Finite Group Databasebeta

← #13173343

Subgroups — #13173343

Order 1728.

800 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C23×C3⋊#157C211yesmaximal, characteristic
2C2×#88994C211yesmaximal, characteristic
3C2×C4⋊(C3⋊#157)C212yesmaximal, complemented
4C3×C22⋊C4×C3⋊D3C211yesmaximal, characteristic
5#11723106C212yesmaximal, complemented
6#10593746—31nomaximal
7C23×C3×C3⋊D3C2211yescharacteristic
8C22×C3⋊#157—22no—
9C22×C3⋊#157—24no—
10C22×C3⋊#157C2211yescharacteristic
11C22×C3⋊#157C2211yescharacteristic
12C22×C3⋊#157C2212yes—
13C2×C3×C4×C3⋊D3C2212yes—
14#88994—22no—
15C4⋊(C3⋊#157)—24no—
16C33×C22⋊C4C411yescharacteristic, complemented
17C3×#3757C411yescharacteristic, complemented
18C3×#3752C412yescomplemented
19C23×#157—31no—
20C2×C22⋊#157—31no—
21C2×C4⋊#157—32no—
22C22⋊C4×C3⋊D3D311yescharacteristic
23C3×D3×C22⋊C4—62no—
24#63781—32no—
25C23×C33C2×C411yescharacteristic
26C22×C3×C3⋊D3—22no—
27C22×C3×C3⋊D3—22no—
28C22×C3×C3⋊D3C2×C411yescharacteristic
29C22×C3×C3⋊D3D412yes—
30C22×C3×C3⋊D3Q812yes—
31C22×C3×C3⋊D3C2311yescharacteristic
32C2×C3⋊#157—28no—
33C2×C3⋊#157—22no—
34C2×C3⋊#157—22no—
35C2×C3⋊#157—22no—
36C2×C3⋊#157—21no—
37C2×C3⋊#157—21no—
38C2×C3⋊#157—22no—
39C2×C3⋊#157D414yes—
40C2×C33×C4C2×C412yescomplemented
41C2×C3×C3⋊Dic3C2×C412yescomplemented
42C3×C4×C3⋊D3—22no—
43C3×C4×C3⋊D3—22no—
44#4950—91nomaximal
45C23×C3⋊D3C2×D311yescharacteristic
46C23×C3×D3—62no—
47C22×#157—32no—
48C22×#157—31no—
49C22×#157—31no—
50C22×#157—64no—
51C22×#157—62no—
52C2×C4×C3⋊D3C2×D312yes—
53C2×C3×C4×D3—64no—
54C22⋊#157—62no—
55C4⋊#157—64no—
56C32×C22⋊C4—22no—
57C32×C22⋊C4—42no—
58C32×C22⋊C4Dic311yescharacteristic, complemented
59#3757Dic311yescharacteristic, complemented
60#3752Dic312yescomplemented
61C3×#217—62no—
62C3×(C2×D3)⋊C4—64no—
63C22×C33—22no—
64C22×C33C4⋊C412yescomplemented
65C22×C33C22×C411yescharacteristic
66C2×C3×C3⋊D3—22no—
67C2×C3×C3⋊D3—22no—
68C2×C3×C3⋊D3—24no—
69C2×C3×C3⋊D3—22no—
70C2×C3×C3⋊D3C4⋊C412yescomplemented
71C2×C3×C3⋊D3C2×D412yes—
72C2×C3×C3⋊D3C2×D411yescharacteristic
73C2×C3×C3⋊D3C2×Q811yescharacteristic
74C2×C3×C3⋊D3#4112yes—
75C3⋊#157—24no—
76C3⋊#157—41no—
77C3⋊#157—41no—
78C33×C4—22no—
79C3×C3⋊Dic3—22no—
80C23×Dic3—91no—
81C2×C3×C22⋊C4—91no—
82C2×#217—91no—
83C2×C4⋊Dic3—92no—
84D3×C22⋊C4—62no—
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—62no—
97C22×C3×D3—64no—
98C22×C3×D3—64no—
99C22×C3×D3—124no—
100C22×C3×D3—124no—
101C2×#157—34no—
102C2×#157—68no—
103C2×#157—62no—
104C2×#157—62no—
105C2×#157—61no—
106C2×#157—61no—
107C2×#157—62no—
108C2×#157—62no—
109C2×C32×C4—24no—
110C2×C32×C4—44no—
111C2×C32×C4C2×Dic312yescomplemented
112C2×C3⋊Dic3C2×Dic312yescomplemented
113C4×C3⋊D3—22no—
114C4×C3⋊D3—22no—
115C2×C3×Dic3—64no—
116C3×C4×D3—124no—
117C3×C4×D3—124no—
118#385—271noSylow
119C2×C33—22no—
120C2×C33C2×C4⋊C411yescharacteristic
121C2×C33C4×D412yes—
122C3×C3⋊D3—22no—
123C3×C3⋊D3#12011yescharacteristic
124C3×C3⋊D3C22⋊Q812yes—
125C3×C3⋊D3#12311yescharacteristic
126C24×C3—91no—
127C23×D3—62no—
128C22×C3×C4—92no—
129C22×Dic3—92no—
130C22×Dic3—91no—
131C22×Dic3—91no—
132C22×Dic3—184no—
133C22×Dic3—182no—
134C2×C4×D3—64no—
135C3×C22⋊C4—22no—
136C3×C22⋊C4—42no—
137C3×C22⋊C4—91no—
138C3×C22⋊C4—92no—
139C3×C22⋊C4#15711yescharacteristic, complemented
140#217—62no—
141#217—182no—
142(C2×D3)⋊C4—64no—
143C4⋊Dic3—184no—
144C22×C32—22no—
145C22×C32—24no—
146C22×C32—22no—
147C22×C32—42no—
148C22×C32—44no—
149C22×C32—44no—
150C22×C32—84no—
151C22×C32C4⋊Dic312yescomplemented
152C22×C32C22×Dic311yescharacteristic
153C2×C3⋊D3—22no—
154C2×C3⋊D3—24no—
155C2×C3⋊D3—22no—
156C2×C3⋊D3—22no—
157C2×C3⋊D3C4⋊Dic312yescomplemented
158C2×C3⋊D3C2×Dic611yescharacteristic
159C2×C3⋊D3C2×D1211yescharacteristic
160C2×C3⋊D3D3×D412yes—
161C2×C3⋊D3#23712yes—
162C2×C3×D3—64no—
163C2×C3×D3—62no—
164C2×C3×D3—62no—
165C2×C3×D3—64no—
166C2×C3×D3—64no—
167C2×C3×D3—124no—
168C2×C3×D3—128no—
169C2×C3×D3—124no—
170C2×C3×D3—124no—
171#157—64no—
172#157—121no—
173#157—121no—
174C32×C4—22no—
175C32×C4—44no—
176C32×C4—84no—
177C3⋊Dic3—22no—
178C3×Dic3—124no—
179C23×C4—271no—
180C2×C4⋊C4—272no—
181C2×C22⋊C4—91no—
182C2×C22⋊C4—271no—
183#95—272no—
184C33#38511yescharacteristic, complemented, Sylow
185C23×C3—22no—
186C23×C3—42no—
187C23×C3—91no—
188C23×C3—91no—
189C23×C3—92no—
190C23×C3—92no—
191C23×C3—182no—
192C23×C3—182no—
193C23×C3C2×#15711yescharacteristic
194C22×D3—62no—
195C22×D3—64no—
196C22×D3—64no—
197C22×D3—62no—
198C22×D3—124no—
199C22×D3—124no—
200C2×C3×C4—24no—
201C2×C3×C4—44no—
202C2×C3×C4—92no—
203C2×C3×C4—182no—
204C2×C3×C4—182no—
205C2×C3×C4C2×#15712yescomplemented
206C2×Dic3—64no—
207C2×Dic3—94no—
208C2×Dic3—182no—
209C2×Dic3—182no—
210C2×Dic3—188no—
211C2×Dic3—182no—
212C2×Dic3—181no—
213C2×Dic3—181no—
214C2×Dic3—182no—
215C4×D3—124no—
216C4×D3—124no—
217C2×C32—22no—
218C2×C32—24no—
219C2×C32—22no—
220C2×C32—42no—
221C2×C32—44no—
222C2×C32—44no—
223C2×C32—84no—
224C2×C32C2×C4⋊Dic311yescharacteristic
225C2×C32D4×Dic312yes—
226C3⋊D3—22no—
227C3⋊D3#87812yes—
228C3⋊D3#88211yescharacteristic
229C3⋊D3#88611yescharacteristic
230C3×D3—62no—
231C3×D3—64no—
232C3×D3—62no—
233C3×D3—124no—
234C24—91no—
235C22×C4—92no—
236C22×C4—272no—
237C22×C4—271no—
238C22×C4—271no—
239C22×C4—544no—
240C22×C4—542no—
241C4⋊C4—544no—
242C22⋊C4—91no—
243C22⋊C4—92no—
244C22⋊C4—542no—
245C22⋊C4C3⋊#15711yescharacteristic, complemented
246C22×C3—22no—
247C22×C3—22no—
248C22×C3—24no—
249C22×C3—44no—
250C22×C3—42no—
251C22×C3—44no—
252C22×C3—84no—
253C22×C3—91no—
254C22×C3—91no—
255C22×C3—92no—
256C22×C3—92no—
257C22×C3—92no—
258C22×C3—182no—
259C22×C3—182no—
260C22×C3—184no—
261C22×C3—182no—
262C22×C3C4⋊#15712yescomplemented
263C22×C3C22×#15711yescharacteristic
264C2×D3—64no—
265C2×D3—64no—
266C2×D3—62no—
267C2×D3—62no—
268C2×D3—64no—
269C2×D3—124no—
270C2×D3—128no—
271C2×D3—124no—
272C2×D3—124no—
273C3×C4—22no—
274C3×C4—44no—
275C3×C4—84no—
276C3×C4—182no—
277Dic3—124no—
278Dic3—184no—
279Dic3—361no—
280Dic3—361no—
281C32—22no—
282C32—42no—
283C32#495011yescharacteristic, complemented
284C23—91no—
285C23—91no—
286C23—92no—
287C23—92no—
288C23—182no—
289C23—182no—
290C23C2×C3⋊#15711yescharacteristic
291C2×C4—92no—
292C2×C4—182no—
293C2×C4—182no—
294C2×C4—274no—
295C2×C4—542no—
296C2×C4—542no—
297C2×C4—548no—
298C2×C4—541no—
299C2×C4—541no—
300C2×C4—542no—
301C2×C4—542no—
302C2×C4C2×C3⋊#15712yescomplemented
303C2×C3—22no—
304C2×C3—22no—
305C2×C3—24no—
306C2×C3—44no—
307C2×C3—42no—
308C2×C3—44no—
309C2×C3—84no—
310C2×C3—91no—
311C2×C3—91no—
312C2×C3—92no—
313C2×C3—182no—
314C2×C3C2×C4⋊#15711yescharacteristic
315C2×C3D4×#15712yes—
316D3—64no—
317D3—62no—
318D3—62no—
319D3—124no—
320C22—22no—
321C22—91no—
322C22—91no—
323C22—92no—
324C22—92no—
325C22—92no—
326C22—182no—
327C22—182no—
328C22—184no—
329C22—182no—
330C22C4⋊(C3⋊#157)12yescomplemented
331C22C22×C3⋊#15711yescharacteristic
332C4—22no—
333C4—182no—
334C4—544no—
335C4—1081no—
336C4—1081no—
337C3—22no—
338C3—42no—
339C3#1059374611yescharacteristic, complemented
340C2—22no—
341C2—91no—
342C2—91no—
343C2—92no—
344C2—182no—
345C2C2×C4⋊(C3⋊#157)11yescharacteristic
346C2D4×C3⋊#15712yes—
subgroupG/H
center, FrattiniC22C22×C3⋊#157
commutator, socleC22×C33C22×C4
FittingC33×C22⋊C4C4
prime radicalC23×C3×C3⋊D3C22
cosocleC22×C3×C3⋊D3C23
hypercenterC22⋊C4C3⋊#157
hypocenterC33#385
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
5. #11723106C2semidirect
16. C33×C22⋊C4C4semidirect
17. C3×#3757C4semidirect
18. C3×#3752C4semidirect
40. C2×C33×C4C2×C4semidirect
41. C2×C3×C3⋊Dic3C2×C4semidirect
58. C32×C22⋊C4Dic3semidirect
59. #3757Dic3semidirect
60. #3752Dic3semidirect
64. C22×C33C4⋊C4semidirect
70. C2×C3×C3⋊D3C4⋊C4semidirect
111. C2×C32×C4C2×Dic3semidirect
112. C2×C3⋊Dic3C2×Dic3semidirect
139. C3×C22⋊C4#157semidirect
151. C22×C32C4⋊Dic3semidirect
157. C2×C3⋊D3C4⋊Dic3semidirect
184. C33#385semidirect
205. C2×C3×C4C2×#157semidirect
245. C22⋊C4C3⋊#157semidirect
262. C22×C3C4⋊#157semidirect
283. C32#4950semidirect
302. C2×C4C2×C3⋊#157semidirect
330. C22C4⋊(C3⋊#157)semidirect
339. C3#10593746semidirect

Maximal subgroups

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

subgroupindex
C23×C3⋊#1572
C2×#889942
C2×C4⋊(C3⋊#157)2
C3×C22⋊C4×C3⋊D32
#117231062
#105937463
#49509