Finite Group Databasebeta

← #13166000

Subgroups — #13166000

Order 1728.

520 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 62 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
1C4×C3⋊(C3⋊S4)C211yesmaximal, characteristic
2C3×A4×C3⋊C8C211yesmaximal, characteristic, complemented
3C3×C3⋊(A4⋊C8)C211yesmaximal, characteristic, complemented
4#10593634—31nomaximal
5#10593176—33nomaximal
6#10591157—31nomaximal
7C2×C3⋊(C3⋊S4)C411yescharacteristic
8C32×C4×A4C2211yescharacteristic
9C3⋊(C3⋊SL(2,ℤ/4))C411yescharacteristic
10#88788—41nomaximal
11C4×C3⋊S4—31no—
12C4×C3⋊S4—33no—
13C4×C3⋊S4—61no—
14C4×C3⋊S4—63no—
15C4×#638—31no—
16C22×C3×C3⋊C8D311yescharacteristic, complemented
17C3×C8×A4—31no—
18C2×#3712—31no—
19A4×C3⋊C8D313yescomplemented
20C3⋊(A4⋊C8)D311yescharacteristic, complemented
21C3×A4⋊C8—33no—
22C3×#830—31no—
23#63572—31no—
24#63557—31no—
25#63555—31no—
26C2×C32×A4C2×C411yescharacteristic
27C3⋊(C3⋊S4)—21no—
28C4×C3⋊(C3⋊D3)—41no—
29C3×C3⋊(C3⋊C8)—41no—
30C32×C3⋊C8—41no—
31#5400—93no—
32#5111—91no—
33#4730—91no—
34C2×C3⋊S4—31no—
35C2×C3⋊S4—33no—
36C2×C3⋊S4—61no—
37C2×C3⋊S4—63no—
38C22×C32×C4C2×D311yescharacteristic
39C2×#638—31no—
40C2×C4×C3⋊D3—31no—
41C3×C4×A4—21no—
42C3×C4×A4—23no—
43C3×C4×A4C2×D313yes—
44C3×C4×A4C2×D311yescharacteristic
45C3⋊SL(2,ℤ/4)—31no—
46C3⋊SL(2,ℤ/4)—33no—
47C3⋊SL(2,ℤ/4)—61no—
48C3⋊SL(2,ℤ/4)—63no—
49#3757—31no—
50#3752—31no—
51#3750—31no—
52C4×C3⋊Dic3—31no—
53#3743—121no—
54C2×C3×C3⋊C8—31no—
55C2×C3×C3⋊C8—31no—
56C2×C3×C3⋊C8—31no—
57C2×C3×C3⋊C8—31no—
58#3712—61no—
59#3712—61no—
60#3712—123no—
61C2×C3⋊(C3⋊D3)—41no—
62C32×A4M1611yescharacteristic, complemented
63C33×C4—41no—
64C3⋊(C3⋊Dic3)—41no—
65C4×S4—93no—
66C4×S4—183no—
67C22×C3×C8—31no—
68C4×#67—91no—
69C4×#67—91no—
70C4×#67—181no—
71C22×C3⋊C8C3⋊D311yescharacteristic, complemented
72C2×ΓL(1,𝔽25)—91no—
73C2×ΓL(1,𝔽25)—91no—
74C8×A4—33no—
75A4⋊C8—33no—
76C3×C22⋊C8—91no—
77#830—31no—
78(C2×D3)⋊C8—91no—
79(C2×D3)⋊C8—91no—
80#815—91no—
81#815—91no—
82Dic3⋊C8—91no—
83Dic3⋊C8—91no—
84C23×C32C4×D311yescharacteristic
85C22×C3⋊D3—31no—
86C2×C3×A4—21no—
87C2×C3×A4—23no—
88C2×C3×A4C4×D313yes—
89C2×C3×A4C4×D311yescharacteristic
90C3⋊S4—66no—
91C3⋊S4—62no—
92C3⋊S4—61no—
93C3⋊S4—63no—
94C2×C32×C4—31no—
95C2×C32×C4—31no—
96#638—61no—
97#638—61no—
98C2×C3⋊Dic3—31no—
99C2×C3⋊Dic3—31no—
100C2×C3⋊Dic3—31no—
101C4×C3⋊D3—61no—
102C4×C3⋊D3—61no—
103C4×C3⋊D3—121no—
104C4×C3⋊D3—123no—
105C4×C3⋊D3—241no—
106C4×C3⋊D3—243no—
107C32×C8—121no—
108C3⋊(C3⋊C8)—41no—
109C3×C3⋊C8—31no—
110C3×C3⋊C8—43no—
111C3×C3⋊C8—61no—
112C3×C3⋊C8—61no—
113C3×C3⋊C8—123no—
114C3×C3⋊C8S411yescharacteristic, complemented
115#435—271noSylow
116C2×C33—41no—
117C3⋊(C3⋊D3)—81no—
118C2×S4—93no—
119C2×S4—183no—
120C22×C3×C4—21no—
121C22×C3×C4D3211yescharacteristic
122C22×C3×C4C2×C3⋊D311yescharacteristic
123C2×#67—91no—
124C2×#67—91no—
125C2×#67—181no—
126C2×C4×D3—91no—
127C2×C4×D3—91no—
128C2×C4×D3—181no—
129C4×A4—23no—
130C4×A4D3213yes—
131SL(2,ℤ/4)—93no—
132SL(2,ℤ/4)—183no—
133C2×C3×C8—91no—
134C2×C3×C8—91no—
135C2×C3×C8—91no—
136C2×C3×C8—91no—
137#217—91no—
138#217—91no—
139#217—181no—
140(C2×D3)⋊C4—91no—
141(C2×D3)⋊C4—91no—
142(C2×D3)⋊C4—181no—
143Dic3⋊C4—91no—
144Dic3⋊C4—91no—
145Dic3⋊C4—181no—
146C4×Dic3—91no—
147C4×Dic3—91no—
148C4×Dic3—181no—
149C2×C3⋊C8—31no—
150C2×C3⋊C8—31no—
151C2×C3⋊C8—31no—
152C2×C3⋊C8—31no—
153ΓL(1,𝔽25)—181no—
154ΓL(1,𝔽25)—181no—
155ΓL(1,𝔽25)—181no—
156ΓL(1,𝔽25)—181no—
157ΓL(1,𝔽25)—363no—
158C22×C32—31no—
159C22×C32—31no—
160C22×C32ΓL(1,𝔽25)11yescharacteristic, complemented
161C2×C3⋊D3—61no—
162C2×C3⋊D3—61no—
163C2×C3⋊D3—61no—
164C2×C3⋊D3—121no—
165C2×C3⋊D3—123no—
166C2×C3⋊D3—241no—
167C2×C3⋊D3—243no—
168C3×A4—21no—
169C3×A4—23no—
170C3×A4ΓL(1,𝔽25)13yescomplemented
171C3×A4ΓL(1,𝔽25)11yescharacteristic, complemented
172C32×C4—31no—
173C32×C4—41no—
174C32×C4—43no—
175C32×C4—83no—
176C32×C4—81no—
177C32×C4C2×S411yescharacteristic
178C3⋊Dic3—61no—
179C3⋊Dic3—61no—
180C3⋊Dic3—61no—
181C3⋊Dic3—121no—
182C3⋊Dic3—123no—
183C3⋊Dic3—241no—
184C3⋊Dic3—243no—
185C2×M16—271no—
186C22×C8—31no—
187C4×D4—271no—
188C4⋊C8—271no—
189C22⋊C8—91no—
190C22⋊C8—271no—
191#97—271no—
192C33—41noSylow
193C23×C3—21no—
194C23×C3#62411yescharacteristic
195C23×C3C4×C3⋊D311yescharacteristic
196C22×D3—91no—
197C22×D3—91no—
198C22×D3—181no—
199C2×A4—23no—
200C2×A4#62413yes—
201S4—186no—
202S4—183no—
203C2×C3×C4—31no—
204C2×C3×C4—31no—
205C2×C3×C4—31no—
206C2×C3×C4—31no—
207C2×C3×C4—61no—
208C2×C3×C4—61no—
209#67—182no—
210#67—181no—
211#67—181no—
212#67—182no—
213#67—181no—
214#67—181no—
215C2×Dic3—91no—
216C2×Dic3—91no—
217C2×Dic3—91no—
218C2×Dic3—91no—
219C2×Dic3—91no—
220C2×Dic3—91no—
221C2×Dic3—181no—
222C2×Dic3—181no—
223C2×Dic3—181no—
224C4×D3—181no—
225C4×D3—181no—
226C4×D3—181no—
227C4×D3—181no—
228C4×D3—361no—
229C4×D3—363no—
230C4×D3—361no—
231C4×D3—723no—
232C3×C8—31no—
233C3×C8—91no—
234C3×C8—123no—
235C3×C8—181no—
236C3×C8—181no—
237C3⋊C8—31no—
238C3⋊C8—61no—
239C3⋊C8—61no—
240C3⋊C8—123no—
241C3⋊C8C3⋊S411yescharacteristic, complemented
242C2×C32—31no—
243C2×C32—31no—
244C2×C32—41no—
245C2×C32—43no—
246C2×C32—83no—
247C2×C32—81no—
248C2×C32C4×S411yescharacteristic
249C3⋊D3—121no—
250C3⋊D3—243no—
251C3⋊D3—241no—
252C3⋊D3—246no—
253C3⋊D3—242no—
254C2×D4—271no—
255C22×C4—271no—
256C22×C4D3×C3⋊D311yescharacteristic
257M16—541no—
258M16—541no—
259C2×C8—91no—
260C2×C8—91no—
261C2×C8—91no—
262C2×C8—91no—
263C4⋊C4—271no—
264C22⋊C4—271no—
265C22⋊C4—271no—
266C42—271no—
267C22×C3—21no—
268C22×C3—31no—
269C22×C3—31no—
270C22×C3—31no—
271C22×C3—31no—
272C22×C3—61no—
273C22×C3—61no—
274C22×C3#371211yescharacteristic, complemented
275C22×C3#374311yescharacteristic, complemented
276C2×D3—181no—
277C2×D3—181no—
278C2×D3—181no—
279C2×D3—181no—
280C2×D3—182no—
281C2×D3—181no—
282C2×D3—181no—
283C2×D3—361no—
284C2×D3—361no—
285C2×D3—363no—
286C2×D3—723no—
287A4—23no—
288A4#371213yescomplemented
289C3×C4—21no—
290C3×C4—31no—
291C3×C4—31no—
292C3×C4—43no—
293C3×C4—61no—
294C3×C4—83no—
295C3×C4D3×S411yescharacteristic
296C3×C4C2×C3⋊S411yescharacteristic
297Dic3—181no—
298Dic3—181no—
299Dic3—181no—
300Dic3—181no—
301Dic3—182no—
302Dic3—181no—
303Dic3—181no—
304Dic3—361no—
305Dic3—363no—
306Dic3—361no—
307Dic3—723no—
308C32—41no—
309C32—43no—
310C32—83no—
311C32—81no—
312C32#540011yescharacteristic, complemented
313C23—271no—
314C23#628911yescharacteristic
315D4—541no—
316D4—541no—
317C2×C4—31no—
318C2×C4—31no—
319C2×C4—271no—
320C2×C4—271no—
321C2×C4—271no—
322C2×C4—541no—
323C2×C4—541no—
324C8—31no—
325C8—91no—
326C8—181no—
327C8—181no—
328C2×C3—21no—
329C2×C3—31no—
330C2×C3—31no—
331C2×C3—31no—
332C2×C3—31no—
333C2×C3—43no—
334C2×C3—61no—
335C2×C3—61no—
336C2×C3—83no—
337C2×C3(C3⋊S4)⋊C411yescharacteristic
338C2×C3C4×C3⋊S411yescharacteristic
339D3—362no—
340D3—361no—
341D3—361no—
342D3—723no—
343D3—726no—
344C22—31no—
345C22—31no—
346C22—541no—
347C22—541no—
348C22—541no—
349C22#8878811yescharacteristic, complemented
350C4—31no—
351C4—541no—
352C4—541no—
353C4—541no—
354C4D3×C3⋊S411yescharacteristic
355C3—21no—
356C3—43no—
357C3—83no—
358C3#1059317611yescharacteristic, complemented
359C3#1059363411yescharacteristic, complemented
360C2—31no—
361C2—31no—
362C2—1081no—
363C2#1172420111yescharacteristic
subgroupG/H
center, Frattini, hypercenterC4D3×C3⋊S4
commutatorC2×C32×A4C2×C4
FittingC22×C32×C4C2×D3
socleC23×C32C4×D3
prime radicalC2×C3⋊(C3⋊S4)C4
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
16. C22×C3×C3⋊C8D3semidirect
19. A4×C3⋊C8D3semidirect
20. C3⋊(A4⋊C8)D3semidirect
62. C32×A4M16semidirect
71. C22×C3⋊C8C3⋊D3semidirect
114. C3×C3⋊C8S4semidirect
160. C22×C32ΓL(1,𝔽25)semidirect
170. C3×A4ΓL(1,𝔽25)semidirect
171. C3×A4ΓL(1,𝔽25)semidirect
241. C3⋊C8C3⋊S4semidirect
274. C22×C3#3712semidirect
275. C22×C3#3743semidirect
288. A4#3712semidirect
312. C32#5400semidirect
349. C22#88788semidirect
358. C3#10593176semidirect
359. C3#10593634semidirect

Maximal subgroups

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

subgroupindex
C4×C3⋊(C3⋊S4)2
C3×A4×C3⋊C82
C3×C3⋊(A4⋊C8)2
#105936343
#105931763
#105911573
#887884