Finite Group Databasebeta

← C3×#10593651 #13173019

Subgroups — C3×#10593651 #13173019

Order 1728.

506 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C32×Q8×A4C211yesmaximal, characteristic, complemented
2C3×C4⋊(C3⋊S4)C211yesmaximal, characteristic
3C3×C3⋊(A4⋊C8)C211yesmaximal, characteristic, complemented
4#10593651C311yesmaximal, characteristic, direct product factor
5C3×#5417—33nomaximal
6C3×#5227—31nomaximal
7C2×C3×C3⋊S4—21no—
8C32×C4×A4—21no—
9C32×C4×A4C2211yescharacteristic
10C3×#3755—41nomaximal
11C22×C32×Q8D311yescharacteristic, complemented
12C3×Q8×A4—21no—
13C3×Q8×A4—23no—
14C3×Q8×A4D313yescomplemented
15C3×Q8×A4C2×C311yescharacteristic, complemented
16C4⋊(C3⋊S4)C2×C311yescharacteristic
17C3×C4⋊S4—33no—
18C2×C3×#215—31no—
19C2×C3×#215—31no—
20C3×#930—31no—
21C3⋊(A4⋊C8)C2×C311yescharacteristic, complemented
22C3×A4⋊C8—33no—
23C3×#835—31no—
24C3×#835—31no—
25C3×#830—31no—
26C2×C32×A4D411yescharacteristic
27C3×C3⋊S4—41no—
28C33×Q8—41no—
29C3×C4⋊(C3⋊D3)—41no—
30C3×C3⋊(C3⋊C8)—41no—
31#5417—33no—
32C3×#448—91no—
33#5227—31no—
34C2×C3⋊S4—21no—
35C2×C3×S4—63no—
36C2×C32×Q8—31no—
37C2×C32×Q8—31no—
38C2×C32×Q8—61no—
39C22×C32×C4—21no—
40C22×C32×C4C2×D311yescharacteristic
41C2×C3×#67—61no—
42C2×C3×D12—31no—
43C3×C4×A4—22no—
44C3×C4×A4—26no—
45C3×C4×A4—21no—
46C3×C4×A4—23no—
47C3×C4×A4—23no—
48C3×C4×A4—21no—
49C3×C4×A4C2×D313yes—
50C3×C4×A4C22×C311yescharacteristic
51#3755—41no—
52C3×#215—62no—
53C3×#215—61no—
54C3×#215—61no—
55C3×#215—123no—
56C3×(C2×D3)⋊C4—61no—
57C3×C4⋊Dic3—31no—
58C2×C3×C3⋊C8—31no—
59C2×C3×C3⋊C8—31no—
60C2×C3×C3⋊D3—81no—
61C32×A4SD1611yescharacteristic, complemented
62C33×C4—41no—
63C33×C4—81no—
64C22×C3×Q8—21no—
65C22×C3×Q8C3×D311yescharacteristic, complemented
66C22×C3×Q8C3⋊D311yescharacteristic, complemented
67Q8×A4—23no—
68Q8×A4C3×D313yescomplemented
69C4⋊S4—33no—
70C2×C3×SD16—91no—
71C2×C3×SD16—91no—
72C3×#121—91no—
73C2×#215—31no—
74C2×#215—31no—
75#930—31no—
76A4⋊C8—33no—
77C3×#103—91no—
78C3×#103—91no—
79C3×C22⋊C8—91no—
80#835—31no—
81#835—31no—
82#830—31no—
83C23×C32#6711yescharacteristic
84C22×C3×D3—61no—
85C2×C3×A4—21no—
86C2×C3×A4—23no—
87C2×C3×A4#6713yes—
88C2×C3×A4C3×D411yescharacteristic
89C3⋊S4—41no—
90C3×S4—123no—
91C32×Q8—32no—
92C32×Q8—31no—
93C32×Q8—41no—
94C32×Q8—43no—
95C32×Q8—61no—
96C32×Q8—83no—
97C32×Q8—81no—
98C32×Q8S411yescharacteristic, complemented
99C2×C32×C4—31no—
100C2×C32×C4—31no—
101C2×C32×C4—61no—
102C2×C32×C4—61no—
103C4⋊(C3⋊D3)—41no—
104C3×#67—121no—
105C3×#67—121no—
106C2×C3×Dic3—61no—
107C3×D12—61no—
108C3×D12—61no—
109C3×D12—123no—
110C3⋊(C3⋊C8)—41no—
111C3×C3⋊C8—61no—
112C3×C3⋊C8—61no—
113C3×C3⋊C8—123no—
114#448—91noSylow
115C2×C33—41no—
116C3×C3⋊D3—161no—
117C2×S4—63no—
118C2×C3×Q8—31no—
119C2×C3×Q8—31no—
120C2×C3×Q8—31no—
121C2×C3×Q8—31no—
122C2×C3×Q8—61no—
123C2×C3×Q8—61no—
124C2×C3×Q8—61no—
125C2×C3×Q8—61no—
126C2×C3×Q8—62no—
127C2×C3×D4—91no—
128C2×C3×D4—181no—
129C22×C3×C4—22no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4C2×C3×D311yescharacteristic
134C22×C3×C4C2×C3⋊D311yescharacteristic
135C2×#67—61no—
136C2×D12—31no—
137C4×A4—26no—
138C4×A4—23no—
139C4×A4—23no—
140C4×A4C2×C3×D313yes—
141C3×SD16—182no—
142C3×SD16—181no—
143C3×SD16—181no—
144C2×C3×C8—91no—
145C2×C3×C8—91no—
146C3×C4⋊C4—91no—
147C3×C22⋊C4—181no—
148#215—62no—
149#215—61no—
150#215—61no—
151#215—123no—
152(C2×D3)⋊C4—61no—
153C4⋊Dic3—31no—
154C2×C3⋊C8—31no—
155C2×C3⋊C8—31no—
156C22×C32—31no—
157C22×C32—31no—
158C22×C32#21511yescharacteristic, complemented
159C2×C3⋊D3—81no—
160C2×C3×D3—121no—
161C2×C3×D3—121no—
162C2×C3×D3—121no—
163C2×C3×D3—243no—
164C3×A4—21no—
165C3×A4—23no—
166C3×A4#21513yescomplemented
167C3×A4C3×SD1611yescharacteristic, complemented
168C32×C4—21no—
169C32×C4—31no—
170C32×C4—41no—
171C32×C4—43no—
172C32×C4—61no—
173C32×C4—83no—
174C32×C4—81no—
175C32×C4—83no—
176C32×C4—86no—
177C32×C4—82no—
178C32×C4—81no—
179C32×C4C2×S411yescharacteristic
180C3×Dic3—121no—
181C22×Q8C3×C3⋊D311yescharacteristic, complemented
182C2×SD16—91no—
183C2×SD16—91no—
184#121—91no—
185#103—91no—
186#103—91no—
187C22⋊C8—91no—
188C33—41noSylow
189C23×C3—21no—
190C23×C3—181no—
191C23×C3C3×#6711yescharacteristic
192C23×C3#63811yescharacteristic
193C22×D3—61no—
194C2×A4—23no—
195C2×A4C3×#6713yes—
196S4—123no—
197C3×Q8—21no—
198C3×Q8—32no—
199C3×Q8—32no—
200C3×Q8—31no—
201C3×Q8—31no—
202C3×Q8—43no—
203C3×Q8—61no—
204C3×Q8—62no—
205C3×Q8—61no—
206C3×Q8—61no—
207C3×Q8—62no—
208C3×Q8—83no—
209C3×Q8C3×S411yescharacteristic, complemented
210C3×Q8C3⋊S411yescharacteristic, complemented
211C3×D4—181no—
212C3×D4—181no—
213C3×D4—361no—
214C3×D4—361no—
215C2×C3×C4—31no—
216C2×C3×C4—31no—
217C2×C3×C4—31no—
218C2×C3×C4—31no—
219C2×C3×C4—61no—
220C2×C3×C4—61no—
221C2×C3×C4—61no—
222C2×C3×C4—61no—
223C2×C3×C4—62no—
224C2×C3×C4—61no—
225C2×C3×C4—61no—
226C2×C3×C4—62no—
227C2×C3×C4—181no—
228#67—121no—
229#67—121no—
230C2×Dic3—61no—
231D12—61no—
232D12—61no—
233D12—123no—
234C3×C8—181no—
235C3×C8—181no—
236C3⋊C8—61no—
237C3⋊C8—61no—
238C3⋊C8—123no—
239C2×C32—31no—
240C2×C32—31no—
241C2×C32—41no—
242C2×C32—43no—
243C2×C32—83no—
244C2×C32—81no—
245C2×C32GL(2,ℤ/4)11yescharacteristic
246C3⋊D3—161no—
247C3×D3—241no—
248C3×D3—483no—
249C2×Q8—31no—
250C2×Q8—31no—
251C2×Q8—61no—
252C2×D4—91no—
253C2×D4—181no—
254C22×C4—21no—
255C22×C4C2×C3×C3⋊D311yescharacteristic
256SD16—182no—
257SD16—181no—
258SD16—181no—
259C2×C8—91no—
260C2×C8—91no—
261C4⋊C4—91no—
262C22⋊C4—181no—
263C22×C3—21no—
264C22×C3—31no—
265C22×C3—31no—
266C22×C3—31no—
267C22×C3—31no—
268C22×C3—61no—
269C22×C3—61no—
270C22×C3—361no—
271C22×C3—361no—
272C22×C3—361no—
273C22×C3C3×#21511yescharacteristic, complemented
274C22×C3#375511yescharacteristic, complemented
275C2×D3—121no—
276C2×D3—121no—
277C2×D3—121no—
278C2×D3—243no—
279A4—23no—
280A4C3×#21513yescomplemented
281C3×C4—22no—
282C3×C4—21no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—31no—
286C3×C4—31no—
287C3×C4—43no—
288C3×C4—61no—
289C3×C4—62no—
290C3×C4—61no—
291C3×C4—61no—
292C3×C4—83no—
293C3×C4—86no—
294C3×C4—83no—
295C3×C4—361no—
296C3×C4C2×C3×S411yescharacteristic
297C3×C4C2×C3⋊S411yescharacteristic
298Dic3—121no—
299C32—41no—
300C32—43no—
301C32—83no—
302C32—81no—
303C32#541711yescharacteristic, complemented
304C23—181no—
305C23C3×#63811yescharacteristic
306Q8—32no—
307Q8—31no—
308Q8—61no—
309Q8C3×C3⋊S411yescharacteristic, complemented
310D4—181no—
311D4—181no—
312D4—361no—
313D4—361no—
314C2×C4—31no—
315C2×C4—31no—
316C2×C4—61no—
317C2×C4—61no—
318C2×C4—181no—
319C8—181no—
320C8—181no—
321C2×C3—21no—
322C2×C3—31no—
323C2×C3—31no—
324C2×C3—31no—
325C2×C3—31no—
326C2×C3—43no—
327C2×C3—61no—
328C2×C3—61no—
329C2×C3—83no—
330C2×C3—721no—
331C2×C3C3×GL(2,ℤ/4)11yescharacteristic
332C2×C3#6426911yescharacteristic
333D3—241no—
334D3—483no—
335C22—31no—
336C22—31no—
337C22—361no—
338C22—361no—
339C22—361no—
340C22C3×#375511yescharacteristic, complemented
341C4—21no—
342C4—31no—
343C4—61no—
344C4—361no—
345C4C2×C3×C3⋊S411yescharacteristic
346C3—21no—
347C3—43no—
348C3—83no—
349C3C3×#541711yescharacteristic, complemented
350C3#1059365111yescharacteristic, direct product factor
351C2—31no—
352C2—31no—
353C2—721no—
354C2C3×#6426911yescharacteristic
subgroupG/H
centerC2×C3#64269
commutator, cosocleC3×C4×A4C22×C3
FrattiniC4C2×C3×C3⋊S4
FittingC22×C32×Q8D3
socleC23×C32#67
prime radicalC3×C4⋊(C3⋊S4)C2
hypercenterC3×Q8C3⋊S4
hypocenterC3×A4C3×SD16
LCSG ▷ C3×C4×A4 ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×Q8

Complements

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

NH ≅ G/Ntype
1. C32×Q8×A4C2semidirect
3. C3×C3⋊(A4⋊C8)C2semidirect
4. #10593651C3direct (N × H)
11. C22×C32×Q8D3semidirect
14. C3×Q8×A4D3semidirect
15. C3×Q8×A4C2×C3semidirect
21. C3⋊(A4⋊C8)C2×C3semidirect
61. C32×A4SD16semidirect
65. C22×C3×Q8C3×D3semidirect
66. C22×C3×Q8C3⋊D3semidirect
68. Q8×A4C3×D3semidirect
98. C32×Q8S4semidirect
158. C22×C32#215semidirect
166. C3×A4#215semidirect
167. C3×A4C3×SD16semidirect
181. C22×Q8C3×C3⋊D3semidirect
209. C3×Q8C3×S4semidirect
210. C3×Q8C3⋊S4semidirect
273. C22×C3C3×#215semidirect
274. C22×C3#3755semidirect
280. A4C3×#215semidirect
303. C32#5417semidirect
309. Q8C3×C3⋊S4semidirect
340. C22C3×#3755semidirect
349. C3C3×#5417semidirect
350. C3#10593651direct (N × H)

Maximal subgroups

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

subgroupindex
C32×Q8×A42
C3×C4⋊(C3⋊S4)2
C3×C3⋊(A4⋊C8)2
#105936513
C3×#54173
C3×#52273
C3×#37554