Finite Group Databasebeta

← #13144760

Subgroups — #13144760

Order 1728.

682 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 113 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×Q16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#11721864C211yesmaximal, characteristic
3#11721861C211yesmaximal, characteristic, complemented
4C2×#88550C211yesmaximal, characteristic, complemented
5C2×#88549C211yesmaximal, characteristic
6C2×#88528C211yesmaximal, characteristic
7#11720548C211yesmaximal, characteristic, complemented
8C3×#5190—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12C2×#6229C2211yescharacteristic
13Q16×H(3,𝔽3)—22no—
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15#88550—22no—
16#88549—22no—
17#88542—22no—
18#88540—22no—
19C4×#1084C2211yescharacteristic
20C2×#6180C2211yescharacteristic
21#88528—22no—
22C2×C32×Q16D314yescomplemented
23C3×#947—34no—
24C3×#944—34no—
25C2×C3×#216—34no—
26C2×C3×#215—34no—
27C2×C3×#204—34no—
28C3×#815—34no—
29C22×#272—21no—
30Q8×H(3,𝔽3)—21no—
31Q8×H(3,𝔽3)—21no—
32Q8×H(3,𝔽3)—21no—
33Q8×H(3,𝔽3)—21no—
34C2×C4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)—21no—
36C2×C4×H(3,𝔽3)C2311yescharacteristic
37C2×#1084—21no—
38C2×#1084D412yes—
39C4⋊#272—21no—
40C4⋊#272—21no—
41#6229—21no—
42#6229—21no—
43C8×H(3,𝔽3)D412yes—
44#6180D412yes—
45C3×#497—91no—
46#5190—34no—
47C2×C32×Q8C2×D314yes—
48C2×C32×Q8C2×D314yes—
49C2×C3×D12—34no—
50C2×C3×Dic6—34no—
51C32×Q16—28no—
52C2×C32×C8C2×D314yes—
53C3×#216—68no—
54C3×#215—68no—
55C3×(C2×D3)⋊C4—68no—
56C3×Dic3⋊C4—68no—
57C3×C4×Dic3—34no—
58C2×C3×C3⋊C8—34no—
59C3×#204—68no—
60C22×H(3,𝔽3)C2×D411yescharacteristic
61C2×#272—41no—
62C2×#272—42no—
63C4×H(3,𝔽3)—41no—
64C4×H(3,𝔽3)—41no—
65C4×H(3,𝔽3)C2×D411yescharacteristic
66C4×H(3,𝔽3)C2×D411yescharacteristic
67#1084—22no—
68#1084—41no—
69C2×C3×Q16—34no—
70C2×C3×Q16—91no—
71C2×C3×Q16C3⋊D311yescharacteristic
72C2×C3×SD16—91no—
73C2×C3×SD16—91no—
74C3×C4⋊Q8—91no—
75C3×#124—91no—
76#947—34no—
77#944—34no—
78C2×#216—34no—
79C2×#215—34no—
80C2×#204—34no—
81C3×#97—91no—
82#815—34no—
83C22×C3×D3—64no—
84C32×Q8—24no—
85C32×Q8—24no—
86C32×Q8—24no—
87C32×Q8—24no—
88C2×C32×C4—24no—
89C2×C32×C4—24no—
90C2×C32×C4C22×D314yes—
91C2×C3×Dic3—38no—
92C2×C3×Dic3—64no—
93C3×D12—64no—
94C3×D12—64no—
95C3×Dic6—64no—
96C3×Dic6—64no—
97C32×C8#6718yes—
98C3×C3⋊C8—38no—
99#497—91noSylow
100C2×H(3,𝔽3)#12711yescharacteristic
101C2×H(3,𝔽3)#13712yes—
102#272—81no—
103C2×C3×Q8—34no—
104C2×C3×Q8—34no—
105C2×C3×Q8—91no—
106C2×C3×Q8C2×C3⋊D311yescharacteristic
107C2×C3×Q8C2×C3⋊D311yescharacteristic
108C2×C3×D4—91no—
109C2×D12—34no—
110C2×Dic6—34no—
111C3×Q16—22no—
112C3×Q16—68no—
113C3×Q16—182no—
114C3×SD16—182no—
115C3×SD16—182no—
116C2×C3×C8—34no—
117C2×C3×C8—91no—
118C2×C3×C8C2×C3⋊D311yescharacteristic
119C3×C4⋊C4—182no—
120C3×C22⋊C4—182no—
121C3×C42—91no—
122#216—68no—
123#215—68no—
124(C2×D3)⋊C4—68no—
125Dic3⋊C4—68no—
126C4×Dic3—34no—
127C2×C3⋊C8—34no—
128#204—68no—
129C22×C32D3×D414yes—
130C2×C3×D3—128no—
131C2×C3×D3—124no—
132C32×C4—44no—
133C32×C4—44no—
134C32×C4D3×D414yes—
135C32×C4C2×#6714yes—
136C3×Dic3—68no—
137C3×Dic3—124no—
138C2×Q16—91no—
139C2×Q16#27211yescharacteristic, complemented
140C2×SD16—91no—
141C2×SD16—91no—
142C4⋊Q8—91no—
143#124—91no—
144#97—91no—
145H(3,𝔽3)#49711yescharacteristic, complemented, Sylow
146C23×C3—181no—
147C22×D3—64no—
148C3×Q8—21no—
149C3×Q8—21no—
150C3×Q8—21no—
151C3×Q8—21no—
152C3×Q8—64no—
153C3×Q8—64no—
154C3×Q8—64no—
155C3×Q8—64no—
156C3×Q8—181no—
157C3×Q8—181no—
158C3×D4—181no—
159C3×D4—181no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—34no—
163C2×C3×C4—64no—
164C2×C3×C4—64no—
165C2×C3×C4—92no—
166C2×C3×C4—181no—
167C2×C3×C4C22×C3⋊D311yescharacteristic
168C2×Dic3—38no—
169C2×Dic3—64no—
170D12—64no—
171D12—64no—
172Dic6—64no—
173Dic6—64no—
174C3×C8—38no—
175C3×C8—92no—
176C3×C8#63812yes—
177C3⋊C8—38no—
178C2×C32#91818yes—
179C2×C32#94014yes—
180C3×D3—244no—
181C2×Q8—91no—
182C2×Q8C2×#27211yescharacteristic
183C2×Q8C2×#27211yescharacteristic
184C2×D4—91no—
185Q16—22no—
186Q16—182no—
187SD16—182no—
188SD16—182no—
189C2×C8—91no—
190C2×C8C2×#27211yescharacteristic
191C4⋊C4—182no—
192C22⋊C4—182no—
193C42—91no—
194C22×C3—34no—
195C22×C3—362no—
196C22×C3—361no—
197C22×C3D4×C3⋊D311yescharacteristic
198C2×D3—124no—
199C2×D3—128no—
200C3×C4—34no—
201C3×C4—34no—
202C3×C4—41no—
203C3×C4—41no—
204C3×C4—124no—
205C3×C4—124no—
206C3×C4—182no—
207C3×C4—361no—
208C3×C4D4×C3⋊D311yescharacteristic
209C3×C4C2×#63811yescharacteristic
210Dic3—68no—
211Dic3—124no—
212C32#519014yescomplemented
213C23—181no—
214Q8—21no—
215Q8—21no—
216Q8—21no—
217Q8—21no—
218Q8—181no—
219Q8—181no—
220D4—181no—
221D4—181no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—92no—
225C2×C4—181no—
226C2×C4C22×#27211yescharacteristic
227C8—92no—
228C8#623512yes—
229C2×C3—38no—
230C2×C3—34no—
231C2×C3—721no—
232C2×C3#6412712yes—
233C2×C3#6414911yescharacteristic
234D3—244no—
235C22—362no—
236C22—361no—
237C22D4×#27211yescharacteristic
238C4—41no—
239C4—41no—
240C4—182no—
241C4—361no—
242C4D4×#27211yescharacteristic
243C4C2×#623511yescharacteristic
244C3—34no—
245C3#1059288911yescharacteristic
246C2—721no—
247C2#1172180712yes—
248C2#1172185011yescharacteristic
subgroupG/H
center, socleC22×C3D4×C3⋊D3
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingC2×Q16×H(3,𝔽3)C2
prime radicalC2×C4⋊#272C22
hypercenterC2×C3×Q16C3⋊D3
hypocenterH(3,𝔽3)#497
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q16

Complements

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

NH ≅ G/Ntype
1. C2×Q16×H(3,𝔽3)C2semidirect
3. #11721861C2semidirect
4. C2×#88550C2semidirect
7. #11720548C2semidirect
22. C2×C32×Q16D3semidirect
139. C2×Q16#272semidirect
145. H(3,𝔽3)#497semidirect
212. C32#5190semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×H(3,𝔽3)2
#117218642
#117218612
C2×#885502
C2×#885492
C2×#885282
#117205482
C3×#51903