Finite Group Databasebeta

← #13144784

Subgroups — #13144784

Order 1728.

594 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 97 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)×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
2#11721870C211yesmaximal, characteristic, complemented
3#11721869C211yesmaximal, characteristic, complemented
4C2×#88545C211yesmaximal, characteristic, complemented
5#11720579C211yesmaximal, characteristic, complemented
6#11720560C211yesmaximal, characteristic, complemented
7#11720559C211yesmaximal, characteristic, complemented
8C3×#5199—34nomaximal
9#41×H(3,𝔽3)C2211yescharacteristic
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊#272C2211yescharacteristic
12SD16×H(3,𝔽3)—21no—
13D8×H(3,𝔽3)—21no—
14M16×H(3,𝔽3)C2211yescharacteristic
15#88549—21no—
16#88545—21no—
17#88545—21no—
18#88545—21no—
19#88538—21no—
20#88538C2211yescharacteristic
21#88538C2211yescharacteristic
22C2×#6180—21no—
23C2×#6180C2211yescharacteristic
24C32×AGL(1,ℤ/8)D314yescomplemented
25C3×#949—34no—
26C3×#948—34no—
27C2×C3×#213—34no—
28C3×#833—34no—
29C3×#823—34no—
30C3×#822—34no—
31C23×H(3,𝔽3)—21no—
32C22×#272—21no—
33Q8×H(3,𝔽3)D411yescharacteristic
34D4×H(3,𝔽3)—21no—
35D4×H(3,𝔽3)—21no—
36D4×H(3,𝔽3)—21no—
37D4×H(3,𝔽3)D411yescharacteristic
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)C2311yescharacteristic
40C4⋊#272—21no—
41C4⋊#272—21no—
42C8×H(3,𝔽3)—21no—
43#6180—21no—
44#6180—21no—
45#6180D411yescharacteristic
46#6180D411yescharacteristic
47C3×#472—91no—
48#5199—34no—
49C32×#41C2×D314yes—
50C2×C32×D4C2×D314yes—
51C2×C3×D12—34no—
52C32×SD16—24no—
53C32×D8—24no—
54C32×M16C2×D314yes—
55C3×#215—64no—
56C3×#213—64no—
57C3×#213—64no—
58C3×#213—64no—
59C3×#208—34no—
60C3×#208—34no—
61C3×#208—64no—
62C2×C3×C3⋊C8—34no—
63C2×C3×C3⋊C8—64no—
64C22×H(3,𝔽3)—21no—
65C22×H(3,𝔽3)—41no—
66C22×H(3,𝔽3)—81no—
67C22×H(3,𝔽3)#4111yescharacteristic
68C2×#272—41no—
69C2×#272—81no—
70C4×H(3,𝔽3)—21no—
71C4×H(3,𝔽3)C2×D411yescharacteristic
72C4×H(3,𝔽3)C2×D411yescharacteristic
73C3×AGL(1,ℤ/8)—34no—
74C3×AGL(1,ℤ/8)—91no—
75C3×AGL(1,ℤ/8)C3⋊D311yescharacteristic
76C2×C3×D8—91no—
77C3×#131—91no—
78#949—34no—
79#948—34no—
80C2×#213—34no—
81C3×#108—91no—
82C3×#100—91no—
83C3×#100—91no—
84#833—34no—
85#823—34no—
86#822—34no—
87C23×C32—24no—
88C22×C3×D3—64no—
89C32×Q8#6714yes—
90C32×D4—24no—
91C32×D4—24no—
92C32×D4—24no—
93C32×D4#6714yes—
94C2×C32×C4—24no—
95C2×C32×C4C22×D314yes—
96C3×D12—64no—
97C3×D12—64no—
98C32×C8—24no—
99C3×C3⋊C8—34no—
100C3×C3⋊C8—34no—
101C3×C3⋊C8—64no—
102C3×C3⋊C8—64no—
103#472—91noSylow
104C2×H(3,𝔽3)—21no—
105C2×H(3,𝔽3)—41no—
106C2×H(3,𝔽3)—81no—
107C2×H(3,𝔽3)#12111yescharacteristic
108#272—81no—
109C3×#41—34no—
110C3×#41C2×C3⋊D311yescharacteristic
111C2×C3×D4—34no—
112C2×C3×D4—91no—
113C2×C3×D4C2×C3⋊D311yescharacteristic
114C2×D12—34no—
115C3×SD16—21no—
116C3×SD16—64no—
117C3×SD16—181no—
118C3×D8—21no—
119C3×D8—64no—
120C3×D8—181no—
121C3×D8—181no—
122C3×D8—181no—
123C3×M16—34no—
124C3×M16—91no—
125C3×M16—91no—
126C3×M16—181no—
127C3×M16C2×C3⋊D311yescharacteristic
128C2×C3×C8—91no—
129C2×C3×C8—181no—
130#215—64no—
131#213—64no—
132#213—64no—
133#213—64no—
134#208—34no—
135#208—34no—
136#208—64no—
137C2×C3⋊C8—34no—
138C2×C3⋊C8—64no—
139C22×C32—24no—
140C22×C32—44no—
141C22×C32—84no—
142C22×C32#23714yes—
143C2×C3×D3—124no—
144C2×C3×D3—244no—
145C32×C4—24no—
146C32×C4D3×D414yes—
147C32×C4C2×#6714yes—
148AGL(1,ℤ/8)—91no—
149AGL(1,ℤ/8)#27211yescharacteristic, complemented
150C2×D8—91no—
151#131—91no—
152#108—91no—
153#100—91no—
154#100—91no—
155H(3,𝔽3)#47211yescharacteristic, complemented, Sylow
156C23×C3—21no—
157C23×C3—64no—
158C23×C3—181no—
159C22×D3—64no—
160C3×Q8—34no—
161C3×Q8#63811yescharacteristic
162C3×D4—21no—
163C3×D4—21no—
164C3×D4—21no—
165C3×D4—34no—
166C3×D4—64no—
167C3×D4—64no—
168C3×D4—64no—
169C3×D4—181no—
170C3×D4—181no—
171C3×D4#63811yescharacteristic
172C2×C3×C4—21no—
173C2×C3×C4—34no—
174C2×C3×C4—64no—
175C2×C3×C4C22×C3⋊D311yescharacteristic
176D12—64no—
177D12—64no—
178C3×C8—21no—
179C3×C8—64no—
180C3×C8—91no—
181C3×C8—91no—
182C3×C8—181no—
183C3×C8—181no—
184C3⋊C8—34no—
185C3⋊C8—34no—
186C3⋊C8—64no—
187C3⋊C8—64no—
188C2×C32—24no—
189C2×C32—44no—
190C2×C32—84no—
191C2×C32#93914yes—
192C3×D3—244no—
193#41C2×#27211yescharacteristic
194C2×D4—91no—
195C2×D4C2×#27211yescharacteristic
196SD16—21no—
197SD16—181no—
198D8—21no—
199D8—181no—
200D8—181no—
201D8—181no—
202M16—91no—
203M16—91no—
204M16—181no—
205M16C2×#27211yescharacteristic
206C2×C8—91no—
207C2×C8—181no—
208C22×C3—21no—
209C22×C3—34no—
210C22×C3—41no—
211C22×C3—64no—
212C22×C3—81no—
213C22×C3—124no—
214C22×C3—244no—
215C22×C3—361no—
216C22×C3—721no—
217C22×C3#383011yescharacteristic
218C2×D3—124no—
219C2×D3—244no—
220C3×C4—21no—
221C3×C4—34no—
222C3×C4—34no—
223C3×C4—64no—
224C3×C4D4×C3⋊D311yescharacteristic
225C3×C4C2×#63811yescharacteristic
226C32#519914yescomplemented
227C23—21no—
228C23—181no—
229Q8#623511yescharacteristic
230D4—21no—
231D4—21no—
232D4—21no—
233D4—181no—
234D4—181no—
235D4#623511yescharacteristic
236C2×C4—21no—
237C2×C4C22×#27211yescharacteristic
238C8—21no—
239C8—91no—
240C8—91no—
241C8—181no—
242C8—181no—
243C2×C3—21no—
244C2×C3—34no—
245C2×C3—41no—
246C2×C3—64no—
247C2×C3—81no—
248C2×C3—124no—
249C2×C3—244no—
250C2×C3—721no—
251C2×C3#6414811yescharacteristic
252D3—244no—
253C22—21no—
254C22—41no—
255C22—81no—
256C22—361no—
257C22—721no—
258C22#8874411yescharacteristic
259C4—21no—
260C4D4×#27211yescharacteristic
261C4C2×#623511yescharacteristic
262C3—34no—
263C3#1059289811yescharacteristic
264C2—21no—
265C2—41no—
266C2—81no—
267C2—721no—
268C2#1172184911yescharacteristic
subgroupG/H
center, socleC2×C3#64148
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×AGL(1,ℤ/8)C2
hypercenterC3×AGL(1,ℤ/8)C3⋊D3
hypocenterH(3,𝔽3)#472
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×AGL(1,ℤ/8)C2semidirect
2. #11721870C2semidirect
3. #11721869C2semidirect
4. C2×#88545C2semidirect
5. #11720579C2semidirect
6. #11720560C2semidirect
7. #11720559C2semidirect
24. C32×AGL(1,ℤ/8)D3semidirect
149. AGL(1,ℤ/8)#272semidirect
155. H(3,𝔽3)#472semidirect
226. C32#5199semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×AGL(1,ℤ/8)2
#117218702
#117218692
C2×#885452
#117205792
#117205602
#117205592
C3×#51993