Finite Group Databasebeta

← #13166011

Subgroups — #13166011

Order 1728.

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

normal subgroups only

This group has 462 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, complemented
2C3×C4⋊(C3⋊S4)C211yesmaximal, characteristic
3C3⋊(C3⋊(A4⋊C8))C211yesmaximal, characteristic, complemented
4#10593651—31nomaximal
5#10593187—33nomaximal
6#10591171—31nomaximal
7C2×C3×C3⋊S4—21no—
8C32×C4×A4C2211yescharacteristic
9C3×Dic3×A4—21no—
10#88793—41nomaximal
11C22×C3×Dic6D311yescharacteristic, complemented
12C3×Q8×A4—31no—
13A4×Dic6D313yescomplemented
14C4⋊(C3⋊S4)D311yescharacteristic
15C3×C4⋊S4—33no—
16C3×#930—31no—
17C2×#3715—31no—
18C2×#3715—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×C3⋊S4—41no—
28C3×C4⋊(C3⋊D3)—41no—
29C32×Dic6—41no—
30C3⋊(C3⋊(C3⋊C8))—41no—
31#5417—93no—
32#5227—91no—
33#5040—91no—
34C2×C3⋊S4—21no—
35C2×C3×S4—63no—
36C22×C32×C4C2×D311yescharacteristic
37C2×C3×#67—61no—
38C22×C3×Dic3—21no—
39C2×C3×D12—31no—
40C2×C3×Dic6—31no—
41C2×C3×Dic6—31no—
42C2×C3×Dic6—61no—
43C3×C4×A4—21no—
44C3×C4×A4—23no—
45C3×C4×A4—61no—
46C3×C4×A4C2×D313yes—
47C3×C4×A4C2×D311yescharacteristic
48Dic3×A4—23no—
49#3755—121no—
50C2×C3⋊(C3⋊C8)—31no—
51C2×C3⋊(C3⋊C8)—31no—
52C3×(C2×D3)⋊C4—61no—
53C3×C4⋊Dic3—31no—
54#3715—62no—
55#3715—61no—
56#3715—61no—
57#3715—123no—
58C2×C3×C3⋊D3—81no—
59C32×A4SD1611yescharacteristic, complemented
60C33×C4—41no—
61C32×Dic3—81no—
62C22×C3×Q8—31no—
63C22×Dic6C3⋊D311yescharacteristic, complemented
64Q8×A4—33no—
65C4⋊S4—33no—
66C3×#121—91no—
67C2×#215—91no—
68C2×#215—91no—
69C2×#214—91no—
70C2×#214—91no—
71#930—31no—
72A4⋊C8—93no—
73A4⋊C8—183no—
74#835—91no—
75#835—91no—
76#830—91no—
77#830—91no—
78#830—181no—
79#810—91no—
80#810—91no—
81C23×C32#6711yescharacteristic
82C22×C3×D3—61no—
83C2×C3×A4—21no—
84C2×C3×A4—23no—
85C2×C3×A4#6713yes—
86C2×C3×A4#6711yescharacteristic
87C3⋊S4—41no—
88C3×S4—123no—
89C32×Q8—121no—
90C2×C32×C4—31no—
91C2×C32×C4—31no—
92C4⋊(C3⋊D3)—41no—
93C3×#67—121no—
94C3×#67—121no—
95C2×C3×Dic3—61no—
96C2×C3×Dic3—61no—
97C2×C3×Dic3—61no—
98C3×D12—61no—
99C3×D12—61no—
100C3×D12—123no—
101C3×Dic6—32no—
102C3×Dic6—31no—
103C3×Dic6—43no—
104C3×Dic6—61no—
105C3×Dic6S411yescharacteristic, complemented
106C3⋊(C3⋊C8)—61no—
107C3⋊(C3⋊C8)—61no—
108C3⋊(C3⋊C8)—121no—
109C3⋊(C3⋊C8)—123no—
110C3⋊(C3⋊C8)—241no—
111C3⋊(C3⋊C8)—243no—
112#448—271noSylow
113C2×C33—41no—
114C3×C3⋊D3—161no—
115C2×S4—63no—
116C2×C3×Q8—91no—
117C2×C3×Q8—91no—
118C2×C3×Q8—181no—
119C2×C3×D4—91no—
120C2×C3×D4—181no—
121C22×C3×C4—21no—
122C22×C3×C4—61no—
123C22×C3×C4D3211yescharacteristic
124C22×C3×C4C2×C3⋊D311yescharacteristic
125C2×#67—61no—
126C22×Dic3—21no—
127C2×D12—31no—
128C2×Dic6—31no—
129C2×Dic6—31no—
130C2×Dic6—61no—
131C4×A4—23no—
132C4×A4—63no—
133C4×A4D3213yes—
134C3×C4⋊C4—91no—
135C3×C22⋊C4—181no—
136#215—181no—
137#215—182no—
138#215—181no—
139#215—363no—
140#214—182no—
141#214—181no—
142#214—181no—
143(C2×D3)⋊C4—61no—
144C4⋊Dic3—31no—
145C2×C3⋊C8—91no—
146C2×C3⋊C8—91no—
147C2×C3⋊C8—91no—
148C2×C3⋊C8—91no—
149C2×C3⋊C8—181no—
150C2×C3⋊C8—181no—
151C22×C32—31no—
152C22×C32—31no—
153C22×C32#21511yescharacteristic, complemented
154C2×C3⋊D3—81no—
155C2×C3×D3—121no—
156C2×C3×D3—121no—
157C2×C3×D3—121no—
158C2×C3×D3—243no—
159C3×A4—21no—
160C3×A4—23no—
161C3×A4#21411yescharacteristic, complemented
162C3×A4#21513yescomplemented
163C32×C4—31no—
164C32×C4—41no—
165C32×C4—43no—
166C32×C4—83no—
167C32×C4—81no—
168C32×C4—241no—
169C32×C4C2×S411yescharacteristic
170C3×Dic3—21no—
171C3×Dic3—61no—
172C3×Dic3—83no—
173C3×Dic3—121no—
174C22×Q8—31no—
175C2×SD16—271no—
176C2×SD16—271no—
177#121—91no—
178#103—271no—
179#103—271no—
180C22⋊C8—271no—
181C33—41noSylow
182C23×C3—21no—
183C23×C3—181no—
184C23×C3#62511yescharacteristic
185C23×C3#63811yescharacteristic
186C22×D3—61no—
187C2×A4—23no—
188C2×A4#62513yes—
189S4—123no—
190C3×Q8—31no—
191C3×Q8—92no—
192C3×Q8—91no—
193C3×Q8—123no—
194C3×Q8—181no—
195C3×D4—181no—
196C3×D4—181no—
197C3×D4—361no—
198C3×D4—361no—
199C2×C3×C4—31no—
200C2×C3×C4—31no—
201C2×C3×C4—31no—
202C2×C3×C4—31no—
203C2×C3×C4—61no—
204C2×C3×C4—61no—
205C2×C3×C4—181no—
206C2×C3×C4—181no—
207C2×C3×C4—181no—
208#67—121no—
209#67—121no—
210C2×Dic3—61no—
211C2×Dic3—61no—
212C2×Dic3—61no—
213D12—61no—
214D12—61no—
215D12—123no—
216Dic6—32no—
217Dic6—31no—
218Dic6—61no—
219Dic6C3⋊S411yescharacteristic, complemented
220C3⋊C8—181no—
221C3⋊C8—181no—
222C3⋊C8—181no—
223C3⋊C8—181no—
224C3⋊C8—361no—
225C3⋊C8—363no—
226C3⋊C8—361no—
227C3⋊C8—723no—
228C2×C32—31no—
229C2×C32—31no—
230C2×C32—41no—
231C2×C32—43no—
232C2×C32—83no—
233C2×C32—81no—
234C2×C32GL(2,ℤ/4)11yescharacteristic
235C3⋊D3—161no—
236C3×D3—241no—
237C3×D3—483no—
238C2×Q8—91no—
239C2×Q8—91no—
240C2×Q8—181no—
241C2×D4—91no—
242C2×D4—181no—
243C22×C4—61no—
244C22×C4D3×C3⋊D311yescharacteristic
245SD16—541no—
246SD16—542no—
247SD16—541no—
248C2×C8—271no—
249C2×C8—271no—
250C4⋊C4—91no—
251C22⋊C4—181no—
252C22×C3—21no—
253C22×C3—31no—
254C22×C3—31no—
255C22×C3—31no—
256C22×C3—31no—
257C22×C3—61no—
258C22×C3—61no—
259C22×C3—361no—
260C22×C3—361no—
261C22×C3—361no—
262C22×C3#371511yescharacteristic, complemented
263C22×C3#375511yescharacteristic, complemented
264C2×D3—121no—
265C2×D3—121no—
266C2×D3—121no—
267C2×D3—243no—
268A4—23no—
269A4#371513yescomplemented
270C3×C4—21no—
271C3×C4—31no—
272C3×C4—31no—
273C3×C4—43no—
274C3×C4—61no—
275C3×C4—61no—
276C3×C4—83no—
277C3×C4—181no—
278C3×C4—243no—
279C3×C4—361no—
280C3×C4D3×S411yescharacteristic
281C3×C4C2×C3⋊S411yescharacteristic
282Dic3—21no—
283Dic3—61no—
284Dic3—121no—
285C32—41no—
286C32—43no—
287C32—83no—
288C32—81no—
289C32#541711yescharacteristic, complemented
290C23—181no—
291C23#629011yescharacteristic
292Q8—31no—
293Q8—92no—
294Q8—91no—
295Q8—181no—
296D4—181no—
297D4—181no—
298D4—361no—
299D4—361no—
300C2×C4—31no—
301C2×C4—31no—
302C2×C4—181no—
303C2×C4—181no—
304C2×C4—181no—
305C8—541no—
306C8—541no—
307C2×C3—21no—
308C2×C3—31no—
309C2×C3—31no—
310C2×C3—31no—
311C2×C3—31no—
312C2×C3—43no—
313C2×C3—61no—
314C2×C3—61no—
315C2×C3—83no—
316C2×C3—721no—
317C2×C3#6420911yescharacteristic
318C2×C3#6426911yescharacteristic
319D3—241no—
320D3—483no—
321C22—31no—
322C22—31no—
323C22—361no—
324C22—361no—
325C22—361no—
326C22#8879311yescharacteristic, complemented
327C4—31no—
328C4—61no—
329C4—181no—
330C4—361no—
331C4D3×C3⋊S411yescharacteristic
332C3—21no—
333C3—43no—
334C3—83no—
335C3#1059318711yescharacteristic, complemented
336C3#1059365111yescharacteristic, complemented
337C2—31no—
338C2—31no—
339C2—721no—
340C2#1172420411yescharacteristic
subgroupG/H
centerC2#11724204
commutator, cosocleC32×C4×A4C22
Frattini, hypercenterC4D3×C3⋊S4
FittingC22×C32×C4C2×D3
socleC23×C32#67
prime radicalC3×C4⋊(C3⋊S4)C2
hypocenterC32×A4SD16
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
1. C3×A4×Dic6C2semidirect
3. C3⋊(C3⋊(A4⋊C8))C2semidirect
11. C22×C3×Dic6D3semidirect
13. A4×Dic6D3semidirect
59. C32×A4SD16semidirect
63. C22×Dic6C3⋊D3semidirect
105. C3×Dic6S4semidirect
153. C22×C32#215semidirect
161. C3×A4#214semidirect
162. C3×A4#215semidirect
219. Dic6C3⋊S4semidirect
262. C22×C3#3715semidirect
263. C22×C3#3755semidirect
269. A4#3715semidirect
289. C32#5417semidirect
326. C22#88793semidirect
335. C3#10593187semidirect
336. C3#10593651semidirect

Maximal subgroups

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

subgroupindex
C3×A4×Dic62
C3×C4⋊(C3⋊S4)2
C3⋊(C3⋊(A4⋊C8))2
#105936513
#105931873
#105911713
#887934