Finite Group Databasebeta

← #13163814

Subgroups — #13163814

Order 1728.

2128 conjugacy classes of subgroups; only the 407 normal subgroups are shown.

normal subgroups only

This group has 2128 conjugacy classes of subgroups — too many to draw. Only its 407 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×#89078C211yesmaximal, characteristic
2C2×C33×SD16C211yesmaximal, characteristic, complemented
3#11723987C211yesmaximal, characteristic, complemented
4C2×#88863C211yesmaximal, characteristic, complemented
5#11722202C211yesmaximal, characteristic, complemented
6#11722190C211yesmaximal, characteristic, complemented
7#11722186C211yesmaximal, characteristic, complemented
8#10592872—313nomaximal
9C2×C33×Q8C2211yescharacteristic
10C2×C33×D4C2211yescharacteristic
11C2×#6312—21no—
12C22×C3⋊(C3⋊Dic3)—21no—
13#89078—22no—
14#89078—22no—
15C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
16C2×#6308C2211yescharacteristic
17C33×SD16—22no—
18C2×C33×C8C2211yescharacteristic
19#88863—22no—
20#88859—21no—
21C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
22#88857—21no—
23C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
24C2×#3830—313no—
25C2×C32×SD16D3113yescomplemented
26#64155—313no—
27C2×#3756—313no—
28#63659—313no—
29#63647—313no—
30#63643—313no—
31C23×C33—21no—
32C22×C3⋊(C3⋊D3)D411yescharacteristic
33C33×Q8—21no—
34C33×Q8—21no—
35C33×D4—21no—
36C33×D4D412yes—
37C2×C33×C4—21no—
38C2×C33×C4C2311yescharacteristic
39#6312—24no—
40C2×C3⋊(C3⋊Dic3)—24no—
41C2×C3⋊(C3⋊Dic3)—21no—
42C2×C3⋊(C3⋊Dic3)—21no—
43C2×C3⋊(C3⋊Dic3)D411yescharacteristic
44C4×C3⋊(C3⋊D3)—21no—
45C4×C3⋊(C3⋊D3)—21no—
46#6308—21no—
47#6308D412yes—
48C33×C8—21no—
49C3⋊(C3⋊(C3⋊C8))—21no—
50#5173—913no—
51C2×C32×Q8C2×D3113yes—
52C2×C32×D4C2×D3113yes—
53C2×#638—613no—
54C22×C3⋊Dic3—613no—
55#3830—626no—
56#3830—626no—
57C2×C4×C3⋊D3—313no—
58C2×#634—313no—
59C32×SD16—226no—
60C2×C32×C8C2×D3113yes—
61#3756—626no—
62#3752—613no—
63C4⋊(C3⋊Dic3)—313no—
64#3750—613no—
65C2×C3⋊(C3⋊C8)—313no—
66C22×C33—22no—
67C22×C33—41no—
68C22×C33—41no—
69C22×C33C2×D411yescharacteristic
70C2×C3⋊(C3⋊D3)—21no—
71C2×C3⋊(C3⋊D3)—21no—
72C2×C3⋊(C3⋊D3)—21no—
73C33×C4—41no—
74C33×C4C2×D411yescharacteristic
75C33×C4C2×D411yescharacteristic
76C3⋊(C3⋊Dic3)—22no—
77C3⋊(C3⋊Dic3)—21no—
78C3⋊(C3⋊Dic3)—41no—
79C2×#237—913no—
80C2×C3×SD16C3⋊D3113yescomplemented
81#946—913no—
82C2×#216—913no—
83#832—913no—
84(C2×D3)⋊C8—913no—
85#816—913no—
86C23×C32—213no—
87C22×C3⋊D3—313no—
88C32×Q8—213no—
89C32×Q8—213no—
90C32×D4—213no—
91C32×D4#67126yes—
92C2×C32×C4—213no—
93C2×C32×C4C22×D3113yes—
94#638—652no—
95C2×C3⋊Dic3—313no—
96C2×C3⋊Dic3—652no—
97C2×C3⋊Dic3—613no—
98C2×C3⋊Dic3—613no—
99C4×C3⋊D3—613no—
100C4×C3⋊D3—613no—
101#634—326no—
102#634—613no—
103C32×C8—213no—
104C3⋊(C3⋊C8)—613no—
105#452—271noSylow
106C2×C33—42no—
107C2×C33#12011yescharacteristic
108C2×C33#13511yescharacteristic
109C2×C33#13711yescharacteristic
110C3⋊(C3⋊D3)—41no—
111C2×C3×Q8C2×C3⋊D3113yes—
112C2×C3×D4C2×C3⋊D3113yes—
113C2×#67—1813no—
114C22×Dic3—1813no—
115#237—1826no—
116#237—1826no—
117C2×C4×D3—913no—
118C2×Dic6—913no—
119C3×SD16—226no—
120C2×C3×C8C2×C3⋊D3113yes—
121#216—1826no—
122(C2×D3)⋊C4—1813no—
123C4⋊Dic3—913no—
124Dic3⋊C4—1813no—
125C2×C3⋊C8—913no—
126C22×C32—226no—
127C22×C32—413no—
128C22×C32—413no—
129C22×C32D3×D4113yes—
130C2×C3⋊D3—613no—
131C2×C3⋊D3—613no—
132C2×C3⋊D3—613no—
133C32×C4—413no—
134C32×C4D3×D4113yes—
135C32×C4C2×#67113yes—
136C3⋊Dic3—626no—
137C3⋊Dic3—613no—
138C3⋊Dic3—1213no—
139C2×#41—271no—
140C2×Q16—271no—
141C2×SD16C3⋊(C3⋊D3)11yescharacteristic, complemented
142C22⋊Q8—271no—
143#103—271no—
144#102—271no—
145C22⋊C8—271no—
146C33#45211yescharacteristic, complemented, Sylow
147C23×C3—213no—
148C22×D3—913no—
149C3×Q8—213no—
150C3×Q8—213no—
151C3×D4—213no—
152C3×D4#638126yes—
153C2×C3×C4—213no—
154C2×C3×C4C22×C3⋊D3113yes—
155#67—1852no—
156C2×Dic3—913no—
157C2×Dic3—1852no—
158C2×Dic3—1813no—
159C2×Dic3—1813no—
160C4×D3—1813no—
161C4×D3—1813no—
162Dic6—926no—
163Dic6—1813no—
164C3×C8—213no—
165C3⋊C8—1813no—
166C2×C32—426no—
167C2×C32#915113yes—
168C2×C32#916113yes—
169C2×C32#937113yes—
170C3⋊D3—1213no—
171#41—542no—
172#41—542no—
173C2×Q8—271no—
174C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
175C2×D4—541no—
176C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic
177C22×C4—271no—
178C22×C4—541no—
179Q16—542no—
180SD16—22no—
181C2×C8—271no—
182C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic
183C4⋊C4—271no—
184C4⋊C4—541no—
185C22⋊C4—541no—
186C22×C3—226no—
187C22×C3—413no—
188C22×C3—413no—
189C22×C3D4×C3⋊D3113yes—
190C2×D3—1813no—
191C2×D3—1813no—
192C2×D3—1813no—
193C3×C4—413no—
194C3×C4D4×C3⋊D3113yes—
195C3×C4C2×#638113yes—
196Dic3—1826no—
197Dic3—1813no—
198Dic3—3613no—
199C32#5173113yescomplemented
200C23—21no—
201C23—271no—
202Q8—21no—
203Q8—21no—
204Q8—272no—
205Q8—541no—
206D4—21no—
207D4—544no—
208D4#631212yes—
209C2×C4—21no—
210C2×C4—271no—
211C2×C4—544no—
212C2×C4—541no—
213C2×C4—541no—
214C2×C4—541no—
215C2×C4—541no—
216C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
217C8—21no—
218C8—541no—
219C2×C3—426no—
220C2×C3#64124113yes—
221C2×C3#64125113yes—
222C2×C3#64146113yes—
223D3—3613no—
224C22—22no—
225C22—41no—
226C22—41no—
227C22—541no—
228C22—541no—
229C22—541no—
230C22D4×C3⋊(C3⋊D3)11yescharacteristic
231C4—41no—
232C4—541no—
233C4—542no—
234C4—1081no—
235C4D4×C3⋊(C3⋊D3)11yescharacteristic
236C4C2×#631211yescharacteristic
237C3#10592872113yescomplemented
238C2—42no—
239C2—1081no—
240C2#1172395611yescharacteristic
241C2#1172395711yescharacteristic
242C2#1172397811yescharacteristic
subgroupG/H
centerC22D4×C3⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC2×C33×SD16C2
socleC22×C33C2×D4
prime radicalC2×#89078C2
hypercenterC2×SD16C3⋊(C3⋊D3)
hypocenterC33#452
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
2. C2×C33×SD16C2semidirect
3. #11723987C2semidirect
4. C2×#88863C2semidirect
5. #11722202C2semidirect
6. #11722190C2semidirect
7. #11722186C2semidirect
25. C2×C32×SD16D3semidirect
80. C2×C3×SD16C3⋊D3semidirect
141. C2×SD16C3⋊(C3⋊D3)semidirect
146. C33#452semidirect
199. C32#5173semidirect
237. C3#10592872semidirect

Maximal subgroups

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

subgroupindex
C2×#890782
C2×C33×SD162
#117239872
C2×#888632
#117222022
#117221902
#117221862
#105928723