Finite Group Databasebeta

← #13138527

Subgroups — #13138527

Order 1728.

410 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C2×C9×#204C211yesmaximal, characteristic
2#11721025C211yesmaximal, characteristic
3#11720911C211yesmaximal, characteristic, complemented
4C8×C3⋊Dic9C211yesmaximal, characteristic, complemented
5C3×#63377C211yesmaximal, characteristic, complemented
6#11720297C211yesmaximal, characteristic, complemented
7#11720294C211yesmaximal, characteristic
8#10590775—31nomaximal
9SD16×Dic9—31nomaximal
10C2×C9×D12C2211yescharacteristic
11C2×C9×Dic6C2211yescharacteristic
12C2×D3×Dic9—21no—
13C2×C3×C8×C9C2211yescharacteristic
14C4×C3⋊Dic9C2211yescharacteristic
15C2×C3⋊(C9⋊C8)C2211yescharacteristic
16C3×C4⋊Dic9C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18C9×#204C414yescomplemented
19(C2×D3)⋊Dic9—21no—
20Dic3⋊Dic9—21no—
21Dic3×Dic9—21no—
22C2×C3×#204D311yescharacteristic
23#63898—31no—
24#63843—31no—
25C8×C3⋊Dic3—31no—
26C3×#817—31no—
27#63565—31no—
28#63562—31no—
29C2×C9×SD16—31no—
30Q8×Dic9—31no—
31D4×Dic9—31no—
32#63395—31no—
33#63392—31no—
34#63377D311yescharacteristic, complemented
35C8×Dic9—31no—
36C8×Dic9—61no—
37C22×D3×C9—21no—
38C2×C3×C4×C9C2311yescharacteristic
39C2×C3⋊Dic9D411yescharacteristic
40C2×C3⋊Dic9D411yescharacteristic
41C2×C9×Dic3—21no—
42C2×C3×Dic9—21no—
43C2×C3×Dic9—21no—
44C9×D12—21no—
45C9×D12C2×C412yes—
46C9×Dic6—21no—
47C9×Dic6C2×C412yescomplemented
48D3×Dic9—42no—
49C3×C8×C9C2×C412yescomplemented
50C3⋊(C9⋊C8)—21no—
51Dic3×SD16—91no—
52#4852—91no—
53C2×C3×D12C2×D311yescharacteristic
54C2×C3×Dic6C2×D311yescharacteristic
55C2×D3×Dic3—61no—
56C2×C32×C8C2×D311yescharacteristic
57C4×C3⋊Dic3—31no—
58C2×C3⋊(C3⋊C8)—31no—
59C3×C4⋊Dic3—31no—
60C3×C4⋊Dic3—31no—
61C3×#204Dic314yes—
62Dic3⋊Dic3—61no—
63(C2×D3)⋊Dic3—61no—
64Dic32—61no—
65C2×Q8×C9—31no—
66C2×D4×C9—31no—
67C22×Dic9—61no—
68C9×SD16—34no—
69C2×C8×C9—21no—
70C2×C8×C9C2×D311yescharacteristic
71#3676—61no—
72C4⋊Dic9—61no—
73C4⋊Dic9C2×D311yescharacteristic
74C4⋊Dic9C2×D311yescharacteristic
75C4×Dic9—31no—
76C4×Dic9—61no—
77C4×Dic9—61no—
78C2×C9⋊C8—31no—
79C2×C9⋊C8—61no—
80C22×C3×C9C2×D411yescharacteristic
81C2×D3×C9—22no—
82C2×D3×C9—41no—
83C2×D3×C9—41no—
84C3×C4×C9C22×C411yescharacteristic
85C3×C4×C9#4111yescharacteristic
86C3⋊Dic9—21no—
87C3⋊Dic9SD1612yescomplemented
88C9×Dic3—22no—
89C3×Dic9—41no—
90C3×Dic9—41no—
91C2×C3×SD16—31no—
92Q8×Dic3—91no—
93D4×Dic3—91no—
94C2×#204D911yescharacteristic
95#893—91no—
96#887—91no—
97C3×#106—91no—
98#835—91no—
99#832—91no—
100#817—31no—
101C8×Dic3—91no—
102C8×Dic3—91no—
103C8×Dic3—181no—
104#810—91no—
105#809—91no—
106C22×C3×D3—21no—
107C2×C32×C4C22×D311yescharacteristic
108C2×C3⋊Dic3—31no—
109C2×C3⋊Dic3—31no—
110C2×C3×Dic3—21no—
111C2×C3×Dic3—61no—
112C2×C3×Dic3—61no—
113C3×D12—21no—
114C3×D12C2×Dic312yes—
115C3×Dic6—21no—
116C3×Dic6C2×Dic312yes—
117D3×Dic3—122no—
118C23×C9—61no—
119C32×C8C2×Dic312yes—
120C3⋊(C3⋊C8)—61no—
121Q8×C9—32no—
122Q8×C9—61no—
123D4×C9—32no—
124D4×C9—61no—
125C2×C4×C9—21no—
126C2×C4×C9—61no—
127C2×C4×C9C22×D311yescharacteristic
128C2×Dic9—21no—
129C2×Dic9—21no—
130C2×Dic9—31no—
131C2×Dic9—31no—
132C2×Dic9—61no—
133C2×Dic9—61no—
134C2×Dic9—122no—
135C8×C9—22no—
136C8×C9C4×D312yescomplemented
137C9⋊C8—62no—
138C9⋊C8—61no—
139C4×SD16—271noSylow
140C2×C3×C9C4×D411yescharacteristic
141C2×C3×C9C2×SD1611yescharacteristic
142C2×C3×C9#13511yescharacteristic
143D3×C9—42no—
144C2×C3×Q8—31no—
145C2×C3×D4—31no—
146C22×Dic3—181no—
147C2×D12C2×D911yescharacteristic
148C2×C4×D3—181no—
149C2×Dic6C2×D911yescharacteristic
150C3×SD16—34no—
151C2×C3×C8—21no—
152C2×C3×C8C2×D911yescharacteristic
153C2×C3×C8D3211yescharacteristic
154C3×C4⋊C4—91no—
155C3×C4⋊C4—91no—
156#217—181no—
157(C2×D3)⋊C4—181no—
158C4⋊Dic3—31no—
159C4⋊Dic3—31no—
160C4⋊Dic3—181no—
161Dic3⋊C4—181no—
162C4×Dic3—91no—
163C4×Dic3—91no—
164C4×Dic3—181no—
165C4×Dic3—181no—
166C4×Dic3—181no—
167C2×C3⋊C8—91no—
168C2×C3⋊C8—91no—
169C2×C3⋊C8—181no—
170#204Dic914yescomplemented
171C22×C32D3×D411yescharacteristic
172C2×C3×D3—22no—
173C2×C3×D3—41no—
174C2×C3×D3—41no—
175C32×C4#23711yescharacteristic
176C32×C4C22×Dic311yescharacteristic
177C3⋊Dic3—32no—
178C3⋊Dic3—61no—
179C3×Dic3—22no—
180C3×Dic3—121no—
181C3×Dic3—121no—
182C22×C9—21no—
183C22×C9—62no—
184C22×C9—121no—
185C22×C9—121no—
186C22×C9D3×D411yescharacteristic
187C4×C9—21no—
188C4×C9—21no—
189C4×C9—62no—
190C4×C9C2×C4×D311yescharacteristic
191C4×C9#23911yescharacteristic
192Dic9—32no—
193Dic9—41no—
194Dic9—41no—
195Dic9—62no—
196Dic9—62no—
197Dic9—61no—
198C2×SD16—31no—
199C4×Q8—271no—
200C4×D4—271no—
201#106—91no—
202#103—271no—
203#102—271no—
204C4×C8—271no—
205C3×C9C4×SD1611yescharacteristic, complemented, Sylow
206C23×C3—61no—
207C22×D3—21no—
208C3×Q8—32no—
209C3×Q8—61no—
210C3×D4—32no—
211C3×D4—61no—
212C2×C3×C4—21no—
213C2×C3×C4—61no—
214C2×C3×C4—181no—
215C2×C3×C4—181no—
216C2×C3×C4C22×D911yescharacteristic
217C2×C3×C4C2×D3211yescharacteristic
218C2×Dic3—21no—
219C2×Dic3—61no—
220C2×Dic3—61no—
221C2×Dic3—91no—
222C2×Dic3—91no—
223C2×Dic3—91no—
224C2×Dic3—91no—
225C2×Dic3—181no—
226C2×Dic3—181no—
227C2×Dic3—362no—
228D12—21no—
229D12C2×Dic912yes—
230C4×D3—362no—
231Dic6—21no—
232Dic6C2×Dic912yescomplemented
233C3×C8—22no—
234C3×C8C2×Dic912yescomplemented
235C3×C8D3×Dic312yes—
236C3⋊C8—182no—
237C3⋊C8—181no—
238C3⋊C8—181no—
239C2×C32D3×SD1611yescharacteristic
240C2×C32#91611yescharacteristic
241C2×C32D4×Dic311yescharacteristic
242C3×D3—42no—
243C2×C9—21no—
244C2×C9—21no—
245C2×C9—21no—
246C2×C9—122no—
247C2×C9#89311yescharacteristic
248C2×C9D3×SD1611yescharacteristic
249C2×C9#91611yescharacteristic
250C2×Q8—31no—
251C2×D4—31no—
252C22×C4—541no—
253SD16—34no—
254C2×C8—271no—
255C2×C8D3×D911yescharacteristic
256C4⋊C4—91no—
257C4⋊C4—91no—
258C4⋊C4—541no—
259C22⋊C4—541no—
260C42—271no—
261C42—541no—
262C22×C3—21no—
263C22×C3—62no—
264C22×C3—121no—
265C22×C3—121no—
266C22×C3D4×D911yescharacteristic
267C22×C3#380211yescharacteristic
268C2×D3—22no—
269C2×D3—41no—
270C2×D3—41no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—62no—
274C3×C4—361no—
275C3×C4—361no—
276C3×C4#369911yescharacteristic
277C3×C4C22×Dic911yescharacteristic
278C3×C4#379611yescharacteristic
279C3×C4C2×D3×Dic311yescharacteristic
280Dic3—22no—
281Dic3—92no—
282Dic3—92no—
283Dic3—121no—
284Dic3—121no—
285Dic3—182no—
286Dic3—182no—
287Dic3—181no—
288Dic3—181no—
289C32Dic3×SD1611yescharacteristic
290C9—21no—
291C9#485211yescharacteristic, complemented
292C23—61no—
293Q8—32no—
294Q8—61no—
295D4—32no—
296D4—61no—
297C2×C4—61no—
298C2×C4—181no—
299C2×C4—181no—
300C2×C4—271no—
301C2×C4—271no—
302C2×C4—1082no—
303C2×C4C2×D3×D911yescharacteristic
304C8—541no—
305C8D3×Dic912yescomplemented
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—122no—
310C2×C3SD16×D911yescharacteristic
311C2×C3#6347811yescharacteristic
312C2×C3D4×Dic911yescharacteristic
313C2×C3#6379611yescharacteristic
314C2×C3#6380911yescharacteristic
315C2×C3#6389811yescharacteristic
316D3—42no—
317C22—62no—
318C22—121no—
319C22—121no—
320C22#8864611yescharacteristic
321C4—62no—
322C4—272no—
323C4—361no—
324C4—361no—
325C4—541no—
326C4#8863911yescharacteristic
327C4C2×D3×Dic911yescharacteristic
328C3—21no—
329C3SD16×Dic911yescharacteristic, complemented
330C3#1059077511yescharacteristic
331C2—122no—
332C2#1172081711yescharacteristic
333C2#1172082311yescharacteristic
334C2#1172102511yescharacteristic
subgroupG/H
centerC22#88646
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32D3×D4
prime radicalC2×C3×D12C2×D3
cosocleC2×C3×C4×C9C23
hypercenterC2×C8D3×D9
hypocenterC3×C9C4×SD16
LCSG ▷ 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
3. #11720911C2semidirect
4. C8×C3⋊Dic9C2semidirect
5. C3×#63377C2semidirect
6. #11720297C2semidirect
18. C9×#204C4semidirect
34. #63377D3semidirect
47. C9×Dic6C2×C4semidirect
49. C3×C8×C9C2×C4semidirect
87. C3⋊Dic9SD16semidirect
136. C8×C9C4×D3semidirect
170. #204Dic9semidirect
205. C3×C9C4×SD16semidirect
232. Dic6C2×Dic9semidirect
234. C3×C8C2×Dic9semidirect
291. C9#4852semidirect
305. C8D3×Dic9semidirect
329. C3SD16×Dic9semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2042
#117210252
#117209112
C8×C3⋊Dic92
C3×#633772
#117202972
#117202942
#105907753
SD16×Dic93