Finite Group Databasebeta

← #13137532

Subgroups — #13137532

Order 1728.

356 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 85 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#11721744C211yesmaximal, characteristic
2#11720910C211yesmaximal, characteristic
3C3×#63368C211yesmaximal, characteristic
4C9×#808C211yesmaximal, characteristic
5#11720276C211yesmaximal, characteristic
6#11720273C211yesmaximal, characteristic
7#11720264C211yesmaximal, characteristic
8#10590497—31nomaximal
9#10588534—31nomaximal
10C2×#6226—21no—
11C2×C3×Dic18C2211yescharacteristic
12C3×C9×C4⋊C4C2211yescharacteristic
13C4⋊(C3⋊Dic9)C2211yescharacteristic
14#88534—21no—
15C4×C3⋊Dic9C2211yescharacteristic
16C9×C4⋊Dic3C2211yescharacteristic
17C2×C9×C3⋊C8C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19Dic9⋊Dic3—21no—
20Dic3×Dic9—21no—
21#64097—31no—
22#63843—31no—
23C3×#810—31no—
24C3×#808D311yescharacteristic
25#63573—31no—
26#63572—31no—
27#63566—31no—
28#63450—31no—
29#63450—61no—
30#63449—31no—
31C9×#106—31no—
32#63376—31no—
33Dic9⋊C8—31no—
34#63368D311yescharacteristic
35#63367—31no—
36C2×C3×C4×C9—21no—
37C2×C3×C4×C9C2311yescharacteristic
38C2×C3⋊Dic9—21no—
39C2×C3⋊Dic9D411yescharacteristic
40C2×C3⋊Dic9D411yescharacteristic
41#6226—42no—
42C2×C9×Dic3—21no—
43C2×C3×Dic9—21no—
44C3×Dic18—21no—
45C3×Dic18Q812yescomplemented
46C9×C3⋊C8—21no—
47C3×C9⋊C8—21no—
48#4854—91no—
49#4791—91no—
50C2×#634—61no—
51C2×C3×Dic6—31no—
52C32×C4⋊C4C2×D311yescharacteristic
53C4⋊(C3⋊Dic3)—31no—
54#3750—61no—
55C4×C3⋊Dic3—31no—
56C3×C4⋊Dic3C2×D311yescharacteristic
57C2×C3×C3⋊C8—31no—
58C2×C3×C3⋊C8C2×D311yescharacteristic
59Dic3⋊Dic3—61no—
60Dic32—61no—
61C2×Dic18—61no—
62C2×Dic18—62no—
63C2×Dic18C2×D311yescharacteristic
64C2×C8×C9—31no—
65C9×C4⋊C4—21no—
66C9×C4⋊C4—31no—
67C9×C4⋊C4C2×D311yescharacteristic
68C4⋊Dic9—31no—
69C4⋊Dic9—61no—
70Dic9⋊C4—61no—
71Dic9⋊C4—61no—
72Dic9⋊C4—62no—
73C4×Dic9—31no—
74C4×Dic9—61no—
75C4×Dic9—61no—
76C2×C9⋊C8C2×D311yescharacteristic
77C22×C3×C9C2×D411yescharacteristic
78C3×C4×C9—41no—
79C3×C4×C9C2×Q811yescharacteristic
80C3×C4×C9#4111yescharacteristic
81C3⋊Dic9—21no—
82C3⋊Dic9—41no—
83C3⋊Dic9SD1612yes—
84C9×Dic3—41no—
85C3×Dic9—22no—
86Q8×Dic3—91no—
87#888—91no—
88#888—91no—
89#888—181no—
90#887—91no—
91C3×#106—31no—
92C3×#103—91no—
93#835—91no—
94#817—91no—
95#816—91no—
96Dic3⋊C8—91no—
97Dic3⋊C8—91no—
98#810—31no—
99#808—91no—
100#808D911yescharacteristic
101C2×C32×C4—21no—
102C2×C32×C4C22×D311yescharacteristic
103C2×C3⋊Dic3—31no—
104C2×C3⋊Dic3—31no—
105C2×C3⋊Dic3—61no—
106#634—122no—
107C2×C3×Dic3—21no—
108C2×C3×Dic3—61no—
109C3×Dic6—32no—
110C3×Dic6—61no—
111C3×C3⋊C8—21no—
112C3×C3⋊C8—61no—
113C2×C4×C9—22no—
114C2×C4×C9—21no—
115C2×C4×C9—21no—
116C2×C4×C9—61no—
117C2×C4×C9C22×D311yescharacteristic
118C2×Dic9—21no—
119C2×Dic9—31no—
120C2×Dic9—31no—
121C2×Dic9—61no—
122C2×Dic9—61no—
123C2×Dic9—62no—
124C2×Dic9—61no—
125Dic18—21no—
126Dic18—122no—
127Dic18—124no—
128Dic18Dic612yescomplemented
129C8×C9—61no—
130C9⋊C8—21no—
131#475—271noSylow
132C2×C3×C9C22⋊Q811yescharacteristic
133C2×C3×C9C2×SD1611yescharacteristic
134C2×C3×C9#13711yescharacteristic
135C2×C3×Q8—91no—
136C2×Dic6—31no—
137C2×Dic6—181no—
138C2×Dic6—181no—
139C2×Dic6—182no—
140C2×C3×C8—31no—
141C2×C3×C8—91no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—31no—
144C3×C4⋊C4C2×D911yescharacteristic
145C3×C4⋊C4D3211yescharacteristic
146C4⋊Dic3—91no—
147C4⋊Dic3—91no—
148C4⋊Dic3—181no—
149C4⋊Dic3—181no—
150C4⋊Dic3C2×D911yescharacteristic
151Dic3⋊C4—181no—
152Dic3⋊C4—181no—
153Dic3⋊C4—181no—
154Dic3⋊C4—182no—
155C4×Dic3—91no—
156C4×Dic3—91no—
157C4×Dic3—181no—
158C4×Dic3—181no—
159C4×Dic3—181no—
160C2×C3⋊C8—31no—
161C2×C3⋊C8C2×D911yescharacteristic
162C22×C32D3×D411yescharacteristic
163C32×C4—41no—
164C32×C4#23511yescharacteristic
165C32×C4D3×Q811yescharacteristic
166C3⋊Dic3—32no—
167C3⋊Dic3—61no—
168C3⋊Dic3—121no—
169C3×Dic3—41no—
170C3×Dic3—62no—
171C22×C9—21no—
172C22×C9D3×D411yescharacteristic
173C4×C9—21no—
174C4×C9—21no—
175C4×C9—42no—
176C4×C9—41no—
177C4×C9—121no—
178C4×C9C2×Dic611yescharacteristic
179C4×C9#23711yescharacteristic
180Dic9—22no—
181Dic9—32no—
182Dic9—62no—
183Dic9—62no—
184Dic9—61no—
185Dic9—122no—
186Dic9—121no—
187C4⋊Q8—271no—
188C4×Q8—271no—
189#106—31no—
190#106—271no—
191C4⋊C8—271no—
192#103—91no—
193#103—271no—
194C3×C9#47511yescharacteristic, complemented, Sylow
195C3×Q8—92no—
196C3×Q8—181no—
197C2×C3×C4—22no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—61no—
202C2×C3×C4—181no—
203C2×C3×C4C22×D911yescharacteristic
204C2×C3×C4C2×D3211yescharacteristic
205C2×Dic3—21no—
206C2×Dic3—61no—
207C2×Dic3—91no—
208C2×Dic3—91no—
209C2×Dic3—91no—
210C2×Dic3—91no—
211C2×Dic3—181no—
212C2×Dic3—181no—
213C2×Dic3—182no—
214C2×Dic3—181no—
215C2×Dic3—181no—
216Dic6—32no—
217Dic6—61no—
218Dic6—362no—
219Dic6—362no—
220Dic6—364no—
221C3×C8—61no—
222C3×C8—181no—
223C3⋊C8—21no—
224C3⋊C8—61no—
225C2×C32#89611yescharacteristic
226C2×C32D3×SD1611yescharacteristic
227C2×C32#91511yescharacteristic
228C2×C9—21no—
229C2×C9—21no—
230C2×C9—21no—
231C2×C9#87811yescharacteristic
232C2×C9D3×SD1611yescharacteristic
233C2×C9#91811yescharacteristic
234C2×Q8—91no—
235C2×Q8—541no—
236C2×C8—31no—
237C2×C8—91no—
238C4⋊C4—31no—
239C4⋊C4—271no—
240C4⋊C4—541no—
241C4⋊C4—541no—
242C4⋊C4D3×D911yescharacteristic
243C42—271no—
244C42—541no—
245C22×C3—21no—
246C22×C3D4×D911yescharacteristic
247C22×C3#381111yescharacteristic
248C3×C4—21no—
249C3×C4—21no—
250C3×C4—42no—
251C3×C4—41no—
252C3×C4—41no—
253C3×C4—121no—
254C3×C4—182no—
255C3×C4#369711yescharacteristic
256C3×C4Q8×D911yescharacteristic
257C3×C4D3×Dic611yescharacteristic
258C3×C4#379511yescharacteristic
259Dic3—41no—
260Dic3—62no—
261Dic3—92no—
262Dic3—92no—
263Dic3—182no—
264Dic3—182no—
265Dic3—181no—
266Dic3—181no—
267Dic3—362no—
268Dic3—361no—
269Dic3—361no—
270C32#485411yescharacteristic
271C9—21no—
272C9#479111yescharacteristic, complemented
273Q8—92no—
274Q8—181no—
275Q8—1082no—
276C2×C4—21no—
277C2×C4—61no—
278C2×C4—181no—
279C2×C4—271no—
280C2×C4—271no—
281C2×C4—541no—
282C2×C4C2×D3×D911yescharacteristic
283C8—61no—
284C8—181no—
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3#6345811yescharacteristic
289C2×C3SD16×D911yescharacteristic
290C2×C3#6347711yescharacteristic
291C2×C3#6389611yescharacteristic
292C2×C3#6393011yescharacteristic
293C2×C3#6395111yescharacteristic
294C22#8866811yescharacteristic
295C4—41no—
296C4—121no—
297C4—182no—
298C4—272no—
299C4—541no—
300C4—1081no—
301C4D9×Dic611yescharacteristic
302C4#8864111yescharacteristic
303C3—21no—
304C3#1058853411yescharacteristic, complemented
305C3#1059049711yescharacteristic
306C2#1172099611yescharacteristic
307C2#1172118211yescharacteristic
308C2#1172121711yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC3×C9×C4⋊C4C22
socle, prime radicalC22×C32D3×D4
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#475
LCSG ▷ C2×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
45. C3×Dic18Q8semidirect
128. Dic18Dic6semidirect
194. C3×C9#475semidirect
272. C9#4791semidirect
304. C3#10588534semidirect

Maximal subgroups

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

subgroupindex
#117217442
#117209102
C3×#633682
C9×#8082
#117202762
#117202732
#117202642
#105904973
#105885343