Finite Group Databasebeta

← #13159404

Subgroups — #13159404

Order 1728.

940 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 940 conjugacy classes of subgroups — too many to draw. Only its 166 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
1#11723928C211yesmaximal, characteristic
2#11723410C211yesmaximal, characteristic
3C2×#88800C211yesmaximal, characteristic
4C3×#63637C211yesmaximal, characteristic
5C32×#807C211yesmaximal, characteristic
6#11722041C211yesmaximal, characteristic
7#11722018C211yesmaximal, characteristic
8#10592572—31nomaximal
9#10590482—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C33×C4⋊C4C2211yescharacteristic
13#88857—21no—
14C4×C3⋊(C3⋊Dic3)—21no—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88807—21no—
20Dic3×C3⋊Dic3—21no—
21#88800C414yescomplemented
22C2×#3746—31no—
23#64096—31no—
24#64096—31no—
25#64096—34no—
26#64096—64no—
27#63843—34no—
28C2×#3719—34no—
29C32×#107—31no—
30C8×C3⋊Dic3—31no—
31#63637—31no—
32#63637D311yescharacteristic
33C3×#810—34no—
34C3×#807D314yes—
35#63567—34no—
36#63553—34no—
37C2×C33×C4—21no—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3⋊(C3⋊Dic3)D411yescharacteristic
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
42#6308—21no—
43#6308C2×C412yes—
44C2×C3×C3⋊Dic3—21no—
45C3×#634—21no—
46C3×#634C2×C412yes—
47C2×C32×Dic3—21no—
48C3×C3⋊(C3⋊C8)—21no—
49C32×C3⋊C8C2×C412yes—
50#4873—94no—
51#4790—91no—
52C2×#634—31no—
53C2×#634—34no—
54C2×#634—64no—
55C2×#634C2×D311yescharacteristic
56C2×C3×Dic6—34no—
57C2×C32×C8—31no—
58C32×C4⋊C4—24no—
59C32×C4⋊C4—31no—
60C32×C4⋊C4C2×D314yes—
61C32×C4⋊C4C2×D311yescharacteristic
62#3750—61no—
63#3750—61no—
64#3750—64no—
65#3750—68no—
66C4×C3⋊Dic3—31no—
67C4×C3⋊Dic3—34no—
68C4×C3⋊Dic3—61no—
69C4×C3⋊Dic3—61no—
70C4×C3⋊Dic3—64no—
71C4×C3⋊Dic3—64no—
72C4×C3⋊Dic3—68no—
73C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
74#3746—34no—
75C3×C4⋊Dic3C2×D314yes—
76C2×C3×C3⋊C8—34no—
77C2×C3×C3⋊C8C2×D314yes—
78Dic3⋊Dic3—64no—
79Dic32—64no—
80#3719—316no—
81C22×C33C2×D411yescharacteristic
82C33×C4—41no—
83C33×C4C22×C411yescharacteristic
84C33×C4#4111yescharacteristic
85C3⋊(C3⋊Dic3)—22no—
86C3⋊(C3⋊Dic3)—21no—
87C3⋊(C3⋊Dic3)Q1612yescomplemented
88C3×C3⋊Dic3—22no—
89C32×Dic3—41no—
90Q8×Dic3—91no—
91C2×#216—91no—
92C2×Dic12—94no—
93#887—94no—
94#887—94no—
95#887—91no—
96#887—184no—
97C3×#107—34no—
98C3×#103—91no—
99#816—91no—
100C8×Dic3—94no—
101C8×Dic3—91no—
102#810—34no—
103#810—94no—
104#807C3⋊D311yescharacteristic
105C2×C32×C4—28no—
106C2×C32×C4—24no—
107C2×C32×C4—21no—
108C2×C32×C4—24no—
109C2×C32×C4—61no—
110C2×C32×C4C22×D314yes—
111C2×C32×C4C22×D311yescharacteristic
112C2×C3⋊Dic3—21no—
113C2×C3⋊Dic3—31no—
114C2×C3⋊Dic3—31no—
115C2×C3⋊Dic3—34no—
116C2×C3⋊Dic3—34no—
117C2×C3⋊Dic3—64no—
118C2×C3⋊Dic3—64no—
119C2×C3⋊Dic3—68no—
120C2×C3⋊Dic3—64no—
121C2×C3⋊Dic3—61no—
122#634—21no—
123#634—32no—
124#634—38no—
125#634—68no—
126#634—68no—
127#634—61no—
128#634—64no—
129#634C4×D312yes—
130C2×C3×Dic3—24no—
131C2×C3×Dic3—64no—
132C3×Dic6—38no—
133C3×Dic6—64no—
134C32×C8—32no—
135C3⋊(C3⋊C8)—21no—
136C3×C3⋊C8—64no—
137C3×C3⋊C8C4×D318yes—
138C4×Q16—271noSylow
139C2×C33C4×D411yescharacteristic
140C2×C33C2×Q1611yescharacteristic
141C2×C33#13511yescharacteristic
142C2×C3×Q8—91no—
143C2×Dic6—34no—
144C2×Dic6—94no—
145C2×Dic6—91no—
146C2×Dic6—184no—
147C2×C3×C8—34no—
148C2×C3×C8—91no—
149C3×C4⋊C4—24no—
150C3×C4⋊C4—34no—
151C3×C4⋊C4D3214yes—
152C3×C4⋊C4C2×C3⋊D311yescharacteristic
153#216—94no—
154C4⋊Dic3—181no—
155C4⋊Dic3C2×C3⋊D311yescharacteristic
156Dic3⋊C4—184no—
157Dic3⋊C4—184no—
158Dic3⋊C4—181no—
159Dic3⋊C4—188no—
160C4×Dic3—94no—
161C4×Dic3—91no—
162C4×Dic3—184no—
163C4×Dic3—181no—
164C4×Dic3—184no—
165C4×Dic3—184no—
166C4×Dic3—181no—
167C4×Dic3—188no—
168C2×C3⋊C8—34no—
169C2×C3⋊C8C2×C3⋊D311yescharacteristic
170Dic12—916no—
171C22×C32—24no—
172C22×C32D3×D414yes—
173C22×C32D3×D411yescharacteristic
174C32×C4—24no—
175C32×C4—24no—
176C32×C4—48no—
177C32×C4—44no—
178C32×C4—41no—
179C32×C4—121no—
180C32×C4C2×C4×D314yes—
181C32×C4C2×C4×D311yescharacteristic
182C32×C4#23711yescharacteristic
183C32×C4#23914yes—
184C3⋊Dic3—22no—
185C3⋊Dic3—32no—
186C3⋊Dic3—38no—
187C3⋊Dic3—68no—
188C3⋊Dic3—616no—
189C3⋊Dic3—62no—
190C3⋊Dic3—68no—
191C3⋊Dic3—61no—
192C3⋊Dic3—64no—
193C3⋊Dic3—124no—
194C3×Dic3—44no—
195C3×Dic3—68no—
196C2×Q16—271no—
197C4×Q8—271no—
198C4×Q8—271no—
199#107—31no—
200#103—91no—
201#103—271no—
202C4×C8—271no—
203C33C4×Q1611yescharacteristic, complemented, Sylow
204C3×Q8—92no—
205C3×Q8—181no—
206C2×C3×C4—28no—
207C2×C3×C4—24no—
208C2×C3×C4—21no—
209C2×C3×C4—24no—
210C2×C3×C4—64no—
211C2×C3×C4—181no—
212C2×C3×C4C2×D3214yes—
213C2×C3×C4C22×C3⋊D311yescharacteristic
214C2×Dic3—21no—
215C2×Dic3—64no—
216C2×Dic3—94no—
217C2×Dic3—91no—
218C2×Dic3—94no—
219C2×Dic3—91no—
220C2×Dic3—184no—
221C2×Dic3—184no—
222C2×Dic3—188no—
223C2×Dic3—184no—
224C2×Dic3—181no—
225Dic6—38no—
226Dic6—64no—
227Dic6—98no—
228Dic6—92no—
229Dic6—188no—
230Dic6—188no—
231Dic6—184no—
232Dic6—181no—
233C3×C8—38no—
234C3×C8—181no—
235C3⋊C8—64no—
236C3⋊C8C4×C3⋊D312yes—
237C2×C32—24no—
238C2×C32—24no—
239C2×C32—24no—
240C2×C32#67⋊C411yescharacteristic
241C2×C32#89314yes—
242C2×C32#91214yes—
243C2×C32#91611yescharacteristic
244C2×C32D3×Q1614yes—
245C2×C32D3×Q1611yescharacteristic
246C2×Q8—91no—
247C2×Q8—271no—
248Q16—274no—
249C2×C8—31no—
250C2×C8—91no—
251C4⋊C4—31no—
252C4⋊C4—541no—
253C4⋊C4—541no—
254C4⋊C4D3×C3⋊D311yescharacteristic
255C42—271no—
256C42—541no—
257C42—541no—
258C22×C3—24no—
259C22×C3#381114yes—
260C22×C3D4×C3⋊D311yescharacteristic
261C3×C4—24no—
262C3×C4—24no—
263C3×C4—48no—
264C3×C4—41no—
265C3×C4—44no—
266C3×C4—124no—
267C3×C4—182no—
268C3×C4#379614yes—
269C3×C4C4×D3214yes—
270C3×C4C2×C4×C3⋊D311yescharacteristic
271C3×C4#383211yescharacteristic
272Dic3—41no—
273Dic3—68no—
274Dic3—92no—
275Dic3—98no—
276Dic3—188no—
277Dic3—1816no—
278Dic3—182no—
279Dic3—181no—
280Dic3—188no—
281Dic3—184no—
282Dic3—364no—
283C32—24no—
284C32#479011yescharacteristic, complemented
285C32#487314yescomplemented
286Q8—92no—
287Q8—181no—
288Q8—272no—
289Q8—541no—
290C2×C4—21no—
291C2×C4—61no—
292C2×C4—181no—
293C2×C4—271no—
294C2×C4—271no—
295C2×C4—541no—
296C2×C4C2×D3×C3⋊D311yescharacteristic
297C8—32no—
298C8—181no—
299C2×C3—24no—
300C2×C3—24no—
301C2×C3—24no—
302C2×C3#6388114yes—
303C2×C3#6393614yes—
304C2×C3#6394414yes—
305C2×C3#6410211yescharacteristic
306C2×C3#6412111yescharacteristic
307C2×C3Q16×C3⋊D311yescharacteristic
308C22#8903911yescharacteristic
309C4—41no—
310C4—121no—
311C4—182no—
312C4—272no—
313C4—541no—
314C4—542no—
315C4#8901711yescharacteristic
316C4C4×D3×C3⋊D311yescharacteristic
317C3—24no—
318C3#1059048214yescomplemented
319C3#1059257211yescharacteristic, complemented
320C2#1172347211yescharacteristic
321C2#1172356311yescharacteristic
322C2#1172357711yescharacteristic
subgroupG/H
centerC22#89039
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. #88800C4semidirect
87. C3⋊(C3⋊Dic3)Q16semidirect
203. C33C4×Q16semidirect
284. C32#4790semidirect
285. C32#4873semidirect
318. C3#10590482semidirect
319. C3#10592572semidirect

Maximal subgroups

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

subgroupindex
#117239282
#117234102
C2×#888002
C3×#636372
C32×#8072
#117220412
#117220182
#105925723
#105904823