Finite Group Databasebeta

← #13137563

Subgroups — #13137563

Order 1728.

338 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 89 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#11720937C211yesmaximal, characteristic
3C9×#808C211yesmaximal, characteristic
4C3×#63366C211yesmaximal, characteristic
5#11720279C211yesmaximal, characteristic
6#11720275C211yesmaximal, characteristic
7#11720264C211yesmaximal, characteristic
8#10590546—31nomaximal
9#10588536—31nomaximal
10C2×#6226—21no—
11C3×C9×C4⋊C4C2211yescharacteristic
12C4⋊(C3⋊Dic9)C2211yescharacteristic
13#88534—21no—
14C4×C3⋊Dic9C2211yescharacteristic
15C9×C4⋊Dic3C2211yescharacteristic
16C3×C4⋊Dic9C2211yescharacteristic
17C2×C9×C3⋊C8C2211yescharacteristic
18C2×C3×C9⋊C8C2211yescharacteristic
19Dic3⋊Dic9—21no—
20Dic9⋊Dic3—21no—
21#64097—31no—
22#63872—31no—
23C3×#808D311yescharacteristic
24C3×#807—31no—
25#63575—31no—
26#63573—31no—
27#63555—31no—
28#63452—31no—
29#63450—31no—
30#63450—61no—
31C9×#106—31no—
32#63378—31no—
33#63375—31no—
34#63367—31no—
35#63366D311yescharacteristic
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—
44C9×C3⋊C8Q812yescomplemented
45C3×C9⋊C8Q812yescomplemented
46#4876—91no—
47#4856—91no—
48C2×#634—61no—
49C32×C4⋊C4C2×D311yescharacteristic
50C4⋊(C3⋊Dic3)—31no—
51#3750—61no—
52C4×C3⋊Dic3—31no—
53C3×C4⋊Dic3—31no—
54C3×C4⋊Dic3C2×D311yescharacteristic
55C2×C3×C3⋊C8—31no—
56C2×C3×C3⋊C8C2×D311yescharacteristic
57Dic3⋊Dic3—61no—
58Dic3⋊Dic3—61no—
59C2×Dic18—61no—
60C2×Dic18—62no—
61C2×C8×C9—31no—
62C9×C4⋊C4—21no—
63C9×C4⋊C4—31no—
64C9×C4⋊C4C2×D311yescharacteristic
65C4⋊Dic9—31no—
66C4⋊Dic9—61no—
67C4⋊Dic9—61no—
68C4⋊Dic9C2×D311yescharacteristic
69Dic9⋊C4—61no—
70Dic9⋊C4—61no—
71Dic9⋊C4—62no—
72C4×Dic9—31no—
73C4×Dic9—61no—
74C2×C9⋊C8C2×D311yescharacteristic
75C22×C3×C9C2×D411yescharacteristic
76C3×C4×C9—41no—
77C3×C4×C9C2×Q811yescharacteristic
78C3×C4×C9C2×Q811yescharacteristic
79C3⋊Dic9—21no—
80C3⋊Dic9—21no—
81C3⋊Dic9—41no—
82C9×Dic3—41no—
83C3×Dic9—41no—
84#890—91no—
85#890—91no—
86#888—91no—
87#888—91no—
88#888—181no—
89C3×#107—91no—
90C3×#106—31no—
91#818—91no—
92#817—91no—
93#815—91no—
94#815—91no—
95#808—91no—
96#808D911yescharacteristic
97#807—31no—
98#807—91no—
99C2×C32×C4—21no—
100C2×C32×C4C22×D311yescharacteristic
101C2×C3⋊Dic3—31no—
102C2×C3⋊Dic3—31no—
103C2×C3⋊Dic3—61no—
104#634—122no—
105C2×C3×Dic3—21no—
106C2×C3×Dic3—61no—
107C3×C3⋊C8—32no—
108C3×C3⋊C8Dic612yes—
109C2×C4×C9—22no—
110C2×C4×C9—21no—
111C2×C4×C9—21no—
112C2×C4×C9—61no—
113C2×C4×C9C22×D311yescharacteristic
114C2×Dic9—21no—
115C2×Dic9—31no—
116C2×Dic9—31no—
117C2×Dic9—61no—
118C2×Dic9—61no—
119C2×Dic9—62no—
120C2×Dic9—61no—
121Dic18—124no—
122Dic18—122no—
123C8×C9—32no—
124C9⋊C8Dic612yescomplemented
125#501—271noSylow
126C2×C3×C9C4⋊Q811yescharacteristic
127C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
128C2×C3×C9#13711yescharacteristic
129C2×Dic6—181no—
130C2×Dic6—181no—
131C2×Dic6—182no—
132C2×C3×C8—31no—
133C2×C3×C8—91no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—31no—
136C3×C4⋊C4—91no—
137C3×C4⋊C4C2×D911yescharacteristic
138C3×C4⋊C4D3211yescharacteristic
139C4⋊Dic3—31no—
140C4⋊Dic3—91no—
141C4⋊Dic3—91no—
142C4⋊Dic3—181no—
143C4⋊Dic3—181no—
144C4⋊Dic3—181no—
145C4⋊Dic3C2×D911yescharacteristic
146Dic3⋊C4—181no—
147Dic3⋊C4—181no—
148Dic3⋊C4—181no—
149Dic3⋊C4—181no—
150Dic3⋊C4—182no—
151C4×Dic3—91no—
152C4×Dic3—91no—
153C4×Dic3—181no—
154C2×C3⋊C8—31no—
155C2×C3⋊C8C2×D911yescharacteristic
156C22×C32D3×D411yescharacteristic
157C32×C4—41no—
158C32×C4C2×Dic611yescharacteristic
159C32×C4D3×Q811yescharacteristic
160C3⋊Dic3—61no—
161C3⋊Dic3—61no—
162C3⋊Dic3—121no—
163C3×Dic3—41no—
164C3×Dic3—121no—
165C22×C9—21no—
166C22×C9D3×D411yescharacteristic
167C4×C9—21no—
168C4×C9—21no—
169C4×C9—42no—
170C4×C9—41no—
171C4×C9—121no—
172C4×C9C2×Dic611yescharacteristic
173C4×C9D3×Q811yescharacteristic
174Dic9—41no—
175Dic9—62no—
176Dic9—62no—
177Dic9—61no—
178Dic9—61no—
179Dic9—122no—
180Dic9—121no—
181C4⋊Q8—271no—
182#125—271no—
183#107—91no—
184#107—271no—
185#106—31no—
186#106—271no—
187#97—271no—
188C3×C9#50111yescharacteristic, complemented, Sylow
189C2×C3×C4—22no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—61no—
194C2×C3×C4—181no—
195C2×C3×C4C22×D911yescharacteristic
196C2×C3×C4C2×D3211yescharacteristic
197C2×Dic3—21no—
198C2×Dic3—61no—
199C2×Dic3—91no—
200C2×Dic3—91no—
201C2×Dic3—91no—
202C2×Dic3—91no—
203C2×Dic3—181no—
204C2×Dic3—181no—
205C2×Dic3—182no—
206C2×Dic3—181no—
207C2×Dic3—181no—
208Dic6—362no—
209Dic6—364no—
210Dic6—362no—
211C3×C8—32no—
212C3×C8—92no—
213C3⋊C8—32no—
214C3⋊C8Dic1812yescomplemented
215C2×C32#88811yescharacteristic
216C2×C32#91411yescharacteristic
217C2×C32#91511yescharacteristic
218C2×C9—21no—
219C2×C9—21no—
220C2×C9—21no—
221C2×C9#88811yescharacteristic
222C2×C9#91111yescharacteristic
223C2×C9#91811yescharacteristic
224C2×Q8—541no—
225C2×C8—31no—
226C2×C8—91no—
227C4⋊C4—31no—
228C4⋊C4—91no—
229C4⋊C4—271no—
230C4⋊C4—541no—
231C4⋊C4—541no—
232C4⋊C4—541no—
233C4⋊C4D3×D911yescharacteristic
234C42—271no—
235C22×C3—21no—
236C22×C3D4×D911yescharacteristic
237C22×C3#381111yescharacteristic
238C3×C4—21no—
239C3×C4—21no—
240C3×C4—42no—
241C3×C4—41no—
242C3×C4—41no—
243C3×C4—121no—
244C3×C4—361no—
245C3×C4C2×Dic1811yescharacteristic
246C3×C4Q8×D911yescharacteristic
247C3×C4D3×Dic611yescharacteristic
248C3×C4D3×Dic611yescharacteristic
249Dic3—41no—
250Dic3—121no—
251Dic3—182no—
252Dic3—182no—
253Dic3—181no—
254Dic3—181no—
255Dic3—181no—
256Dic3—181no—
257Dic3—362no—
258Dic3—361no—
259Dic3—361no—
260C32#485611yescharacteristic
261C9—21no—
262C9#487611yescharacteristic, complemented
263Q8—1082no—
264C2×C4—21no—
265C2×C4—61no—
266C2×C4—181no—
267C2×C4—271no—
268C2×C4—271no—
269C2×C4—541no—
270C2×C4C2×D3×D911yescharacteristic
271C8—32no—
272C8—92no—
273C2×C3—21no—
274C2×C3—21no—
275C2×C3—21no—
276C2×C3#6345011yescharacteristic
277C2×C3#6347611yescharacteristic
278C2×C3#6347711yescharacteristic
279C2×C3#6384011yescharacteristic
280C2×C3#6393711yescharacteristic
281C2×C3#6395111yescharacteristic
282C22#8866811yescharacteristic
283C4—41no—
284C4—121no—
285C4—361no—
286C4—541no—
287C4—541no—
288C4—1081no—
289C4D9×Dic611yescharacteristic
290C4D3×Dic1811yescharacteristic
291C3—21no—
292C3#1058853611yescharacteristic, complemented
293C3#1059054611yescharacteristic
294C2#1172090511yescharacteristic
295C2#1172118711yescharacteristic
296C2#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#501
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
44. C9×C3⋊C8Q8semidirect
45. C3×C9⋊C8Q8semidirect
124. C9⋊C8Dic6semidirect
188. C3×C9#501semidirect
214. C3⋊C8Dic18semidirect
262. C9#4876semidirect
292. C3#10588536semidirect

Maximal subgroups

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

subgroupindex
#117217442
#117209372
C9×#8082
C3×#633662
#117202792
#117202752
#117202642
#105905463
#105885363