Finite Group Databasebeta

← #13159913

Subgroups — #13159913

Order 1728.

940 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 940 conjugacy classes of subgroups — too many to draw. Only its 158 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×C8×C3⋊(C3⋊D3)C211yesmaximal, characteristic
2#11723358C212yesmaximal
3C3×#63646C211yesmaximal, characteristic, complemented
4C32×#819C211yesmaximal, characteristic, complemented
5#11722059C212yesmaximal, complemented
6C3⋊D3×#108—31nomaximal
7#10590913—34nomaximal
8C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
9C2×C33×C8C2211yescharacteristic
10C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
11C8×C3⋊(C3⋊D3)C412yes—
12C8×C3⋊(C3⋊D3)C412yes—
13C3×#3748C2212yes—
14C32×#208C2212yes—
15#88788—22no—
16#88785—22no—
17M16×C3⋊D3—32no—
18C2×C8×C3⋊D3—31no—
19C2×C8×C3⋊D3—34no—
20C2×C8×C3⋊D3—64no—
21#63818—38no—
22C32×#108—31no—
23#63649—32no—
24#63646D311yescharacteristic, complemented
25C3×#819—34no—
26C3×#819D314yescomplemented
27#63578—38no—
28C22×C3⋊(C3⋊D3)Q811yescharacteristic
29C2×C33×C4C2311yescharacteristic
30C2×C3⋊(C3⋊Dic3)Q811yescharacteristic
31C4×C3⋊(C3⋊D3)C2×C411yescharacteristic
32C4×C3⋊(C3⋊D3)C2×C411yescharacteristic
33C4×C3⋊(C3⋊D3)D411yescharacteristic
34C4×C3⋊(C3⋊D3)D411yescharacteristic
35C33×C8C2×C412yes—
36C3⋊(C3⋊(C3⋊C8))C2×C412yes—
37C3×C3⋊(C3⋊C8)—22no—
38C32×C3⋊C8—22no—
39D3×#108—94no—
40D3×#108—91no—
41C2×C4×C3⋊D3—31no—
42C2×C4×C3⋊D3—34no—
43C2×C4×C3⋊D3—64no—
44C32×M16—32no—
45C2×C32×C8—24no—
46C2×C32×C8C2×D314yes—
47C2×C32×C8C2×D311yescharacteristic
48#3748C2×D312yes—
49C2×C3⋊(C3⋊C8)—31no—
50C2×C3⋊(C3⋊C8)—34no—
51C2×C3⋊(C3⋊C8)—64no—
52#3743—62no—
53C8×C3⋊D3—32no—
54C8×C3⋊D3—32no—
55C8×C3⋊D3—38no—
56C8×C3⋊D3—38no—
57C8×C3⋊D3—62no—
58C8×C3⋊D3—68no—
59C8×C3⋊D3—68no—
60C3×#208—38no—
61C3×#208C2×D318yes—
62#3712—68no—
63#3710—68no—
64C22×C33C2×Q811yescharacteristic
65C2×C3⋊(C3⋊D3)—22no—
66C2×C3⋊(C3⋊D3)C4⋊C411yescharacteristic
67C2×C3⋊(C3⋊D3)C4⋊C411yescharacteristic
68C33×C4C22×C411yescharacteristic
69C33×C4C2×D411yescharacteristic
70C3⋊(C3⋊Dic3)C4⋊C411yescharacteristic
71C3⋊(C3⋊Dic3)C4⋊C411yescharacteristic
72D3×M16—98no—
73D3×M16—92no—
74C2×D3×C8—94no—
75C2×D3×C8—91no—
76C2×D3×C8—184no—
77C3×#108—34no—
78C3×#108—91no—
79#822—98no—
80#822—92no—
81#819—34no—
82#819C3⋊D311yescharacteristic, complemented
83C22×C3⋊D3—31no—
84C22×C3⋊D3—34no—
85C22×C3⋊D3—64no—
86C2×C32×C4—24no—
87C2×C32×C4C22×D314yes—
88C2×C32×C4C22×D311yescharacteristic
89C2×C3⋊Dic3—31no—
90C2×C3⋊Dic3—34no—
91C2×C3⋊Dic3—64no—
92C4×C3⋊D3—31no—
93C4×C3⋊D3—31no—
94C4×C3⋊D3—31no—
95C4×C3⋊D3—31no—
96C4×C3⋊D3—34no—
97C4×C3⋊D3—34no—
98C4×C3⋊D3—34no—
99C4×C3⋊D3—34no—
100C4×C3⋊D3—64no—
101C4×C3⋊D3—64no—
102C4×C3⋊D3—64no—
103C4×C3⋊D3—64no—
104C32×C8—28no—
105C32×C8—62no—
106C32×C8C4×D318yes—
107C32×C8C4×D312yes—
108C3⋊(C3⋊C8)—22no—
109C3⋊(C3⋊C8)—32no—
110C3⋊(C3⋊C8)—38no—
111C3⋊(C3⋊C8)—68no—
112C3×C3⋊C8—28no—
113C3×C3⋊C8—68no—
114C2×#108—271noSylow
115C2×C33—21no—
116C2×C33C2×C4⋊C411yescharacteristic
117C3⋊(C3⋊D3)—21no—
118C3⋊(C3⋊D3)#10812yescomplemented
119C2×C4×D3—94no—
120C2×C4×D3—91no—
121C2×C4×D3—184no—
122C3×M16—38no—
123C3×M16—92no—
124C2×C3×C8—24no—
125C2×C3×C8D3214yes—
126C2×C3×C8C2×C3⋊D311yescharacteristic
127#208—38no—
128#208C2×C3⋊D312yes—
129C2×C3⋊C8—94no—
130C2×C3⋊C8—91no—
131C2×C3⋊C8—184no—
132ΓL(1,𝔽25)—182no—
133ΓL(1,𝔽25)—188no—
134D3×C8—98no—
135D3×C8—98no—
136D3×C8—92no—
137D3×C8—92no—
138D3×C8—188no—
139D3×C8—188no—
140D3×C8—182no—
141D3×C8—188no—
142C22×C32—24no—
143C22×C32D3×Q814yes—
144C22×C32D3×Q811yescharacteristic
145C2×C3⋊D3—31no—
146C2×C3⋊D3—31no—
147C2×C3⋊D3—34no—
148C2×C3⋊D3—34no—
149C2×C3⋊D3—64no—
150C2×C3⋊D3—64no—
151C2×C3⋊D3—62no—
152C2×C3⋊D3—68no—
153C2×C3⋊D3—616no—
154C32×C4—24no—
155C32×C4—24no—
156C32×C4C2×C4×D314yes—
157C32×C4C2×C4×D311yescharacteristic
158C32×C4D3×D414yes—
159C32×C4D3×D411yescharacteristic
160C3⋊Dic3—31no—
161C3⋊Dic3—31no—
162C3⋊Dic3—34no—
163C3⋊Dic3—34no—
164C3⋊Dic3—64no—
165C3⋊Dic3—64no—
166C2×M16—272no—
167C22×C8—271no—
168#108—31no—
169#108—91no—
170#108—272no—
171C33C2×#10811yescharacteristic, complemented, Sylow
172C22×D3—94no—
173C22×D3—91no—
174C22×D3—184no—
175C2×C3×C4—24no—
176C2×C3×C4C2×D3214yes—
177C2×C3×C4C22×C3⋊D311yescharacteristic
178C2×Dic3—94no—
179C2×Dic3—91no—
180C2×Dic3—184no—
181C4×D3—94no—
182C4×D3—94no—
183C4×D3—94no—
184C4×D3—94no—
185C4×D3—91no—
186C4×D3—91no—
187C4×D3—91no—
188C4×D3—91no—
189C4×D3—184no—
190C4×D3—184no—
191C4×D3—184no—
192C4×D3—184no—
193C3×C8—28no—
194C3×C8—68no—
195C3×C8—182no—
196C3×C8#62418yes—
197C3×C8C4×C3⋊D312yes—
198C3⋊C8—22no—
199C3⋊C8—68no—
200C3⋊C8—98no—
201C3⋊C8—92no—
202C3⋊C8—188no—
203C2×C32—28no—
204C2×C32—24no—
205C2×C32—21no—
206C2×C32—24no—
207C2×C32D3×C4⋊C414yes—
208C2×C32D3×C4⋊C411yescharacteristic
209C3⋊D3—32no—
210C3⋊D3—38no—
211C3⋊D3—68no—
212C3⋊D3—68no—
213C3⋊D3—64no—
214C3⋊D3—61no—
215C22×C4—271no—
216M16—32no—
217M16—92no—
218M16—542no—
219C2×C8—271no—
220C2×C8—272no—
221C2×C8—272no—
222C2×C8—542no—
223C2×C8D3×C3⋊D311yescharacteristic
224C22×C3—24no—
225C22×C3#379714yes—
226C22×C3Q8×C3⋊D311yescharacteristic
227C2×D3—91no—
228C2×D3—94no—
229C2×D3—91no—
230C2×D3—94no—
231C2×D3—184no—
232C2×D3—184no—
233C2×D3—182no—
234C2×D3—188no—
235C2×D3—1816no—
236C3×C4—24no—
237C3×C4—24no—
238C3×C4#380214yes—
239C3×C4C2×#62414yes—
240C3×C4C2×C4×C3⋊D311yescharacteristic
241C3×C4D4×C3⋊D311yescharacteristic
242Dic3—91no—
243Dic3—91no—
244Dic3—94no—
245Dic3—94no—
246Dic3—184no—
247Dic3—184no—
248C32—24no—
249C32D3×#10814yescomplemented
250C32D3×#10811yescharacteristic, complemented
251C23—271no—
252C2×C4—271no—
253C2×C4—271no—
254C2×C4—271no—
255C2×C4—271no—
256C2×C4—271no—
257C2×C4C2×D3×C3⋊D311yescharacteristic
258C8—62no—
259C8—182no—
260C8—272no—
261C8#628912yes—
262C2×C3—28no—
263C2×C3—21no—
264C2×C3—24no—
265C2×C3—24no—
266C2×C3#6390014yes—
267C2×C3C4⋊C4×C3⋊D311yescharacteristic
268D3—92no—
269D3—98no—
270D3—188no—
271D3—188no—
272D3—181no—
273D3—184no—
274C22—271no—
275C22—271no—
276C22—542no—
277C22#8901911yescharacteristic
278C4—271no—
279C4—271no—
280C4#8902611yescharacteristic
281C4C2×#628911yescharacteristic
282C3—24no—
283C3#1059091314yescomplemented
284C3C3⋊D3×#10811yescharacteristic, complemented
285C2—21no—
286C2—272no—
287C2—541no—
288C2#1172350311yescharacteristic
subgroupG/H
centerC4#89026
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC2×C33C2×C4⋊C4
prime radicalC22×C3⋊(C3⋊D3)Q8
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33C2×#108
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. C3×#63646C2semidirect
4. C32×#819C2semidirect
5. #11722059C2semidirect
24. #63646D3semidirect
26. C3×#819D3semidirect
82. #819C3⋊D3semidirect
118. C3⋊(C3⋊D3)#108semidirect
171. C33C2×#108semidirect
249. C32D3×#108semidirect
250. C32D3×#108semidirect
283. C3#10590913semidirect
284. C3C3⋊D3×#108semidirect

Maximal subgroups

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

subgroupindex
C2×C8×C3⋊(C3⋊D3)2
#117233582
C3×#636462
C32×#8192
#117220592
C3⋊D3×#1083
#105909133