Finite Group Databasebeta

← #13163841

Subgroups — #13163841

Order 1728.

2044 conjugacy classes of subgroups; only the 351 normal subgroups are shown.

normal subgroups only

This group has 2044 conjugacy classes of subgroups — too many to draw. Only its 351 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
1#11724685C211yesmaximal, characteristic, complemented
2C33×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
3#11723977C211yesmaximal, characteristic, complemented
4#11723973C211yesmaximal, characteristic, complemented
5#11722207C211yesmaximal, characteristic, complemented
6#11722195C211yesmaximal, characteristic, complemented
7#11722194C211yesmaximal, characteristic, complemented
8#10592900—313nomaximal
9C33×#41C2211yescharacteristic
10C2×C33×D4C2211yescharacteristic
11Q8×C3⋊(C3⋊D3)—21no—
12#89078—21no—
13#89078—22no—
14#89076—21no—
15#89076C2211yescharacteristic
16C2×#6308—21no—
17C2×#6308C2211yescharacteristic
18C33×SD16—21no—
19C33×D8—21no—
20C33×M16C2211yescharacteristic
21#88864—22no—
22#88861—21no—
23#88860—21no—
24C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
25#88855C2211yescharacteristic
26#64367—313no—
27C32×AGL(1,ℤ/8)D3113yescomplemented
28#64145—313no—
29#64141—313no—
30#63664—313no—
31#63652—313no—
32#63651—313no—
33C23×C33—21no—
34C33×Q8D411yescharacteristic
35C33×D4—21no—
36C33×D4—21no—
37C33×D4—21no—
38C33×D4D411yescharacteristic
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41#6312—22no—
42#6312—21no—
43C2×C3⋊(C3⋊Dic3)—22no—
44C2×C3⋊(C3⋊Dic3)—21no—
45C2×C3⋊(C3⋊Dic3)D412yes—
46C4⋊(C3⋊(C3⋊D3))D411yescharacteristic
47C4×C3⋊(C3⋊D3)—22no—
48C4×C3⋊(C3⋊D3)—21no—
49#6308—22no—
50#6308—21no—
51#6308—21no—
52#6308D411yescharacteristic
53C33×C8—21no—
54C3⋊(C3⋊(C3⋊C8))—21no—
55#5201—913no—
56C32×#41C2×D3113yes—
57C2×C32×D4C2×D3113yes—
58Q8×C3⋊D3—613no—
59#3830—626no—
60#3830—613no—
61#3828—313no—
62#3828—613no—
63C2×#634—313no—
64C2×#634—613no—
65C32×SD16—213no—
66C32×D8—213no—
67C32×M16C2×D3113yes—
68#3757—626no—
69#3754—613no—
70#3753—613no—
71C4×C3⋊Dic3—313no—
72#3748—313no—
73C22×C33—21no—
74C22×C33—42no—
75C22×C33—41no—
76C22×C33C2×D411yescharacteristic
77C2×C3⋊(C3⋊D3)—21no—
78C33×C4—21no—
79C33×C4C2×D411yescharacteristic
80C33×C4C2×D411yescharacteristic
81C3⋊(C3⋊Dic3)—22no—
82C3⋊(C3⋊Dic3)—21no—
83C3⋊(C3⋊Dic3)—41no—
84#1010—913no—
85C3×AGL(1,ℤ/8)C3⋊D3113yescomplemented
86#936—913no—
87#932—913no—
88#837—913no—
89#825—913no—
90#824—913no—
91C23×C32—213no—
92C32×Q8#67113yes—
93C32×D4—213no—
94C32×D4—213no—
95C32×D4—213no—
96C32×D4#67113yes—
97C2×C32×C4—213no—
98C2×C32×C4C22×D3113yes—
99#638—626no—
100#638—613no—
101C2×C3⋊Dic3—326no—
102C2×C3⋊Dic3—626no—
103C2×C3⋊Dic3—613no—
104C4⋊(C3⋊D3)—313no—
105C4×C3⋊D3—626no—
106C4×C3⋊D3—613no—
107#634—313no—
108#634—626no—
109#634—613no—
110#634—613no—
111C32×C8—213no—
112C3⋊(C3⋊C8)—613no—
113#454—271noSylow
114C2×C33—21no—
115C2×C33—41no—
116C2×C33—81no—
117C2×C33#12011yescharacteristic
118C3⋊(C3⋊D3)—41no—
119C3×#41C2×C3⋊D3113yes—
120C2×C3×D4C2×C3⋊D3113yes—
121D3×Q8—1813no—
122#237—1826no—
123#237—1813no—
124#235—913no—
125#235—1813no—
126C2×Dic6—913no—
127C2×Dic6—1813no—
128C3×SD16—213no—
129C3×D8—213no—
130C3×M16C2×C3⋊D3113yes—
131#217—1826no—
132#214—1813no—
133#213—1813no—
134C4×Dic3—913no—
135#208—913no—
136C22×C32—213no—
137C22×C32—426no—
138C22×C32—413no—
139C22×C32D3×D4113yes—
140C2×C3⋊D3—613no—
141C32×C4—213no—
142C32×C4D3×D4113yes—
143C32×C4C2×#67113yes—
144C3⋊Dic3—613no—
145C3⋊Dic3—626no—
146C3⋊Dic3—1213no—
1472−1+4—271no—
148AGL(1,ℤ/8)—271no—
149AGL(1,ℤ/8)C3⋊(C3⋊D3)11yescharacteristic, complemented
150#124—271no—
151#104—271no—
152#104—271no—
153#101—271no—
154C33#45411yescharacteristic, complemented, Sylow
155C23×C3—213no—
156C3×Q8#638113yes—
157C3×D4—213no—
158C3×D4—213no—
159C3×D4—213no—
160C3×D4#638113yes—
161C2×C3×C4—213no—
162C2×C3×C4C22×C3⋊D3113yes—
163#67—1826no—
164#67—1813no—
165C2×Dic3—926no—
166C2×Dic3—1826no—
167C2×Dic3—1813no—
168D12—913no—
169C4×D3—1826no—
170C4×D3—1813no—
171Dic6—913no—
172Dic6—1826no—
173Dic6—1813no—
174Dic6—1813no—
175C3×C8—213no—
176C3⋊C8—1813no—
177C2×C32—213no—
178C2×C32—413no—
179C2×C32—813no—
180C2×C32#937113yes—
181C3⋊D3—1213no—
182#41—271no—
183#41—541no—
184#41—542no—
185#41—541no—
186#41C2×C3⋊(C3⋊D3)11yescharacteristic
187C2×Q8—271no—
188C2×Q8—541no—
189C2×Q8—541no—
190C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic
191SD16—21no—
192SD16—541no—
193D8—21no—
194D8—541no—
195M16—271no—
196M16C2×C3⋊(C3⋊D3)11yescharacteristic
197C22⋊C4—542no—
198C42—271no—
199C22×C3—213no—
200C22×C3—426no—
201C22×C3—413no—
202C22×C3D4×C3⋊D3113yes—
203C2×D3—1813no—
204C3×C4—213no—
205C3×C4D4×C3⋊D3113yes—
206C3×C4C2×#638113yes—
207Dic3—1813no—
208Dic3—1826no—
209Dic3—3613no—
210C32#5201113yescomplemented
211C23—21no—
212Q8—271no—
213Q8—542no—
214Q8—541no—
215Q8—541no—
216Q8#631211yescharacteristic
217D4—21no—
218D4—21no—
219D4—21no—
220D4—271no—
221D4—542no—
222D4—541no—
223D4#631211yescharacteristic
224C2×C4—21no—
225C2×C4—272no—
226C2×C4—542no—
227C2×C4—542no—
228C2×C4—541no—
229C2×C4—541no—
230C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
231C8—21no—
232C8—541no—
233C2×C3—213no—
234C2×C3—413no—
235C2×C3—813no—
236C2×C3#64146113yes—
237D3—3613no—
238C22—21no—
239C22—42no—
240C22—41no—
241C22—541no—
242C22D4×C3⋊(C3⋊D3)11yescharacteristic
243C4—21no—
244C4—542no—
245C4—541no—
246C4—1081no—
247C4D4×C3⋊(C3⋊D3)11yescharacteristic
248C4C2×#631211yescharacteristic
249C3#10592900113yescomplemented
250C2—21no—
251C2—41no—
252C2—81no—
253C2—1081no—
254C2#1172397811yescharacteristic
subgroupG/H
centerC2#11723978
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×AGL(1,ℤ/8)C2
socleC2×C33#120
hypercenterAGL(1,ℤ/8)C3⋊(C3⋊D3)
hypocenterC33#454
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #11724685C2semidirect
2. C33×AGL(1,ℤ/8)C2semidirect
3. #11723977C2semidirect
4. #11723973C2semidirect
5. #11722207C2semidirect
6. #11722195C2semidirect
7. #11722194C2semidirect
27. C32×AGL(1,ℤ/8)D3semidirect
85. C3×AGL(1,ℤ/8)C3⋊D3semidirect
149. AGL(1,ℤ/8)C3⋊(C3⋊D3)semidirect
154. C33#454semidirect
210. C32#5201semidirect
249. C3#10592900semidirect

Maximal subgroups

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

subgroupindex
#117246852
C33×AGL(1,ℤ/8)2
#117239772
#117239732
#117222072
#117221952
#117221942
#105929003