Finite Group Databasebeta

← #13158790

Subgroups — #13158790

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
1C32×#872C211yesmaximal, characteristic
2C4⋊(C3⋊(C3⋊(C3⋊C8)))C211yesmaximal, characteristic, complemented
3C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
4(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
5(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
6#11722019C211yesmaximal, characteristic, complemented
7Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10592699—31nomaximal
9#10590253—34nomaximal
10C2×C32×C4×D3C2211yescharacteristic
11C33×C42C2211yescharacteristic
12C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×(C2×D3)⋊C4C411yescharacteristic
17C32×(C2×D3)⋊C4C411yescharacteristic
18C32×Dic3⋊C4C411yescharacteristic
19C32×Dic3⋊C4C411yescharacteristic
20C32×C4×Dic3C2211yescharacteristic
21C32×#117—31no—
22C3×#872D314yes—
23#63657—31no—
24#63657—31no—
25C4⋊(C3⋊(C3⋊C8))—31no—
26C4⋊(C3⋊(C3⋊C8))—34no—
27C4⋊(C3⋊(C3⋊C8))—64no—
28C4⋊(C3⋊(C3⋊C8))D311yescharacteristic, complemented
29#63630—31no—
30C4×C3⋊(C3⋊C8)—31no—
31C3×C4⋊(C3⋊C8)—34no—
32(C2×D3)⋊(C3⋊C8)—34no—
33(C2×D3)⋊(C3⋊C8)—34no—
34#63554—34no—
35Dic3×C3⋊C8—34no—
36C22×C32×D3C2×C411yescharacteristic
37C2×C33×C4C2×C411yescharacteristic
38C2×C33×C4C2×C411yescharacteristic
39C2×C33×C4C2311yescharacteristic
40C2×C32×Dic3C2×C411yescharacteristic
41C2×C32×Dic3C2×C411yescharacteristic
42C2×C32×Dic3C2×C411yescharacteristic
43C32×C4×D3—21no—
44C32×C4×D3—21no—
45C3⋊(C3⋊(C3⋊C8))—21no—
46C3⋊(C3⋊(C3⋊C8))—21no—
47C3×C3⋊(C3⋊C8)—21no—
48C3×C3⋊(C3⋊C8)—21no—
49#5000—94no—
50#4840—91no—
51C22×C32×C4—31no—
52C2×C3×C4×D3C2×D314yes—
53C32×C4⋊C4—31no—
54C32×C4⋊C4—31no—
55C32×C22⋊C4—31no—
56C32×C22⋊C4—31no—
57C32×C42—24no—
58C32×C42—31no—
59C32×C42C2×D311yescharacteristic
60C32×C42C2×D314yes—
61C2×C3⋊(C3⋊C8)—31no—
62C2×C3⋊(C3⋊C8)—31no—
63C2×C3⋊(C3⋊C8)—34no—
64C2×C3⋊(C3⋊C8)—34no—
65C2×C3⋊(C3⋊C8)—64no—
66C2×C3⋊(C3⋊C8)—64no—
67C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
68C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
69C3×(C2×D3)⋊C4Dic314yes—
70C3×(C2×D3)⋊C4Dic314yes—
71C3×Dic3⋊C4Dic314yes—
72C3×Dic3⋊C4Dic314yes—
73C3×C4×Dic3C2×D314yes—
74C2×C3×C3⋊C8—34no—
75C2×C3×C3⋊C8—34no—
76C22×C33C22×C411yescharacteristic
77C2×C32×D3—21no—
78C2×C32×D3—21no—
79C2×C32×D3—21no—
80C33×C4—21no—
81C33×C4—21no—
82C33×C4#4111yescharacteristic
83C33×C4#4111yescharacteristic
84C32×Dic3—21no—
85C32×Dic3—21no—
86C32×Dic3—21no—
87C3×#117—34no—
88#872C3⋊D311yescharacteristic
89C3×C4⋊C8—91no—
90#830—94no—
91#830—94no—
92(C2×D3)⋊C8—91no—
93(C2×D3)⋊C8—91no—
94#815—91no—
95C8×Dic3—91no—
96C4⋊(C3⋊C8)—34no—
97C4⋊(C3⋊C8)—94no—
98C4⋊(C3⋊C8)—91no—
99C4⋊(C3⋊C8)—184no—
100#803—94no—
101C4×C3⋊C8—94no—
102C23×C32—31no—
103C22×C3×D3C2×Dic314yes—
104C2×C32×C4—24no—
105C2×C32×C4—24no—
106C2×C32×C4—24no—
107C2×C32×C4—31no—
108C2×C32×C4—31no—
109C2×C32×C4—31no—
110C2×C32×C4—61no—
111C2×C32×C4—61no—
112C2×C32×C4C4×D311yescharacteristic
113C2×C32×C4C4×D311yescharacteristic
114C2×C32×C4C2×Dic314yes—
115C2×C32×C4C2×Dic314yes—
116C2×C32×C4C22×D314yes—
117C2×C32×C4C22×D311yescharacteristic
118C2×C3×Dic3C2×Dic314yes—
119C2×C3×Dic3C2×Dic314yes—
120C2×C3×Dic3C2×Dic314yes—
121C3×C4×D3—24no—
122C3×C4×D3—24no—
123C3⋊(C3⋊C8)—21no—
124C3⋊(C3⋊C8)—21no—
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—64no—
132C3×C3⋊C8—64no—
133#433—271noSylow
134C2×C33#11711yescharacteristic
135C2×C33#13111yescharacteristic
136C2×C33#13111yescharacteristic
137C32×D3—41no—
138C22×C3×C4—34no—
139C2×C4×D3C2×C3⋊D311yescharacteristic
140C2×C3×C8—91no—
141C2×C3×C8—91no—
142C3×C4⋊C4—34no—
143C3×C4⋊C4—34no—
144C3×C22⋊C4—34no—
145C3×C22⋊C4—34no—
146C3×C42—24no—
147C3×C42—34no—
148C3×C42D3214yes—
149C3×C42C2×C3⋊D311yescharacteristic
150(C2×D3)⋊C4C3⋊Dic311yescharacteristic
151(C2×D3)⋊C4C3⋊Dic311yescharacteristic
152Dic3⋊C4C3⋊Dic311yescharacteristic
153Dic3⋊C4C3⋊Dic311yescharacteristic
154C4×Dic3C2×C3⋊D311yescharacteristic
155C2×C3⋊C8—34no—
156C2×C3⋊C8—34no—
157C2×C3⋊C8—94no—
158C2×C3⋊C8—94no—
159C2×C3⋊C8—91no—
160C2×C3⋊C8—91no—
161C2×C3⋊C8—184no—
162C2×C3⋊C8—184no—
163C22×C32—24no—
164C22×C32—61no—
165C22×C32—61no—
166C22×C32—61no—
167C22×C32C2×C4×D311yescharacteristic
168C22×C32C22×Dic314yes—
169C2×C3×D3—24no—
170C2×C3×D3—24no—
171C2×C3×D3—24no—
172C32×C4—28no—
173C32×C4—21no—
174C32×C4—21no—
175C32×C4—24no—
176C32×C4—24no—
177C32×C4—24no—
178C32×C4—24no—
179C32×C4—44no—
180C32×C4—61no—
181C32×C4—61no—
182C32×C4—61no—
183C32×C4#23514yes—
184C32×C4#23514yes—
185C32×C4#23711yescharacteristic
186C32×C4#23911yescharacteristic
187C3×Dic3—24no—
188C3×Dic3—24no—
189C3×Dic3—24no—
190#117—31no—
191C4⋊C8—91no—
192C4⋊C8—271no—
193C22⋊C8—271no—
194C22⋊C8—271no—
195#97—271no—
196C4×C8—271no—
197C33#43311yescharacteristic, complemented, Sylow
198C23×C3—34no—
199C22×D3C2×C3⋊Dic311yescharacteristic
200C2×C3×C4—24no—
201C2×C3×C4—24no—
202C2×C3×C4—24no—
203C2×C3×C4—34no—
204C2×C3×C4—34no—
205C2×C3×C4—34no—
206C2×C3×C4—64no—
207C2×C3×C4—64no—
208C2×C3×C4D3×Dic314yes—
209C2×C3×C4D3×Dic314yes—
210C2×C3×C4C2×C3⋊Dic311yescharacteristic
211C2×C3×C4C2×C3⋊Dic311yescharacteristic
212C2×C3×C4C2×D3214yes—
213C2×C3×C4C22×C3⋊D311yescharacteristic
214C2×Dic3C2×C3⋊Dic311yescharacteristic
215C2×Dic3C2×C3⋊Dic311yescharacteristic
216C2×Dic3C2×C3⋊Dic311yescharacteristic
217C4×D3—21no—
218C4×D3—21no—
219C3×C8—181no—
220C3×C8—181no—
221C3⋊C8—64no—
222C3⋊C8—64no—
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—121no—
233C2×C32#89211yescharacteristic
234C2×C32#90111yescharacteristic
235C2×C32#90711yescharacteristic
236C2×C32#92614yes—
237C2×C32#94814yes—
238C2×C32#94814yes—
239C3×D3—44no—
240C22×C4—31no—
241C2×C8—91no—
242C2×C8—91no—
243C2×C8—271no—
244C2×C8—271no—
245C4⋊C4—31no—
246C4⋊C4—31no—
247C22⋊C4—31no—
248C22⋊C4—31no—
249C42—31no—
250C42D3×C3⋊D311yescharacteristic
251C22×C3—24no—
252C22×C3—64no—
253C22×C3—64no—
254C22×C3—64no—
255C22×C3C2×D3×Dic314yes—
256C22×C3C22×C3⋊Dic311yescharacteristic
257C2×D3—21no—
258C2×D3—21no—
259C2×D3—21no—
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—64no—
269C3×C4—64no—
270C3×C4—64no—
271C3×C4#379514yes—
272C3×C4#379914yes—
273C3×C4#382811yescharacteristic
274C3×C4#382811yescharacteristic
275Dic3—21no—
276Dic3—21no—
277Dic3—21no—
278C32—24no—
279C32#484011yescharacteristic, complemented
280C32#500014yescomplemented
281C23—31no—
282C2×C4—31no—
283C2×C4—31no—
284C2×C4—31no—
285C2×C4—61no—
286C2×C4—61no—
287C2×C4D3×C3⋊Dic311yescharacteristic
288C2×C4D3×C3⋊Dic311yescharacteristic
289C2×C4C2×D3×C3⋊D311yescharacteristic
290C8—181no—
291C8—181no—
292C8—541no—
293C8—541no—
294C2×C3—24no—
295C2×C3—24no—
296C2×C3—24no—
297C2×C3—124no—
298C2×C3#6381414yes—
299C2×C3#6381514yes—
300C2×C3#6384114yes—
301C2×C3#6413511yescharacteristic
302C2×C3#6415711yescharacteristic
303C2×C3#6415711yescharacteristic
304D3—41no—
305C22—61no—
306C22—61no—
307C22—61no—
308C22C2×D3×C3⋊Dic311yescharacteristic
309C4—21no—
310C4—21no—
311C4—61no—
312C4—61no—
313C4—61no—
314C4#8902111yescharacteristic
315C4#8902211yescharacteristic
316C3—24no—
317C3#1059025314yescomplemented
318C3#1059269911yescharacteristic, complemented
319C2—121no—
320C2#1172334911yescharacteristic
321C2#1172335011yescharacteristic
322C2#1172342411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutator, socleC22×C33C22×C4
FittingC33×C42C22
prime radicalC22×C32×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. C3×C4⋊(C3⋊(C3⋊C8))C2semidirect
6. #11722019C2semidirect
7. Dic3×C3⋊(C3⋊C8)C2semidirect
28. C4⋊(C3⋊(C3⋊C8))D3semidirect
197. C33#433semidirect
279. C32#4840semidirect
280. C32#5000semidirect
317. C3#10590253semidirect
318. C3#10592699semidirect

Maximal subgroups

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

subgroupindex
C32×#8722
C4⋊(C3⋊(C3⋊(C3⋊C8)))2
C3×C4⋊(C3⋊(C3⋊C8))2
(C2×D3)⋊(C3⋊(C3⋊C8))2
#117220192
Dic3×C3⋊(C3⋊C8)2
#105926993
#105902533