Finite Group Databasebeta

← #13140885

Subgroups — #13140885

Order 1728.

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

normal subgroups only

This group has 448 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
1Q8×#1084C211yesmaximal, characteristic
2C2×#3756⋊C3C212yesmaximal
3#11721099C211yesmaximal, characteristic
4#11720309C212yesmaximal
5#11720304C211yesmaximal, characteristic
6#10591625—32nomaximal
7C2×Q8×H(3,𝔽3)C2211yescharacteristic
8C2×#634⋊C3C2212yes—
9C4⋊#1084—21no—
10C4⋊#1084C2211yescharacteristic
11C4×#1084—21no—
12C4×#1084C2211yescharacteristic
13#3756⋊C3—24no—
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15#88449—21no—
16#88449—21no—
17C2×#3756D312yes—
18C3×Q8×Dic3—32no—
19C3×Q8×Dic3—61no—
20C2×C3×#216—32no—
21#63866—32no—
22Dic3⋊(C3⋊C8)—32no—
23#63567—32no—
24#63565—32no—
25Q8×H(3,𝔽3)—21no—
26Q8×H(3,𝔽3)D412yes—
27C2×C4×H(3,𝔽3)—21no—
28C2×C4×H(3,𝔽3)C2311yescharacteristic
29C2×#1084—21no—
30C2×#1084D411yescharacteristic
31C2×#1084D411yescharacteristic
32C2×(C3⋊Dic3)⋊C3—22no—
33#634⋊C3—22no—
34#634⋊C3—22no—
35(C3⋊(C3⋊C8))⋊C3—22no—
36#5182—92no—
37#5031—91no—
38C2×C32×Q8—21no—
39C2×C32×Q8C2×D312yes—
40C2×#634C2×D312yes—
41C2×C3×Dic6—32no—
42#3756—24no—
43C2×C3⋊(C3⋊C8)C2×D312yes—
44C3×#216—64no—
45C3×C4⋊Dic3—32no—
46C3×C4⋊Dic3—62no—
47C3×C4⋊Dic3—61no—
48C3×C4⋊Dic3—62no—
49C3×C4×Dic3—32no—
50C3×C4×Dic3—62no—
51C3×C4×Dic3—61no—
52C3×C4×Dic3—62no—
53C2×C3×C3⋊C8—32no—
54Dic3⋊Dic3—62no—
55Dic3⋊Dic3—62no—
56C22×H(3,𝔽3)C2×D411yescharacteristic
57C4×H(3,𝔽3)—22no—
58C4×H(3,𝔽3)C2×D411yescharacteristic
59C4×H(3,𝔽3)#4111yescharacteristic
60#1084—21no—
61#1084—41no—
62#1084Q1612yescomplemented
63(C3⋊Dic3)⋊C3—42no—
64C2×C3×Q16—92no—
65C3×C4×Q8—91no—
66Q8×Dic3—32no—
67Q8×Dic3—61no—
68#944—92no—
69C2×#216—32no—
70C2×#216—92no—
71#869—91no—
72#835—92no—
73#816—92no—
74Dic3⋊C8—92no—
75#810—92no—
76C4⋊(C3⋊C8)—91no—
77C32×Q8—22no—
78C32×Q8—22no—
79C32×Q8—22no—
80C32×Q8#6714yes—
81C2×C32×C4—22no—
82C2×C32×C4—22no—
83C2×C32×C4—21no—
84C2×C32×C4C22×D312yes—
85C2×C3⋊Dic3—22no—
86#634—22no—
87#634—22no—
88C2×C3×Dic3—32no—
89C2×C3×Dic3—32no—
90C2×C3×Dic3—61no—
91C2×C3×Dic3—61no—
92C2×C3×Dic3—62no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—62no—
95C3×Dic6—62no—
96C3×Dic6—62no—
97C3⋊(C3⋊C8)—22no—
98C3×C3⋊C8—62no—
99#462—271noSylow
100C2×H(3,𝔽3)#12111yescharacteristic
101C2×H(3,𝔽3)C2×Q1611yescharacteristic
102C2×H(3,𝔽3)#13711yescharacteristic
103C2×C3×Q8—32no—
104C2×C3×Q8—61no—
105C2×C3×Q8—92no—
106C2×C3×Q8D3211yescharacteristic
107C2×Dic6—32no—
108C2×Dic6—92no—
109C3×Q16—184no—
110C2×C3×C8—92no—
111C3×C4⋊C4—91no—
112C3×C4⋊C4—181no—
113C3×C42—91no—
114C3×C42—181no—
115#216—64no—
116#216—184no—
117C4⋊Dic3—32no—
118C4⋊Dic3—61no—
119C4⋊Dic3—62no—
120C4⋊Dic3—62no—
121C4⋊Dic3—181no—
122C4⋊Dic3—181no—
123Dic3⋊C4—182no—
124Dic3⋊C4—182no—
125C4×Dic3—32no—
126C4×Dic3—61no—
127C4×Dic3—62no—
128C4×Dic3—62no—
129C2×C3⋊C8—32no—
130C2×C3⋊C8—92no—
131C22×C32—21no—
132C22×C32D3×D412yes—
133C32×C4—24no—
134C32×C4—24no—
135C32×C4—21no—
136C32×C4—21no—
137C32×C4#23712yes—
138C32×C4C2×#6712yes—
139C3⋊Dic3—42no—
140C3×Dic3—34no—
141C3×Dic3—62no—
142C3×Dic3—62no—
143C3×Dic3—121no—
144C3×Dic3—122no—
145C3×Dic3—122no—
146C3×Dic3—122no—
147C2×Q16—92no—
148C4⋊Q8—271no—
149C4×Q8—91no—
150C4⋊C8—271no—
151#103—272no—
152H(3,𝔽3)#46211yescharacteristic, complemented, Sylow
153C3×Q8—21no—
154C3×Q8—34no—
155C3×Q8—62no—
156C3×Q8—62no—
157C3×Q8—62no—
158C3×Q8—182no—
159C3×Q8—182no—
160C3×Q8#62512yes—
161C2×C3×C4—21no—
162C2×C3×C4—32no—
163C2×C3×C4—62no—
164C2×C3×C4—62no—
165C2×C3×C4—61no—
166C2×C3×C4—91no—
167C2×C3×C4—91no—
168C2×C3×C4—182no—
169C2×C3×C4—181no—
170C2×C3×C4C2×D3211yescharacteristic
171C2×Dic3—32no—
172C2×Dic3—32no—
173C2×Dic3—61no—
174C2×Dic3—61no—
175C2×Dic3—62no—
176C2×Dic3—62no—
177C2×Dic3—62no—
178C2×Dic3—182no—
179Dic6—62no—
180Dic6—62no—
181Dic6—182no—
182Dic6—182no—
183C3×C8—182no—
184C3⋊C8—62no—
185C3⋊C8—182no—
186C2×C32—21no—
187C2×C32—21no—
188C2×C32—21no—
189C2×C32D3×Q1612yes—
190C2×C32#91812yes—
191C2×C32#93912yes—
192C2×Q8—92no—
193C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
194Q16—184no—
195C2×C8—92no—
196C4⋊C4—91no—
197C4⋊C4—181no—
198C4⋊C4—541no—
199C4⋊C4—541no—
200C42—91no—
201C42—181no—
202C22×C3—32no—
203C22×C3—61no—
204C22×C3#381111yescharacteristic
205C3×C4—22no—
206C3×C4—32no—
207C3×C4—32no—
208C3×C4—64no—
209C3×C4—64no—
210C3×C4—61no—
211C3×C4—61no—
212C3×C4—92no—
213C3×C4—181no—
214C3×C4—362no—
215C3×C4—361no—
216C3×C4#380511yescharacteristic
217C3×C4C2×#62511yescharacteristic
218Dic3—34no—
219Dic3—62no—
220Dic3—62no—
221Dic3—121no—
222Dic3—122no—
223Dic3—122no—
224Dic3—122no—
225Dic3—362no—
226C32—21no—
227C32#518212yescomplemented
228Q8—21no—
229Q8—182no—
230Q8—182no—
231Q8#619812yes—
232C2×C4—21no—
233C2×C4—91no—
234C2×C4—91no—
235C2×C4—181no—
236C2×C4—182no—
237C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
238C8—182no—
239C2×C3—32no—
240C2×C3—32no—
241C2×C3—32no—
242C2×C3—61no—
243C2×C3—61no—
244C2×C3—61no—
245C2×C3#6394411yescharacteristic
246C2×C3#6394511yescharacteristic
247C2×C3#6398011yescharacteristic
248C22#8867711yescharacteristic
249C4—22no—
250C4—92no—
251C4—181no—
252C4—361no—
253C4—362no—
254C4#8867211yescharacteristic
255C4C2×#619811yescharacteristic
256C3—32no—
257C3—61no—
258C3#1059163311yescharacteristic
259C2#1172122611yescharacteristic
260C2#1172122711yescharacteristic
261C2#1172131811yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#462
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
62. #1084Q16semidirect
152. H(3,𝔽3)#462semidirect
227. C32#5182semidirect

Maximal subgroups

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

subgroupindex
Q8×#10842
C2×#3756⋊C32
#117210992
#117203092
#117203042
#105916253