Finite Group Databasebeta

← #13140386

Subgroups — #13140386

Order 1728.

496 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C2×C3×#3699C211yesmaximal, characteristic
2D9×#208C212yesmaximal, complemented
3#11720575C211yesmaximal, characteristic, complemented
4C9×#833C211yesmaximal, characteristic, complemented
5#11720289C212yesmaximal, complemented
6#10591501—31nomaximal
7#10588425—31nomaximal
8C2×C3×D4×C9C2211yescharacteristic
9C2×C3×#611C412yes—
10C22×C3×Dic9C412yes—
11C3×#3699—22no—
12C3×#3699—22no—
13C2×C3×C4×D9C2211yescharacteristic
14C2×C3×Dic18C2211yescharacteristic
15#88532C2212yes—
16C9×#208C2212yes—
17#88413—22no—
18D9×C3⋊C8—22no—
19C2×C3×#237—31no—
20D3×#208—32no—
21C2×#3699D311yescharacteristic
22#63660—31no—
23C3×#833D311yescharacteristic
24#63560—32no—
25M16×D9—32no—
26C9×#100—31no—
27#63391—31no—
28#63391—61no—
29#63382—32no—
30C23×C3×C9C2×C412yes—
31C22×C3×D9C2×C411yescharacteristic
32C3×D4×C9—22no—
33C2×C3×C4×C9C2311yescharacteristic
34C3×#611—24no—
35C2×C3×Dic9—22no—
36C2×C3×Dic9—22no—
37C2×C3×Dic9C2×C412yes—
38C2×C3×Dic9C2×C411yescharacteristic
39C3×C4×D9D412yes—
40C3×C4×D9D412yes—
41C3×Dic18—22no—
42C3⋊(C9⋊C8)—22no—
43C9×C3⋊C8—22no—
44#5237—91no—
45#4745—91no—
46C2×C32×D4C2×D311yescharacteristic
47C2×C3×#67—32no—
48C22×C3×Dic3—32no—
49C3×#237—62no—
50C3×#237—62no—
51C2×C3×C4×D3—31no—
52C2×C3×Dic6—31no—
53#3748—32no—
54C3×#208C2×D312yes—
55#3711—62no—
56D3×C3⋊C8—62no—
57C2×D4×C9—21no—
58C2×D4×C9C2×D311yescharacteristic
59C2×#611Dic312yes—
60C22×Dic9Dic312yes—
61#3699—22no—
62#3699—22no—
63C2×C4×D9C2×D311yescharacteristic
64C2×Dic18C2×D311yescharacteristic
65C9×M16—32no—
66#3667—32no—
67#3667—62no—
68#3663—62no—
69C8×D9—62no—
70C22×C3×C9—22no—
71C22×C3×C9—42no—
72C22×C3×C9C22×C411yescharacteristic
73C2×C3×D9—41no—
74C2×C3×D9C22⋊C412yes—
75C3×C4×C9C2×D412yes—
76C3×Dic9—22no—
77C3×Dic9C22⋊C411yescharacteristic
78C3×Dic9C22⋊C411yescharacteristic
79C2×C3×#41—91no—
80C2×#237—31no—
81C2×#208—92no—
82D3×M16—92no—
83C3×#100—31no—
84#836—92no—
85#833—91no—
86#833—91no—
87#833—181no—
88#833D911yescharacteristic, complemented
89#824—92no—
90C23×C32C4×D312yes—
91C22×C3×D3—31no—
92C32×D4—22no—
93C2×C32×C4C22×D311yescharacteristic
94C3×#67—64no—
95C2×C3×Dic3—31no—
96C2×C3×Dic3—32no—
97C2×C3×Dic3—62no—
98C2×C3×Dic3—62no—
99C3×C4×D3—32no—
100C3×C4×D3—32no—
101C3×Dic6—62no—
102C23×C9—22no—
103C23×C9C2×Dic312yes—
104C22×D9C2×Dic311yescharacteristic
105C3⋊(C3⋊C8)—62no—
106C3×C3⋊C8—22no—
107D4×C9—22no—
108D4×C9—42no—
109C2×C4×C9—21no—
110C2×C4×C9C22×D311yescharacteristic
111#611—24no—
112C2×Dic9—22no—
113C2×Dic9—22no—
114C2×Dic9C2×Dic312yes—
115C2×Dic9C2×Dic311yescharacteristic
116C4×D9#6712yes—
117C4×D9#6712yes—
118Dic18—22no—
119C8×C9—62no—
120C9⋊C8—62no—
121C9⋊C8—122no—
122#413—271noSylow
123C2×C3×C9—21no—
124C2×C3×C9—42no—
125C2×C3×C9C2×C22⋊C411yescharacteristic
126C3×D9—22no—
127C3×#41—182no—
128C3×#41—182no—
129C2×C3×Q8—91no—
130C2×C3×D4—21no—
131C2×C3×D4—92no—
132C2×C3×D4C2×D911yescharacteristic
133C2×C3×D4D3211yescharacteristic
134C22×C3×C4—91no—
135C22×C3×C4—92no—
136C2×#67—32no—
137C22×Dic3—32no—
138#237—62no—
139#237—62no—
140C2×C4×D3—31no—
141C2×Dic6—31no—
142C3×M16—32no—
143#208—92no—
144#208—92no—
145#208—182no—
146#208—182no—
147#208C2×D912yes—
148C2×C3⋊C8—182no—
149ΓL(1,𝔽25)—182no—
150D3×C8—182no—
151C22×C32—22no—
152C22×C32—42no—
153C22×C32C2×C4×D311yescharacteristic
154C2×C3×D3—32no—
155C2×C3×D3—121no—
156C32×C4D3×D412yes—
157C3×Dic3—31no—
158C3×Dic3—31no—
159C3×Dic3—62no—
160C22×C9—22no—
161C22×C9—21no—
162C22×C9—42no—
163C22×C9—42no—
164C22×C9—82no—
165C22×C9C22×Dic311yescharacteristic
166C2×D9—41no—
167C2×D9#21712yes—
168C4×C9—22no—
169C4×C9C2×#6712yes—
170Dic9—22no—
171Dic9#21711yescharacteristic
172Dic9#21711yescharacteristic
173C2×#41—91no—
174C2×M16—272no—
175#101—272no—
176#100—31no—
177#100—271no—
178C3×C9#41311yescharacteristic, complemented, Sylow
179C23×C3—22no—
180C23×C3—91no—
181C23×C3C4×D912yes—
182C23×C3D3×Dic312yes—
183C22×D3—31no—
184C3×Q8—182no—
185C3×D4—22no—
186C3×D4—22no—
187C3×D4—42no—
188C3×D4—184no—
189C2×C3×C4—21no—
190C2×C3×C4—91no—
191C2×C3×C4—92no—
192C2×C3×C4—92no—
193C2×C3×C4—92no—
194C2×C3×C4—182no—
195C2×C3×C4—182no—
196C2×C3×C4C22×D911yescharacteristic
197C2×C3×C4C2×D3211yescharacteristic
198#67—64no—
199C2×Dic3—31no—
200C2×Dic3—32no—
201C2×Dic3—62no—
202C2×Dic3—62no—
203C4×D3—32no—
204C4×D3—32no—
205Dic6—62no—
206C3×C8—62no—
207C3⋊C8—22no—
208C3⋊C8—182no—
209C3⋊C8—182no—
210C3⋊C8—362no—
211C2×C32—21no—
212C2×C32—42no—
213C2×C32D3×C22⋊C411yescharacteristic
214C3×D3—62no—
215C2×C9—21no—
216C2×C9—21no—
217C2×C9—41no—
218C2×C9—42no—
219C2×C9—82no—
220C2×C9C2×#21711yescharacteristic
221D9—22no—
222#41—182no—
223#41—182no—
224C2×Q8—91no—
225C2×D4—92no—
226C2×D4D3×D911yescharacteristic
227C22×C4—91no—
228C22×C4—92no—
229M16—32no—
230M16—272no—
231M16—542no—
232C2×C8—542no—
233C22×C3—22no—
234C22×C3—22no—
235C22×C3—21no—
236C22×C3—42no—
237C22×C3—42no—
238C22×C3—42no—
239C22×C3—82no—
240C22×C3—92no—
241C22×C3—361no—
242C22×C3C2×C4×D911yescharacteristic
243C22×C3C2×D3×Dic311yescharacteristic
244C2×D3—32no—
245C2×D3—121no—
246C3×C4—22no—
247C3×C4—91no—
248C3×C4—91no—
249C3×C4—182no—
250C3×C4D4×D912yes—
251C3×C4D3×#6712yes—
252Dic3—31no—
253Dic3—31no—
254Dic3—62no—
255C32#474511yescharacteristic
256C9—21no—
257C9#523711yescharacteristic, complemented
258C23—91no—
259C23Dic3×D912yes—
260Q8—182no—
261D4—22no—
262D4—184no—
263C2×C4—91no—
264C2×C4—92no—
265C2×C4—92no—
266C2×C4—92no—
267C2×C4—182no—
268C2×C4—182no—
269C2×C4C2×D3×D911yescharacteristic
270C8—62no—
271C8—542no—
272C2×C3—21no—
273C2×C3—21no—
274C2×C3—21no—
275C2×C3—41no—
276C2×C3—42no—
277C2×C3—42no—
278C2×C3—82no—
279C2×C3—182no—
280C2×C3C22⋊C4×D911yescharacteristic
281C2×C3D3×#21711yescharacteristic
282D3—62no—
283C22—22no—
284C22—42no—
285C22—92no—
286C22—361no—
287C22C2×Dic3×D911yescharacteristic
288C4—91no—
289C4—91no—
290C4—182no—
291C4D9×#6712yes—
292C3—21no—
293C3#1058842511yescharacteristic, complemented
294C3#1059150111yescharacteristic
295C2—21no—
296C2—42no—
297C2—182no—
298C2D9×#21711yescharacteristic
subgroupG/H
centerC2D9×#217
commutatorC22×C3×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC2×C32D3×C22⋊C4
prime radicalC2×C3×#3699C2
cosocleC2×C3×C4×C9C23
hypercenterC2×D4D3×D9
hypocenterC3×C9#413
LCSG ▷ C22×C3×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. D9×#208C2semidirect
3. #11720575C2semidirect
4. C9×#833C2semidirect
5. #11720289C2semidirect
88. #833D9semidirect
178. C3×C9#413semidirect
257. C9#5237semidirect
293. C3#10588425semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#36992
D9×#2082
#117205752
C9×#8332
#117202892
#105915013
#105884253