Finite Group Databasebeta

← #13140459

Subgroups — #13140459

Order 1728.

534 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 93 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#11721840C211yesmaximal, characteristic
2C2×C9×#214C211yesmaximal, characteristic, complemented
3C2×C3×#3672C211yesmaximal, characteristic, complemented
4#11720930C211yesmaximal, characteristic, complemented
5C2×#88426C211yesmaximal, characteristic, complemented
6C2×#88416C211yesmaximal, characteristic, complemented
7#11720271C211yesmaximal, characteristic, complemented
8#10591492—31nomaximal
9#10588855—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C2×#6233—22no—
12C2×C4⋊(C3⋊D9)C2211yescharacteristic
13C2×C9×Dic6C2211yescharacteristic
14C2×C3×Dic18C2211yescharacteristic
15C4×C3⋊Dic9C2211yescharacteristic
16C9×#214—22no—
17C3×#3672—22no—
18C2×C9×C3⋊C8C2211yescharacteristic
19C2×C3×C9⋊C8C2211yescharacteristic
20#88442—22no—
21#88426—22no—
22#88416—22no—
23#64149—31no—
24C2×C3×#214—31no—
25C2×C3×#214D311yescharacteristic
26#63873—31no—
27C2×#3717—31no—
28C2×#3717—31no—
29#63553—31no—
30C2×C9×SD16—31no—
31#63506—31no—
32C2×#3675—31no—
33#63502—31no—
34#63502—61no—
35C2×#3672D311yescharacteristic, complemented
36C2×#3664—31no—
37C8×Dic9—31no—
38C23×C3×C9—21no—
39C22×C3⋊D9—21no—
40C3×D4×C9—21no—
41C3×D4×C9—21no—
42C2×C3×C4×C9C2311yescharacteristic
43#6233—44no—
44C2×C3⋊Dic9D412yes—
45C4⋊(C3⋊D9)—21no—
46C4⋊(C3⋊D9)—21no—
47C2×C9×Dic3—21no—
48C2×C3×Dic9—21no—
49C9×Dic6—21no—
50C9×Dic6—21no—
51C3×Dic18—21no—
52C3×Dic18—21no—
53C9×C3⋊C8D412yescomplemented
54C3×C9⋊C8D412yescomplemented
55#5175—91no—
56#5175—91no—
57C2×C32×D4C2×D311yescharacteristic
58C2×#638—62no—
59C2×C4⋊(C3⋊D3)—31no—
60C2×C3×Dic6—31no—
61C2×C3×Dic6C2×D311yescharacteristic
62C4×C3⋊Dic3—31no—
63C3×#214—22no—
64C3×#214—62no—
65C2×C3×C3⋊C8—31no—
66C2×C3×C3⋊C8C2×D311yescharacteristic
67#3724—62no—
68#3717—62no—
69#3717—62no—
70C2×Q8×C9—31no—
71C2×D4×C9—21no—
72C2×D4×C9C2×D311yescharacteristic
73C2×#611—62no—
74C2×#611—64no—
75C2×D36—31no—
76C2×D36—61no—
77C2×Dic18C2×D311yescharacteristic
78C9×SD16—62no—
79C2×C8×C9—31no—
80#3675—62no—
81#3672—22no—
82Dic9⋊C4—62no—
83C4×Dic9—31no—
84C4×Dic9—61no—
85C2×C9⋊C8C2×D311yescharacteristic
86#3664—62no—
87C22×C3×C9—41no—
88C22×C3×C9—42no—
89C22×C3×C9C2×D411yescharacteristic
90C2×C3⋊D9—41no—
91C2×C3⋊D9—42no—
92C3×C4×C9C2×D411yescharacteristic
93C3×C4×C9C2×D411yescharacteristic
94C3⋊Dic9SD1614yescomplemented
95C9×Dic3—41no—
96C3×Dic9—41no—
97C2×C3×SD16—31no—
98C2×C3×SD16—91no—
99#944—91no—
100#944—91no—
101C2×#215—91no—
102C2×#215—91no—
103#940—91no—
104#940—91no—
105#940—181no—
106C2×#214—31no—
107C2×#214D911yescharacteristic, complemented
108C2×#204—91no—
109C2×#204—91no—
110C8×Dic3—91no—
111C8×Dic3—91no—
112C23×C32—21no—
113C22×C3⋊D3—61no—
114C32×D4—21no—
115C32×D4—21no—
116C2×C32×C4C22×D311yescharacteristic
117#638—124no—
118C2×C3⋊Dic3—32no—
119C4⋊(C3⋊D3)—61no—
120C4⋊(C3⋊D3)—61no—
121C2×C3×Dic3—21no—
122C2×C3×Dic3—61no—
123C3×Dic6—21no—
124C3×Dic6—21no—
125C3×Dic6—61no—
126C3×Dic6—61no—
127C23×C9—22no—
128C23×C9—21no—
129C22×D9—62no—
130C22×D9—61no—
131C3×C3⋊C8—32no—
132C3×C3⋊C8#6712yes—
133Q8×C9—61no—
134Q8×C9—61no—
135D4×C9—21no—
136D4×C9—21no—
137D4×C9—41no—
138D4×C9—41no—
139C2×C4×C9—21no—
140C2×C4×C9—61no—
141C2×C4×C9C22×D311yescharacteristic
142#611—124no—
143#611—128no—
144C2×Dic9—21no—
145C2×Dic9—32no—
146C2×Dic9—62no—
147D36—61no—
148D36—61no—
149D36—121no—
150D36—121no—
151Dic18—21no—
152Dic18—21no—
153C8×C9—32no—
154C9⋊C8#6712yescomplemented
155#492—271noSylow
156C2×C3×C9—81no—
157C2×C3×C9#12711yescharacteristic
158C2×C3×C9C2×SD1612yes—
159C3⋊D9—81no—
160C2×C3×Q8—31no—
161C2×C3×Q8—91no—
162C2×C3×D4—21no—
163C2×C3×D4C2×D911yescharacteristic
164C2×C3×D4D3211yescharacteristic
165C2×#67—182no—
166C2×#67—182no—
167C2×#67—184no—
168C2×D12—91no—
169C2×D12—91no—
170C2×D12—181no—
171C2×Dic6—31no—
172C2×Dic6C2×D911yescharacteristic
173C3×SD16—62no—
174C3×SD16—182no—
175C2×C3×C8—31no—
176C2×C3×C8—91no—
177#215—182no—
178#215—182no—
179#214—22no—
180#214—62no—
181Dic3⋊C4—182no—
182Dic3⋊C4—182no—
183C4×Dic3—91no—
184C4×Dic3—91no—
185C4×Dic3—181no—
186C2×C3⋊C8—31no—
187C2×C3⋊C8C2×D911yescharacteristic
188#204—182no—
189#204—182no—
190C22×C32—42no—
191C22×C32—41no—
192C22×C32D3×D411yescharacteristic
193C2×C3⋊D3—122no—
194C2×C3⋊D3—121no—
195C32×C4D3×D411yescharacteristic
196C32×C4C2×#6711yescharacteristic
197C3⋊Dic3—34no—
198C3×Dic3—41no—
199C3×Dic3—121no—
200C22×C9—21no—
201C22×C9—44no—
202C22×C9—42no—
203C22×C9—42no—
204C22×C9—41no—
205C22×C9D3×D411yescharacteristic
206C2×D9—122no—
207C2×D9—124no—
208C2×D9—122no—
209C2×D9—121no—
210C4×C9—21no—
211C4×C9—21no—
212C4×C9—121no—
213C4×C9D3×D411yescharacteristic
214C4×C9C2×#6711yescharacteristic
215Dic9—34no—
216Dic9—41no—
217Dic9—64no—
218C2×SD16—31no—
219C2×SD16—91no—
220C2×SD16—271no—
221C2×SD16—271no—
222C4⋊Q8—271no—
223#127—271no—
224C4×C8—271no—
225C3×C9#49211yescharacteristic, complemented, Sylow
226C23×C3—22no—
227C23×C3—21no—
228C23×C3—21no—
229C22×D3—182no—
230C22×D3—181no—
231C22×D3—181no—
232C3×Q8—61no—
233C3×Q8—61no—
234C3×Q8—181no—
235C3×Q8—181no—
236C3×D4—21no—
237C3×D4—21no—
238C3×D4—21no—
239C3×D4—21no—
240C3×D4—41no—
241C3×D4—41no—
242C2×C3×C4—21no—
243C2×C3×C4—61no—
244C2×C3×C4—181no—
245C2×C3×C4C22×D911yescharacteristic
246C2×C3×C4C2×D3211yescharacteristic
247#67—364no—
248#67—364no—
249#67—368no—
250C2×Dic3—21no—
251C2×Dic3—61no—
252C2×Dic3—92no—
253C2×Dic3—92no—
254C2×Dic3—182no—
255D12—181no—
256D12—181no—
257D12—181no—
258D12—181no—
259D12—361no—
260D12—361no—
261Dic6—21no—
262Dic6—21no—
263Dic6—61no—
264Dic6—61no—
265C3×C8—32no—
266C3×C8—92no—
267C3⋊C8—32no—
268C3⋊C8#61112yescomplemented
269C2×C32—81no—
270C2×C32D3×SD1612yes—
271C2×C32#94011yescharacteristic
272C3⋊D3—241no—
273C2×C9—22no—
274C2×C9—21no—
275C2×C9—82no—
276C2×C9—81no—
277C2×C9D3×SD1612yes—
278C2×C9#94011yescharacteristic
279D9—242no—
280D9—241no—
281C2×Q8—31no—
282C2×Q8—91no—
283C2×D4—271no—
284C2×D4—542no—
285C2×D4D3×D911yescharacteristic
286SD16—62no—
287SD16—182no—
288SD16—542no—
289SD16—542no—
290C2×C8—31no—
291C2×C8—91no—
292C4⋊C4—542no—
293C42—271no—
294C22×C3—21no—
295C22×C3—44no—
296C22×C3—42no—
297C22×C3—41no—
298C22×C3—42no—
299C22×C3—41no—
300C22×C3—42no—
301C22×C3D4×D911yescharacteristic
302C22×C3#381111yescharacteristic
303C2×D3—362no—
304C2×D3—364no—
305C2×D3—361no—
306C2×D3—362no—
307C2×D3—361no—
308C2×D3—362no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—121no—
312C3×C4—361no—
313C3×C4D4×D911yescharacteristic
314C3×C4C2×#61111yescharacteristic
315C3×C4D3×#6711yescharacteristic
316C3×C4D3×#6711yescharacteristic
317Dic3—41no—
318Dic3—94no—
319Dic3—94no—
320Dic3—121no—
321Dic3—184no—
322C32#517511yescharacteristic
323C9—21no—
324C9#517511yescharacteristic, complemented
325C23—21no—
326C23—541no—
327Q8—61no—
328Q8—61no—
329Q8—181no—
330Q8—181no—
331D4—21no—
332D4—21no—
333D4—541no—
334D4—541no—
335D4—1084no—
336C2×C4—61no—
337C2×C4—181no—
338C2×C4—272no—
339C2×C4C2×D3×D911yescharacteristic
340C8—32no—
341C8—92no—
342C2×C3—21no—
343C2×C3—22no—
344C2×C3—82no—
345C2×C3—81no—
346C2×C3—81no—
347C2×C3SD16×D912yes—
348C2×C3#6350211yescharacteristic
349C2×C3#6393012yes—
350C2×C3#6397911yescharacteristic
351D3—722no—
352D3—721no—
353D3—721no—
354C22—41no—
355C22—42no—
356C22—1081no—
357C22—1082no—
358C22#8866811yescharacteristic
359C4—121no—
360C4—274no—
361C4—361no—
362C4D3×#61111yescharacteristic
363C4D9×#6711yescharacteristic
364C3—21no—
365C3#1058885511yescharacteristic, complemented
366C3#1059149211yescharacteristic
367C2—81no—
368C2—2161no—
369C2#1172118212yes—
370C2#1172128911yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
prime radical#11721840C2
hypercenterC2×D4D3×D9
hypocenterC3×C9#492
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×C9×#214C2semidirect
3. C2×C3×#3672C2semidirect
4. #11720930C2semidirect
5. C2×#88426C2semidirect
6. C2×#88416C2semidirect
7. #11720271C2semidirect
35. C2×#3672D3semidirect
53. C9×C3⋊C8D4semidirect
54. C3×C9⋊C8D4semidirect
94. C3⋊Dic9SD16semidirect
107. C2×#214D9semidirect
154. C9⋊C8#67semidirect
225. C3×C9#492semidirect
268. C3⋊C8#611semidirect
324. C9#5175semidirect
365. C3#10588855semidirect

Maximal subgroups

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

subgroupindex
#117218402
C2×C9×#2142
C2×C3×#36722
#117209302
C2×#884262
C2×#884162
#117202712
#105914923
#105888553