Finite Group Databasebeta

← #13138877

Subgroups — #13138877

Order 1728.

523 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 523 conjugacy classes of subgroups — too many to draw. Only its 69 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
1C2×#88527C211yesmaximal, characteristic, complemented
2#11721159C212yesmaximal, complemented
3C2×#88432C211yesmaximal, characteristic, complemented
4#63648⋊C3C212yesmaximal, complemented
5#11720314C211yesmaximal, characteristic, complemented
6#10590827—32nomaximal
7C2×C4×#272C2211yescharacteristic
8C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
9C2×#6200—22no—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11C2×#6180C2211yescharacteristic
12#88527—22no—
13C4⋊((C3⋊Dic3)⋊C3)C2212yes—
14#88450—22no—
15#88432—22no—
16C2×C3×ΓL(1,𝔽25)—32no—
17C2×C3×ΓL(1,𝔽25)—61no—
18#63909—32no—
19#63908—32no—
20C2×#3714—32no—
21#63648D312yescomplemented
22C3×#821—32no—
23#63573—32no—
24C22×#272D411yescharacteristic
25C22×(C3⋊D3)⋊C3—22no—
26C2×C4×H(3,𝔽3)C2311yescharacteristic
27C2×#1084D411yescharacteristic
28C4×#272—21no—
29C4×#272—21no—
30C2×(C3⋊Dic3)⋊C3—22no—
31C4⋊((C3⋊D3)⋊C3)—22no—
32C4⋊((C3⋊D3)⋊C3)—22no—
33#6200—44no—
34C8×H(3,𝔽3)—21no—
35#6180D412yescomplemented
36#5135—91no—
37#4782—92no—
38C2×C4⋊(C3⋊D3)C2×D312yes—
39C2×C3×D12—32no—
40C2×C3×C4×D3—32no—
41C2×C3×C4×D3—61no—
42C2×#626—62no—
43C2×#625—62no—
44C2×C32×C8—21no—
45C2×C32×C8C2×D312yes—
46C4⋊(C3⋊Dic3)C2×D312yes—
47C3×C4⋊Dic3—32no—
48C2×C3×C3⋊C8—32no—
49C2×C3×C3⋊C8—61no—
50C3×ΓL(1,𝔽25)—64no—
51C3×ΓL(1,𝔽25)—64no—
52#3722—62no—
53(C2×D3)⋊Dic3—62no—
54#3714—64no—
55C22×H(3,𝔽3)C2×D411yescharacteristic
56C2×#272—21no—
57C2×#272—22no—
58C2×(C3⋊D3)⋊C3—42no—
59C2×(C3⋊D3)⋊C3—42no—
60C2×(C3⋊D3)⋊C3—42no—
61C4×H(3,𝔽3)C2×D411yescharacteristic
62C4×H(3,𝔽3)#4111yescharacteristic
63#1084—21no—
64(C3⋊Dic3)⋊C3—42no—
65C2×C3×M16—91no—
66#938—92no—
67C2×#213—92no—
68#930—92no—
69C2×D24—91no—
70C2×ΓL(1,𝔽25)—32no—
71C2×ΓL(1,𝔽25)—61no—
72#895—92no—
73C3×#102—92no—
74#821—32no—
75#821—92no—
76#817—91no—
77#808—92no—
78C22×C3⋊D3—22no—
79C22×C3×D3—32no—
80C22×C3×D3—61no—
81C22×C3×D3—62no—
82C2×C32×C4—21no—
83C2×C32×C4C22×D312yes—
84C2×C3⋊Dic3—22no—
85C4⋊(C3⋊D3)—22no—
86C4⋊(C3⋊D3)—22no—
87C2×C3×Dic3—32no—
88C2×C3×Dic3—61no—
89C2×C3×Dic3—62no—
90C3×D12—62no—
91C3×D12—62no—
92C3×C4×D3—62no—
93C3×C4×D3—62no—
94C3×C4×D3—62no—
95C3×C4×D3—62no—
96#626—124no—
97#625—124no—
98C32×C8—22no—
99C32×C8—22no—
100C3×C3⋊C8—34no—
101C3×C3⋊C8—62no—
102#469—271noSylow
103C2×H(3,𝔽3)#12111yescharacteristic
104C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
105#272—41no—
106(C3⋊D3)⋊C3—82no—
107C2×C3×D4—92no—
108C22×C3×C4—91no—
109C2×#67—182no—
110C2×#67—182no—
111C2×D12—32no—
112C2×D12—92no—
113C2×D12—182no—
114C2×C4×D3—32no—
115C2×C4×D3—61no—
116C3×M16—182no—
117C2×C3×C8—32no—
118C2×C3×C8—61no—
119C2×C3×C8—91no—
120C2×C3×C8D3211yescharacteristic
121C3×C4⋊C4—92no—
122#217—182no—
123#213—184no—
124(C2×D3)⋊C4—182no—
125(C2×D3)⋊C4—182no—
126C4⋊Dic3—32no—
127C4⋊Dic3—92no—
128C2×C3⋊C8—32no—
129C2×C3⋊C8—61no—
130D24—182no—
131ΓL(1,𝔽25)—64no—
132ΓL(1,𝔽25)—64no—
133C22×C32—21no—
134C22×C32D3×D412yes—
135C2×C3⋊D3—42no—
136C2×C3⋊D3—42no—
137C2×C3⋊D3—42no—
138C2×C3×D3—62no—
139C2×C3×D3—62no—
140C2×C3×D3—62no—
141C2×C3×D3—62no—
142C2×C3×D3—122no—
143C2×C3×D3—122no—
144C2×C3×D3—122no—
145C2×C3×D3—122no—
146C32×C4—21no—
147C32×C4—21no—
148C32×C4#23512yes—
149C32×C4D3×D412yes—
150C3⋊Dic3—42no—
151C3×Dic3—62no—
152C3×Dic3—62no—
153C3×Dic3—122no—
154C2×D8—271no—
155C2×M16—91no—
156#121—272no—
157#106—271no—
158#102—92no—
159H(3,𝔽3)#46911yescharacteristic, complemented, Sylow
160C23×C3—91no—
161C23×C3—182no—
162C22×D3—32no—
163C22×D3—61no—
164C22×D3—62no—
165C22×D3—182no—
166C3×D4—182no—
167C3×D4—182no—
168C2×C3×C4—32no—
169C2×C3×C4—61no—
170C2×C3×C4—91no—
171C2×C3×C4—181no—
172C2×C3×C4—181no—
173C2×C3×C4—182no—
174C2×C3×C4C2×D3211yescharacteristic
175#67—364no—
176#67—364no—
177C2×Dic3—32no—
178C2×Dic3—61no—
179C2×Dic3—62no—
180C2×Dic3—182no—
181D12—62no—
182D12—62no—
183D12—182no—
184D12—182no—
185D12—364no—
186C4×D3—62no—
187C4×D3—62no—
188C4×D3—62no—
189C4×D3—62no—
190C3×C8—21no—
191C3×C8—62no—
192C3×C8—62no—
193C3×C8—92no—
194C3⋊C8—34no—
195C3⋊C8—62no—
196C2×C32—22no—
197C2×C32—21no—
198C2×C32#88412yes—
199C2×C32#91112yes—
200C2×C32#91412yes—
201C3⋊D3—82no—
202C3×D3—122no—
203C3×D3—122no—
204C3×D3—242no—
205C2×D4—92no—
206C2×D4—542no—
207C22×C4—91no—
208D8—542no—
209M16—182no—
210C2×C8—91no—
211C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
212C4⋊C4—92no—
213C22⋊C4—542no—
214C22×C3—32no—
215C22×C3—61no—
216C22×C3—182no—
217C22×C3—181no—
218C22×C3—362no—
219C22×C3—362no—
220C22×C3—362no—
221C22×C3#380211yescharacteristic
222C2×D3—62no—
223C2×D3—62no—
224C2×D3—62no—
225C2×D3—62no—
226C2×D3—122no—
227C2×D3—122no—
228C2×D3—122no—
229C2×D3—122no—
230C2×D3—362no—
231C2×D3—362no—
232C2×D3—362no—
233C3×C4—32no—
234C3×C4—32no—
235C3×C4—61no—
236C3×C4—61no—
237C3×C4—181no—
238C3×C4—362no—
239C3×C4#379811yescharacteristic
240C3×C4#381111yescharacteristic
241Dic3—62no—
242Dic3—62no—
243Dic3—122no—
244Dic3—362no—
245C32—21no—
246C32#478212yescomplemented
247C23—91no—
248C23—182no—
249D4—182no—
250D4—182no—
251D4—1084no—
252C2×C4—91no—
253C2×C4—181no—
254C2×C4—181no—
255C2×C4—182no—
256C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
257C8—21no—
258C8—92no—
259C2×C3—32no—
260C2×C3—32no—
261C2×C3—32no—
262C2×C3—62no—
263C2×C3—61no—
264C2×C3—361no—
265C2×C3—722no—
266C2×C3#6379812yes—
267C2×C3#6391211yescharacteristic
268D3—122no—
269D3—122no—
270D3—242no—
271D3—722no—
272C22—182no—
273C22—181no—
274C22—362no—
275C22—362no—
276C22—362no—
277C22#8865511yescharacteristic
278C4—181no—
279C4—362no—
280C4#8865011yescharacteristic
281C4#8867711yescharacteristic
282C3—32no—
283C3—61no—
284C3#1059084011yescharacteristic
285C2—361no—
286C2—722no—
287C2#1172083912yes—
288C2#1172114211yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#469
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
1. C2×#88527C2semidirect
2. #11721159C2semidirect
3. C2×#88432C2semidirect
4. #63648⋊C3C2semidirect
5. #11720314C2semidirect
21. #63648D3semidirect
35. #6180D4semidirect
159. H(3,𝔽3)#469semidirect
246. C32#4782semidirect

Maximal subgroups

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

subgroupindex
C2×#885272
#117211592
C2×#884322
#63648⋊C32
#117203142
#105908273