Finite Group Databasebeta

← #13166007

Subgroups — #13166007

Order 1728.

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

normal subgroups only

This group has 510 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×D12C211yesmaximal, characteristic, complemented
2C3×C4⋊(C3⋊S4)C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊(A4⋊C8))C211yesmaximal, characteristic, complemented
4#10593649—31nomaximal
5#10593183—33nomaximal
6#10591160—31nomaximal
7C2×C3×D3×A4—21no—
8C2×C3×C3⋊S4—21no—
9C32×C4×A4C2211yescharacteristic
10#88789—41nomaximal
11C22×C3×D12D311yescharacteristic, complemented
12C3×D4×A4—31no—
13A4×D12D313yescomplemented
14C4⋊(C3⋊S4)D311yescharacteristic, complemented
15C3×C4⋊S4—33no—
16C3×#930—31no—
17C2×#3713—31no—
18C2×#3713—31no—
19C3⋊(A4⋊C8)—31no—
20C3⋊(A4⋊C8)—33no—
21C3⋊(A4⋊C8)—61no—
22C3⋊(A4⋊C8)—63no—
23#63657—31no—
24#63562—31no—
25#63562—31no—
26C2×C32×A4D411yescharacteristic
27C3×D3×A4—41no—
28C3×C3⋊S4—41no—
29C3×C4⋊(C3⋊D3)—41no—
30C32×D12—41no—
31C3⋊(C3⋊(C3⋊C8))—41no—
32#5415—93no—
33#5217—91no—
34#5036—91no—
35C23×C3×D3—21no—
36C22×C3×A4—61no—
37C2×D3×A4—23no—
38C2×C3⋊S4—21no—
39C2×C3×S4—63no—
40C22×C32×C4C2×D311yescharacteristic
41C2×C3×#67—61no—
42C2×C3×D12—31no—
43C2×C3×D12—31no—
44C2×C3×D12—31no—
45C2×C3×D12—61no—
46C3×C4×A4—21no—
47C3×C4×A4—23no—
48C3×C4×A4C2×D313yes—
49C3×C4×A4C2×D311yescharacteristic
50#3753—121no—
51C2×C3⋊(C3⋊C8)—31no—
52C2×C3⋊(C3⋊C8)—31no—
53C3×(C2×D3)⋊C4—61no—
54C3×C4⋊Dic3—31no—
55#3713—62no—
56#3713—61no—
57#3713—61no—
58#3713—123no—
59C2×C3×C3⋊D3—81no—
60C2×C32×D3—81no—
61C32×A4D811yescharacteristic, complemented
62C33×C4—41no—
63C22×C3×D4—31no—
64C22×D12C3⋊D311yescharacteristic, complemented
65D4×A4—33no—
66C4⋊S4—33no—
67C3×#121—91no—
68C2×#213—91no—
69C2×#213—91no—
70C2×#213—91no—
71C2×#213—91no—
72#930—31no—
73A4⋊C8—93no—
74A4⋊C8—183no—
75#832—91no—
76#832—91no—
77#830—91no—
78#830—91no—
79#830—181no—
80#809—91no—
81#809—91no—
82C23×C32#6711yescharacteristic
83C22×C3×D3—41no—
84C22×C3×D3—61no—
85C22×C3×D3—61no—
86C22×C3×D3—61no—
87C22×C3×D3—121no—
88C2×C3×A4—21no—
89C2×C3×A4—23no—
90C2×C3×A4—121no—
91C2×C3×A4#6713yes—
92C2×C3×A4#6711yescharacteristic
93D3×A4—43no—
94C3⋊S4—41no—
95C3×S4—123no—
96C32×D4—121no—
97C2×C32×C4—31no—
98C2×C32×C4—31no—
99C4⋊(C3⋊D3)—41no—
100C3×#67—121no—
101C3×#67—121no—
102C2×C3×Dic3—61no—
103C3×D12—32no—
104C3×D12—31no—
105C3×D12—43no—
106C3×D12—61no—
107C3×D12—61no—
108C3×D12—61no—
109C3×D12—123no—
110C3×D12S411yescharacteristic, complemented
111C3⋊(C3⋊C8)—61no—
112C3⋊(C3⋊C8)—61no—
113C3⋊(C3⋊C8)—121no—
114C3⋊(C3⋊C8)—123no—
115C3⋊(C3⋊C8)—241no—
116C3⋊(C3⋊C8)—243no—
117#447—271noSylow
118C2×C33—41no—
119C3×C3⋊D3—161no—
120C32×D3—161no—
121C24×C3—61no—
122C23×D3—21no—
123C22×A4—63no—
124C2×S4—63no—
125C2×C3×D4—91no—
126C2×C3×D4—91no—
127C2×C3×D4—91no—
128C2×C3×D4—181no—
129C2×C3×D4—181no—
130C22×C3×C4—21no—
131C22×C3×C4D3211yescharacteristic
132C22×C3×C4C2×C3⋊D311yescharacteristic
133C2×#67—61no—
134C2×D12—31no—
135C2×D12—31no—
136C2×D12—31no—
137C2×D12—61no—
138C4×A4—23no—
139C4×A4D3213yes—
140C3×C4⋊C4—91no—
141C3×C22⋊C4—181no—
142#213—182no—
143#213—181no—
144#213—181no—
145#213—182no—
146#213—181no—
147#213—181no—
148#213—363no—
149(C2×D3)⋊C4—61no—
150C4⋊Dic3—31no—
151C2×C3⋊C8—91no—
152C2×C3⋊C8—91no—
153C2×C3⋊C8—91no—
154C2×C3⋊C8—91no—
155C2×C3⋊C8—181no—
156C2×C3⋊C8—181no—
157C22×C32—31no—
158C22×C32—31no—
159C22×C32—241no—
160C22×C32#21311yescharacteristic, complemented
161C2×C3⋊D3—81no—
162C2×C3×D3—21no—
163C2×C3×D3—61no—
164C2×C3×D3—83no—
165C2×C3×D3—121no—
166C2×C3×D3—121no—
167C2×C3×D3—121no—
168C2×C3×D3—121no—
169C2×C3×D3—121no—
170C2×C3×D3—121no—
171C2×C3×D3—121no—
172C2×C3×D3—243no—
173C3×A4—21no—
174C3×A4—23no—
175C3×A4#21313yescomplemented
176C3×A4#21311yescharacteristic, complemented
177C32×C4—31no—
178C32×C4—41no—
179C32×C4—43no—
180C32×C4—83no—
181C32×C4—81no—
182C32×C4C2×S411yescharacteristic
183C3×Dic3—121no—
184C22×D4—31no—
185C2×D8—271no—
186C2×D8—271no—
187#121—91no—
188#102—271no—
189#102—271no—
190C22⋊C8—271no—
191C33—41noSylow
192C23×C3—21no—
193C23×C3—121no—
194C23×C3—181no—
195C23×C3—181no—
196C23×C3—181no—
197C23×C3—361no—
198C23×C3#62511yescharacteristic
199C23×C3#63811yescharacteristic
200C22×D3—41no—
201C22×D3—61no—
202C22×D3—61no—
203C22×D3—61no—
204C22×D3—121no—
205C2×A4—23no—
206C2×A4—123no—
207C2×A4#62513yes—
208S4—123no—
209C3×D4—31no—
210C3×D4—92no—
211C3×D4—91no—
212C3×D4—123no—
213C3×D4—181no—
214C3×D4—181no—
215C3×D4—181no—
216C3×D4—361no—
217C3×D4—361no—
218C2×C3×C4—31no—
219C2×C3×C4—31no—
220C2×C3×C4—31no—
221C2×C3×C4—31no—
222C2×C3×C4—61no—
223C2×C3×C4—61no—
224C2×C3×C4—181no—
225#67—121no—
226#67—121no—
227C2×Dic3—61no—
228D12—32no—
229D12—31no—
230D12—61no—
231D12—61no—
232D12—61no—
233D12—123no—
234D12C3⋊S411yescharacteristic, complemented
235C3⋊C8—181no—
236C3⋊C8—181no—
237C3⋊C8—181no—
238C3⋊C8—181no—
239C3⋊C8—361no—
240C3⋊C8—363no—
241C3⋊C8—361no—
242C3⋊C8—723no—
243C2×C32—31no—
244C2×C32—31no—
245C2×C32—41no—
246C2×C32—43no—
247C2×C32—83no—
248C2×C32—81no—
249C2×C32—481no—
250C2×C32GL(2,ℤ/4)11yescharacteristic
251C3⋊D3—161no—
252C3×D3—41no—
253C3×D3—121no—
254C3×D3—163no—
255C3×D3—241no—
256C3×D3—483no—
257C24—61no—
258C2×D4—91no—
259C2×D4—91no—
260C2×D4—91no—
261C2×D4—181no—
262C2×D4—181no—
263C22×C4D3×C3⋊D311yescharacteristic
264D8—541no—
265D8—542no—
266D8—541no—
267C2×C8—271no—
268C2×C8—271no—
269C4⋊C4—91no—
270C22⋊C4—181no—
271C22×C3—21no—
272C22×C3—31no—
273C22×C3—31no—
274C22×C3—31no—
275C22×C3—31no—
276C22×C3—61no—
277C22×C3—61no—
278C22×C3—61no—
279C22×C3—181no—
280C22×C3—243no—
281C22×C3—361no—
282C22×C3—361no—
283C22×C3—361no—
284C22×C3—361no—
285C22×C3—361no—
286C22×C3—361no—
287C22×C3—361no—
288C22×C3#371311yescharacteristic, complemented
289C22×C3#375311yescharacteristic, complemented
290C2×D3—21no—
291C2×D3—61no—
292C2×D3—121no—
293C2×D3—121no—
294C2×D3—121no—
295C2×D3—121no—
296C2×D3—121no—
297C2×D3—121no—
298C2×D3—121no—
299C2×D3—243no—
300A4—23no—
301A4#371313yescomplemented
302C3×C4—21no—
303C3×C4—31no—
304C3×C4—31no—
305C3×C4—43no—
306C3×C4—61no—
307C3×C4—83no—
308C3×C4—361no—
309C3×C4D3×S411yescharacteristic
310C3×C4C2×C3⋊S411yescharacteristic
311Dic3—121no—
312C32—41no—
313C32—43no—
314C32—83no—
315C32—81no—
316C32#541511yescharacteristic, complemented
317C23—121no—
318C23—181no—
319C23—181no—
320C23—181no—
321C23—361no—
322C23#629011yescharacteristic
323D4—31no—
324D4—92no—
325D4—91no—
326D4—181no—
327D4—181no—
328D4—181no—
329D4—361no—
330D4—361no—
331C2×C4—31no—
332C2×C4—31no—
333C2×C4—181no—
334C8—541no—
335C8—541no—
336C2×C3—21no—
337C2×C3—31no—
338C2×C3—31no—
339C2×C3—31no—
340C2×C3—31no—
341C2×C3—43no—
342C2×C3—61no—
343C2×C3—61no—
344C2×C3—83no—
345C2×C3—121no—
346C2×C3—361no—
347C2×C3—483no—
348C2×C3—721no—
349C2×C3#6420911yescharacteristic
350C2×C3#6426911yescharacteristic
351D3—41no—
352D3—121no—
353D3—241no—
354D3—483no—
355C22—31no—
356C22—31no—
357C22—61no—
358C22—181no—
359C22—361no—
360C22—361no—
361C22—361no—
362C22—361no—
363C22—361no—
364C22—361no—
365C22—361no—
366C22#8878911yescharacteristic, complemented
367C4—31no—
368C4—361no—
369C4D3×C3⋊S411yescharacteristic
370C3—21no—
371C3—43no—
372C3—83no—
373C3#1059318311yescharacteristic, complemented
374C3#1059364911yescharacteristic, complemented
375C2—31no—
376C2—31no—
377C2—121no—
378C2—361no—
379C2—721no—
380C2#1172420411yescharacteristic
subgroupG/H
centerC2#11724204
commutator, cosocleC32×C4×A4C22
Frattini, hypercenterC4D3×C3⋊S4
FittingC22×C32×C4C2×D3
socleC23×C32#67
hypocenterC32×A4D8
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×D12C2semidirect
2. C3×C4⋊(C3⋊S4)C2semidirect
3. C3⋊(C3⋊(A4⋊C8))C2semidirect
11. C22×C3×D12D3semidirect
13. A4×D12D3semidirect
14. C4⋊(C3⋊S4)D3semidirect
61. C32×A4D8semidirect
64. C22×D12C3⋊D3semidirect
110. C3×D12S4semidirect
160. C22×C32#213semidirect
175. C3×A4#213semidirect
176. C3×A4#213semidirect
234. D12C3⋊S4semidirect
288. C22×C3#3713semidirect
289. C22×C3#3753semidirect
301. A4#3713semidirect
316. C32#5415semidirect
366. C22#88789semidirect
373. C3#10593183semidirect
374. C3#10593649semidirect

Maximal subgroups

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

subgroupindex
C3×A4×D122
C3×C4⋊(C3⋊S4)2
C3⋊(C3⋊(A4⋊C8))2
#105936493
#105931833
#105911603
#887894