Finite Group Databasebeta

← #13137546

Subgroups — #13137546

Order 1728.

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

normal subgroups only

This group has 326 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#11721746C211yesmaximal, characteristic
2#11720971C211yesmaximal, characteristic
3C9×#807C211yesmaximal, characteristic
4C3×#63366C211yesmaximal, characteristic
5#11720278C211yesmaximal, characteristic
6#11720275C211yesmaximal, characteristic
7#11720264C211yesmaximal, characteristic
8#10590563—31nomaximal
9#10588556—31nomaximal
10C2×#6189—21no—
11C3×C9×C4⋊C4C2211yescharacteristic
12C4⋊(C3⋊Dic9)—21no—
13C4⋊(C3⋊Dic9)C2211yescharacteristic
14#88534—21no—
15C4×C3⋊Dic9C2211yescharacteristic
16C9×C4⋊Dic3C2211yescharacteristic
17C3×C4⋊Dic9C2211yescharacteristic
18C2×C9×C3⋊C8C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20#88442—21no—
21#64099—31no—
22#63919—31no—
23C3×#807—31no—
24C3×#807D311yescharacteristic
25#63573—31no—
26#63573—31no—
27#63555—31no—
28#63452—31no—
29#63452—61no—
30#63450—31no—
31C9×#107—31no—
32#63377—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
41C2×C9×Dic3—21no—
42C2×C3×Dic9—21no—
43#6189—42no—
44C9×C3⋊C8Q812yescomplemented
45C3×C9⋊C8Q812yescomplemented
46#4876—91no—
47#4876—91no—
48C2×#627—61no—
49C32×C4⋊C4C2×D311yescharacteristic
50C4⋊(C3⋊Dic3)—31no—
51C4⋊(C3⋊Dic3)—61no—
52#3750—61no—
53C4×C3⋊Dic3—31no—
54C3×C4⋊Dic3—31no—
55C3×C4⋊Dic3C2×D311yescharacteristic
56C2×C3×C3⋊C8—31no—
57C2×C3×C3⋊C8C2×D311yescharacteristic
58#3724—61no—
59C2×Dic18—61no—
60C2×C8×C9—31no—
61C9×C4⋊C4—21no—
62C9×C4⋊C4—31no—
63C9×C4⋊C4C2×D311yescharacteristic
64C4⋊Dic9—31no—
65C4⋊Dic9—61no—
66C4⋊Dic9—61no—
67C4⋊Dic9—62no—
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#890—181no—
87#888—91no—
88#888—91no—
89C3×#107—31no—
90C3×#107—91no—
91#817—91no—
92#817—91no—
93#815—91no—
94#815—91no—
95#808—91no—
96#808—91no—
97#807—31no—
98#807D911yescharacteristic
99C2×C32×C4—21no—
100C2×C32×C4C22×D311yescharacteristic
101C2×C3⋊Dic3—31no—
102C2×C3⋊Dic3—31no—
103C2×C3⋊Dic3—61no—
104C2×C3×Dic3—21no—
105C2×C3×Dic3—61no—
106#627—122no—
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—122no—
122C8×C9—32no—
123C9⋊C8Dic612yescomplemented
124#501—271noSylow
125C2×C3×C9C4⋊Q811yescharacteristic
126C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
127C2×C3×C9#13711yescharacteristic
128C2×Dic6—181no—
129C2×Dic6—181no—
130C2×C3×C8—31no—
131C2×C3×C8—91no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—31no—
134C3×C4⋊C4—91no—
135C3×C4⋊C4C2×D911yescharacteristic
136C3×C4⋊C4D3211yescharacteristic
137C4⋊Dic3—31no—
138C4⋊Dic3—91no—
139C4⋊Dic3—91no—
140C4⋊Dic3—181no—
141C4⋊Dic3—181no—
142C4⋊Dic3—181no—
143C4⋊Dic3—182no—
144C4⋊Dic3C2×D911yescharacteristic
145Dic3⋊C4—181no—
146Dic3⋊C4—181no—
147Dic3⋊C4—181no—
148Dic3⋊C4—181no—
149Dic3⋊C4—182no—
150C4×Dic3—91no—
151C4×Dic3—91no—
152C4×Dic3—181no—
153C2×C3⋊C8—31no—
154C2×C3⋊C8C2×D911yescharacteristic
155C22×C32D3×D411yescharacteristic
156C32×C4—41no—
157C32×C4C2×Dic611yescharacteristic
158C32×C4D3×Q811yescharacteristic
159C3⋊Dic3—61no—
160C3⋊Dic3—61no—
161C3⋊Dic3—121no—
162C3×Dic3—41no—
163C3×Dic3—121no—
164C22×C9—21no—
165C22×C9D3×D411yescharacteristic
166C4×C9—21no—
167C4×C9—21no—
168C4×C9—42no—
169C4×C9—41no—
170C4×C9—121no—
171C4×C9C2×Dic611yescharacteristic
172C4×C9D3×Q811yescharacteristic
173Dic9—41no—
174Dic9—61no—
175Dic9—61no—
176Dic9—121no—
177Dic9—121no—
178Dic9—122no—
179Dic9—121no—
180C4⋊Q8—271no—
181#125—271no—
182#107—31no—
183#107—91no—
184#106—271no—
185#106—271no—
186#97—271no—
187C3×C9#50111yescharacteristic, complemented, Sylow
188C2×C3×C4—22no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—61no—
193C2×C3×C4—181no—
194C2×C3×C4C22×D911yescharacteristic
195C2×C3×C4C2×D3211yescharacteristic
196C2×Dic3—21no—
197C2×Dic3—61no—
198C2×Dic3—91no—
199C2×Dic3—91no—
200C2×Dic3—91no—
201C2×Dic3—91no—
202C2×Dic3—181no—
203C2×Dic3—181no—
204C2×Dic3—182no—
205C2×Dic3—181no—
206C2×Dic3—181no—
207Dic6—362no—
208Dic6—362no—
209C3×C8—32no—
210C3×C8—92no—
211C3⋊C8—32no—
212C3⋊C8Dic1812yescomplemented
213C2×C32#88811yescharacteristic
214C2×C32#91111yescharacteristic
215C2×C32#91811yescharacteristic
216C2×C9—21no—
217C2×C9—21no—
218C2×C9—21no—
219C2×C9#88811yescharacteristic
220C2×C9#91111yescharacteristic
221C2×C9#91811yescharacteristic
222C2×Q8—541no—
223C2×C8—31no—
224C2×C8—91no—
225C4⋊C4—31no—
226C4⋊C4—91no—
227C4⋊C4—271no—
228C4⋊C4—541no—
229C4⋊C4—541no—
230C4⋊C4—541no—
231C4⋊C4D3×D911yescharacteristic
232C42—271no—
233C22×C3—21no—
234C22×C3D4×D911yescharacteristic
235C22×C3#381111yescharacteristic
236C3×C4—21no—
237C3×C4—21no—
238C3×C4—42no—
239C3×C4—41no—
240C3×C4—41no—
241C3×C4—121no—
242C3×C4—361no—
243C3×C4C2×Dic1811yescharacteristic
244C3×C4Q8×D911yescharacteristic
245C3×C4D3×Dic611yescharacteristic
246C3×C4D3×Dic611yescharacteristic
247Dic3—41no—
248Dic3—121no—
249Dic3—181no—
250Dic3—181no—
251Dic3—181no—
252Dic3—181no—
253Dic3—361no—
254Dic3—361no—
255Dic3—362no—
256Dic3—361no—
257Dic3—361no—
258C32#487611yescharacteristic
259C9—21no—
260C9#487611yescharacteristic, complemented
261Q8—1082no—
262C2×C4—21no—
263C2×C4—61no—
264C2×C4—181no—
265C2×C4—271no—
266C2×C4—271no—
267C2×C4—541no—
268C2×C4C2×D3×D911yescharacteristic
269C8—32no—
270C8—92no—
271C2×C3—21no—
272C2×C3—21no—
273C2×C3—21no—
274C2×C3#6345011yescharacteristic
275C2×C3#6347311yescharacteristic
276C2×C3#6348011yescharacteristic
277C2×C3#6384011yescharacteristic
278C2×C3#6392711yescharacteristic
279C2×C3#6394511yescharacteristic
280C22#8866811yescharacteristic
281C4—41no—
282C4—121no—
283C4—361no—
284C4—541no—
285C4—541no—
286C4—1081no—
287C4D9×Dic611yescharacteristic
288C4D3×Dic1811yescharacteristic
289C3—21no—
290C3#1058855611yescharacteristic, complemented
291C3#1059056311yescharacteristic
292C2#1172090511yescharacteristic
293C2#1172117511yescharacteristic
294C2#1172122111yescharacteristic
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
123. C9⋊C8Dic6semidirect
187. C3×C9#501semidirect
212. C3⋊C8Dic18semidirect
260. C9#4876semidirect
290. C3#10588556semidirect

Maximal subgroups

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

subgroupindex
#117217462
#117209712
C9×#8072
C3×#633662
#117202782
#117202752
#117202642
#105905633
#105885563