Finite Group Databasebeta

← #13137646

Subgroups — #13137646

Order 1728.

408 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C3×#63455C211yesmaximal, characteristic
2#11720913C211yesmaximal, characteristic, complemented
3C2×#88415C211yesmaximal, characteristic
4#11720527C211yesmaximal, characteristic, complemented
5C9×#808C211yesmaximal, characteristic, complemented
6C3⋊C8×Dic9C211yesmaximal, characteristic, complemented
7#11720266C211yesmaximal, characteristic
8#10590479—31nomaximal
9#10588532—31nomaximal
10C2×#6226C2211yescharacteristic
11C2×C3×D36C2211yescharacteristic
12C2×C3×C4×D9—21no—
13C3×C9×C4⋊C4C2211yescharacteristic
14C2×C3⋊(C9⋊C8)C2211yescharacteristic
15C3×(C2×D9)⋊C4—21no—
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4×Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19#88442—21no—
20Dic3×Dic9—21no—
21#88415C414yescomplemented
22C3×#893—31no—
23#63844—31no—
24C2×#3716—31no—
25#63637—31no—
26C3×#808D311yescharacteristic
27#63562—31no—
28Dic3×C3⋊C8—31no—
29C2×#3664—31no—
30#63455D311yescharacteristic
31#63449—31no—
32C9×#106—31no—
33C8×Dic9—31no—
34#63369—31no—
35#63368—31no—
36#63368—61no—
37C22×C3×D9—21no—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9C2311yescharacteristic
40C2×C3⋊Dic9—21no—
41#6226—21no—
42#6226C2×C412yescomplemented
43C2×C9×Dic3—21no—
44C2×C3×Dic9D411yescharacteristic
45C2×C3×Dic9D411yescharacteristic
46C3×D36—21no—
47C3×D36C2×C412yes—
48C3×C4×D9—42no—
49C3⋊(C9⋊C8)—21no—
50C9×C3⋊C8C2×C412yescomplemented
51C4×#214—91no—
52#4852—91no—
53C2×#634—31no—
54C2×C3×D12—31no—
55C2×C3×C4×D3—61no—
56C32×C4⋊C4C2×D311yescharacteristic
57C2×C3⋊(C3⋊C8)—31no—
58C3×(C2×D3)⋊C4—61no—
59C3×C4⋊Dic3C2×D311yescharacteristic
60C3×C4×Dic3—31no—
61C2×C3×C3⋊C8C2×D311yescharacteristic
62#3724—61no—
63Dic32—61no—
64#3716—34no—
65C2×D36C2×D311yescharacteristic
66C2×C4×D9—21no—
67C2×Dic18—31no—
68C2×Dic18—61no—
69C2×C8×C9—31no—
70C9×C4⋊C4—21no—
71C9×C4⋊C4—31no—
72C9×C4⋊C4C2×D311yescharacteristic
73(C2×D9)⋊C4—21no—
74Dic9⋊C4—61no—
75C4×Dic9—61no—
76C4×Dic9C2×D311yescharacteristic
77C2×C9⋊C8—31no—
78C2×C9⋊C8—61no—
79#3664—34no—
80C22×C3×C9C2×D411yescharacteristic
81C2×C3×D9—22no—
82C2×C3×D9—41no—
83C2×C3×D9—41no—
84C3×C4×C9—41no—
85C3×C4×C9C22×C411yescharacteristic
86C3×C4×C9#4111yescharacteristic
87C3⋊Dic9—22no—
88C9×Dic3—41no—
89C3×Dic9—21no—
90C3×Dic9SD1612yescomplemented
91C3×C4×D4—91no—
92C2×#214—91no—
93C2×#204—91no—
94#893—31no—
95#887—91no—
96C4×Dic6—91no—
97C3×#106—31no—
98#832—91no—
99C8×Dic3—91no—
100#810—91no—
101#810—91no—
102#810—181no—
103#809—91no—
104#808D911yescharacteristic, complemented
105C4×C3⋊C8—91no—
106C22×C3×D3—61no—
107C2×C32×C4—21no—
108C2×C32×C4C22×D311yescharacteristic
109C2×C3⋊Dic3—61no—
110#634—32no—
111#634—61no—
112C2×C3×Dic3—21no—
113C2×C3×Dic3—31no—
114C2×C3×Dic3—31no—
115C3×D12—32no—
116C3×D12—61no—
117C3×C4×D3—122no—
118C22×D9—21no—
119C3⋊(C3⋊C8)—61no—
120C3×C3⋊C8C4×D312yes—
121C2×C4×C9—21no—
122C2×C4×C9—21no—
123C2×C4×C9—41no—
124C2×C4×C9—61no—
125C2×C4×C9C22×D311yescharacteristic
126C2×Dic9—61no—
127C2×Dic9—121no—
128C2×Dic9#6711yescharacteristic
129C2×Dic9#6711yescharacteristic
130D36—21no—
131D36C4×D312yes—
132C4×D9—42no—
133Dic18—32no—
134Dic18—62no—
135Dic18—61no—
136Dic18—121no—
137C8×C9—32no—
138C9⋊C8—61no—
139C9⋊C8—121no—
140C4×SD16—271noSylow
141C2×C3×C9C4×D411yescharacteristic
142C2×C3×C9C2×SD1611yescharacteristic
143C2×C3×C9#13511yescharacteristic
144C3×D9—42no—
145C2×C3×D4—91no—
146C22×C3×C4—181no—
147C2×D12—31no—
148C2×C4×D3—61no—
149C2×Dic6—91no—
150C2×Dic6—91no—
151C2×Dic6—181no—
152C2×C3×C8—31no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—31no—
155C3×C4⋊C4C2×D911yescharacteristic
156C3×C4⋊C4D3211yescharacteristic
157C3×C22⋊C4—181no—
158C3×C42—91no—
159#214—94no—
160(C2×D3)⋊C4—61no—
161C4⋊Dic3C2×D911yescharacteristic
162Dic3⋊C4—181no—
163Dic3⋊C4—181no—
164C4×Dic3—31no—
165C4×Dic3—181no—
166C4×Dic3—181no—
167C2×C3⋊C8—91no—
168C2×C3⋊C8—91no—
169C2×C3⋊C8—181no—
170C2×C3⋊C8C2×D911yescharacteristic
171#204—94no—
172C22×C32D3×D411yescharacteristic
173C2×C3×D3—62no—
174C2×C3×D3—121no—
175C2×C3×D3—121no—
176C32×C4—41no—
177C32×C4C2×C4×D311yescharacteristic
178C32×C4#23911yescharacteristic
179C3⋊Dic3—62no—
180C3×Dic3—32no—
181C3×Dic3—41no—
182C3×Dic3—61no—
183C22×C9—21no—
184C22×C9C2×#6711yescharacteristic
185C2×D9—22no—
186C2×D9—41no—
187C2×D9—41no—
188C4×C9—21no—
189C4×C9—21no—
190C4×C9—41no—
191C4×C9—81no—
192C4×C9—121no—
193C4×C9C2×C4×D311yescharacteristic
194C4×C9#23511yescharacteristic
195Dic9—21no—
196Dic9—62no—
197Dic9—122no—
198Dic9#21412yescomplemented
199C2×SD16—271no—
200C4×Q8—271no—
201C4×D4—91no—
202#106—31no—
203#103—271no—
204#102—271no—
205C4×C8—271no—
206C3×C9C4×SD1611yescharacteristic, complemented, Sylow
207C23×C3—181no—
208C22×D3—61no—
209C3×D4—92no—
210C3×D4—181no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—41no—
215C2×C3×C4—61no—
216C2×C3×C4—91no—
217C2×C3×C4—91no—
218C2×C3×C4—362no—
219C2×C3×C4C22×D911yescharacteristic
220C2×C3×C4C2×D3211yescharacteristic
221C2×Dic3—21no—
222C2×Dic3—31no—
223C2×Dic3—31no—
224C2×Dic3—181no—
225C2×Dic3—181no—
226C2×Dic3—361no—
227D12—32no—
228D12—61no—
229C4×D3—122no—
230Dic6—92no—
231Dic6—92no—
232Dic6—182no—
233Dic6—181no—
234Dic6—181no—
235Dic6—361no—
236C3×C8—32no—
237C3⋊C8—181no—
238C3⋊C8—181no—
239C3⋊C8—361no—
240C3⋊C8C4×D912yescomplemented
241C2×C32#89311yescharacteristic
242C2×C32D3×SD1611yescharacteristic
243C2×C32#91611yescharacteristic
244C3×D3—122no—
245C2×C9—21no—
246C2×C9—21no—
247C2×C9—21no—
248C2×C9C4×#6711yescharacteristic
249C2×C9C2×#21411yescharacteristic
250C2×C9#95011yescharacteristic
251D9—42no—
252C2×Q8—271no—
253C2×D4—91no—
254C22×C4—181no—
255SD16—274no—
256C2×C8—31no—
257C2×C8—271no—
258C4⋊C4—31no—
259C4⋊C4—541no—
260C4⋊C4D3×D911yescharacteristic
261C22⋊C4—181no—
262C42—91no—
263C42—541no—
264C22×C3—21no—
265C22×C3—182no—
266C22×C3—361no—
267C22×C3—361no—
268C22×C3D4×D911yescharacteristic
269C22×C3D3×#6711yescharacteristic
270C2×D3—62no—
271C2×D3—121no—
272C2×D3—121no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4—81no—
278C3×C4—92no—
279C3×C4—121no—
280C3×C4—181no—
281C3×C4C2×C4×D911yescharacteristic
282C3×C4#370111yescharacteristic
283C3×C4#379911yescharacteristic
284C3×C4C4×D3211yescharacteristic
285Dic3—32no—
286Dic3—41no—
287Dic3—61no—
288Dic3—182no—
289Dic3—182no—
290Dic3—362no—
291C32#485211yescharacteristic
292C9—21no—
293C9C4×#21411yescharacteristic, complemented
294C23—181no—
295Q8—272no—
296Q8—541no—
297D4—92no—
298D4—181no—
299C2×C4—21no—
300C2×C4—61no—
301C2×C4—91no—
302C2×C4—91no—
303C2×C4—362no—
304C2×C4—541no—
305C2×C4C2×D3×D911yescharacteristic
306C8—32no—
307C8—541no—
308C2×C3—21no—
309C2×C3—21no—
310C2×C3—21no—
311C2×C3—362no—
312C2×C3#6345511yescharacteristic
313C2×C3SD16×D911yescharacteristic
314C2×C3#6347811yescharacteristic
315C2×C3#6390411yescharacteristic
316C2×C3D3×#21411yescharacteristic
317C2×C3#6394711yescharacteristic
318D3—122no—
319C22—182no—
320C22—361no—
321C22—361no—
322C22D9×#6711yescharacteristic
323C4—41no—
324C4—92no—
325C4—121no—
326C4—181no—
327C4—542no—
328C4#8863511yescharacteristic
329C4C4×D3×D911yescharacteristic
330C3—21no—
331C3#1058853211yescharacteristic, complemented
332C3#1059047911yescharacteristic
333C2—362no—
334C2#1172100211yescharacteristic
335C2D9×#21411yescharacteristic
336C2#1172121511yescharacteristic
subgroupG/H
centerC22D9×#67
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32D3×D4
prime radicalC2×C3×D36C22
cosocleC2×C3×C4×C9C23
hypercenterC4⋊C4D3×D9
hypocenterC3×C9C4×SD16
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
2. #11720913C2semidirect
4. #11720527C2semidirect
5. C9×#808C2semidirect
6. C3⋊C8×Dic9C2semidirect
21. #88415C4semidirect
42. #6226C2×C4semidirect
50. C9×C3⋊C8C2×C4semidirect
90. C3×Dic9SD16semidirect
104. #808D9semidirect
198. Dic9#214semidirect
206. C3×C9C4×SD16semidirect
240. C3⋊C8C4×D9semidirect
293. C9C4×#214semidirect
331. C3#10588532semidirect

Maximal subgroups

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

subgroupindex
C3×#634552
#117209132
C2×#884152
#117205272
C9×#8082
C3⋊C8×Dic92
#117202662
#105904793
#105885323