Finite Group Databasebeta

← #13165999

Subgroups — #13165999

Order 1728.

442 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 64 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
1C3×C4×C3⋊S4C211yesmaximal, characteristic
2C3×A4×C3⋊C8C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊(A4⋊C8))C211yesmaximal, characteristic, complemented
4#10593634—31nomaximal
5#10593175—33nomaximal
6#10591155—31nomaximal
7C2×C3×C3⋊S4C411yescharacteristic
8C32×C4×A4C2211yescharacteristic
9C3×C3⋊SL(2,ℤ/4)C411yescharacteristic
10#88787—41nomaximal
11C4×C3⋊S4D311yescharacteristic
12C3×C4×S4—33no—
13C3×C4×#67—31no—
14C22×C3×C3⋊C8D311yescharacteristic, complemented
15C3×C8×A4—31no—
16C2×#3711—31no—
17A4×C3⋊C8D313yescomplemented
18C3⋊(A4⋊C8)—31no—
19C3⋊(A4⋊C8)—33no—
20C3⋊(A4⋊C8)—61no—
21C3⋊(A4⋊C8)—63no—
22#63657—31no—
23Dic3⋊(C3⋊C8)—31no—
24(C2×D3)⋊(C3⋊C8)—31no—
25#63554—31no—
26C2×C32×A4C2×C411yescharacteristic
27C3×C3⋊S4—21no—
28C3×C4×C3⋊D3—41no—
29C3⋊(C3⋊(C3⋊C8))—41no—
30C32×C3⋊C8—41no—
31#5400—93no—
32#5111—91no—
33#5011—91no—
34C2×C3⋊S4Dic311yescharacteristic
35C2×C3×S4—33no—
36C22×C32×C4C2×D311yescharacteristic
37C2×C3×#67—31no—
38C2×C3×C4×D3—31no—
39C3×C4×A4—21no—
40C3×C4×A4—23no—
41C3×C4×A4C2×D313yes—
42C3×C4×A4C2×D311yescharacteristic
43C3⋊SL(2,ℤ/4)Dic311yescharacteristic
44C3×SL(2,ℤ/4)—33no—
45C2×C3⋊(C3⋊C8)—31no—
46C2×C3⋊(C3⋊C8)—31no—
47#3743—121no—
48C3×#217—31no—
49C3×(C2×D3)⋊C4—31no—
50C3×Dic3⋊C4—31no—
51C3×C4×Dic3—31no—
52C2×C3×C3⋊C8—31no—
53C2×C3×C3⋊C8—31no—
54#3711—61no—
55#3711—61no—
56#3711—123no—
57C2×C3×C3⋊D3—41no—
58C32×A4M1611yescharacteristic, complemented
59C33×C4—41no—
60C3×C3⋊Dic3—41no—
61C4×S4—33no—
62C22×C3×C8—31no—
63C3×C4×D4—91no—
64C4×#67—31no—
65C2×#208—91no—
66C22×C3⋊C8C3⋊D311yescharacteristic, complemented
67C2×ΓL(1,𝔽25)—91no—
68C8×A4—33no—
69A4⋊C8—93no—
70A4⋊C8—183no—
71#830—91no—
72#830—91no—
73#830—91no—
74#830—181no—
75(C2×D3)⋊C8—91no—
76#815—91no—
77Dic3⋊C8—91no—
78C4⋊(C3⋊C8)—91no—
79#803—91no—
80C23×C32C4×D311yescharacteristic
81C22×C3×D3—31no—
82C2×C3×A4—21no—
83C2×C3×A4—23no—
84C2×C3×A4C4×D313yes—
85C2×C3×A4C2×Dic311yescharacteristic
86C3⋊S4—21no—
87C3×S4—63no—
88C2×C32×C4—31no—
89C2×C32×C4—31no—
90C4×C3⋊D3—41no—
91C3×#67—61no—
92C3×#67—61no—
93C2×C3×Dic3—31no—
94C2×C3×Dic3—31no—
95C2×C3×Dic3—31no—
96C3×C4×D3—61no—
97C3×C4×D3—61no—
98C3×C4×D3—123no—
99C32×C8—121no—
100C3⋊(C3⋊C8)—61no—
101C3⋊(C3⋊C8)—61no—
102C3⋊(C3⋊C8)—121no—
103C3⋊(C3⋊C8)—123no—
104C3⋊(C3⋊C8)—241no—
105C3⋊(C3⋊C8)—243no—
106C3×C3⋊C8—31no—
107C3×C3⋊C8—43no—
108C3×C3⋊C8S411yescharacteristic, complemented
109#435—271noSylow
110C2×C33—41no—
111C3×C3⋊D3—81no—
112C2×S4—33no—
113C2×C3×D4—91no—
114C22×C3×C4—21no—
115C22×C3×C4—91no—
116C22×C3×C4D3211yescharacteristic
117C22×C3×C4C2×C3⋊D311yescharacteristic
118C2×#67—31no—
119C2×C4×D3—31no—
120C4×A4—23no—
121C4×A4D3213yes—
122SL(2,ℤ/4)—33no—
123C2×C3×C8—91no—
124C2×C3×C8—91no—
125C3×C4⋊C4—91no—
126C3×C22⋊C4—91no—
127C3×C22⋊C4—91no—
128C3×C42—91no—
129#217—31no—
130(C2×D3)⋊C4—31no—
131Dic3⋊C4—31no—
132C4×Dic3—31no—
133#208—181no—
134#208—181no—
135C2×C3⋊C8—31no—
136C2×C3⋊C8—31no—
137C2×C3⋊C8—91no—
138C2×C3⋊C8—91no—
139C2×C3⋊C8—91no—
140C2×C3⋊C8—91no—
141C2×C3⋊C8—181no—
142C2×C3⋊C8—181no—
143ΓL(1,𝔽25)—181no—
144ΓL(1,𝔽25)—181no—
145ΓL(1,𝔽25)—363no—
146C22×C32—31no—
147C22×C32—31no—
148C22×C32ΓL(1,𝔽25)11yescharacteristic, complemented
149C2×C3⋊D3—41no—
150C2×C3×D3—61no—
151C2×C3×D3—61no—
152C2×C3×D3—61no—
153C2×C3×D3—123no—
154C3×A4—21no—
155C3×A4—23no—
156C3×A4ΓL(1,𝔽25)13yescomplemented
157C3×A4#20811yescharacteristic, complemented
158C32×C4—31no—
159C32×C4—41no—
160C32×C4—43no—
161C32×C4—83no—
162C32×C4—81no—
163C32×C4C2×S411yescharacteristic
164C3⋊Dic3—41no—
165C3×Dic3—61no—
166C3×Dic3—61no—
167C3×Dic3—61no—
168C3×Dic3—123no—
169C2×M16—271no—
170C22×C8—31no—
171C4×D4—91no—
172C4⋊C8—271no—
173C22⋊C8—271no—
174C22⋊C8—271no—
175#97—271no—
176C33—41noSylow
177C23×C3—21no—
178C23×C3—91no—
179C23×C3D3×Dic311yescharacteristic
180C23×C3C4×C3⋊D311yescharacteristic
181C22×D3—31no—
182C2×A4—23no—
183C2×A4D3×Dic313yes—
184S4—63no—
185C3×D4—181no—
186C3×D4—181no—
187C2×C3×C4—31no—
188C2×C3×C4—31no—
189C2×C3×C4—31no—
190C2×C3×C4—31no—
191C2×C3×C4—61no—
192C2×C3×C4—61no—
193C2×C3×C4—91no—
194C2×C3×C4—91no—
195C2×C3×C4—91no—
196C2×C3×C4—181no—
197C2×C3×C4—181no—
198#67—61no—
199#67—61no—
200C2×Dic3—31no—
201C2×Dic3—31no—
202C2×Dic3—31no—
203C4×D3—61no—
204C4×D3—61no—
205C4×D3—123no—
206C3×C8—31no—
207C3×C8—91no—
208C3×C8—123no—
209C3⋊C8—31no—
210C3⋊C8—181no—
211C3⋊C8—181no—
212C3⋊C8—181no—
213C3⋊C8—181no—
214C3⋊C8—361no—
215C3⋊C8—363no—
216C3⋊C8—361no—
217C3⋊C8—723no—
218C3⋊C8C3⋊S411yescharacteristic, complemented
219C2×C32—31no—
220C2×C32—31no—
221C2×C32—41no—
222C2×C32—43no—
223C2×C32—83no—
224C2×C32—81no—
225C2×C32C4×S411yescharacteristic
226C3⋊D3—81no—
227C3×D3—121no—
228C3×D3—243no—
229C2×D4—91no—
230C22×C4—91no—
231C22×C4D3×C3⋊D311yescharacteristic
232M16—541no—
233M16—541no—
234C2×C8—91no—
235C2×C8—91no—
236C2×C8—271no—
237C2×C8—271no—
238C4⋊C4—91no—
239C22⋊C4—91no—
240C22⋊C4—91no—
241C42—91no—
242C22×C3—21no—
243C22×C3—31no—
244C22×C3—31no—
245C22×C3—31no—
246C22×C3—31no—
247C22×C3—61no—
248C22×C3—61no—
249C22×C3—181no—
250C22×C3—181no—
251C22×C3—181no—
252C22×C3#371111yescharacteristic, complemented
253C22×C3#374311yescharacteristic, complemented
254C2×D3—61no—
255C2×D3—61no—
256C2×D3—61no—
257C2×D3—123no—
258A4—23no—
259A4#371113yescomplemented
260C3×C4—21no—
261C3×C4—31no—
262C3×C4—31no—
263C3×C4—43no—
264C3×C4—61no—
265C3×C4—83no—
266C3×C4—181no—
267C3×C4—181no—
268C3×C4—181no—
269C3×C4D3×S411yescharacteristic
270C3×C4C2×C3⋊S411yescharacteristic
271Dic3—61no—
272Dic3—61no—
273Dic3—61no—
274Dic3—123no—
275C32—41no—
276C32—43no—
277C32—83no—
278C32—81no—
279C32#540011yescharacteristic, complemented
280C23—91no—
281C23Dic3×C3⋊D311yescharacteristic
282D4—181no—
283D4—181no—
284C2×C4—31no—
285C2×C4—31no—
286C2×C4—91no—
287C2×C4—91no—
288C2×C4—91no—
289C2×C4—181no—
290C2×C4—181no—
291C8—31no—
292C8—91no—
293C8—541no—
294C8—541no—
295C2×C3—21no—
296C2×C3—31no—
297C2×C3—31no—
298C2×C3—31no—
299C2×C3—31no—
300C2×C3—43no—
301C2×C3—61no—
302C2×C3—61no—
303C2×C3—83no—
304C2×C3—361no—
305C2×C3Dic3×S411yescharacteristic
306C2×C3C4×C3⋊S411yescharacteristic
307D3—121no—
308D3—243no—
309C22—31no—
310C22—31no—
311C22—181no—
312C22—181no—
313C22—181no—
314C22#8878711yescharacteristic, complemented
315C4—31no—
316C4—181no—
317C4—181no—
318C4—181no—
319C4D3×C3⋊S411yescharacteristic
320C3—21no—
321C3—43no—
322C3—83no—
323C3#1059317511yescharacteristic, complemented
324C3#1059363411yescharacteristic, complemented
325C2—31no—
326C2—31no—
327C2—361no—
328C2Dic3×C3⋊S411yescharacteristic
subgroupG/H
center, Frattini, hypercenterC4D3×C3⋊S4
commutatorC2×C32×A4C2×C4
FittingC22×C32×C4C2×D3
socleC23×C32C4×D3
prime radicalC2×C3×C3⋊S4C4
cosocleC32×C4×A4C22
hypocenterC32×A4M16
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. C3×A4×C3⋊C8C2semidirect
3. C3⋊(C3⋊(A4⋊C8))C2semidirect
14. C22×C3×C3⋊C8D3semidirect
17. A4×C3⋊C8D3semidirect
58. C32×A4M16semidirect
66. C22×C3⋊C8C3⋊D3semidirect
108. C3×C3⋊C8S4semidirect
148. C22×C32ΓL(1,𝔽25)semidirect
156. C3×A4ΓL(1,𝔽25)semidirect
157. C3×A4#208semidirect
218. C3⋊C8C3⋊S4semidirect
252. C22×C3#3711semidirect
253. C22×C3#3743semidirect
259. A4#3711semidirect
279. C32#5400semidirect
314. C22#88787semidirect
323. C3#10593175semidirect
324. C3#10593634semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C3⋊S42
C3×A4×C3⋊C82
C3⋊(C3⋊(A4⋊C8))2
#105936343
#105931753
#105911553
#887874