Finite Group Databasebeta

← #13135100

Subgroups — #13135100

Order 1728.

307 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 307 conjugacy classes of subgroups — too many to draw. Only its 63 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×#88529C211yesmaximal, characteristic
2#63645⋊C3C211yesmaximal, characteristic, complemented
3C2×(C3⋊(C3⋊C16))⋊C3C211yesmaximal, characteristic, complemented
4#10589115—31nomaximal
5#10589114—31nomaximal
6C2×C4⋊#272—21no—
7C2×C8×H(3,𝔽3)C2211yescharacteristic
8#88529—21no—
9#88529C412yescomplemented
10C4⋊((C3⋊Dic3)⋊C3)—21no—
11(C3⋊(C3⋊C16))⋊C3—21no—
12C2×C3×D24—31no—
13C2×C3×D24—31no—
14C2×C3×D24—61no—
15#63645D311yescharacteristic, complemented
16C2×C3⋊(C3⋊C16)D311yescharacteristic, complemented
17C3×#818—31no—
18C2×C3×C3⋊C16—31no—
19C22×#272—41no—
20C2×C4×H(3,𝔽3)D411yescharacteristic
21C4⋊#272—22no—
22C4⋊#272—41no—
23C2×(C3⋊Dic3)⋊C3—41no—
24C8×H(3,𝔽3)C2×C411yescharacteristic
25C8×H(3,𝔽3)D411yescharacteristic
26#4562—91no—
27#4509—91no—
28#4491—91no—
29C2×C3×D12—62no—
30C2×C3×D12—61no—
31C2×C3×D12—61no—
32C2×C32×C8—21no—
33C2×C32×C8C2×D311yescharacteristic
34C2×C32×C8C2×D311yescharacteristic
35C4⋊(C3⋊Dic3)—21no—
36C3×C4⋊Dic3—61no—
37C3×D24—32no—
38C3×D24—32no—
39C3×D24—62no—
40C3×D24—62no—
41C3×D24—61no—
42C3×D24—61no—
43C3⋊(C3⋊C16)—21no—
44C3×C3⋊C16—61no—
45C22×H(3,𝔽3)D811yescharacteristic
46C2×#272—42no—
47C2×#272—81no—
48C2×#272—81no—
49C4×H(3,𝔽3)C22⋊C411yescharacteristic
50C4×H(3,𝔽3)SD1611yescharacteristic
51(C3⋊Dic3)⋊C3—81no—
52C2×C3×D8—91no—
53C2×D24—31no—
54C2×D24—31no—
55C2×D24—61no—
56C2×C3×C16—91no—
57C3×#107—91no—
58#818—31no—
59#818—91no—
60C2×C3⋊C16—31no—
61C2×C3⋊C16—91no—
62C22×C3×D3—122no—
63C22×C3×D3—121no—
64C22×C3×D3—121no—
65C2×C32×C4—21no—
66C2×C32×C4D1211yescharacteristic
67C2×C32×C4#6711yescharacteristic
68C2×C3⋊Dic3—41no—
69C2×C3×Dic3—121no—
70C3×D12—64no—
71C3×D12—62no—
72C3×D12—62no—
73C3×D12—122no—
74C3×D12—121no—
75C3×D12—121no—
76C32×C8—21no—
77C32×C8—21no—
78C32×C8C4×D311yescharacteristic
79C32×C8C4×D311yescharacteristic
80C32×C8D1211yescharacteristic
81C32×C8#6711yescharacteristic
82#357—271noSylow
83C2×H(3,𝔽3)#10211yescharacteristic
84C2×H(3,𝔽3)D1611yescharacteristic
85C2×H(3,𝔽3)SD3211yescharacteristic
86#272—82no—
87C2×C3×D4—181no—
88C2×D12—61no—
89C2×D12—61no—
90C2×D12—62no—
91C3×D8—92no—
92C3×D8—181no—
93C2×C3×C8—31no—
94C2×C3×C8—31no—
95C2×C3×C8—61no—
96C2×C3×C8D3211yescharacteristic
97C3×C4⋊C4—181no—
98C4⋊Dic3—61no—
99C4⋊Dic3—181no—
100D24—32no—
101D24—32no—
102D24—62no—
103D24—61no—
104D24—61no—
105D24—62no—
106C3×C16—181no—
107C3⋊C16—61no—
108C3⋊C16—181no—
109C22×C32—21no—
110C22×C32D2411yescharacteristic
111C22×C32#21311yescharacteristic
112C2×C3×D3—124no—
113C2×C3×D3—122no—
114C2×C3×D3—122no—
115C2×C3×D3—242no—
116C2×C3×D3—242no—
117C2×C3×D3—241no—
118C2×C3×D3—241no—
119C2×C3×D3—241no—
120C2×C3×D3—241no—
121C32×C4—21no—
122C32×C4—21no—
123C32×C4#20411yescharacteristic
124C32×C4(C2×D3)⋊C411yescharacteristic
125C32×C4(C2×D3)⋊C411yescharacteristic
126C32×C4#21511yescharacteristic
127C3⋊Dic3—81no—
128C3×Dic3—241no—
129C2×D8—91no—
130C2×C16—91no—
131#107—91no—
132H(3,𝔽3)#35711yescharacteristic, complemented, Sylow
133C23×C3—361no—
134C22×D3—121no—
135C22×D3—121no—
136C22×D3—122no—
137C3×D4—182no—
138C3×D4—361no—
139C2×C3×C4—31no—
140C2×C3×C4—31no—
141C2×C3×C4—61no—
142C2×C3×C4—361no—
143C2×C3×C4#62611yescharacteristic
144C2×Dic3—121no—
145C2×Dic3—361no—
146D12—62no—
147D12—62no—
148D12—64no—
149D12—121no—
150D12—121no—
151D12—122no—
152C3×C8—31no—
153C3×C8—31no—
154C3×C8—31no—
155C3×C8—31no—
156C3×C8—61no—
157C3×C8—61no—
158C3×C8#62411yescharacteristic
159C3×C8#62611yescharacteristic
160C2×C32—21no—
161C2×C32—21no—
162C2×C32—21no—
163C2×C32D4811yescharacteristic
164C2×C32#80011yescharacteristic
165C2×C32#80911yescharacteristic
166C2×C32#82111yescharacteristic
167C2×C32#82611yescharacteristic
168C2×C32#82811yescharacteristic
169C3×D3—244no—
170C3×D3—242no—
171C3×D3—242no—
172C2×D4—181no—
173D8—92no—
174D8—181no—
175C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
176C4⋊C4—181no—
177C16—181no—
178C22×C3—31no—
179C22×C3—31no—
180C22×C3—61no—
181C22×C3—362no—
182C22×C3—721no—
183C22×C3—721no—
184C22×C3#371411yescharacteristic
185C2×D3—122no—
186C2×D3—122no—
187C2×D3—124no—
188C2×D3—241no—
189C2×D3—241no—
190C2×D3—241no—
191C2×D3—241no—
192C2×D3—242no—
193C2×D3—242no—
194C3×C4—31no—
195C3×C4—31no—
196C3×C4—31no—
197C3×C4—31no—
198C3×C4—61no—
199C3×C4—61no—
200C3×C4—721no—
201C3×C4#371711yescharacteristic
202C3×C4#372211yescharacteristic
203Dic3—241no—
204Dic3—721no—
205C32—21no—
206C32#449111yescharacteristic, complemented
207C32#450911yescharacteristic, complemented
208C23—361no—
209D4—182no—
210D4—361no—
211C2×C4—361no—
212C2×C4#620011yescharacteristic
213C8#619911yescharacteristic
214C8#620011yescharacteristic
215C2×C3—31no—
216C2×C3—31no—
217C2×C3—31no—
218C2×C3—31no—
219C2×C3—31no—
220C2×C3—31no—
221C2×C3—61no—
222C2×C3—61no—
223C2×C3—61no—
224C2×C3—722no—
225C2×C3#6354611yescharacteristic
226C2×C3#6354911yescharacteristic
227C2×C3#6356411yescharacteristic
228D3—242no—
229D3—242no—
230D3—244no—
231C22—362no—
232C22—721no—
233C22—721no—
234C22#8843611yescharacteristic
235C4—721no—
236C4#8843711yescharacteristic
237C4#8845111yescharacteristic
238C3—31no—
239C3—31no—
240C3—61no—
241C3#1058911611yescharacteristic
242C2—722no—
243C2#1172025311yescharacteristic
244C2#1172025411yescharacteristic
245C2#1172032411yescharacteristic
subgroupG/H
centerC22#88436
commutatorC8×H(3,𝔽3)C2×C4
FrattiniC2×C3×C8D32
Fitting, cosocleC2×C8×H(3,𝔽3)C22
socleC22×C3#3714
prime radicalC2×#88529C2
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#357
LCSG ▷ C8×H(3,𝔽3) ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #63645⋊C3C2semidirect
3. C2×(C3⋊(C3⋊C16))⋊C3C2semidirect
9. #88529C4semidirect
15. #63645D3semidirect
16. C2×C3⋊(C3⋊C16)D3semidirect
132. H(3,𝔽3)#357semidirect
206. C32#4491semidirect
207. C32#4509semidirect

Maximal subgroups

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

subgroupindex
C2×#885292
#63645⋊C32
C2×(C3⋊(C3⋊C16))⋊C32
#105891153
#105891143