Finite Group Databasebeta

← #13137870

Subgroups — #13137870

Order 1728.

371 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 75 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#11721758C211yesmaximal, characteristic
2#11721117C211yesmaximal, characteristic
3#63634⋊C3C212yesmaximal
4#11720307C212yesmaximal
5#11720303C211yesmaximal, characteristic
6#10590418—32nomaximal
7C2×#6196—21no—
8C4⋊C4×H(3,𝔽3)C2211yescharacteristic
9C4⋊#1084—21no—
10C4⋊#1084C2211yescharacteristic
11#88540—21no—
12C4×#1084C2211yescharacteristic
13C4⋊((C3⋊Dic3)⋊C3)C2212yes—
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15#88449—21no—
16C3×#890—32no—
17C3×#890—61no—
18#63918—32no—
19#63634D312yes—
20C3×#807—32no—
21#63574—32no—
22#63573—32no—
23#63554—32no—
24C2×C4×H(3,𝔽3)—21no—
25C2×C4×H(3,𝔽3)C2311yescharacteristic
26C2×#1084—21no—
27C2×#1084D411yescharacteristic
28C2×#1084D411yescharacteristic
29C2×(C3⋊Dic3)⋊C3—22no—
30#6196—42no—
31(C3⋊(C3⋊C8))⋊C3Q814yescomplemented
32#5064—91no—
33#4876—92no—
34C2×#627—62no—
35C32×C4⋊C4—21no—
36C32×C4⋊C4C2×D312yes—
37C4⋊(C3⋊Dic3)C2×D312yes—
38C2×C3⋊(C3⋊C8)C2×D312yes—
39C3×C4⋊Dic3—32no—
40C3×C4⋊Dic3—32no—
41C3×C4⋊Dic3—62no—
42C3×C4⋊Dic3—61no—
43C3×C4⋊Dic3—62no—
44C3×Dic3⋊C4—62no—
45C3×Dic3⋊C4—62no—
46C3×C4×Dic3—32no—
47C3×C4×Dic3—61no—
48C2×C3×C3⋊C8—32no—
49Dic3⋊Dic3—62no—
50C22×H(3,𝔽3)C2×D411yescharacteristic
51C4×H(3,𝔽3)—41no—
52C4×H(3,𝔽3)C2×Q812yes—
53#1084—21no—
54#1084—21no—
55#1084—41no—
56(C3⋊Dic3)⋊C3—42no—
57C3×#125—91no—
58#890—32no—
59#890—61no—
60#888—92no—
61#869—91no—
62C3×#107—92no—
63#817—92no—
64#815—92no—
65#808—92no—
66#808—92no—
67#807—32no—
68#807—92no—
69#803—91no—
70C2×C32×C4—22no—
71C2×C32×C4—22no—
72C2×C32×C4—21no—
73C2×C32×C4C22×D312yes—
74C2×C3⋊Dic3—22no—
75C2×C3×Dic3—32no—
76C2×C3×Dic3—32no—
77C2×C3×Dic3—61no—
78C2×C3×Dic3—61no—
79C2×C3×Dic3—62no—
80C2×C3×Dic3—62no—
81C2×C3×Dic3—62no—
82#627—124no—
83C3⋊(C3⋊C8)Dic614yescomplemented
84C3×C3⋊C8—34no—
85#501—271noSylow
86C2×H(3,𝔽3)C4⋊Q811yescharacteristic
87C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
88C2×H(3,𝔽3)#13711yescharacteristic
89C2×Dic6—182no—
90C2×Dic6—181no—
91C2×C3×C8—92no—
92C3×C4⋊C4—32no—
93C3×C4⋊C4—61no—
94C3×C4⋊C4—92no—
95C3×C4⋊C4—91no—
96C3×C4⋊C4—181no—
97C3×C4⋊C4—181no—
98C3×C4⋊C4D3211yescharacteristic
99C3×C42—91no—
100C4⋊Dic3—32no—
101C4⋊Dic3—32no—
102C4⋊Dic3—61no—
103C4⋊Dic3—62no—
104C4⋊Dic3—62no—
105C4⋊Dic3—92no—
106C4⋊Dic3—181no—
107Dic3⋊C4—62no—
108Dic3⋊C4—62no—
109Dic3⋊C4—182no—
110C4×Dic3—32no—
111C4×Dic3—61no—
112C2×C3⋊C8—32no—
113C2×C3⋊C8—92no—
114C22×C32—21no—
115C22×C32D3×D412yes—
116C32×C4—22no—
117C32×C4—42no—
118C32×C4—42no—
119C32×C4C2×Dic612yes—
120C32×C4D3×Q812yes—
121C3⋊Dic3—42no—
122C3×Dic3—62no—
123C3×Dic3—62no—
124C3×Dic3—121no—
125C3×Dic3—121no—
126C3×Dic3—122no—
127C3×Dic3—122no—
128C3×Dic3—122no—
129C4⋊Q8—271no—
130#125—91no—
131#107—92no—
132#106—272no—
133#97—271no—
134H(3,𝔽3)#50111yescharacteristic, complemented, Sylow
135C2×C3×C4—21no—
136C2×C3×C4—32no—
137C2×C3×C4—62no—
138C2×C3×C4—62no—
139C2×C3×C4—61no—
140C2×C3×C4—91no—
141C2×C3×C4—91no—
142C2×C3×C4—182no—
143C2×C3×C4—181no—
144C2×C3×C4C2×D3211yescharacteristic
145C2×Dic3—32no—
146C2×Dic3—32no—
147C2×Dic3—61no—
148C2×Dic3—61no—
149C2×Dic3—62no—
150C2×Dic3—62no—
151C2×Dic3—62no—
152C2×Dic3—182no—
153Dic6—364no—
154Dic6—362no—
155C3×C8—94no—
156C3⋊C8—34no—
157C3⋊C8—94no—
158C2×C32—21no—
159C2×C32—21no—
160C2×C32—21no—
161C2×C32#88812yes—
162C2×C32#91112yes—
163C2×C32#91812yes—
164C2×Q8—541no—
165C2×C8—92no—
166C4⋊C4—91no—
167C4⋊C4—92no—
168C4⋊C4—181no—
169C4⋊C4—181no—
170C4⋊C4—541no—
171C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
172C42—91no—
173C22×C3—32no—
174C22×C3—61no—
175C22×C3#381111yescharacteristic
176C3×C4—32no—
177C3×C4—32no—
178C3×C4—41no—
179C3×C4—62no—
180C3×C4—122no—
181C3×C4—122no—
182C3×C4—181no—
183C3×C4—181no—
184C3×C4—362no—
185C3×C4—361no—
186C3×C4D3×Dic612yes—
187Dic3—62no—
188Dic3—62no—
189Dic3—121no—
190Dic3—121no—
191Dic3—122no—
192Dic3—122no—
193Dic3—122no—
194Dic3—362no—
195C32—21no—
196C32#487612yescomplemented
197Q8—1082no—
198C2×C4—21no—
199C2×C4—91no—
200C2×C4—91no—
201C2×C4—181no—
202C2×C4—182no—
203C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
204C8—94no—
205C2×C3—32no—
206C2×C3—32no—
207C2×C3—32no—
208C2×C3—61no—
209C2×C3—61no—
210C2×C3—61no—
211C2×C3#6384011yescharacteristic
212C2×C3#6392711yescharacteristic
213C2×C3#6394511yescharacteristic
214C22#8867711yescharacteristic
215C4—41no—
216C4—181no—
217C4—181no—
218C4—361no—
219C4—362no—
220C4((C3⋊D3)⋊C3)⋊Q812yes—
221C3—32no—
222C3—61no—
223C3#1059056311yescharacteristic
224C2#1172108011yescharacteristic
225C2#1172119111yescharacteristic
226C2#1172122711yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#501
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
31. (C3⋊(C3⋊C8))⋊C3Q8semidirect
83. C3⋊(C3⋊C8)Dic6semidirect
134. H(3,𝔽3)#501semidirect
196. C32#4876semidirect

Maximal subgroups

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

subgroupindex
#117217582
#117211172
#63634⋊C32
#117203072
#117203032
#105904183