Finite Group Databasebeta

← #13135218

Subgroups — #13135218

Order 1728.

558 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88551C211yesmaximal, characteristic
2C2×C9×#217C211yesmaximal, characteristic
3C2×C3×#3676C211yesmaximal, characteristic
4#10589187—31nomaximal
5#10588204—31nomaximal
6C24×C3×C9C2211yescharacteristic
7C22×C3⋊Dic9C412yescomplemented
8C22×C9×Dic3—21no—
9C22×C3×Dic9—21no—
10#88551—21no—
11#88551—21no—
12C9×#217—21no—
13C9×#217C412yescomplemented
14C3×#3676—21no—
15C3×#3676C412yescomplemented
16C2×#3757—31no—
17C2×C3×#217—31no—
18C2×C3×#217D311yescharacteristic
19C2×C9×C22⋊C4—31no—
20C2×#3676—31no—
21C2×#3676—61no—
22C2×#3676D311yescharacteristic
23C23×C3×C9—42no—
24C23×C3×C9C2×C411yescharacteristic
25C23×C3×C9C2×C411yescharacteristic
26C23×C3×C9C2×C411yescharacteristic
27C23×C3×C9D411yescharacteristic
28C23×C3×C9D411yescharacteristic
29C23×C3×C9D411yescharacteristic
30C23×C3×C9Q811yescharacteristic
31C2×C3⋊Dic9—22no—
32C2×C3⋊Dic9—42no—
33C2×C9×Dic3—22no—
34C2×C9×Dic3—42no—
35C2×C3×Dic9—22no—
36C2×C3×Dic9—42no—
37#4527—91no—
38#4527—91no—
39C24×C32C2×D311yescharacteristic
40C22×C3⋊Dic3—32no—
41C22×C3×Dic3—21no—
42C22×C3×Dic3—61no—
43C24×C9—21no—
44C24×C9C2×D311yescharacteristic
45#3757—61no—
46#3757—61no—
47C3×#217—21no—
48C3×#217—32no—
49C3×#217—61no—
50C3×#217Dic312yes—
51C22×C4×C9—61no—
52C22×Dic9—21no—
53C22×Dic9—32no—
54C22×Dic9—62no—
55C9×C22⋊C4—32no—
56C9×C22⋊C4—61no—
57#3676—21no—
58#3676—62no—
59#3676—62no—
60#3676—61no—
61#3676—61no—
62#3676Dic312yescomplemented
63C22×C3×C9—22no—
64C22×C3×C9—22no—
65C22×C3×C9—22no—
66C22×C3×C9—41no—
67C22×C3×C9—41no—
68C22×C3×C9—41no—
69C22×C3×C9—41no—
70C22×C3×C9C4211yescharacteristic
71C22×C3×C9C22⋊C411yescharacteristic
72C22×C3×C9C22⋊C411yescharacteristic
73C22×C3×C9C22⋊C411yescharacteristic
74C22×C3×C9C4⋊C411yescharacteristic
75C22×C3×C9C4⋊C411yescharacteristic
76C22×C3×C9C4⋊C411yescharacteristic
77C3⋊Dic9—42no—
78C9×Dic3—42no—
79C3×Dic9—42no—
80C2×C3×C22⋊C4—31no—
81C2×C3×C22⋊C4—91no—
82C2×#217—31no—
83C2×#217—91no—
84C2×#217—91no—
85C2×#217—181no—
86C2×#217D911yescharacteristic
87C23×C32—42no—
88C23×C32Dic611yescharacteristic
89C23×C32C4×D311yescharacteristic
90C23×C32C4×D311yescharacteristic
91C23×C32D1211yescharacteristic
92C23×C32C2×Dic311yescharacteristic
93C23×C32#6711yescharacteristic
94C23×C32#6711yescharacteristic
95C2×C3⋊Dic3—62no—
96C2×C3⋊Dic3—122no—
97C2×C3×Dic3—22no—
98C2×C3×Dic3—42no—
99C2×C3×Dic3—62no—
100C2×C3×Dic3—122no—
101C23×C9—21no—
102C23×C9—21no—
103C23×C9—21no—
104C23×C9—21no—
105C23×C9—21no—
106C23×C9—21no—
107C23×C9—21no—
108C23×C9—42no—
109C23×C9—44no—
110C23×C9Dic611yescharacteristic
111C23×C9C4×D311yescharacteristic
112C23×C9C4×D311yescharacteristic
113C23×C9D1211yescharacteristic
114C23×C9C2×Dic311yescharacteristic
115C23×C9#6711yescharacteristic
116C23×C9#6711yescharacteristic
117C2×C4×C9—62no—
118C2×C4×C9—122no—
119C2×Dic9—22no—
120C2×Dic9—42no—
121C2×Dic9—64no—
122C2×Dic9—62no—
123C2×Dic9—124no—
124C2×Dic9—122no—
125#342—271noSylow
126C2×C3×C9—21no—
127C2×C3×C9—21no—
128C2×C3×C9—21no—
129C2×C3×C9—21no—
130C2×C3×C9—21no—
131C2×C3×C9—21no—
132C2×C3×C9#9511yescharacteristic
133C2×C3×C9#9912yes—
134C24×C3—21no—
135C24×C3C2×D911yescharacteristic
136C24×C3D3211yescharacteristic
137C22×C3×C4—61no—
138C22×C3×C4—181no—
139C22×Dic3—21no—
140C22×Dic3—61no—
141C22×Dic3—92no—
142C22×Dic3—92no—
143C22×Dic3—182no—
144C3×C22⋊C4—32no—
145C3×C22⋊C4—61no—
146C3×C22⋊C4—92no—
147C3×C22⋊C4—181no—
148#217—21no—
149#217—32no—
150#217—61no—
151#217—182no—
152#217—182no—
153#217—181no—
154#217—181no—
155#217—181no—
156#217—181no—
157#217Dic912yescomplemented
158C22×C32—22no—
159C22×C32—22no—
160C22×C32—22no—
161C22×C32—41no—
162C22×C32—41no—
163C22×C32—41no—
164C22×C32—41no—
165C22×C32C4×Dic311yescharacteristic
166C22×C32Dic3⋊C411yescharacteristic
167C22×C32Dic3⋊C411yescharacteristic
168C22×C32C4⋊Dic311yescharacteristic
169C22×C32(C2×D3)⋊C411yescharacteristic
170C22×C32(C2×D3)⋊C411yescharacteristic
171C22×C32#21711yescharacteristic
172C3⋊Dic3—122no—
173C3×Dic3—42no—
174C3×Dic3—122no—
175C22×C9—24no—
176C22×C9—22no—
177C22×C9—22no—
178C22×C9—22no—
179C22×C9—21no—
180C22×C9—21no—
181C22×C9—21no—
182C22×C9—21no—
183C22×C9—21no—
184C22×C9—21no—
185C22×C9—21no—
186C22×C9—42no—
187C22×C9—42no—
188C22×C9—41no—
189C22×C9—41no—
190C22×C9—41no—
191C22×C9—41no—
192C22×C9—42no—
193C22×C9—42no—
194C22×C9—42no—
195C22×C9—42no—
196C22×C9C4×Dic311yescharacteristic
197C22×C9Dic3⋊C411yescharacteristic
198C22×C9Dic3⋊C411yescharacteristic
199C22×C9C4⋊Dic311yescharacteristic
200C22×C9(C2×D3)⋊C411yescharacteristic
201C22×C9(C2×D3)⋊C411yescharacteristic
202C22×C9#21711yescharacteristic
203C4×C9—122no—
204Dic9—42no—
205Dic9—124no—
206Dic9—122no—
207C2×C22⋊C4—31no—
208C2×C22⋊C4—91no—
209C2×C22⋊C4—271no—
210C3×C9#34211yescharacteristic, complemented, Sylow
211C23×C3—21no—
212C23×C3—21no—
213C23×C3—21no—
214C23×C3—21no—
215C23×C3—21no—
216C23×C3—21no—
217C23×C3—21no—
218C23×C3—42no—
219C23×C3—42no—
220C23×C3—44no—
221C23×C3Dic1811yescharacteristic
222C23×C3C4×D911yescharacteristic
223C23×C3C4×D911yescharacteristic
224C23×C3D3611yescharacteristic
225C23×C3C2×Dic911yescharacteristic
226C23×C3#61111yescharacteristic
227C23×C3#61111yescharacteristic
228C23×C3D3×Dic311yescharacteristic
229C23×C3D3×Dic311yescharacteristic
230C23×C3#62411yescharacteristic
231C23×C3#62511yescharacteristic
232C23×C3#62611yescharacteristic
233C23×C3#62611yescharacteristic
234C23×C3#62711yescharacteristic
235C2×C3×C4—62no—
236C2×C3×C4—122no—
237C2×C3×C4—182no—
238C2×C3×C4—362no—
239C2×Dic3—22no—
240C2×Dic3—42no—
241C2×Dic3—62no—
242C2×Dic3—122no—
243C2×Dic3—184no—
244C2×Dic3—182no—
245C2×Dic3—182no—
246C2×Dic3—364no—
247C2×Dic3—362no—
248C2×Dic3—362no—
249C2×C32—21no—
250C2×C32—21no—
251C2×C32—21no—
252C2×C32—21no—
253C2×C32—21no—
254C2×C32—21no—
255C2×C32#80612yes—
256C2×C32C3⋊#9511yescharacteristic
257C2×C9—22no—
258C2×C9—22no—
259C2×C9—21no—
260C2×C9—21no—
261C2×C9—21no—
262C2×C9—21no—
263C2×C9—21no—
264C2×C9—21no—
265C2×C9—21no—
266C2×C9—22no—
267C2×C9—41no—
268C2×C9—41no—
269C2×C9—41no—
270C2×C9—41no—
271C2×C9#80612yes—
272C2×C9C3⋊#9511yescharacteristic
273C24D3×D911yescharacteristic
274C22×C4—61no—
275C22×C4—181no—
276C22×C4—272no—
277C22⋊C4—32no—
278C22⋊C4—61no—
279C22⋊C4—92no—
280C22⋊C4—181no—
281C22⋊C4—541no—
282C22⋊C4—541no—
283C22×C3—24no—
284C22×C3—22no—
285C22×C3—22no—
286C22×C3—22no—
287C22×C3—22no—
288C22×C3—22no—
289C22×C3—22no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—42no—
298C22×C3—42no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3—42no—
308C22×C3—42no—
309C22×C3—42no—
310C22×C3—42no—
311C22×C3C4×Dic911yescharacteristic
312C22×C3Dic9⋊C411yescharacteristic
313C22×C3Dic9⋊C411yescharacteristic
314C22×C3C4⋊Dic911yescharacteristic
315C22×C3(C2×D9)⋊C411yescharacteristic
316C22×C3(C2×D9)⋊C411yescharacteristic
317C22×C3#367611yescharacteristic
318C22×C3Dic3211yescharacteristic
319C22×C3(C2×D3)⋊Dic311yescharacteristic
320C22×C3(C2×D3)⋊Dic311yescharacteristic
321C22×C3#372211yescharacteristic
322C22×C3Dic3⋊Dic311yescharacteristic
323C22×C3Dic3⋊Dic311yescharacteristic
324C22×C3#372411yescharacteristic
325C3×C4—122no—
326C3×C4—362no—
327Dic3—42no—
328Dic3—122no—
329Dic3—364no—
330Dic3—362no—
331Dic3—362no—
332C32#452711yescharacteristic
333C9—21no—
334C9#452711yescharacteristic, complemented
335C23—42no—
336C23#618911yescharacteristic
337C23Dic3×D911yescharacteristic
338C23(C3⋊D9)⋊C411yescharacteristic
339C23#619211yescharacteristic
340C23D3×Dic911yescharacteristic
341C23#619411yescharacteristic
342C23#619511yescharacteristic
343C2×C4—62no—
344C2×C4—122no—
345C2×C4—182no—
346C2×C4—362no—
347C2×C4—542no—
348C2×C4—1082no—
349C2×C3—22no—
350C2×C3—22no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—21no—
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—22no—
364C2×C3—21no—
365C2×C3—41no—
366C2×C3—41no—
367C2×C3—41no—
368C2×C3—41no—
369C2×C3#6336512yes—
370C2×C3C9⋊#9511yescharacteristic
371C2×C3#6357911yescharacteristic
372C2×C3#6358112yes—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22Dic3×Dic911yescharacteristic
381C22Dic9⋊Dic311yescharacteristic
382C22#8844211yescharacteristic
383C22Dic3⋊Dic911yescharacteristic
384C22(C2×D9)⋊Dic311yescharacteristic
385C22#8844511yescharacteristic
386C22(C2×D3)⋊Dic911yescharacteristic
387C4—122no—
388C4—362no—
389C4—1082no—
390C3—21no—
391C3#1058820411yescharacteristic, complemented
392C3#1058918711yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—21no—
397C2—21no—
398C2—21no—
399C2#1172033012yes—
400C2#1172033111yescharacteristic
subgroupG/H
centerC22#88445
commutatorC22×C3×C9C42
FrattiniC24×C3D32
Fitting, cosocleC24×C3×C9C22
socleC22×C32(C2×D3)⋊C4
prime radicalC24×C32C2×D3
hypercenterC24D3×D9
hypocenterC3×C9#342
LCSG ▷ C22×C3×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
7. C22×C3⋊Dic9C4semidirect
13. C9×#217C4semidirect
15. C3×#3676C4semidirect
62. #3676Dic3semidirect
157. #217Dic9semidirect
210. C3×C9#342semidirect
334. C9#4527semidirect
391. C3#10588204semidirect

Maximal subgroups

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

subgroupindex
C2×#885512
C2×C9×#2172
C2×C3×#36762
#105891873
#105882043