Finite Group Databasebeta

← #13137869

Subgroups — #13137869

Order 1728.

406 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 71 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#11721755C211yesmaximal, characteristic
2#11721117C211yesmaximal, characteristic
3#63634⋊C3C212yesmaximal
4#11720326C212yesmaximal
5#11720304C211yesmaximal, characteristic
6#10590557—32nomaximal
7C2×#6229C2211yescharacteristic
8C2×#6196—21no—
9C4⋊C4×H(3,𝔽3)C2211yescharacteristic
10#88540—21no—
11C4×#1084—21no—
12C4×#1084C2211yescharacteristic
13C4⋊((C3⋊Dic3)⋊C3)C2212yes—
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15#88449—21no—
16C3×#887—32no—
17C3×#887—61no—
18#63918—32no—
19#63634D312yes—
20C3×#807—32no—
21Dic3⋊(C3⋊C8)—32no—
22#63566—32no—
23#63565—32no—
24C2×C4×H(3,𝔽3)—21no—
25C2×C4×H(3,𝔽3)C2311yescharacteristic
26C2×#1084—21no—
27C2×#1084D411yescharacteristic
28C2×#1084D411yescharacteristic
29#6229—21no—
30#6229Q812yescomplemented
31C2×(C3⋊Dic3)⋊C3—22no—
32#6196—42no—
33(C3⋊(C3⋊C8))⋊C3—22no—
34#5021—91no—
35#4875—92no—
36C2×C3×Dic6—32no—
37C2×C3×Dic6—61no—
38C2×#627—62no—
39C32×C4⋊C4—21no—
40C32×C4⋊C4C2×D312yes—
41C4⋊(C3⋊Dic3)C2×D312yes—
42C2×C3⋊(C3⋊C8)C2×D312yes—
43C3×C4⋊Dic3—32no—
44C3×Dic3⋊C4—62no—
45C3×Dic3⋊C4—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)—41no—
54C4×H(3,𝔽3)C2×Q811yescharacteristic
55C4×H(3,𝔽3)#4111yescharacteristic
56#1084—22no—
57#1084—21no—
58#1084Q1612yes—
59(C3⋊Dic3)⋊C3—42no—
60C3×C4×Q8—91no—
61#888—92no—
62#887—32no—
63#887—61no—
64#869—91no—
65C3×#107—92no—
66#835—92no—
67#816—92no—
68Dic3⋊C8—92no—
69#810—92no—
70#807—32no—
71#807—92no—
72C4⋊(C3⋊C8)—91no—
73C2×C32×C4—22no—
74C2×C32×C4—22no—
75C2×C32×C4—21no—
76C2×C32×C4C22×D312yes—
77C2×C3⋊Dic3—22no—
78C2×C3×Dic3—32no—
79C2×C3×Dic3—32no—
80C2×C3×Dic3—61no—
81C2×C3×Dic3—61no—
82C2×C3×Dic3—62no—
83C2×C3×Dic3—62no—
84C2×C3×Dic3—62no—
85C3×Dic6—34no—
86C3×Dic6—62no—
87C3×Dic6—62no—
88C3×Dic6—62no—
89#627—124no—
90C3⋊(C3⋊C8)—22no—
91C3×C3⋊C8—62no—
92#477—271noSylow
93C2×H(3,𝔽3)C22⋊Q811yescharacteristic
94C2×H(3,𝔽3)C2×Q1611yescharacteristic
95C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
96C2×C3×Q8—91no—
97C2×Dic6—32no—
98C2×Dic6—61no—
99C2×Dic6—182no—
100C2×Dic6—181no—
101C2×C3×C8—92no—
102C3×C4⋊C4—32no—
103C3×C4⋊C4—61no—
104C3×C4⋊C4—92no—
105C3×C4⋊C4—181no—
106C3×C4⋊C4D3211yescharacteristic
107C3×C42—91no—
108C3×C42—181no—
109C4⋊Dic3—32no—
110C4⋊Dic3—92no—
111C4⋊Dic3—181no—
112Dic3⋊C4—62no—
113Dic3⋊C4—62no—
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—21no—
124C32×C4—21no—
125C32×C4—42no—
126C32×C4—42no—
127C32×C4#23512yes—
128C32×C4D3×Q812yes—
129C3⋊Dic3—42no—
130C3×Dic3—34no—
131C3×Dic3—62no—
132C3×Dic3—64no—
133C3×Dic3—64no—
134C3×Dic3—62no—
135C3×Dic3—121no—
136C3×Dic3—122no—
137C4⋊Q8—271no—
138C4×Q8—91no—
139#107—92no—
140C4⋊C8—271no—
141#103—272no—
142H(3,𝔽3)#47711yescharacteristic, complemented, Sylow
143C3×Q8—92no—
144C3×Q8—181no—
145C2×C3×C4—21no—
146C2×C3×C4—32no—
147C2×C3×C4—62no—
148C2×C3×C4—62no—
149C2×C3×C4—61no—
150C2×C3×C4—91no—
151C2×C3×C4—91no—
152C2×C3×C4—181no—
153C2×C3×C4—182no—
154C2×C3×C4C2×D3211yescharacteristic
155C2×Dic3—32no—
156C2×Dic3—32no—
157C2×Dic3—61no—
158C2×Dic3—61no—
159C2×Dic3—62no—
160C2×Dic3—62no—
161C2×Dic3—62no—
162C2×Dic3—182no—
163Dic6—34no—
164Dic6—62no—
165Dic6—62no—
166Dic6—62no—
167Dic6—364no—
168Dic6—362no—
169C3×C8—182no—
170C3⋊C8—62no—
171C3⋊C8—182no—
172C2×C32—21no—
173C2×C32—21no—
174C2×C32—21no—
175C2×C32#89612yes—
176C2×C32#91112yes—
177C2×C32D3×Q1612yes—
178C2×Q8—91no—
179C2×Q8—541no—
180C2×C8—92no—
181C4⋊C4—92no—
182C4⋊C4—181no—
183C4⋊C4—541no—
184C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
185C42—91no—
186C42—181no—
187C22×C3—32no—
188C22×C3—61no—
189C22×C3#381111yescharacteristic
190C3×C4—32no—
191C3×C4—32no—
192C3×C4—41no—
193C3×C4—61no—
194C3×C4—61no—
195C3×C4—92no—
196C3×C4—122no—
197C3×C4—122no—
198C3×C4—182no—
199C3×C4—181no—
200C3×C4—362no—
201C3×C4#379711yescharacteristic
202C3×C4#379811yescharacteristic
203Dic3—34no—
204Dic3—62no—
205Dic3—64no—
206Dic3—62no—
207Dic3—64no—
208Dic3—121no—
209Dic3—122no—
210Dic3—362no—
211C32—21no—
212C32#487512yescomplemented
213Q8—92no—
214Q8—181no—
215Q8—1082no—
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#6386511yescharacteristic
230C2×C3#6392711yescharacteristic
231C2×C3#6394411yescharacteristic
232C22#8867711yescharacteristic
233C4—41no—
234C4—92no—
235C4—182no—
236C4—181no—
237C4—362no—
238C4#8865011yescharacteristic
239C4#8865111yescharacteristic
240C3—32no—
241C3—61no—
242C3#1059043211yescharacteristic
243C2#1172109611yescharacteristic
244C2#1172119111yescharacteristic
245C2#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)#477
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
30. #6229Q8semidirect
142. H(3,𝔽3)#477semidirect
212. C32#4875semidirect

Maximal subgroups

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

subgroupindex
#117217552
#117211172
#63634⋊C32
#117203262
#117203042
#105905573