Finite Group Databasebeta

← #13137660

Subgroups — #13137660

Order 1728.

400 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 97 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#11721743C211yesmaximal, characteristic
2#11720910C211yesmaximal, characteristic
3C2×#88418C211yesmaximal, characteristic
4C3×#63368C211yesmaximal, characteristic
5C9×#807C211yesmaximal, characteristic
6#11720271C211yesmaximal, characteristic
7#11720269C211yesmaximal, characteristic
8#10590482—31nomaximal
9#10588553—31nomaximal
10C2×#6226C2211yescharacteristic
11C2×C3×Dic18C2211yescharacteristic
12C3×C9×C4⋊C4C2211yescharacteristic
13#88534—21no—
14C4×C3⋊Dic9—21no—
15C4×C3⋊Dic9C2211yescharacteristic
16C9×C4⋊Dic3C2211yescharacteristic
17C2×C9×C3⋊C8C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19Dic9⋊Dic3—21no—
20Dic3×Dic9—21no—
21#88418C414yescomplemented
22#64096—31no—
23#63843—31no—
24C2×#3719—31no—
25C3×#810—31no—
26C3×#807D311yescharacteristic
27#63567—31no—
28#63553—31no—
29C2×Dic36—31no—
30#63449—31no—
31#63449—31no—
32#63449—61no—
33C9×#107—31no—
34C8×Dic9—31no—
35#63368—31no—
36#63368D311yescharacteristic
37C2×C3×C4×C9—21no—
38C2×C3×C4×C9C2311yescharacteristic
39C2×C3⋊Dic9—21no—
40C2×C3⋊Dic9D411yescharacteristic
41C2×C3⋊Dic9D411yescharacteristic
42#6226—21no—
43#6226C2×C412yes—
44C2×C9×Dic3—21no—
45C2×C3×Dic9—21no—
46C3×Dic18—21no—
47C3×Dic18C2×C412yes—
48C9×C3⋊C8C2×C412yes—
49C3×C9⋊C8—21no—
50#4873—91no—
51#4790—91no—
52C2×#634—31no—
53C2×C3×Dic6—31no—
54C32×C4⋊C4C2×D311yescharacteristic
55#3750—61no—
56C4×C3⋊Dic3—31no—
57C4×C3⋊Dic3—61no—
58C3×C4⋊Dic3C2×D311yescharacteristic
59C2×C3×C3⋊C8—31no—
60C2×C3×C3⋊C8C2×D311yescharacteristic
61Dic3⋊Dic3—61no—
62Dic32—61no—
63#3719—34no—
64C2×Dic18—31no—
65C2×Dic18—61no—
66C2×Dic18C2×D311yescharacteristic
67C2×C8×C9—31no—
68C9×C4⋊C4—21no—
69C9×C4⋊C4—31no—
70C9×C4⋊C4C2×D311yescharacteristic
71Dic9⋊C4—61no—
72Dic9⋊C4—61no—
73Dic9⋊C4—62no—
74C4×Dic9—31no—
75C4×Dic9—61no—
76C4×Dic9—61no—
77C4×Dic9—61no—
78C4×Dic9—62no—
79C2×C9⋊C8C2×D311yescharacteristic
80Dic36—34no—
81C22×C3×C9C2×D411yescharacteristic
82C3×C4×C9—41no—
83C3×C4×C9C22×C411yescharacteristic
84C3×C4×C9#4111yescharacteristic
85C3⋊Dic9—22no—
86C3⋊Dic9—21no—
87C3⋊Dic9Q1612yescomplemented
88C9×Dic3—41no—
89C3×Dic9—22no—
90Q8×Dic3—91no—
91C2×#216—91no—
92C2×Dic12—91no—
93#887—91no—
94#887—91no—
95#887—91no—
96#887—181no—
97C3×#107—31no—
98C3×#103—91no—
99#816—91no—
100C8×Dic3—91no—
101C8×Dic3—91no—
102#810—31no—
103#810—91no—
104#807D911yescharacteristic
105C2×C32×C4—21no—
106C2×C32×C4C22×D311yescharacteristic
107C2×C3⋊Dic3—31no—
108C2×C3⋊Dic3—31no—
109C2×C3⋊Dic3—61no—
110#634—32no—
111#634—61no—
112C2×C3×Dic3—21no—
113C2×C3×Dic3—61no—
114C3×Dic6—32no—
115C3×Dic6—61no—
116C3×C3⋊C8—61no—
117C3×C3⋊C8C4×D312yes—
118C2×C4×C9—22no—
119C2×C4×C9—21no—
120C2×C4×C9—21no—
121C2×C4×C9—61no—
122C2×C4×C9C22×D311yescharacteristic
123C2×Dic9—21no—
124C2×Dic9—31no—
125C2×Dic9—31no—
126C2×Dic9—61no—
127C2×Dic9—61no—
128C2×Dic9—62no—
129C2×Dic9—61no—
130Dic18—21no—
131Dic18—32no—
132Dic18—62no—
133Dic18—62no—
134Dic18—61no—
135Dic18C4×D312yes—
136C8×C9—32no—
137C9⋊C8—21no—
138C4×Q16—271noSylow
139C2×C3×C9C4×D411yescharacteristic
140C2×C3×C9C2×Q1611yescharacteristic
141C2×C3×C9#13511yescharacteristic
142C2×C3×Q8—91no—
143C2×Dic6—31no—
144C2×Dic6—91no—
145C2×Dic6—91no—
146C2×Dic6—181no—
147C2×C3×C8—31no—
148C2×C3×C8—91no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—31no—
151C3×C4⋊C4C2×D911yescharacteristic
152C3×C4⋊C4D3211yescharacteristic
153#216—94no—
154C4⋊Dic3—181no—
155C4⋊Dic3C2×D911yescharacteristic
156Dic3⋊C4—181no—
157Dic3⋊C4—181no—
158Dic3⋊C4—181no—
159Dic3⋊C4—182no—
160C4×Dic3—91no—
161C4×Dic3—91no—
162C4×Dic3—181no—
163C4×Dic3—181no—
164C4×Dic3—181no—
165C4×Dic3—181no—
166C4×Dic3—181no—
167C4×Dic3—182no—
168C2×C3⋊C8—31no—
169C2×C3⋊C8C2×D911yescharacteristic
170Dic12—94no—
171C22×C32D3×D411yescharacteristic
172C32×C4—41no—
173C32×C4C2×C4×D311yescharacteristic
174C32×C4#23911yescharacteristic
175C3⋊Dic3—32no—
176C3⋊Dic3—62no—
177C3⋊Dic3—61no—
178C3×Dic3—41no—
179C3×Dic3—62no—
180C22×C9—21no—
181C22×C9D3×D411yescharacteristic
182C4×C9—21no—
183C4×C9—21no—
184C4×C9—42no—
185C4×C9—41no—
186C4×C9—121no—
187C4×C9C2×C4×D311yescharacteristic
188C4×C9#23711yescharacteristic
189Dic9—22no—
190Dic9—32no—
191Dic9—62no—
192Dic9—64no—
193Dic9—62no—
194Dic9—61no—
195Dic9—121no—
196C2×Q16—271no—
197C4×Q8—271no—
198C4×Q8—271no—
199#107—31no—
200#103—91no—
201#103—271no—
202C4×C8—271no—
203C3×C9C4×Q1611yescharacteristic, complemented, Sylow
204C3×Q8—92no—
205C3×Q8—181no—
206C2×C3×C4—22no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—61no—
211C2×C3×C4—181no—
212C2×C3×C4C22×D911yescharacteristic
213C2×C3×C4C2×D3211yescharacteristic
214C2×Dic3—21no—
215C2×Dic3—61no—
216C2×Dic3—91no—
217C2×Dic3—91no—
218C2×Dic3—91no—
219C2×Dic3—91no—
220C2×Dic3—181no—
221C2×Dic3—181no—
222C2×Dic3—182no—
223C2×Dic3—181no—
224C2×Dic3—181no—
225Dic6—32no—
226Dic6—61no—
227Dic6—92no—
228Dic6—92no—
229Dic6—182no—
230Dic6—182no—
231Dic6—181no—
232Dic6—181no—
233C3×C8—32no—
234C3×C8—181no—
235C3⋊C8—61no—
236C3⋊C8C4×D912yes—
237C2×C32#89311yescharacteristic
238C2×C32#91211yescharacteristic
239C2×C32D3×Q1611yescharacteristic
240C2×C9—21no—
241C2×C9—21no—
242C2×C9—21no—
243C2×C9#67⋊C411yescharacteristic
244C2×C9#91611yescharacteristic
245C2×C9D3×Q1611yescharacteristic
246C2×Q8—91no—
247C2×Q8—271no—
248Q16—274no—
249C2×C8—31no—
250C2×C8—91no—
251C4⋊C4—31no—
252C4⋊C4—541no—
253C4⋊C4—541no—
254C4⋊C4D3×D911yescharacteristic
255C42—271no—
256C42—541no—
257C42—541no—
258C22×C3—21no—
259C22×C3D4×D911yescharacteristic
260C22×C3#381111yescharacteristic
261C3×C4—21no—
262C3×C4—21no—
263C3×C4—42no—
264C3×C4—41no—
265C3×C4—41no—
266C3×C4—121no—
267C3×C4—182no—
268C3×C4C2×C4×D911yescharacteristic
269C3×C4#370111yescharacteristic
270C3×C4#379611yescharacteristic
271C3×C4C4×D3211yescharacteristic
272Dic3—41no—
273Dic3—62no—
274Dic3—92no—
275Dic3—92no—
276Dic3—182no—
277Dic3—184no—
278Dic3—182no—
279Dic3—181no—
280Dic3—182no—
281Dic3—181no—
282Dic3—361no—
283C32#487311yescharacteristic
284C9—21no—
285C9#479011yescharacteristic, complemented
286Q8—92no—
287Q8—181no—
288Q8—272no—
289Q8—541no—
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×C3#6345511yescharacteristic
303C2×C3#6347411yescharacteristic
304C2×C3Q16×D911yescharacteristic
305C2×C3#6388111yescharacteristic
306C2×C3#6393611yescharacteristic
307C2×C3#6394411yescharacteristic
308C22#8866811yescharacteristic
309C4—41no—
310C4—121no—
311C4—182no—
312C4—272no—
313C4—542no—
314C4—541no—
315C4#8863411yescharacteristic
316C4C4×D3×D911yescharacteristic
317C3—21no—
318C3#1058855311yescharacteristic, complemented
319C3#1059048211yescharacteristic
320C2#1172097811yescharacteristic
321C2#1172117811yescharacteristic
322C2#1172122011yescharacteristic
subgroupG/H
centerC22#88668
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socle, prime radicalC22×C32D3×D4
cosocleC2×C3×C4×C9C23
hypercenterC4⋊C4D3×D9
hypocenterC3×C9C4×Q16
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
21. #88418C4semidirect
87. C3⋊Dic9Q16semidirect
203. C3×C9C4×Q16semidirect
285. C9#4790semidirect
318. C3#10588553semidirect

Maximal subgroups

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

subgroupindex
#117217432
#117209102
C2×#884182
C3×#633682
C9×#8072
#117202712
#117202692
#105904823
#105885533