Finite Group Databasebeta

← #10592595⋊C3 #13142270

Subgroups — #10592595⋊C3 #13142270

Order 1728.

380 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 86 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#64118⋊C3C212yesmaximal
2#64110⋊C3C211yesmaximal, characteristic
3C2×#3746⋊C3C211yesmaximal, characteristic, complemented
4H(3,𝔽3)×#108C211yesmaximal, characteristic, complemented
5#63651⋊C3C212yesmaximal, complemented
6#10592595C311yesmaximal, characteristic, complemented
7C3×#4896—31nomaximal
8#3828⋊C3C2211yescharacteristic
9C2×#634⋊C3C2212yes—
10M16×H(3,𝔽3)C2212yes—
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12#3748⋊C3C2211yescharacteristic
13#3744⋊C3—22no—
14#3743⋊C3—21no—
15C8×(C3⋊D3)⋊C3—21no—
16#3746⋊C3—22no—
17#3746⋊C3—21no—
18#3746⋊C3—21no—
19#64118C2×C312yes—
20C2×#3746C2×C311yescharacteristic, complemented
21#64110C2×C311yescharacteristic
22C3×#909—32no—
23C2×C3×Dic12—31no—
24C3×#901—31no—
25C32×#108—21no—
26C32×#108D311yescharacteristic, complemented
27C32×#108C2×C311yescharacteristic, complemented
28#63651C2×C312yescomplemented
29C3×#824—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)—22no—
39C8×H(3,𝔽3)D411yescharacteristic
40C8×H(3,𝔽3)D411yescharacteristic
41(C3⋊(C3⋊C8))⋊C3—21no—
42C3×#473—91no—
43#4896—31no—
44#4896—91no—
45#3828C22×C311yescharacteristic
46C2×#634C22×C312yes—
47C3×#235—31no—
48C2×C3×Dic6—32no—
49C32×M16—22no—
50C32×M16C2×D312yes—
51C32×M16C22×C312yes—
52C2×C32×C8—21no—
53C2×C32×C8C2×D311yescharacteristic
54C2×C32×C8C22×C311yescharacteristic
55#3748C22×C311yescharacteristic
56#3746—22no—
57#3746—21no—
58#3746—21no—
59#3744—22no—
60#3743—21no—
61C8×C3⋊D3—21no—
62C3×#208—31no—
63C3×Dic12—62no—
64C3×Dic12—61no—
65C3×Dic12—61no—
66C3×#204—62no—
67C3×ΓL(1,𝔽25)—61no—
68C3×D3×C8—61no—
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—
75C3×#137—92no—
76C2×C3×Q16—91no—
77C3×#131—91no—
78#909—32no—
79#909—92no—
80C2×Dic12—31no—
81C2×Dic12—91no—
82#901—31no—
83#901—91no—
84C3×#108—31no—
85C3×#108—31no—
86C3×#108—61no—
87C3×#108C3×D311yescharacteristic
88C3×#101—92no—
89#824—32no—
90#824—92no—
91C2×C32×C4—21no—
92C2×C32×C4C22×D311yescharacteristic
93C2×C32×C4C23×C311yescharacteristic
94#638—21no—
95C2×C3⋊Dic3—22no—
96C4⋊(C3⋊D3)C3×D411yescharacteristic
97C4×C3⋊D3—21no—
98#634—22no—
99#634—22no—
100#634C3×D411yescharacteristic
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—22no—
109C32×C8—22no—
110C32×C8—21no—
111C32×C8—21no—
112C32×C8—42no—
113C32×C8D1211yescharacteristic
114C32×C8D1211yescharacteristic
115C32×C8C3×D411yescharacteristic
116C32×C8C3×D411yescharacteristic
117C3⋊(C3⋊C8)—21no—
118C3×C3⋊C8—61no—
119#473—91noSylow
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×Q16—182no—
132C3×SD16—182no—
133C3×M16—32no—
134C3×M16—32no—
135C3×M16—62no—
136C3×M16—91no—
137C3×M16—181no—
138C3×M16C2×C3×D312yes—
139C2×C3×C8—31no—
140C2×C3×C8—31no—
141C2×C3×C8—61no—
142C2×C3×C8—181no—
143C2×C3×C8C2×C3×D311yescharacteristic
144#208—31no—
145#208—91no—
146Dic12—62no—
147Dic12—61no—
148Dic12—61no—
149Dic12—181no—
150Dic12—181no—
151Dic12—182no—
152#204—62no—
153#204—182no—
154ΓL(1,𝔽25)—61no—
155ΓL(1,𝔽25)—181no—
156D3×C8—61no—
157D3×C8—181no—
158C22×C32—21no—
159C22×C32#23911yescharacteristic
160C22×C32C3×#4111yescharacteristic
161C2×C3⋊D3—21no—
162C2×C3×D3—61no—
163C32×C4—21no—
164C32×C4—21no—
165C32×C4C2×D1211yescharacteristic
166C32×C4D3×D411yescharacteristic
167C32×C4C2×C3×D411yescharacteristic
168C32×C4C2×C3×D411yescharacteristic
169C3⋊Dic3—21no—
170C3⋊Dic3—42no—
171C3×Dic3—61no—
172C3×Dic3—122no—
173#137—92no—
174C2×Q16—91no—
175#131—91no—
176#108(C3⋊D3)⋊C311yescharacteristic, complemented
177#101—92no—
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×C4C22×C3×D311yescharacteristic
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—22no—
205C3×C8—31no—
206C3×C8—31no—
207C3×C8—31no—
208C3×C8—31no—
209C3×C8—62no—
210C3×C8—62no—
211C3×C8—61no—
212C3×C8—61no—
213C3×C8—122no—
214C3×C8—181no—
215C3×C8C3×D1211yescharacteristic
216C3×C8C3×D1211yescharacteristic
217C3⋊C8—61no—
218C3⋊C8—181no—
219C2×C32—21no—
220C2×C32—21no—
221C2×C32—21no—
222C2×C32—41no—
223C2×C32#89511yescharacteristic
224C2×C32C3×#12111yescharacteristic
225C3⋊D3—41no—
226C3×D3—121no—
227#41—91no—
228C2×Q8—92no—
229Q16—182no—
230Q16—181no—
231Q16—181no—
232SD16—182no—
233M16—91no—
234M16—181no—
235M16C2×(C3⋊D3)⋊C312yes—
236C2×C8—181no—
237C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
238C22×C3—31no—
239C22×C3—31no—
240C22×C3—61no—
241C22×C3—181no—
242C22×C3C3×#23911yescharacteristic
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×C4C2×C3×D1211yescharacteristic
254C3×C4C3×D3×D411yescharacteristic
255Dic3—61no—
256Dic3—122no—
257Dic3—181no—
258Dic3—362no—
259C32—21no—
260C32#489611yescharacteristic, complemented
261C32C3×#47311yescharacteristic, complemented
262Q8—91no—
263Q8—182no—
264Q8—182no—
265D4—91no—
266D4—181no—
267C2×C4—181no—
268C2×C4—182no—
269C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
270C8—22no—
271C8—181no—
272C8C4⋊((C3⋊D3)⋊C3)11yescharacteristic
273C8C4⋊((C3⋊D3)⋊C3)11yescharacteristic
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×C3C3×#89511yescharacteristic
283D3—121no—
284D3—361no—
285C22—181no—
286C22#8872211yescharacteristic
287C4—181no—
288C4—362no—
289C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
290C4D4×(C3⋊D3)⋊C311yescharacteristic
291C3—31no—
292C3—31no—
293C3—61no—
294C3C3×#489611yescharacteristic
295C2—21no—
296C2—361no—
297C2#64104⋊C311yescharacteristic
subgroupG/H
centerC2#64104⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#108C2
socleC2×C3C3×#895
prime radical#3828⋊C3C22
hypercenter#108(C3⋊D3)⋊C3
hypocenterC32C3×#473
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #108

Complements

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

NH ≅ G/Ntype
3. C2×#3746⋊C3C2semidirect
4. H(3,𝔽3)×#108C2semidirect
5. #63651⋊C3C2semidirect
6. #10592595C3semidirect
20. C2×#3746C2×C3semidirect
26. C32×#108D3semidirect
27. C32×#108C2×C3semidirect
28. #63651C2×C3semidirect
176. #108(C3⋊D3)⋊C3semidirect
178. H(3,𝔽3)#473semidirect
260. C32#4896semidirect
261. C32C3×#473semidirect

Maximal subgroups

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

subgroupindex
#64118⋊C32
#64110⋊C32
C2×#3746⋊C32
H(3,𝔽3)×#1082
#63651⋊C32
#105925953
C3×#48963