Finite Group Databasebeta

← #13154847

Subgroups — #13154847

Order 1728.

560 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3⋊#622C211yesmaximal, characteristic
2C22×C3×#622C211yesmaximal, characteristic
3C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic
4#10590207—31nomaximal
5C22×C3⋊(C3⋊Dic3)C411yescharacteristic
6C22×C3×C3⋊Dic3C2211yescharacteristic
7C22×C32×Dic3C411yescharacteristic
8C2×C3⋊#622—21no—
9C2×C3⋊#622—21no—
10C2×C3⋊#622C412yescomplemented
11C2×C3×#622—21no—
12C2×C3×#622—21no—
13C2×C3×#622C412yescomplemented
14Dic3×C3⋊Dic3—21no—
15Dic3×C3⋊Dic3—21no—
16C22×#622—31no—
17C22×#622D311yescharacteristic
18C2×C4×C3⋊Dic3—31no—
19C2×Dic32—62no—
20C23×C33C2×C411yescharacteristic
21C2×C3⋊(C3⋊Dic3)—21no—
22C2×C3⋊(C3⋊Dic3)—21no—
23C2×C3⋊(C3⋊Dic3)C812yescomplemented
24C2×C3×C3⋊Dic3C2×C411yescharacteristic
25C2×C3×C3⋊Dic3C2×C411yescharacteristic
26C2×C3×C3⋊Dic3D411yescharacteristic
27C2×C3×C3⋊Dic3D411yescharacteristic
28C2×C3×C3⋊Dic3D411yescharacteristic
29C2×C3×C3⋊Dic3Q811yescharacteristic
30C2×C32×Dic3—21no—
31C2×C32×Dic3—21no—
32C2×C32×Dic3C812yescomplemented
33C3⋊#622—22no—
34C3×#622—22no—
35#4550—91nomaximal
36C22×C32×C4—31no—
37C22×C3⋊Dic3—31no—
38C22×C3⋊Dic3—62no—
39C22×C3⋊Dic3—122no—
40C22×C3⋊Dic3C2×D311yescharacteristic
41C22×C3×Dic3—22no—
42C22×C3×Dic3—62no—
43C2×#622—21no—
44C2×#622—21no—
45C2×#622—32no—
46C2×#622—61no—
47C2×#622—61no—
48C2×#622Dic312yescomplemented
49C4×C3⋊Dic3—61no—
50C4×C3⋊Dic3—61no—
51Dic32—64no—
52Dic32—64no—
53C22×C33C4211yescharacteristic
54C22×C33C22⋊C411yescharacteristic
55C22×C33C4⋊C411yescharacteristic
56C22×C33C2×C811yescharacteristic
57C22×C33C2×C811yescharacteristic
58C22×C33M1611yescharacteristic
59C22×C33M1611yescharacteristic
60C3⋊(C3⋊Dic3)—22no—
61C3×C3⋊Dic3C22⋊C411yescharacteristic
62C3×C3⋊Dic3C22⋊C411yescharacteristic
63C3×C3⋊Dic3C4⋊C411yescharacteristic
64C3×C3⋊Dic3C4⋊C411yescharacteristic
65C32×Dic3—22no—
66C22×C3×C8—91no—
67C2×C4×Dic3—91no—
68C2×C4×Dic3—182no—
69C22×C3⋊C8—91no—
70C23×C32—22no—
71C23×C32—42no—
72C23×C32C4×D311yescharacteristic
73C2×C32×C4—32no—
74C2×C32×C4—61no—
75C2×C32×C4—61no—
76C2×C3⋊Dic3—32no—
77C2×C3⋊Dic3—64no—
78C2×C3⋊Dic3—61no—
79C2×C3⋊Dic3—61no—
80C2×C3⋊Dic3—64no—
81C2×C3⋊Dic3—64no—
82C2×C3⋊Dic3—124no—
83C2×C3⋊Dic3—124no—
84C2×C3⋊Dic3—124no—
85C2×C3⋊Dic3Dic611yescharacteristic
86C2×C3⋊Dic3C4×D311yescharacteristic
87C2×C3⋊Dic3D1211yescharacteristic
88C2×C3⋊Dic3C2×Dic311yescharacteristic
89C2×C3⋊Dic3#6711yescharacteristic
90C2×C3⋊Dic3#6711yescharacteristic
91C2×C3×Dic3—24no—
92C2×C3×Dic3—24no—
93C2×C3×Dic3—24no—
94C2×C3×Dic3—62no—
95C2×C3×Dic3—62no—
96C2×C3×Dic3—62no—
97C2×C3×Dic3—62no—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—62no—
100#622—22no—
101#622—62no—
102#336—271noSylow
103C2×C33#9511yescharacteristic
104C2×C33C4×C811yescharacteristic
105C2×C33#9711yescharacteristic
106C2×C33C22⋊C811yescharacteristic
107C2×C33C22⋊C811yescharacteristic
108C2×C33C4⋊C811yescharacteristic
109C2×C33C4⋊C811yescharacteristic
110C22×C3×C4—62no—
111C22×C3×C4—91no—
112C22×Dic3—62no—
113C22×Dic3—91no—
114C22×Dic3—182no—
115C22×Dic3—362no—
116C22×Dic3#15711yescharacteristic
117C2×C3×C8—92no—
118C2×C3×C8—181no—
119C2×C3×C8—181no—
120C4×Dic3—181no—
121C4×Dic3—181no—
122C4×Dic3—184no—
123C4×Dic3—184no—
124C2×C3⋊C8—92no—
125C2×C3⋊C8—181no—
126C2×C3⋊C8—181no—
127C22×C32—22no—
128C22×C32—22no—
129C22×C32—22no—
130C22×C32—22no—
131C22×C32—22no—
132C22×C32—22no—
133C22×C32—22no—
134C22×C32—42no—
135C22×C32—42no—
136C22×C32—42no—
137C22×C32—42no—
138C22×C32—42no—
139C22×C32—42no—
140C22×C32—42no—
141C22×C32D3×C811yescharacteristic
142C22×C32D3×C811yescharacteristic
143C22×C32ΓL(1,𝔽25)11yescharacteristic
144C22×C32ΓL(1,𝔽25)11yescharacteristic
145C22×C32C4×Dic311yescharacteristic
146C22×C32Dic3⋊C411yescharacteristic
147C22×C32(C2×D3)⋊C411yescharacteristic
148C32×C4—62no—
149C3⋊Dic3—62no—
150C3⋊Dic3—68no—
151C3⋊Dic3—128no—
152C3⋊Dic3Dic3⋊C411yescharacteristic
153C3⋊Dic3C4⋊Dic311yescharacteristic
154C3⋊Dic3(C2×D3)⋊C411yescharacteristic
155C3⋊Dic3#21711yescharacteristic
156C3×Dic3—28no—
157C3×Dic3—62no—
158C3×Dic3—62no—
159C3×Dic3—62no—
160C3×Dic3—62no—
161C22×C8—91no—
162C22×C8—271no—
163C2×C42—271no—
164C33#33611yescharacteristic, complemented, Sylow
165C23×C3—22no—
166C23×C3—42no—
167C23×C3C2×#15711yescharacteristic
168C2×C3×C4—64no—
169C2×C3×C4—64no—
170C2×C3×C4—64no—
171C2×C3×C4—91no—
172C2×C3×C4—91no—
173C2×C3×C4—91no—
174C2×C3×C4—91no—
175C2×C3×C4—91no—
176C2×C3×C4—91no—
177C2×Dic3—21no—
178C2×Dic3—21no—
179C2×Dic3—62no—
180C2×Dic3—62no—
181C2×Dic3—62no—
182C2×Dic3—62no—
183C2×Dic3—62no—
184C2×Dic3—62no—
185C2×Dic3—92no—
186C2×Dic3—184no—
187C2×Dic3—184no—
188C2×Dic3—184no—
189C2×Dic3—181no—
190C2×Dic3—181no—
191C2×Dic3—364no—
192C2×Dic3—364no—
193C2×Dic3—364no—
194C2×Dic3#62212yescomplemented
195C3×C8—182no—
196C3⋊C8—182no—
197C2×C32—22no—
198C2×C32—22no—
199C2×C32—22no—
200C2×C32—22no—
201C2×C32—22no—
202C2×C32—22no—
203C2×C32—22no—
204C2×C32—42no—
205C2×C32—42no—
206C2×C32—42no—
207C2×C32—42no—
208C2×C32—42no—
209C2×C32—42no—
210C2×C32—42no—
211C2×C32C8×Dic311yescharacteristic
212C2×C32Dic3⋊C811yescharacteristic
213C2×C32Dic3⋊C811yescharacteristic
214C2×C32#81511yescharacteristic
215C2×C32(C2×D3)⋊C811yescharacteristic
216C2×C32(C2×D3)⋊C811yescharacteristic
217C2×C32C3⋊#9511yescharacteristic
218C22×C4—31no—
219C22×C4—91no—
220C22×C4—271no—
221C2×C8—92no—
222C2×C8—181no—
223C2×C8—181no—
224C2×C8—272no—
225C2×C8—541no—
226C2×C8—541no—
227C42—541no—
228C42—541no—
229C22×C3—22no—
230C22×C3—22no—
231C22×C3—22no—
232C22×C3—22no—
233C22×C3—22no—
234C22×C3—22no—
235C22×C3—22no—
236C22×C3—42no—
237C22×C3—42no—
238C22×C3—42no—
239C22×C3—42no—
240C22×C3—42no—
241C22×C3—42no—
242C22×C3—42no—
243C22×C3#378711yescharacteristic
244C22×C3#378811yescharacteristic
245C22×C3C4×#15711yescharacteristic
246C22×C3C4⋊#15711yescharacteristic
247C22×C3C2×#62211yescharacteristic
248C22×C3#379211yescharacteristic
249C22×C3C22⋊#15711yescharacteristic
250C3×C4—68no—
251C3×C4—91no—
252C3×C4—91no—
253C3×C4—91no—
254C3×C4—91no—
255Dic3—22no—
256Dic3—62no—
257Dic3—62no—
258Dic3—62no—
259Dic3—62no—
260Dic3—188no—
261Dic3—182no—
262Dic3—368no—
263C32—22no—
264C32—42no—
265C32#455011yescharacteristic, complemented
266C23D3×#15711yescharacteristic
267C2×C4—32no—
268C2×C4—61no—
269C2×C4—61no—
270C2×C4—91no—
271C2×C4—91no—
272C2×C4—91no—
273C2×C4—91no—
274C2×C4—91no—
275C2×C4—91no—
276C2×C4—272no—
277C2×C4—541no—
278C2×C4—541no—
279C8—182no—
280C8—542no—
281C2×C3—22no—
282C2×C3—22no—
283C2×C3—22no—
284C2×C3—22no—
285C2×C3—22no—
286C2×C3—22no—
287C2×C3—22no—
288C2×C3—42no—
289C2×C3—42no—
290C2×C3—42no—
291C2×C3—42no—
292C2×C3—42no—
293C2×C3—42no—
294C2×C3—42no—
295C2×C3C4×#62211yescharacteristic
296C2×C3C4⋊#62211yescharacteristic
297C2×C3#6377711yescharacteristic
298C2×C3#6377811yescharacteristic
299C2×C3#6377911yescharacteristic
300C2×C3#6378111yescharacteristic
301C2×C3C22⋊#62211yescharacteristic
302C22Dic3×#15711yescharacteristic
303C22(C2×D3)⋊#15711yescharacteristic
304C22Dic3⋊#15711yescharacteristic
305C22D3×#62211yescharacteristic
306C22#8892411yescharacteristic
307C22#8892511yescharacteristic
308C22#8892611yescharacteristic
309C4—62no—
310C4—91no—
311C4—91no—
312C4—91no—
313C4—91no—
314C4—542no—
315C3—22no—
316C3—42no—
317C3#1059020711yescharacteristic, complemented
318C2#1172272211yescharacteristic
319C2Dic3×#62211yescharacteristic
320C2#1172272411yescharacteristic
321C2(C2×D3)⋊#62211yescharacteristic
322C2#1172272611yescharacteristic
323C2Dic3⋊#62211yescharacteristic
324C2#1172273211yescharacteristic
subgroupG/H
center, Frattini, hypercenterC23D3×#157
commutatorC2×C33C4×C8
Fitting, socle, prime radicalC23×C33C2×C4
cosocleC22×C3×C3⋊Dic3C22
hypocenterC33#336
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
10. C2×C3⋊#622C4semidirect
13. C2×C3×#622C4semidirect
23. C2×C3⋊(C3⋊Dic3)C8semidirect
32. C2×C32×Dic3C8semidirect
48. C2×#622Dic3semidirect
164. C33#336semidirect
194. C2×Dic3#622semidirect
265. C32#4550semidirect
317. C3#10590207semidirect

Maximal subgroups

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

subgroupindex
C22×C3⋊#6222
C22×C3×#6222
C2×Dic3×C3⋊Dic32
#105902073
#45509