Finite Group Databasebeta

← #13159398

Subgroups — #13159398

Order 1728.

804 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 170 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
1C3×#64096C211yesmaximal, characteristic
2Dic6×C3⋊Dic3C211yesmaximal, characteristic
3C2×#88798C211yesmaximal, characteristic
4#11722177C211yesmaximal, characteristic
5C32×#810C211yesmaximal, characteristic
6#11722040C211yesmaximal, characteristic
7C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic
8#10592489—31nomaximal
9#10590476—34nomaximal
10C2×C3×#634C2211yescharacteristic
11C2×C32×Dic6C2211yescharacteristic
12C33×C4⋊C4C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×#3750—21no—
16C3×C4×C3⋊Dic3—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19Dic3⋊(C3⋊Dic3)—21no—
20Dic3×C3⋊Dic3—21no—
21#88798C414yescomplemented
22Q8×C3⋊Dic3—31no—
23C2×#3756—31no—
24#64096D311yescharacteristic
25C3×#887—34no—
26Dic3×Dic6—34no—
27C2×#3718—34no—
28C32×#103—31no—
29#63643—31no—
30C8×C3⋊Dic3—31no—
31#63634—31no—
32#63634—34no—
33#63634—64no—
34C3×#810D314yes—
35#63566—34no—
36Dic3×C3⋊C8—34no—
37C2×C33×C4—21no—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3×C3⋊Dic3—21no—
41C2×C3×C3⋊Dic3D411yescharacteristic
42C2×C3×C3⋊Dic3D411yescharacteristic
43C3×#634—21no—
44C3×#634C2×C412yes—
45C2×C32×Dic3—21no—
46C32×Dic6—21no—
47C32×Dic6C2×C412yes—
48C3⋊(C3⋊(C3⋊C8))C2×C412yes—
49C32×C3⋊C8—21no—
50C4×#216—91no—
51#4790—94no—
52C2×C32×Q8—31no—
53C2×#634C2×D311yescharacteristic
54C2×C3×Dic6—34no—
55C2×C3×Dic6C2×D314yes—
56C2×C32×C8—31no—
57C32×C4⋊C4—24no—
58C32×C4⋊C4C2×D314yes—
59C32×C4⋊C4C2×D311yescharacteristic
60#3756—34no—
61C4⋊(C3⋊Dic3)—31no—
62C4⋊(C3⋊Dic3)—34no—
63C4⋊(C3⋊Dic3)—61no—
64C4⋊(C3⋊Dic3)—64no—
65#3750—21no—
66C4×C3⋊Dic3—21no—
67C4×C3⋊Dic3—61no—
68C4×C3⋊Dic3C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)—31no—
70C2×C3⋊(C3⋊C8)—34no—
71C2×C3⋊(C3⋊C8)—64no—
72C3×Dic3⋊C4—64no—
73C3×C4×Dic3—34no—
74C3×C4×Dic3—64no—
75C2×C3×C3⋊C8C2×D314yes—
76Dic3⋊Dic3—64no—
77Dic32—64no—
78#3718—316no—
79C22×C33C2×D411yescharacteristic
80C33×C4—41no—
81C33×C4C22×C411yescharacteristic
82C33×C4#4111yescharacteristic
83C3⋊(C3⋊Dic3)—41no—
84C3×C3⋊Dic3—22no—
85C3×C3⋊Dic3—21no—
86C3×C3⋊Dic3Q1612yescomplemented
87C32×Dic3—22no—
88C3×C4×Q8—91no—
89Q8×Dic3—94no—
90C2×#216—94no—
91C2×#216—91no—
92#887—34no—
93C4×Dic6—91no—
94C3×#103—34no—
95#835—91no—
96#816—94no—
97C8×Dic3—94no—
98#810C3⋊D311yescharacteristic
99#807—94no—
100#807—91no—
101#807—184no—
102C4×C3⋊C8—91no—
103C32×Q8—32no—
104C32×Q8—61no—
105C2×C32×C4—21no—
106C2×C32×C4—24no—
107C2×C32×C4—24no—
108C2×C32×C4—44no—
109C2×C32×C4—61no—
110C2×C32×C4C22×D314yes—
111C2×C32×C4C22×D311yescharacteristic
112C2×C3⋊Dic3—21no—
113C2×C3⋊Dic3—61no—
114C2×C3⋊Dic3—64no—
115C2×C3⋊Dic3—124no—
116C2×C3⋊Dic3#6711yescharacteristic
117C2×C3⋊Dic3#6711yescharacteristic
118#634—21no—
119#634C4×D312yes—
120C2×C3×Dic3—24no—
121C2×C3×Dic3—34no—
122C2×C3×Dic3—34no—
123C2×C3×Dic3—64no—
124C3×Dic6—24no—
125C3×Dic6—38no—
126C3×Dic6—64no—
127C3×Dic6C4×D318yes—
128C32×C8—61no—
129C3⋊(C3⋊C8)—32no—
130C3⋊(C3⋊C8)—38no—
131C3⋊(C3⋊C8)—68no—
132C3×C3⋊C8—24no—
133C4×Q16—271noSylow
134C2×C33C4×D411yescharacteristic
135C2×C33C2×Q1611yescharacteristic
136C2×C33#13511yescharacteristic
137C2×C3×Q8—34no—
138C2×C3×Q8—91no—
139C2×Dic6—34no—
140C2×Dic6C2×C3⋊D311yescharacteristic
141C2×C3×C8—34no—
142C3×C4⋊C4—24no—
143C3×C4⋊C4—181no—
144C3×C4⋊C4D3214yes—
145C3×C4⋊C4C2×C3⋊D311yescharacteristic
146C3×C42—91no—
147C3×C42—181no—
148#216—916no—
149#216—94no—
150C4⋊Dic3—94no—
151C4⋊Dic3—91no—
152C4⋊Dic3—184no—
153C4⋊Dic3—184no—
154Dic3⋊C4—64no—
155Dic3⋊C4—181no—
156C4×Dic3—34no—
157C4×Dic3—64no—
158C4×Dic3—181no—
159C4×Dic3—184no—
160C2×C3⋊C8—94no—
161C2×C3⋊C8—91no—
162C2×C3⋊C8—184no—
163C2×C3⋊C8C2×C3⋊D311yescharacteristic
164C22×C32—24no—
165C22×C32D3×D414yes—
166C22×C32C2×#6711yescharacteristic
167C32×C4—24no—
168C32×C4—24no—
169C32×C4—41no—
170C32×C4—44no—
171C32×C4—62no—
172C32×C4—84no—
173C32×C4C2×C4×D314yes—
174C32×C4C2×C4×D311yescharacteristic
175C32×C4#23511yescharacteristic
176C32×C4#23714yes—
177C3⋊Dic3—22no—
178C3⋊Dic3—21no—
179C3⋊Dic3—121no—
180C3⋊Dic3—124no—
181C3⋊Dic3—244no—
182C3⋊Dic3#21612yescomplemented
183C3×Dic3—28no—
184C3×Dic3—38no—
185C3×Dic3—68no—
186C3×Dic3—64no—
187C2×Q16—271no—
188C4×Q8—91no—
189C4×Q8—271no—
190#107—271no—
191#103—31no—
192#103—271no—
193C4×C8—271no—
194C33C4×Q1611yescharacteristic, complemented, Sylow
195C3×Q8—38no—
196C3×Q8—64no—
197C3×Q8—92no—
198C3×Q8—181no—
199C2×C3×C4—24no—
200C2×C3×C4—21no—
201C2×C3×C4—24no—
202C2×C3×C4—44no—
203C2×C3×C4—64no—
204C2×C3×C4—91no—
205C2×C3×C4—91no—
206C2×C3×C4—181no—
207C2×C3×C4C2×D3214yes—
208C2×C3×C4C22×C3⋊D311yescharacteristic
209C2×Dic3—21no—
210C2×Dic3—34no—
211C2×Dic3—34no—
212C2×Dic3—64no—
213C2×Dic3—184no—
214C2×Dic3—181no—
215C2×Dic3—364no—
216Dic6—21no—
217Dic6—38no—
218Dic6—64no—
219Dic6C4×C3⋊D312yes—
220C3×C8—64no—
221C3⋊C8—21no—
222C3⋊C8—98no—
223C3⋊C8—92no—
224C3⋊C8—188no—
225C2×C32—24no—
226C2×C32—24no—
227C2×C32—24no—
228C2×C32#67⋊C414yes—
229C2×C32#91614yes—
230C2×C32D3×Q1614yes—
231C2×C32C4×#6711yescharacteristic
232C2×C32C2×#21611yescharacteristic
233C2×C32#95011yescharacteristic
234C2×Q8—31no—
235C2×Q8—91no—
236Q16—274no—
237C2×C8—31no—
238C2×C8—271no—
239C4⋊C4—181no—
240C4⋊C4—271no—
241C4⋊C4—541no—
242C4⋊C4D3×C3⋊D311yescharacteristic
243C42—91no—
244C42—181no—
245C42—541no—
246C22×C3—24no—
247C22×C3D3×#6714yes—
248C22×C3D4×C3⋊D311yescharacteristic
249C3×C4—24no—
250C3×C4—24no—
251C3×C4—41no—
252C3×C4—44no—
253C3×C4—68no—
254C3×C4—84no—
255C3×C4—92no—
256C3×C4—182no—
257C3×C4—181no—
258C3×C4#379514yes—
259C3×C4C4×D3214yes—
260C3×C4C2×C4×C3⋊D311yescharacteristic
261C3×C4#383011yescharacteristic
262Dic3—22no—
263Dic3—38no—
264Dic3—68no—
265Dic3—64no—
266Dic3—361no—
267Dic3—364no—
268Dic3—724no—
269C32—24no—
270C32#479014yescomplemented
271C32C4×#21611yescharacteristic, complemented
272Q8—32no—
273Q8—61no—
274Q8—92no—
275Q8—181no—
276C2×C4—21no—
277C2×C4—61no—
278C2×C4—91no—
279C2×C4—91no—
280C2×C4—181no—
281C2×C4—541no—
282C2×C4C2×D3×C3⋊D311yescharacteristic
283C8—61no—
284C8—272no—
285C2×C3—24no—
286C2×C3—24no—
287C2×C3—24no—
288C2×C3#6387914yes—
289C2×C3#6393514yes—
290C2×C3D3×#21614yes—
291C2×C3#6409011yescharacteristic
292C2×C3#6412511yescharacteristic
293C2×C3Q16×C3⋊D311yescharacteristic
294C22C3⋊D3×#6711yescharacteristic
295C4—41no—
296C4—62no—
297C4—92no—
298C4—181no—
299C4—182no—
300C4—1081no—
301C4#8901411yescharacteristic
302C4C4×D3×C3⋊D311yescharacteristic
303C3—24no—
304C3#1059047614yescomplemented
305C3#1059248911yescharacteristic, complemented
306C2#1172350711yescharacteristic
307C2#1172356111yescharacteristic
308C2C3⋊D3×#21611yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33C4×Q16
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
21. #88798C4semidirect
86. C3×C3⋊Dic3Q16semidirect
182. C3⋊Dic3#216semidirect
194. C33C4×Q16semidirect
270. C32#4790semidirect
271. C32C4×#216semidirect
304. C3#10590476semidirect
305. C3#10592489semidirect

Maximal subgroups

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

subgroupindex
C3×#640962
Dic6×C3⋊Dic32
C2×#887982
#117221772
C32×#8102
#117220402
C3⋊C8×C3⋊Dic32
#105924893
#105904763