Finite Group Databasebeta

← #13140711

Subgroups — #13140711

Order 1728.

528 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 121 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×#3701C211yesmaximal, characteristic
2D9×C4⋊Dic3C211yesmaximal, characteristic, complemented
3C2×#88413C211yesmaximal, characteristic, complemented
4#11720582C211yesmaximal, characteristic, complemented
5C9×#835C211yesmaximal, characteristic, complemented
6#11720283C211yesmaximal, characteristic, complemented
7#11720266C211yesmaximal, characteristic, complemented
8#10591550—31nomaximal
9#10588483—31nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11C3×#3701—22no—
12C3×#3701C414yescomplemented
13C2×C3×D36—21no—
14C2×C3×D36C2211yescharacteristic
15C2×C3×C4×D9—21no—
16C2×C3×C4×D9C2211yescharacteristic
17C2×Dic3×D9—21no—
18C4⋊(C3⋊Dic9)C2211yescharacteristic
19C2×C3⋊(C9⋊C8)C2211yescharacteristic
20C9×C4⋊Dic3C2211yescharacteristic
21C2×C9×C3⋊C8C2211yescharacteristic
22Dic9⋊Dic3—21no—
23#88413—22no—
24C2×C3×#239—31no—
25D3×C4⋊Dic3—31no—
26C2×#3711—31no—
27C2×#3701D311yescharacteristic
28#63662—31no—
29C3×#835D311yescharacteristic
30#63563—31no—
31#63562—31no—
32C2×#3663—31no—
33C4⋊C4×D9—31no—
34C9×#103—31no—
35#63395—31no—
36#63395—61no—
37#63380—31no—
38#63369—31no—
39C22×C3×D9—21no—
40C22×C3×D9D411yescharacteristic
41C3×Q8×C9—21no—
42C3×Q8×C9C2×C412yescomplemented
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9C2311yescharacteristic
45C2×C3⋊Dic9—21no—
46C2×C9×Dic3—21no—
47C2×C3×Dic9D411yescharacteristic
48C3×D36—24no—
49C3×D36—21no—
50C3×D36C2×C412yescomplemented
51C3×C4×D9—24no—
52C3×C4×D9C2×C412yes—
53C3×C4×D9D412yes—
54Dic3×D9—42no—
55C3⋊(C9⋊C8)—21no—
56C9×C3⋊C8—21no—
57#5232—91no—
58#4803—91no—
59C2×C32×Q8C2×D311yescharacteristic
60C3×#239—34no—
61C3×#239—62no—
62C2×C3×D12—31no—
63C2×C3×D12—61no—
64C2×C3×C4×D3—31no—
65C2×C3×C4×D3—61no—
66C2×D3×Dic3—61no—
67C4⋊(C3⋊Dic3)—31no—
68C2×C3⋊(C3⋊C8)—31no—
69C3×C4⋊Dic3C2×D311yescharacteristic
70C2×C3×C3⋊C8C2×D311yescharacteristic
71Dic3⋊Dic3—61no—
72#3711—62no—
73C2×Q8×C9—21no—
74C2×Q8×C9C2×D311yescharacteristic
75#3701—22no—
76#3701Dic314yescomplemented
77C2×D36—21no—
78C2×D36C2×D311yescharacteristic
79C2×C4×D9—21no—
80C2×C4×D9—61no—
81C2×C4×D9C2×D311yescharacteristic
82C2×C8×C9—31no—
83C9×C4⋊C4—31no—
84C4⋊Dic9—31no—
85C4⋊Dic9—61no—
86Dic9⋊C4—61no—
87C2×C9⋊C8—31no—
88C2×C9⋊C8—61no—
89#3663—62no—
90C22×C3×C9C2×D411yescharacteristic
91C2×C3×D9—21no—
92C2×C3×D9—21no—
93C2×C3×D9—22no—
94C2×C3×D9—41no—
95C2×C3×D9—41no—
96C2×C3×D9C22⋊C412yes—
97C3×C4×C9—22no—
98C3×C4×C9C22×C411yescharacteristic
99C3×C4×C9C2×D411yescharacteristic
100C3⋊Dic9—41no—
101C9×Dic3—41no—
102C3×Dic9C22⋊C412yes—
103C2×C3×#41—91no—
104C2×#239—31no—
105C2×C4⋊Dic3—91no—
106C2×#208—91no—
107C2×ΓL(1,𝔽25)—91no—
108D3×C4⋊C4—91no—
109C3×#103—31no—
110#835—91no—
111#835—91no—
112#835—181no—
113#835D911yescharacteristic, complemented
114#832—91no—
115#832—91no—
116#821—91no—
117#809—91no—
118C22×C3×D3—31no—
119C22×C3×D3—61no—
120C32×Q8—21no—
121C32×Q8C4×D312yes—
122C2×C32×C4—21no—
123C2×C32×C4C22×D311yescharacteristic
124C2×C3⋊Dic3—61no—
125C2×C3×Dic3—21no—
126C2×C3×Dic3—31no—
127C3×D12—32no—
128C3×D12—64no—
129C3×D12—61no—
130C3×C4×D3—32no—
131C3×C4×D3—32no—
132C3×C4×D3—64no—
133D3×Dic3—122no—
134C22×D9—21no—
135C22×D9#6711yescharacteristic
136C3⋊(C3⋊C8)—61no—
137C3×C3⋊C8—21no—
138Q8×C9—21no—
139Q8×C9—22no—
140Q8×C9—41no—
141Q8×C9C2×Dic312yescomplemented
142C2×C4×C9—21no—
143C2×C4×C9—21no—
144C2×C4×C9—41no—
145C2×C4×C9—61no—
146C2×C4×C9C22×D311yescharacteristic
147C2×Dic9—61no—
148C2×Dic9—121no—
149C2×Dic9#6711yescharacteristic
150D36—24no—
151D36—21no—
152D36C2×Dic312yescomplemented
153C4×D9—24no—
154C4×D9—122no—
155C4×D9C2×Dic312yes—
156C4×D9#6712yes—
157C8×C9—61no—
158C9⋊C8—61no—
159C9⋊C8—121no—
160#417—271noSylow
161C2×C3×C9C2×C22⋊C411yescharacteristic
162C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
163C2×C3×C9#13711yescharacteristic
164C3×D9—22no—
165C3×D9—42no—
166C3×#41—94no—
167C3×#41—182no—
168C2×C3×Q8—21no—
169C2×C3×Q8C2×D911yescharacteristic
170C2×C3×Q8D3211yescharacteristic
171C2×C3×D4—91no—
172C2×C3×D4—181no—
173C22×C3×C4—91no—
174C22×C3×C4—181no—
175C22×Dic3—181no—
176#239—34no—
177#239—62no—
178C2×D12—31no—
179C2×D12—61no—
180C2×C4×D3—31no—
181C2×C4×D3—61no—
182C2×C4×D3—181no—
183C2×C3×C8—31no—
184C3×C4⋊C4—31no—
185C4⋊Dic3—91no—
186C4⋊Dic3—91no—
187C4⋊Dic3—181no—
188C4⋊Dic3—181no—
189C4⋊Dic3C2×D911yescharacteristic
190Dic3⋊C4—181no—
191#208—182no—
192C2×C3⋊C8—91no—
193C2×C3⋊C8—91no—
194C2×C3⋊C8—181no—
195C2×C3⋊C8C2×D911yescharacteristic
196ΓL(1,𝔽25)—182no—
197C22×C32D3×D411yescharacteristic
198C2×C3×D3—32no—
199C2×C3×D3—61no—
200C2×C3×D3—61no—
201C2×C3×D3—62no—
202C2×C3×D3—121no—
203C2×C3×D3—121no—
204C32×C4—22no—
205C32×C4C2×C4×D311yescharacteristic
206C32×C4D3×D411yescharacteristic
207C3⋊Dic3—121no—
208C3×Dic3—32no—
209C3×Dic3—41no—
210C22×C9—21no—
211C22×C9C2×#6711yescharacteristic
212C2×D9—21no—
213C2×D9—21no—
214C2×D9—22no—
215C2×D9—41no—
216C2×D9—41no—
217C2×D9#21712yes—
218C4×C9—22no—
219C4×C9—21no—
220C4×C9—21no—
221C4×C9—42no—
222C4×C9—121no—
223C4×C9C22×Dic311yescharacteristic
224C4×C9C2×#6711yescharacteristic
225Dic9—121no—
226Dic9—241no—
227Dic9#21712yes—
228C2×#41—91no—
229C2×M16—271no—
230C2×C4⋊C4—271no—
231#103—31no—
232#103—271no—
233#102—271no—
234#102—271no—
235C3×C9#41711yescharacteristic, complemented, Sylow
236C23×C3—91no—
237C23×C3—181no—
238C22×D3—31no—
239C22×D3—61no—
240C3×Q8—21no—
241C3×Q8—21no—
242C3×Q8—22no—
243C3×Q8—41no—
244C3×Q8C4×D912yescomplemented
245C3×Q8D3×Dic312yes—
246C3×D4—92no—
247C3×D4—184no—
248C3×D4—181no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—41no—
253C2×C3×C4—61no—
254C2×C3×C4—91no—
255C2×C3×C4—92no—
256C2×C3×C4—92no—
257C2×C3×C4—184no—
258C2×C3×C4C22×D911yescharacteristic
259C2×C3×C4C2×D3211yescharacteristic
260C2×Dic3—21no—
261C2×Dic3—31no—
262C2×Dic3—181no—
263C2×Dic3—181no—
264C2×Dic3—361no—
265C2×Dic3—362no—
266D12—32no—
267D12—64no—
268D12—61no—
269C4×D3—32no—
270C4×D3—32no—
271C4×D3—64no—
272C4×D3—362no—
273C3×C8—61no—
274C3⋊C8—21no—
275C3⋊C8—181no—
276C3⋊C8—181no—
277C3⋊C8—361no—
278C2×C32D3×C22⋊C411yescharacteristic
279C2×C32#91411yescharacteristic
280C2×C32#91811yescharacteristic
281C3×D3—62no—
282C3×D3—122no—
283C2×C9—21no—
284C2×C9—21no—
285C2×C9—21no—
286C2×C9#94911yescharacteristic
287C2×C9#95111yescharacteristic
288C2×C9C2×#21711yescharacteristic
289D9—22no—
290D9—42no—
291#41—94no—
292#41—182no—
293C2×Q8D3×D911yescharacteristic
294C2×D4—91no—
295C2×D4—181no—
296C22×C4—91no—
297C22×C4—181no—
298C22×C4—541no—
299M16—542no—
300C2×C8—31no—
301C2×C8—271no—
302C4⋊C4—31no—
303C4⋊C4—271no—
304C4⋊C4—541no—
305C22×C3—21no—
306C22×C3—92no—
307C22×C3—181no—
308C22×C3—181no—
309C22×C3—182no—
310C22×C3—361no—
311C22×C3—361no—
312C22×C3D4×D911yescharacteristic
313C22×C3D3×#6711yescharacteristic
314C2×D3—32no—
315C2×D3—61no—
316C2×D3—61no—
317C2×D3—62no—
318C2×D3—121no—
319C2×D3—121no—
320C3×C4—22no—
321C3×C4—22no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—42no—
325C3×C4—92no—
326C3×C4—121no—
327C3×C4C2×C4×D911yescharacteristic
328C3×C4D4×D911yescharacteristic
329C3×C4C2×D3×Dic311yescharacteristic
330C3×C4D3×#6711yescharacteristic
331Dic3—32no—
332Dic3—41no—
333Dic3—361no—
334Dic3—361no—
335Dic3—721no—
336C32#480311yescharacteristic
337C9—21no—
338C9#523211yescharacteristic, complemented
339C23—91no—
340C23—181no—
341Q8—21no—
342Q8Dic3×D912yescomplemented
343D4—92no—
344D4—184no—
345D4—181no—
346C2×C4—21no—
347C2×C4—61no—
348C2×C4—91no—
349C2×C4—92no—
350C2×C4—92no—
351C2×C4—184no—
352C2×C4—541no—
353C2×C4—1082no—
354C2×C4C2×D3×D911yescharacteristic
355C8—61no—
356C8—541no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—182no—
361C2×C3—362no—
362C2×C3C22⋊C4×D911yescharacteristic
363C2×C3#6347611yescharacteristic
364C2×C3#6348011yescharacteristic
365C2×C3#6393911yescharacteristic
366C2×C3#6394311yescharacteristic
367C2×C3D3×#21711yescharacteristic
368D3—62no—
369D3—122no—
370C22—92no—
371C22—181no—
372C22—181no—
373C22—182no—
374C22—361no—
375C22—361no—
376C22D9×#6711yescharacteristic
377C4—22no—
378C4—92no—
379C4—121no—
380C4—1081no—
381C4C2×Dic3×D911yescharacteristic
382C4D9×#6711yescharacteristic
383C3—21no—
384C3#1058848311yescharacteristic, complemented
385C3#1059155011yescharacteristic
386C2—182no—
387C2—362no—
388C2#1172121111yescharacteristic
389C2#1172121911yescharacteristic
390C2D9×#21711yescharacteristic
subgroupG/H
centerC22D9×#67
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C3×Q8×C9C22
socleC22×C32D3×D4
prime radicalC2×C3×#3701C2
cosocleC2×C3×C4×C9C23
hypercenterC2×Q8D3×D9
hypocenterC3×C9#417
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. D9×C4⋊Dic3C2semidirect
3. C2×#88413C2semidirect
4. #11720582C2semidirect
5. C9×#835C2semidirect
6. #11720283C2semidirect
7. #11720266C2semidirect
12. C3×#3701C4semidirect
42. C3×Q8×C9C2×C4semidirect
50. C3×D36C2×C4semidirect
76. #3701Dic3semidirect
113. #835D9semidirect
141. Q8×C9C2×Dic3semidirect
152. D36C2×Dic3semidirect
235. C3×C9#417semidirect
244. C3×Q8C4×D9semidirect
338. C9#5232semidirect
342. Q8Dic3×D9semidirect
384. C3#10588483semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37012
D9×C4⋊Dic32
C2×#884132
#117205822
C9×#8352
#117202832
#117202662
#105915503
#105884833