Finite Group Databasebeta

← #13137866

Subgroups — #13137866

Order 1728.

411 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 411 conjugacy classes of subgroups — too many to draw. Only its 79 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#11721756C211yesmaximal, characteristic
2#11721117C211yesmaximal, characteristic
3#63634⋊C3C212yesmaximal
4#11720306C212yesmaximal
5#11720302C211yesmaximal, characteristic
6#10590411—32nomaximal
7C2×#6229—21no—
8C2×#6196—21no—
9C4⋊C4×H(3,𝔽3)C2211yescharacteristic
10C4⋊#1084C2211yescharacteristic
11#88540—21no—
12C4×#1084C2211yescharacteristic
13C4⋊((C3⋊Dic3)⋊C3)C2212yes—
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15#88449—21no—
16C3×#888—32no—
17C3×#888—61no—
18#63918—32no—
19#63634D312yes—
20C3×#807—32no—
21#63576—32no—
22#63575—32no—
23Dic3×C3⋊C8—32no—
24C2×C4×H(3,𝔽3)—21no—
25C2×C4×H(3,𝔽3)C2311yescharacteristic
26C2×#1084—21no—
27C2×#1084D411yescharacteristic
28C2×#1084D411yescharacteristic
29#6229—42no—
30C2×(C3⋊Dic3)⋊C3—22no—
31#6196—42no—
32(C3⋊(C3⋊C8))⋊C3Q814yescomplemented
33#5078—91no—
34#4874—92no—
35C2×C3×Dic6—62no—
36C2×C3×Dic6—62no—
37C2×#627—62no—
38C32×C4⋊C4—21no—
39C32×C4⋊C4C2×D312yes—
40C4⋊(C3⋊Dic3)C2×D312yes—
41C2×C3⋊(C3⋊C8)C2×D312yes—
42C3×C4⋊Dic3—32no—
43C3×C4⋊Dic3—32no—
44C3×C4⋊Dic3—61no—
45C3×Dic3⋊C4—62no—
46C3×Dic3⋊C4—62no—
47C3×C4×Dic3—32no—
48C3×C4×Dic3—61no—
49C2×C3×C3⋊C8—32no—
50Dic3⋊Dic3—62no—
51C22×H(3,𝔽3)C2×D411yescharacteristic
52C4×H(3,𝔽3)—41no—
53C4×H(3,𝔽3)C2×Q812yes—
54#1084—41no—
55#1084D812yes—
56#1084Q1612yes—
57(C3⋊Dic3)⋊C3—42no—
58C3×C4⋊Q8—91no—
59#888—32no—
60#888—61no—
61#888—92no—
62#869—91no—
63C3×#107—92no—
64#818—92no—
65C8×Dic3—92no—
66#807—32no—
67#807—92no—
68#807—92no—
69#807—92no—
70C4×C3⋊C8—91no—
71C2×C32×C4—22no—
72C2×C32×C4—22no—
73C2×C32×C4—21no—
74C2×C32×C4C22×D312yes—
75C2×C3⋊Dic3—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—
82C2×C3×Dic3—62no—
83C3×Dic6—124no—
84C3×Dic6—124no—
85#627—124no—
86C3⋊(C3⋊C8)Dic614yescomplemented
87C3×C3⋊C8—34no—
88#500—271noSylow
89C2×H(3,𝔽3)C4⋊Q811yescharacteristic
90C2×H(3,𝔽3)C2×D811yescharacteristic
91C2×H(3,𝔽3)C2×Q1611yescharacteristic
92C2×C3×Q8—181no—
93C2×Dic6—62no—
94C2×Dic6—62no—
95C2×Dic6—182no—
96C2×Dic6—181no—
97C2×C3×C8—92no—
98C3×C4⋊C4—32no—
99C3×C4⋊C4—61no—
100C3×C4⋊C4—92no—
101C3×C4⋊C4—91no—
102C3×C4⋊C4—181no—
103C3×C4⋊C4D3211yescharacteristic
104C3×C42—91no—
105C4⋊Dic3—32no—
106C4⋊Dic3—32no—
107C4⋊Dic3—61no—
108C4⋊Dic3—92no—
109C4⋊Dic3—181no—
110Dic3⋊C4—62no—
111Dic3⋊C4—62no—
112Dic3⋊C4—182no—
113C4×Dic3—32no—
114C4×Dic3—61no—
115C2×C3⋊C8—32no—
116C2×C3⋊C8—92no—
117C22×C32—21no—
118C22×C32D3×D412yes—
119C32×C4—22no—
120C32×C4—42no—
121C32×C4—42no—
122C32×C4C2×Dic612yes—
123C32×C4D3×Q812yes—
124C3⋊Dic3—42no—
125C3×Dic3—34no—
126C3×Dic3—34no—
127C3×Dic3—62no—
128C3×Dic3—62no—
129C3×Dic3—122no—
130C3×Dic3—122no—
131C3×Dic3—122no—
132C4⋊Q8—91no—
133C4⋊Q8—271no—
134#107—92no—
135#107—272no—
136C4×C8—271no—
137H(3,𝔽3)#50011yescharacteristic, complemented, Sylow
138C3×Q8—362no—
139C2×C3×C4—21no—
140C2×C3×C4—32no—
141C2×C3×C4—62no—
142C2×C3×C4—62no—
143C2×C3×C4—61no—
144C2×C3×C4—91no—
145C2×C3×C4—91no—
146C2×C3×C4—182no—
147C2×C3×C4—181no—
148C2×C3×C4C2×D3211yescharacteristic
149C2×Dic3—32no—
150C2×Dic3—32no—
151C2×Dic3—61no—
152C2×Dic3—61no—
153C2×Dic3—62no—
154C2×Dic3—62no—
155C2×Dic3—62no—
156C2×Dic3—182no—
157Dic6—124no—
158Dic6—124no—
159Dic6—364no—
160Dic6—362no—
161C3×C8—94no—
162C3⋊C8—34no—
163C3⋊C8—94no—
164C2×C32—21no—
165C2×C32—21no—
166C2×C32—21no—
167C2×C32#88812yes—
168C2×C32D3×D812yes—
169C2×C32D3×Q1612yes—
170C2×Q8—181no—
171C2×Q8—541no—
172C2×C8—92no—
173C4⋊C4—92no—
174C4⋊C4—91no—
175C4⋊C4—181no—
176C4⋊C4—541no—
177C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
178C42—91no—
179C22×C3—32no—
180C22×C3—61no—
181C22×C3#381111yescharacteristic
182C3×C4—32no—
183C3×C4—32no—
184C3×C4—41no—
185C3×C4—62no—
186C3×C4—92no—
187C3×C4—92no—
188C3×C4—122no—
189C3×C4—122no—
190C3×C4—362no—
191C3×C4—361no—
192C3×C4D3×Dic612yes—
193Dic3—34no—
194Dic3—34no—
195Dic3—62no—
196Dic3—62no—
197Dic3—122no—
198Dic3—122no—
199Dic3—122no—
200Dic3—362no—
201C32—21no—
202C32#487412yescomplemented
203Q8—362no—
204Q8—1082no—
205C2×C4—21no—
206C2×C4—91no—
207C2×C4—91no—
208C2×C4—181no—
209C2×C4—182no—
210C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
211C8—94no—
212C2×C3—32no—
213C2×C3—32no—
214C2×C3—32no—
215C2×C3—61no—
216C2×C3—61no—
217C2×C3—61no—
218C2×C3#6384011yescharacteristic
219C2×C3#6392611yescharacteristic
220C2×C3#6394411yescharacteristic
221C22#8867711yescharacteristic
222C4—41no—
223C4—92no—
224C4—92no—
225C4—361no—
226C4—362no—
227C4((C3⋊D3)⋊C3)⋊Q812yes—
228C3—32no—
229C3—61no—
230C3#1059041511yescharacteristic
231C2#1172108011yescharacteristic
232C2#1172119011yescharacteristic
233C2#1172122611yescharacteristic
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)#500
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
32. (C3⋊(C3⋊C8))⋊C3Q8semidirect
86. C3⋊(C3⋊C8)Dic6semidirect
137. H(3,𝔽3)#500semidirect
202. C32#4874semidirect

Maximal subgroups

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

subgroupindex
#117217562
#117211172
#63634⋊C32
#117203062
#117203022
#105904113