Finite Group Databasebeta

← #13138893

Subgroups — #13138893

Order 1728.

447 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 447 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C2×C8×#272C211yesmaximal, characteristic
2#11721065C212yesmaximal
3#63646⋊C3C212yesmaximal, complemented
4#11720316C212yesmaximal, complemented
5D3×#819—32nomaximal
6C2×C4×#272C2211yescharacteristic
7C2×C8×H(3,𝔽3)C2211yescharacteristic
8C2×#6180C2211yescharacteristic
9C8×#272C412yes—
10C8×#272C412yes—
11#3748⋊C3C2214yes—
12#88435—22no—
13#88434—22no—
14C2×C3×D3×C8—32no—
15C2×C3×D3×C8—61no—
16D3×#208—34no—
17#63646D312yescomplemented
18C3×#819—32no—
19#63577—34no—
20C22×#272Q811yescharacteristic
21C2×C4×H(3,𝔽3)C2311yescharacteristic
22C2×#1084Q811yescharacteristic
23C4×#272C2×C411yescharacteristic
24C4×#272C2×C411yescharacteristic
25C4×#272D411yescharacteristic
26C4×#272D411yescharacteristic
27C8×H(3,𝔽3)C2×C412yes—
28#6180C2×C412yes—
29(C3⋊(C3⋊C8))⋊C3—24no—
30C2×#819—91no—
31D3×#108—92no—
32C2×C3×C4×D3—32no—
33C2×C3×C4×D3—61no—
34C2×C32×C8—21no—
35C2×C32×C8C2×D312yes—
36#3748C2×D314yes—
37C3×#208—34no—
38C2×C3×C3⋊C8—32no—
39C2×C3×C3⋊C8—61no—
40C3×D3×C8—34no—
41C3×D3×C8—34no—
42C3×D3×C8—62no—
43C3×D3×C8—62no—
44#3711—64no—
45D3×C3⋊C8—64no—
46C22×H(3,𝔽3)C2×Q811yescharacteristic
47C2×#272—22no—
48C2×#272C4⋊C411yescharacteristic
49C2×#272C4⋊C411yescharacteristic
50C4×H(3,𝔽3)C22×C411yescharacteristic
51C4×H(3,𝔽3)C2×D411yescharacteristic
52#1084C4⋊C411yescharacteristic
53#1084C4⋊C411yescharacteristic
54C22×C3×C8—91no—
55C2×#208—92no—
56D3×M16—94no—
57C2×D3×C8—32no—
58C2×D3×C8—61no—
59C3×#108—92no—
60#822—94no—
61#819—32no—
62#819—92no—
63#819—92no—
64C22×C3×D3—32no—
65C22×C3×D3—61no—
66C2×C32×C4—21no—
67C2×C32×C4C22×D312yes—
68C2×C3×Dic3—32no—
69C2×C3×Dic3—61no—
70C3×C4×D3—32no—
71C3×C4×D3—32no—
72C3×C4×D3—32no—
73C3×C4×D3—32no—
74C3×C4×D3—61no—
75C3×C4×D3—61no—
76C3×C4×D3—61no—
77C3×C4×D3—61no—
78C32×C8—22no—
79C32×C8C4×D314yes—
80C3⋊(C3⋊C8)—24no—
81C3×C3⋊C8—34no—
82C3×C3⋊C8—62no—
83C3×C3⋊C8—64no—
84C2×#108—271noSylow
85C2×H(3,𝔽3)—21no—
86C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
87#272—21no—
88#272#10812yescomplemented
89C22×C3×C4—91no—
90C2×C4×D3—32no—
91C2×C4×D3—61no—
92C3×M16—94no—
93C2×C3×C8—32no—
94C2×C3×C8—61no—
95C2×C3×C8—91no—
96C2×C3×C8—92no—
97C2×C3×C8—92no—
98C2×C3×C8D3211yescharacteristic
99#208—34no—
100#208—94no—
101#208—182no—
102C2×C3⋊C8—32no—
103C2×C3⋊C8—61no—
104C2×C3⋊C8—182no—
105ΓL(1,𝔽25)—184no—
106D3×C8—34no—
107D3×C8—34no—
108D3×C8—62no—
109D3×C8—62no—
110D3×C8—184no—
111C22×C32—21no—
112C22×C32D3×Q812yes—
113C2×C3×D3—32no—
114C2×C3×D3—32no—
115C2×C3×D3—61no—
116C2×C3×D3—61no—
117C2×C3×D3—64no—
118C2×C3×D3—64no—
119C32×C4—21no—
120C32×C4—21no—
121C32×C4C2×C4×D312yes—
122C32×C4D3×D412yes—
123C3×Dic3—32no—
124C3×Dic3—32no—
125C3×Dic3—61no—
126C3×Dic3—61no—
127C2×M16—272no—
128C22×C8—91no—
129#108—92no—
130#108—272no—
131H(3,𝔽3)C2×#10811yescharacteristic, complemented, Sylow
132C23×C3—91no—
133C22×D3—32no—
134C22×D3—61no—
135C2×C3×C4—32no—
136C2×C3×C4—61no—
137C2×C3×C4—91no—
138C2×C3×C4—91no—
139C2×C3×C4—91no—
140C2×C3×C4—91no—
141C2×C3×C4—91no—
142C2×C3×C4C2×D3211yescharacteristic
143C2×Dic3—32no—
144C2×Dic3—61no—
145C4×D3—32no—
146C4×D3—32no—
147C4×D3—32no—
148C4×D3—32no—
149C4×D3—61no—
150C4×D3—61no—
151C4×D3—61no—
152C4×D3—61no—
153C3×C8—34no—
154C3×C8—62no—
155C3×C8—92no—
156C3×C8—184no—
157C3×C8#62412yes—
158C3⋊C8—34no—
159C3⋊C8—62no—
160C3⋊C8—64no—
161C3⋊C8—184no—
162C2×C32—22no—
163C2×C32—22no—
164C2×C32—21no—
165C2×C32D3×C4⋊C412yes—
166C3×D3—34no—
167C3×D3—62no—
168C3×D3—62no—
169C3×D3—62no—
170C22×C4—91no—
171M16—94no—
172M16—542no—
173C2×C8—91no—
174C2×C8—92no—
175C2×C8—92no—
176C2×C8—542no—
177C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
178C22×C3—32no—
179C22×C3—61no—
180C22×C3—91no—
181C22×C3—91no—
182C22×C3—182no—
183C22×C3#379711yescharacteristic
184C2×D3—32no—
185C2×D3—32no—
186C2×D3—61no—
187C2×D3—61no—
188C2×D3—64no—
189C2×D3—64no—
190C3×C4—32no—
191C3×C4—32no—
192C3×C4—61no—
193C3×C4—61no—
194C3×C4—91no—
195C3×C4—91no—
196C3×C4#380211yescharacteristic
197C3×C4C2×#62411yescharacteristic
198Dic3—32no—
199Dic3—32no—
200Dic3—61no—
201Dic3—61no—
202C32—21no—
203C32D3×#10812yescomplemented
204C23—91no—
205C2×C4—91no—
206C2×C4—91no—
207C2×C4—91no—
208C2×C4—91no—
209C2×C4—91no—
210C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
211C8—92no—
212C8—184no—
213C8#619912yes—
214C2×C3—21no—
215C2×C3—32no—
216C2×C3—62no—
217C2×C3—62no—
218C2×C3—61no—
219C2×C3—92no—
220C2×C3—181no—
221C2×C3#6390011yescharacteristic
222D3—34no—
223D3—62no—
224D3—62no—
225D3—62no—
226C22—91no—
227C22—91no—
228C22—182no—
229C22#8865111yescharacteristic
230C4—91no—
231C4—91no—
232C4#8865511yescharacteristic
233C4C2×#619911yescharacteristic
234C3—32no—
235C3—61no—
236C3#1059091311yescharacteristic
237C2—21no—
238C2—92no—
239C2—181no—
240C2#1172115211yescharacteristic
subgroupG/H
centerC4#88655
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC2×C3#63900
prime radicalC22×#272Q8
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#108
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
3. #63646⋊C3C2semidirect
4. #11720316C2semidirect
17. #63646D3semidirect
88. #272#108semidirect
131. H(3,𝔽3)C2×#108semidirect
203. C32D3×#108semidirect

Maximal subgroups

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

subgroupindex
C2×C8×#2722
#117210652
#63646⋊C32
#117203162
D3×#8193