Finite Group Databasebeta

← #13140841

Subgroups — #13140841

Order 1728.

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

normal subgroups only

This group has 400 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
1Q8×C3⋊Dic9C211yesmaximal, characteristic
2C2×C9×#216C211yesmaximal, characteristic
3#11720911C211yesmaximal, characteristic
4C3×#63395C211yesmaximal, characteristic
5#11720298C211yesmaximal, characteristic
6#11720279C211yesmaximal, characteristic
7#11720271C211yesmaximal, characteristic
8#10591581—31nomaximal
9Q16×Dic9—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C2×C9×Dic6C2211yescharacteristic
12C4⋊(C3⋊Dic9)—21no—
13C4⋊(C3⋊Dic9)C2211yescharacteristic
14C4×C3⋊Dic9—21no—
15C4×C3⋊Dic9C2211yescharacteristic
16C9×#216C414yescomplemented
17C3×C4⋊Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20Dic3⋊Dic9—21no—
21Dic3×Dic9—21no—
22Q8×C3⋊Dic3—31no—
23C2×C3×#216D311yescharacteristic
24#63843—31no—
25C3×#835—31no—
26#63575—31no—
27#63566—31no—
28#63553—31no—
29C2×C9×Q16—31no—
30Q8×Dic9—31no—
31Q8×Dic9—31no—
32Q8×Dic9—61no—
33#63395—31no—
34#63395D311yescharacteristic
35#63378—31no—
36C8×Dic9—31no—
37C3×Q8×C9—21no—
38C3×Q8×C9C2×C412yes—
39C2×C3×C4×C9—21no—
40C2×C3×C4×C9C2311yescharacteristic
41C2×C3⋊Dic9—21no—
42C2×C3⋊Dic9D411yescharacteristic
43C2×C3⋊Dic9D411yescharacteristic
44C2×C9×Dic3—21no—
45C2×C3×Dic9—21no—
46C9×Dic6—21no—
47C9×Dic6C2×C412yes—
48C9×C3⋊C8C2×C412yes—
49C3×C9⋊C8—21no—
50Dic3×Q16—91no—
51#4790—91no—
52C2×C32×Q8C2×D311yescharacteristic
53C2×C3×Dic6C2×D311yescharacteristic
54C4⋊(C3⋊Dic3)—31no—
55C4⋊(C3⋊Dic3)—61no—
56C4×C3⋊Dic3—31no—
57C4×C3⋊Dic3—61no—
58C3×#216Dic314yes—
59C3×C4⋊Dic3—31no—
60C2×C3×C3⋊C8—31no—
61C2×C3×C3⋊C8C2×D311yescharacteristic
62Dic3⋊Dic3—61no—
63Dic32—61no—
64C2×Q8×C9—21no—
65C2×Q8×C9—31no—
66C2×Q8×C9C2×D311yescharacteristic
67C9×Q16—34no—
68C2×C8×C9—31no—
69C4⋊Dic9—31no—
70C4⋊Dic9—61no—
71C4⋊Dic9—61no—
72C4⋊Dic9—61no—
73C4⋊Dic9—62no—
74C4⋊Dic9C2×D311yescharacteristic
75C4×Dic9—31no—
76C4×Dic9—61no—
77C4×Dic9—61no—
78C4×Dic9—61no—
79C4×Dic9—62no—
80C2×C9⋊C8C2×D311yescharacteristic
81C22×C3×C9C2×D411yescharacteristic
82C3×C4×C9—22no—
83C3×C4×C9C22×C411yescharacteristic
84C3×C4×C9#4111yescharacteristic
85C3⋊Dic9—21no—
86C3⋊Dic9—41no—
87C3⋊Dic9Q1612yescomplemented
88C9×Dic3—22no—
89C3×Dic9—41no—
90C2×C3×Q16—31no—
91Q8×Dic3—91no—
92Q8×Dic3—91no—
93Q8×Dic3—91no—
94Q8×Dic3—181no—
95C2×#216D911yescharacteristic
96#887—91no—
97C3×#103—91no—
98#835—31no—
99#835—91no—
100#818—91no—
101#816—91no—
102C8×Dic3—91no—
103C8×Dic3—91no—
104#807—91no—
105C32×Q8—21no—
106C32×Q8C2×Dic312yes—
107C2×C32×C4—21no—
108C2×C32×C4C22×D311yescharacteristic
109C2×C3⋊Dic3—31no—
110C2×C3⋊Dic3—31no—
111C2×C3⋊Dic3—61no—
112C2×C3×Dic3—21no—
113C2×C3×Dic3—61no—
114C3×Dic6—21no—
115C3×Dic6C2×Dic312yes—
116C3×C3⋊C8—61no—
117C3×C3⋊C8C2×Dic312yes—
118Q8×C9—22no—
119Q8×C9—21no—
120Q8×C9—22no—
121Q8×C9—32no—
122Q8×C9—61no—
123Q8×C9C4×D312yes—
124C2×C4×C9—22no—
125C2×C4×C9—21no—
126C2×C4×C9—21no—
127C2×C4×C9—61no—
128C2×C4×C9C22×D311yescharacteristic
129C2×Dic9—21no—
130C2×Dic9—31no—
131C2×Dic9—31no—
132C2×Dic9—61no—
133C2×Dic9—61no—
134C2×Dic9—62no—
135C2×Dic9—61no—
136C8×C9—32no—
137C9⋊C8—21no—
138C4×Q16—271noSylow
139C2×C3×C9C4×D411yescharacteristic
140C2×C3×C9C2×Q1611yescharacteristic
141C2×C3×C9#13511yescharacteristic
142C2×C3×Q8—21no—
143C2×C3×Q8—31no—
144C2×C3×Q8C2×D911yescharacteristic
145C2×C3×Q8D3211yescharacteristic
146C2×Dic6C2×D911yescharacteristic
147C3×Q16—34no—
148C2×C3×C8—31no—
149C2×C3×C8—91no—
150C3×C4⋊C4—91no—
151#216Dic914yescomplemented
152C4⋊Dic3—31no—
153C4⋊Dic3—91no—
154C4⋊Dic3—91no—
155C4⋊Dic3—181no—
156C4⋊Dic3—181no—
157C4⋊Dic3—181no—
158C4⋊Dic3—181no—
159C4⋊Dic3—182no—
160Dic3⋊C4—181no—
161C4×Dic3—91no—
162C4×Dic3—91no—
163C4×Dic3—181no—
164C4×Dic3—181no—
165C4×Dic3—181no—
166C4×Dic3—181no—
167C4×Dic3—181no—
168C4×Dic3—182no—
169C2×C3⋊C8—31no—
170C2×C3⋊C8C2×D911yescharacteristic
171C22×C32D3×D411yescharacteristic
172C32×C4—22no—
173C32×C4#23711yescharacteristic
174C32×C4C22×Dic311yescharacteristic
175C3⋊Dic3—32no—
176C3⋊Dic3—61no—
177C3⋊Dic3—121no—
178C3×Dic3—22no—
179C3×Dic3—121no—
180C22×C9—21no—
181C22×C9D3×D411yescharacteristic
182C4×C9—24no—
183C4×C9—22no—
184C4×C9—21no—
185C4×C9—21no—
186C4×C9—62no—
187C4×C9C2×C4×D311yescharacteristic
188C4×C9#23711yescharacteristic
189Dic9—32no—
190Dic9—41no—
191Dic9—62no—
192Dic9—61no—
193Dic9—121no—
194Dic9—122no—
195Dic9—121no—
196C2×Q16—31no—
197C4×Q8—271no—
198C4×Q8—271no—
199#107—271no—
200#103—91no—
201#103—271no—
202C4×C8—271no—
203C3×C9C4×Q1611yescharacteristic, complemented, Sylow
204C3×Q8—22no—
205C3×Q8—21no—
206C3×Q8—21no—
207C3×Q8—22no—
208C3×Q8—32no—
209C3×Q8—61no—
210C3×Q8C2×Dic912yes—
211C3×Q8D3×Dic312yes—
212C2×C3×C4—22no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—61no—
217C2×C3×C4—181no—
218C2×C3×C4C22×D911yescharacteristic
219C2×C3×C4C2×D3211yescharacteristic
220C2×Dic3—21no—
221C2×Dic3—61no—
222C2×Dic3—91no—
223C2×Dic3—91no—
224C2×Dic3—91no—
225C2×Dic3—91no—
226C2×Dic3—181no—
227C2×Dic3—181no—
228C2×Dic3—182no—
229C2×Dic3—181no—
230C2×Dic3—181no—
231Dic6—21no—
232Dic6C2×Dic912yes—
233C3×C8—32no—
234C3×C8—181no—
235C3⋊C8—61no—
236C3⋊C8C2×Dic912yes—
237C2×C32D3×Q1611yescharacteristic
238C2×C32#91911yescharacteristic
239C2×C32D4×Dic311yescharacteristic
240C2×C9—21no—
241C2×C9—21no—
242C2×C9—21no—
243C2×C9#67⋊C411yescharacteristic
244C2×C9#91611yescharacteristic
245C2×C9D3×Q1611yescharacteristic
246C2×Q8—31no—
247C2×Q8D3×D911yescharacteristic
248Q16—34no—
249C2×C8—31no—
250C2×C8—91no—
251C4⋊C4—91no—
252C4⋊C4—271no—
253C4⋊C4—541no—
254C4⋊C4—541no—
255C42—271no—
256C42—541no—
257C42—541no—
258C22×C3—21no—
259C22×C3D4×D911yescharacteristic
260C22×C3#381111yescharacteristic
261C3×C4—24no—
262C3×C4—22no—
263C3×C4—22no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—62no—
267C3×C4—361no—
268C3×C4#369911yescharacteristic
269C3×C4C22×Dic911yescharacteristic
270C3×C4C2×D3×Dic311yescharacteristic
271C3×C4#380511yescharacteristic
272Dic3—22no—
273Dic3—92no—
274Dic3—92no—
275Dic3—121no—
276Dic3—182no—
277Dic3—181no—
278Dic3—181no—
279Dic3—361no—
280Dic3—362no—
281Dic3—361no—
282Dic3—361no—
283C32Dic3×Q1611yescharacteristic
284C9—21no—
285C9#479011yescharacteristic, complemented
286Q8—21no—
287Q8—32no—
288Q8—61no—
289Q8D3×Dic912yes—
290C2×C4—21no—
291C2×C4—61no—
292C2×C4—181no—
293C2×C4—271no—
294C2×C4—271no—
295C2×C4—541no—
296C2×C4C2×D3×D911yescharacteristic
297C8—32no—
298C8—181no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3Q16×D911yescharacteristic
303C2×C3#6348111yescharacteristic
304C2×C3D4×Dic911yescharacteristic
305C2×C3#6394411yescharacteristic
306C2×C3#6395011yescharacteristic
307C2×C3#67⋊Dic311yescharacteristic
308C22#8866811yescharacteristic
309C4—22no—
310C4—62no—
311C4—272no—
312C4—361no—
313C4—541no—
314C4—1081no—
315C4C2×D3×Dic911yescharacteristic
316C4#8866311yescharacteristic
317C3—21no—
318C3Q16×Dic911yescharacteristic, complemented
319C3#1059158111yescharacteristic
320C2#1172122011yescharacteristic
321C2#1172122411yescharacteristic
322C2#67⋊Dic911yescharacteristic
subgroupG/H
centerC22#88668
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socle, prime radicalC22×C32D3×D4
cosocleC2×C3×C4×C9C23
hypercenterC2×Q8D3×D9
hypocenterC3×C9C4×Q16
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
16. C9×#216C4semidirect
87. C3⋊Dic9Q16semidirect
151. #216Dic9semidirect
203. C3×C9C4×Q16semidirect
285. C9#4790semidirect
318. C3Q16×Dic9semidirect

Maximal subgroups

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

subgroupindex
Q8×C3⋊Dic92
C2×C9×#2162
#117209112
C3×#633952
#117202982
#117202792
#117202712
#105915813
Q16×Dic93