Finite Group Databasebeta

← #13140437

Subgroups — #13140437

Order 1728.

496 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 117 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
1D4×C3⋊Dic9C211yesmaximal, characteristic, complemented
2C2×C9×#213C211yesmaximal, characteristic
3#11721025C211yesmaximal, characteristic, complemented
4C3×#63392C211yesmaximal, characteristic, complemented
5#11720295C211yesmaximal, characteristic, complemented
6#11720279C211yesmaximal, characteristic, complemented
7#11720271C211yesmaximal, characteristic, complemented
8#10591435—31nomaximal
9D8×Dic9—31nomaximal
10C2×C3×D4×C9C2211yescharacteristic
11C22×C3⋊Dic9—21no—
12C2×C9×D12C2211yescharacteristic
13C2×D3×Dic9—21no—
14#88551—21no—
15C4⋊(C3⋊Dic9)C2211yescharacteristic
16C4×C3⋊Dic9C2211yescharacteristic
17C9×#213C414yescomplemented
18C3×C4⋊Dic9C2211yescharacteristic
19C2×C9×C3⋊C8C2211yescharacteristic
20C2×C3×C9⋊C8C2211yescharacteristic
21(C2×D3)⋊Dic9—21no—
22D4×C3⋊Dic3—31no—
23C2×C3×#213D311yescharacteristic
24#63898—31no—
25C3×#832—31no—
26#63575—31no—
27#63563—31no—
28#63553—31no—
29C2×C9×D8—31no—
30D4×Dic9—31no—
31D4×Dic9—31no—
32D4×Dic9—61no—
33#63392—31no—
34#63392D311yescharacteristic, complemented
35#63378—31no—
36C8×Dic9—31no—
37C23×C3×C9—21no—
38C22×D3×C9—21no—
39C3×D4×C9—21no—
40C3×D4×C9C2×C412yescomplemented
41C2×C3×C4×C9C2311yescharacteristic
42C2×C3⋊Dic9—21no—
43C2×C3⋊Dic9—42no—
44C2×C3⋊Dic9D411yescharacteristic
45C2×C3⋊Dic9D411yescharacteristic
46C2×C3×Dic9—21no—
47C9×D12—21no—
48C9×D12C2×C412yescomplemented
49D3×Dic9—42no—
50C9×C3⋊C8C2×C412yescomplemented
51C3×C9⋊C8—21no—
52Dic3×D8—91no—
53#4756—91no—
54C2×C32×D4C2×D311yescharacteristic
55C22×C3⋊Dic3—61no—
56C2×C3×D12C2×D311yescharacteristic
57C2×D3×Dic3—61no—
58#3757—61no—
59C4⋊(C3⋊Dic3)—31no—
60C4×C3⋊Dic3—31no—
61C3×#213Dic314yes—
62C3×C4⋊Dic3—31no—
63C2×C3×C3⋊C8—31no—
64C2×C3×C3⋊C8C2×D311yescharacteristic
65(C2×D3)⋊Dic3—61no—
66C2×D4×C9—21no—
67C2×D4×C9—31no—
68C2×D4×C9C2×D311yescharacteristic
69C22×Dic9—61no—
70C22×Dic9—61no—
71C22×Dic9—62no—
72C9×D8—34no—
73C2×C8×C9—31no—
74#3676—61no—
75#3676—61no—
76#3676—62no—
77C4⋊Dic9—31no—
78C4⋊Dic9—61no—
79C4⋊Dic9C2×D311yescharacteristic
80C4×Dic9—31no—
81C4×Dic9—61no—
82C2×C9⋊C8C2×D311yescharacteristic
83C22×C3×C9—22no—
84C22×C3×C9—41no—
85C22×C3×C9—41no—
86C22×C3×C9C2×D411yescharacteristic
87C2×D3×C9—22no—
88C2×D3×C9—41no—
89C2×D3×C9—41no—
90C3×C4×C9C22×C411yescharacteristic
91C3×C4×C9#4111yescharacteristic
92C3⋊Dic9—21no—
93C3⋊Dic9—41no—
94C3⋊Dic9D812yescomplemented
95C3×Dic9—41no—
96C2×C3×D8—31no—
97D4×Dic3—91no—
98D4×Dic3—91no—
99D4×Dic3—91no—
100D4×Dic3—181no—
101C2×#213D911yescharacteristic
102#893—91no—
103C3×#102—91no—
104#832—31no—
105#832—91no—
106#821—91no—
107#818—91no—
108C8×Dic3—91no—
109C8×Dic3—91no—
110#807—91no—
111C23×C32—21no—
112C22×C3×D3—21no—
113C32×D4—21no—
114C32×D4C2×Dic312yes—
115C2×C32×C4C22×D311yescharacteristic
116C2×C3⋊Dic3—31no—
117C2×C3⋊Dic3—31no—
118C2×C3⋊Dic3—61no—
119C2×C3⋊Dic3—122no—
120C2×C3×Dic3—61no—
121C3×D12—21no—
122C3×D12C2×Dic312yes—
123D3×Dic3—122no—
124C23×C9—22no—
125C23×C9—21no—
126C23×C9—61no—
127C3×C3⋊C8—61no—
128C3×C3⋊C8C2×Dic312yes—
129D4×C9—22no—
130D4×C9—21no—
131D4×C9—22no—
132D4×C9—32no—
133D4×C9—61no—
134D4×C9C4×D312yescomplemented
135C2×C4×C9—21no—
136C2×C4×C9C22×D311yescharacteristic
137C2×Dic9—21no—
138C2×Dic9—31no—
139C2×Dic9—31no—
140C2×Dic9—61no—
141C2×Dic9—61no—
142C2×Dic9—62no—
143C2×Dic9—61no—
144C2×Dic9—122no—
145C2×Dic9—122no—
146C2×Dic9—124no—
147C8×C9—32no—
148C9⋊C8—21no—
149C4×D8—271noSylow
150C2×C3×C9—42no—
151C2×C3×C9C4×D411yescharacteristic
152C2×C3×C9C2×D811yescharacteristic
153C2×C3×C9#13511yescharacteristic
154D3×C9—42no—
155C2×C3×D4—21no—
156C2×C3×D4—31no—
157C2×C3×D4C2×D911yescharacteristic
158C2×C3×D4D3211yescharacteristic
159C22×Dic3—181no—
160C22×Dic3—181no—
161C22×Dic3—181no—
162C22×Dic3—182no—
163C2×D12C2×D911yescharacteristic
164C2×C4×D3—181no—
165C3×D8—34no—
166C2×C3×C8—31no—
167C2×C3×C8—91no—
168C3×C4⋊C4—91no—
169#217—181no—
170#217—181no—
171#217—181no—
172#217—182no—
173#213Dic914yescomplemented
174(C2×D3)⋊C4—181no—
175C4⋊Dic3—31no—
176C4⋊Dic3—91no—
177C4⋊Dic3—91no—
178C4⋊Dic3—181no—
179C4×Dic3—91no—
180C4×Dic3—91no—
181C4×Dic3—181no—
182C2×C3⋊C8—31no—
183C2×C3⋊C8C2×D911yescharacteristic
184C22×C32—22no—
185C22×C32—41no—
186C22×C32—41no—
187C22×C32D3×D411yescharacteristic
188C2×C3×D3—22no—
189C2×C3×D3—41no—
190C2×C3×D3—41no—
191C32×C4#23711yescharacteristic
192C32×C4C22×Dic311yescharacteristic
193C3⋊Dic3—32no—
194C3⋊Dic3—61no—
195C3⋊Dic3—121no—
196C3×Dic3—121no—
197C22×C9—24no—
198C22×C9—22no—
199C22×C9—21no—
200C22×C9—42no—
201C22×C9—42no—
202C22×C9—41no—
203C22×C9—41no—
204C22×C9—62no—
205C22×C9—121no—
206C22×C9—121no—
207C22×C9D3×D411yescharacteristic
208C4×C9—21no—
209C4×C9—21no—
210C4×C9C2×C4×D311yescharacteristic
211C4×C9#23711yescharacteristic
212Dic9—32no—
213Dic9—41no—
214Dic9—62no—
215Dic9—61no—
216Dic9—121no—
217Dic9—122no—
218Dic9—121no—
219C2×D8—31no—
220C4×D4—271no—
221C4×D4—271no—
222#107—271no—
223#102—91no—
224#102—271no—
225C4×C8—271no—
226C3×C9C4×D811yescharacteristic, complemented, Sylow
227C23×C3—22no—
228C23×C3—21no—
229C23×C3—21no—
230C23×C3—61no—
231C22×D3—21no—
232C3×D4—22no—
233C3×D4—21no—
234C3×D4—21no—
235C3×D4—22no—
236C3×D4—32no—
237C3×D4—61no—
238C3×D4C2×Dic912yescomplemented
239C3×D4D3×Dic312yes—
240C2×C3×C4—21no—
241C2×C3×C4—181no—
242C2×C3×C4C22×D911yescharacteristic
243C2×C3×C4C2×D3211yescharacteristic
244C2×Dic3—61no—
245C2×Dic3—91no—
246C2×Dic3—91no—
247C2×Dic3—91no—
248C2×Dic3—91no—
249C2×Dic3—181no—
250C2×Dic3—181no—
251C2×Dic3—182no—
252C2×Dic3—181no—
253C2×Dic3—181no—
254C2×Dic3—362no—
255C2×Dic3—362no—
256C2×Dic3—362no—
257C2×Dic3—364no—
258D12—21no—
259D12C2×Dic912yescomplemented
260C4×D3—362no—
261C3×C8—32no—
262C3×C8—181no—
263C3⋊C8—61no—
264C3⋊C8C2×Dic912yescomplemented
265C2×C32—42no—
266C2×C32D3×D811yescharacteristic
267C2×C32#91211yescharacteristic
268C2×C32D4×Dic311yescharacteristic
269C3×D3—42no—
270C2×C9—21no—
271C2×C9—21no—
272C2×C9—21no—
273C2×C9—44no—
274C2×C9—42no—
275C2×C9—122no—
276C2×C9#67⋊C411yescharacteristic
277C2×C9D3×D811yescharacteristic
278C2×C9#91611yescharacteristic
279C2×D4—31no—
280C2×D4D3×D911yescharacteristic
281C22×C4—541no—
282C22×C4—541no—
283D8—34no—
284C2×C8—31no—
285C2×C8—91no—
286C4⋊C4—91no—
287C4⋊C4—271no—
288C22⋊C4—541no—
289C22⋊C4—541no—
290C42—271no—
291C22×C3—24no—
292C22×C3—22no—
293C22×C3—22no—
294C22×C3—21no—
295C22×C3—42no—
296C22×C3—42no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—62no—
302C22×C3—121no—
303C22×C3—121no—
304C22×C3D4×D911yescharacteristic
305C22×C3#381111yescharacteristic
306C2×D3—22no—
307C2×D3—41no—
308C2×D3—41no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—361no—
312C3×C4#369911yescharacteristic
313C3×C4C22×Dic911yescharacteristic
314C3×C4C2×D3×Dic311yescharacteristic
315C3×C4#380511yescharacteristic
316Dic3—92no—
317Dic3—92no—
318Dic3—121no—
319Dic3—182no—
320Dic3—181no—
321Dic3—181no—
322Dic3—361no—
323Dic3—362no—
324Dic3—361no—
325Dic3—361no—
326C32Dic3×D811yescharacteristic
327C9—21no—
328C9#475611yescharacteristic, complemented
329C23—21no—
330C23—61no—
331D4—21no—
332D4—32no—
333D4—61no—
334D4D3×Dic912yescomplemented
335C2×C4—181no—
336C2×C4—271no—
337C2×C4—271no—
338C2×C4—541no—
339C2×C4—1082no—
340C2×C4—1082no—
341C2×C4C2×D3×D911yescharacteristic
342C8—32no—
343C8—181no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—44no—
348C2×C3—42no—
349C2×C3—42no—
350C2×C3—122no—
351C2×C3D8×D911yescharacteristic
352C2×C3#6347411yescharacteristic
353C2×C3D4×Dic911yescharacteristic
354C2×C3#6392611yescharacteristic
355C2×C3#6393611yescharacteristic
356C2×C3#67⋊Dic311yescharacteristic
357D3—42no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—62no—
362C22—121no—
363C22—121no—
364C22#8866811yescharacteristic
365C4—272no—
366C4—361no—
367C4—541no—
368C4—1081no—
369C4C2×D3×Dic911yescharacteristic
370C4#8866311yescharacteristic
371C3—21no—
372C3D8×Dic911yescharacteristic, complemented
373C3#1059143511yescharacteristic
374C2—42no—
375C2—122no—
376C2#1172117411yescharacteristic
377C2#1172117811yescharacteristic
378C2#67⋊Dic911yescharacteristic
subgroupG/H
centerC22#88668
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
prime radicalC2×C3×#213D3
cosocleC2×C3×C4×C9C23
hypercenterC2×D4D3×D9
hypocenterC3×C9C4×D8
LCSG ▷ 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
1. D4×C3⋊Dic9C2semidirect
3. #11721025C2semidirect
4. C3×#63392C2semidirect
5. #11720295C2semidirect
6. #11720279C2semidirect
7. #11720271C2semidirect
17. C9×#213C4semidirect
34. #63392D3semidirect
40. C3×D4×C9C2×C4semidirect
48. C9×D12C2×C4semidirect
50. C9×C3⋊C8C2×C4semidirect
94. C3⋊Dic9D8semidirect
134. D4×C9C4×D3semidirect
173. #213Dic9semidirect
226. C3×C9C4×D8semidirect
238. C3×D4C2×Dic9semidirect
259. D12C2×Dic9semidirect
264. C3⋊C8C2×Dic9semidirect
328. C9#4756semidirect
334. D4D3×Dic9semidirect
372. C3D8×Dic9semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊Dic92
C2×C9×#2132
#117210252
C3×#633922
#117202952
#117202792
#117202712
#105914353
D8×Dic93