Finite Group Databasebeta

← #13161329

Subgroups — #13161329

Order 1728.

1004 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 1004 conjugacy classes of subgroups — too many to draw. Only its 164 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
1C32×#1007C211yesmaximal, characteristic
2C3×#64153C211yesmaximal, characteristic, complemented
3#11723373C212yesmaximal, complemented
4#11722206C211yesmaximal, characteristic, complemented
5#11722044C212yesmaximal, complemented
6#10592945—31nomaximal
7#10591648—34nomaximal
8C2×C33×Q8C2211yescharacteristic
9C2×C3×#634C2211yescharacteristic
10C32×#239—22no—
11C32×D3×Q8—22no—
12C32×#235—22no—
13C32×#235C2212yes—
14#88855C2212yes—
15C3×#3750—22no—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17#88798—22no—
18#88792—22no—
19C32×2−1+4—31no—
20C3×#1007D314yes—
21#64160—32no—
22#64153D311yescharacteristic, complemented
23C3×#944—34no—
24#63827—38no—
25#63664—32no—
26#63663—31no—
27#63663—34no—
28#63663—64no—
29#63569—38no—
30C33×Q8—22no—
31C2×C33×C4D412yes—
32C2×C33×C4C2311yescharacteristic
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3—21no—
35C3×#634—22no—
36C32×#67—22no—
37C32×D12—22no—
38C32×D12D412yes—
39C32×C4×D3—24no—
40C32×C4×D3—22no—
41C32×Dic6—22no—
42C32×Dic6D412yes—
43C3⋊(C3⋊(C3⋊C8))—22no—
44#5246—94no—
45#5095—91no—
46C32×#41—32no—
47C32×#41—62no—
48C32×#41—62no—
49C2×C32×Q8—24no—
50C2×C32×Q8—62no—
51C2×C32×Q8C2×D314yes—
52C2×C32×Q8C2×D311yescharacteristic
53C2×#634C2×D311yescharacteristic
54C3×#239—28no—
55C3×D3×Q8—28no—
56C3×#235—28no—
57C3×#235C2×D318yes—
58C2×C3×Dic6—34no—
59#3756—62no—
60#3754—62no—
61#3750—22no—
62C4×C3⋊Dic3C2×D311yescharacteristic
63#3748—32no—
64#3748—38no—
65#3748—68no—
66C3×Dic3⋊C4—68no—
67C3×C4×Dic3—34no—
68#3718—68no—
69#3715—68no—
70C22×C33C2×D411yescharacteristic
71C2×C32×D3—22no—
72C33×C4—22no—
73C33×C4C2×D412yes—
74C3×C3⋊Dic3—22no—
75C3×C3⋊Dic3—41no—
76C32×Dic3—22no—
77C3×2−1+4—34no—
78#1007C3⋊D311yescharacteristic
79C3×C4⋊Q8—91no—
80#951—98no—
81#944—34no—
82#942—92no—
83#837—98no—
84#836—94no—
85#836—91no—
86#836—184no—
87#805—92no—
88C32×Q8—22no—
89C32×Q8—28no—
90C32×Q8—32no—
91C32×Q8—48no—
92C32×Q8—62no—
93C32×D4—32no—
94C32×D4—62no—
95C32×D4—62no—
96C2×C32×C4—28no—
97C2×C32×C4—24no—
98C2×C32×C4—64no—
99C2×C32×C4—62no—
100C2×C32×C4#6718yes—
101C2×C32×C4#6712yes—
102C2×C32×C4C22×D314yes—
103C2×C32×C4C22×D311yescharacteristic
104C2×C3⋊Dic3—21no—
105C2×C3⋊Dic3—21no—
106#634—22no—
107C3×#67—28no—
108C2×C3×Dic3—64no—
109C2×C3×Dic3—64no—
110C3×D12—28no—
111C3×D12#6718yes—
112C3×C4×D3—216no—
113C3×C4×D3—28no—
114C3×Dic6—28no—
115C3×Dic6—68no—
116C3×Dic6#6718yes—
117C3⋊(C3⋊C8)—62no—
118C3⋊(C3⋊C8)—68no—
119C3⋊(C3⋊C8)—128no—
120#456—271noSylow
121C2×C33—21no—
122C2×C33#12011yescharacteristic
123C32×D3—42no—
124C3×#41—38no—
125C3×#41—68no—
126C3×#41—68no—
127C2×C3×Q8—24no—
128C2×C3×Q8—68no—
129C2×C3×Q8—91no—
130C2×C3×Q8D3214yes—
131C2×C3×Q8C2×C3⋊D311yescharacteristic
132#239—22no—
133D3×Q8—22no—
134#235—22no—
135#235C2×C3⋊D312yes—
136C2×Dic6—34no—
137C3×C4⋊C4—182no—
138C3×C42—91no—
139#216—188no—
140#216—182no—
141#215—182no—
142#214—188no—
143Dic3⋊C4—68no—
144C4×Dic3—34no—
145#208—98no—
146#208—92no—
147#208—188no—
148C22×C32—24no—
149C22×C32—62no—
150C22×C32C2×#6714yes—
151C22×C32C2×#6711yescharacteristic
152C2×C3×D3—28no—
153C32×C4—22no—
154C32×C4—28no—
155C32×C4—28no—
156C32×C4—48no—
157C32×C4—62no—
158C32×C4D3×D412yes—
159C32×C4C2×#6718yes—
160C3⋊Dic3—22no—
161C3⋊Dic3—41no—
162C3×Dic3—28no—
163C3×Dic3—68no—
164C3×Dic3—124no—
1652−1+4—31no—
166#137—272no—
167C4⋊Q8—91no—
168#104—272no—
169#101—271no—
170C33#45611yescharacteristic, complemented, Sylow
171C3×Q8—28no—
172C3×Q8—22no—
173C3×Q8—38no—
174C3×Q8—48no—
175C3×Q8—68no—
176C3×Q8—182no—
177C3×D4—38no—
178C3×D4—68no—
179C3×D4—68no—
180C2×C3×C4—28no—
181C2×C3×C4—24no—
182C2×C3×C4—616no—
183C2×C3×C4—68no—
184C2×C3×C4—181no—
185C2×C3×C4—181no—
186C2×C3×C4#62518yes—
187C2×C3×C4#63812yes—
188C2×C3×C4C2×D3214yes—
189C2×C3×C4C22×C3⋊D311yescharacteristic
190#67—22no—
191C2×Dic3—64no—
192C2×Dic3—64no—
193D12—22no—
194D12#63812yes—
195C4×D3—24no—
196C4×D3—22no—
197Dic6—22no—
198Dic6—68no—
199Dic6#63812yes—
200C3⋊C8—188no—
201C3⋊C8—182no—
202C3⋊C8—368no—
203C2×C32—21no—
204C2×C32—24no—
205C2×C32—24no—
206C2×C32—44no—
207C2×C32—122no—
208C2×C32#93711yescharacteristic
209C2×C32#95314yes—
210C3×D3—48no—
211#41—32no—
212#41—62no—
213#41—62no—
214C2×Q8—62no—
215C2×Q8—91no—
216C2×Q8D3×C3⋊D311yescharacteristic
217Q16—542no—
218SD16—542no—
219M16—272no—
220C4⋊C4—182no—
221C42—91no—
222C22×C3—24no—
223C22×C3—68no—
224C22×C3C2×#62514yes—
225C22×C3C2×#63811yescharacteristic
226C2×D3—22no—
227C3×C4—22no—
228C3×C4—28no—
229C3×C4—28no—
230C3×C4—48no—
231C3×C4—68no—
232C3×C4—182no—
233C3×C4—361no—
234C3×C4D3×#6718yes—
235C3×C4C2×#63812yes—
236Dic3—22no—
237Dic3—68no—
238Dic3—124no—
239C32—24no—
240C32#509511yescharacteristic, complemented
241C32#524614yescomplemented
242Q8—22no—
243Q8—32no—
244Q8—62no—
245Q8—182no—
246D4—32no—
247D4—62no—
248D4—62no—
249C2×C4—64no—
250C2×C4—62no—
251C2×C4—181no—
252C2×C4—181no—
253C2×C4#629012yes—
254C2×C4C2×D3×C3⋊D311yescharacteristic
255C8—542no—
256C2×C3—24no—
257C2×C3—21no—
258C2×C3—24no—
259C2×C3—44no—
260C2×C3—128no—
261C2×C3#6397614yes—
262C2×C3#6416211yescharacteristic
263D3—42no—
264C22—62no—
265C22C2×#629011yescharacteristic
266C4—22no—
267C4—62no—
268C4—182no—
269C4—361no—
270C4D3×#63812yes—
271C3—24no—
272C3#1059164814yescomplemented
273C3#1059294511yescharacteristic, complemented
274C2—21no—
275C2—122no—
276C2#1172362911yescharacteristic
subgroupG/H
centerC2#11723629
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC2×C33#120
prime radicalC32×#1007C2
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#456
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C3×#64153C2semidirect
3. #11723373C2semidirect
4. #11722206C2semidirect
5. #11722044C2semidirect
22. #64153D3semidirect
170. C33#456semidirect
240. C32#5095semidirect
241. C32#5246semidirect
272. C3#10591648semidirect
273. C3#10592945semidirect

Maximal subgroups

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

subgroupindex
C32×#10072
C3×#641532
#117233732
#117222062
#117220442
#105929453
#105916483