Finite Group Databasebeta

← #82563⋊C3 #11921282

Subgroups — #82563⋊C3 #11921282

Order 1152.

407 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 407 conjugacy classes of subgroups — too many to draw. Only its 57 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
1C2×C3⋊#994C211yesmaximal, characteristic
2C3×#520⋊C3C211yesmaximal, characteristic, complemented
3C2×C3⋊(#131⋊C3)C211yesmaximal, characteristic, complemented
4#82563C311yesmaximal, characteristic, complemented
5#2897⋊C3—31nomaximal
6C2×C3×#41⋊C3C2211yescharacteristic
7C3⋊#994—21no—
8C3⋊#994C412yescomplemented
9C2×C3⋊(#41⋊C3)—21no—
10C3×C4×SL(2,𝔽3)—21no—
11C3⋊(#131⋊C3)—21no—
12C3×#810—41nomaximal
13C2×#1009C2×C311yescharacteristic
14C2×#994—31no—
15C3×#520C2×C311yescharacteristic
16C2×#948C2×C311yescharacteristic
17C2×#131⋊C3—31no—
18#520⋊C3—21no—
19#520⋊C3D311yescharacteristic, complemented
20#5002—31no—
21#5001—31no—
22#4966—31no—
23C2×#809—31no—
24C2×C3×Dic6—41no—
25C3×#41⋊C3C2×C411yescharacteristic
26C3×#41⋊C3D411yescharacteristic
27C2×C3×SL(2,𝔽3)D411yescharacteristic
28C3⋊(#41⋊C3)—22no—
29C32×C4⋊C4—41no—
30C2×C3×C3⋊C8—41no—
31#2897—31noSylow
32C2×C3×#41C22×C311yescharacteristic
33#1009—21no—
34#1009—32no—
35#1009—61no—
36#1009C3×C412yescomplemented
37C2×#239—21no—
38C2×D3×D4—61no—
39C2×#235—31no—
40C22×D12—31no—
41#994—32no—
42#994—61no—
43C2×#41⋊C3—21no—
44C2×#41⋊C3—61no—
45C2×#41⋊C3C2×D311yescharacteristic
46C3×C4×Q8—21no—
47C3×C4×D4—61no—
48C3×#117—31no—
49C2×C3×C4⋊C4—31no—
50#948—21no—
51#948—61no—
52C2×#208—31no—
53C22×C3⋊C8—31no—
54#131⋊C3—61no—
55C4×SL(2,𝔽3)—22no—
56C4×SL(2,𝔽3)—21no—
57C3×#103—121no—
58#810—31no—
59#810A411yescharacteristic
60#809—32no—
61#809—31no—
62#809—31no—
63C2×C32×C4—41no—
64C2×C32×C4—81no—
65C2×C3×Dic3—81no—
66C3×Dic6—42no—
67C3×Dic6—81no—
68C3×SL(2,𝔽3)C22⋊C411yescharacteristic
69C3×C3⋊C8—81no—
70C2×H(5,𝔽2)—31no—
71C2×#131—31no—
72#520C3×D311yescharacteristic
73#418—91no—
74#417—91no—
75#416—91no—
76C2×#102—91no—
77C23×D3—61no—
78C3×#41—31no—
79C3×#41—31no—
80C3×#41C2×C3×C411yescharacteristic
81C3×#41C3×D411yescharacteristic
82C2×C3×Q8—121no—
83C2×C3×Q8C3×D411yescharacteristic
84C2×C3×D4—31no—
85C22×C3×C4—31no—
86C22×C3×C4—61no—
87C2×#67—61no—
88#239—22no—
89#239—62no—
90D3×D4—64no—
91D3×D4—62no—
92D3×D4—62no—
93#235—32no—
94#235—32no—
95#235—61no—
96#235—61no—
97C2×D12—32no—
98C2×D12—31no—
99C2×D12—32no—
100C2×D12—61no—
101C2×D12—61no—
102C2×D12—61no—
103C2×C4×D3—61no—
104C2×Dic6C2×A411yescharacteristic
105#41⋊C3—21no—
106#41⋊C3—21no—
107#41⋊C3—62no—
108#41⋊C3C4×D311yescharacteristic
109#41⋊C3D1211yescharacteristic
110C2×SL(2,𝔽3)—21no—
111C2×SL(2,𝔽3)#6711yescharacteristic
112C2×C3×C8—121no—
113C3×C4⋊C4—31no—
114C3×C4⋊C4—41no—
115C3×C4⋊C4—61no—
116C3×C4⋊C4—81no—
117C3×C4⋊C4C2×A411yescharacteristic
118C3×C22⋊C4—61no—
119C3×C42—61no—
120#208—61no—
121#208—61no—
122C2×C3⋊C8—31no—
123C2×C3⋊C8—61no—
124C2×C3⋊C8—61no—
125C2×C3⋊C8C2×A411yescharacteristic
126C22×C32—41no—
127C32×C4—41no—
128C32×C4—41no—
129C32×C4—161no—
130C3×Dic3—82no—
131H(5,𝔽2)—32no—
132H(5,𝔽2)—61no—
133H(5,𝔽2)—92no—
134H(5,𝔽2)—181no—
135C2×#41—61no—
136C2×#41—91no—
137C2×#41C2×C3×D311yescharacteristic
138C22×D4—91no—
139C22×D4—181no—
140#131—61no—
141#131—181no—
142C2×M16—91no—
143C22×C8—91no—
144C4×Q8—21no—
145C4×D4—61no—
146#117—31no—
147C2×C4⋊C4—31no—
148#103—31no—
149#103—91no—
150#102—91no—
151#102—92no—
152#102—91no—
153C23×C3—31no—
154C22×D3—61no—
155C22×D3—62no—
156C22×D3—62no—
157C22×D3—241no—
158C22×D3—241no—
159C3×Q8—31no—
160C3×Q8—122no—
161C3×Q8—241no—
162C3×Q8C3×C22⋊C411yescharacteristic
163C3×D4—31no—
164C3×D4—31no—
165C3×D4—32no—
166C2×C3×C4—21no—
167C2×C3×C4—31no—
168C2×C3×C4—31no—
169C2×C3×C4—31no—
170C2×C3×C4—31no—
171C2×C3×C4—31no—
172C2×C3×C4—41no—
173C2×C3×C4—61no—
174C2×C3×C4—81no—
175C2×C3×C4—82no—
176C2×C3×C4—81no—
177C2×C3×C4—241no—
178C2×C3×C4—241no—
179C2×C3×C4C22×A411yescharacteristic
180#67—62no—
181#67—62no—
182C2×Dic3—21no—
183D12—32no—
184D12—62no—
185D12—62no—
186D12—61no—
187C4×D3—62no—
188C4×D3—62no—
189Dic6—21no—
190Dic6C4×A412yes—
191SL(2,𝔽3)—21no—
192SL(2,𝔽3)(C2×D3)⋊C411yescharacteristic
193C3×C8—241no—
194C3⋊C8—21no—
195C3⋊C8—61no—
196C2×C32—41no—
197C2×C32—41no—
198C2×C32—41no—
199C24—181no—
200#41—31no—
201#41—31no—
202#41—62no—
203#41—92no—
204#41—92no—
205#41—181no—
206#41—181no—
207#41—182no—
208#41C3×C4×D311yescharacteristic
209#41C3×D1211yescharacteristic
210C2×Q8—31no—
211C2×Q8C3×#6711yescharacteristic
212C2×D4—31no—
213C2×D4—92no—
214C2×D4—92no—
215C2×D4—91no—
216C2×D4—181no—
217C2×D4—181no—
218C2×D4—181no—
219C2×D4—181no—
220C2×D4—184no—
221C2×D4—182no—
222C2×D4—182no—
223C22×C4—31no—
224C22×C4—61no—
225C22×C4—181no—
226M16—181no—
227M16—181no—
228C2×C8—31no—
229C2×C8—91no—
230C2×C8—181no—
231C2×C8—181no—
232C4⋊C4—31no—
233C4⋊C4—61no—
234C4⋊C4D3×A411yescharacteristic
235C22⋊C4—61no—
236C42—61no—
237C22×C3—31no—
238C22×C3—31no—
239C22×C3—41no—
240C22×C3—61no—
241C22×C3—61no—
242C22×C3—81no—
243C22×C3D4×A411yescharacteristic
244C2×D3—62no—
245C2×D3—121no—
246C2×D3—121no—
247C2×D3—242no—
248C2×D3—242no—
249C3×C4—31no—
250C3×C4—31no—
251C3×C4—41no—
252C3×C4—41no—
253C3×C4—41no—
254C3×C4—81no—
255C3×C4—81no—
256C3×C4—121no—
257C3×C4—161no—
258C3×C4—162no—
259C3×C4—242no—
260C3×C4C2×C4×A411yescharacteristic
261C3×C4D4×A411yescharacteristic
262Dic3—22no—
263C32—41noSylow
264C23—31no—
265C23—182no—
266C23—182no—
267C23—181no—
268C23—721no—
269C23—721no—
270Q8—31no—
271Q8—32no—
272Q8—61no—
273Q8C3×(C2×D3)⋊C411yescharacteristic
274D4—31no—
275D4—31no—
276D4—32no—
277D4—92no—
278D4—182no—
279D4—182no—
280D4—182no—
281D4—182no—
282D4—181no—
283C2×C4—21no—
284C2×C4—31no—
285C2×C4—31no—
286C2×C4—31no—
287C2×C4—31no—
288C2×C4—31no—
289C2×C4—61no—
290C2×C4—61no—
291C2×C4—182no—
292C2×C4—182no—
293C2×C4—241no—
294C2×C4C2×D3×A411yescharacteristic
295C8—61no—
296C8—181no—
297C2×C3—41no—
298C2×C3—41no—
299C2×C3—41no—
300C2×C3—61no—
301C2×C3—61no—
302C2×C3—81no—
303C2×C3—81no—
304C2×C3—81no—
305C2×C3A4×C22⋊C411yescharacteristic
306C2×C3#576⋊C311yescharacteristic
307C2×C3#577⋊C311yescharacteristic
308D3—122no—
309C22—31no—
310C22—31no—
311C22—61no—
312C22—61no—
313C22—182no—
314C22—361no—
315C22—361no—
316C22—722no—
317C22—722no—
318C22A4×#6711yescharacteristic
319C4—31no—
320C4—31no—
321C4—41no—
322C4—62no—
323C4—121no—
324C4C4×D3×A411yescharacteristic
325C4A4×D1211yescharacteristic
326C3—41no—
327C3—81no—
328C3#2897⋊C311yescharacteristic, complemented
329C2—61no—
330C2—61no—
331C2—362no—
332C2A4×(C2×D3)⋊C411yescharacteristic
333C2#5835⋊C311yescharacteristic
334C2#5837⋊C311yescharacteristic
subgroupG/H
centerC22A4×#67
commutatorC3×#41C2×C3×C4
FrattiniC2×C4C2×D3×A4
FittingC3×#520C2×C3
socleC22×C3D4×A4
prime radicalC2×C3⋊#994C2
cosocleC2×C3×#41C22×C3
hypercenterC4⋊C4D3×A4
hypocenterC3×Q8C3×C22⋊C4
LCSG ▷ C3×#41 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#520⋊C3C2semidirect
3. C2×C3⋊(#131⋊C3)C2semidirect
4. #82563C3semidirect
8. C3⋊#994C4semidirect
19. #520⋊C3D3semidirect
36. #1009C3×C4semidirect
328. C3#2897⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊#9942
C3×#520⋊C32
C2×C3⋊(#131⋊C3)2
#825633
#2897⋊C33
C3×#8104