Finite Group Databasebeta

← #13141063

Subgroups — #13141063

Order 1728.

440 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 67 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#11721869C211yesmaximal, characteristic, complemented
2#64158⋊C3C212yesmaximal, complemented
3C2×#88432C211yesmaximal, characteristic, complemented
4#11720320C212yesmaximal, complemented
5#11720316C211yesmaximal, characteristic, complemented
6#10591677—32nomaximal
7#41×H(3,𝔽3)C2211yescharacteristic
8C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
9#88538—21no—
10#88538C2211yescharacteristic
11C2×#6180—21no—
12C2×#6180C2211yescharacteristic
13#3755⋊C3—22no—
14#3753⋊C3—22no—
15#3748⋊C3C2212yes—
16#88432—21no—
17#88432—21no—
18#64158D312yescomplemented
19C3×#949—32no—
20C3×#948—32no—
21C3×#948—61no—
22C2×#3714—32no—
23#63577—32no—
24#63559—32no—
25#63558—32no—
26C22×(C3⋊D3)⋊C3—22no—
27Q8×H(3,𝔽3)D411yescharacteristic
28D4×H(3,𝔽3)—21no—
29D4×H(3,𝔽3)D411yescharacteristic
30C2×C4×H(3,𝔽3)—21no—
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C4⋊((C3⋊D3)⋊C3)—22no—
33C4⋊((C3⋊D3)⋊C3)—22no—
34#6180—21no—
35#6180D411yescharacteristic
36#6180D411yescharacteristic
37(C3⋊(C3⋊C8))⋊C3—22no—
38#5199—92no—
39#5142—91no—
40C32×#41—21no—
41C32×#41C2×D312yes—
42C2×C4⋊(C3⋊D3)C2×D312yes—
43C2×C3×D12—32no—
44#3755—22no—
45#3753—22no—
46#3748C2×D312yes—
47C3×#215—62no—
48C3×#213—62no—
49C3×#208—32no—
50C3×#208—32no—
51C3×#208—62no—
52C3×#208—61no—
53C3×#208—62no—
54C2×C3×C3⋊C8—32no—
55C2×C3×C3⋊C8—62no—
56C2×C3×C3⋊C8—61no—
57C2×C3×C3⋊C8—62no—
58#3714—62no—
59#3714—62no—
60C22×H(3,𝔽3)—21no—
61C22×H(3,𝔽3)#4111yescharacteristic
62C2×(C3⋊D3)⋊C3—42no—
63C2×(C3⋊D3)⋊C3—82no—
64C4×H(3,𝔽3)—21no—
65C4×H(3,𝔽3)C2×D411yescharacteristic
66C4×H(3,𝔽3)C2×D411yescharacteristic
67C3×AGL(1,ℤ/8)—92no—
68C3×#131—91no—
69#949—32no—
70#949—92no—
71#948—32no—
72#948—61no—
73C2×#213—92no—
74C2×D24—91no—
75#833—92no—
76#823—92no—
77#823—92no—
78#822—92no—
79#819—91no—
80C22×C3⋊D3—22no—
81C22×C3×D3—62no—
82C32×Q8—21no—
83C32×Q8#6712yes—
84C32×D4—22no—
85C32×D4—22no—
86C32×D4—21no—
87C32×D4#6712yes—
88C2×C32×C4—22no—
89C2×C32×C4—22no—
90C2×C32×C4—21no—
91C2×C32×C4C22×D312yes—
92C4⋊(C3⋊D3)—22no—
93C4⋊(C3⋊D3)—22no—
94C3×D12—62no—
95C3×D12—62no—
96C3⋊(C3⋊C8)—22no—
97C3×C3⋊C8—32no—
98C3×C3⋊C8—32no—
99C3×C3⋊C8—61no—
100C3×C3⋊C8—61no—
101C3×C3⋊C8—62no—
102C3×C3⋊C8—62no—
103C3×C3⋊C8—62no—
104#472—271noSylow
105C2×H(3,𝔽3)—21no—
106C2×H(3,𝔽3)—41no—
107C2×H(3,𝔽3)#12111yescharacteristic
108(C3⋊D3)⋊C3—82no—
109C3×#41—32no—
110C3×#41—61no—
111C3×#41D3211yescharacteristic
112C2×C3×D4—92no—
113C2×D12—32no—
114C2×D12—92no—
115C3×SD16—182no—
116C3×D8—182no—
117C3×M16—92no—
118C3×M16—91no—
119C3×M16—181no—
120C2×C3×C8—91no—
121C2×C3×C8—181no—
122#215—62no—
123#215—182no—
124#213—62no—
125#213—182no—
126#213—182no—
127#213—182no—
128#208—32no—
129#208—32no—
130#208—61no—
131#208—62no—
132#208—62no—
133#208—92no—
134C2×C3⋊C8—32no—
135C2×C3⋊C8—61no—
136C2×C3⋊C8—62no—
137C2×C3⋊C8—62no—
138D24—181no—
139D24—181no—
140C22×C32—22no—
141C22×C32—22no—
142C22×C32—21no—
143C22×C32#23712yes—
144C2×C3⋊D3—42no—
145C2×C3⋊D3—82no—
146C2×C3×D3—122no—
147C2×C3×D3—242no—
148C32×C4—22no—
149C32×C4—22no—
150C32×C4—21no—
151C32×C4—21no—
152C32×C4D3×D412yes—
153C32×C4C2×#6712yes—
154AGL(1,ℤ/8)—92no—
155C2×D8—271no—
156#131—91no—
157#108—271no—
158#100—272no—
159H(3,𝔽3)#47211yescharacteristic, complemented, Sylow
160C23×C3—182no—
161C22×D3—62no—
162C22×D3—182no—
163C3×Q8—32no—
164C3×Q8—61no—
165C3×Q8#62511yescharacteristic
166C3×D4—21no—
167C3×D4—32no—
168C3×D4—62no—
169C3×D4—62no—
170C3×D4—61no—
171C3×D4—182no—
172C3×D4—182no—
173C3×D4#62511yescharacteristic
174C2×C3×C4—21no—
175C2×C3×C4—32no—
176C2×C3×C4—62no—
177C2×C3×C4—62no—
178C2×C3×C4—61no—
179C2×C3×C4C2×D3211yescharacteristic
180D12—62no—
181D12—62no—
182D12—182no—
183D12—182no—
184C3×C8—91no—
185C3×C8—91no—
186C3×C8—182no—
187C3×C8—181no—
188C3⋊C8—32no—
189C3⋊C8—32no—
190C3⋊C8—61no—
191C3⋊C8—61no—
192C3⋊C8—62no—
193C3⋊C8—62no—
194C3⋊C8—62no—
195C3⋊C8—182no—
196C2×C32—22no—
197C2×C32—21no—
198C2×C32—41no—
199C2×C32—42no—
200C2×C32—42no—
201C2×C32#93912yes—
202C3⋊D3—82no—
203C3×D3—242no—
204#41((C3⋊D3)⋊C3)⋊C211yescharacteristic
205C2×D4—92no—
206SD16—182no—
207D8—182no—
208D8—541no—
209D8—541no—
210M16—91no—
211M16—92no—
212M16—181no—
213C2×C8—91no—
214C2×C8—181no—
215C22×C3—21no—
216C22×C3—32no—
217C22×C3—62no—
218C22×C3—62no—
219C22×C3—61no—
220C22×C3—362no—
221C22×C3—722no—
222C22×C3#380511yescharacteristic
223C2×D3—122no—
224C2×D3—242no—
225C2×D3—362no—
226C2×D3—722no—
227C3×C4—21no—
228C3×C4—32no—
229C3×C4—32no—
230C3×C4—62no—
231C3×C4—62no—
232C3×C4—61no—
233C3×C4—61no—
234C3×C4C2×#62511yescharacteristic
235C3×C4#381111yescharacteristic
236C32—21no—
237C32#519912yescomplemented
238C23—182no—
239Q8#619811yescharacteristic
240D4—21no—
241D4—182no—
242D4—182no—
243D4#619811yescharacteristic
244C2×C4—21no—
245C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
246C8—91no—
247C8—91no—
248C8—182no—
249C8—181no—
250C2×C3—21no—
251C2×C3—32no—
252C2×C3—41no—
253C2×C3—62no—
254C2×C3—61no—
255C2×C3—121no—
256C2×C3—122no—
257C2×C3—122no—
258C2×C3—722no—
259C2×C3#6398011yescharacteristic
260D3—242no—
261D3—722no—
262C22—21no—
263C22—362no—
264C22—722no—
265C22#8867211yescharacteristic
266C4—21no—
267C4C2×#619811yescharacteristic
268C4#8867711yescharacteristic
269C3—32no—
270C3—61no—
271C3#1059167811yescharacteristic
272C2—21no—
273C2—41no—
274C2—722no—
275C2#1172131811yescharacteristic
subgroupG/H
centerC2#11721318
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
Fitting#41×H(3,𝔽3)C22
socleC2×C3#63980
hypercenter#41((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#472
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #11721869C2semidirect
2. #64158⋊C3C2semidirect
3. C2×#88432C2semidirect
4. #11720320C2semidirect
5. #11720316C2semidirect
18. #64158D3semidirect
159. H(3,𝔽3)#472semidirect
237. C32#5199semidirect

Maximal subgroups

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

subgroupindex
#117218692
#64158⋊C32
C2×#884322
#117203202
#117203162
#105916773