Finite Group Databasebeta

← #13137964

Subgroups — #13137964

Order 1728.

489 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 489 conjugacy classes of subgroups — too many to draw. Only its 87 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#11721760C211yesmaximal, characteristic
2#11721152C211yesmaximal, characteristic
3C2×#88435C211yesmaximal, characteristic
4#63634⋊C3C212yesmaximal, complemented
5#11720307C212yesmaximal, complemented
6#10590436—32nomaximal
7C2×C4×#272C2211yescharacteristic
8C2×#6199—21no—
9C4⋊C4×H(3,𝔽3)C2211yescharacteristic
10#88542—21no—
11C4⋊#1084C2211yescharacteristic
12C4×#1084—21no—
13C4⋊((C3⋊Dic3)⋊C3)C2212yes—
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15#88449—21no—
16#88435C414yescomplemented
17C3×#892—32no—
18C3×#892—61no—
19D3×C4⋊Dic3—32no—
20C2×#3711—32no—
21#63634D312yescomplemented
22C3×#807—32no—
23#63574—32no—
24#63573—32no—
25C22×#272D411yescharacteristic
26C2×C4×H(3,𝔽3)—21no—
27C2×C4×H(3,𝔽3)C2311yescharacteristic
28C2×#1084—21no—
29C2×#1084D411yescharacteristic
30C4×#272C2×C412yes—
31C4×#272Q812yes—
32C2×(C3⋊Dic3)⋊C3—22no—
33#6199—42no—
34(C3⋊(C3⋊C8))⋊C3C2×C414yescomplemented
35#4995—91no—
36#4881—92no—
37C2×C3×C4×D3—32no—
38C2×C3×C4×D3—61no—
39C2×D3×Dic3—62no—
40C32×C4⋊C4—21no—
41C32×C4⋊C4C2×D312yes—
42C4⋊(C3⋊Dic3)C2×D312yes—
43C2×C3⋊(C3⋊C8)C2×D312yes—
44C3×(C2×D3)⋊C4—62no—
45C3×(C2×D3)⋊C4—62no—
46C3×C4⋊Dic3—32no—
47C3×C4⋊Dic3—32no—
48C3×C4⋊Dic3—61no—
49C3×C4×Dic3—62no—
50C3×C4×Dic3—62no—
51C2×C3×C3⋊C8—32no—
52Dic3⋊Dic3—62no—
53#3711—38no—
54C22×H(3,𝔽3)C2×D411yescharacteristic
55C2×#272—21no—
56C2×#272—21no—
57C2×#272C4⋊C412yes—
58C4×H(3,𝔽3)—41no—
59C4×H(3,𝔽3)C22×C411yescharacteristic
60C4×H(3,𝔽3)C2×Q811yescharacteristic
61#1084—41no—
62#1084C4⋊C412yes—
63(C3⋊Dic3)⋊C3—42no—
64C3×#117—91no—
65C2×C4⋊Dic3—91no—
66C2×#208—91no—
67C2×ΓL(1,𝔽25)—92no—
68#892—32no—
69#892—61no—
70D3×C4⋊C4—92no—
71C3×#107—92no—
72#817—92no—
73#808—92no—
74#808—92no—
75#807—32no—
76#807—92no—
77C22×C3×D3—32no—
78C22×C3×D3—61no—
79C2×C32×C4—22no—
80C2×C32×C4—22no—
81C2×C32×C4—21no—
82C2×C32×C4C22×D312yes—
83C2×C3⋊Dic3—22no—
84C2×C3×Dic3—32no—
85C2×C3×Dic3—61no—
86C2×C3×Dic3—62no—
87C2×C3×Dic3—62no—
88C2×C3×Dic3—62no—
89C3×C4×D3—34no—
90C3×C4×D3—34no—
91C3×C4×D3—62no—
92C3×C4×D3—62no—
93D3×Dic3—124no—
94C3⋊(C3⋊C8)C4×D314yescomplemented
95C3×C3⋊C8—34no—
96#428—271noSylow
97C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
98C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
99C2×H(3,𝔽3)#13711yescharacteristic
100#272—22no—
101C22×C3×C4—91no—
102C22×Dic3—181no—
103C2×C4×D3—32no—
104C2×C4×D3—61no—
105C2×C4×D3—182no—
106C2×C3×C8—92no—
107C3×C4⋊C4—32no—
108C3×C4⋊C4—61no—
109C3×C4⋊C4—92no—
110C3×C4⋊C4—91no—
111C3×C4⋊C4D3211yescharacteristic
112C3×C22⋊C4—181no—
113C3×C42—181no—
114(C2×D3)⋊C4—62no—
115(C2×D3)⋊C4—62no—
116C4⋊Dic3—32no—
117C4⋊Dic3—32no—
118C4⋊Dic3—61no—
119C4⋊Dic3—92no—
120C4⋊Dic3—181no—
121Dic3⋊C4—182no—
122C4×Dic3—62no—
123C4×Dic3—62no—
124#208—94no—
125C2×C3⋊C8—32no—
126C2×C3⋊C8—92no—
127ΓL(1,𝔽25)—98no—
128C22×C32—21no—
129C22×C32D3×D412yes—
130C2×C3×D3—34no—
131C2×C3×D3—62no—
132C2×C3×D3—62no—
133C2×C3×D3—62no—
134C2×C3×D3—121no—
135C2×C3×D3—121no—
136C32×C4—21no—
137C32×C4—21no—
138C32×C4—42no—
139C32×C4—42no—
140C32×C4C2×C4×D312yes—
141C32×C4D3×Q812yes—
142C3⋊Dic3—42no—
143C3×Dic3—34no—
144C3×Dic3—62no—
145C3×Dic3—122no—
146C3×Dic3—122no—
147C3×Dic3—122no—
148C2×M16—271no—
149#117—91no—
150C2×C4⋊C4—271no—
151#107—92no—
152#106—272no—
153H(3,𝔽3)#42811yescharacteristic, complemented, Sylow
154C23×C3—91no—
155C22×D3—32no—
156C22×D3—61no—
157C2×C3×C4—21no—
158C2×C3×C4—32no—
159C2×C3×C4—62no—
160C2×C3×C4—62no—
161C2×C3×C4—61no—
162C2×C3×C4—91no—
163C2×C3×C4—92no—
164C2×C3×C4—92no—
165C2×C3×C4—182no—
166C2×C3×C4—181no—
167C2×C3×C4C2×D3211yescharacteristic
168C2×Dic3—32no—
169C2×Dic3—61no—
170C2×Dic3—62no—
171C2×Dic3—62no—
172C2×Dic3—62no—
173C2×Dic3—182no—
174C2×Dic3—362no—
175C4×D3—34no—
176C4×D3—34no—
177C4×D3—62no—
178C4×D3—62no—
179C4×D3—364no—
180C3×C8—94no—
181C3⋊C8—34no—
182C3⋊C8—94no—
183C2×C32—21no—
184C2×C32—21no—
185C2×C32—21no—
186C2×C32D3×C4⋊C412yes—
187C2×C32#91112yes—
188C2×C32#91812yes—
189C3×D3—64no—
190C3×D3—122no—
191C22×C4—91no—
192C22×C4—541no—
193M16—274no—
194C2×C8—92no—
195C4⋊C4—91no—
196C4⋊C4—92no—
197C4⋊C4—541no—
198C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
199C22⋊C4—181no—
200C42—181no—
201C22×C3—32no—
202C22×C3—61no—
203C22×C3—92no—
204C22×C3—181no—
205C22×C3—181no—
206C22×C3#381111yescharacteristic
207C2×D3—34no—
208C2×D3—62no—
209C2×D3—62no—
210C2×D3—62no—
211C2×D3—121no—
212C2×D3—121no—
213C3×C4—32no—
214C3×C4—32no—
215C3×C4—41no—
216C3×C4—61no—
217C3×C4—61no—
218C3×C4—92no—
219C3×C4—122no—
220C3×C4—122no—
221C3×C4—362no—
222C3×C4—361no—
223C3×C4#379711yescharacteristic
224C3×C4C4×D3211yescharacteristic
225Dic3—34no—
226Dic3—62no—
227Dic3—122no—
228Dic3—122no—
229Dic3—122no—
230Dic3—362no—
231C32—21no—
232C32#488112yescomplemented
233C23—91no—
234C2×C4—21no—
235C2×C4—91no—
236C2×C4—92no—
237C2×C4—92no—
238C2×C4—181no—
239C2×C4—182no—
240C2×C4—1082no—
241C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
242C8—94no—
243C2×C3—32no—
244C2×C3—32no—
245C2×C3—32no—
246C2×C3—61no—
247C2×C3—61no—
248C2×C3—61no—
249C2×C3—182no—
250C2×C3#6388311yescharacteristic
251C2×C3#6392711yescharacteristic
252C2×C3#6394511yescharacteristic
253D3—64no—
254D3—122no—
255C22—92no—
256C22—181no—
257C22—181no—
258C22#8867711yescharacteristic
259C4—41no—
260C4—92no—
261C4—361no—
262C4—362no—
263C4#8865111yescharacteristic
264C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
265C3—32no—
266C3—61no—
267C3#1059058211yescharacteristic
268C2—182no—
269C2#1172114711yescharacteristic
270C2#1172119111yescharacteristic
271C2#1172122711yescharacteristic
subgroupG/H
centerC22#88677
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC22×#272D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#428
LCSG ▷ 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
4. #63634⋊C3C2semidirect
5. #11720307C2semidirect
16. #88435C4semidirect
21. #63634D3semidirect
34. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
94. C3⋊(C3⋊C8)C4×D3semidirect
153. H(3,𝔽3)#428semidirect
232. C32#4881semidirect

Maximal subgroups

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

subgroupindex
#117217602
#117211522
C2×#884352
#63634⋊C32
#117203072
#105904363