Finite Group Databasebeta

← #13138801

Subgroups — #13138801

Order 1728.

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

normal subgroups only

This group has 457 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#11721144C211yesmaximal, characteristic
2#11721117C211yesmaximal, characteristic, complemented
3C8×#1084C211yesmaximal, characteristic, complemented
4#63648⋊C3C212yesmaximal, complemented
5#11720318C212yesmaximal, complemented
6#10590878—32nomaximal
7C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
8C2×#6198—21no—
9C2×#6198—21no—
10C2×#6196—21no—
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C4×#1084C2211yescharacteristic
13C2×#6180C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)C2212yes—
15#88449—21no—
16#63918—32no—
17#63911—32no—
18#63648D312yescomplemented
19C3×#821—32no—
20C3×C8×Dic3—32no—
21C3×C8×Dic3—61no—
22#63564—32no—
23#63563—32no—
24C22×(C3⋊D3)⋊C3—22no—
25C2×C4×H(3,𝔽3)C2311yescharacteristic
26C2×#1084D411yescharacteristic
27C2×#1084D411yescharacteristic
28C2×(C3⋊Dic3)⋊C3—22no—
29C4⋊((C3⋊D3)⋊C3)—22no—
30C4⋊((C3⋊D3)⋊C3)—22no—
31#6198—42no—
32#6198—42no—
33#6196—42no—
34C8×H(3,𝔽3)—21no—
35#6180—21no—
36#4760—92no—
37#4694—91no—
38C2×C4⋊(C3⋊D3)C2×D312yes—
39C2×C3×D12—32no—
40C2×#627—62no—
41C2×#626—62no—
42C2×#626—62no—
43C2×C32×C8—21no—
44C2×C32×C8C2×D312yes—
45C4⋊(C3⋊Dic3)C2×D312yes—
46C3×C4⋊Dic3—32no—
47C3×C4×Dic3—32no—
48C3×C4×Dic3—61no—
49C2×C3×C3⋊C8—32no—
50C2×C3×C3⋊C8—61no—
51Dic3⋊Dic3—62no—
52C22×H(3,𝔽3)C2×D411yescharacteristic
53C2×(C3⋊D3)⋊C3—42no—
54C2×(C3⋊D3)⋊C3—42no—
55C2×(C3⋊D3)⋊C3—42no—
56C4×H(3,𝔽3)#4111yescharacteristic
57C4×H(3,𝔽3)#4111yescharacteristic
58#1084D812yes—
59#1084SD1612yes—
60(C3⋊Dic3)⋊C3—42no—
61#940—92no—
62#888—92no—
63#874—91no—
64#869—91no—
65C3×#102—92no—
66C3×C4×C8—91no—
67#832—92no—
68#821—32no—
69#821—92no—
70#821—92no—
71C8×Dic3—32no—
72C8×Dic3—61no—
73#809—92no—
74C22×C3⋊D3—22no—
75C22×C3×D3—62no—
76C2×C32×C4—21no—
77C2×C32×C4C22×D312yes—
78C2×C3⋊Dic3—22no—
79C4⋊(C3⋊D3)—22no—
80C4⋊(C3⋊D3)—22no—
81C2×C3×Dic3—32no—
82C2×C3×Dic3—32no—
83C2×C3×Dic3—61no—
84C2×C3×Dic3—61no—
85C2×C3×Dic3—62no—
86C3×D12—62no—
87C3×D12—62no—
88#627—124no—
89#626—124no—
90#626—124no—
91C32×C8—22no—
92C32×C8—22no—
93C3×C3⋊C8—62no—
94C3×C3⋊C8—62no—
95#486—271noSylow
96C2×H(3,𝔽3)#12411yescharacteristic
97C2×H(3,𝔽3)C2×D811yescharacteristic
98C2×H(3,𝔽3)C2×SD1611yescharacteristic
99(C3⋊D3)⋊C3—82no—
100C2×C3×D4—92no—
101C2×#67—182no—
102C2×#67—182no—
103C2×D12—32no—
104C2×D12—92no—
105C2×D12—181no—
106C2×D12—181no—
107C2×Dic6—182no—
108C2×Dic6—181no—
109C2×C3×C8—32no—
110C2×C3×C8—61no—
111C2×C3×C8—91no—
112C2×C3×C8D3211yescharacteristic
113C3×C4⋊C4—92no—
114C3×C42—91no—
115C4⋊Dic3—32no—
116C4⋊Dic3—92no—
117C4⋊Dic3—181no—
118Dic3⋊C4—182no—
119C4×Dic3—32no—
120C4×Dic3—61no—
121C2×C3⋊C8—32no—
122C2×C3⋊C8—61no—
123C22×C32—21no—
124C22×C32D3×D412yes—
125C2×C3⋊D3—42no—
126C2×C3⋊D3—42no—
127C2×C3⋊D3—42no—
128C2×C3×D3—122no—
129C2×C3×D3—122no—
130C2×C3×D3—122no—
131C32×C4—21no—
132C32×C4—21no—
133C32×C4#23512yes—
134C32×C4#23712yes—
135C3⋊Dic3—42no—
136C3×Dic3—34no—
137C3×Dic3—34no—
138C3×Dic3—62no—
139C3×Dic3—62no—
140C3×Dic3—122no—
141C4⋊Q8—271no—
142#127—271no—
143#102—92no—
144#102—272no—
145C4×C8—91no—
146H(3,𝔽3)#48611yescharacteristic, complemented, Sylow
147C23×C3—182no—
148C22×D3—62no—
149C22×D3—182no—
150C3×D4—182no—
151C3×D4—182no—
152C2×C3×C4—32no—
153C2×C3×C4—61no—
154C2×C3×C4—91no—
155C2×C3×C4—91no—
156C2×C3×C4—182no—
157C2×C3×C4C2×D3211yescharacteristic
158#67—364no—
159#67—364no—
160C2×Dic3—32no—
161C2×Dic3—32no—
162C2×Dic3—61no—
163C2×Dic3—61no—
164C2×Dic3—62no—
165C2×Dic3—182no—
166D12—62no—
167D12—62no—
168D12—182no—
169D12—182no—
170D12—362no—
171D12—362no—
172Dic6—364no—
173Dic6—362no—
174C3×C8—21no—
175C3×C8—62no—
176C3×C8—62no—
177C3×C8—181no—
178C3⋊C8—62no—
179C3⋊C8—62no—
180C2×C32—21no—
181C2×C32—21no—
182C2×C32—21no—
183C2×C32#88512yes—
184C2×C32D3×D812yes—
185C2×C32D3×SD1612yes—
186C3⋊D3—82no—
187C3×D3—242no—
188C2×Q8—541no—
189C2×D4—92no—
190C2×D4—541no—
191C2×D4—541no—
192C2×C8—91no—
193C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
194C4⋊C4—92no—
195C4⋊C4—541no—
196C42—91no—
197C22×C3—32no—
198C22×C3—61no—
199C22×C3—362no—
200C22×C3—362no—
201C22×C3—362no—
202C22×C3#380211yescharacteristic
203C2×D3—122no—
204C2×D3—122no—
205C2×D3—122no—
206C2×D3—362no—
207C2×D3—362no—
208C2×D3—362no—
209C3×C4—32no—
210C3×C4—32no—
211C3×C4—61no—
212C3×C4—61no—
213C3×C4—92no—
214C3×C4—92no—
215C3×C4—362no—
216C3×C4#379811yescharacteristic
217C3×C4#380511yescharacteristic
218Dic3—34no—
219Dic3—34no—
220Dic3—62no—
221Dic3—62no—
222Dic3—122no—
223Dic3—362no—
224C32—21no—
225C32#476012yescomplemented
226C23—182no—
227Q8—1082no—
228D4—182no—
229D4—182no—
230D4—1082no—
231D4—1082no—
232C2×C4—91no—
233C2×C4—91no—
234C2×C4—182no—
235C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
236C8—21no—
237C8—181no—
238C2×C3—32no—
239C2×C3—32no—
240C2×C3—32no—
241C2×C3—61no—
242C2×C3—61no—
243C2×C3—61no—
244C2×C3—722no—
245C2×C3#6379611yescharacteristic
246C2×C3#6379711yescharacteristic
247C2×C3#6391511yescharacteristic
248D3—242no—
249D3—722no—
250C22—362no—
251C22—362no—
252C22—362no—
253C22#8865511yescharacteristic
254C4—92no—
255C4—92no—
256C4—362no—
257C4#8865011yescharacteristic
258C4#8867211yescharacteristic
259C3—32no—
260C3—61no—
261C3#1059088511yescharacteristic
262C2—722no—
263C2#1172083711yescharacteristic
264C2#1172083811yescharacteristic
265C2#1172114511yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radical#11721144C2
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#486
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #11721117C2semidirect
3. C8×#1084C2semidirect
4. #63648⋊C3C2semidirect
5. #11720318C2semidirect
18. #63648D3semidirect
146. H(3,𝔽3)#486semidirect
225. C32#4760semidirect

Maximal subgroups

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

subgroupindex
#117211442
#117211172
C8×#10842
#63648⋊C32
#117203182
#105908783