Finite Group Databasebeta

← #13173409

Subgroups — #13173409

Order 1728.

1048 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 129 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
1C22×C3×C4×C3⋊D3C211yesmaximal, characteristic
2C2×#88994C212yesmaximal
3C2×C4⋊(C3⋊#157)C212yesmaximal, complemented
4#11723106C212yesmaximal, complemented
5#10593812—31nomaximal
6C23×C3×C3⋊D3C2211yescharacteristic
7C22×C3⋊#157C2214yes—
8C22×C33×C4C411yescharacteristic, complemented
9C22×C3×C3⋊Dic3C411yescharacteristic, complemented
10C2×C3×C4×C3⋊D3—24no—
11C2×C3×C4×C3⋊D3C411yescharacteristic, complemented
12C2×C3×C4×C3⋊D3C411yescharacteristic, complemented
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15#88994—24no—
16C4⋊(C3⋊#157)—24no—
17C22×C4×C3⋊D3D311yescharacteristic
18C22×C3×C4×D3—62no—
19C2×C22⋊#157—32no—
20C2×C4⋊#157—32no—
21#63781—32no—
22C23×C33C2×C411yescharacteristic
23C22×C3×C3⋊D3—22no—
24C22×C3×C3⋊D3—22no—
25C22×C3×C3⋊D3C2×C411yescharacteristic
26C22×C3×C3⋊D3D412yes—
27C22×C3×C3⋊D3Q812yes—
28C22×C3×C3⋊D3C2311yescharacteristic
29C2×C3⋊#157—24no—
30C2×C3⋊#157—24no—
31C2×C3⋊#157—24no—
32C2×C33×C4—22no—
33C2×C33×C4C2×C411yescharacteristic
34C2×C33×C4C2×C411yescharacteristic
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3C2×C411yescharacteristic
37C2×C3×C3⋊Dic3C2×C411yescharacteristic
38C3×C4×C3⋊D3—22no—
39C3×C4×C3⋊D3—22no—
40C3×C4×C3⋊D3—22no—
41C3×C4×C3⋊D3D414yes—
42#5212—91nomaximal
43C23×C3⋊D3C2×D311yescharacteristic
44C23×C3×D3—62no—
45C22×#157—34no—
46C22×C32×C4—22no—
47C22×C32×C4—42no—
48C22×C32×C4Dic311yescharacteristic, complemented
49C22×C3⋊Dic3Dic311yescharacteristic, complemented
50C2×C4×C3⋊D3—24no—
51C2×C4×C3⋊D3Dic311yescharacteristic, complemented
52C2×C4×C3⋊D3Dic311yescharacteristic, complemented
53C2×C4×C3⋊D3C2×D311yescharacteristic
54C2×C4×C3⋊D3C2×D311yescharacteristic
55C22×C3×Dic3—62no—
56C2×C3×C4×D3—62no—
57C2×C3×C4×D3—62no—
58C2×C3×C4×D3—62no—
59C2×C3×C4×D3—62no—
60C2×C3×C4×D3—616no—
61C22⋊#157—64no—
62C4⋊#157—64no—
63C22×C33—21no—
64C22×C33—21no—
65C22×C33C4⋊C412yescomplemented
66C22×C33C22×C411yescharacteristic
67C2×C3×C3⋊D3—21no—
68C2×C3×C3⋊D3—21no—
69C2×C3×C3⋊D3—24no—
70C2×C3×C3⋊D3—24no—
71C2×C3×C3⋊D3C4⋊C412yescomplemented
72C2×C3×C3⋊D3C2×D411yescharacteristic
73C2×C3×C3⋊D3C2×D412yes—
74C2×C3×C3⋊D3C2×Q811yescharacteristic
75C2×C3×C3⋊D3#4112yes—
76C3⋊#157—44no—
77C33×C4—21no—
78C33×C4C22⋊C412yescomplemented
79C3×C3⋊Dic3—21no—
80C3×C3⋊Dic3C22⋊C412yescomplemented
81C23×C3×C4—91no—
82C22×C4×D3—62no—
83C2×#217—92no—
84C2×C4⋊Dic3—92no—
85C3⋊#95—92no—
86C23×C32—22no—
87C23×C32—42no—
88C23×C32C2×Dic311yescharacteristic
89C22×C3⋊D3—22no—
90C22×C3⋊D3—22no—
91C22×C3⋊D3Dic612yes—
92C22×C3⋊D3D1212yes—
93C22×C3⋊D3C2×Dic311yescharacteristic
94C22×C3⋊D3C22×D311yescharacteristic
95C22×C3×D3—62no—
96C22×C3×D3—64no—
97C22×C3×D3—64no—
98C22×C3×D3—62no—
99C22×C3×D3—68no—
100C22×C3×D3—68no—
101C2×#157—64no—
102C2×#157—64no—
103C2×#157—64no—
104C2×C32×C4—22no—
105C2×C32×C4—28no—
106C2×C32×C4—22no—
107C2×C32×C4—22no—
108C2×C32×C4—48no—
109C2×C32×C4—42no—
110C2×C32×C4—42no—
111C2×C32×C4C2×Dic311yescharacteristic
112C2×C32×C4C2×Dic311yescharacteristic
113C2×C3⋊Dic3—22no—
114C2×C3⋊Dic3C2×Dic311yescharacteristic
115C2×C3⋊Dic3C2×Dic311yescharacteristic
116C4×C3⋊D3—22no—
117C4×C3⋊D3—22no—
118C4×C3⋊D3—22no—
119C4×C3⋊D3#6714yes—
120C2×C3×Dic3—62no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—68no—
123C3×C4×D3—68no—
124C3×C4×D3—68no—
125C3×C4×D3—68no—
126C3×C4×D3—68no—
127#380—271noSylow
128C2×C33—22no—
129C2×C33C2×C22⋊C411yescharacteristic
130C2×C33C2×C4⋊C411yescharacteristic
131C2×C33#11711yescharacteristic
132C3×C3⋊D3—22no—
133C3×C3⋊D3#12112yes—
134C3×C3⋊D3C22⋊Q812yes—
135C24×C3—91no—
136C23×D3—62no—
137C22×C3×C4—22no—
138C22×C3×C4—42no—
139C22×C3×C4—91no—
140C22×C3×C4—91no—
141C22×C3×C4—91no—
142C22×C3×C4—91no—
143C22×C3×C4—91no—
144C22×C3×C4—184no—
145C22×C3×C4#15711yescharacteristic, complemented
146C22×Dic3—62no—
147C22×Dic3—94no—
148C2×C4×D3—62no—
149C2×C4×D3—62no—
150C2×C4×D3—62no—
151C2×C4×D3—62no—
152C2×C4×D3—616no—
153#217—184no—
154C4⋊Dic3—184no—
155C22×C32—21no—
156C22×C32—21no—
157C22×C32—24no—
158C22×C32—24no—
159C22×C32—22no—
160C22×C32—24no—
161C22×C32—44no—
162C22×C32—44no—
163C22×C32—44no—
164C22×C32—42no—
165C22×C32C4⋊Dic312yescomplemented
166C22×C32C22×Dic311yescharacteristic
167C2×C3⋊D3—21no—
168C2×C3⋊D3—21no—
169C2×C3⋊D3—24no—
170C2×C3⋊D3—24no—
171C2×C3⋊D3C4⋊Dic312yescomplemented
172C2×C3⋊D3C2×Dic611yescharacteristic
173C2×C3⋊D3C2×D1211yescharacteristic
174C2×C3⋊D3#23512yes—
175C2×C3⋊D3C2×#6712yes—
176C2×C3×D3—64no—
177C2×C3×D3—64no—
178C2×C3×D3—62no—
179C2×C3×D3—62no—
180C2×C3×D3—64no—
181C2×C3×D3—64no—
182C2×C3×D3—616no—
183C2×C3×D3—616no—
184C2×C3×D3—64no—
185#157—124no—
186C32×C4—21no—
187C32×C4—24no—
188C32×C4—24no—
189C32×C4—44no—
190C32×C4—44no—
191C32×C4#21712yescomplemented
192C3⋊Dic3—21no—
193C3⋊Dic3#21712yescomplemented
194C3×Dic3—64no—
195C3×Dic3—64no—
196C23×C4—91no—
197C2×C4⋊C4—272no—
198C2×C22⋊C4—272no—
199#95—272no—
200C33#38011yescharacteristic, complemented, Sylow
201C23×C3—22no—
202C23×C3—42no—
203C23×C3—91no—
204C23×C3—91no—
205C23×C3—92no—
206C23×C3—92no—
207C23×C3—182no—
208C23×C3—182no—
209C23×C3C2×#15711yescharacteristic
210C22×D3—64no—
211C22×D3—64no—
212C22×D3—62no—
213C22×D3—62no—
214C22×D3—68no—
215C22×D3—68no—
216C2×C3×C4—28no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—22no—
220C2×C3×C4—48no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4—91no—
224C2×C3×C4—94no—
225C2×C3×C4—91no—
226C2×C3×C4—182no—
227C2×C3×C4—182no—
228C2×C3×C4—182no—
229C2×C3×C4—182no—
230C2×C3×C4C2×#15711yescharacteristic
231C2×C3×C4C2×#15711yescharacteristic
232C2×Dic3—62no—
233C2×Dic3—62no—
234C2×Dic3—68no—
235C2×Dic3—184no—
236C2×Dic3—184no—
237C2×Dic3—184no—
238C4×D3—68no—
239C4×D3—68no—
240C4×D3—68no—
241C4×D3—68no—
242C2×C32—22no—
243C2×C32—28no—
244C2×C32—22no—
245C2×C32—22no—
246C2×C32—22no—
247C2×C32—48no—
248C2×C32—42no—
249C2×C32—42no—
250C2×C32—42no—
251C2×C32C2×C4⋊Dic311yescharacteristic
252C2×C32#92611yescharacteristic
253C2×C32C2×#21711yescharacteristic
254C3⋊D3—22no—
255C3⋊D3#92412yes—
256C3⋊D3#93012yes—
257C3×D3—64no—
258C3×D3—64no—
259C3×D3—68no—
260C24—91no—
261C22×C4—91no—
262C22×C4—91no—
263C22×C4—91no—
264C22×C4—91no—
265C22×C4—91no—
266C22×C4—184no—
267C22×C4—274no—
268C22×C4C3⋊#15711yescharacteristic, complemented
269C4⋊C4—544no—
270C22⋊C4—544no—
271C22×C3—24no—
272C22×C3—24no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—22no—
276C22×C3—24no—
277C22×C3—44no—
278C22×C3—44no—
279C22×C3—42no—
280C22×C3—44no—
281C22×C3—91no—
282C22×C3—91no—
283C22×C3—92no—
284C22×C3—92no—
285C22×C3—92no—
286C22×C3—184no—
287C22×C3—184no—
288C22×C3—181no—
289C22×C3—181no—
290C22×C3C4⋊#15712yescomplemented
291C22×C3C22×#15711yescharacteristic
292C2×D3—64no—
293C2×D3—64no—
294C2×D3—62no—
295C2×D3—62no—
296C2×D3—64no—
297C2×D3—64no—
298C2×D3—616no—
299C2×D3—616no—
300C2×D3—64no—
301C3×C4—24no—
302C3×C4—21no—
303C3×C4—24no—
304C3×C4—44no—
305C3×C4—44no—
306C3×C4—92no—
307C3×C4—181no—
308C3×C4C22⋊#15712yescomplemented
309Dic3—64no—
310Dic3—64no—
311Dic3—364no—
312C32—22no—
313C32—42no—
314C32#521211yescharacteristic, complemented
315C23—91no—
316C23—91no—
317C23—92no—
318C23—92no—
319C23—182no—
320C23—182no—
321C23C2×C3⋊#15711yescharacteristic
322C2×C4—22no—
323C2×C4—91no—
324C2×C4—94no—
325C2×C4—91no—
326C2×C4—182no—
327C2×C4—182no—
328C2×C4—182no—
329C2×C4—182no—
330C2×C4—544no—
331C2×C4—544no—
332C2×C4—544no—
333C2×C4C2×C3⋊#15711yescharacteristic
334C2×C4C2×C3⋊#15711yescharacteristic
335C2×C3—28no—
336C2×C3—22no—
337C2×C3—22no—
338C2×C3—22no—
339C2×C3—22no—
340C2×C3—48no—
341C2×C3—42no—
342C2×C3—42no—
343C2×C3—42no—
344C2×C3—92no—
345C2×C3—92no—
346C2×C3—182no—
347C2×C3C2×C4⋊#15711yescharacteristic
348C2×C3#6428611yescharacteristic
349C2×C3C2×C22⋊#15711yescharacteristic
350D3—64no—
351D3—64no—
352D3—68no—
353C22—21no—
354C22—21no—
355C22—91no—
356C22—91no—
357C22—92no—
358C22—92no—
359C22—92no—
360C22—184no—
361C22—184no—
362C22—181no—
363C22—181no—
364C22C4⋊(C3⋊#157)12yescomplemented
365C22C22×C3⋊#15711yescharacteristic
366C4—21no—
367C4—92no—
368C4—181no—
369C4—1084no—
370C4#8899412yescomplemented
371C3—22no—
372C3—42no—
373C3#1059381211yescharacteristic, complemented
374C2—22no—
375C2—92no—
376C2—92no—
377C2—182no—
378C2C2×C4⋊(C3⋊#157)11yescharacteristic
379C2#1172451411yescharacteristic
380C2C2×#8899411yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊#157
commutator, socleC22×C33C22×C4
FittingC22×C33×C4C4
prime radicalC23×C3×C3⋊D3C22
cosocleC22×C3×C3⋊D3C23
hypercenterC22×C4C3⋊#157
hypocenterC33#380
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. C2×C4⋊(C3⋊#157)C2semidirect
4. #11723106C2semidirect
8. C22×C33×C4C4semidirect
9. C22×C3×C3⋊Dic3C4semidirect
11. C2×C3×C4×C3⋊D3C4semidirect
12. C2×C3×C4×C3⋊D3C4semidirect
48. C22×C32×C4Dic3semidirect
49. C22×C3⋊Dic3Dic3semidirect
51. C2×C4×C3⋊D3Dic3semidirect
52. C2×C4×C3⋊D3Dic3semidirect
65. C22×C33C4⋊C4semidirect
71. C2×C3×C3⋊D3C4⋊C4semidirect
78. C33×C4C22⋊C4semidirect
80. C3×C3⋊Dic3C22⋊C4semidirect
145. C22×C3×C4#157semidirect
165. C22×C32C4⋊Dic3semidirect
171. C2×C3⋊D3C4⋊Dic3semidirect
191. C32×C4#217semidirect
193. C3⋊Dic3#217semidirect
200. C33#380semidirect
268. C22×C4C3⋊#157semidirect
290. C22×C3C4⋊#157semidirect
308. C3×C4C22⋊#157semidirect
314. C32#5212semidirect
364. C22C4⋊(C3⋊#157)semidirect
370. C4#88994semidirect
373. C3#10593812semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×C3⋊D32
C2×#889942
C2×C4⋊(C3⋊#157)2
#117231062
#105938123
#52129