Finite Group Databasebeta

← #13138891

Subgroups — #13138891

Order 1728.

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

normal subgroups only

This group has 408 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×#3745⋊C3C211yesmaximal, characteristic, complemented
2#64110⋊C3C211yesmaximal, characteristic, complemented
3#11721061C212yesmaximal, complemented
4#11720552C211yesmaximal, characteristic, complemented
5#11720320C212yesmaximal, complemented
6#10590929—31nomaximal
7#10590927—31nomaximal
8#3828⋊C3C2211yescharacteristic
9C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11#88538C2212yes—
12#3748⋊C3C2211yescharacteristic
13#3745⋊C3—21no—
14#3745⋊C3—21no—
15#3743⋊C3—21no—
16C8×(C3⋊D3)⋊C3—21no—
17#88432—22no—
18#88431—22no—
19C2×#3745D311yescharacteristic, complemented
20#64110D311yescharacteristic, complemented
21C2×C3×D24—31no—
22C3×#901—31no—
23#63830—32no—
24#63825—32no—
25C3×#819—31no—
26C3×#819—31no—
27C3×#819—61no—
28#63559—32no—
29#63558—32no—
30C22×(C3⋊D3)⋊C3—22no—
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32#638⋊C3—21no—
33C4⋊((C3⋊D3)⋊C3)—22no—
34C4⋊((C3⋊D3)⋊C3)—22no—
35C4⋊((C3⋊D3)⋊C3)D411yescharacteristic
36C4×(C3⋊D3)⋊C3—21no—
37#634⋊C3D411yescharacteristic
38C8×H(3,𝔽3)D411yescharacteristic
39C8×H(3,𝔽3)D411yescharacteristic
40#6180—22no—
41(C3⋊(C3⋊C8))⋊C3—21no—
42#5160—91no—
43#5142—91no—
44#4897—91no—
45#3828C2×D311yescharacteristic
46C2×C4⋊(C3⋊D3)C2×D312yes—
47C3×#235—31no—
48C2×C3×D12—32no—
49C2×C32×C8—21no—
50C2×C32×C8C2×D311yescharacteristic
51C2×C32×C8C2×D311yescharacteristic
52#3748C2×D311yescharacteristic
53#3745—21no—
54#3745—21no—
55#3743—21no—
56C8×C3⋊D3—21no—
57C3×#208—31no—
58C3×#208—32no—
59C3×#208—32no—
60C3×#208—62no—
61C3×D24—61no—
62C3×D24—61no—
63C3×ΓL(1,𝔽25)—61no—
64C3×D3×C8—61no—
65#3717—62no—
66#3716—62no—
67#3714—62no—
68#3714—62no—
69C22×H(3,𝔽3)#4111yescharacteristic
70C2×(C3⋊D3)⋊C3—21no—
71C2×(C3⋊D3)⋊C3—42no—
72C2×(C3⋊D3)⋊C3—82no—
73C4×H(3,𝔽3)C2×D411yescharacteristic
74C4×H(3,𝔽3)C2×D411yescharacteristic
75(C3⋊Dic3)⋊C3—21no—
76C2×C3×D8—91no—
77C3×#131—91no—
78#949—92no—
79#932—92no—
80#908—92no—
81C2×D24—31no—
82C2×D24—91no—
83#901—31no—
84#901—91no—
85C3×#108—91no—
86#833—92no—
87#823—92no—
88#823—92no—
89#819—31no—
90#819—31no—
91#819—61no—
92C22×C3⋊D3—22no—
93C22×C3×D3—62no—
94C2×C32×C4—21no—
95C2×C32×C4C22×D311yescharacteristic
96C2×C32×C4C22×D311yescharacteristic
97#638—21no—
98C4⋊(C3⋊D3)—22no—
99C4⋊(C3⋊D3)—22no—
100C4⋊(C3⋊D3)D1211yescharacteristic
101C4×C3⋊D3—21no—
102#634D1211yescharacteristic
103C3×#67—61no—
104C3×D12—31no—
105C3×D12—62no—
106C3×D12—62no—
107C3×C4×D3—61no—
108C3×Dic6—31no—
109C32×C8—21no—
110C32×C8—21no—
111C32×C8#6711yescharacteristic
112C32×C8#6711yescharacteristic
113C32×C8#6711yescharacteristic
114C32×C8#6711yescharacteristic
115C3⋊(C3⋊C8)—21no—
116C3×C3⋊C8—62no—
117C3×C3⋊C8—62no—
118C3×C3⋊C8—61no—
119C3×C3⋊C8—122no—
120#472—271noSylow
121C2×H(3,𝔽3)—21no—
122C2×H(3,𝔽3)#12111yescharacteristic
123(C3⋊D3)⋊C3—41no—
124(C3⋊D3)⋊C3—82no—
125C3×#41—91no—
126C2×C3×D4—92no—
127#235—31no—
128#235—91no—
129C2×D12—32no—
130C2×D12—92no—
131C3×D8—181no—
132C3×D8—181no—
133C3×M16—91no—
134C3×M16—92no—
135C3×M16—181no—
136C2×C3×C8—31no—
137C2×C3×C8—31no—
138C2×C3×C8—61no—
139C2×C3×C8—181no—
140C2×C3×C8D3211yescharacteristic
141#215—182no—
142#214—182no—
143#213—182no—
144#213—182no—
145#208—32no—
146#208—32no—
147#208—31no—
148#208—62no—
149#208—91no—
150D24—61no—
151D24—61no—
152D24—181no—
153D24—181no—
154D24—182no—
155#204—182no—
156ΓL(1,𝔽25)—61no—
157ΓL(1,𝔽25)—181no—
158D3×C8—61no—
159D3×C8—181no—
160C22×C32—21no—
161C22×C32#23511yescharacteristic
162C22×C32#23711yescharacteristic
163C2×C3⋊D3—21no—
164C2×C3⋊D3—42no—
165C2×C3⋊D3—82no—
166C2×C3×D3—61no—
167C2×C3×D3—122no—
168C2×C3×D3—242no—
169C32×C4—21no—
170C32×C4—21no—
171C32×C4C2×D1211yescharacteristic
172C32×C4D3×D411yescharacteristic
173C32×C4C2×#6711yescharacteristic
174C32×C4C2×#6711yescharacteristic
175C3⋊Dic3—21no—
176C3×Dic3—61no—
177AGL(1,ℤ/8)—272no—
178C2×D8—91no—
179#131—91no—
180#108—91no—
181#100—272no—
182H(3,𝔽3)#47211yescharacteristic, complemented, Sylow
183C23×C3—182no—
184C22×D3—62no—
185C22×D3—182no—
186C3×Q8—91no—
187C3×D4—91no—
188C3×D4—182no—
189C3×D4—182no—
190C3×D4—181no—
191C2×C3×C4—31no—
192C2×C3×C4—31no—
193C2×C3×C4—61no—
194C2×C3×C4—181no—
195C2×C3×C4C2×D3211yescharacteristic
196#67—61no—
197#67—181no—
198D12—31no—
199D12—62no—
200D12—62no—
201D12—91no—
202D12—182no—
203D12—182no—
204C4×D3—61no—
205C4×D3—181no—
206Dic6—31no—
207Dic6—91no—
208C3×C8—31no—
209C3×C8—31no—
210C3×C8—31no—
211C3×C8—31no—
212C3×C8—61no—
213C3×C8—61no—
214C3×C8—181no—
215C3×C8—182no—
216C3×C8#62511yescharacteristic
217C3×C8#62511yescharacteristic
218C3⋊C8—62no—
219C3⋊C8—62no—
220C3⋊C8—61no—
221C3⋊C8—122no—
222C3⋊C8—181no—
223C2×C32—21no—
224C2×C32—21no—
225C2×C32—21no—
226C2×C32—41no—
227C2×C32#93011yescharacteristic
228C2×C32#93811yescharacteristic
229C3⋊D3—41no—
230C3⋊D3—82no—
231C3×D3—121no—
232C3×D3—242no—
233#41—91no—
234C2×D4—92no—
235SD16—542no—
236D8—181no—
237D8—181no—
238D8—542no—
239M16—91no—
240M16—92no—
241M16—181no—
242C2×C8—181no—
243C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
244C22×C3—31no—
245C22×C3—31no—
246C22×C3—61no—
247C22×C3—181no—
248C22×C3—362no—
249C22×C3—722no—
250C22×C3#379511yescharacteristic
251C2×D3—61no—
252C2×D3—122no—
253C2×D3—181no—
254C2×D3—242no—
255C2×D3—362no—
256C2×D3—722no—
257C3×C4—31no—
258C3×C4—31no—
259C3×C4—31no—
260C3×C4—31no—
261C3×C4—61no—
262C3×C4—61no—
263C3×C4—181no—
264C3×C4D3×D1211yescharacteristic
265C3×C4C2×#62511yescharacteristic
266Dic3—61no—
267Dic3—181no—
268C32—21no—
269C32#514211yescharacteristic, complemented
270C32#516011yescharacteristic, complemented
271C23—182no—
272Q8—91no—
273D4—91no—
274D4—181no—
275D4—182no—
276D4—182no—
277C2×C4—181no—
278C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
279C8—182no—
280C8—181no—
281C8#619811yescharacteristic
282C8#619811yescharacteristic
283C2×C3—21no—
284C2×C3—31no—
285C2×C3—31no—
286C2×C3—61no—
287C2×C3—61no—
288C2×C3—61no—
289C2×C3—121no—
290C2×C3—361no—
291C2×C3—722no—
292C2×C3#6390811yescharacteristic
293D3—121no—
294D3—242no—
295D3—361no—
296D3—722no—
297C22—181no—
298C22—362no—
299C22—722no—
300C22#8864811yescharacteristic
301C4—181no—
302C4#8865411yescharacteristic
303C4C2×#619811yescharacteristic
304C3—31no—
305C3—31no—
306C3—61no—
307C3#1059092611yescharacteristic
308C2—21no—
309C2—361no—
310C2—722no—
311C2#1172114111yescharacteristic
subgroupG/H
centerC2#11721141
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC2×C3#63908
hypercenterC2×C8((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 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×#3745⋊C3C2semidirect
2. #64110⋊C3C2semidirect
3. #11721061C2semidirect
4. #11720552C2semidirect
5. #11720320C2semidirect
19. C2×#3745D3semidirect
20. #64110D3semidirect
182. H(3,𝔽3)#472semidirect
269. C32#5142semidirect
270. C32#5160semidirect

Maximal subgroups

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

subgroupindex
C2×#3745⋊C32
#64110⋊C32
#117210612
#117205522
#117203202
#105909293
#105909273