Finite Group Databasebeta

← #13158808

Subgroups — #13158808

Order 1728.

728 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64081C211yesmaximal, characteristic
2C4⋊(C3⋊(C3⋊(C3⋊C8)))C211yesmaximal, characteristic, complemented
3C32×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4#11722023C211yesmaximal, characteristic
5#11722023C211yesmaximal, characteristic
6#11722020C211yesmaximal, characteristic, complemented
7C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
8#10592539—31nomaximal
9#10590253—34nomaximal
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C33×C42C2211yescharacteristic
12C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×#3752C411yescharacteristic
15C3×#3752C411yescharacteristic
16C3×#3750C411yescharacteristic
17C3×#3750C411yescharacteristic
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#64081D311yescharacteristic
22C3×#872—34no—
23C32×C4⋊C8—31no—
24#63647—31no—
25#63647—31no—
26#63642—31no—
27C8×C3⋊Dic3—31no—
28C4⋊(C3⋊(C3⋊C8))—31no—
29C4⋊(C3⋊(C3⋊C8))—34no—
30C4⋊(C3⋊(C3⋊C8))—64no—
31C3×C4⋊(C3⋊C8)D314yescomplemented
32(C2×D3)⋊(C3⋊C8)—34no—
33(C2×D3)⋊(C3⋊C8)—34no—
34#63554—34no—
35Dic3×C3⋊C8—34no—
36C22×C3×C3⋊D3C2×C411yescharacteristic
37C2×C33×C4C2×C411yescharacteristic
38C2×C33×C4C2×C411yescharacteristic
39C2×C33×C4C2311yescharacteristic
40C2×C3×C3⋊Dic3C2×C411yescharacteristic
41C2×C3×C3⋊Dic3C2×C411yescharacteristic
42C2×C3×C3⋊Dic3C2×C411yescharacteristic
43C3×C4×C3⋊D3—21no—
44C3×C4×C3⋊D3—21no—
45C3⋊(C3⋊(C3⋊C8))—21no—
46C3⋊(C3⋊(C3⋊C8))—21no—
47C32×C3⋊C8—21no—
48C32×C3⋊C8—21no—
49#5000—91no—
50#4840—94no—
51C2×C4×C3⋊D3C2×D311yescharacteristic
52C2×C3×C4×D3—34no—
53C2×C32×C8—31no—
54C2×C32×C8—31no—
55C32×C42—24no—
56C32×C42C2×D314yes—
57C32×C42C2×D311yescharacteristic
58#3752Dic311yescharacteristic
59#3752Dic311yescharacteristic
60#3750Dic311yescharacteristic
61#3750Dic311yescharacteristic
62C4×C3⋊Dic3C2×D311yescharacteristic
63C2×C3⋊(C3⋊C8)—31no—
64C2×C3⋊(C3⋊C8)—31no—
65C2×C3⋊(C3⋊C8)—34no—
66C2×C3⋊(C3⋊C8)—34no—
67C2×C3⋊(C3⋊C8)—64no—
68C2×C3⋊(C3⋊C8)—64no—
69C3×(C2×D3)⋊C4—34no—
70C3×(C2×D3)⋊C4—34no—
71C3×Dic3⋊C4—34no—
72C3×Dic3⋊C4—34no—
73C3×C4×Dic3—34no—
74C2×C3×C3⋊C8C2×D314yes—
75C2×C3×C3⋊C8C2×D314yes—
76C22×C33C22×C411yescharacteristic
77C2×C3×C3⋊D3—21no—
78C2×C3×C3⋊D3—21no—
79C2×C3×C3⋊D3—21no—
80C33×C4—21no—
81C33×C4—21no—
82C33×C4#4111yescharacteristic
83C33×C4#4111yescharacteristic
84C3×C3⋊Dic3—21no—
85C3×C3⋊Dic3—21no—
86C3×C3⋊Dic3—21no—
87C3×#117—91no—
88#872—34no—
89C3×C4⋊C8—34no—
90#830—91no—
91#830—91no—
92(C2×D3)⋊C8—94no—
93(C2×D3)⋊C8—94no—
94#815—94no—
95C8×Dic3—94no—
96C4⋊(C3⋊C8)—94no—
97C4⋊(C3⋊C8)—91no—
98C4⋊(C3⋊C8)—184no—
99C4⋊(C3⋊C8)C3⋊D311yescharacteristic, complemented
100#803—91no—
101C4×C3⋊C8—91no—
102C22×C3⋊D3C2×Dic311yescharacteristic
103C22×C3×D3—34no—
104C2×C32×C4—24no—
105C2×C32×C4—24no—
106C2×C32×C4—24no—
107C2×C32×C4C4×D314yes—
108C2×C32×C4C4×D314yes—
109C2×C32×C4C2×Dic311yescharacteristic
110C2×C32×C4C2×Dic311yescharacteristic
111C2×C32×C4C22×D314yes—
112C2×C32×C4C22×D311yescharacteristic
113C2×C3⋊Dic3C2×Dic311yescharacteristic
114C2×C3⋊Dic3C2×Dic311yescharacteristic
115C2×C3⋊Dic3C2×Dic311yescharacteristic
116C4×C3⋊D3—21no—
117C4×C3⋊D3—21no—
118C2×C3×Dic3—34no—
119C2×C3×Dic3—34no—
120C2×C3×Dic3—34no—
121C3×C4×D3—64no—
122C3×C4×D3—64no—
123C32×C8—61no—
124C32×C8—61no—
125C3⋊(C3⋊C8)—61no—
126C3⋊(C3⋊C8)—61no—
127C3⋊(C3⋊C8)—64no—
128C3⋊(C3⋊C8)—64no—
129C3⋊(C3⋊C8)—124no—
130C3⋊(C3⋊C8)—124no—
131C3×C3⋊C8—24no—
132C3×C3⋊C8—24no—
133#433—271noSylow
134C2×C33#11711yescharacteristic
135C2×C33#13111yescharacteristic
136C2×C33#13111yescharacteristic
137C3×C3⋊D3—41no—
138C22×C3×C4—91no—
139C2×C4×D3—34no—
140C2×C3×C8—34no—
141C2×C3×C8—34no—
142C3×C4⋊C4—91no—
143C3×C4⋊C4—91no—
144C3×C22⋊C4—91no—
145C3×C22⋊C4—91no—
146C3×C42—24no—
147C3×C42—91no—
148C3×C42D3214yes—
149C3×C42C2×C3⋊D311yescharacteristic
150(C2×D3)⋊C4—34no—
151(C2×D3)⋊C4—34no—
152Dic3⋊C4—34no—
153Dic3⋊C4—34no—
154C4×Dic3—34no—
155C2×C3⋊C8—94no—
156C2×C3⋊C8—94no—
157C2×C3⋊C8—91no—
158C2×C3⋊C8—91no—
159C2×C3⋊C8—184no—
160C2×C3⋊C8—184no—
161C2×C3⋊C8C2×C3⋊D311yescharacteristic
162C2×C3⋊C8C2×C3⋊D311yescharacteristic
163C22×C32—24no—
164C22×C32C2×C4×D314yes—
165C22×C32C22×Dic311yescharacteristic
166C2×C3⋊D3—21no—
167C2×C3⋊D3—21no—
168C2×C3⋊D3—21no—
169C2×C3×D3—64no—
170C2×C3×D3—64no—
171C2×C3×D3—64no—
172C32×C4—28no—
173C32×C4—24no—
174C32×C4—21no—
175C32×C4—21no—
176C32×C4—24no—
177C32×C4—24no—
178C32×C4—24no—
179C32×C4—44no—
180C32×C4#23511yescharacteristic
181C32×C4#23511yescharacteristic
182C32×C4#23714yes—
183C32×C4#23914yes—
184C3⋊Dic3—21no—
185C3⋊Dic3—21no—
186C3⋊Dic3—21no—
187C3×Dic3—64no—
188C3×Dic3—64no—
189C3×Dic3—64no—
190#117—91no—
191C4⋊C8—31no—
192C4⋊C8—271no—
193C22⋊C8—271no—
194C22⋊C8—271no—
195#97—271no—
196C4×C8—271no—
197C33#43311yescharacteristic, complemented, Sylow
198C23×C3—91no—
199C22×D3—34no—
200C2×C3×C4—24no—
201C2×C3×C4—24no—
202C2×C3×C4—24no—
203C2×C3×C4—91no—
204C2×C3×C4—91no—
205C2×C3×C4—91no—
206C2×C3×C4—181no—
207C2×C3×C4—181no—
208C2×C3×C4D3×Dic314yes—
209C2×C3×C4D3×Dic314yes—
210C2×C3×C4C4×C3⋊D311yescharacteristic
211C2×C3×C4C4×C3⋊D311yescharacteristic
212C2×C3×C4C2×D3214yes—
213C2×C3×C4C22×C3⋊D311yescharacteristic
214C2×Dic3—34no—
215C2×Dic3—34no—
216C2×Dic3—34no—
217C4×D3—64no—
218C4×D3—64no—
219C3×C8—64no—
220C3×C8—64no—
221C3⋊C8—21no—
222C3⋊C8—21no—
223C3⋊C8—184no—
224C3⋊C8—184no—
225C3⋊C8—181no—
226C3⋊C8—181no—
227C3⋊C8—364no—
228C3⋊C8—364no—
229C2×C32—24no—
230C2×C32—24no—
231C2×C32—24no—
232C2×C32#89214yes—
233C2×C32#90114yes—
234C2×C32#90714yes—
235C2×C32#92611yescharacteristic
236C2×C32#94811yescharacteristic
237C2×C32#94811yescharacteristic
238C3⋊D3—41no—
239C3×D3—124no—
240C22×C4—91no—
241C2×C8—31no—
242C2×C8—31no—
243C2×C8—271no—
244C2×C8—271no—
245C4⋊C4—91no—
246C4⋊C4—91no—
247C22⋊C4—91no—
248C22⋊C4—91no—
249C42—91no—
250C42D3×C3⋊D311yescharacteristic
251C22×C3—24no—
252C22×C3—181no—
253C22×C3—181no—
254C22×C3—181no—
255C22×C3C2×D3×Dic314yes—
256C22×C3C2×C4×C3⋊D311yescharacteristic
257C2×D3—64no—
258C2×D3—64no—
259C2×D3—64no—
260C3×C4—28no—
261C3×C4—24no—
262C3×C4—24no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—24no—
266C3×C4—24no—
267C3×C4—44no—
268C3×C4—181no—
269C3×C4—181no—
270C3×C4—181no—
271C3×C4#379514yes—
272C3×C4#379914yes—
273C3×C4#383011yescharacteristic
274C3×C4#383211yescharacteristic
275Dic3—64no—
276Dic3—64no—
277Dic3—64no—
278C32—24no—
279C32#484014yescomplemented
280C32#500011yescharacteristic, complemented
281C23—91no—
282C2×C4—91no—
283C2×C4—91no—
284C2×C4—91no—
285C2×C4—181no—
286C2×C4—181no—
287C2×C4Dic3×C3⋊D311yescharacteristic
288C2×C4Dic3×C3⋊D311yescharacteristic
289C2×C4C2×D3×C3⋊D311yescharacteristic
290C8—61no—
291C8—61no—
292C8—541no—
293C8—541no—
294C2×C3—24no—
295C2×C3—24no—
296C2×C3—24no—
297C2×C3—361no—
298C2×C3#6381414yes—
299C2×C3#6381514yes—
300C2×C3#6384114yes—
301C2×C3#6410111yescharacteristic
302C2×C3#6411011yescharacteristic
303C2×C3#6411611yescharacteristic
304D3—124no—
305C22—181no—
306C22—181no—
307C22—181no—
308C22C2×Dic3×C3⋊D311yescharacteristic
309C4—21no—
310C4—21no—
311C4—181no—
312C4—181no—
313C4—181no—
314C4#8901411yescharacteristic
315C4#8901811yescharacteristic
316C3—24no—
317C3#1059025314yescomplemented
318C3#1059253911yescharacteristic, complemented
319C2—361no—
320C2#1172335211yescharacteristic
321C2#1172335411yescharacteristic
322C2#1172340411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutator, socleC22×C33C22×C4
FittingC33×C42C22
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C33×C4C23
hypercenterC42D3×C3⋊D3
hypocenterC33#433
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C4⋊(C3⋊(C3⋊(C3⋊C8)))C2semidirect
3. C32×C4⋊(C3⋊C8)C2semidirect
6. #11722020C2semidirect
7. C3⋊C8×C3⋊Dic3C2semidirect
31. C3×C4⋊(C3⋊C8)D3semidirect
99. C4⋊(C3⋊C8)C3⋊D3semidirect
197. C33#433semidirect
279. C32#4840semidirect
280. C32#5000semidirect
317. C3#10590253semidirect
318. C3#10592539semidirect

Maximal subgroups

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

subgroupindex
C3×#640812
C4⋊(C3⋊(C3⋊(C3⋊C8)))2
C32×C4⋊(C3⋊C8)2
#117220232
#117220202
C3⋊C8×C3⋊Dic32
#105925393
#105902533