Finite Group Databasebeta

← #13144503

Subgroups — #13144503

Order 1728.

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

normal subgroups only

This group has 715 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
1H(3,𝔽3)×#124C211yesmaximal, characteristic, complemented
2C2×#88550C211yesmaximal, characteristic, complemented
3C2×#88549C211yesmaximal, characteristic, complemented
4C2×#88546C211yesmaximal, characteristic, complemented
5C2×#88545C211yesmaximal, characteristic
6#11721721C211yesmaximal, characteristic, complemented
7C4×#6180C211yesmaximal, characteristic, complemented
8C3×#5059—34nomaximal
9C2×Q8×H(3,𝔽3)C2211yescharacteristic
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12C2×#6229C2211yescharacteristic
13C22⋊C4×H(3,𝔽3)—22no—
14C42×H(3,𝔽3)C2211yescharacteristic
15#88550—22no—
16#88549—22no—
17#88546—22no—
18#88545—22no—
19#88542—22no—
20C2×#6180C2211yescharacteristic
21C2×#6180C2211yescharacteristic
22C32×#124D314yescomplemented
23C2×C3×#216—34no—
24C2×C3×#215—34no—
25C2×C3×#214—34no—
26C2×C3×#213—34no—
27C3×#875—34no—
28C3×C4×C3⋊C8—34no—
29C23×H(3,𝔽3)—21no—
30C22×#272—21no—
31Q8×H(3,𝔽3)—21no—
32Q8×H(3,𝔽3)—21no—
33D4×H(3,𝔽3)—21no—
34D4×H(3,𝔽3)—21no—
35C2×C4×H(3,𝔽3)—21no—
36C2×C4×H(3,𝔽3)D412yes—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C2×#1084—21no—
39C4⋊#272—21no—
40C4⋊#272—21no—
41#6229—21no—
42#6229—21no—
43#6180D412yes—
44#6180D412yes—
45C3×#495—91no—
46#5059—34no—
47C2×C32×Q8C2×D314yes—
48C2×C32×D4C2×D314yes—
49C2×C3×D12—34no—
50C2×C3×Dic6—34no—
51C32×C22⋊C4—28no—
52C32×C42C2×D314yes—
53C3×#216—68no—
54C3×#215—68no—
55C3×#214—68no—
56C3×#213—68no—
57C3×(C2×D3)⋊C4—68no—
58C2×C3×C3⋊C8—34no—
59C2×C3×C3⋊C8—34no—
60C22×H(3,𝔽3)—41no—
61C22×H(3,𝔽3)—42no—
62C22×H(3,𝔽3)C2×D411yescharacteristic
63C2×#272—41no—
64C2×#272—42no—
65C4×H(3,𝔽3)—22no—
66C4×H(3,𝔽3)—41no—
67C4×H(3,𝔽3)C2×D411yescharacteristic
68C4×H(3,𝔽3)C2×D411yescharacteristic
69#1084—41no—
70C2×C3×Q16—91no—
71C2×C3×SD16—91no—
72C2×C3×SD16—91no—
73C2×C3×D8—91no—
74C3×#124—34no—
75C3×#124—91no—
76C3×#124C3⋊D311yescharacteristic
77C2×#216—34no—
78C2×#215—34no—
79C2×#214—34no—
80C2×#213—34no—
81#875—34no—
82C3×C4×C8—91no—
83C4×C3⋊C8—34no—
84C23×C32—24no—
85C22×C3×D3—64no—
86C32×Q8—24no—
87C32×Q8—24no—
88C32×D4—24no—
89C32×D4—24no—
90C2×C32×C4—24no—
91C2×C32×C4#6718yes—
92C2×C32×C4C22×D314yes—
93C2×C3×Dic3—64no—
94C3×D12—64no—
95C3×D12—64no—
96C3×Dic6—64no—
97C3×Dic6—64no—
98C3×C3⋊C8—38no—
99C3×C3⋊C8—38no—
100#495—91noSylow
101C2×H(3,𝔽3)—81no—
102C2×H(3,𝔽3)#12711yescharacteristic
103C2×H(3,𝔽3)#13512yes—
104#272—81no—
105C2×C3×Q8—34no—
106C2×C3×Q8—91no—
107C2×C3×Q8C2×C3⋊D311yescharacteristic
108C2×C3×D4—34no—
109C2×C3×D4—91no—
110C2×C3×D4C2×C3⋊D311yescharacteristic
111C2×D12—34no—
112C2×Dic6—34no—
113C3×Q16—182no—
114C3×SD16—182no—
115C3×SD16—182no—
116C3×D8—182no—
117C2×C3×C8—91no—
118C2×C3×C8—91no—
119C3×C22⋊C4—22no—
120C3×C22⋊C4—68no—
121C3×C22⋊C4—182no—
122C3×C42—34no—
123C3×C42C2×C3⋊D311yescharacteristic
124#216—68no—
125#215—68no—
126#214—68no—
127#213—68no—
128(C2×D3)⋊C4—68no—
129C2×C3⋊C8—34no—
130C2×C3⋊C8—34no—
131C22×C32—44no—
132C22×C32—48no—
133C22×C32C2×#6714yes—
134C2×C3×D3—124no—
135C2×C3×D3—128no—
136C32×C4—28no—
137C32×C4—44no—
138C32×C4D3×D414yes—
139C32×C4D3×D414yes—
140C3×Dic3—124no—
141C2×Q16—91no—
142C2×SD16—91no—
143C2×SD16—91no—
144C2×D8—91no—
145#124—91no—
146#124#27211yescharacteristic, complemented
147C4×C8—91no—
148H(3,𝔽3)#49511yescharacteristic, complemented, Sylow
149C23×C3—21no—
150C23×C3—64no—
151C23×C3—181no—
152C22×D3—64no—
153C3×Q8—21no—
154C3×Q8—21no—
155C3×Q8—64no—
156C3×Q8—64no—
157C3×Q8—181no—
158C3×Q8—181no—
159C3×D4—21no—
160C3×D4—21no—
161C3×D4—64no—
162C3×D4—64no—
163C3×D4—181no—
164C3×D4—181no—
165C2×C3×C4—21no—
166C2×C3×C4—38no—
167C2×C3×C4—34no—
168C2×C3×C4—64no—
169C2×C3×C4—181no—
170C2×C3×C4#63812yes—
171C2×C3×C4C22×C3⋊D311yescharacteristic
172C2×Dic3—64no—
173D12—64no—
174D12—64no—
175Dic6—64no—
176Dic6—64no—
177C3×C8—92no—
178C3×C8—92no—
179C3⋊C8—38no—
180C3⋊C8—38no—
181C2×C32—84no—
182C2×C32#94014yes—
183C2×C32#95018yes—
184C3×D3—244no—
185C2×Q8—91no—
186C2×Q8C2×#27211yescharacteristic
187C2×D4—91no—
188C2×D4C2×#27211yescharacteristic
189Q16—182no—
190SD16—182no—
191SD16—182no—
192D8—182no—
193C2×C8—91no—
194C2×C8—91no—
195C22⋊C4—22no—
196C22⋊C4—182no—
197C42C2×#27211yescharacteristic
198C22×C3—34no—
199C22×C3—41no—
200C22×C3—42no—
201C22×C3—124no—
202C22×C3—128no—
203C22×C3—361no—
204C22×C3—362no—
205C22×C3C2×#63811yescharacteristic
206C2×D3—124no—
207C2×D3—128no—
208C3×C4—22no—
209C3×C4—34no—
210C3×C4—34no—
211C3×C4—41no—
212C3×C4—68no—
213C3×C4—124no—
214C3×C4—361no—
215C3×C4D4×C3⋊D311yescharacteristic
216C3×C4D4×C3⋊D311yescharacteristic
217Dic3—124no—
218C32#505914yescomplemented
219C23—21no—
220C23—181no—
221Q8—21no—
222Q8—21no—
223Q8—181no—
224Q8—181no—
225D4—21no—
226D4—21no—
227D4—181no—
228D4—181no—
229C2×C4—21no—
230C2×C4—181no—
231C2×C4#623512yes—
232C2×C4C22×#27211yescharacteristic
233C8—92no—
234C8—92no—
235C2×C3—34no—
236C2×C3—38no—
237C2×C3—81no—
238C2×C3—244no—
239C2×C3—721no—
240C2×C3#6414911yescharacteristic
241C2×C3#6415912yes—
242D3—244no—
243C22—41no—
244C22—42no—
245C22—361no—
246C22—362no—
247C22C2×#623511yescharacteristic
248C4—22no—
249C4—41no—
250C4—361no—
251C4D4×#27211yescharacteristic
252C4D4×#27211yescharacteristic
253C3—34no—
254C3#1059275811yescharacteristic
255C2—81no—
256C2—721no—
257C2#1172185011yescharacteristic
258C2#1172187112yes—
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#124C2
prime radicalC2×#88545C2
hypercenterC3×#124C3⋊D3
hypocenterH(3,𝔽3)#495
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#124

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#124C2semidirect
2. C2×#88550C2semidirect
3. C2×#88549C2semidirect
4. C2×#88546C2semidirect
6. #11721721C2semidirect
7. C4×#6180C2semidirect
22. C32×#124D3semidirect
146. #124#272semidirect
148. H(3,𝔽3)#495semidirect
218. C32#5059semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1242
C2×#885502
C2×#885492
C2×#885462
C2×#885452
#117217212
C4×#61802
C3×#50593