Finite Group Databasebeta

← #13166071

Subgroups — #13166071

Order 1728.

402 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 52 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×C4×C3⋊S4C211yesmaximal, characteristic
2A4×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
3C3×C3⋊(A4⋊C8)C211yesmaximal, characteristic, complemented
4(C3⋊S4)⋊C8—31nomaximal
5C3⋊C8×S4—31nomaximal
6#67⋊(C3⋊C8)—31nomaximal
7C2×C3×C3⋊S4C411yescharacteristic
8C32×C4×A4C2211yescharacteristic
9C3×C3⋊SL(2,ℤ/4)C411yescharacteristic
10#88808—41nomaximal
11C4×C3⋊S4D311yescharacteristic
12C3×C4×S4—31no—
13C3×C4×S4—61no—
14C22×C3⋊(C3⋊C8)D311yescharacteristic, complemented
15C3×C4×#67—31no—
16C2×D3×C3⋊C8—31no—
17A4×C3⋊C8—31no—
18A4×C3⋊C8—31no—
19A4×C3⋊C8—61no—
20C3⋊(A4⋊C8)D311yescharacteristic, complemented
21C3×A4⋊C8—31no—
22C3×A4⋊C8—61no—
23C3×#830—31no—
24Dic3⋊(C3⋊C8)—31no—
25(C2×D3)⋊(C3⋊C8)—31no—
26Dic3×C3⋊C8—31no—
27C2×C32×A4C2×C411yescharacteristic
28C3×C3⋊S4C812yescomplemented
29C3×C4×C3⋊D3—41no—
30C3×C3⋊(C3⋊C8)—41no—
31C3×C3⋊(C3⋊C8)—41no—
32C8×S4—91no—
33D4×C3⋊C8—91no—
34#67⋊C8—91no—
35C2×C3⋊S4Dic311yescharacteristic
36C2×C3×S4—31no—
37C2×C3×S4—61no—
38C22×C32×C4C2×D311yescharacteristic
39C2×C3×#67—31no—
40C2×C3×C4×D3—31no—
41C3×C4×A4—21no—
42C3×C4×A4—21no—
43C3×C4×A4—21no—
44C3×C4×A4—41no—
45C3×C4×A4C2×D311yescharacteristic
46C3×C4×A4C2×D311yescharacteristic
47C3⋊SL(2,ℤ/4)Dic311yescharacteristic
48C3×SL(2,ℤ/4)—31no—
49C3×SL(2,ℤ/4)—61no—
50C2×C3⋊(C3⋊C8)—31no—
51C2×C3⋊(C3⋊C8)—31no—
52C3×#217—31no—
53C3×(C2×D3)⋊C4—31no—
54C3×Dic3⋊C4—31no—
55C3×C4×Dic3—31no—
56C2×C3×C3⋊C8—31no—
57C2×C3×C3⋊C8—31no—
58#3710—121no—
59D3×C3⋊C8—61no—
60D3×C3⋊C8—61no—
61D3×C3⋊C8—121no—
62C2×C3×C3⋊D3—41no—
63C32×A4C2×C811yescharacteristic, complemented
64C33×C4—41no—
65C3×C3⋊Dic3—41no—
66C4×S4—31no—
67C4×S4—61no—
68C3×C4×D4—91no—
69C4×#67—31no—
70C22×C3⋊C8—31no—
71C22×C3⋊C8—31no—
72C22×C3⋊C8—61no—
73C22×C3⋊C8—91no—
74C2×D3×C8—91no—
75C8×A4—91no—
76A4⋊C8—31no—
77A4⋊C8—61no—
78C3×C22⋊C8—91no—
79#830—31no—
80#830—91no—
81(C2×D3)⋊C8—91no—
82Dic3⋊C8—91no—
83C8×Dic3—91no—
84C4⋊(C3⋊C8)—91no—
85C4×C3⋊C8—91no—
86C23×C32C4×D311yescharacteristic
87C22×C3×D3—31no—
88C2×C3×A4—21no—
89C2×C3×A4—21no—
90C2×C3×A4—21no—
91C2×C3×A4—41no—
92C2×C3×A4C4×D311yescharacteristic
93C2×C3×A4C2×Dic311yescharacteristic
94C3⋊S4C3⋊C812yescomplemented
95C3×S4—32no—
96C3×S4—62no—
97C2×C32×C4—31no—
98C2×C32×C4—31no—
99C4×C3⋊D3—41no—
100C3×#67—32no—
101C3×#67—32no—
102C2×C3×Dic3—31no—
103C2×C3×Dic3—31no—
104C2×C3×Dic3—31no—
105C3×C4×D3—61no—
106C3×C4×D3—61no—
107C3×C4×D3—121no—
108C3×C4×D3—241no—
109C3⋊(C3⋊C8)—31no—
110C3⋊(C3⋊C8)—41no—
111C3⋊(C3⋊C8)S411yescharacteristic, complemented
112C3×C3⋊C8—61no—
113C3×C3⋊C8—61no—
114C3×C3⋊C8—121no—
115C3×C3⋊C8—121no—
116C3×C3⋊C8—121no—
117C3×C3⋊C8—241no—
118C3×C3⋊C8—241no—
119C8×D4—271noSylow
120C2×C33—41no—
121C3×C3⋊D3—42no—
122C2×S4—31no—
123C2×S4—61no—
124C2×C3×D4—91no—
125C22×C3×C4—21no—
126C22×C3×C4—91no—
127C22×C3×C4D3211yescharacteristic
128C22×C3×C4D3211yescharacteristic
129C2×#67—31no—
130C2×C4×D3—31no—
131C4×A4—21no—
132C4×A4—21no—
133C4×A4—41no—
134C4×A4D3211yescharacteristic
135SL(2,ℤ/4)—31no—
136SL(2,ℤ/4)—61no—
137C2×C3×C8—91no—
138C2×C3×C8—91no—
139C3×C4⋊C4—91no—
140C3×C22⋊C4—91no—
141C3×C22⋊C4—91no—
142C3×C42—91no—
143#217—31no—
144(C2×D3)⋊C4—31no—
145Dic3⋊C4—31no—
146C4×Dic3—31no—
147C2×C3⋊C8—31no—
148C2×C3⋊C8—31no—
149C2×C3⋊C8—91no—
150C2×C3⋊C8—91no—
151C2×C3⋊C8—91no—
152C2×C3⋊C8—91no—
153C2×C3⋊C8—181no—
154C2×C3⋊C8—181no—
155C2×C3⋊C8—181no—
156C2×C3⋊C8—181no—
157D3×C8—181no—
158D3×C8—181no—
159D3×C8—361no—
160C22×C32—31no—
161C22×C32—31no—
162C22×C32D3×C811yescharacteristic, complemented
163C2×C3⋊D3—41no—
164C2×C3×D3—32no—
165C2×C3×D3—61no—
166C2×C3×D3—61no—
167C2×C3×D3—121no—
168C2×C3×D3—241no—
169C3×A4—21no—
170C3×A4—21no—
171C3×A4—21no—
172C3×A4—41no—
173C3×A4D3×C811yescharacteristic, complemented
174C3×A4C2×C3⋊C811yescharacteristic, complemented
175C32×C4—31no—
176C32×C4—41no—
177C32×C4—41no—
178C32×C4—81no—
179C32×C4—81no—
180C32×C4—81no—
181C32×C4—161no—
182C32×C4C2×S411yescharacteristic
183C3⋊Dic3—41no—
184C3×Dic3—32no—
185C3×Dic3—61no—
186C3×Dic3—61no—
187C3×Dic3—121no—
188C3×Dic3—241no—
189C22×C8—91no—
190C22×C8—271no—
191C4×D4—91no—
192C4⋊C8—271no—
193C22⋊C8—91no—
194C22⋊C8—271no—
195C4×C8—271no—
196C33—41noSylow
197C23×C3—21no—
198C23×C3—91no—
199C23×C3D3×Dic311yescharacteristic
200C23×C3#62411yescharacteristic
201C22×D3—31no—
202C2×A4—21no—
203C2×A4—21no—
204C2×A4—41no—
205C2×A4D3×Dic311yescharacteristic
206S4—32no—
207S4—62no—
208C3×D4—92no—
209C3×D4—92no—
210C2×C3×C4—31no—
211C2×C3×C4—31no—
212C2×C3×C4—31no—
213C2×C3×C4—31no—
214C2×C3×C4—61no—
215C2×C3×C4—61no—
216C2×C3×C4—91no—
217C2×C3×C4—91no—
218C2×C3×C4—91no—
219C2×C3×C4—181no—
220C2×C3×C4—181no—
221#67—32no—
222#67—32no—
223C2×Dic3—31no—
224C2×Dic3—31no—
225C2×Dic3—31no—
226C4×D3—61no—
227C4×D3—61no—
228C4×D3—121no—
229C4×D3—241no—
230C3×C8—181no—
231C3×C8—181no—
232C3×C8—361no—
233C3⋊C8—31no—
234C3⋊C8—31no—
235C3⋊C8—61no—
236C3⋊C8—61no—
237C3⋊C8—61no—
238C3⋊C8—91no—
239C3⋊C8—91no—
240C3⋊C8—121no—
241C3⋊C8—181no—
242C3⋊C8—241no—
243C2×C32—31no—
244C2×C32—31no—
245C2×C32—41no—
246C2×C32—41no—
247C2×C32—81no—
248C2×C32—81no—
249C2×C32—81no—
250C2×C32—161no—
251C2×C32C4×S411yescharacteristic
252C3⋊D3—42no—
253C3×D3—62no—
254C3×D3—122no—
255C3×D3—242no—
256C2×D4—91no—
257C22×C4—91no—
258C22×C4#111311yescharacteristic
259C2×C8—91no—
260C2×C8—91no—
261C2×C8—271no—
262C2×C8—271no—
263C2×C8—541no—
264C2×C8—541no—
265C4⋊C4—91no—
266C22⋊C4—91no—
267C22⋊C4—91no—
268C42—91no—
269C22×C3—21no—
270C22×C3—31no—
271C22×C3—31no—
272C22×C3—31no—
273C22×C3—31no—
274C22×C3—61no—
275C22×C3—61no—
276C22×C3—92no—
277C22×C3—181no—
278C22×C3—181no—
279C22×C3D3×C3⋊C811yescharacteristic, complemented
280C22×C3#371011yescharacteristic, complemented
281C2×D3—32no—
282C2×D3—61no—
283C2×D3—61no—
284C2×D3—121no—
285C2×D3—241no—
286A4—21no—
287A4—21no—
288A4—41no—
289A4D3×C3⋊C811yescharacteristic, complemented
290C3×C4—21no—
291C3×C4—31no—
292C3×C4—31no—
293C3×C4—41no—
294C3×C4—61no—
295C3×C4—81no—
296C3×C4—81no—
297C3×C4—92no—
298C3×C4—161no—
299C3×C4—181no—
300C3×C4—181no—
301C3×C4D3×S411yescharacteristic
302C3×C4D3×S411yescharacteristic
303Dic3—32no—
304Dic3—61no—
305Dic3—61no—
306Dic3—121no—
307Dic3—241no—
308C32—41no—
309C32—41no—
310C32—81no—
311C32—81no—
312C32—81no—
313C32—161no—
314C32C8×S411yescharacteristic, complemented
315C23—91no—
316C23#629411yescharacteristic
317D4—92no—
318D4—92no—
319C2×C4—31no—
320C2×C4—31no—
321C2×C4—91no—
322C2×C4—91no—
323C2×C4—91no—
324C2×C4—181no—
325C2×C4—181no—
326C8—91no—
327C8—181no—
328C8—181no—
329C8—271no—
330C2×C3—21no—
331C2×C3—31no—
332C2×C3—31no—
333C2×C3—31no—
334C2×C3—31no—
335C2×C3—41no—
336C2×C3—61no—
337C2×C3—61no—
338C2×C3—81no—
339C2×C3—81no—
340C2×C3—161no—
341C2×C3—182no—
342C2×C3Dic3×S411yescharacteristic
343C2×C3(C3⋊S4)⋊C411yescharacteristic
344D3—62no—
345D3—122no—
346D3—242no—
347C22—31no—
348C22—31no—
349C22—92no—
350C22—181no—
351C22—181no—
352C22#8880811yescharacteristic, complemented
353C4—31no—
354C4—92no—
355C4—181no—
356C4—181no—
357C4#8910111yescharacteristic
358C3—21no—
359C3—41no—
360C3—81no—
361C3—81no—
362C3—161no—
363C3C3⋊C8×S411yescharacteristic, complemented
364C3(C3⋊S4)⋊C811yescharacteristic, complemented
365C2—31no—
366C2—31no—
367C2—182no—
368C2#1172421311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC4#89101
commutator, hypocenterC32×A4C2×C8
FittingC22×C32×C4C2×D3
socleC23×C32C4×D3
prime radicalC2×C3×C3⋊S4C4
cosocleC32×C4×A4C22
LCSG ▷ C32×A4
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. A4×C3⋊(C3⋊C8)C2semidirect
3. C3×C3⋊(A4⋊C8)C2semidirect
14. C22×C3⋊(C3⋊C8)D3semidirect
20. C3⋊(A4⋊C8)D3semidirect
28. C3×C3⋊S4C8semidirect
63. C32×A4C2×C8semidirect
94. C3⋊S4C3⋊C8semidirect
111. C3⋊(C3⋊C8)S4semidirect
162. C22×C32D3×C8semidirect
173. C3×A4D3×C8semidirect
174. C3×A4C2×C3⋊C8semidirect
279. C22×C3D3×C3⋊C8semidirect
280. C22×C3#3710semidirect
289. A4D3×C3⋊C8semidirect
314. C32C8×S4semidirect
352. C22#88808semidirect
363. C3C3⋊C8×S4semidirect
364. C3(C3⋊S4)⋊C8semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C3⋊S42
A4×C3⋊(C3⋊C8)2
C3×C3⋊(A4⋊C8)2
(C3⋊S4)⋊C83
C3⋊C8×S43
#67⋊(C3⋊C8)3
#888084