Finite Group Databasebeta

← #13144355

Subgroups — #13144355

Order 1728.

825 conjugacy classes of subgroups.

The subgroup lattice has 315 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic
2C2×C4⋊#1084C211yesmaximal, characteristic
3C2×#88540C211yesmaximal, characteristic
4C2×#88540C211yesmaximal, characteristic
5#11720574C211yesmaximal, characteristic
6#11720574C211yesmaximal, characteristic
7#11720574C211yesmaximal, characteristic
8C3×#4982—34nomaximal
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×#1084C2211yescharacteristic
13C22×#1084C2211yescharacteristic
14C22×#1084C2211yescharacteristic
15C22×#1084C2211yescharacteristic
16C4⋊C4×H(3,𝔽3)—22no—
17C4⋊#1084—22no—
18#88540—22no—
19#88540—22no—
20C2×C32×C4⋊C4D314yes—
21C2×C3×C4⋊Dic3—34no—
22C2×C3×Dic3⋊C4—34no—
23C2×C3×Dic3⋊C4—34no—
24C3×C3⋊#95—34no—
25C3×C3⋊#95—34no—
26C3×C3⋊#95—34no—
27C23×H(3,𝔽3)C2311yescharacteristic
28C2×C4×H(3,𝔽3)—21no—
29C2×C4×H(3,𝔽3)—21no—
30C2×C4×H(3,𝔽3)—21no—
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)—21no—
33C2×C4×H(3,𝔽3)—21no—
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)D412yes—
36C2×C4×H(3,𝔽3)Q812yes—
37C2×#1084—21no—
38C2×#1084—21no—
39C2×#1084—21no—
40C2×#1084—21no—
41C2×#1084—21no—
42C2×#1084—21no—
43C2×#1084—21no—
44C2×#1084—21no—
45C2×#1084—21no—
46C2×#1084—21no—
47C2×#1084D412yes—
48C2×#1084Q812yes—
49C3×#398—91no—
50#4982—34no—
51C22×C32×C4C2×D314yes—
52C22×C32×C4C2×D314yes—
53C22×C32×C4C2×D314yes—
54C22×C3×Dic3—34no—
55C22×C3×Dic3—34no—
56C22×C3×Dic3—34no—
57C22×C3×Dic3—34no—
58C32×C4⋊C4—28no—
59C3×C4⋊Dic3—68no—
60C3×Dic3⋊C4—68no—
61C3×Dic3⋊C4—68no—
62C22×H(3,𝔽3)C2×D411yescharacteristic
63C22×H(3,𝔽3)C2×D411yescharacteristic
64C22×H(3,𝔽3)C2×Q811yescharacteristic
65C22×H(3,𝔽3)C2×Q811yescharacteristic
66C22×H(3,𝔽3)#4111yescharacteristic
67C22×H(3,𝔽3)#4111yescharacteristic
68C22×H(3,𝔽3)#4111yescharacteristic
69C4×H(3,𝔽3)—22no—
70C4×H(3,𝔽3)—22no—
71C4×H(3,𝔽3)—41no—
72#1084—22no—
73#1084—22no—
74#1084—41no—
75#1084—41no—
76C2×C3×C4⋊C4—34no—
77C2×C3×C4⋊C4—91no—
78C2×C3×C4⋊C4—91no—
79C2×C3×C4⋊C4—91no—
80C2×C3×C4⋊C4C3⋊D311yescharacteristic
81C2×C4⋊Dic3—34no—
82C2×Dic3⋊C4—34no—
83C2×Dic3⋊C4—34no—
84C3×#95—91no—
85C3×#95—91no—
86C3×#95—91no—
87C3⋊#95—34no—
88C3⋊#95—34no—
89C3⋊#95—34no—
90C23×C32C22×D314yes—
91C2×C32×C4—24no—
92C2×C32×C4—24no—
93C2×C32×C4—24no—
94C2×C32×C4—24no—
95C2×C32×C4—24no—
96C2×C32×C4—24no—
97C2×C32×C4—24no—
98C2×C32×C4Dic618yes—
99C2×C32×C4#6718yes—
100C2×C3×Dic3—38no—
101C2×C3×Dic3—38no—
102C2×C3×Dic3—64no—
103C2×C3×Dic3—64no—
104C2×C3×Dic3—64no—
105C2×C3×Dic3—64no—
106C2×C3×Dic3—64no—
107C2×C3×Dic3—64no—
108C2×C3×Dic3—64no—
109C2×C3×Dic3—64no—
110C2×C3×Dic3—64no—
111C2×C3×Dic3—64no—
112#398—91noSylow
113C2×H(3,𝔽3)#12111yescharacteristic
114C2×H(3,𝔽3)C22⋊Q811yescharacteristic
115C2×H(3,𝔽3)C22⋊Q811yescharacteristic
116C2×H(3,𝔽3)#12311yescharacteristic
117C2×H(3,𝔽3)#12511yescharacteristic
118C2×H(3,𝔽3)#12511yescharacteristic
119C2×H(3,𝔽3)C4⋊Q811yescharacteristic
120C22×C3×C4—34no—
121C22×C3×C4—34no—
122C22×C3×C4—34no—
123C22×C3×C4—91no—
124C22×C3×C4—91no—
125C22×C3×C4—91no—
126C22×C3×C4—91no—
127C22×C3×C4C2×C3⋊D311yescharacteristic
128C22×C3×C4C2×C3⋊D311yescharacteristic
129C22×C3×C4C2×C3⋊D311yescharacteristic
130C22×Dic3—34no—
131C22×Dic3—34no—
132C22×Dic3—34no—
133C22×Dic3—34no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—68no—
136C3×C4⋊C4—182no—
137C3×C4⋊C4—182no—
138C3×C4⋊C4—182no—
139C4⋊Dic3—68no—
140Dic3⋊C4—68no—
141Dic3⋊C4—68no—
142C22×C32C2×Dic614yes—
143C22×C32#23514yes—
144C22×C32D3×D414yes—
145C22×C32#23714yes—
146C22×C32D3×Q814yes—
147C22×C32#23914yes—
148C22×C32C2×#6714yes—
149C32×C4—28no—
150C32×C4—28no—
151C32×C4—44no—
152C3×Dic3—68no—
153C3×Dic3—68no—
154C3×Dic3—124no—
155C3×Dic3—124no—
156C2×C4⋊C4—91no—
157C2×C4⋊C4—91no—
158C2×C4⋊C4—91no—
159C2×C4⋊C4#27211yescharacteristic
160#95—91no—
161#95—91no—
162#95—91no—
163H(3,𝔽3)#39811yescharacteristic, complemented, Sylow
164C23×C3—34no—
165C23×C3C22×C3⋊D311yescharacteristic
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—38no—
174C2×C3×C4—38no—
175C2×C3×C4—64no—
176C2×C3×C4—64no—
177C2×C3×C4—64no—
178C2×C3×C4—64no—
179C2×C3×C4—64no—
180C2×C3×C4—64no—
181C2×C3×C4—64no—
182C2×C3×C4—92no—
183C2×C3×C4—92no—
184C2×C3×C4—181no—
185C2×C3×C4—181no—
186C2×C3×C4—181no—
187C2×C3×C4—181no—
188C2×C3×C4—181no—
189C2×C3×C4—181no—
190C2×C3×C4—181no—
191C2×C3×C4—181no—
192C2×C3×C4—181no—
193C2×C3×C4—181no—
194C2×C3×C4#63412yes—
195C2×C3×C4#63812yes—
196C2×Dic3—38no—
197C2×Dic3—38no—
198C2×Dic3—64no—
199C2×Dic3—64no—
200C2×Dic3—64no—
201C2×Dic3—64no—
202C2×Dic3—64no—
203C2×Dic3—64no—
204C2×Dic3—64no—
205C2×Dic3—64no—
206C2×Dic3—64no—
207C2×Dic3—64no—
208C2×C32#88814yes—
209C2×C32#88914yes—
210C2×C32#89014yes—
211C2×C32#89414yes—
212C2×C32#92414yes—
213C2×C32#93914yes—
214C2×C32#94614yes—
215C22×C4—91no—
216C22×C4—91no—
217C22×C4—91no—
218C22×C4—91no—
219C22×C4C2×#27211yescharacteristic
220C22×C4C2×#27211yescharacteristic
221C22×C4C2×#27211yescharacteristic
222C4⋊C4—22no—
223C4⋊C4—182no—
224C4⋊C4—182no—
225C4⋊C4—182no—
226C22×C3—34no—
227C22×C3—34no—
228C22×C3—34no—
229C22×C3—34no—
230C22×C3—34no—
231C22×C3—34no—
232C22×C3—34no—
233C22×C3C2×#63411yescharacteristic
234C22×C3#382811yescharacteristic
235C22×C3D4×C3⋊D311yescharacteristic
236C22×C3#383011yescharacteristic
237C22×C3Q8×C3⋊D311yescharacteristic
238C22×C3#383211yescharacteristic
239C22×C3C2×#63811yescharacteristic
240C3×C4—22no—
241C3×C4—22no—
242C3×C4—41no—
243C3×C4—68no—
244C3×C4—68no—
245C3×C4—124no—
246C3×C4—182no—
247C3×C4—182no—
248C3×C4—361no—
249C3×C4—361no—
250Dic3—68no—
251Dic3—68no—
252Dic3—124no—
253Dic3—124no—
254C32#498214yescomplemented
255C23C22×#27211yescharacteristic
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—92no—
264C2×C4—92no—
265C2×C4—181no—
266C2×C4—181no—
267C2×C4—181no—
268C2×C4—181no—
269C2×C4—181no—
270C2×C4—181no—
271C2×C4—181no—
272C2×C4—181no—
273C2×C4—181no—
274C2×C4—181no—
275C2×C4#622912yes—
276C2×C4#623512yes—
277C2×C3—34no—
278C2×C3—34no—
279C2×C3—34no—
280C2×C3—34no—
281C2×C3—34no—
282C2×C3—34no—
283C2×C3—34no—
284C2×C3#6409711yescharacteristic
285C2×C3#6409811yescharacteristic
286C2×C3#6409911yescharacteristic
287C2×C3#6410311yescharacteristic
288C2×C3#6413311yescharacteristic
289C2×C3#6414811yescharacteristic
290C2×C3#6415511yescharacteristic
291C22C2×#622911yescharacteristic
292C22#8874011yescharacteristic
293C22D4×#27211yescharacteristic
294C22#8874411yescharacteristic
295C22Q8×#27211yescharacteristic
296C22#8874811yescharacteristic
297C22C2×#623511yescharacteristic
298C4—22no—
299C4—22no—
300C4—41no—
301C4—182no—
302C4—182no—
303C4—361no—
304C4—361no—
305C3—34no—
306C3#1059268111yescharacteristic
307C2#1172175611yescharacteristic
308C2#1172175711yescharacteristic
309C2#1172175811yescharacteristic
310C2#1172176211yescharacteristic
311C2#1172182411yescharacteristic
312C2#1172184911yescharacteristic
313C2#1172186311yescharacteristic
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutator, prime radical, cosocleC23×H(3,𝔽3)C23
FittingC2×C4⋊C4×H(3,𝔽3)C2
hypercenterC2×C3×C4⋊C4C3⋊D3
hypocenterH(3,𝔽3)#398
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
163. H(3,𝔽3)#398semidirect
254. C32#4982semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×H(3,𝔽3)2
C2×C4⋊#10842
C2×#885402
#117205742
C3×#49823