Finite Group Databasebeta

← #13137876

Subgroups — #13137876

Order 1728.

371 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 67 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#11721099C211yesmaximal, characteristic
3#63637⋊C3C212yesmaximal
4#11720309C212yesmaximal
5#11720303C211yesmaximal, characteristic
6#10590562—32nomaximal
7C2×#634⋊C3C2212yes—
8C4⋊C4×H(3,𝔽3)C2211yescharacteristic
9C4⋊#1084—21no—
10C4⋊#1084C2211yescharacteristic
11#88540—21no—
12C4×#1084C2211yescharacteristic
13C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
14#88449—21no—
15#88449—21no—
16C3×#890—32no—
17C3×#890—61no—
18#63866—32no—
19#63637D312yes—
20C3×#810—32no—
21#63567—32no—
22#63565—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#634⋊C3—22no—
31#634⋊C3—22no—
32(C3⋊(C3⋊C8))⋊C3—22no—
33#5069—91no—
34#4797—92no—
35C2×#634C2×D312yes—
36C2×C3×Dic6—32no—
37C32×C4⋊C4—21no—
38C32×C4⋊C4C2×D312yes—
39C2×C3⋊(C3⋊C8)C2×D312yes—
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—
50Dic3⋊Dic3—62no—
51C22×H(3,𝔽3)C2×D411yescharacteristic
52C4×H(3,𝔽3)—41no—
53C4×H(3,𝔽3)#4112yes—
54#1084—21no—
55#1084—21no—
56#1084—41no—
57(C3⋊Dic3)⋊C3—42no—
58C3×#125—91no—
59#944—92no—
60#890—32no—
61#890—61no—
62#869—91no—
63C3×#103—92no—
64#835—92no—
65#816—92no—
66#815—92no—
67#810—32no—
68#810—92no—
69#810—92no—
70#803—91no—
71C2×C32×C4—22no—
72C2×C32×C4—22no—
73C2×C32×C4—21no—
74C2×C32×C4C22×D312yes—
75C2×C3⋊Dic3—22no—
76#634—22no—
77#634—22no—
78C2×C3×Dic3—32no—
79C2×C3×Dic3—32no—
80C2×C3×Dic3—61no—
81C2×C3×Dic3—61no—
82C2×C3×Dic3—62no—
83C2×C3×Dic3—62no—
84C2×C3×Dic3—62no—
85C3×Dic6—62no—
86C3×Dic6—62no—
87C3⋊(C3⋊C8)—22no—
88C3×C3⋊C8—62no—
89#491—271noSylow
90C2×H(3,𝔽3)#12411yescharacteristic
91C2×H(3,𝔽3)#13711yescharacteristic
92C2×H(3,𝔽3)#13711yescharacteristic
93C2×C3×Q8—92no—
94C2×Dic6—32no—
95C2×Dic6—92no—
96C2×C3×C8—92no—
97C3×C4⋊C4—32no—
98C3×C4⋊C4—61no—
99C3×C4⋊C4—91no—
100C3×C4⋊C4—181no—
101C3×C4⋊C4—181no—
102C3×C4⋊C4D3211yescharacteristic
103C3×C42—91no—
104C4⋊Dic3—32no—
105C4⋊Dic3—61no—
106C4⋊Dic3—62no—
107C4⋊Dic3—62no—
108C4⋊Dic3—181no—
109C4⋊Dic3—181no—
110Dic3⋊C4—62no—
111Dic3⋊C4—62no—
112Dic3⋊C4—182no—
113Dic3⋊C4—182no—
114C4×Dic3—32no—
115C4×Dic3—61no—
116C2×C3⋊C8—32no—
117C2×C3⋊C8—92no—
118C22×C32—21no—
119C22×C32D3×D412yes—
120C32×C4—22no—
121C32×C4—42no—
122C32×C4—42no—
123C32×C4#23512yes—
124C32×C4#23712yes—
125C3⋊Dic3—42no—
126C3×Dic3—62no—
127C3×Dic3—62no—
128C3×Dic3—121no—
129C3×Dic3—121no—
130C3×Dic3—122no—
131C3×Dic3—122no—
132C3×Dic3—122no—
133C4⋊Q8—271no—
134#125—91no—
135#103—92no—
136#103—272no—
137#97—271no—
138H(3,𝔽3)#49111yescharacteristic, complemented, Sylow
139C3×Q8—182no—
140C3×Q8—182no—
141C2×C3×C4—21no—
142C2×C3×C4—32no—
143C2×C3×C4—62no—
144C2×C3×C4—62no—
145C2×C3×C4—61no—
146C2×C3×C4—91no—
147C2×C3×C4—91no—
148C2×C3×C4—182no—
149C2×C3×C4—181no—
150C2×C3×C4C2×D3211yescharacteristic
151C2×Dic3—32no—
152C2×Dic3—32no—
153C2×Dic3—61no—
154C2×Dic3—61no—
155C2×Dic3—62no—
156C2×Dic3—62no—
157C2×Dic3—62no—
158C2×Dic3—182no—
159Dic6—62no—
160Dic6—62no—
161Dic6—182no—
162Dic6—182no—
163C3×C8—182no—
164C3⋊C8—62no—
165C3⋊C8—182no—
166C2×C32—21no—
167C2×C32—21no—
168C2×C32—21no—
169C2×C32#88512yes—
170C2×C32#91512yes—
171C2×C32#91812yes—
172C2×Q8—92no—
173C2×C8—92no—
174C4⋊C4—91no—
175C4⋊C4—181no—
176C4⋊C4—181no—
177C4⋊C4—541no—
178C4⋊C4—541no—
179C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
180C42—91no—
181C22×C3—32no—
182C22×C3—61no—
183C22×C3#381111yescharacteristic
184C3×C4—32no—
185C3×C4—32no—
186C3×C4—41no—
187C3×C4—62no—
188C3×C4—122no—
189C3×C4—122no—
190C3×C4—181no—
191C3×C4—181no—
192C3×C4—362no—
193C3×C4—361no—
194C3×C4#379512yes—
195Dic3—62no—
196Dic3—62no—
197Dic3—121no—
198Dic3—121no—
199Dic3—122no—
200Dic3—122no—
201Dic3—122no—
202Dic3—362no—
203C32—21no—
204C32#479712yescomplemented
205Q8—182no—
206Q8—182no—
207C2×C4—21no—
208C2×C4—91no—
209C2×C4—91no—
210C2×C4—181no—
211C2×C4—182no—
212C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
213C8—182no—
214C2×C3—32no—
215C2×C3—32no—
216C2×C3—32no—
217C2×C3—61no—
218C2×C3—61no—
219C2×C3—61no—
220C2×C3#6391311yescharacteristic
221C2×C3#6393111yescharacteristic
222C2×C3#6394511yescharacteristic
223C22#8867711yescharacteristic
224C4—41no—
225C4—181no—
226C4—181no—
227C4—361no—
228C4—362no—
229C4#8864812yes—
230C3—32no—
231C3—61no—
232C3#1059056811yescharacteristic
233C2#1172114311yescharacteristic
234C2#1172118911yescharacteristic
235C2#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)#491
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
138. H(3,𝔽3)#491semidirect
204. C32#4797semidirect

Maximal subgroups

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

subgroupindex
#117217582
#117210992
#63637⋊C32
#117203092
#117203032
#105905623