Finite Group Databasebeta

← #13140883

Subgroups — #13140883

Order 1728.

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

normal subgroups only

This group has 406 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
2#11721117C211yesmaximal, characteristic
3#63662⋊C3C212yesmaximal
4#11720307C212yesmaximal
5#11720304C211yesmaximal, characteristic
6#10591544—32nomaximal
7C2×Q8×H(3,𝔽3)C2211yescharacteristic
8C2×#6196—21no—
9C4⋊#1084—21no—
10C4⋊#1084C2211yescharacteristic
11C4×#1084—21no—
12C4×#1084C2211yescharacteristic
13C4⋊((C3⋊Dic3)⋊C3)C2212yes—
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15#88449—21no—
16C3×Q8×Dic3—32no—
17C3×Q8×Dic3—61no—
18#63918—32no—
19#63662D312yes—
20C3×#835—32no—
21#63574—32no—
22#63573—32no—
23Dic3⋊(C3⋊C8)—32no—
24Q8×H(3,𝔽3)—21no—
25Q8×H(3,𝔽3)Q812yescomplemented
26C2×C4×H(3,𝔽3)—21no—
27C2×C4×H(3,𝔽3)C2311yescharacteristic
28C2×#1084—21no—
29C2×#1084D411yescharacteristic
30C2×#1084D411yescharacteristic
31C2×(C3⋊Dic3)⋊C3—22no—
32#6196—42no—
33(C3⋊(C3⋊C8))⋊C3—22no—
34#5020—91no—
35#4791—92no—
36C2×C32×Q8—21no—
37C2×C32×Q8C2×D312yes—
38C2×#627—62no—
39C4⋊(C3⋊Dic3)C2×D312yes—
40C2×C3⋊(C3⋊C8)C2×D312yes—
41C3×C4⋊Dic3—32no—
42C3×C4⋊Dic3—32no—
43C3×C4⋊Dic3—62no—
44C3×C4⋊Dic3—61no—
45C3×C4⋊Dic3—62no—
46C3×C4×Dic3—32no—
47C3×C4×Dic3—62no—
48C3×C4×Dic3—61no—
49C3×C4×Dic3—62no—
50C2×C3×C3⋊C8—32no—
51Dic3⋊Dic3—62no—
52C22×H(3,𝔽3)C2×D411yescharacteristic
53C4×H(3,𝔽3)—22no—
54C4×H(3,𝔽3)C2×Q811yescharacteristic
55C4×H(3,𝔽3)#4111yescharacteristic
56#1084—21no—
57#1084—41no—
58#1084SD1612yes—
59(C3⋊Dic3)⋊C3—42no—
60C3×C4×Q8—91no—
61Q8×Dic3—32no—
62Q8×Dic3—61no—
63#888—92no—
64#869—91no—
65C3×#103—92no—
66#835—32no—
67#835—92no—
68#817—92no—
69Dic3⋊C8—92no—
70#808—92no—
71#808—92no—
72C4⋊(C3⋊C8)—91no—
73C32×Q8—22no—
74C32×Q8—22no—
75C32×Q8—22no—
76C32×Q8Dic614yescomplemented
77C2×C32×C4—22no—
78C2×C32×C4—22no—
79C2×C32×C4—21no—
80C2×C32×C4C22×D312yes—
81C2×C3⋊Dic3—22no—
82C2×C3×Dic3—32no—
83C2×C3×Dic3—32no—
84C2×C3×Dic3—61no—
85C2×C3×Dic3—61no—
86C2×C3×Dic3—62no—
87C2×C3×Dic3—62no—
88C2×C3×Dic3—62no—
89#627—124no—
90C3⋊(C3⋊C8)—22no—
91C3×C3⋊C8—62no—
92#475—271noSylow
93C2×H(3,𝔽3)C22⋊Q811yescharacteristic
94C2×H(3,𝔽3)C2×SD1611yescharacteristic
95C2×H(3,𝔽3)#13711yescharacteristic
96C2×C3×Q8—32no—
97C2×C3×Q8—61no—
98C2×C3×Q8D3211yescharacteristic
99C2×Dic6—182no—
100C2×Dic6—181no—
101C2×C3×C8—92no—
102C3×C4⋊C4—92no—
103C3×C4⋊C4—91no—
104C3×C4⋊C4—181no—
105C3×C42—91no—
106C3×C42—181no—
107C4⋊Dic3—32no—
108C4⋊Dic3—32no—
109C4⋊Dic3—61no—
110C4⋊Dic3—62no—
111C4⋊Dic3—62no—
112C4⋊Dic3—92no—
113C4⋊Dic3—181no—
114Dic3⋊C4—182no—
115C4×Dic3—32no—
116C4×Dic3—61no—
117C4×Dic3—62no—
118C4×Dic3—62no—
119C2×C3⋊C8—32no—
120C2×C3⋊C8—92no—
121C22×C32—21no—
122C22×C32D3×D412yes—
123C32×C4—24no—
124C32×C4—24no—
125C32×C4—21no—
126C32×C4—21no—
127C32×C4C2×Dic612yes—
128C32×C4#23712yes—
129C3⋊Dic3—42no—
130C3×Dic3—34no—
131C3×Dic3—62no—
132C3×Dic3—62no—
133C3×Dic3—121no—
134C3×Dic3—122no—
135C3×Dic3—122no—
136C3×Dic3—122no—
137C4⋊Q8—271no—
138C4×Q8—91no—
139#106—272no—
140C4⋊C8—271no—
141#103—92no—
142H(3,𝔽3)#47511yescharacteristic, complemented, Sylow
143C3×Q8—21no—
144C3×Q8—34no—
145C3×Q8—62no—
146C3×Q8—62no—
147C3×Q8—62no—
148C3×Q8#62712yes—
149C2×C3×C4—21no—
150C2×C3×C4—32no—
151C2×C3×C4—62no—
152C2×C3×C4—62no—
153C2×C3×C4—61no—
154C2×C3×C4—91no—
155C2×C3×C4—91no—
156C2×C3×C4—182no—
157C2×C3×C4—181no—
158C2×C3×C4C2×D3211yescharacteristic
159C2×Dic3—32no—
160C2×Dic3—32no—
161C2×Dic3—61no—
162C2×Dic3—61no—
163C2×Dic3—62no—
164C2×Dic3—62no—
165C2×Dic3—62no—
166C2×Dic3—182no—
167Dic6—364no—
168Dic6—362no—
169C3×C8—182no—
170C3⋊C8—62no—
171C3⋊C8—182no—
172C2×C32—21no—
173C2×C32—21no—
174C2×C32—21no—
175C2×C32#87812yes—
176C2×C32D3×SD1612yes—
177C2×C32#91812yes—
178C2×Q8—541no—
179C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
180C2×C8—92no—
181C4⋊C4—92no—
182C4⋊C4—91no—
183C4⋊C4—181no—
184C4⋊C4—541no—
185C42—91no—
186C42—181no—
187C22×C3—32no—
188C22×C3—61no—
189C22×C3#381111yescharacteristic
190C3×C4—22no—
191C3×C4—32no—
192C3×C4—32no—
193C3×C4—64no—
194C3×C4—64no—
195C3×C4—61no—
196C3×C4—61no—
197C3×C4—92no—
198C3×C4—181no—
199C3×C4—362no—
200C3×C4—361no—
201C3×C4#380511yescharacteristic
202C3×C4C2×#62711yescharacteristic
203Dic3—34no—
204Dic3—62no—
205Dic3—62no—
206Dic3—121no—
207Dic3—122no—
208Dic3—122no—
209Dic3—122no—
210Dic3—362no—
211C32—21no—
212C32#479112yescomplemented
213Q8—21no—
214Q8—1082no—
215Q8#619612yescomplemented
216C2×C4—21no—
217C2×C4—91no—
218C2×C4—91no—
219C2×C4—181no—
220C2×C4—182no—
221C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
222C8—182no—
223C2×C3—32no—
224C2×C3—32no—
225C2×C3—32no—
226C2×C3—61no—
227C2×C3—61no—
228C2×C3—61no—
229C2×C3#6394011yescharacteristic
230C2×C3#6394511yescharacteristic
231C2×C3#6398211yescharacteristic
232C22#8867711yescharacteristic
233C4—22no—
234C4—92no—
235C4—181no—
236C4—361no—
237C4—362no—
238C4C2×#619611yescharacteristic
239C4#8867211yescharacteristic
240C3—32no—
241C3—61no—
242C3#1059163011yescharacteristic
243C2#1172122711yescharacteristic
244C2#1172122811yescharacteristic
245C2#1172130911yescharacteristic
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)#475
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
25. Q8×H(3,𝔽3)Q8semidirect
76. C32×Q8Dic6semidirect
142. H(3,𝔽3)#475semidirect
212. C32#4791semidirect
215. Q8#6196semidirect

Maximal subgroups

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

subgroupindex
Q8×#10842
#117211172
#63662⋊C32
#117203072
#117203042
#105915443