Finite Group Databasebeta

← #13161178

Subgroups — #13161178

Order 1728.

940 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 940 conjugacy classes of subgroups — too many to draw. Only its 184 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
1Q8×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic
2C2×C3×#3756C211yesmaximal, characteristic
3#11723410C211yesmaximal, characteristic
4C32×#835C211yesmaximal, characteristic
5#11722057C211yesmaximal, characteristic
6#11722040C211yesmaximal, characteristic
7#11722018C211yesmaximal, characteristic
8#10592489—31nomaximal
9#10591581—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C4⋊(C3⋊(C3⋊Dic3))—21no—
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C4×C3⋊(C3⋊Dic3)—21no—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#3756C414yescomplemented
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88807—21no—
21Dic3×C3⋊Dic3—21no—
22Q8×C3⋊Dic3—31no—
23Q8×C3⋊Dic3—34no—
24Q8×C3⋊Dic3—64no—
25C2×#3756D311yescharacteristic
26#64096—31no—
27C2×C3×#216—34no—
28#63843—34no—
29C32×#103—31no—
30#63643—31no—
31C8×C3⋊Dic3—31no—
32#63634—31no—
33C3×#835D314yes—
34#63575—34no—
35#63566—34no—
36#63553—34no—
37C33×Q8—21no—
38C33×Q8C2×C412yes—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42C2×C3⋊(C3⋊Dic3)D411yescharacteristic
43C2×C3⋊(C3⋊Dic3)D411yescharacteristic
44C2×C3×C3⋊Dic3—21no—
45C3×#634—21no—
46C3×#634C2×C412yes—
47C2×C32×Dic3—21no—
48C3×C3⋊(C3⋊C8)C2×C412yes—
49C32×C3⋊C8—21no—
50Dic3×Q16—91no—
51#4790—94no—
52C2×C32×Q8—24no—
53C2×C32×Q8C2×D314yes—
54C2×C32×Q8C2×D311yescharacteristic
55C2×#634C2×D311yescharacteristic
56C2×C3×Dic6—34no—
57C2×C32×C8—31no—
58C32×C4⋊C4—31no—
59#3756Dic314yescomplemented
60C4⋊(C3⋊Dic3)—31no—
61C4⋊(C3⋊Dic3)—34no—
62C4⋊(C3⋊Dic3)—61no—
63C4⋊(C3⋊Dic3)—64no—
64C4⋊(C3⋊Dic3)—64no—
65C4⋊(C3⋊Dic3)—68no—
66#3750—61no—
67C4×C3⋊Dic3—31no—
68C4×C3⋊Dic3—34no—
69C4×C3⋊Dic3—61no—
70C4×C3⋊Dic3—61no—
71C4×C3⋊Dic3—64no—
72C4×C3⋊Dic3—64no—
73C4×C3⋊Dic3—68no—
74C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
75C3×#216—316no—
76C3×C4⋊Dic3C2×D314yes—
77C2×C3×C3⋊C8—34no—
78C2×C3×C3⋊C8C2×D314yes—
79Dic3⋊Dic3—64no—
80Dic32—64no—
81C22×C33C2×D411yescharacteristic
82C33×C4—22no—
83C33×C4C22×C411yescharacteristic
84C33×C4#4111yescharacteristic
85C3⋊(C3⋊Dic3)—21no—
86C3⋊(C3⋊Dic3)—41no—
87C3⋊(C3⋊Dic3)Q1612yescomplemented
88C3×C3⋊Dic3—22no—
89C32×Dic3—41no—
90C2×C3×Q16—91no—
91Q8×Dic3—94no—
92Q8×Dic3—91no—
93Q8×Dic3—91no—
94Q8×Dic3—184no—
95C2×#216—34no—
96#887—94no—
97C3×#103—34no—
98#835—91no—
99#835C3⋊D311yescharacteristic
100#818—91no—
101#816—94no—
102C8×Dic3—94no—
103C8×Dic3—91no—
104#807—94no—
105C32×Q8—28no—
106C32×Q8—24no—
107C32×Q8—21no—
108C32×Q8—28no—
109C32×Q8C4×D318yes—
110C32×Q8C2×Dic312yes—
111C2×C32×C4—28no—
112C2×C32×C4—24no—
113C2×C32×C4—21no—
114C2×C32×C4—24no—
115C2×C32×C4—61no—
116C2×C32×C4C22×D314yes—
117C2×C32×C4C22×D311yescharacteristic
118C2×C3⋊Dic3—21no—
119C2×C3⋊Dic3—31no—
120C2×C3⋊Dic3—31no—
121C2×C3⋊Dic3—34no—
122C2×C3⋊Dic3—34no—
123C2×C3⋊Dic3—64no—
124C2×C3⋊Dic3—64no—
125C2×C3⋊Dic3—68no—
126C2×C3⋊Dic3—64no—
127C2×C3⋊Dic3—61no—
128#634—21no—
129#634C2×Dic312yes—
130C2×C3×Dic3—24no—
131C2×C3×Dic3—64no—
132C3×Dic6—38no—
133C3×Dic6—64no—
134C32×C8—61no—
135C3⋊(C3⋊C8)C2×Dic312yes—
136C3×C3⋊C8—24no—
137C3×C3⋊C8—38no—
138C4×Q16—271noSylow
139C2×C33C4×D411yescharacteristic
140C2×C33C2×Q1611yescharacteristic
141C2×C33#13511yescharacteristic
142C2×C3×Q8—24no—
143C2×C3×Q8—91no—
144C2×C3×Q8D3214yes—
145C2×C3×Q8C2×C3⋊D311yescharacteristic
146C2×Dic6—34no—
147C3×Q16—94no—
148C2×C3×C8—34no—
149C2×C3×C8—91no—
150C3×C4⋊C4—34no—
151#216—316no—
152C4⋊Dic3—94no—
153C4⋊Dic3—91no—
154C4⋊Dic3—184no—
155C4⋊Dic3—181no—
156C4⋊Dic3—184no—
157C4⋊Dic3—181no—
158C4⋊Dic3—188no—
159C4⋊Dic3C2×C3⋊D311yescharacteristic
160Dic3⋊C4—184no—
161C4×Dic3—94no—
162C4×Dic3—91no—
163C4×Dic3—184no—
164C4×Dic3—181no—
165C4×Dic3—184no—
166C4×Dic3—184no—
167C4×Dic3—181no—
168C4×Dic3—188no—
169C2×C3⋊C8—34no—
170C2×C3⋊C8C2×C3⋊D311yescharacteristic
171C22×C32—24no—
172C22×C32D3×D414yes—
173C22×C32D3×D411yescharacteristic
174C32×C4—216no—
175C32×C4—28no—
176C32×C4—22no—
177C32×C4—24no—
178C32×C4—24no—
179C32×C4—121no—
180C32×C4C2×C4×D314yes—
181C32×C4#23714yes—
182C32×C4#23711yescharacteristic
183C32×C4C22×Dic311yescharacteristic
184C3⋊Dic3—22no—
185C3⋊Dic3—32no—
186C3⋊Dic3—38no—
187C3⋊Dic3—68no—
188C3⋊Dic3—61no—
189C3⋊Dic3—64no—
190C3⋊Dic3—124no—
191C3⋊Dic3—128no—
192C3⋊Dic3—124no—
193C3⋊Dic3—121no—
194C3×Dic3—44no—
195C3×Dic3—68no—
196C2×Q16—91no—
197C4×Q8—271no—
198C4×Q8—271no—
199#107—271no—
200#103—31no—
201#103—271no—
202C4×C8—271no—
203C33C4×Q1611yescharacteristic, complemented, Sylow
204C3×Q8—28no—
205C3×Q8—24no—
206C3×Q8—21no—
207C3×Q8—28no—
208C3×Q8—92no—
209C3×Q8—181no—
210C3×Q8D3×Dic318yes—
211C3×Q8C4×C3⋊D312yes—
212C2×C3×C4—28no—
213C2×C3×C4—24no—
214C2×C3×C4—21no—
215C2×C3×C4—24no—
216C2×C3×C4—64no—
217C2×C3×C4—181no—
218C2×C3×C4C2×D3214yes—
219C2×C3×C4C22×C3⋊D311yescharacteristic
220C2×Dic3—21no—
221C2×Dic3—64no—
222C2×Dic3—94no—
223C2×Dic3—91no—
224C2×Dic3—94no—
225C2×Dic3—91no—
226C2×Dic3—184no—
227C2×Dic3—184no—
228C2×Dic3—188no—
229C2×Dic3—184no—
230C2×Dic3—181no—
231Dic6—38no—
232Dic6—64no—
233C3×C8—64no—
234C3×C8—92no—
235C3⋊C8—21no—
236C3⋊C8—38no—
237C2×C32—24no—
238C2×C32—24no—
239C2×C32—24no—
240C2×C32#67⋊C414yes—
241C2×C32#91614yes—
242C2×C32D3×Q1614yes—
243C2×C32D3×Q1611yescharacteristic
244C2×C32#91911yescharacteristic
245C2×C32D4×Dic311yescharacteristic
246C2×Q8—91no—
247C2×Q8D3×C3⋊D311yescharacteristic
248Q16—94no—
249C2×C8—31no—
250C2×C8—91no—
251C4⋊C4—31no—
252C4⋊C4—271no—
253C4⋊C4—541no—
254C4⋊C4—541no—
255C42—271no—
256C42—541no—
257C42—541no—
258C22×C3—24no—
259C22×C3#381114yes—
260C22×C3D4×C3⋊D311yescharacteristic
261C3×C4—216no—
262C3×C4—22no—
263C3×C4—28no—
264C3×C4—24no—
265C3×C4—24no—
266C3×C4—124no—
267C3×C4—182no—
268C3×C4C2×D3×Dic314yes—
269C3×C4#380514yes—
270C3×C4C2×C4×C3⋊D311yescharacteristic
271C3×C4#383011yescharacteristic
272Dic3—41no—
273Dic3—68no—
274Dic3—92no—
275Dic3—98no—
276Dic3—188no—
277Dic3—181no—
278Dic3—184no—
279Dic3—364no—
280Dic3—368no—
281Dic3—361no—
282Dic3—364no—
283C32—24no—
284C32#479014yescomplemented
285C32Dic3×Q1611yescharacteristic, complemented
286Q8—21no—
287Q8—92no—
288Q8—181no—
289Q8Dic3×C3⋊D312yes—
290C2×C4—21no—
291C2×C4—61no—
292C2×C4—181no—
293C2×C4—271no—
294C2×C4—271no—
295C2×C4—541no—
296C2×C4C2×D3×C3⋊D311yescharacteristic
297C8—61no—
298C8—92no—
299C2×C3—24no—
300C2×C3—24no—
301C2×C3—24no—
302C2×C3#6394414yes—
303C2×C3#6395014yes—
304C2×C3#67⋊Dic314yes—
305C2×C3#6409011yescharacteristic
306C2×C3#6412511yescharacteristic
307C2×C3Q16×C3⋊D311yescharacteristic
308C22#8903911yescharacteristic
309C4—22no—
310C4—121no—
311C4—182no—
312C4—272no—
313C4—541no—
314C4—1081no—
315C4#8902911yescharacteristic
316C4C2×Dic3×C3⋊D311yescharacteristic
317C3—24no—
318C3#1059158114yescomplemented
319C3#1059248911yescharacteristic, complemented
320C2#1172357711yescharacteristic
321C2#1172358711yescharacteristic
322C2#1172362611yescharacteristic
subgroupG/H
centerC22#89039
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
cosocleC2×C33×C4C23
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33C4×Q16
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
16. C3×#3756C4semidirect
59. #3756Dic3semidirect
87. C3⋊(C3⋊Dic3)Q16semidirect
203. C33C4×Q16semidirect
284. C32#4790semidirect
285. C32Dic3×Q16semidirect
318. C3#10591581semidirect
319. C3#10592489semidirect

Maximal subgroups

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

subgroupindex
Q8×C3⋊(C3⋊Dic3)2
C2×C3×#37562
#117234102
C32×#8352
#117220572
#117220402
#117220182
#105924893
#105915813