Finite Group Databasebeta

← #13138754

Subgroups — #13138754

Order 1728.

387 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 387 conjugacy classes of subgroups — too many to draw. Only its 71 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#11721117C211yesmaximal, characteristic
2#11721099C211yesmaximal, characteristic
3C8×#1084C211yesmaximal, characteristic
4#63643⋊C3C212yesmaximal
5#11720313C212yesmaximal
6#10590890—32nomaximal
7C2×#634⋊C3C2212yes—
8C2×#6196—21no—
9C2×C8×H(3,𝔽3)C2211yescharacteristic
10C4×#1084C2211yescharacteristic
11C2×#6180C2211yescharacteristic
12C4⋊((C3⋊Dic3)⋊C3)C2212yes—
13#88449—21no—
14#88449—21no—
15#88449—21no—
16#63918—32no—
17#63866—32no—
18#63643D312yes—
19C3×#816—32no—
20C3×C8×Dic3—32no—
21C3×C8×Dic3—61no—
22#63567—32no—
23#63566—32no—
24C2×C4×H(3,𝔽3)C2311yescharacteristic
25C2×#1084D411yescharacteristic
26C2×#1084D411yescharacteristic
27C2×(C3⋊Dic3)⋊C3—22no—
28C2×(C3⋊Dic3)⋊C3—22no—
29#634⋊C3—22no—
30#634⋊C3—22no—
31#6196—42no—
32C8×H(3,𝔽3)—21no—
33#6180—21no—
34#4792—92no—
35#4681—91no—
36C2×#634C2×D312yes—
37C2×C3×Dic6—32no—
38C2×#627—62no—
39C2×C32×C8—21no—
40C2×C32×C8C2×D312yes—
41C4⋊(C3⋊Dic3)C2×D312yes—
42C3×C4⋊Dic3—32no—
43C3×C4×Dic3—32no—
44C3×C4×Dic3—61no—
45C2×C3×C3⋊C8—32no—
46C2×C3×C3⋊C8—61no—
47Dic3⋊Dic3—62no—
48Dic3⋊Dic3—62no—
49Dic3⋊Dic3—62no—
50C22×H(3,𝔽3)C2×D411yescharacteristic
51C4×H(3,𝔽3)#4111yescharacteristic
52C4×H(3,𝔽3)#4111yescharacteristic
53#1084SD1612yes—
54#1084Q1612yes—
55(C3⋊Dic3)⋊C3—42no—
56(C3⋊Dic3)⋊C3—42no—
57#944—92no—
58#888—92no—
59#869—91no—
60#869—91no—
61C3×#103—92no—
62C3×C4×C8—91no—
63#835—92no—
64#816—32no—
65#816—92no—
66#816—92no—
67C8×Dic3—32no—
68C8×Dic3—61no—
69#810—92no—
70C2×C32×C4—21no—
71C2×C32×C4C22×D312yes—
72C2×C3⋊Dic3—22no—
73C2×C3⋊Dic3—22no—
74#634—22no—
75#634—22no—
76C2×C3×Dic3—32no—
77C2×C3×Dic3—32no—
78C2×C3×Dic3—61no—
79C2×C3×Dic3—61no—
80C2×C3×Dic3—62no—
81C2×C3×Dic3—62no—
82C3×Dic6—62no—
83C3×Dic6—62no—
84#627—124no—
85C32×C8—22no—
86C32×C8—22no—
87C3×C3⋊C8—62no—
88C3×C3⋊C8—62no—
89#487—271noSylow
90C2×H(3,𝔽3)#12411yescharacteristic
91C2×H(3,𝔽3)C2×SD1611yescharacteristic
92C2×H(3,𝔽3)C2×Q1611yescharacteristic
93C2×C3×Q8—92no—
94C2×Dic6—32no—
95C2×Dic6—92no—
96C2×Dic6—182no—
97C2×Dic6—181no—
98C2×C3×C8—32no—
99C2×C3×C8—61no—
100C2×C3×C8—91no—
101C2×C3×C8D3211yescharacteristic
102C3×C4⋊C4—92no—
103C3×C42—91no—
104C4⋊Dic3—32no—
105C4⋊Dic3—92no—
106C4⋊Dic3—181no—
107C4⋊Dic3—181no—
108C4⋊Dic3—181no—
109Dic3⋊C4—182no—
110Dic3⋊C4—182no—
111Dic3⋊C4—182no—
112C4×Dic3—32no—
113C4×Dic3—61no—
114C2×C3⋊C8—32no—
115C2×C3⋊C8—61no—
116C22×C32—21no—
117C22×C32D3×D412yes—
118C32×C4—21no—
119C32×C4—21no—
120C32×C4#23512yes—
121C32×C4#23712yes—
122C3⋊Dic3—42no—
123C3⋊Dic3—42no—
124C3×Dic3—34no—
125C3×Dic3—34no—
126C3×Dic3—62no—
127C3×Dic3—62no—
128C3×Dic3—122no—
129C3×Dic3—122no—
130C4⋊Q8—271no—
131C4⋊Q8—271no—
132#103—92no—
133#103—272no—
134C4×C8—91no—
135H(3,𝔽3)#48711yescharacteristic, complemented, Sylow
136C3×Q8—182no—
137C3×Q8—182no—
138C2×C3×C4—32no—
139C2×C3×C4—61no—
140C2×C3×C4—91no—
141C2×C3×C4—91no—
142C2×C3×C4—182no—
143C2×C3×C4—182no—
144C2×C3×C4C2×D3211yescharacteristic
145C2×Dic3—32no—
146C2×Dic3—32no—
147C2×Dic3—61no—
148C2×Dic3—61no—
149C2×Dic3—62no—
150C2×Dic3—62no—
151C2×Dic3—182no—
152C2×Dic3—182no—
153Dic6—62no—
154Dic6—62no—
155Dic6—182no—
156Dic6—182no—
157Dic6—364no—
158Dic6—362no—
159C3×C8—21no—
160C3×C8—62no—
161C3×C8—62no—
162C3×C8—181no—
163C3⋊C8—62no—
164C3⋊C8—62no—
165C2×C32—21no—
166C2×C32—21no—
167C2×C32—21no—
168C2×C32#88512yes—
169C2×C32D3×SD1612yes—
170C2×C32D3×Q1612yes—
171C2×Q8—92no—
172C2×Q8—541no—
173C2×C8—91no—
174C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
175C4⋊C4—92no—
176C4⋊C4—541no—
177C4⋊C4—541no—
178C4⋊C4—541no—
179C42—91no—
180C22×C3—32no—
181C22×C3—61no—
182C22×C3#380211yescharacteristic
183C3×C4—32no—
184C3×C4—32no—
185C3×C4—61no—
186C3×C4—61no—
187C3×C4—92no—
188C3×C4—92no—
189C3×C4—362no—
190C3×C4—362no—
191C3×C4#379811yescharacteristic
192C3×C4#380511yescharacteristic
193Dic3—34no—
194Dic3—34no—
195Dic3—62no—
196Dic3—62no—
197Dic3—122no—
198Dic3—122no—
199Dic3—362no—
200Dic3—362no—
201C32—21no—
202C32#479212yescomplemented
203Q8—182no—
204Q8—182no—
205Q8—1082no—
206C2×C4—91no—
207C2×C4—91no—
208C2×C4—182no—
209C2×C4—182no—
210C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
211C8—21no—
212C8—181no—
213C2×C3—32no—
214C2×C3—32no—
215C2×C3—32no—
216C2×C3—61no—
217C2×C3—61no—
218C2×C3—61no—
219C2×C3#6379611yescharacteristic
220C2×C3#6380211yescharacteristic
221C2×C3#6391511yescharacteristic
222C22#8865511yescharacteristic
223C4—92no—
224C4—92no—
225C4—362no—
226C4—362no—
227C4#8865011yescharacteristic
228C4#8867211yescharacteristic
229C3—32no—
230C3—61no—
231C3#1059090111yescharacteristic
232C2#1172083311yescharacteristic
233C2#1172083711yescharacteristic
234C2#1172114511yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#487
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
135. H(3,𝔽3)#487semidirect
202. C32#4792semidirect

Maximal subgroups

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

subgroupindex
#117211172
#117210992
C8×#10842
#63643⋊C32
#117203132
#105908903