Finite Group Databasebeta

← #13143976

Subgroups — #13143976

Order 1728.

810 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 167 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×#88746C211yesmaximal, characteristic
2C2×#88523C211yesmaximal, characteristic, complemented
3C4⋊C4×C3⋊D9C211yesmaximal, characteristic, complemented
4C3×C9×#103C211yesmaximal, characteristic, complemented
5#11720582C211yesmaximal, characteristic, complemented
6#11720543C211yesmaximal, characteristic, complemented
7#11720528C211yesmaximal, characteristic, complemented
8#10592502—31nomaximal
9#10588483—33nomaximal
10C2×C3×Q8×C9C2211yescharacteristic
11#88746—22no—
12#88746C414yescomplemented
13C2×C4⋊(C3⋊D9)—21no—
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×C4×C3⋊D9—21no—
16C2×C4×C3⋊D9—21no—
17C2×C4×C3⋊D9C2211yescharacteristic
18C2×C3×C8×C9C2211yescharacteristic
19C3×C9×C4⋊C4C2211yescharacteristic
20C4⋊(C3⋊Dic9)C2211yescharacteristic
21#88534—21no—
22C2×C3⋊(C9⋊C8)C2211yescharacteristic
23#88523—22no—
24C2×#3832—31no—
25C2×#3743—31no—
26C4⋊C4×C3⋊D3—31no—
27C2×#3701—33no—
28C32×#103D311yescharacteristic
29#63662—31no—
30#63648—31no—
31#63636—31no—
32C2×#3663—33no—
33C4⋊C4×D9—33no—
34C9×#103D313yescomplemented
35#63395—33no—
36#63380—33no—
37#63369—33no—
38C22×C3⋊D9—21no—
39C22×C3⋊D9D411yescharacteristic
40C3×Q8×C9—21no—
41C3×Q8×C9C2×C412yescomplemented
42C2×C3×C4×C9—21no—
43C2×C3×C4×C9—21no—
44C2×C3×C4×C9C2311yescharacteristic
45C2×C3⋊Dic9—21no—
46C2×C3⋊Dic9D411yescharacteristic
47C4⋊(C3⋊D9)—24no—
48C4⋊(C3⋊D9)—21no—
49C4⋊(C3⋊D9)C2×C412yescomplemented
50C4×C3⋊D9—24no—
51C4×C3⋊D9—42no—
52C4×C3⋊D9C2×C412yes—
53C4×C3⋊D9D412yes—
54C3×C8×C9—21no—
55C3⋊(C9⋊C8)—21no—
56#4803—93no—
57#4803—91no—
58C2×C32×Q8C2×D311yescharacteristic
59#3832—34no—
60#3832—62no—
61C2×C4⋊(C3⋊D3)—31no—
62C2×C4⋊(C3⋊D3)—61no—
63C2×C4×C3⋊D3—31no—
64C2×C4×C3⋊D3—61no—
65C2×C4×C3⋊D3—61no—
66C2×C32×C8C2×D311yescharacteristic
67C32×C4⋊C4C2×D311yescharacteristic
68C4⋊(C3⋊Dic3)—31no—
69#3750—61no—
70C2×C3⋊(C3⋊C8)—31no—
71#3743—62no—
72C2×Q8×C9C2×D313yes—
73#3701—312no—
74#3701—66no—
75C2×D36—33no—
76C2×D36—63no—
77C2×C4×D9—33no—
78C2×C4×D9—63no—
79C2×C4×D9—63no—
80C2×C8×C9C2×D313yes—
81C9×C4⋊C4C2×D313yes—
82C4⋊Dic9—33no—
83Dic9⋊C4—63no—
84C2×C9⋊C8—33no—
85#3663—66no—
86C22×C3×C9C2×D411yescharacteristic
87C2×C3⋊D9—21no—
88C2×C3⋊D9—21no—
89C2×C3⋊D9—22no—
90C2×C3⋊D9—41no—
91C2×C3⋊D9—41no—
92C2×C3⋊D9C22⋊C412yes—
93C3×C4×C9—22no—
94C3×C4×C9—41no—
95C3×C4×C9C22×C411yescharacteristic
96C3×C4×C9C2×D411yescharacteristic
97C3⋊Dic9—41no—
98C3⋊Dic9C22⋊C412yes—
99C2×#239—93no—
100C2×#239—91no—
101C2×ΓL(1,𝔽25)—93no—
102C2×ΓL(1,𝔽25)—91no—
103D3×C4⋊C4—93no—
104D3×C4⋊C4—91no—
105C3×#103D913yescomplemented
106C3×#103C3⋊D311yescharacteristic
107#835—93no—
108#835—91no—
109#821—93no—
110#821—91no—
111#809—93no—
112#809—91no—
113C22×C3⋊D3—31no—
114C22×C3⋊D3—61no—
115C32×Q8—21no—
116C32×Q8C4×D312yes—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4C22×D311yescharacteristic
120C2×C3⋊Dic3—31no—
121C2×C3⋊Dic3—61no—
122C4⋊(C3⋊D3)—32no—
123C4⋊(C3⋊D3)—64no—
124C4⋊(C3⋊D3)—61no—
125C4×C3⋊D3—32no—
126C4×C3⋊D3—32no—
127C4×C3⋊D3—64no—
128C4×C3⋊D3—122no—
129C22×D9—33no—
130C22×D9—63no—
131C32×C8—21no—
132C3⋊(C3⋊C8)—61no—
133Q8×C9—23no—
134Q8×C9C4×D316yescomplemented
135C2×C4×C9—23no—
136C2×C4×C9—23no—
137C2×C4×C9C22×D313yes—
138C2×Dic9—33no—
139C2×Dic9—63no—
140D36—36no—
141D36—612no—
142D36—63no—
143C4×D9—36no—
144C4×D9—36no—
145C4×D9—612no—
146C4×D9—126no—
147C8×C9—23no—
148C9⋊C8—63no—
149#417—271noSylow
150C2×C3×C9C2×C22⋊C411yescharacteristic
151C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
152C2×C3×C9#13711yescharacteristic
153C3⋊D9—22no—
154C3⋊D9—42no—
155C2×C3×Q8C2×D913yes—
156C2×C3×Q8C2×C3⋊D311yescharacteristic
157#239—912no—
158#239—94no—
159#239—182no—
160#239—186no—
161C2×D12—93no—
162C2×D12—91no—
163C2×D12—181no—
164C2×D12—183no—
165C2×C4×D3—93no—
166C2×C4×D3—91no—
167C2×C4×D3—181no—
168C2×C4×D3—183no—
169C2×C4×D3—181no—
170C2×C4×D3—183no—
171C2×C3×C8C2×D913yes—
172C2×C3×C8C2×C3⋊D311yescharacteristic
173C3×C4⋊C4C2×D913yes—
174C3×C4⋊C4C2×C3⋊D311yescharacteristic
175C4⋊Dic3—93no—
176C4⋊Dic3—91no—
177Dic3⋊C4—181no—
178Dic3⋊C4—183no—
179C2×C3⋊C8—93no—
180C2×C3⋊C8—91no—
181ΓL(1,𝔽25)—182no—
182ΓL(1,𝔽25)—186no—
183C22×C32D3×D411yescharacteristic
184C2×C3⋊D3—32no—
185C2×C3⋊D3—61no—
186C2×C3⋊D3—61no—
187C2×C3⋊D3—62no—
188C2×C3⋊D3—121no—
189C2×C3⋊D3—121no—
190C32×C4—22no—
191C32×C4—41no—
192C32×C4C2×C4×D311yescharacteristic
193C32×C4D3×D411yescharacteristic
194C3⋊Dic3—32no—
195C3⋊Dic3—121no—
196C22×C9D3×D413yes—
197C2×D9—36no—
198C2×D9—63no—
199C2×D9—63no—
200C2×D9—66no—
201C2×D9—123no—
202C2×D9—123no—
203C4×C9—26no—
204C4×C9—43no—
205C4×C9C2×C4×D313yes—
206C4×C9D3×D413yes—
207Dic9—36no—
208Dic9—123no—
209C2×#41—271no—
210C2×M16—271no—
211C2×C4⋊C4—271no—
212#103—271no—
213#103C3⋊D911yescharacteristic, complemented
214#102—271no—
215#102—271no—
216C3×C9#41711yescharacteristic, complemented, Sylow
217C22×D3—93no—
218C22×D3—91no—
219C22×D3—183no—
220C22×D3—181no—
221C3×Q8—23no—
222C3×Q8—21no—
223C3×Q8C4×D916yescomplemented
224C3×Q8C4×C3⋊D312yes—
225C2×C3×C4—23no—
226C2×C3×C4—21no—
227C2×C3×C4—23no—
228C2×C3×C4—21no—
229C2×C3×C4C22×D913yes—
230C2×C3×C4C22×C3⋊D311yescharacteristic
231C2×Dic3—93no—
232C2×Dic3—91no—
233C2×Dic3—183no—
234C2×Dic3—181no—
235D12—96no—
236D12—92no—
237D12—184no—
238D12—1812no—
239D12—183no—
240D12—181no—
241C4×D3—96no—
242C4×D3—96no—
243C4×D3—92no—
244C4×D3—92no—
245C4×D3—184no—
246C4×D3—1812no—
247C4×D3—362no—
248C4×D3—366no—
249C3×C8—23no—
250C3×C8—21no—
251C3⋊C8—183no—
252C3⋊C8—181no—
253C2×C32D3×C22⋊C411yescharacteristic
254C2×C32#91411yescharacteristic
255C2×C32#91811yescharacteristic
256C3⋊D3—62no—
257C3⋊D3—122no—
258C2×C9D3×C22⋊C413yes—
259C2×C9#91413yes—
260C2×C9#91813yes—
261D9—66no—
262D9—126no—
263#41—274no—
264#41—542no—
265C2×Q8C2×C3⋊D911yescharacteristic
266C2×D4—271no—
267C2×D4—541no—
268C22×C4—271no—
269C22×C4—541no—
270C22×C4—541no—
271M16—542no—
272C2×C8—271no—
273C2×C8C2×C3⋊D911yescharacteristic
274C4⋊C4—271no—
275C4⋊C4—541no—
276C4⋊C4C2×C3⋊D911yescharacteristic
277C22×C3D4×D913yes—
278C22×C3D4×C3⋊D311yescharacteristic
279C2×D3—92no—
280C2×D3—96no—
281C2×D3—181no—
282C2×D3—181no—
283C2×D3—183no—
284C2×D3—183no—
285C2×D3—182no—
286C2×D3—186no—
287C2×D3—361no—
288C2×D3—361no—
289C2×D3—363no—
290C2×D3—363no—
291C3×C4—22no—
292C3×C4—26no—
293C3×C4—41no—
294C3×C4—43no—
295C3×C4C2×C4×D913yes—
296C3×C4D4×D913yes—
297C3×C4C2×C4×C3⋊D311yescharacteristic
298C3×C4D4×C3⋊D311yescharacteristic
299Dic3—92no—
300Dic3—96no—
301Dic3—361no—
302Dic3—363no—
303C32#480311yescharacteristic
304C9#480313yescomplemented
305C23—271no—
306C23—541no—
307Q8—21no—
308Q8C4×C3⋊D912yescomplemented
309D4—272no—
310D4—544no—
311D4—541no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—271no—
315C2×C4—272no—
316C2×C4—272no—
317C2×C4—544no—
318C2×C4—541no—
319C2×C4—1082no—
320C2×C4C22×C3⋊D911yescharacteristic
321C8—21no—
322C8—541no—
323C2×C3C22⋊C4×D913yes—
324C2×C3#6347613yes—
325C2×C3#6348013yes—
326C2×C3C22⋊C4×C3⋊D311yescharacteristic
327C2×C3#6412311yescharacteristic
328C2×C3#6412711yescharacteristic
329D3—186no—
330D3—182no—
331D3—366no—
332D3—362no—
333C22—272no—
334C22—541no—
335C22—541no—
336C22—542no—
337C22—1081no—
338C22—1081no—
339C22D4×C3⋊D911yescharacteristic
340C4—22no—
341C4—41no—
342C4—272no—
343C4—1081no—
344C4C2×C4×C3⋊D911yescharacteristic
345C4D4×C3⋊D911yescharacteristic
346C3#1058848313yescomplemented
347C3#1059250211yescharacteristic
348C2—542no—
349C2—1082no—
350C2C22⋊C4×C3⋊D911yescharacteristic
351C2#1172179611yescharacteristic
352C2#1172180411yescharacteristic
subgroupG/H
centerC22D4×C3⋊D9
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingC3×C9×#103C2
socleC22×C32D3×D4
prime radicalC2×#88746C2
cosocleC2×C3×C4×C9C23
hypercenter#103C3⋊D9
hypocenterC3×C9#417
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
2. C2×#88523C2semidirect
3. C4⋊C4×C3⋊D9C2semidirect
4. C3×C9×#103C2semidirect
5. #11720582C2semidirect
6. #11720543C2semidirect
7. #11720528C2semidirect
12. #88746C4semidirect
34. C9×#103D3semidirect
41. C3×Q8×C9C2×C4semidirect
49. C4⋊(C3⋊D9)C2×C4semidirect
105. C3×#103D9semidirect
134. Q8×C9C4×D3semidirect
213. #103C3⋊D9semidirect
216. C3×C9#417semidirect
223. C3×Q8C4×D9semidirect
304. C9#4803semidirect
308. Q8C4×C3⋊D9semidirect
346. C3#10588483semidirect

Maximal subgroups

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

subgroupindex
C2×#887462
C2×#885232
C4⋊C4×C3⋊D92
C3×C9×#1032
#117205822
#117205432
#117205282
#105925023
#105884833