Finite Group Databasebeta

← #13144850

Subgroups — #13144850

Order 1728.

770 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C9×#41C211yesmaximal, characteristic
2#11721815C211yesmaximal, characteristic, complemented
3C2×#88532C211yesmaximal, characteristic, complemented
4#11720585C214yesmaximal, complemented
5#10592935—31nomaximal
6#10588914—33nomaximal
7C3×C9×#41—21no—
8C3×C9×#41—21no—
9C3×C9×#41C412yescomplemented
10C3×C9×#41C2212yes—
11C2×C3×Q8×C9C411yescharacteristic, complemented
12C2×C3×D4×C9—21no—
13C2×C3×D4×C9C411yescharacteristic, complemented
14C22×C3×C4×C9—21no—
15C22×C3×C4×C9C2211yescharacteristic
16#88551—21no—
17C4⋊(C3⋊Dic9)—21no—
18C4×C3⋊Dic9C2212yescomplemented
19#88532—21no—
20#88532C2212yescomplemented
21C2×C3⋊(C9⋊C8)—21no—
22C2×C32×#41D311yescharacteristic
23#64135—31no—
24C2×#3748—31no—
25C2×C9×#41D313yes—
26#63664—34no—
27#63488—33no—
28C2×#3667—33no—
29#63396—312no—
30C23×C3×C9—21no—
31C23×C3×C9D411yescharacteristic
32C3×Q8×C9—21no—
33C3×Q8×C9C2×C412yes—
34C3×D4×C9—22no—
35C3×D4×C9—22no—
36C3×D4×C9—21no—
37C3×D4×C9C2×C412yes—
38C2×C3×C4×C9—21no—
39C2×C3×C4×C9—22no—
40C2×C3×C4×C9—22no—
41C2×C3×C4×C9C2×C411yescharacteristic
42C2×C3×C4×C9C2×C411yescharacteristic
43C2×C3×C4×C9D411yescharacteristic
44C2×C3×C4×C9D411yescharacteristic
45C2×C3×C4×C9D411yescharacteristic
46C2×C3×C4×C9C2311yescharacteristic
47C2×C3⋊Dic9—22no—
48C3⋊(C9⋊C8)—22no—
49#5236—93no—
50#5236—91no—
51C32×#41—21no—
52C32×#41—21no—
53C32×#41Dic312yes—
54C32×#41C2×D312yes—
55C2×C32×Q8Dic311yescharacteristic
56C2×C32×D4—21no—
57C2×C32×D4Dic311yescharacteristic
58C22×C32×C4—21no—
59C22×C32×C4C2×D311yescharacteristic
60#3757—61no—
61C4⋊(C3⋊Dic3)—61no—
62C4×C3⋊Dic3—32no—
63#3748—32no—
64#3748—61no—
65C2×C3⋊(C3⋊C8)—61no—
66C9×#41—23no—
67C9×#41—23no—
68C9×#41Dic316yescomplemented
69C9×#41C2×D316yes—
70C2×Q8×C9Dic313yescomplemented
71C2×D4×C9—23no—
72C2×D4×C9Dic313yescomplemented
73C22×C4×C9—23no—
74C22×C4×C9C2×D313yes—
75#3676—63no—
76C4⋊Dic9—63no—
77C4×Dic9—36no—
78#3667—36no—
79#3667—63no—
80C2×C9⋊C8—63no—
81C22×C3×C9—22no—
82C22×C3×C9—41no—
83C22×C3×C9—42no—
84C22×C3×C9C22⋊C411yescharacteristic
85C22×C3×C9C22⋊C411yescharacteristic
86C22×C3×C9C2×D411yescharacteristic
87C3×C4×C9—22no—
88C3×C4×C9C22⋊C411yescharacteristic
89C3×C4×C9C22⋊C411yescharacteristic
90C3×C4×C9C22×C411yescharacteristic
91C3×C4×C9C2×D411yescharacteristic
92C3⋊Dic9—42no—
93C2×C3×#41D913yes—
94C2×C3×#41C3⋊D311yescharacteristic
95#926—93no—
96#926—91no—
97C2×#208—93no—
98C2×#208—91no—
99#837—912no—
100#837—94no—
101C23×C32—21no—
102C23×C32#6711yescharacteristic
103C32×Q8—21no—
104C32×Q8C2×Dic312yes—
105C32×D4—22no—
106C32×D4—22no—
107C32×D4—21no—
108C32×D4C2×Dic312yes—
109C2×C32×C4—21no—
110C2×C32×C4—22no—
111C2×C32×C4—22no—
112C2×C32×C4C2×Dic311yescharacteristic
113C2×C32×C4C2×Dic311yescharacteristic
114C2×C32×C4#6711yescharacteristic
115C2×C32×C4#6711yescharacteristic
116C2×C32×C4#6711yescharacteristic
117C2×C32×C4C22×D311yescharacteristic
118C2×C3⋊Dic3—62no—
119C23×C9—23no—
120C23×C9#6713yes—
121C3⋊(C3⋊C8)—62no—
122Q8×C9—23no—
123Q8×C9C2×Dic316yes—
124D4×C9—26no—
125D4×C9—26no—
126D4×C9—23no—
127D4×C9C2×Dic316yes—
128C2×C4×C9—23no—
129C2×C4×C9—26no—
130C2×C4×C9—26no—
131C2×C4×C9C2×Dic313yes—
132C2×C4×C9C2×Dic313yes—
133C2×C4×C9#6713yes—
134C2×C4×C9#6713yes—
135C2×C4×C9#6713yes—
136C2×C4×C9C22×D313yes—
137C2×Dic9—66no—
138C9⋊C8—66no—
139#421—271noSylow
140C2×C3×C9—21no—
141C2×C3×C9—21no—
142C2×C3×C9—21no—
143C2×C3×C9—42no—
144C2×C3×C9C2×C22⋊C411yescharacteristic
145C3×#41—23no—
146C3×#41—23no—
147C3×#41—21no—
148C3×#41—21no—
149C3×#41Dic916yescomplemented
150C3×#41C2×D916yes—
151C3×#41C3⋊Dic312yes—
152C3×#41C2×C3⋊D312yes—
153C2×C3×Q8Dic913yescomplemented
154C2×C3×Q8C3⋊Dic311yescharacteristic
155C2×C3×D4—23no—
156C2×C3×D4—21no—
157C2×C3×D4Dic913yescomplemented
158C2×C3×D4C3⋊Dic311yescharacteristic
159C22×C3×C4—23no—
160C22×C3×C4—21no—
161C22×C3×C4C2×D913yes—
162C22×C3×C4C2×C3⋊D311yescharacteristic
163#217—183no—
164#217—181no—
165C4⋊Dic3—183no—
166C4⋊Dic3—181no—
167C4×Dic3—96no—
168C4×Dic3—92no—
169#208—96no—
170#208—92no—
171#208—183no—
172#208—181no—
173C2×C3⋊C8—183no—
174C2×C3⋊C8—181no—
175C22×C32—22no—
176C22×C32—41no—
177C22×C32—42no—
178C22×C32#21711yescharacteristic
179C22×C32#21711yescharacteristic
180C22×C32C2×#6711yescharacteristic
181C32×C4—22no—
182C32×C4#21711yescharacteristic
183C32×C4#21711yescharacteristic
184C32×C4C22×Dic311yescharacteristic
185C32×C4C2×#6711yescharacteristic
186C3⋊Dic3—122no—
187C22×C9—26no—
188C22×C9—43no—
189C22×C9—46no—
190C22×C9#21713yes—
191C22×C9#21713yes—
192C22×C9C2×#6713yes—
193C4×C9—26no—
194C4×C9#21713yes—
195C4×C9#21713yes—
196C4×C9C22×Dic313yes—
197C4×C9C2×#6713yes—
198Dic9—126no—
199C2×#41C3⋊D911yescharacteristic
200C2×M16—271no—
201#117—271no—
202#104—274no—
203C3×C9#42111yescharacteristic, complemented, Sylow
204C23×C3—23no—
205C23×C3—21no—
206C23×C3#61113yes—
207C23×C3#63811yescharacteristic
208C3×Q8—23no—
209C3×Q8—21no—
210C3×Q8C2×Dic916yes—
211C3×Q8C2×C3⋊Dic312yes—
212C3×D4—26no—
213C3×D4—26no—
214C3×D4—22no—
215C3×D4—22no—
216C3×D4—23no—
217C3×D4—21no—
218C3×D4C2×Dic916yes—
219C3×D4C2×C3⋊Dic312yes—
220C2×C3×C4—23no—
221C2×C3×C4—21no—
222C2×C3×C4—26no—
223C2×C3×C4—26no—
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4C2×Dic913yes—
227C2×C3×C4C2×Dic913yes—
228C2×C3×C4#61113yes—
229C2×C3×C4#61113yes—
230C2×C3×C4#61113yes—
231C2×C3×C4C22×D913yes—
232C2×C3×C4C2×C3⋊Dic311yescharacteristic
233C2×C3×C4C2×C3⋊Dic311yescharacteristic
234C2×C3×C4#63811yescharacteristic
235C2×C3×C4#63811yescharacteristic
236C2×C3×C4#63811yescharacteristic
237C2×C3×C4C22×C3⋊D311yescharacteristic
238C2×Dic3—186no—
239C2×Dic3—182no—
240C3⋊C8—186no—
241C3⋊C8—182no—
242C2×C32—21no—
243C2×C32—21no—
244C2×C32—21no—
245C2×C32—42no—
246C2×C32C2×#21711yescharacteristic
247C2×C9—23no—
248C2×C9—23no—
249C2×C9—23no—
250C2×C9—46no—
251C2×C9C2×#21713yes—
252#41—21no—
253#41—21no—
254#41C3⋊Dic912yescomplemented
255#41C2×C3⋊D912yes—
256C2×Q8C3⋊Dic911yescharacteristic, complemented
257C2×D4—21no—
258C2×D4C3⋊Dic911yescharacteristic, complemented
259C22×C4—21no—
260C22×C4C2×C3⋊D911yescharacteristic
261M16—272no—
262M16—541no—
263C2×C8—541no—
264C4⋊C4—541no—
265C22⋊C4—541no—
266C42—272no—
267C22×C3—26no—
268C22×C3—22no—
269C22×C3—43no—
270C22×C3—41no—
271C22×C3—42no—
272C22×C3—46no—
273C22×C3#367613yes—
274C22×C3#367613yes—
275C22×C3C2×#61113yes—
276C22×C3#375711yescharacteristic
277C22×C3#375711yescharacteristic
278C22×C3C2×#63811yescharacteristic
279C3×C4—26no—
280C3×C4—22no—
281C3×C4#367613yes—
282C3×C4#367613yes—
283C3×C4C22×Dic913yes—
284C3×C4C2×#61113yes—
285C3×C4#375711yescharacteristic
286C3×C4#375711yescharacteristic
287C3×C4C22×C3⋊Dic311yescharacteristic
288C3×C4C2×#63811yescharacteristic
289Dic3—362no—
290Dic3—366no—
291C32#523611yescharacteristic
292C9#523613yescomplemented
293C23—21no—
294C23#623311yescharacteristic
295Q8—21no—
296Q8C2×C3⋊Dic912yes—
297D4—22no—
298D4—22no—
299D4—21no—
300D4C2×C3⋊Dic912yes—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—542no—
305C2×C4C2×C3⋊Dic911yescharacteristic
306C2×C4C2×C3⋊Dic911yescharacteristic
307C2×C4#623311yescharacteristic
308C2×C4#623311yescharacteristic
309C2×C4#623311yescharacteristic
310C2×C4C22×C3⋊D911yescharacteristic
311C8—542no—
312C2×C3—21no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—23no—
316C2×C3—23no—
317C2×C3—23no—
318C2×C3—42no—
319C2×C3—46no—
320C2×C3C2×#367613yes—
321C2×C3C2×#375711yescharacteristic
322C22—22no—
323C22—41no—
324C22—42no—
325C22#8855111yescharacteristic
326C22#8855111yescharacteristic
327C22C2×#623311yescharacteristic
328C4—22no—
329C4—1082no—
330C4#8855111yescharacteristic
331C4#8855111yescharacteristic
332C4C22×C3⋊Dic911yescharacteristic
333C4C2×#623311yescharacteristic
334C3#1058891413yescomplemented
335C3#1059293511yescharacteristic
336C2—21no—
337C2—21no—
338C2—21no—
339C2—42no—
340C2C2×#8855111yescharacteristic
subgroupG/H
centerC4C2×#6233
commutatorC3×C4×C9C22×C4
FrattiniC2×C3×C4C22×C3⋊D3
FittingC2×C3×C9×#41C2
socleC2×C32C2×#217
prime radicalC2×C32×#41D3
cosocleC2×C3×C4×C9C23
hypercenterC2×#41C3⋊D9
hypocenterC3×C9#421
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. #11721815C2semidirect
3. C2×#88532C2semidirect
4. #11720585C2semidirect
9. C3×C9×#41C4semidirect
11. C2×C3×Q8×C9C4semidirect
13. C2×C3×D4×C9C4semidirect
18. C4×C3⋊Dic9C22semidirect
20. #88532C22semidirect
68. C9×#41Dic3semidirect
70. C2×Q8×C9Dic3semidirect
72. C2×D4×C9Dic3semidirect
149. C3×#41Dic9semidirect
153. C2×C3×Q8Dic9semidirect
157. C2×C3×D4Dic9semidirect
203. C3×C9#421semidirect
254. #41C3⋊Dic9semidirect
256. C2×Q8C3⋊Dic9semidirect
258. C2×D4C3⋊Dic9semidirect
292. C9#5236semidirect
334. C3#10588914semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C9×#412
#117218152
C2×#885322
#117205852
#105929353
#105889143