Finite Group Databasebeta

← #13159377

Subgroups — #13159377

Order 1728.

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

normal subgroups only

This group has 888 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
1C3×#64102C211yesmaximal, characteristic
2#11723406C211yesmaximal, characteristic, complemented
3C2×#88796C211yesmaximal, characteristic
4#11722180C211yesmaximal, characteristic, complemented
5C32×#807C211yesmaximal, characteristic, complemented
6#11722033C211yesmaximal, characteristic
7#11722020C211yesmaximal, characteristic, complemented
8#10592588—31nomaximal
9#10590590—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3—21no—
13C33×C4⋊C4C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×#3752—21no—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88806—21no—
20Dic3×C3⋊Dic3—21no—
21#88796C414yescomplemented
22C2×#3744—31no—
23#64102D311yescharacteristic
24#64096—31no—
25C3×#893—34no—
26#63844—34no—
27C2×#3716—34no—
28C32×#107—31no—
29#63642—31no—
30#63637—31no—
31#63637—34no—
32#63637—64no—
33#63636—31no—
34C3×#807D314yescomplemented
35#63562—34no—
36#63554—34no—
37C22×C3×C3⋊D3—21no—
38C2×C33×C4—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41#6308—21no—
42#6308C2×C412yescomplemented
43C2×C3×C3⋊Dic3D411yescharacteristic
44C2×C3×C3⋊Dic3D411yescharacteristic
45C3×C4⋊(C3⋊D3)—21no—
46C3×C4⋊(C3⋊D3)C2×C412yes—
47C3×C4×C3⋊D3—42no—
48C2×C32×Dic3—21no—
49C3⋊(C3⋊(C3⋊C8))—21no—
50C32×C3⋊C8C2×C412yescomplemented
51#5018—91no—
52#4889—94no—
53C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
54C2×C4×C3⋊D3—21no—
55C2×#634—31no—
56C2×#634—34no—
57C2×#634—64no—
58C2×C3×D12—34no—
59C2×C3×C4×D3—64no—
60C2×C32×C8—31no—
61C32×C4⋊C4—24no—
62C32×C4⋊C4—31no—
63C32×C4⋊C4C2×D314yes—
64C32×C4⋊C4C2×D311yescharacteristic
65#3752—21no—
66#3750—61no—
67C4×C3⋊Dic3—61no—
68C4×C3⋊Dic3C2×D311yescharacteristic
69C2×C3⋊(C3⋊C8)—31no—
70C2×C3⋊(C3⋊C8)—34no—
71C2×C3⋊(C3⋊C8)—64no—
72#3744—34no—
73C3×(C2×D3)⋊C4—64no—
74C3×C4⋊Dic3C2×D314yes—
75C3×C4×Dic3—34no—
76C2×C3×C3⋊C8C2×D314yes—
77#3724—64no—
78Dic32—64no—
79#3716—316no—
80C22×C33C2×D411yescharacteristic
81C2×C3×C3⋊D3—22no—
82C2×C3×C3⋊D3—41no—
83C2×C3×C3⋊D3—41no—
84C33×C4—41no—
85C33×C4C22×C411yescharacteristic
86C33×C4#4111yescharacteristic
87C3⋊(C3⋊Dic3)—22no—
88C3×C3⋊Dic3—21no—
89C3×C3⋊Dic3—21no—
90C32×Dic3—41no—
91C3×C4×D4—91no—
92C2×#214—91no—
93C2×#204—94no—
94#893—34no—
95#887—94no—
96C4×Dic6—91no—
97C3×#107—34no—
98#832—91no—
99#815—94no—
100#810—94no—
101#810—91no—
102#810—184no—
103#809—94no—
104#807C3⋊D311yescharacteristic, complemented
105#803—91no—
106C22×C3⋊D3—21no—
107C22×C3×D3—64no—
108C2×C32×C4—21no—
109C2×C32×C4—24no—
110C2×C32×C4—24no—
111C2×C32×C4—44no—
112C2×C32×C4—61no—
113C2×C32×C4C22×D314yes—
114C2×C32×C4C22×D311yescharacteristic
115C2×C3⋊Dic3—61no—
116C2×C3⋊Dic3—64no—
117C2×C3⋊Dic3—124no—
118C2×C3⋊Dic3#6711yescharacteristic
119C2×C3⋊Dic3#6711yescharacteristic
120C4⋊(C3⋊D3)—21no—
121C4⋊(C3⋊D3)C4×D312yes—
122C4×C3⋊D3—42no—
123#634—32no—
124#634—38no—
125#634—68no—
126#634—61no—
127#634—64no—
128#634—124no—
129C2×C3×Dic3—24no—
130C2×C3×Dic3—34no—
131C2×C3×Dic3—34no—
132C3×D12—38no—
133C3×D12—64no—
134C3×C4×D3—128no—
135C32×C8—32no—
136C3⋊(C3⋊C8)—61no—
137C3⋊(C3⋊C8)—64no—
138C3⋊(C3⋊C8)—124no—
139C3×C3⋊C8C4×D318yescomplemented
140#440—271noSylow
141C2×C33C4×D411yescharacteristic
142C2×C33AGL(1,ℤ/8)11yescharacteristic
143C2×C33#13711yescharacteristic
144C3×C3⋊D3—42no—
145C2×C3×D4—91no—
146C22×C3×C4—181no—
147C2×D12—34no—
148C2×C4×D3—64no—
149C2×Dic6—94no—
150C2×Dic6—91no—
151C2×Dic6—184no—
152C2×C3×C8—34no—
153C3×C4⋊C4—24no—
154C3×C4⋊C4—34no—
155C3×C4⋊C4D3214yes—
156C3×C4⋊C4C2×C3⋊D311yescharacteristic
157C3×C22⋊C4—181no—
158C3×C42—91no—
159#214—94no—
160(C2×D3)⋊C4—64no—
161C4⋊Dic3C2×C3⋊D311yescharacteristic
162Dic3⋊C4—181no—
163Dic3⋊C4—184no—
164C4×Dic3—34no—
165C4×Dic3—181no—
166C4×Dic3—184no—
167C2×C3⋊C8—94no—
168C2×C3⋊C8—91no—
169C2×C3⋊C8—184no—
170C2×C3⋊C8C2×C3⋊D311yescharacteristic
171#204—916no—
172C22×C32—24no—
173C22×C32D3×D414yes—
174C22×C32C2×#6711yescharacteristic
175C2×C3⋊D3—22no—
176C2×C3⋊D3—41no—
177C2×C3⋊D3—41no—
178C2×C3×D3—68no—
179C2×C3×D3—124no—
180C2×C3×D3—124no—
181C32×C4—24no—
182C32×C4—24no—
183C32×C4—41no—
184C32×C4—44no—
185C32×C4—84no—
186C32×C4—121no—
187C32×C4C2×C4×D314yes—
188C32×C4C2×C4×D311yescharacteristic
189C32×C4#23511yescharacteristic
190C32×C4#23914yes—
191C3⋊Dic3—21no—
192C3⋊Dic3—21no—
193C3⋊Dic3—62no—
194C3⋊Dic3—68no—
195C3⋊Dic3—128no—
196C3×Dic3—44no—
197C3×Dic3—64no—
198C3×Dic3—64no—
199C2×SD16—271no—
200C4×Q8—271no—
201C4×D4—91no—
202#107—31no—
203#103—271no—
204#102—271no—
205#97—271no—
206C33#44011yescharacteristic, complemented, Sylow
207C23×C3—181no—
208C22×D3—64no—
209C3×D4—92no—
210C3×D4—181no—
211C2×C3×C4—24no—
212C2×C3×C4—21no—
213C2×C3×C4—24no—
214C2×C3×C4—44no—
215C2×C3×C4—64no—
216C2×C3×C4—91no—
217C2×C3×C4—91no—
218C2×C3×C4—362no—
219C2×C3×C4C2×D3214yes—
220C2×C3×C4C22×C3⋊D311yescharacteristic
221C2×Dic3—21no—
222C2×Dic3—34no—
223C2×Dic3—34no—
224C2×Dic3—184no—
225C2×Dic3—181no—
226C2×Dic3—364no—
227D12—38no—
228D12—64no—
229C4×D3—128no—
230Dic6—98no—
231Dic6—92no—
232Dic6—188no—
233Dic6—184no—
234Dic6—181no—
235Dic6—364no—
236C3×C8—38no—
237C3⋊C8—184no—
238C3⋊C8—181no—
239C3⋊C8—364no—
240C3⋊C8C4×C3⋊D312yescomplemented
241C2×C32—24no—
242C2×C32—24no—
243C2×C32—24no—
244C2×C32#89314yes—
245C2×C32#91114yes—
246C2×C32#91814yes—
247C2×C32C4×#6711yescharacteristic
248C2×C32#93211yescharacteristic
249C2×C32#95111yescharacteristic
250C3⋊D3—42no—
251C3×D3—128no—
252C2×Q8—271no—
253C2×D4—91no—
254C22×C4—181no—
255SD16—274no—
256C2×C8—31no—
257C2×C8—271no—
258C4⋊C4—31no—
259C4⋊C4—541no—
260C4⋊C4D3×C3⋊D311yescharacteristic
261C22⋊C4—181no—
262C42—91no—
263C42—541no—
264C22×C3—24no—
265C22×C3—182no—
266C22×C3—361no—
267C22×C3—361no—
268C22×C3D3×#6714yes—
269C22×C3D4×C3⋊D311yescharacteristic
270C2×D3—68no—
271C2×D3—124no—
272C2×D3—124no—
273C3×C4—24no—
274C3×C4—24no—
275C3×C4—41no—
276C3×C4—44no—
277C3×C4—84no—
278C3×C4—124no—
279C3×C4—181no—
280C3×C4—181no—
281C3×C4#379914yes—
282C3×C4C4×D3214yes—
283C3×C4C2×C4×C3⋊D311yescharacteristic
284C3×C4#383211yescharacteristic
285Dic3—41no—
286Dic3—64no—
287Dic3—64no—
288Dic3—182no—
289Dic3—188no—
290Dic3—368no—
291C32—24no—
292C32#488914yescomplemented
293C32#501811yescharacteristic, complemented
294C23—181no—
295Q8—272no—
296Q8—541no—
297D4—92no—
298D4—181no—
299C2×C4—21no—
300C2×C4—61no—
301C2×C4—91no—
302C2×C4—91no—
303C2×C4—362no—
304C2×C4—541no—
305C2×C4C2×D3×C3⋊D311yescharacteristic
306C8—32no—
307C8—541no—
308C2×C3—24no—
309C2×C3—24no—
310C2×C3—24no—
311C2×C3—362no—
312C2×C3#6390414yes—
313C2×C3#6393214yes—
314C2×C3#6394314yes—
315C2×C3#6410211yescharacteristic
316C2×C3#6412011yescharacteristic
317C2×C3#6412711yescharacteristic
318D3—128no—
319C22—182no—
320C22—361no—
321C22—361no—
322C22C3⋊D3×#6711yescharacteristic
323C4—41no—
324C4—121no—
325C4—181no—
326C4—181no—
327C4—542no—
328C4#8901811yescharacteristic
329C4C4×D3×C3⋊D311yescharacteristic
330C3—24no—
331C3#1059059014yescomplemented
332C3#1059258811yescharacteristic, complemented
333C2—362no—
334C2#1172351211yescharacteristic
335C2#1172356011yescharacteristic
336C2#1172357611yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#440
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
2. #11723406C2semidirect
4. #11722180C2semidirect
5. C32×#807C2semidirect
7. #11722020C2semidirect
21. #88796C4semidirect
34. C3×#807D3semidirect
42. #6308C2×C4semidirect
50. C32×C3⋊C8C2×C4semidirect
104. #807C3⋊D3semidirect
139. C3×C3⋊C8C4×D3semidirect
206. C33#440semidirect
240. C3⋊C8C4×C3⋊D3semidirect
292. C32#4889semidirect
293. C32#5018semidirect
331. C3#10590590semidirect
332. C3#10592588semidirect

Maximal subgroups

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

subgroupindex
C3×#641022
#117234062
C2×#887962
#117221802
C32×#8072
#117220332
#117220202
#105925883
#105905903