Finite Group Databasebeta

← #13137367

Subgroups — #13137367

Order 1728.

411 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 411 conjugacy classes of subgroups — too many to draw. Only its 95 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#11721715C211yesmaximal, characteristic
2#64078⋊C3C211yesmaximal, characteristic
3C4×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic
4#11720326C214yesmaximal
5#10590290—31nomaximal
6#10590287—31nomaximal
7C2×#6229C2212yes—
8C2×#634⋊C3—21no—
9C42×H(3,𝔽3)C2211yescharacteristic
10C4⋊#1084—21no—
11C4⋊#1084—21no—
12C4⋊((C3⋊Dic3)⋊C3)—21no—
13C4⋊((C3⋊Dic3)⋊C3)C2212yes—
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15#64078D311yescharacteristic
16C3×#869—31no—
17C3×#869—31no—
18C3×#869—31no—
19C3×#869—61no—
20C4×C3⋊(C3⋊C8)D311yescharacteristic
21C3×C4×C3⋊C8—31no—
22#63566—34no—
23#63565—34no—
24C2×C4×H(3,𝔽3)D411yescharacteristic
25C2×C4×H(3,𝔽3)D411yescharacteristic
26C2×C4×H(3,𝔽3)C2311yescharacteristic
27C2×#1084—22no—
28#6229—24no—
29C2×(C3⋊Dic3)⋊C3—22no—
30#634⋊C3—42no—
31(C3⋊(C3⋊C8))⋊C3—22no—
32#5093—91no—
33#5079—91no—
34#4681—91no—
35C2×#634—21no—
36C2×C3×Dic6—32no—
37C2×C3×Dic6—32no—
38C2×C3×Dic6—62no—
39C2×C3×Dic6—61no—
40C32×C42—21no—
41C32×C42C2×D311yescharacteristic
42C32×C42C2×D311yescharacteristic
43C4⋊(C3⋊Dic3)—21no—
44C4⋊(C3⋊Dic3)C2×D312yes—
45C2×C3⋊(C3⋊C8)C2×D312yes—
46C3×C4⋊Dic3—32no—
47C3×C4⋊Dic3—62no—
48C3×C4⋊Dic3—62no—
49C3×C4⋊Dic3—61no—
50C3×C4⋊Dic3—61no—
51C3×C4⋊Dic3—61no—
52C3×C4⋊Dic3—61no—
53C3×C4⋊Dic3—61no—
54C2×C3×C3⋊C8—32no—
55C22×H(3,𝔽3)C2×D411yescharacteristic
56C4×H(3,𝔽3)SD1612yes—
57C4×H(3,𝔽3)Q1612yes—
58C4×H(3,𝔽3)#4112yes—
59#1084—42no—
60(C3⋊Dic3)⋊C3—42no—
61C3×C4⋊Q8—91no—
62C3×C4⋊Q8—91no—
63#869—31no—
64#869—31no—
65#869—31no—
66#869—61no—
67#869—91no—
68C3×C4×C8—91no—
69#835—94no—
70#816—94no—
71#810—94no—
72C4×C3⋊C8—31no—
73C4×C3⋊C8—91no—
74C2×C32×C4—21no—
75C2×C32×C4—21no—
76C2×C32×C4—21no—
77C2×C32×C4D1211yescharacteristic
78C2×C32×C4D1211yescharacteristic
79C2×C32×C4#6711yescharacteristic
80C2×C32×C4#6711yescharacteristic
81C2×C32×C4C22×D311yescharacteristic
82C2×C32×C4C22×D311yescharacteristic
83C2×C3⋊Dic3—22no—
84#634—42no—
85C2×C3×Dic3—64no—
86C2×C3×Dic3—62no—
87C2×C3×Dic3—62no—
88C2×C3×Dic3—62no—
89C3×Dic6—64no—
90C3×Dic6—64no—
91C3×Dic6—124no—
92C3×Dic6—122no—
93C3⋊(C3⋊C8)—22no—
94C3×C3⋊C8—62no—
95#487—271noSylow
96C2×H(3,𝔽3)#12411yescharacteristic
97C2×H(3,𝔽3)C2×SD1611yescharacteristic
98C2×H(3,𝔽3)C2×Q1611yescharacteristic
99C2×C3×Q8—92no—
100C2×C3×Q8—181no—
101C2×Dic6—32no—
102C2×Dic6—32no—
103C2×Dic6—62no—
104C2×Dic6—61no—
105C2×Dic6—181no—
106C2×C3×C8—92no—
107C3×C4⋊C4—92no—
108C3×C4⋊C4—181no—
109C3×C4⋊C4—181no—
110C3×C4⋊C4—181no—
111C3×C42—31no—
112C3×C42—31no—
113C3×C42—61no—
114C3×C42D3211yescharacteristic
115C4⋊Dic3—32no—
116C4⋊Dic3—61no—
117C4⋊Dic3—61no—
118C4⋊Dic3—61no—
119C4⋊Dic3—61no—
120C4⋊Dic3—62no—
121C4⋊Dic3—62no—
122C4⋊Dic3—61no—
123C4⋊Dic3—92no—
124C4⋊Dic3—181no—
125C2×C3⋊C8—32no—
126C2×C3⋊C8—92no—
127C22×C32—21no—
128C22×C32C2×D1211yescharacteristic
129C22×C32C2×#6711yescharacteristic
130C32×C4—22no—
131C32×C4—22no—
132C32×C4—22no—
133C32×C4#20412yes—
134C32×C4Dic1212yes—
135C32×C4#21412yes—
136C32×C4#21612yes—
137C32×C4#23512yes—
138C32×C4#23912yes—
139C3⋊Dic3—42no—
140C3×Dic3—124no—
141C3×Dic3—122no—
142C3×Dic3—122no—
143C3×Dic3—122no—
144C4⋊Q8—91no—
145C4⋊Q8—91no—
146#103—274no—
147C4×C8—91no—
148H(3,𝔽3)#48711yescharacteristic, complemented, Sylow
149C3×Q8—184no—
150C3×Q8—362no—
151C2×C3×C4—31no—
152C2×C3×C4—31no—
153C2×C3×C4—31no—
154C2×C3×C4—31no—
155C2×C3×C4—31no—
156C2×C3×C4—31no—
157C2×C3×C4—61no—
158C2×C3×C4—61no—
159C2×C3×C4—61no—
160C2×C3×C4—182no—
161C2×C3×C4—182no—
162C2×C3×C4#62611yescharacteristic
163C2×C3×C4#62611yescharacteristic
164C2×C3×C4C2×D3211yescharacteristic
165C2×Dic3—62no—
166C2×Dic3—62no—
167C2×Dic3—64no—
168C2×Dic3—62no—
169C2×Dic3—182no—
170Dic6—64no—
171Dic6—64no—
172Dic6—124no—
173Dic6—122no—
174Dic6—362no—
175C3×C8—182no—
176C3⋊C8—62no—
177C3⋊C8—182no—
178C2×C32—21no—
179C2×C32—21no—
180C2×C32—21no—
181C2×C32#87511yescharacteristic
182C2×C32C2×#20411yescharacteristic
183C2×C32C2×Dic1211yescharacteristic
184C2×C32C2×#21411yescharacteristic
185C2×C32C2×#21611yescharacteristic
186C2×C32#94711yescharacteristic
187C2×Q8—92no—
188C2×Q8—181no—
189C2×C8—92no—
190C4⋊C4—92no—
191C4⋊C4—181no—
192C4⋊C4—181no—
193C4⋊C4—181no—
194C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
195C22×C3—31no—
196C22×C3—31no—
197C22×C3—61no—
198C22×C3C2×#62611yescharacteristic
199C3×C4—32no—
200C3×C4—32no—
201C3×C4—32no—
202C3×C4—32no—
203C3×C4—32no—
204C3×C4—32no—
205C3×C4—62no—
206C3×C4—62no—
207C3×C4—62no—
208C3×C4—362no—
209C3×C4—362no—
210C3×C4#371612yes—
211C3×C4#371912yes—
212C3×C4#379912yes—
213Dic3—122no—
214Dic3—122no—
215Dic3—124no—
216Dic3—122no—
217Dic3—362no—
218C32—21no—
219C32#468111yescharacteristic, complemented
220C32#509311yescharacteristic, complemented
221Q8—184no—
222Q8—362no—
223C2×C4—182no—
224C2×C4—182no—
225C2×C4#620011yescharacteristic
226C2×C4#620011yescharacteristic
227C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
228C8—182no—
229C2×C3—31no—
230C2×C3—31no—
231C2×C3—31no—
232C2×C3—31no—
233C2×C3—31no—
234C2×C3—31no—
235C2×C3—61no—
236C2×C3—61no—
237C2×C3—61no—
238C2×C3C2×#371611yescharacteristic
239C2×C3C2×#371911yescharacteristic
240C2×C3#6386411yescharacteristic
241C22C2×#620011yescharacteristic
242C4—362no—
243C4—362no—
244C4#8843812yes—
245C4#8843912yes—
246C4#8865212yes—
247C3—31no—
248C3—31no—
249C3—61no—
250C3#1059028811yescharacteristic
251C2C2×#8843811yescharacteristic
252C2C2×#8843911yescharacteristic
253C2#1172111411yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#487
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
148. H(3,𝔽3)#487semidirect
219. C32#4681semidirect
220. C32#5093semidirect

Maximal subgroups

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

subgroupindex
#117217152
#64078⋊C32
C4×(C3⋊(C3⋊C8))⋊C32
#117203262
#105902903
#105902873