Finite Group Databasebeta

← #13139175

Subgroups — #13139175

Order 1728.

425 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 425 conjugacy classes of subgroups — too many to draw. Only its 77 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
1M16×#272C211yesmaximal, characteristic
2C2×#88435C211yesmaximal, characteristic
3#11721065C211yesmaximal, characteristic
4#63649⋊C3C212yesmaximal, complemented
5#11720319C212yesmaximal, complemented
6#10590973—32nomaximal
7C2×C4×#272C2211yescharacteristic
8M16×H(3,𝔽3)C2211yescharacteristic
9#88538C2211yescharacteristic
10#88527—21no—
11C8×#272—21no—
12#3748⋊C3C2212yes—
13C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
14#88435—21no—
15#88435C412yes—
16#88435C412yes—
17#88434—21no—
18C3×D3×M16—32no—
19C3×D3×M16—61no—
20C2×#3711—32no—
21D3×#208—32no—
22#63649D312yescomplemented
23C3×#822—32no—
24#63578—32no—
25#63577—32no—
26C22×#272Q811yescharacteristic
27C2×C4×H(3,𝔽3)C2311yescharacteristic
28C2×#1084Q811yescharacteristic
29C4×#272C2×C411yescharacteristic
30C4×#272C2×C411yescharacteristic
31C4×#272D411yescharacteristic
32C4×#272D411yescharacteristic
33C8×H(3,𝔽3)—21no—
34#6180—21no—
35(C3⋊(C3⋊C8))⋊C3—22no—
36(C3⋊(C3⋊C8))⋊C3C2×C414yes—
37#5124—91no—
38#4893—92no—
39C2×C3×C4×D3—32no—
40C2×C3×C4×D3—61no—
41C32×M16—21no—
42C32×M16C2×D312yes—
43#3748C2×D312yes—
44C2×C3⋊(C3⋊C8)C2×D312yes—
45C3×#208—32no—
46C3×#208—32no—
47C3×#208—61no—
48C2×C3×C3⋊C8—32no—
49C3×ΓL(1,𝔽25)—62no—
50C3×ΓL(1,𝔽25)—62no—
51C3×D3×C8—62no—
52C3×D3×C8—62no—
53#3711—34no—
54#3711—34no—
55#3711—62no—
56D3×C3⋊C8—62no—
57C22×H(3,𝔽3)C2×Q811yescharacteristic
58C2×#272—41no—
59C2×#272C4⋊C411yescharacteristic
60C2×#272C4⋊C411yescharacteristic
61C4×H(3,𝔽3)C22×C411yescharacteristic
62C4×H(3,𝔽3)C2×D411yescharacteristic
63#1084C4⋊C411yescharacteristic
64#1084C4⋊C411yescharacteristic
65C2×C3×M16—91no—
66C2×#208—91no—
67C2×#208—91no—
68D3×M16—32no—
69D3×M16—61no—
70D3×M16—92no—
71C2×ΓL(1,𝔽25)—92no—
72C3×#108—92no—
73#822—32no—
74#822—92no—
75#822—92no—
76#822—92no—
77#819—92no—
78C22×C3×D3—32no—
79C22×C3×D3—61no—
80C2×C32×C4—21no—
81C2×C32×C4C22×D312yes—
82C2×C3×Dic3—32no—
83C2×C3×Dic3—61no—
84C3×C4×D3—32no—
85C3×C4×D3—32no—
86C3×C4×D3—32no—
87C3×C4×D3—32no—
88C3×C4×D3—61no—
89C3×C4×D3—61no—
90C3×C4×D3—61no—
91C3×C4×D3—61no—
92C32×C8—22no—
93C32×C8—22no—
94C3⋊(C3⋊C8)—22no—
95C3⋊(C3⋊C8)C4×D314yes—
96C3×C3⋊C8—34no—
97C3×C3⋊C8—62no—
98C3×C3⋊C8—62no—
99C3×C3⋊C8—62no—
100#430—271noSylow
101C2×H(3,𝔽3)—21no—
102C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
103#272—21no—
104#272—21no—
105C22×C3×C4—91no—
106C2×C4×D3—32no—
107C2×C4×D3—61no—
108C3×M16—32no—
109C3×M16—61no—
110C3×M16—92no—
111C3×M16—91no—
112C3×M16—181no—
113C3×M16D3211yescharacteristic
114C2×C3×C8—92no—
115C2×C3×C8—181no—
116#208—32no—
117#208—32no—
118#208—61no—
119#208—92no—
120#208—92no—
121#208—92no—
122#208—181no—
123C2×C3⋊C8—32no—
124C2×C3⋊C8—92no—
125C2×C3⋊C8—181no—
126ΓL(1,𝔽25)—62no—
127ΓL(1,𝔽25)—62no—
128ΓL(1,𝔽25)—94no—
129ΓL(1,𝔽25)—94no—
130ΓL(1,𝔽25)—182no—
131D3×C8—62no—
132D3×C8—62no—
133D3×C8—182no—
134C22×C32—21no—
135C22×C32D3×Q812yes—
136C2×C3×D3—32no—
137C2×C3×D3—32no—
138C2×C3×D3—61no—
139C2×C3×D3—61no—
140C2×C3×D3—122no—
141C2×C3×D3—122no—
142C32×C4—21no—
143C32×C4—21no—
144C32×C4C2×C4×D312yes—
145C32×C4D3×D412yes—
146C3×Dic3—32no—
147C3×Dic3—32no—
148C3×Dic3—61no—
149C3×Dic3—61no—
150C2×M16—91no—
151C2×M16—271no—
152C2×M16—271no—
153#108—92no—
154#108—272no—
155H(3,𝔽3)#43011yescharacteristic, complemented, Sylow
156C23×C3—91no—
157C22×D3—32no—
158C22×D3—61no—
159C2×C3×C4—32no—
160C2×C3×C4—61no—
161C2×C3×C4—91no—
162C2×C3×C4—91no—
163C2×C3×C4—91no—
164C2×C3×C4—91no—
165C2×C3×C4—91no—
166C2×C3×C4C2×D3211yescharacteristic
167C2×Dic3—32no—
168C2×Dic3—61no—
169C4×D3—32no—
170C4×D3—32no—
171C4×D3—32no—
172C4×D3—32no—
173C4×D3—61no—
174C4×D3—61no—
175C4×D3—61no—
176C4×D3—61no—
177C3×C8—21no—
178C3×C8—62no—
179C3×C8—62no—
180C3×C8—94no—
181C3×C8—182no—
182C3×C8—181no—
183C3⋊C8—34no—
184C3⋊C8—62no—
185C3⋊C8—62no—
186C3⋊C8—62no—
187C3⋊C8—94no—
188C3⋊C8—182no—
189C2×C32—22no—
190C2×C32—21no—
191C2×C32—41no—
192C2×C32D3×C4⋊C412yes—
193C3×D3—62no—
194C3×D3—62no—
195C3×D3—62no—
196C3×D3—121no—
197C22×C4—91no—
198M16—92no—
199M16—91no—
200M16—181no—
201M16—272no—
202M16—272no—
203M16—541no—
204M16((C3⋊D3)⋊C3)⋊C211yescharacteristic
205C2×C8—92no—
206C2×C8—181no—
207C2×C8—541no—
208C22×C3—32no—
209C22×C3—61no—
210C22×C3—91no—
211C22×C3—91no—
212C22×C3—361no—
213C22×C3#379711yescharacteristic
214C2×D3—32no—
215C2×D3—32no—
216C2×D3—61no—
217C2×D3—61no—
218C2×D3—122no—
219C2×D3—122no—
220C3×C4—32no—
221C3×C4—32no—
222C3×C4—61no—
223C3×C4—61no—
224C3×C4—91no—
225C3×C4—91no—
226C3×C4C4×D3211yescharacteristic
227C3×C4#381111yescharacteristic
228Dic3—32no—
229Dic3—32no—
230Dic3—61no—
231Dic3—61no—
232C32—21no—
233C32#489312yescomplemented
234C23—91no—
235C2×C4—91no—
236C2×C4—91no—
237C2×C4—91no—
238C2×C4—91no—
239C2×C4—91no—
240C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
241C8—21no—
242C8—94no—
243C8—181no—
244C8—182no—
245C2×C3—21no—
246C2×C3—32no—
247C2×C3—62no—
248C2×C3—61no—
249C2×C3—121no—
250C2×C3—181no—
251C2×C3—181no—
252C2×C3#6388311yescharacteristic
253D3—62no—
254D3—62no—
255D3—62no—
256D3—121no—
257C22—91no—
258C22—91no—
259C22—361no—
260C22#8865111yescharacteristic
261C4—91no—
262C4—91no—
263C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
264C4#8867711yescharacteristic
265C3—32no—
266C3—61no—
267C3#1059097411yescharacteristic
268C2—21no—
269C2—181no—
270C2—181no—
271C2#1172114711yescharacteristic
subgroupG/H
centerC4#88677
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingM16×H(3,𝔽3)C22
socleC2×C3#63883
prime radicalC22×#272Q8
cosocleC2×C4×H(3,𝔽3)C23
hypercenterM16((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#430
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
4. #63649⋊C3C2semidirect
5. #11720319C2semidirect
22. #63649D3semidirect
155. H(3,𝔽3)#430semidirect
233. C32#4893semidirect

Maximal subgroups

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

subgroupindex
M16×#2722
C2×#884352
#117210652
#63649⋊C32
#117203192
#105909733