Finite Group Databasebeta

← #13143722

Subgroups — #13143722

Order 1728.

737 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 737 conjugacy classes of subgroups — too many to draw. Only its 167 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
1C2×C4⋊#1084C211yesmaximal, characteristic
2C2×#88540C211yesmaximal, characteristic
3H(3,𝔽3)×#95C211yesmaximal, characteristic
4#11720574C211yesmaximal, characteristic
5#11720574C211yesmaximal, characteristic
6#11720574C211yesmaximal, characteristic
7#11720574C211yesmaximal, characteristic
8C3×#4660—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⋊#1084—22no—
17#88540—22no—
18C2×C3×C4⋊Dic3—34no—
19C2×C3×Dic3⋊C4—34no—
20C32×#95D314yes—
21C3×C3⋊#95—34no—
22C3×C3⋊#95—34no—
23C3×C3⋊#95—34no—
24C3×C3⋊#95—34no—
25C23×H(3,𝔽3)C2311yescharacteristic
26C2×C4×H(3,𝔽3)—21no—
27C2×C4×H(3,𝔽3)—21no—
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)Q812yes—
35C2×#1084—21no—
36C2×#1084—21no—
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×#1084D412yes—
47C3×#400—91no—
48#4660—34no—
49C22×C32×C4C2×D314yes—
50C22×C32×C4C2×D314yes—
51C22×C32×C4C2×D314yes—
52C22×C3×Dic3—34no—
53C22×C3×Dic3—34no—
54C22×C3×Dic3—34no—
55C22×C3×Dic3—34no—
56C3×C4⋊Dic3—68no—
57C3×Dic3⋊C4—68no—
58C22×H(3,𝔽3)C2×D411yescharacteristic
59C22×H(3,𝔽3)C2×Q811yescharacteristic
60C22×H(3,𝔽3)#4111yescharacteristic
61C22×H(3,𝔽3)#4111yescharacteristic
62C22×H(3,𝔽3)#4111yescharacteristic
63C22×H(3,𝔽3)#4111yescharacteristic
64C22×H(3,𝔽3)#4111yescharacteristic
65C4×H(3,𝔽3)—22no—
66C4×H(3,𝔽3)—41no—
67C4×H(3,𝔽3)—41no—
68#1084—22no—
69#1084—41no—
70#1084—41no—
71#1084—41no—
72C2×C3×C4⋊C4—91no—
73C2×C3×C4⋊C4—91no—
74C2×C4⋊Dic3—34no—
75C2×Dic3⋊C4—34no—
76C3×#95—34no—
77C3×#95—91no—
78C3×#95—91no—
79C3×#95—91no—
80C3×#95—91no—
81C3×#95C3⋊D311yescharacteristic
82C3⋊#95—34no—
83C3⋊#95—34no—
84C3⋊#95—34no—
85C3⋊#95—34no—
86C23×C32C22×D314yes—
87C2×C32×C4—24no—
88C2×C32×C4—24no—
89C2×C32×C4—24no—
90C2×C32×C4—24no—
91C2×C32×C4—24no—
92C2×C32×C4—24no—
93C2×C32×C4—24no—
94C2×C32×C4—24no—
95C2×C32×C4Dic618yes—
96C2×C3×Dic3—38no—
97C2×C3×Dic3—64no—
98C2×C3×Dic3—64no—
99C2×C3×Dic3—64no—
100C2×C3×Dic3—64no—
101C2×C3×Dic3—64no—
102C2×C3×Dic3—64no—
103C2×C3×Dic3—64no—
104C2×C3×Dic3—64no—
105C2×C3×Dic3—64no—
106C2×C3×Dic3—64no—
107C2×C3×Dic3—64no—
108#400—91noSylow
109C2×H(3,𝔽3)C22⋊Q811yescharacteristic
110C2×H(3,𝔽3)#12311yescharacteristic
111C2×H(3,𝔽3)#12411yescharacteristic
112C2×H(3,𝔽3)#12511yescharacteristic
113C2×H(3,𝔽3)#12511yescharacteristic
114C2×H(3,𝔽3)#12611yescharacteristic
115C2×H(3,𝔽3)#12611yescharacteristic
116C22×C3×C4—34no—
117C22×C3×C4—34no—
118C22×C3×C4—34no—
119C22×C3×C4—91no—
120C22×C3×C4—91no—
121C22×C3×C4—91no—
122C22×C3×C4—91no—
123C22×C3×C4C2×C3⋊D311yescharacteristic
124C22×C3×C4C2×C3⋊D311yescharacteristic
125C22×C3×C4C2×C3⋊D311yescharacteristic
126C22×Dic3—34no—
127C22×Dic3—34no—
128C22×Dic3—34no—
129C22×Dic3—34no—
130C3×C4⋊C4—182no—
131C3×C4⋊C4—182no—
132C4⋊Dic3—68no—
133Dic3⋊C4—68no—
134C22×C32C2×Dic614yes—
135C22×C32#23514yes—
136C22×C32#23514yes—
137C22×C32D3×D414yes—
138C22×C32#23714yes—
139C22×C32#23714yes—
140C22×C32#23914yes—
141C32×C4—28no—
142C32×C4—44no—
143C32×C4—44no—
144C3×Dic3—68no—
145C3×Dic3—124no—
146C3×Dic3—124no—
147C3×Dic3—124no—
148C2×C4⋊C4—91no—
149C2×C4⋊C4—91no—
150#95—91no—
151#95—91no—
152#95—91no—
153#95—91no—
154#95#27211yescharacteristic
155H(3,𝔽3)#40011yescharacteristic, complemented, Sylow
156C23×C3—34no—
157C23×C3C22×C3⋊D311yescharacteristic
158C2×C3×C4—21no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—38no—
167C2×C3×C4—64no—
168C2×C3×C4—64no—
169C2×C3×C4—64no—
170C2×C3×C4—64no—
171C2×C3×C4—64no—
172C2×C3×C4—64no—
173C2×C3×C4—64no—
174C2×C3×C4—64no—
175C2×C3×C4—92no—
176C2×C3×C4—181no—
177C2×C3×C4—181no—
178C2×C3×C4—181no—
179C2×C3×C4—181no—
180C2×C3×C4—181no—
181C2×C3×C4—181no—
182C2×C3×C4—181no—
183C2×C3×C4—181no—
184C2×C3×C4—181no—
185C2×C3×C4—181no—
186C2×C3×C4—181no—
187C2×C3×C4#63412yes—
188C2×Dic3—38no—
189C2×Dic3—64no—
190C2×Dic3—64no—
191C2×Dic3—64no—
192C2×Dic3—64no—
193C2×Dic3—64no—
194C2×Dic3—64no—
195C2×Dic3—64no—
196C2×Dic3—64no—
197C2×Dic3—64no—
198C2×Dic3—64no—
199C2×Dic3—64no—
200C2×C32#87014yes—
201C2×C32#87814yes—
202C2×C32#87914yes—
203C2×C32#88514yes—
204C2×C32#89014yes—
205C2×C32#89414yes—
206C2×C32#89814yes—
207C22×C4—91no—
208C22×C4—91no—
209C22×C4—91no—
210C22×C4—91no—
211C22×C4C2×#27211yescharacteristic
212C22×C4C2×#27211yescharacteristic
213C22×C4C2×#27211yescharacteristic
214C4⋊C4—182no—
215C4⋊C4—182no—
216C22×C3—34no—
217C22×C3—34no—
218C22×C3—34no—
219C22×C3—34no—
220C22×C3—34no—
221C22×C3—34no—
222C22×C3—34no—
223C22×C3C2×#63411yescharacteristic
224C22×C3#382811yescharacteristic
225C22×C3#382811yescharacteristic
226C22×C3D4×C3⋊D311yescharacteristic
227C22×C3#383011yescharacteristic
228C22×C3#383011yescharacteristic
229C22×C3#383211yescharacteristic
230C3×C4—22no—
231C3×C4—41no—
232C3×C4—41no—
233C3×C4—68no—
234C3×C4—124no—
235C3×C4—124no—
236C3×C4—182no—
237C3×C4—361no—
238C3×C4—361no—
239C3×C4—361no—
240Dic3—68no—
241Dic3—124no—
242Dic3—124no—
243Dic3—124no—
244C32#466014yescomplemented
245C23C22×#27211yescharacteristic
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—92no—
255C2×C4—181no—
256C2×C4—181no—
257C2×C4—181no—
258C2×C4—181no—
259C2×C4—181no—
260C2×C4—181no—
261C2×C4—181no—
262C2×C4—181no—
263C2×C4—181no—
264C2×C4—181no—
265C2×C4—181no—
266C2×C4#622912yes—
267C2×C3—34no—
268C2×C3—34no—
269C2×C3—34no—
270C2×C3—34no—
271C2×C3—34no—
272C2×C3—34no—
273C2×C3—34no—
274C2×C3#6407911yescharacteristic
275C2×C3#6408711yescharacteristic
276C2×C3#6408811yescharacteristic
277C2×C3#6409411yescharacteristic
278C2×C3#6409911yescharacteristic
279C2×C3#6410311yescharacteristic
280C2×C3#6410711yescharacteristic
281C22C2×#622911yescharacteristic
282C22#8874011yescharacteristic
283C22#8874011yescharacteristic
284C22D4×#27211yescharacteristic
285C22#8874411yescharacteristic
286C22#8874411yescharacteristic
287C22#8874811yescharacteristic
288C4—22no—
289C4—41no—
290C4—41no—
291C4—182no—
292C4—361no—
293C4—361no—
294C4—361no—
295C3—34no—
296C3#1059235911yescharacteristic
297C2#1172171611yescharacteristic
298C2#1172173411yescharacteristic
299C2#1172173511yescharacteristic
300C2#1172174111yescharacteristic
301C2#1172175811yescharacteristic
302C2#1172176211yescharacteristic
303C2#1172176611yescharacteristic
subgroupG/H
center, Frattini, socleC23×C3C22×C3⋊D3
commutator, prime radical, cosocleC23×H(3,𝔽3)C23
FittingH(3,𝔽3)×#95C2
hypercenterC3×#95C3⋊D3
hypocenterH(3,𝔽3)#400
LCSG ▷ C23×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C3×#95

Complements

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

NH ≅ G/Ntype
155. H(3,𝔽3)#400semidirect
244. C32#4660semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#10842
C2×#885402
H(3,𝔽3)×#952
#117205742
C3×#46603