Finite Group Databasebeta

← #13173356

Subgroups — #13173356

Order 1728.

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

normal subgroups only

This group has 576 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
1C2×C4⋊(C3⋊#157)C211yesmaximal, characteristic
2C2×C4⋊(C3⋊#157)C212yesmaximal
3C2×C4×C3⋊#157C211yesmaximal, characteristic
4C3×C4⋊C4×C3⋊D3C211yesmaximal, characteristic
5#11723106C212yesmaximal
6#10593759—31nomaximal
7C22×C3⋊#157C2212yes—
8C22×C3⋊#157C2211yescharacteristic
9C22×C3⋊#157C2211yescharacteristic
10C2×C3×C4×C3⋊D3C2212yes—
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C4⋊(C3⋊#157)—24no—
13C4⋊(C3⋊#157)—22no—
14C4×C3⋊#157—22no—
15C33×C4⋊C4C411yescharacteristic, complemented
16C3×C4⋊(C3⋊Dic3)C411yescharacteristic, complemented
17C3×#3750C412yescomplemented
18C2×C4⋊#157—32no—
19C2×C4⋊#157—31no—
20C2×C4×#157—31no—
21C4⋊C4×C3⋊D3D311yescharacteristic
22C3×D3×C4⋊C4—62no—
23#63781—32no—
24C22×C3×C3⋊D3C2311yescharacteristic
25C2×C3⋊#157—21no—
26C2×C3⋊#157—21no—
27C2×C3⋊#157—21no—
28C2×C3⋊#157—21no—
29C2×C3⋊#157—22no—
30C2×C3⋊#157—22no—
31C2×C3⋊#157—22no—
32C2×C3⋊#157D412yes—
33C2×C3⋊#157Q812yes—
34C2×C33×C4C2×C412yes—
35C2×C33×C4C2×C411yescharacteristic
36C2×C3×C3⋊Dic3C2×C412yes—
37C2×C3×C3⋊Dic3C2×C411yescharacteristic
38C3×C4×C3⋊D3—22no—
39C3×C4×C3⋊D3—22no—
40C3×C4×C3⋊D3D412yes—
41C3×C4×C3⋊D3Q812yes—
42#4984—91nomaximal
43C22×#157—31no—
44C22×#157—31no—
45C22×#157—32no—
46C2×C4×C3⋊D3C2×D312yes—
47C2×C4×C3⋊D3C2×D311yescharacteristic
48C2×C3×C4×D3—62no—
49C2×C3×C4×D3—64no—
50C4⋊#157—64no—
51C4⋊#157—62no—
52C4×#157—62no—
53C32×C4⋊C4—22no—
54C32×C4⋊C4—42no—
55C32×C4⋊C4Dic311yescharacteristic, complemented
56C4⋊(C3⋊Dic3)Dic311yescharacteristic, complemented
57#3750Dic312yescomplemented
58C3×C4⋊Dic3—62no—
59C3×Dic3⋊C4—64no—
60C22×C33C22×C411yescharacteristic
61C2×C3×C3⋊D3C2×D411yescharacteristic
62C2×C3×C3⋊D3C2×D411yescharacteristic
63C2×C3×C3⋊D3C2×Q811yescharacteristic
64C2×C3×C3⋊D3C2×Q811yescharacteristic
65C2×C3×C3⋊D3#4111yescharacteristic
66C2×C3×C3⋊D3#4111yescharacteristic
67C3⋊#157—22no—
68C3⋊#157—22no—
69C3⋊#157—42no—
70C33×C4—22no—
71C33×C4C4⋊C412yescomplemented
72C3×C3⋊Dic3—22no—
73C3×C3⋊Dic3C4⋊C412yescomplemented
74C2×C3×C4⋊C4—91no—
75C2×C4⋊Dic3—92no—
76C2×C4⋊Dic3—91no—
77C2×C4×Dic3—91no—
78D3×C4⋊C4—62no—
79C3⋊#95—92no—
80C22×C3⋊D3C22×D311yescharacteristic
81C22×C3×D3—62no—
82C2×#157—32no—
83C2×#157—32no—
84C2×#157—61no—
85C2×#157—61no—
86C2×#157—61no—
87C2×#157—61no—
88C2×#157—62no—
89C2×#157—62no—
90C2×#157—62no—
91C2×C32×C4—22no—
92C2×C32×C4—24no—
93C2×C32×C4—42no—
94C2×C32×C4—44no—
95C2×C32×C4C2×Dic312yes—
96C2×C32×C4C2×Dic311yescharacteristic
97C2×C3⋊Dic3C2×Dic312yes—
98C2×C3⋊Dic3C2×Dic311yescharacteristic
99C4×C3⋊D3—22no—
100C4×C3⋊D3—22no—
101C4×C3⋊D3Dic612yes—
102C4×C3⋊D3D1212yes—
103C2×C3×Dic3—62no—
104C2×C3×Dic3—64no—
105C3×C4×D3—64no—
106C3×C4×D3—64no—
107C3×C4×D3—124no—
108C3×C4×D3—124no—
109#389—271noSylow
110C2×C33C2×C4⋊C411yescharacteristic
111C2×C33C4×D411yescharacteristic
112C2×C33C4×Q811yescharacteristic
113C3×C3⋊D3#12111yescharacteristic
114C3×C3⋊D3C22⋊Q811yescharacteristic
115C3×C3⋊D3#12511yescharacteristic
116C3×C3⋊D3C4⋊Q811yescharacteristic
117C22×C3×C4—91no—
118C22×C3×C4—92no—
119C22×Dic3—91no—
120C22×Dic3—91no—
121C22×Dic3—92no—
122C2×C4×D3—62no—
123C2×C4×D3—64no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—42no—
126C3×C4⋊C4—92no—
127C3×C4⋊C4—91no—
128C3×C4⋊C4#15711yescharacteristic, complemented
129C4⋊Dic3—62no—
130C4⋊Dic3—184no—
131C4⋊Dic3—182no—
132Dic3⋊C4—64no—
133C4×Dic3—182no—
134C22×C32—22no—
135C22×C32—42no—
136C22×C32C22×Dic311yescharacteristic
137C2×C3⋊D3C2×Dic611yescharacteristic
138C2×C3⋊D3C2×D1211yescharacteristic
139C2×C3⋊D3D3×D411yescharacteristic
140C2×C3⋊D3#23711yescharacteristic
141C2×C3⋊D3D3×Q811yescharacteristic
142C2×C3⋊D3#23911yescharacteristic
143C2×C3×D3—62no—
144C2×C3×D3—62no—
145C2×C3×D3—62no—
146C2×C3×D3—62no—
147C2×C3×D3—62no—
148C2×C3×D3—62no—
149#157—62no—
150#157—62no—
151#157—122no—
152C32×C4—24no—
153C32×C4—22no—
154C32×C4—44no—
155C32×C4—44no—
156C32×C4—84no—
157C32×C4C4⋊Dic312yescomplemented
158C3⋊Dic3—22no—
159C3⋊Dic3C4⋊Dic312yescomplemented
160C3×Dic3—64no—
161C3×Dic3—124no—
162C2×C4⋊C4—91no—
163C2×C4⋊C4—271no—
164C2×C4⋊C4—272no—
165C2×C42—271no—
166#95—272no—
167C33#38911yescharacteristic, complemented, Sylow
168C23×C3—91no—
169C22×D3—62no—
170C2×C3×C4—22no—
171C2×C3×C4—24no—
172C2×C3×C4—42no—
173C2×C3×C4—44no—
174C2×C3×C4—91no—
175C2×C3×C4—92no—
176C2×C3×C4—92no—
177C2×C3×C4—92no—
178C2×C3×C4—182no—
179C2×C3×C4—182no—
180C2×C3×C4C2×#15712yes—
181C2×C3×C4C2×#15711yescharacteristic
182C2×Dic3—62no—
183C2×Dic3—64no—
184C2×Dic3—92no—
185C2×Dic3—92no—
186C2×Dic3—181no—
187C2×Dic3—181no—
188C2×Dic3—181no—
189C2×Dic3—181no—
190C2×Dic3—182no—
191C2×Dic3—182no—
192C2×Dic3—182no—
193C4×D3—64no—
194C4×D3—64no—
195C4×D3—124no—
196C4×D3—124no—
197C2×C32—22no—
198C2×C32—22no—
199C2×C32—22no—
200C2×C32—42no—
201C2×C32—42no—
202C2×C32—42no—
203C2×C32C2×C4⋊Dic311yescharacteristic
204C2×C32D4×Dic311yescharacteristic
205C2×C32Q8×Dic311yescharacteristic
206C3⋊D3#88811yescharacteristic
207C3⋊D3#89011yescharacteristic
208C3⋊D3#89511yescharacteristic
209C3⋊D3#89711yescharacteristic
210C3×D3—62no—
211C3×D3—62no—
212C3×D3—62no—
213C3×D3—62no—
214C22×C4—91no—
215C22×C4—92no—
216C22×C4—271no—
217C22×C4—271no—
218C22×C4—272no—
219C4⋊C4—92no—
220C4⋊C4—91no—
221C4⋊C4—542no—
222C4⋊C4—544no—
223C4⋊C4C3⋊#15711yescharacteristic, complemented
224C42—542no—
225C22×C3—22no—
226C22×C3—42no—
227C22×C3—91no—
228C22×C3—91no—
229C22×C3—91no—
230C22×C3—91no—
231C22×C3—91no—
232C22×C3—91no—
233C22×C3C22×#15711yescharacteristic
234C2×D3—62no—
235C2×D3—62no—
236C2×D3—62no—
237C2×D3—62no—
238C2×D3—62no—
239C2×D3—62no—
240C3×C4—22no—
241C3×C4—24no—
242C3×C4—44no—
243C3×C4—44no—
244C3×C4—84no—
245C3×C4—92no—
246C3×C4—182no—
247C3×C4C4⋊#15712yescomplemented
248Dic3—64no—
249Dic3—124no—
250Dic3—182no—
251Dic3—182no—
252Dic3—362no—
253C32—22no—
254C32—42no—
255C32#498411yescharacteristic, complemented
256C23—91no—
257C2×C4—91no—
258C2×C4—92no—
259C2×C4—92no—
260C2×C4—92no—
261C2×C4—182no—
262C2×C4—182no—
263C2×C4—272no—
264C2×C4—272no—
265C2×C4—541no—
266C2×C4—541no—
267C2×C4—541no—
268C2×C4—541no—
269C2×C4—542no—
270C2×C4—542no—
271C2×C4—542no—
272C2×C4C2×C3⋊#15712yes—
273C2×C4C2×C3⋊#15711yescharacteristic
274C2×C3—22no—
275C2×C3—22no—
276C2×C3—22no—
277C2×C3—42no—
278C2×C3—42no—
279C2×C3—42no—
280C2×C3—91no—
281C2×C3—91no—
282C2×C3—91no—
283C2×C3—91no—
284C2×C3C2×C4⋊#15711yescharacteristic
285C2×C3D4×#15711yescharacteristic
286C2×C3Q8×#15711yescharacteristic
287D3—62no—
288D3—62no—
289D3—62no—
290D3—62no—
291C22—91no—
292C22—91no—
293C22—91no—
294C22—91no—
295C22—91no—
296C22—91no—
297C22C22×C3⋊#15711yescharacteristic
298C4—22no—
299C4—92no—
300C4—182no—
301C4—542no—
302C4—542no—
303C4—1082no—
304C4C4⋊(C3⋊#157)12yescomplemented
305C3—22no—
306C3—42no—
307C3#1059375911yescharacteristic, complemented
308C2—91no—
309C2—91no—
310C2—91no—
311C2—91no—
312C2C2×C4⋊(C3⋊#157)11yescharacteristic
313C2D4×C3⋊#15711yescharacteristic
314C2Q8×C3⋊#15711yescharacteristic
subgroupG/H
center, FrattiniC22C22×C3⋊#157
commutator, socleC22×C33C22×C4
FittingC33×C4⋊C4C4
prime radical, cosocleC22×C3×C3⋊D3C23
hypercenterC4⋊C4C3⋊#157
hypocenterC33#389
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
15. C33×C4⋊C4C4semidirect
16. C3×C4⋊(C3⋊Dic3)C4semidirect
17. C3×#3750C4semidirect
55. C32×C4⋊C4Dic3semidirect
56. C4⋊(C3⋊Dic3)Dic3semidirect
57. #3750Dic3semidirect
71. C33×C4C4⋊C4semidirect
73. C3×C3⋊Dic3C4⋊C4semidirect
128. C3×C4⋊C4#157semidirect
157. C32×C4C4⋊Dic3semidirect
159. C3⋊Dic3C4⋊Dic3semidirect
167. C33#389semidirect
223. C4⋊C4C3⋊#157semidirect
247. C3×C4C4⋊#157semidirect
255. C32#4984semidirect
304. C4C4⋊(C3⋊#157)semidirect
307. C3#10593759semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C3⋊#157)2
C2×C4×C3⋊#1572
C3×C4⋊C4×C3⋊D32
#117231062
#105937593
#49849