Finite Group Databasebeta

← #13159355

Subgroups — #13159355

Order 1728.

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

normal subgroups only

This group has 936 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, complemented
2D12×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C2×#88789C211yesmaximal, characteristic
4#11722178C211yesmaximal, characteristic, complemented
5C32×#809C211yesmaximal, characteristic, complemented
6#11722034C211yesmaximal, characteristic, complemented
7#11722020C211yesmaximal, characteristic, complemented
8#10592484—31nomaximal
9#10590455—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×C4×C3⋊D3—21no—
12C2×C32×D12C2211yescharacteristic
13C2×D3×C3⋊Dic3—21no—
14C33×C4⋊C4C2211yescharacteristic
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×#3752—21no—
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20(C2×D3)⋊(C3⋊Dic3)—21no—
21#88789C414yescomplemented
22D4×C3⋊Dic3—31no—
23C2×#3753—31no—
24#64102D311yescharacteristic, complemented
25C3×#893—34no—
26Dic3×D12—34no—
27C2×#3713—34no—
28C32×#102—31no—
29#63648—31no—
30#63642—31no—
31#63635—31no—
32#63635—34no—
33#63635—64no—
34C3×#809D314yescomplemented
35#63563—34no—
36#63554—34no—
37C22×C3×C3⋊D3—21no—
38C22×C32×D3—21no—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42C2×C3×C3⋊Dic3D411yescharacteristic
43C2×C3×C3⋊Dic3D411yescharacteristic
44C3×C4⋊(C3⋊D3)—21no—
45C3×C4⋊(C3⋊D3)C2×C412yescomplemented
46C3×C4×C3⋊D3—42no—
47C32×D12—21no—
48C32×D12C2×C412yescomplemented
49D3×C3⋊Dic3—42no—
50C3⋊(C3⋊(C3⋊C8))C2×C412yescomplemented
51C32×C3⋊C8—21no—
52#5014—91no—
53#4785—94no—
54C2×C32×D4—31no—
55C22×C3⋊Dic3—61no—
56C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
57C2×C4×C3⋊D3—21no—
58C2×C3×D12—34no—
59C2×C3×D12C2×D314yes—
60C2×C3×C4×D3—64no—
61C2×D3×Dic3—64no—
62C2×C32×C8—31no—
63C32×C4⋊C4—24no—
64C32×C4⋊C4C2×D314yes—
65C32×C4⋊C4C2×D311yescharacteristic
66#3757—61no—
67#3753—34no—
68#3752—21no—
69C4⋊(C3⋊Dic3)—31no—
70C4⋊(C3⋊Dic3)—34no—
71C4⋊(C3⋊Dic3)—64no—
72C4×C3⋊Dic3C2×D311yescharacteristic
73C2×C3⋊(C3⋊C8)—31no—
74C2×C3⋊(C3⋊C8)—34no—
75C2×C3⋊(C3⋊C8)—64no—
76C3×(C2×D3)⋊C4—64no—
77C3×C4×Dic3—34no—
78C2×C3×C3⋊C8C2×D314yes—
79(C2×D3)⋊Dic3—64no—
80#3713—316no—
81C22×C33C2×D411yescharacteristic
82C2×C3×C3⋊D3—22no—
83C2×C3×C3⋊D3—41no—
84C2×C3×C3⋊D3—41no—
85C2×C32×D3—22no—
86C2×C32×D3—41no—
87C2×C32×D3—41no—
88C33×C4—41no—
89C33×C4C22×C411yescharacteristic
90C33×C4#4111yescharacteristic
91C3⋊(C3⋊Dic3)—41no—
92C3×C3⋊Dic3—21no—
93C3×C3⋊Dic3—21no—
94C3×C4×D4—91no—
95D4×Dic3—94no—
96C2×#213—94no—
97C2×#213—91no—
98#893—34no—
99C4×D12—91no—
100C3×#102—34no—
101#832—91no—
102#821—94no—
103#815—94no—
104#809C3⋊D311yescharacteristic, complemented
105#808—94no—
106#808—91no—
107#808—184no—
108#803—91no—
109C23×C32—61no—
110C22×C3⋊D3—21no—
111C22×C3×D3—24no—
112C22×C3×D3—64no—
113C32×D4—32no—
114C32×D4—61no—
115C2×C32×C4—21no—
116C2×C32×C4—24no—
117C2×C32×C4—24no—
118C2×C32×C4—44no—
119C2×C32×C4C22×D314yes—
120C2×C32×C4C22×D311yescharacteristic
121C2×C3⋊Dic3—61no—
122C2×C3⋊Dic3—64no—
123C2×C3⋊Dic3—124no—
124C2×C3⋊Dic3—122no—
125C2×C3⋊Dic3#6711yescharacteristic
126C2×C3⋊Dic3#6711yescharacteristic
127C4⋊(C3⋊D3)—21no—
128C4⋊(C3⋊D3)C4×D312yescomplemented
129C4×C3⋊D3—42no—
130C2×C3×Dic3—34no—
131C2×C3×Dic3—34no—
132C3×D12—24no—
133C3×D12—38no—
134C3×D12—64no—
135C3×D12C4×D318yescomplemented
136C3×C4×D3—128no—
137D3×Dic3—128no—
138C32×C8—61no—
139C3⋊(C3⋊C8)—32no—
140C3⋊(C3⋊C8)—38no—
141C3⋊(C3⋊C8)—68no—
142C3×C3⋊C8—24no—
143#442—271noSylow
144C2×C33C4×D411yescharacteristic
145C2×C33AGL(1,ℤ/8)11yescharacteristic
146C2×C33AGL(1,ℤ/8)11yescharacteristic
147C3×C3⋊D3—42no—
148C32×D3—42no—
149C2×C3×D4—34no—
150C2×C3×D4—91no—
151C22×C3×C4—181no—
152C22×Dic3—184no—
153C2×D12—34no—
154C2×D12C2×C3⋊D311yescharacteristic
155C2×C4×D3—64no—
156C2×C4×D3—181no—
157C2×C3×C8—34no—
158C3×C4⋊C4—24no—
159C3×C4⋊C4D3214yes—
160C3×C4⋊C4C2×C3⋊D311yescharacteristic
161C3×C22⋊C4—181no—
162C3×C42—91no—
163#217—184no—
164#213—916no—
165#213—94no—
166(C2×D3)⋊C4—64no—
167(C2×D3)⋊C4—181no—
168C4⋊Dic3—94no—
169C4⋊Dic3—91no—
170C4⋊Dic3—184no—
171C4×Dic3—34no—
172C2×C3⋊C8—94no—
173C2×C3⋊C8—91no—
174C2×C3⋊C8—184no—
175C2×C3⋊C8C2×C3⋊D311yescharacteristic
176C22×C32—24no—
177C22×C32—62no—
178C22×C32—121no—
179C22×C32—121no—
180C22×C32D3×D414yes—
181C22×C32C2×#6711yescharacteristic
182C2×C3⋊D3—22no—
183C2×C3⋊D3—41no—
184C2×C3⋊D3—41no—
185C2×C3×D3—28no—
186C2×C3×D3—44no—
187C2×C3×D3—44no—
188C2×C3×D3—68no—
189C2×C3×D3—124no—
190C2×C3×D3—124no—
191C32×C4—24no—
192C32×C4—24no—
193C32×C4—41no—
194C32×C4—44no—
195C32×C4—84no—
196C32×C4C2×C4×D314yes—
197C32×C4C2×C4×D311yescharacteristic
198C32×C4#23511yescharacteristic
199C32×C4#23714yes—
200C3⋊Dic3—21no—
201C3⋊Dic3—21no—
202C3⋊Dic3—121no—
203C3⋊Dic3—124no—
204C3⋊Dic3—244no—
205C3×Dic3—64no—
206C3×Dic3—64no—
207C2×D8—271no—
208C4×D4—91no—
209C4×D4—271no—
210#106—271no—
211#102—31no—
212#102—271no—
213#97—271no—
214C33#44211yescharacteristic, complemented, Sylow
215C23×C3—64no—
216C23×C3—181no—
217C22×D3—21no—
218C22×D3—64no—
219C3×D4—38no—
220C3×D4—64no—
221C3×D4—92no—
222C3×D4—181no—
223C2×C3×C4—24no—
224C2×C3×C4—21no—
225C2×C3×C4—24no—
226C2×C3×C4—44no—
227C2×C3×C4—91no—
228C2×C3×C4—91no—
229C2×C3×C4—362no—
230C2×C3×C4C2×D3214yes—
231C2×C3×C4C22×C3⋊D311yescharacteristic
232C2×Dic3—34no—
233C2×Dic3—34no—
234C2×Dic3—184no—
235C2×Dic3—181no—
236C2×Dic3—364no—
237C2×Dic3—368no—
238D12—21no—
239D12—38no—
240D12—64no—
241D12C4×C3⋊D312yescomplemented
242C4×D3—128no—
243C4×D3—362no—
244C3×C8—64no—
245C3⋊C8—21no—
246C3⋊C8—98no—
247C3⋊C8—92no—
248C3⋊C8—188no—
249C2×C32—24no—
250C2×C32—24no—
251C2×C32—24no—
252C2×C32—122no—
253C2×C32#67⋊C414yes—
254C2×C32#91114yes—
255C2×C32#91414yes—
256C2×C32C4×#6711yescharacteristic
257C2×C32#93211yescharacteristic
258C2×C32#94911yescharacteristic
259C3⋊D3—42no—
260C3×D3—48no—
261C3×D3—128no—
262C2×D4—31no—
263C2×D4—91no—
264C22×C4—181no—
265C22×C4—541no—
266D8—274no—
267C2×C8—31no—
268C2×C8—271no—
269C4⋊C4—271no—
270C4⋊C4D3×C3⋊D311yescharacteristic
271C22⋊C4—181no—
272C22⋊C4—541no—
273C42—91no—
274C22×C3—24no—
275C22×C3—68no—
276C22×C3—124no—
277C22×C3—124no—
278C22×C3—182no—
279C22×C3—361no—
280C22×C3—361no—
281C22×C3D3×#6714yes—
282C22×C3D4×C3⋊D311yescharacteristic
283C2×D3—22no—
284C2×D3—41no—
285C2×D3—41no—
286C2×D3—68no—
287C2×D3—124no—
288C2×D3—124no—
289C3×C4—24no—
290C3×C4—24no—
291C3×C4—41no—
292C3×C4—44no—
293C3×C4—84no—
294C3×C4—181no—
295C3×C4—181no—
296C3×C4#379514yes—
297C3×C4C4×D3214yes—
298C3×C4C2×C4×C3⋊D311yescharacteristic
299C3×C4#383011yescharacteristic
300Dic3—64no—
301Dic3—64no—
302Dic3—361no—
303Dic3—364no—
304Dic3—724no—
305C32—24no—
306C32#478514yescomplemented
307C32#501411yescharacteristic, complemented
308C23—61no—
309C23—181no—
310D4—32no—
311D4—61no—
312D4—92no—
313D4—181no—
314C2×C4—21no—
315C2×C4—91no—
316C2×C4—91no—
317C2×C4—362no—
318C2×C4—541no—
319C2×C4—1082no—
320C2×C4C2×D3×C3⋊D311yescharacteristic
321C8—61no—
322C8—272no—
323C2×C3—24no—
324C2×C3—24no—
325C2×C3—24no—
326C2×C3—128no—
327C2×C3—362no—
328C2×C3#6387914yes—
329C2×C3#6393214yes—
330C2×C3#6393914yes—
331C2×C3#6409011yescharacteristic
332C2×C3#6412011yescharacteristic
333C2×C3#6412311yescharacteristic
334D3—42no—
335D3—128no—
336C22—62no—
337C22—121no—
338C22—121no—
339C22—182no—
340C22—361no—
341C22—361no—
342C22C3⋊D3×#6711yescharacteristic
343C4—41no—
344C4—181no—
345C4—181no—
346C4—1081no—
347C4#8901411yescharacteristic
348C4C4×D3×C3⋊D311yescharacteristic
349C3—24no—
350C3#1059045514yescomplemented
351C3#1059248411yescharacteristic, complemented
352C2—122no—
353C2—362no—
354C2#1172350711yescharacteristic
355C2#1172356011yescharacteristic
356C2#1172357011yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×#88789C2
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#442
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
1. C3×#64102C2semidirect
2. D12×C3⋊Dic3C2semidirect
4. #11722178C2semidirect
5. C32×#809C2semidirect
6. #11722034C2semidirect
7. #11722020C2semidirect
21. #88789C4semidirect
24. #64102D3semidirect
34. C3×#809D3semidirect
45. C3×C4⋊(C3⋊D3)C2×C4semidirect
48. C32×D12C2×C4semidirect
50. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
104. #809C3⋊D3semidirect
128. C4⋊(C3⋊D3)C4×D3semidirect
135. C3×D12C4×D3semidirect
214. C33#442semidirect
241. D12C4×C3⋊D3semidirect
306. C32#4785semidirect
307. C32#5014semidirect
350. C3#10590455semidirect
351. C3#10592484semidirect

Maximal subgroups

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

subgroupindex
C3×#641022
D12×C3⋊Dic32
C2×#887892
#117221782
C32×#8092
#117220342
#117220202
#105924843
#105904553