Finite Group Databasebeta

← #13158867

Subgroups — #13158867

Order 1728.

860 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 860 conjugacy classes of subgroups — too many to draw. Only its 212 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#11723910C211yesmaximal, characteristic
2C32×#869C211yesmaximal, characteristic
3C3×C4×C3⋊(C3⋊C8)C211yesmaximal, characteristic
4#11722038C214yesmaximal
5#10592778—31nomaximal
6#10590287—34nomaximal
7C2×#6308—21no—
8C2×C32×Dic6C2212yes—
9C33×C42C2211yescharacteristic
10C4⋊(C3⋊(C3⋊Dic3))—21no—
11C4⋊(C3⋊(C3⋊Dic3))C2212yes—
12C2×C3×C3⋊(C3⋊C8)C2212yes—
13C32×C4⋊Dic3—21no—
14C32×C4⋊Dic3—21no—
15C32×C4⋊Q8—31no—
16#64078—31no—
17#64078—34no—
18#64078—64no—
19C3×#869D314yes—
20#63662—34no—
21C4×C3⋊(C3⋊C8)D311yescharacteristic
22C3×C4×C3⋊C8—34no—
23#63566—316no—
24C2×C33×C4D411yescharacteristic
25C2×C33×C4D411yescharacteristic
26C2×C33×C4C2311yescharacteristic
27C2×C3⋊(C3⋊Dic3)—22no—
28#6308—42no—
29C2×C32×Dic3—22no—
30C32×Dic6—24no—
31C3×C3⋊(C3⋊C8)—22no—
32#5079—94no—
33#4681—91no—
34C2×C32×Q8—32no—
35C2×#634—68no—
36C2×#634—61no—
37C2×#634—64no—
38C2×C3×Dic6C2×D318yes—
39C32×C4⋊C4—61no—
40C32×C4⋊C4—61no—
41C32×C42—24no—
42C32×C42C2×D311yescharacteristic
43C32×C42C2×D314yes—
44C4⋊(C3⋊Dic3)—32no—
45C4⋊(C3⋊Dic3)—38no—
46C4⋊(C3⋊Dic3)—68no—
47C4⋊(C3⋊Dic3)—68no—
48C4⋊(C3⋊Dic3)—61no—
49C4⋊(C3⋊Dic3)—64no—
50C2×C3⋊(C3⋊C8)C2×D312yes—
51C3×C4⋊Dic3—24no—
52C3×C4⋊Dic3—24no—
53C2×C3×C3⋊C8—38no—
54C22×C33C2×D411yescharacteristic
55C33×C4SD1612yes—
56C33×C4Q1612yes—
57C33×C4#4112yes—
58C3⋊(C3⋊Dic3)—42no—
59C32×Dic3—42no—
60C3×C4⋊Q8—34no—
61#869—94no—
62#869—91no—
63#869—184no—
64#869C3⋊D311yescharacteristic
65C3×C4×C8—91no—
66#835—916no—
67#816—94no—
68C4×C3⋊C8—34no—
69C32×Q8—64no—
70C2×C32×C4—24no—
71C2×C32×C4—24no—
72C2×C32×C4—24no—
73C2×C32×C4—62no—
74C2×C32×C4D1211yescharacteristic
75C2×C32×C4D1211yescharacteristic
76C2×C32×C4#6714yes—
77C2×C32×C4#6714yes—
78C2×C32×C4C22×D314yes—
79C2×C32×C4C22×D311yescharacteristic
80C2×C3⋊Dic3—616no—
81C2×C3⋊Dic3—62no—
82C2×C3⋊Dic3—68no—
83#634—1216no—
84#634—122no—
85#634—128no—
86C2×C3×Dic3—28no—
87C3×Dic6—216no—
88C3⋊(C3⋊C8)—22no—
89C3×C3⋊C8—68no—
90#487—271noSylow
91C2×C33#12411yescharacteristic
92C2×C33C2×SD1611yescharacteristic
93C2×C33C2×Q1611yescharacteristic
94C2×C3×Q8—38no—
95C2×Dic6—188no—
96C2×Dic6—184no—
97C2×Dic6—181no—
98C2×Dic6C2×C3⋊D312yes—
99C2×C3×C8—92no—
100C3×C4⋊C4—64no—
101C3×C4⋊C4—64no—
102C3×C42—24no—
103C3×C42D3214yes—
104C3×C42C2×C3⋊D311yescharacteristic
105C4⋊Dic3—21no—
106C4⋊Dic3—21no—
107C4⋊Dic3—98no—
108C4⋊Dic3—92no—
109C4⋊Dic3—188no—
110C4⋊Dic3—188no—
111C4⋊Dic3—184no—
112C4⋊Dic3—181no—
113C2×C3⋊C8—38no—
114C22×C32—24no—
115C22×C32C2×D1211yescharacteristic
116C22×C32C2×#6714yes—
117C32×C4—28no—
118C32×C4—28no—
119C32×C4—28no—
120C32×C4—122no—
121C32×C4#20412yes—
122C32×C4Dic1212yes—
123C32×C4#21518yes—
124C32×C4#21618yes—
125C32×C4#23512yes—
126C32×C4#23718yes—
127C3⋊Dic3—1216no—
128C3⋊Dic3—122no—
129C3⋊Dic3—128no—
130C3×Dic3—48no—
131C4⋊Q8—31no—
132C4⋊Q8—271no—
133#103—274no—
134C4×C8—91no—
135C33#48711yescharacteristic, complemented, Sylow
136C3×Q8—616no—
137C2×C3×C4—24no—
138C2×C3×C4—24no—
139C2×C3×C4—24no—
140C2×C3×C4—68no—
141C2×C3×C4#62614yes—
142C2×C3×C4#62614yes—
143C2×C3×C4#63811yescharacteristic
144C2×C3×C4#63811yescharacteristic
145C2×C3×C4C2×D3214yes—
146C2×C3×C4C22×C3⋊D311yescharacteristic
147C2×Dic3—22no—
148C2×Dic3—1816no—
149C2×Dic3—188no—
150C2×Dic3—182no—
151Dic6—24no—
152Dic6—3616no—
153Dic6—368no—
154Dic6—362no—
155C3×C8—182no—
156C3⋊C8—68no—
157C2×C32—24no—
158C2×C32—24no—
159C2×C32—24no—
160C2×C32#87511yescharacteristic
161C2×C32C2×#20411yescharacteristic
162C2×C32C2×Dic1211yescharacteristic
163C2×C32#93614yes—
164C2×C32C2×#21514yes—
165C2×C32C2×#21614yes—
166C2×Q8—32no—
167C2×Q8—541no—
168C2×C8—92no—
169C4⋊C4—61no—
170C4⋊C4—61no—
171C4⋊C4—272no—
172C4⋊C4—541no—
173C42D3×C3⋊D311yescharacteristic
174C22×C3—24no—
175C22×C3C2×#62614yes—
176C22×C3C2×#63811yescharacteristic
177C3×C4—28no—
178C3×C4—28no—
179C3×C4—28no—
180C3×C4—128no—
181C3×C4#371718yes—
182C3×C4#371918yes—
183C3×C4#375512yes—
184C3×C4#375612yes—
185C3×C4#379518yes—
186C3×C4#383012yes—
187Dic3—42no—
188Dic3—3616no—
189Dic3—368no—
190Dic3—362no—
191C32—24no—
192C32#468111yescharacteristic, complemented
193C32#507914yescomplemented
194Q8—64no—
195Q8—1082no—
196C2×C4—62no—
197C2×C4—542no—
198C2×C4#629111yescharacteristic
199C2×C4#629111yescharacteristic
200C2×C4C2×D3×C3⋊D311yescharacteristic
201C8—182no—
202C2×C3—24no—
203C2×C3—24no—
204C2×C3—24no—
205C2×C3C2×#371714yes—
206C2×C3C2×#371914yes—
207C2×C3#6386014yes—
208C2×C3#6414511yescharacteristic
209C2×C3C2×#375511yescharacteristic
210C2×C3C2×#375611yescharacteristic
211C22C2×#629111yescharacteristic
212C4—122no—
213C4—1082no—
214C4#8879512yes—
215C4#8879912yes—
216C4#8901412yes—
217C3—24no—
218C3#1059028714yescomplemented
219C3#1059277811yescharacteristic, complemented
220C2C2×#8879511yescharacteristic
221C2C2×#8879911yescharacteristic
222C2#1172342911yescharacteristic
subgroupG/H
centerC22C2×#6291
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socle, prime radicalC22×C33C2×D4
hypercenterC42D3×C3⋊D3
hypocenterC33#487
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
135. C33#487semidirect
192. C32#4681semidirect
193. C32#5079semidirect
218. C3#10590287semidirect
219. C3#10592778semidirect

Maximal subgroups

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

subgroupindex
#117239102
C32×#8692
C3×C4×C3⋊(C3⋊C8)2
#117220382
#105927783
#105902873