Finite Group Databasebeta

← #13138542

Subgroups — #13138542

Order 1728.

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

normal subgroups only

This group has 378 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
1#11721036C211yesmaximal, characteristic
2D9×C4⋊Dic3C211yesmaximal, characteristic, complemented
3#11720540C211yesmaximal, characteristic, complemented
4C9×#821C211yesmaximal, characteristic, complemented
5C3×(C2×D9)⋊C8C211yesmaximal, characteristic, complemented
6#11720295C211yesmaximal, characteristic, complemented
7#11720264C211yesmaximal, characteristic, complemented
8#10590676—31nomaximal
9#10588457—31nomaximal
10C2×C9×D12C2211yescharacteristic
11C2×C3×C4×D9C2211yescharacteristic
12C2×#6194—21no—
13C2×Dic3×D9—21no—
14C2×C3×C8×C9C2211yescharacteristic
15C4⋊(C3⋊Dic9)C2211yescharacteristic
16C4⋊(C3⋊Dic9)C2211yescharacteristic
17C9×C4⋊Dic3C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19(C2×D3)⋊Dic9—21no—
20Dic9⋊Dic3—21no—
21#63908—31no—
22D3×C4⋊Dic3—31no—
23#63644—31no—
24C3×#821D311yescharacteristic
25C3×(C2×D3)⋊C8—31no—
26#63573—31no—
27#63563—31no—
28#63500—31no—
29C4⋊C4×D9—31no—
30C9×#102—31no—
31#63392—31no—
32(C2×D9)⋊C8D311yescharacteristic, complemented
33#63377—31no—
34#63377—61no—
35#63367—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—
48Dic3×D9—42no—
49C3×C8×C9—21no—
50C3×C9⋊C8—21no—
51#4777—91no—
52#4735—91no—
53C2×C3×D12C2×D311yescharacteristic
54C2×C3×C4×D3—31no—
55C2×#625—61no—
56C2×D3×Dic3—61no—
57C2×C32×C8C2×D311yescharacteristic
58C4⋊(C3⋊Dic3)—31no—
59C4⋊(C3⋊Dic3)—31no—
60C3×C4⋊Dic3C2×D311yescharacteristic
61C2×C3×C3⋊C8—31no—
62Dic3⋊Dic3—61no—
63(C2×D3)⋊Dic3—61no—
64C2×D4×C9—31no—
65C2×#611—61no—
66C2×C4×D9—61no—
67C2×C4×D9C2×D311yescharacteristic
68C2×C8×C9—21no—
69C2×C8×C9C2×D311yescharacteristic
70C9×C4⋊C4—31no—
71#3676—61no—
72C4⋊Dic9—31no—
73C4⋊Dic9—31no—
74C4⋊Dic9—61no—
75C4⋊Dic9—61no—
76Dic9⋊C4—61no—
77C2×C9⋊C8C2×D311yescharacteristic
78C22×C3×C9C2×D411yescharacteristic
79C2×D3×C9—41no—
80C2×D3×C9—41no—
81C2×D3×C9—41no—
82C2×C3×D9—21no—
83C2×C3×D9—21no—
84C2×C3×D9SD1612yes—
85C3×C4×C9#4111yescharacteristic
86C3×C4×C9#4111yescharacteristic
87C3⋊Dic9—41no—
88C3⋊Dic9—41no—
89C9×Dic3—41no—
90C3×Dic9—21no—
91#938—91no—
92#930—91no—
93C2×C4⋊Dic3—91no—
94D3×C4⋊C4—91no—
95C3×#102—31no—
96C3×C22⋊C8—91no—
97#832—91no—
98#821—91no—
99#821D911yescharacteristic, complemented
100(C2×D3)⋊C8—31no—
101#817—91no—
102#817—91no—
103#817—91no—
104#817—181no—
105#808—91no—
106C22×C3×D3—21no—
107C22×C3×D3—31no—
108C2×C32×C4C22×D311yescharacteristic
109C2×C3⋊Dic3—61no—
110C2×C3⋊Dic3—61no—
111C2×C3×Dic3—21no—
112C2×C3×Dic3—31no—
113C3×D12—21no—
114C3×D12—21no—
115C3×C4×D3—61no—
116C3×C4×D3—61no—
117#625—122no—
118D3×Dic3—122no—
119C23×C9—61no—
120C22×D9D1211yescharacteristic
121C32×C8—21no—
122C3×C3⋊C8—61no—
123D4×C9—61no—
124D4×C9—61no—
125C2×C4×C9—21no—
126C2×C4×C9—61no—
127C2×C4×C9C22×D311yescharacteristic
128#611—122no—
129C2×Dic9—61no—
130C2×Dic9—61no—
131C2×Dic9—121no—
132C2×Dic9—121no—
133C2×Dic9D1211yescharacteristic
134C4×D9—21no—
135C4×D9—21no—
136C4×D9—122no—
137C8×C9—22no—
138C8×C9—21no—
139C9⋊C8—21no—
140#481—271noSylow
141C2×C3×C9#12311yescharacteristic
142C2×C3×C9C2×SD1611yescharacteristic
143C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
144D3×C9—81no—
145C3×D9—22no—
146C2×C3×D4—31no—
147C22×C3×C4—91no—
148C2×#67—181no—
149C2×#67—181no—
150C22×Dic3—181no—
151C2×D12C2×D911yescharacteristic
152C2×C4×D3—31no—
153C2×C4×D3—181no—
154C2×C3×C8—21no—
155C2×C3×C8—91no—
156C2×C3×C8C2×D911yescharacteristic
157C2×C3×C8D3211yescharacteristic
158C3×C4⋊C4—31no—
159#217—181no—
160(C2×D3)⋊C4—181no—
161C4⋊Dic3—91no—
162C4⋊Dic3—91no—
163C4⋊Dic3—91no—
164C4⋊Dic3—91no—
165C4⋊Dic3—181no—
166C4⋊Dic3—181no—
167C4⋊Dic3—181no—
168C4⋊Dic3C2×D911yescharacteristic
169Dic3⋊C4—181no—
170C2×C3⋊C8—31no—
171C22×C32D3×D411yescharacteristic
172C2×C3×D3—32no—
173C2×C3×D3—41no—
174C2×C3×D3—41no—
175C2×C3×D3—41no—
176C2×C3×D3—61no—
177C2×C3×D3—61no—
178C32×C4#23511yescharacteristic
179C32×C4#23711yescharacteristic
180C3⋊Dic3—121no—
181C3⋊Dic3—121no—
182C3×Dic3—41no—
183C3×Dic3—61no—
184C22×C9—21no—
185C22×C9—121no—
186C22×C9—121no—
187C22×C9—121no—
188C22×C9C2×D1211yescharacteristic
189C2×D9—21no—
190C2×D9—21no—
191C2×D9#20412yes—
192C4×C9—21no—
193C4×C9—21no—
194C4×C9—121no—
195C4×C9#23711yescharacteristic
196C4×C9#23711yescharacteristic
197Dic9—21no—
198Dic9—121no—
199Dic9—121no—
200Dic9—241no—
201Dic9—241no—
202#121—271no—
203C2×C4⋊C4—271no—
204#106—271no—
205#106—271no—
206#102—31no—
207#102—271no—
208C22⋊C8—91no—
209C3×C9#48111yescharacteristic, complemented, Sylow
210C23×C3—61no—
211C23×C3—91no—
212C22×D3—21no—
213C22×D3—31no—
214C3×D4—61no—
215C3×D4—61no—
216C2×C3×C4—21no—
217C2×C3×C4—61no—
218C2×C3×C4—91no—
219C2×C3×C4—181no—
220C2×C3×C4—181no—
221C2×C3×C4C22×D911yescharacteristic
222C2×C3×C4C2×D3211yescharacteristic
223#67—362no—
224#67—362no—
225C2×Dic3—21no—
226C2×Dic3—31no—
227C2×Dic3—181no—
228C2×Dic3—181no—
229C2×Dic3—181no—
230C2×Dic3—181no—
231C2×Dic3—361no—
232C2×Dic3—361no—
233C2×Dic3—362no—
234D12—21no—
235D12—21no—
236C4×D3—61no—
237C4×D3—61no—
238C4×D3—362no—
239C3×C8—22no—
240C3×C8—21no—
241C3×C8—21no—
242C3×C8—181no—
243C3⋊C8—61no—
244C2×C32#88311yescharacteristic
245C2×C32#91111yescharacteristic
246C2×C32D3×SD1611yescharacteristic
247C3×D3—62no—
248C3×D3—81no—
249C2×C9—21no—
250C2×C9—21no—
251C2×C9—21no—
252C2×C9—241no—
253C2×C9#88611yescharacteristic
254C2×C9C2×#20411yescharacteristic
255C2×C9#90811yescharacteristic
256D9—22no—
257C2×D4—31no—
258C2×D4—541no—
259C22×C4—91no—
260C22×C4—541no—
261C2×C8—91no—
262C2×C8D3×D911yescharacteristic
263C4⋊C4—31no—
264C4⋊C4—271no—
265C4⋊C4—271no—
266C4⋊C4—541no—
267C22⋊C4—541no—
268C22×C3—21no—
269C22×C3—92no—
270C22×C3—121no—
271C22×C3—121no—
272C22×C3—121no—
273C22×C3—181no—
274C22×C3—181no—
275C22×C3D4×D911yescharacteristic
276C22×C3D3×D1211yescharacteristic
277C2×D3—32no—
278C2×D3—41no—
279C2×D3—41no—
280C2×D3—41no—
281C2×D3—61no—
282C2×D3—61no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—121no—
286C3×C4—181no—
287C3×C4#369711yescharacteristic
288C3×C4#369911yescharacteristic
289C3×C4#379511yescharacteristic
290C3×C4#380511yescharacteristic
291Dic3—41no—
292Dic3—61no—
293Dic3—361no—
294Dic3—361no—
295Dic3—361no—
296Dic3—361no—
297Dic3—721no—
298Dic3—721no—
299C32#477711yescharacteristic
300C9—21no—
301C9#473511yescharacteristic, complemented
302C23—61no—
303C23—91no—
304D4—61no—
305D4—61no—
306D4—1082no—
307C2×C4—61no—
308C2×C4—91no—
309C2×C4—181no—
310C2×C4—181no—
311C2×C4—541no—
312C2×C4—541no—
313C2×C4—1082no—
314C2×C4C2×D3×D911yescharacteristic
315C8—21no—
316C8—181no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—182no—
321C2×C3—241no—
322C2×C3#6344511yescharacteristic
323C2×C3#6347311yescharacteristic
324C2×C3SD16×D911yescharacteristic
325C2×C3D3×#20411yescharacteristic
326C2×C3#6379511yescharacteristic
327C2×C3#6389111yescharacteristic
328D3—62no—
329D3—81no—
330C22—92no—
331C22—121no—
332C22—121no—
333C22—121no—
334C22—181no—
335C22—181no—
336C22D9×D1211yescharacteristic
337C4—121no—
338C4—181no—
339C4—1081no—
340C4—1081no—
341C4#8864111yescharacteristic
342C4#8866311yescharacteristic
343C3—21no—
344C3#1058845711yescharacteristic, complemented
345C3#1059067611yescharacteristic
346C2—182no—
347C2—241no—
348C2D9×#20411yescharacteristic
349C2#1172081611yescharacteristic
350C2#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#481
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. D9×C4⋊Dic3C2semidirect
3. #11720540C2semidirect
4. C9×#821C2semidirect
5. C3×(C2×D9)⋊C8C2semidirect
6. #11720295C2semidirect
7. #11720264C2semidirect
32. (C2×D9)⋊C8D3semidirect
99. #821D9semidirect
209. C3×C9#481semidirect
301. C9#4735semidirect
344. C3#10588457semidirect

Maximal subgroups

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

subgroupindex
#117210362
D9×C4⋊Dic32
#117205402
C9×#8212
C3×(C2×D9)⋊C82
#117202952
#117202642
#105906763
#105884573