Finite Group Databasebeta

← #13166073

Subgroups — #13166073

Order 1728.

382 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 44 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⋊S4)C211yesmaximal, characteristic
2C3×#64259C211yesmaximal, characteristic, complemented
3A4×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4#10593178—31nomaximal
5#10593177—31nomaximal
6#10591179—31nomaximal
7C2×C3×C3⋊S4—21no—
8C32×C4×A4C2211yescharacteristic
9C3×C3⋊SL(2,ℤ/4)—21no—
10#88811—41nomaximal
11C4⋊(C3⋊S4)D311yescharacteristic
12#64259D311yescharacteristic, complemented
13C3×C4⋊S4—31no—
14C3×C4⋊S4—61no—
15C3×A4⋊Q8—31no—
16C3×A4⋊Q8—61no—
17C22×C3⋊(C3⋊C8)D311yescharacteristic, complemented
18C3×#930—31no—
19C3×#924—31no—
20C2×#3715—31no—
21A4×C3⋊C8—31no—
22A4×C3⋊C8—31no—
23A4×C3⋊C8—61no—
24#63574—31no—
25#63565—31no—
26#63562—31no—
27C2×C32×A4D411yescharacteristic
28C3×C3⋊S4—41no—
29C3×C4⋊(C3⋊D3)—41no—
30C3×#634—41no—
31C3×C3⋊(C3⋊C8)—41no—
32#5401—91no—
33#5048—91no—
34#5041—91no—
35C2×C3⋊S4—21no—
36C2×C3×S4—62no—
37C2×C3×S4—61no—
38C22×C32×C4C2×D311yescharacteristic
39C2×C3×#67—61no—
40C2×C3×D12—31no—
41C2×C3×Dic6—31no—
42C3×C4×A4—21no—
43C3×C4×A4—21no—
44C3×C4×A4—21no—
45C3×C4×A4—41no—
46C3×C4×A4C2×D311yescharacteristic
47C3×C4×A4C2×D311yescharacteristic
48C3⋊SL(2,ℤ/4)—21no—
49C3×SL(2,ℤ/4)—62no—
50C3×SL(2,ℤ/4)—61no—
51C2×C3⋊(C3⋊C8)—31no—
52C2×C3⋊(C3⋊C8)—31no—
53C3×#217—61no—
54C3×(C2×D3)⋊C4—61no—
55C3×C4⋊Dic3—31no—
56C3×C4⋊Dic3—31no—
57C3×Dic3⋊C4—61no—
58#3717—121no—
59#3716—121no—
60#3715—61no—
61#3715—61no—
62C2×C3×C3⋊D3—81no—
63C32×A4SD1611yescharacteristic, complemented
64C33×C4—41no—
65C3×C3⋊Dic3—81no—
66C4⋊S4—31no—
67C4⋊S4—61no—
68A4⋊Q8—31no—
69A4⋊Q8—61no—
70C3×C22⋊Q8—91no—
71C3×#121—91no—
72C2×#215—91no—
73C2×#214—91no—
74#930—31no—
75#924—31no—
76C22×C3⋊C8—31no—
77C22×C3⋊C8—31no—
78C22×C3⋊C8—61no—
79C8×A4—91no—
80#835—91no—
81#832—91no—
82#810—91no—
83#809—91no—
84#808—91no—
85#808—91no—
86C23×C32D1211yescharacteristic
87C22×C3×D3—61no—
88C2×C3×A4—21no—
89C2×C3×A4—21no—
90C2×C3×A4—21no—
91C2×C3×A4—41no—
92C2×C3×A4#6711yescharacteristic
93C2×C3×A4#6711yescharacteristic
94C3⋊S4—41no—
95C3×S4—122no—
96C3×S4—121no—
97C2×C32×C4—31no—
98C2×C32×C4—31no—
99C4⋊(C3⋊D3)—41no—
100#634—41no—
101C3×#67—121no—
102C3×#67—121no—
103C2×C3×Dic3—61no—
104C2×C3×Dic3—61no—
105C2×C3×Dic3—61no—
106C3×D12—61no—
107C3×D12—61no—
108C3×D12—121no—
109C3×D12—241no—
110C3×Dic6—61no—
111C3×Dic6—61no—
112C3×Dic6—121no—
113C3×Dic6—241no—
114C3⋊(C3⋊C8)—31no—
115C3⋊(C3⋊C8)S411yescharacteristic, complemented
116C3×C3⋊C8—121no—
117C3×C3⋊C8—121no—
118C3×C3⋊C8—241no—
119#465—271noSylow
120C2×C33—41no—
121C3×C3⋊D3—161no—
122C2×S4—62no—
123C2×S4—61no—
124C2×C3×Q8—91no—
125C2×C3×D4—91no—
126C2×C3×D4—181no—
127C22×C3×C4—21no—
128C22×C3×C4D3211yescharacteristic
129C22×C3×C4D3211yescharacteristic
130C2×#67—61no—
131C2×D12—31no—
132C2×Dic6—31no—
133C4×A4—21no—
134C4×A4—21no—
135C4×A4—41no—
136C4×A4D3211yescharacteristic
137SL(2,ℤ/4)—62no—
138SL(2,ℤ/4)—61no—
139C3×C4⋊C4—91no—
140C3×C4⋊C4—91no—
141C3×C4⋊C4—181no—
142C3×C22⋊C4—181no—
143C3×C22⋊C4—181no—
144#217—61no—
145#215—181no—
146#215—181no—
147#214—181no—
148#214—181no—
149(C2×D3)⋊C4—61no—
150C4⋊Dic3—31no—
151C4⋊Dic3—31no—
152Dic3⋊C4—61no—
153C2×C3⋊C8—91no—
154C2×C3⋊C8—91no—
155C2×C3⋊C8—91no—
156C2×C3⋊C8—91no—
157C2×C3⋊C8—181no—
158C2×C3⋊C8—181no—
159#204—361no—
160C22×C32—31no—
161C22×C32—31no—
162C22×C32#20411yescharacteristic, complemented
163C2×C3⋊D3—81no—
164C2×C3×D3—121no—
165C2×C3×D3—121no—
166C2×C3×D3—121no—
167C2×C3×D3—241no—
168C2×C3×D3—242no—
169C3×A4—21no—
170C3×A4—21no—
171C3×A4—21no—
172C3×A4—41no—
173C3×A4#21411yescharacteristic, complemented
174C3×A4#21511yescharacteristic, 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—81no—
184C3×Dic3—121no—
185C3×Dic3—121no—
186C3×Dic3—121no—
187C3×Dic3—241no—
188C3×Dic3—242no—
189C2×SD16—271no—
190C22×C8—91no—
191C22⋊Q8—91no—
192#121—91no—
193#106—271no—
194#103—271no—
195#102—271no—
196C33—41noSylow
197C23×C3—21no—
198C23×C3—181no—
199C23×C3#62611yescharacteristic
200C23×C3#62611yescharacteristic
201C22×D3—61no—
202C2×A4—21no—
203C2×A4—21no—
204C2×A4—41no—
205C2×A4#62511yescharacteristic
206S4—122no—
207S4—121no—
208C3×Q8—181no—
209C3×Q8—181no—
210C3×D4—181no—
211C3×D4—181no—
212C3×D4—361no—
213C3×D4—361no—
214C2×C3×C4—31no—
215C2×C3×C4—31no—
216C2×C3×C4—31no—
217C2×C3×C4—31no—
218C2×C3×C4—61no—
219C2×C3×C4—61no—
220C2×C3×C4—181no—
221C2×C3×C4—181no—
222C2×C3×C4—181no—
223#67—121no—
224#67—121no—
225C2×Dic3—61no—
226C2×Dic3—61no—
227C2×Dic3—61no—
228D12—61no—
229D12—61no—
230D12—121no—
231D12—241no—
232Dic6—61no—
233Dic6—61no—
234Dic6—121no—
235Dic6—241no—
236C3×C8—361no—
237C3⋊C8—31no—
238C3⋊C8—31no—
239C3⋊C8—61no—
240C3⋊C8—91no—
241C3⋊C8—91no—
242C3⋊C8—181no—
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—161no—
253C3×D3—241no—
254C3×D3—481no—
255C3×D3—482no—
256C2×Q8—91no—
257C2×D4—91no—
258C2×D4—181no—
259C22×C4#111311yescharacteristic
260SD16—541no—
261SD16—541no—
262C2×C8—271no—
263C2×C8—271no—
264C4⋊C4—91no—
265C4⋊C4—91no—
266C4⋊C4—181no—
267C22⋊C4—181no—
268C22⋊C4—181no—
269C22×C3—21no—
270C22×C3—31no—
271C22×C3—31no—
272C22×C3—31no—
273C22×C3—31no—
274C22×C3—61no—
275C22×C3—61no—
276C22×C3—361no—
277C22×C3—361no—
278C22×C3—361no—
279C22×C3#371611yescharacteristic, complemented
280C22×C3#371711yescharacteristic, complemented
281C2×D3—121no—
282C2×D3—121no—
283C2×D3—121no—
284C2×D3—241no—
285C2×D3—242no—
286A4—21no—
287A4—21no—
288A4—41no—
289A4#371511yescharacteristic, complemented
290C3×C4—21no—
291C3×C4—31no—
292C3×C4—31no—
293C3×C4—41no—
294C3×C4—61no—
295C3×C4—81no—
296C3×C4—81no—
297C3×C4—161no—
298C3×C4—361no—
299C3×C4—361no—
300C3×C4—361no—
301C3×C4D3×S411yescharacteristic
302C3×C4D3×S411yescharacteristic
303Dic3—121no—
304Dic3—121no—
305Dic3—121no—
306Dic3—241no—
307Dic3—242no—
308C32—41no—
309C32—41no—
310C32—81no—
311C32—81no—
312C32—81no—
313C32—161no—
314C32#540111yescharacteristic, complemented
315C23—181no—
316C23#629511yescharacteristic
317Q8—181no—
318Q8—181no—
319D4—181no—
320D4—181no—
321D4—361no—
322D4—361no—
323C2×C4—31no—
324C2×C4—31no—
325C2×C4—181no—
326C2×C4—181no—
327C2×C4—181no—
328C8—91no—
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—721no—
342C2×C3#6420711yescharacteristic
343C2×C3#6420711yescharacteristic
344D3—241no—
345D3—481no—
346D3—482no—
347C22—31no—
348C22—31no—
349C22—361no—
350C22—361no—
351C22—361no—
352C22#8881111yescharacteristic, complemented
353C4—31no—
354C4—361no—
355C4—361no—
356C4—361no—
357C4#8910111yescharacteristic
358C3—21no—
359C3—41no—
360C3—81no—
361C3—81no—
362C3—161no—
363C3#1059317711yescharacteristic, complemented
364C3#1059317811yescharacteristic, complemented
365C2—31no—
366C2—31no—
367C2—721no—
368C2#1172421411yescharacteristic
subgroupG/H
centerC2#11724214
commutator, cosocleC32×C4×A4C22
Frattini, hypercenterC4#89101
FittingC22×C32×C4C2×D3
socleC23×C32D12
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
2. C3×#64259C2semidirect
3. A4×C3⋊(C3⋊C8)C2semidirect
12. #64259D3semidirect
17. C22×C3⋊(C3⋊C8)D3semidirect
63. C32×A4SD16semidirect
115. C3⋊(C3⋊C8)S4semidirect
162. C22×C32#204semidirect
173. C3×A4#214semidirect
174. C3×A4#215semidirect
279. C22×C3#3716semidirect
280. C22×C3#3717semidirect
289. A4#3715semidirect
314. C32#5401semidirect
352. C22#88811semidirect
363. C3#10593177semidirect
364. C3#10593178semidirect

Maximal subgroups

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

subgroupindex
C3×C4⋊(C3⋊S4)2
C3×#642592
A4×C3⋊(C3⋊C8)2
#105931783
#105931773
#105911793
#888114