C2×D1211yescharacteristic
186C2×D9—21no—
187C2×D9—21no—
188C2×D9—21no—
189C4×C9—21no—
190C4×C9—21no—
191C4×C9—121no—
192C4×C9#23711yescharacteristic
193C4×C9#23711yescharacteristic
194Dic9—21no—
195Dic9—121no—
196Dic9—121no—
197Dic9—241no—
198Dic9—241no—
199#121—271no—
200#117—271no—
201#107—271no—
202#106—271no—
203#102—31no—
204#102—271no—
205C22⋊C8—91no—
206C3×C9#48211yescharacteristic, complemented, Sylow
207C23×C3—61no—
208C23×C3—91no—
209C22×D3—21no—
210C22×D3—31no—
211C3×D4—61no—
212C3×D4—61no—
213C2×C3×C4—21no—
214C2×C3×C4—61no—
215C2×C3×C4—91no—
216C2×C3×C4—181no—
217C2×C3×C4—181no—
218C2×C3×C4C22×D911yescharacteristic
219C2×C3×C4C2×D3211yescharacteristic
220#67—362no—
221#67—362no—
222C2×Dic3—21no—
223C2×Dic3—31no—
224C2×Dic3—181no—
225C2×Dic3—181no—
226C2×Dic3—181no—
227C2×Dic3—181no—
228C2×Dic3—361no—
229C2×Dic3—361no—
230D12—21no—
231D12—21no—
232C4×D3—61no—
233C4×D3—61no—
234C3×C8—22no—
235C3×C8—21no—
236C3×C8—21no—
237C3×C8—181no—
238C3⋊C8—61no—
239C2×C32#88311yescharacteristic
240C2×C32#91111yescharacteristic
241C2×C32#91611yescharacteristic
242C3×D3—81no—
243C3×D3—121no—
244C2×C9—21no—
245C2×C9—21no—
246C2×C9—21no—
247C2×C9—241no—
248C2×C9#88611yescharacteristic
249C2×C9#90411yescharacteristic
250C2×C9#90811yescharacteristic
251D9—41no—
252C2×D4—31no—
253C2×D4—541no—
254C22×C4—91no—
255C2×C8—91no—
256C2×C8D3×D911yescharacteristic
257C4⋊C4—31no—
258C4⋊C4—271no—
259C4⋊C4—271no—
260C22⋊C4—541no—
261C22⋊C4—541no—
262C42—541no—
263C22×C3—21no—
264C22×C3—121no—
265C22×C3—121no—
266C22×C3—121no—
267C22×C3—181no—
268C22×C3—181no—
269C22×C3—181no—
270C22×C3D4×D911yescharacteristic
271C22×C3D3×D1211yescharacteristic
272C2×D3—41no—
273C2×D3—41no—
274C2×D3—41no—
275C2×D3—61no—
276C2×D3—61no—
277C2×D3—61no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—121no—
281C3×C4—181no—
282C3×C4#369711yescharacteristic
283C3×C4#369911yescharacteristic
284C3×C4#379511yescharacteristic
285C3×C4#380511yescharacteristic
286Dic3—41no—
287Dic3—61no—
288Dic3—361no—
289Dic3—361no—
290Dic3—361no—
291Dic3—361no—
292Dic3—721no—
293Dic3—721no—
294C32#477911yescharacteristic
295C9—21no—
296C9#473711yescharacteristic, complemented
297C23—61no—
298C23—91no—
299D4—61no—
300D4—61no—
301D4—1082no—
302C2×C4—61no—
303C2×C4—91no—
304C2×C4—181no—
305C2×C4—181no—
306C2×C4—541no—
307C2×C4—541no—
308C2×C4C2×D3×D911yescharacteristic
309C8—21no—
310C8—181no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—21no—
314C2×C3—241no—
315C2×C3—361no—
316C2×C3#6344511yescharacteristic
317C2×C3#6347311yescharacteristic
318C2×C3#6347811yescharacteristic
319C2×C3#6379511yescharacteristic
320C2×C3#6380611yescharacteristic
321C2×C3#6389111yescharacteristic
322D3—81no—
323D3—121no—
324C22—121no—
325C22—121no—
326C22—121no—
327C22—181no—
328C22—181no—
329C22—181no—
330C22D9×D1211yescharacteristic
331C4—121no—
332C4—181no—
333C4—1081no—
334C4—1081no—
335C4#8864111yescharacteristic
336C4#8866311yescharacteristic
337C3—21no—
338C3#1058845911yescharacteristic, complemented
339C3#1059083711yescharacteristic
340C2—241no—
341C2—361no—
342C2#1172081611yescharacteristic
343C2#1172082111yescharacteristic
344C2#1172098911yescharacteristic
subgroupG/H
centerC22D9×D12
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radical#11721036C2
hypercenterC2×C8D3×D9
hypocenterC3×C9#482
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11720907C2semidirect
3. #11720540C2semidirect
4. C9×#821C2semidirect
5. C3×(C2×D9)⋊C8C2semidirect
6. #11720295C2semidirect
7. #11720262C2semidirect
32. (C2×D9)⋊C8D3semidirect
98. #821D9semidirect
206. C3×C9#482semidirect
296. C9#4737semidirect
338. C3#10588459semidirect

Maximal subgroups

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

subgroupindex
#117210362
#117209072
#117205402
C9×#8212
C3×(C2×D9)⋊C82
#117202952
#117202622
#105908373
#105884593