Finite Group Databasebeta

← #13138892

Subgroups — #13138892

Order 1728.

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

normal subgroups only

This group has 380 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
1#64110⋊C3C211yesmaximal, characteristic
2C2×#3746⋊C3C211yesmaximal, characteristic, complemented
3#11721058C212yesmaximal
4#11720552C211yesmaximal, characteristic, complemented
5#11720321C212yesmaximal, complemented
6#10590933—31nomaximal
7#10590932—31nomaximal
8#3828⋊C3C2211yescharacteristic
9C2×#634⋊C3C2212yes—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11#88538C2212yes—
12#3748⋊C3C2211yescharacteristic
13#3743⋊C3—21no—
14C8×(C3⋊D3)⋊C3—21no—
15#3746⋊C3—21no—
16#3746⋊C3—21no—
17#88433—22no—
18#88431—22no—
19C2×#3746D311yescharacteristic, complemented
20#64110D311yescharacteristic
21C2×C3×Dic12—31no—
22C3×#901—31no—
23#63833—32no—
24#63831—32no—
25C3×#819—31no—
26C3×#819—31no—
27C3×#819—61no—
28#63561—32no—
29#63560—32no—
30C2×C4×H(3,𝔽3)C2311yescharacteristic
31#638⋊C3—21no—
32C2×(C3⋊Dic3)⋊C3—22no—
33C4⋊((C3⋊D3)⋊C3)D411yescharacteristic
34C4×(C3⋊D3)⋊C3—21no—
35#634⋊C3—22no—
36#634⋊C3—22no—
37#634⋊C3D411yescharacteristic
38C8×H(3,𝔽3)D411yescharacteristic
39C8×H(3,𝔽3)D411yescharacteristic
40#6180—22no—
41(C3⋊(C3⋊C8))⋊C3—21no—
42#5192—91no—
43#5143—91no—
44#4896—91no—
45#3828C2×D311yescharacteristic
46C2×#634C2×D312yes—
47C3×#235—31no—
48C2×C3×Dic6—32no—
49C2×C32×C8—21no—
50C2×C32×C8C2×D311yescharacteristic
51C2×C32×C8C2×D311yescharacteristic
52#3748C2×D311yescharacteristic
53#3746—21no—
54#3746—21no—
55#3743—21no—
56C8×C3⋊D3—21no—
57C3×#208—31no—
58C3×#208—32no—
59C3×#208—32no—
60C3×#208—62no—
61C3×Dic12—61no—
62C3×Dic12—61no—
63C3×ΓL(1,𝔽25)—61no—
64C3×D3×C8—61no—
65#3719—62no—
66#3719—62no—
67#3717—62no—
68#3716—62no—
69C22×H(3,𝔽3)#4111yescharacteristic
70C2×(C3⋊D3)⋊C3—21no—
71C4×H(3,𝔽3)C2×D411yescharacteristic
72C4×H(3,𝔽3)C2×D411yescharacteristic
73(C3⋊Dic3)⋊C3—21no—
74(C3⋊Dic3)⋊C3—42no—
75C2×C3×Q16—91no—
76C3×#131—91no—
77#951—92no—
78#942—92no—
79#909—92no—
80C2×Dic12—31no—
81C2×Dic12—91no—
82#901—31no—
83#901—91no—
84C3×#108—91no—
85#836—92no—
86#824—92no—
87#824—92no—
88#819—31no—
89#819—31no—
90#819—61no—
91C2×C32×C4—21no—
92C2×C32×C4C22×D311yescharacteristic
93C2×C32×C4C22×D311yescharacteristic
94#638—21no—
95C2×C3⋊Dic3—22no—
96C4⋊(C3⋊D3)D1211yescharacteristic
97C4×C3⋊D3—21no—
98#634—22no—
99#634—22no—
100#634D1211yescharacteristic
101C3×#67—61no—
102C2×C3×Dic3—62no—
103C3×D12—31no—
104C3×C4×D3—61no—
105C3×Dic6—31no—
106C3×Dic6—62no—
107C3×Dic6—62no—
108C32×C8—21no—
109C32×C8—21no—
110C32×C8#6711yescharacteristic
111C32×C8#6711yescharacteristic
112C32×C8#6711yescharacteristic
113C32×C8#6711yescharacteristic
114C3⋊(C3⋊C8)—21no—
115C3×C3⋊C8—62no—
116C3×C3⋊C8—62no—
117C3×C3⋊C8—61no—
118C3×C3⋊C8—122no—
119#473—271noSylow
120C2×H(3,𝔽3)—21no—
121C2×H(3,𝔽3)#12111yescharacteristic
122(C3⋊D3)⋊C3—41no—
123C3×#41—91no—
124C2×C3×Q8—92no—
125#235—31no—
126#235—91no—
127C2×Dic6—32no—
128C2×Dic6—92no—
129C3×Q16—181no—
130C3×Q16—181no—
131C3×M16—91no—
132C3×M16—92no—
133C3×M16—181no—
134C2×C3×C8—31no—
135C2×C3×C8—31no—
136C2×C3×C8—61no—
137C2×C3×C8—181no—
138C2×C3×C8D3211yescharacteristic
139#216—182no—
140#216—182no—
141#215—182no—
142#214—182no—
143#208—32no—
144#208—32no—
145#208—31no—
146#208—62no—
147#208—91no—
148Dic12—61no—
149Dic12—61no—
150Dic12—181no—
151Dic12—181no—
152Dic12—182no—
153#204—182no—
154ΓL(1,𝔽25)—61no—
155ΓL(1,𝔽25)—181no—
156D3×C8—61no—
157D3×C8—181no—
158C22×C32—21no—
159C22×C32#23511yescharacteristic
160C22×C32#23711yescharacteristic
161C2×C3⋊D3—21no—
162C2×C3×D3—61no—
163C32×C4—21no—
164C32×C4—21no—
165C32×C4C2×D1211yescharacteristic
166C32×C4D3×D411yescharacteristic
167C32×C4C2×#6711yescharacteristic
168C32×C4C2×#6711yescharacteristic
169C3⋊Dic3—21no—
170C3⋊Dic3—42no—
171C3×Dic3—61no—
172C3×Dic3—122no—
173#137—272no—
174C2×Q16—91no—
175#131—91no—
176#108—91no—
177#101—272no—
178H(3,𝔽3)#47311yescharacteristic, complemented, Sylow
179C3×Q8—91no—
180C3×Q8—182no—
181C3×Q8—182no—
182C3×D4—91no—
183C3×D4—181no—
184C2×C3×C4—31no—
185C2×C3×C4—31no—
186C2×C3×C4—61no—
187C2×C3×C4—181no—
188C2×C3×C4—182no—
189C2×C3×C4C2×D3211yescharacteristic
190#67—61no—
191#67—181no—
192C2×Dic3—62no—
193C2×Dic3—182no—
194D12—31no—
195D12—91no—
196C4×D3—61no—
197C4×D3—181no—
198Dic6—31no—
199Dic6—62no—
200Dic6—62no—
201Dic6—91no—
202Dic6—182no—
203Dic6—182no—
204C3×C8—31no—
205C3×C8—31no—
206C3×C8—31no—
207C3×C8—31no—
208C3×C8—61no—
209C3×C8—61no—
210C3×C8—181no—
211C3×C8—182no—
212C3×C8#62511yescharacteristic
213C3×C8#62511yescharacteristic
214C3⋊C8—62no—
215C3⋊C8—62no—
216C3⋊C8—61no—
217C3⋊C8—122no—
218C3⋊C8—181no—
219C2×C32—21no—
220C2×C32—21no—
221C2×C32—21no—
222C2×C32—41no—
223C2×C32#93011yescharacteristic
224C2×C32#93811yescharacteristic
225C3⋊D3—41no—
226C3×D3—121no—
227#41—91no—
228C2×Q8—92no—
229Q16—181no—
230Q16—181no—
231Q16—542no—
232SD16—542no—
233M16—91no—
234M16—92no—
235M16—181no—
236C2×C8—181no—
237C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
238C22×C3—31no—
239C22×C3—31no—
240C22×C3—61no—
241C22×C3—181no—
242C22×C3#379511yescharacteristic
243C2×D3—61no—
244C2×D3—181no—
245C3×C4—31no—
246C3×C4—31no—
247C3×C4—31no—
248C3×C4—31no—
249C3×C4—61no—
250C3×C4—61no—
251C3×C4—181no—
252C3×C4—362no—
253C3×C4D3×D1211yescharacteristic
254C3×C4C2×#62511yescharacteristic
255Dic3—61no—
256Dic3—122no—
257Dic3—181no—
258Dic3—362no—
259C32—21no—
260C32#514311yescharacteristic, complemented
261C32#519211yescharacteristic, complemented
262Q8—91no—
263Q8—182no—
264Q8—182no—
265D4—91no—
266D4—181no—
267C2×C4—181no—
268C2×C4—182no—
269C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
270C8—182no—
271C8—181no—
272C8#619811yescharacteristic
273C8#619811yescharacteristic
274C2×C3—21no—
275C2×C3—31no—
276C2×C3—31no—
277C2×C3—61no—
278C2×C3—61no—
279C2×C3—61no—
280C2×C3—121no—
281C2×C3—361no—
282C2×C3#6390811yescharacteristic
283D3—121no—
284D3—361no—
285C22—181no—
286C22#8864811yescharacteristic
287C4—181no—
288C4—362no—
289C4#8865411yescharacteristic
290C4C2×#619811yescharacteristic
291C3—31no—
292C3—31no—
293C3—61no—
294C3#1059093011yescharacteristic
295C2—21no—
296C2—361no—
297C2#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
prime radical#3828⋊C3C22
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#473
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
2. C2×#3746⋊C3C2semidirect
4. #11720552C2semidirect
5. #11720321C2semidirect
19. C2×#3746D3semidirect
178. H(3,𝔽3)#473semidirect
260. C32#5143semidirect
261. C32#5192semidirect

Maximal subgroups

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

subgroupindex
#64110⋊C32
C2×#3746⋊C32
#117210582
#117205522
#117203212
#105909333
#105909323