Finite Group Databasebeta

← #13163691

Subgroups — #13163691

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×#89076C211yesmaximal, characteristic
2C33×C22⋊Q8C211yesmaximal, characteristic, complemented
3C2×#88863C211yesmaximal, characteristic, complemented
4C2×#88862C211yesmaximal, characteristic, complemented
5#11722200C211yesmaximal, characteristic, complemented
6#11722180C211yesmaximal, characteristic, complemented
7#11722179C211yesmaximal, characteristic, complemented
8#10592746—313nomaximal
9C2×C33×Q8C2211yescharacteristic
10C22×C33×C4C2211yescharacteristic
11C2×#6312—21no—
12#89076—22no—
13#89076—22no—
14C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)—21no—
16C2×#6308C2211yescharacteristic
17C33×C4⋊C4—21no—
18C33×C4⋊C4C2211yescharacteristic
19C33×C22⋊C4—21no—
20#88863—22no—
21#88862—22no—
22C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
23C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
24C2×#3828—313no—
25C32×C22⋊Q8D3113yescomplemented
26C2×#3756—313no—
27C2×#3755—313no—
28#63657—313no—
29#63637—313no—
30#63636—313no—
31C23×C33D411yescharacteristic
32C22×C3⋊(C3⋊D3)—21no—
33C33×Q8—21no—
34C33×Q8—21no—
35C2×C33×C4—21no—
36C2×C33×C4—21no—
37C2×C33×C4—21no—
38C2×C33×C4—21no—
39C2×C33×C4D411yescharacteristic
40C2×C33×C4C2311yescharacteristic
41#6312—24no—
42C2×C3⋊(C3⋊Dic3)—21no—
43C4⋊(C3⋊(C3⋊D3))—21no—
44C4⋊(C3⋊(C3⋊D3))D412yes—
45C4×C3⋊(C3⋊D3)—24no—
46#6308—21no—
47#6308D412yes—
48C3⋊(C3⋊(C3⋊C8))—21no—
49C3⋊(C3⋊(C3⋊C8))—21no—
50#5047—913no—
51C2×C32×Q8C2×D3113yes—
52C22×C32×C4C2×D3113yes—
53C2×#638—613no—
54#3828—626no—
55#3828—626no—
56C2×C4⋊(C3⋊D3)—313no—
57C2×C4×C3⋊D3—613no—
58C2×#634—313no—
59C32×C4⋊C4—213no—
60C32×C4⋊C4C2×D3113yes—
61C32×C22⋊C4—213no—
62#3756—626no—
63#3755—626no—
64C2×C3⋊(C3⋊C8)—313no—
65C2×C3⋊(C3⋊C8)—313no—
66C22×C33—21no—
67C22×C33—21no—
68C22×C33—21no—
69C22×C33C2×D411yescharacteristic
70C2×C3⋊(C3⋊D3)—22no—
71C2×C3⋊(C3⋊D3)—41no—
72C2×C3⋊(C3⋊D3)—41no—
73C33×C4—21no—
74C33×C4—41no—
75C33×C4—41no—
76C33×C4C2×D411yescharacteristic
77C33×C4C2×D411yescharacteristic
78C3⋊(C3⋊Dic3)—22no—
79C2×#235—913no—
80C3×C22⋊Q8C3⋊D3113yescomplemented
81C2×#216—913no—
82C2×#215—913no—
83#830—913no—
84#810—913no—
85#809—913no—
86C23×C32#67113yes—
87C22×C3⋊D3—613no—
88C32×Q8—213no—
89C32×Q8—213no—
90C2×C32×C4—213no—
91C2×C32×C4—213no—
92C2×C32×C4—213no—
93C2×C32×C4—213no—
94C2×C32×C4#67113yes—
95C2×C32×C4C22×D3113yes—
96#638—652no—
97C2×C3⋊Dic3—613no—
98C4⋊(C3⋊D3)—326no—
99C4⋊(C3⋊D3)—613no—
100C4×C3⋊D3—652no—
101#634—326no—
102#634—613no—
103C3⋊(C3⋊C8)—613no—
104C3⋊(C3⋊C8)—613no—
105#452—271noSylow
106C2×C33—41no—
107C2×C33#12011yescharacteristic
108C2×C33#13511yescharacteristic
109C2×C33#13711yescharacteristic
110C3⋊(C3⋊D3)—42no—
111C2×C3×Q8C2×C3⋊D3113yes—
112C22×C3×C4C2×C3⋊D3113yes—
113C2×#67—1813no—
114#235—1826no—
115#235—1826no—
116C2×D12—913no—
117C2×C4×D3—1813no—
118C2×Dic6—913no—
119C3×C4⋊C4—213no—
120C3×C4⋊C4C2×C3⋊D3113yes—
121C3×C22⋊C4—213no—
122#216—1826no—
123#215—1826no—
124C2×C3⋊C8—913no—
125C2×C3⋊C8—913no—
126C22×C32—213no—
127C22×C32—213no—
128C22×C32—213no—
129C22×C32C2×#67113yes—
130C2×C3⋊D3—626no—
131C2×C3⋊D3—1213no—
132C2×C3⋊D3—1213no—
133C32×C4—213no—
134C32×C4—413no—
135C32×C4—413no—
136C32×C4D3×D4113yes—
137C32×C4D3×D4113yes—
138C3⋊Dic3—626no—
139C2×#41—271no—
140C2×Q16—271no—
141C2×SD16—271no—
142C22⋊Q8C3⋊(C3⋊D3)11yescharacteristic, complemented
143#103—271no—
144#102—271no—
145C22⋊C8—271no—
146C33#45211yescharacteristic, complemented, Sylow
147C23×C3#638113yes—
148C22×D3—1813no—
149C3×Q8—213no—
150C3×Q8—213no—
151C2×C3×C4—213no—
152C2×C3×C4—213no—
153C2×C3×C4—213no—
154C2×C3×C4—213no—
155C2×C3×C4#638113yes—
156C2×C3×C4C22×C3⋊D3113yes—
157#67—1852no—
158C2×Dic3—1813no—
159D12—926no—
160D12—1813no—
161C4×D3—1852no—
162Dic6—926no—
163Dic6—1813no—
164C3⋊C8—1813no—
165C3⋊C8—1813no—
166C2×C32—413no—
167C2×C32#937113yes—
168C2×C32#942113yes—
169C2×C32#950113yes—
170C3⋊D3—1226no—
171#41—542no—
172#41—542no—
173C2×Q8—271no—
174C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
175C2×D4—271no—
176C2×D4—541no—
177C22×C4—541no—
178C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic
179Q16—542no—
180SD16—542no—
181C2×C8—271no—
182C2×C8—271no—
183C4⋊C4—21no—
184C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
185C22⋊C4—21no—
186C22×C3—213no—
187C22×C3—213no—
188C22×C3—213no—
189C22×C3C2×#638113yes—
190C2×D3—1826no—
191C2×D3—3613no—
192C2×D3—3613no—
193C3×C4—213no—
194C3×C4—413no—
195C3×C4—413no—
196C3×C4D4×C3⋊D3113yes—
197C3×C4D4×C3⋊D3113yes—
198Dic3—1826no—
199C32#5047113yescomplemented
200C23—541no—
201C23#631211yescharacteristic
202Q8—21no—
203Q8—21no—
204Q8—272no—
205Q8—541no—
206D4—272no—
207D4—544no—
208D4—541no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—544no—
214C2×C4—541no—
215C2×C4#631211yescharacteristic
216C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
217C8—541no—
218C8—541no—
219C2×C3—413no—
220C2×C3#64146113yes—
221C2×C3#64151113yes—
222C2×C3#64159113yes—
223D3—3626no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—542no—
228C22—1081no—
229C22—1081no—
230C22C2×#631211yescharacteristic
231C4—21no—
232C4—41no—
233C4—41no—
234C4—542no—
235C4D4×C3⋊(C3⋊D3)11yescharacteristic
236C4D4×C3⋊(C3⋊D3)11yescharacteristic
237C3#10592746113yescomplemented
238C2—41no—
239C2—1082no—
240C2#1172397811yescharacteristic
241C2#1172398311yescharacteristic
242C2#1172399111yescharacteristic
subgroupG/H
centerC22C2×#6312
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C22×C3⋊(C3⋊D3)
FittingC33×C22⋊Q8C2
socleC22×C33C2×D4
prime radicalC2×#89076C2
hypercenterC22⋊Q8C3⋊(C3⋊D3)
hypocenterC33#452
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C33×C22⋊Q8C2semidirect
3. C2×#88863C2semidirect
4. C2×#88862C2semidirect
5. #11722200C2semidirect
6. #11722180C2semidirect
7. #11722179C2semidirect
25. C32×C22⋊Q8D3semidirect
80. C3×C22⋊Q8C3⋊D3semidirect
142. C22⋊Q8C3⋊(C3⋊D3)semidirect
146. C33#452semidirect
199. C32#5047semidirect
237. C3#10592746semidirect

Maximal subgroups

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

subgroupindex
C2×#890762
C33×C22⋊Q82
C2×#888632
C2×#888622
#117222002
#117221802
#117221792
#105927463