Finite Group Databasebeta

← #13138472

Subgroups — #13138472

Order 1728.

378 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 89 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#11721037C211yesmaximal, characteristic
2D3×C4⋊Dic9C211yesmaximal, characteristic, complemented
3#11720540C211yesmaximal, characteristic, complemented
4C9×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
5C3×#63380C211yesmaximal, characteristic, complemented
6#11720283C211yesmaximal, characteristic, complemented
7#11720278C211yesmaximal, characteristic, complemented
8#10590676—31nomaximal
9#10588417—31nomaximal
10C2×C4×D3×C9C2211yescharacteristic
11C2×C3×D36C2211yescharacteristic
12C2×#6194—21no—
13C2×D3×Dic9—21no—
14C2×C3×C8×C9C2211yescharacteristic
15C4⋊(C3⋊Dic9)C2211yescharacteristic
16C4⋊(C3⋊Dic9)C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19(C2×D9)⋊Dic3—21no—
20Dic3⋊Dic9—21no—
21#63908—31no—
22D3×C4⋊Dic3—31no—
23#63644—31no—
24C3×#821—31no—
25C3×(C2×D3)⋊C8D311yescharacteristic
26#63573—31no—
27#63563—31no—
28#63492—31no—
29C2×C4⋊Dic9—31no—
30C9×C22⋊C8—31no—
31#63380—31no—
32#63380D311yescharacteristic, complemented
33#63377—31no—
34#63377—31no—
35#63377—61no—
36C22×D3×C9D411yescharacteristic
37C22×C3×D9—21no—
38C2×C3×C4×C9C2311yescharacteristic
39C2×C3⋊Dic9—21no—
40C2×C3⋊Dic9—21no—
41C2×C9×Dic3D411yescharacteristic
42C2×C3×Dic9—21no—
43C4×D3×C9—21no—
44C4×D3×C9—21no—
45C3×D36—21no—
46C3×D36—21no—
47#6194—42no—
48D3×Dic9—42no—
49C3×C8×C9—21no—
50C9×C3⋊C8—21no—
51#4777—91no—
52#4735—91no—
53C2×C3×D12—31no—
54C2×C3×C4×D3C2×D311yescharacteristic
55C2×#625—61no—
56C2×D3×Dic3—61no—
57C2×C32×C8C2×D311yescharacteristic
58C4⋊(C3⋊Dic3)—31no—
59C4⋊(C3⋊Dic3)—31no—
60C3×C4⋊Dic3—31no—
61C2×C3×C3⋊C8C2×D311yescharacteristic
62Dic3⋊Dic3—61no—
63(C2×D3)⋊Dic3—61no—
64C22×C4×C9—31no—
65C2×#611—61no—
66C22×Dic9—61no—
67C2×D36C2×D311yescharacteristic
68C2×C8×C9—21no—
69C2×C8×C9—31no—
70C2×C8×C9C2×D311yescharacteristic
71(C2×D9)⋊C4—61no—
72C4⋊Dic9—31no—
73C4⋊Dic9—31no—
74C4⋊Dic9—61no—
75C4⋊Dic9—61no—
76C4⋊Dic9—61no—
77C4⋊Dic9C2×D311yescharacteristic
78C22×C3×C9C2×D411yescharacteristic
79C2×D3×C9—21no—
80C2×D3×C9—21no—
81C2×D3×C9SD1612yes—
82C2×C3×D9—41no—
83C2×C3×D9—41no—
84C2×C3×D9—41no—
85C3×C4×C9#4111yescharacteristic
86C3×C4×C9#4111yescharacteristic
87C3⋊Dic9—41no—
88C3⋊Dic9—41no—
89C9×Dic3—21no—
90C3×Dic9—41no—
91#938—91no—
92#930—91no—
93C2×C4⋊Dic3—91no—
94D3×C4⋊C4—91no—
95C3×#102—91no—
96C3×C22⋊C8—31no—
97#832—91no—
98#821—31no—
99#821—91no—
100(C2×D3)⋊C8D911yescharacteristic, complemented
101#817—91no—
102#817—91no—
103#817—91no—
104#817—181no—
105#808—91no—
106C22×C3×D3—61no—
107C22×C3×D3D1211yescharacteristic
108C2×C32×C4C22×D311yescharacteristic
109C2×C3⋊Dic3—61no—
110C2×C3⋊Dic3—61no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3D1211yescharacteristic
113C3×D12—61no—
114C3×D12—61no—
115C3×C4×D3—21no—
116C3×C4×D3—21no—
117#625—122no—
118D3×Dic3—122no—
119C23×C9—31no—
120C22×D9—21no—
121C32×C8—21no—
122C3×C3⋊C8—21no—
123C2×C4×C9—21no—
124C2×C4×C9—31no—
125C2×C4×C9—61no—
126C2×C4×C9—61no—
127C2×C4×C9C22×D311yescharacteristic
128#611—122no—
129C2×Dic9—21no—
130C2×Dic9—61no—
131C2×Dic9—61no—
132C2×Dic9—121no—
133C2×Dic9—121no—
134C2×Dic9—122no—
135D36—21no—
136D36—21no—
137C8×C9—22no—
138C8×C9—21no—
139C8×C9—61no—
140#481—271noSylow
141C2×C3×C9#12311yescharacteristic
142C2×C3×C9C2×SD1611yescharacteristic
143C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
144D3×C9—22no—
145C3×D9—81no—
146C2×C3×D4—91no—
147C22×C3×C4—31no—
148C2×#67—181no—
149C2×#67—181no—
150C22×Dic3—181no—
151C2×D12—31no—
152C2×C4×D3—181no—
153C2×C4×D3C2×D911yescharacteristic
154C2×C3×C8—21no—
155C2×C3×C8—31no—
156C2×C3×C8C2×D911yescharacteristic
157C2×C3×C8D3211yescharacteristic
158C3×C4⋊C4—91no—
159#217—181no—
160(C2×D3)⋊C4—181no—
161C4⋊Dic3—31no—
162C4⋊Dic3—91no—
163C4⋊Dic3—91no—
164C4⋊Dic3—91no—
165C4⋊Dic3—91no—
166C4⋊Dic3—181no—
167C4⋊Dic3—181no—
168C4⋊Dic3—181no—
169Dic3⋊C4—181no—
170C2×C3⋊C8C2×D911yescharacteristic
171C22×C32C2×D1211yescharacteristic
172C2×C3×D3—21no—
173C2×C3×D3—21no—
174C2×C3×D3—121no—
175C2×C3×D3—121no—
176C2×C3×D3—121no—
177C2×C3×D3#20412yes—
178C32×C4#23711yescharacteristic
179C32×C4#23711yescharacteristic
180C3⋊Dic3—121no—
181C3⋊Dic3—121no—
182C3×Dic3—21no—
183C3×Dic3—121no—
184C22×C9—21no—
185C22×C9—32no—
186C22×C9—61no—
187C22×C9—61no—
188C22×C9D3×D411yescharacteristic
189C2×D9—41no—
190C2×D9—41no—
191C2×D9—41no—
192C4×C9—21no—
193C4×C9—21no—
194C4×C9—61no—
195C4×C9#23511yescharacteristic
196C4×C9#23711yescharacteristic
197Dic9—41no—
198Dic9—121no—
199Dic9—121no—
200Dic9—241no—
201Dic9—241no—
202#121—271no—
203C2×C4⋊C4—271no—
204#106—271no—
205#106—271no—
206#102—91no—
207#102—271no—
208C22⋊C8—31no—
209C3×C9#48111yescharacteristic, complemented, Sylow
210C23×C3—31no—
211C23×C3—181no—
212C22×D3—61no—
213C22×D3D3611yescharacteristic
214C3×D4—181no—
215C3×D4—181no—
216C2×C3×C4—21no—
217C2×C3×C4—31no—
218C2×C3×C4—61no—
219C2×C3×C4—61no—
220C2×C3×C4—181no—
221C2×C3×C4C22×D911yescharacteristic
222C2×C3×C4C2×D3211yescharacteristic
223#67—362no—
224#67—362no—
225C2×Dic3—61no—
226C2×Dic3—181no—
227C2×Dic3—181no—
228C2×Dic3—181no—
229C2×Dic3—181no—
230C2×Dic3—361no—
231C2×Dic3—361no—
232C2×Dic3—362no—
233C2×Dic3D3611yescharacteristic
234D12—61no—
235D12—61no—
236C4×D3—21no—
237C4×D3—21no—
238C4×D3—362no—
239C3×C8—22no—
240C3×C8—21no—
241C3×C8—21no—
242C3×C8—61no—
243C3⋊C8—21no—
244C2×C32#88611yescharacteristic
245C2×C32C2×#20411yescharacteristic
246C2×C32#90811yescharacteristic
247C3×D3—22no—
248C3×D3—241no—
249C2×C9—21no—
250C2×C9—21no—
251C2×C9—21no—
252C2×C9—62no—
253C2×C9#88311yescharacteristic
254C2×C9#91111yescharacteristic
255C2×C9D3×SD1611yescharacteristic
256D9—81no—
257C2×D4—91no—
258C2×D4—541no—
259C22×C4—31no—
260C22×C4—541no—
261C2×C8—31no—
262C2×C8D3×D911yescharacteristic
263C4⋊C4—91no—
264C4⋊C4—271no—
265C4⋊C4—271no—
266C4⋊C4—541no—
267C22⋊C4—541no—
268C22×C3—21no—
269C22×C3—32no—
270C22×C3—61no—
271C22×C3—61no—
272C22×C3—361no—
273C22×C3—361no—
274C22×C3—361no—
275C22×C3C2×D3611yescharacteristic
276C22×C3D3×D1211yescharacteristic
277C2×D3—21no—
278C2×D3—21no—
279C2×D3—121no—
280C2×D3—121no—
281C2×D3—121no—
282C2×D3#366412yes—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—61no—
286C3×C4—361no—
287C3×C4#369911yescharacteristic
288C3×C4#369911yescharacteristic
289C3×C4#379511yescharacteristic
290C3×C4#380511yescharacteristic
291Dic3—21no—
292Dic3—121no—
293Dic3—361no—
294Dic3—361no—
295Dic3—361no—
296Dic3—361no—
297Dic3—721no—
298Dic3—721no—
299C32#473511yescharacteristic
300C9—21no—
301C9#477711yescharacteristic, complemented
302C23—31no—
303C23—181no—
304D4—181no—
305D4—181no—
306D4—1082no—
307C2×C4—31no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—181no—
311C2×C4—541no—
312C2×C4—541no—
313C2×C4—1082no—
314C2×C4C2×D3×D911yescharacteristic
315C8—21no—
316C8—61no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—62no—
321C2×C3—721no—
322C2×C3#6344811yescharacteristic
323C2×C3C2×#366411yescharacteristic
324C2×C3#6347011yescharacteristic
325C2×C3D3×#20411yescharacteristic
326C2×C3#6379511yescharacteristic
327C2×C3#6389111yescharacteristic
328D3—22no—
329D3—241no—
330C22—32no—
331C22—61no—
332C22—61no—
333C22—361no—
334C22—361no—
335C22—361no—
336C22D3×D3611yescharacteristic
337C4—61no—
338C4—361no—
339C4—1081no—
340C4—1081no—
341C4#8863811yescharacteristic
342C4#8866311yescharacteristic
343C3—21no—
344C3#1058841711yescharacteristic, complemented
345C3#1059067611yescharacteristic
346C2—62no—
347C2—721no—
348C2D3×#366411yescharacteristic
349C2#1172080511yescharacteristic
350C2#1172102111yescharacteristic
subgroupG/H
centerC22D3×D36
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32C2×D12
prime radical#11721037C2
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. D3×C4⋊Dic9C2semidirect
3. #11720540C2semidirect
4. C9×(C2×D3)⋊C8C2semidirect
5. C3×#63380C2semidirect
6. #11720283C2semidirect
7. #11720278C2semidirect
32. #63380D3semidirect
100. (C2×D3)⋊C8D9semidirect
209. C3×C9#481semidirect
301. C9#4777semidirect
344. C3#10588417semidirect

Maximal subgroups

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

subgroupindex
#117210372
D3×C4⋊Dic92
#117205402
C9×(C2×D3)⋊C82
C3×#633802
#117202832
#117202782
#105906763
#105884173