Finite Group Databasebeta

← #13166013

Subgroups — #13166013

Order 1728.

426 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 60 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×A4×Dic6C211yesmaximal, characteristic
2C3×#64259C211yesmaximal, characteristic
3C3⋊(C3⋊(A4⋊C8))C211yesmaximal, characteristic
4#10593650—31nomaximal
5#10593189—33nomaximal
6#10591185—31nomaximal
7C32×C4×A4C2211yescharacteristic
8C3×Dic3×A4—21no—
9C3×C3⋊SL(2,ℤ/4)—21no—
10#88798—41nomaximal
11C22×C3×Dic6D311yescharacteristic
12C3×Q8×A4—31no—
13A4×Dic6D313yes—
14#64259D311yescharacteristic
15C3×A4⋊Q8—33no—
16C3×#924—31no—
17C2×#3718—31no—
18C2×#3718—31no—
19C3⋊(A4⋊C8)—31no—
20C3⋊(A4⋊C8)—33no—
21C3⋊(A4⋊C8)—61no—
22C3⋊(A4⋊C8)—63no—
23#63657—31no—
24#63565—31no—
25#63565—31no—
26C2×C32×A4D411yescharacteristic
27C3×#634—41no—
28C32×Dic6—41no—
29C3⋊(C3⋊(C3⋊C8))—41no—
30#5416—93no—
31#5228—91no—
32#5050—91no—
33C22×C32×C4C2×D311yescharacteristic
34C22×C3×Dic3—21no—
35C2×C3×Dic6—31no—
36C2×C3×Dic6—31no—
37C2×C3×Dic6—31no—
38C2×C3×Dic6—61no—
39C3×C4×A4—21no—
40C3×C4×A4—23no—
41C3×C4×A4—61no—
42C3×C4×A4C2×D313yes—
43C3×C4×A4C2×D311yescharacteristic
44Dic3×A4—23no—
45C3⋊SL(2,ℤ/4)—21no—
46C3×SL(2,ℤ/4)—63no—
47#3756—121no—
48C2×C3⋊(C3⋊C8)—31no—
49C2×C3⋊(C3⋊C8)—31no—
50C3×#217—61no—
51C3×C4⋊Dic3—31no—
52C3×Dic3⋊C4—61no—
53#3718—62no—
54#3718—61no—
55#3718—61no—
56#3718—123no—
57C32×A4Q1611yescharacteristic, complemented
58C33×C4—41no—
59C3×C3⋊Dic3—81no—
60C32×Dic3—81no—
61C22×C3×Q8—31no—
62C22×Dic6C3⋊D311yescharacteristic
63Q8×A4—33no—
64A4⋊Q8—33no—
65C3×C22⋊Q8—91no—
66C2×#216—91no—
67C2×#216—91no—
68C2×#216—91no—
69C2×#216—91no—
70#924—31no—
71A4⋊C8—93no—
72A4⋊C8—183no—
73#835—91no—
74#835—91no—
75#830—91no—
76#830—91no—
77#830—181no—
78#810—91no—
79#810—91no—
80C23×C32#6711yescharacteristic
81C2×C3×A4—21no—
82C2×C3×A4—23no—
83C2×C3×A4#6713yes—
84C2×C3×A4#6711yescharacteristic
85C32×Q8—121no—
86C2×C32×C4—31no—
87C2×C32×C4—31no—
88#634—41no—
89C2×C3×Dic3—61no—
90C2×C3×Dic3—61no—
91C2×C3×Dic3—61no—
92C2×C3×Dic3—61no—
93C3×Dic6—32no—
94C3×Dic6—31no—
95C3×Dic6—43no—
96C3×Dic6—61no—
97C3×Dic6—61no—
98C3×Dic6—61no—
99C3×Dic6—123no—
100C3×Dic6S411yescharacteristic
101C3⋊(C3⋊C8)—61no—
102C3⋊(C3⋊C8)—61no—
103C3⋊(C3⋊C8)—121no—
104C3⋊(C3⋊C8)—123no—
105C3⋊(C3⋊C8)—241no—
106C3⋊(C3⋊C8)—243no—
107#451—271noSylow
108C2×C33—41no—
109C2×C3×Q8—91no—
110C2×C3×Q8—91no—
111C2×C3×Q8—91no—
112C2×C3×Q8—181no—
113C22×C3×C4—21no—
114C22×C3×C4—61no—
115C22×C3×C4D3211yescharacteristic
116C22×C3×C4C2×C3⋊D311yescharacteristic
117C22×Dic3—21no—
118C2×Dic6—31no—
119C2×Dic6—31no—
120C2×Dic6—31no—
121C2×Dic6—61no—
122C4×A4—23no—
123C4×A4—63no—
124C4×A4D3213yes—
125SL(2,ℤ/4)—63no—
126C3×C4⋊C4—91no—
127C3×C4⋊C4—181no—
128C3×C22⋊C4—181no—
129#217—61no—
130#216—182no—
131#216—181no—
132#216—181no—
133#216—182no—
134#216—181no—
135#216—181no—
136#216—363no—
137C4⋊Dic3—31no—
138Dic3⋊C4—61no—
139C2×C3⋊C8—91no—
140C2×C3⋊C8—91no—
141C2×C3⋊C8—91no—
142C2×C3⋊C8—91no—
143C2×C3⋊C8—181no—
144C2×C3⋊C8—181no—
145C22×C32—31no—
146C22×C32—31no—
147C22×C32#21611yescharacteristic, complemented
148C3×A4—21no—
149C3×A4—23no—
150C3×A4#21613yescomplemented
151C3×A4#21611yescharacteristic, complemented
152C32×C4—31no—
153C32×C4—41no—
154C32×C4—43no—
155C32×C4—83no—
156C32×C4—81no—
157C32×C4—241no—
158C32×C4C2×S411yescharacteristic
159C3⋊Dic3—81no—
160C3×Dic3—21no—
161C3×Dic3—61no—
162C3×Dic3—83no—
163C3×Dic3—121no—
164C3×Dic3—121no—
165C3×Dic3—243no—
166C22×Q8—31no—
167C2×Q16—271no—
168C2×Q16—271no—
169C22⋊Q8—91no—
170#103—271no—
171#103—271no—
172C22⋊C8—271no—
173C33—41noSylow
174C23×C3—21no—
175C23×C3#62511yescharacteristic
176C23×C3#63811yescharacteristic
177C2×A4—23no—
178C2×A4#62513yes—
179C3×Q8—31no—
180C3×Q8—92no—
181C3×Q8—91no—
182C3×Q8—123no—
183C3×Q8—181no—
184C3×Q8—181no—
185C3×Q8—181no—
186C2×C3×C4—31no—
187C2×C3×C4—31no—
188C2×C3×C4—31no—
189C2×C3×C4—31no—
190C2×C3×C4—61no—
191C2×C3×C4—61no—
192C2×C3×C4—181no—
193C2×C3×C4—181no—
194C2×C3×C4—181no—
195C2×C3×C4—181no—
196C2×Dic3—61no—
197C2×Dic3—61no—
198C2×Dic3—61no—
199C2×Dic3—61no—
200Dic6—32no—
201Dic6—31no—
202Dic6—61no—
203Dic6—61no—
204Dic6—61no—
205Dic6—123no—
206Dic6C3⋊S411yescharacteristic
207C3⋊C8—181no—
208C3⋊C8—181no—
209C3⋊C8—181no—
210C3⋊C8—181no—
211C3⋊C8—361no—
212C3⋊C8—363no—
213C3⋊C8—361no—
214C3⋊C8—723no—
215C2×C32—31no—
216C2×C32—31no—
217C2×C32—41no—
218C2×C32—43no—
219C2×C32—83no—
220C2×C32—81no—
221C2×C32GL(2,ℤ/4)11yescharacteristic
222C2×Q8—91no—
223C2×Q8—91no—
224C2×Q8—91no—
225C2×Q8—181no—
226C22×C4—61no—
227C22×C4D3×C3⋊D311yescharacteristic
228Q16—541no—
229Q16—542no—
230Q16—541no—
231C2×C8—271no—
232C2×C8—271no—
233C4⋊C4—91no—
234C4⋊C4—181no—
235C22⋊C4—181no—
236C22×C3—21no—
237C22×C3—31no—
238C22×C3—31no—
239C22×C3—31no—
240C22×C3—31no—
241C22×C3—61no—
242C22×C3—61no—
243C22×C3#371811yescharacteristic, complemented
244C22×C3#375611yescharacteristic, complemented
245A4—23no—
246A4#371813yescomplemented
247C3×C4—21no—
248C3×C4—31no—
249C3×C4—31no—
250C3×C4—43no—
251C3×C4—61no—
252C3×C4—61no—
253C3×C4—83no—
254C3×C4—181no—
255C3×C4—243no—
256C3×C4—361no—
257C3×C4—361no—
258C3×C4D3×S411yescharacteristic
259C3×C4C2×C3⋊S411yescharacteristic
260Dic3—21no—
261Dic3—61no—
262Dic3—121no—
263Dic3—121no—
264Dic3—243no—
265C32—41no—
266C32—43no—
267C32—83no—
268C32—81no—
269C32#541611yescharacteristic, complemented
270C23#629011yescharacteristic
271Q8—31no—
272Q8—92no—
273Q8—91no—
274Q8—181no—
275Q8—181no—
276Q8—181no—
277C2×C4—31no—
278C2×C4—31no—
279C2×C4—181no—
280C2×C4—181no—
281C2×C4—181no—
282C2×C4—181no—
283C8—541no—
284C8—541no—
285C2×C3—21no—
286C2×C3—31no—
287C2×C3—31no—
288C2×C3—31no—
289C2×C3—31no—
290C2×C3—43no—
291C2×C3—61no—
292C2×C3—61no—
293C2×C3—83no—
294C2×C3#6420911yescharacteristic
295C2×C3#6426911yescharacteristic
296C22—31no—
297C22—31no—
298C22#8879811yescharacteristic, complemented
299C4—31no—
300C4—61no—
301C4—181no—
302C4—361no—
303C4—361no—
304C4D3×C3⋊S411yescharacteristic
305C3—21no—
306C3—43no—
307C3—83no—
308C3#1059318911yescharacteristic, complemented
309C3#1059365011yescharacteristic, complemented
310C2—31no—
311C2—31no—
312C2#1172420411yescharacteristic
subgroupG/H
centerC2#11724204
commutator, cosocleC32×C4×A4C22
Frattini, hypercenterC4D3×C3⋊S4
FittingC22×C32×C4C2×D3
socleC23×C32#67
prime radicalC2×C32×A4D4
hypocenterC32×A4Q16
LCSG ▷ C32×C4×A4 ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
57. C32×A4Q16semidirect
147. C22×C32#216semidirect
150. C3×A4#216semidirect
151. C3×A4#216semidirect
243. C22×C3#3718semidirect
244. C22×C3#3756semidirect
246. A4#3718semidirect
269. C32#5416semidirect
298. C22#88798semidirect
308. C3#10593189semidirect
309. C3#10593650semidirect

Maximal subgroups

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

subgroupindex
C3×A4×Dic62
C3×#642592
C3⋊(C3⋊(A4⋊C8))2
#105936503
#105931893
#105911853
#887984