Finite Group Databasebeta

← #11797966

Subgroups — #11797966

Order 1152.

337 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 337 conjugacy classes of subgroups — too many to draw. Only its 78 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×#4869C211yesmaximal, characteristic, complemented
2#10590918C211yesmaximal, characteristic, complemented
3#10590633C211yesmaximal, characteristic
4#10589354C211yesmaximal, characteristic, complemented
5C3×#4487C211yesmaximal, characteristic, complemented
6#10589126C211yesmaximal, characteristic, complemented
7#10589105C211yesmaximal, characteristic, complemented
8#70918—31nomaximal
9#69310—31nomaximal
10#64113C2211yescharacteristic
11C2×C3×D24C2211yescharacteristic
12C3×#893—21no—
13#63817—21no—
14C32×#106C2211yescharacteristic
15#63639C2211yescharacteristic
16C3×#819C2211yescharacteristic
17C3×#815C2211yescharacteristic
18C3×#812C2211yescharacteristic
19C3×#809—21no—
20#63570—21no—
21#63548C412yescomplemented
22#63546C412yescomplemented
23C3×#442—31no—
24#5147—31no—
25#4908—31no—
26#4894—31no—
27#4869D311yescharacteristic, complemented
28#4717—31no—
29C3×#365—31no—
30#4563—31no—
31#4512—31no—
32#4495—31no—
33#4488—31no—
34#4488—31no—
35#4488—61no—
36#4487D311yescharacteristic, complemented
37#4466—31no—
38#3828—21no—
39C2×C3×D12—21no—
40C2×C3×C4×D3—41no—
41C2×C32×C8C2311yescharacteristic
42C32×C4⋊C4—21no—
43#3746C2×C411yescharacteristic, complemented
44#3745C2×C411yescharacteristic, complemented
45#3744—21no—
46C3×(C2×D3)⋊C4—41no—
47C3×C4×Dic3D411yescharacteristic
48C3×#208—21no—
49C2×C3×C3⋊C8D411yescharacteristic
50C3×D24—21no—
51C3×D24C2×C412yes—
52#3712—41no—
53#3710—41no—
54C3⋊(C3⋊C16)—21no—
55C3×C3⋊C16C2×C412yescomplemented
56AGL(1,ℤ/16)—91noSylow
57C2×C3×D8—31no—
58C3×C4×D4—61no—
59#907—61no—
60#904—31no—
61#904—31no—
62#904—61no—
63C2×D24C2×D311yescharacteristic
64#901—61no—
65#893—21no—
66C3×M32—31no—
67C3×#108—31no—
68C3×#106—21no—
69C3×#106C2×D311yescharacteristic
70C3×#106C2×D311yescharacteristic
71C3×#102—61no—
72C3×#97—31no—
73#827—32no—
74#826—32no—
75#825—61no—
76#819C2×D311yescharacteristic
77#815C2×D311yescharacteristic
78#812—31no—
79#812—31no—
80#812—61no—
81#812C2×D311yescharacteristic
82#809—21no—
83#805—61no—
84#800—32no—
85D48—32no—
86C22×C3×D3—41no—
87C2×C32×C4—41no—
88C2×C32×C4C2×D411yescharacteristic
89#638—41no—
90C4⋊(C3⋊D3)—21no—
91C4×C3⋊D3—41no—
92#634—21no—
93C2×C3×Dic3—21no—
94C3×D12—22no—
95C3×D12—41no—
96C3×C4×D3—82no—
97C32×C8C22×C411yescharacteristic
98C32×C8#4111yescharacteristic
99C3×C3⋊C8—21no—
100C3×C3⋊C8—41no—
101#509—91no—
102#444—91no—
103#442—31no—
104#365—31no—
105#361—91no—
106#360—91no—
107#347—91no—
108C2×C3×D4—61no—
109C22×C3×C4—121no—
110#235—61no—
111#235—61no—
112#235—121no—
113C2×D12—21no—
114C2×C4×D3—41no—
115C3×D8—32no—
116C3×D8—61no—
117C3×M16—61no—
118C2×C3×C8—21no—
119C2×C3×C8—31no—
120C2×C3×C8C22×D311yescharacteristic
121C2×C3×C8C22×D311yescharacteristic
122C3×C4⋊C4—21no—
123C3×C4⋊C4—21no—
124C3×C4⋊C4—41no—
125C3×C22⋊C4—121no—
126C3×C42—31no—
127(C2×D3)⋊C4—41no—
128C4×Dic3#6711yescharacteristic
129#208—21no—
130C2×C3⋊C8#6711yescharacteristic
131Dic12—31no—
132Dic12—31no—
133Dic12—61no—
134D24—21no—
135D24—31no—
136D24—31no—
137D24—61no—
138D24C4×D312yes—
139#204—61no—
140#204—61no—
141#204—121no—
142ΓL(1,𝔽25)—121no—
143ΓL(1,𝔽25)—121no—
144D3×C8—121no—
145D3×C8—121no—
146C3×C16—32no—
147C3⋊C16—61no—
148C3⋊C16—61no—
149C3⋊C16—121no—
150C3⋊C16C4×D312yescomplemented
151C22×C32AGL(1,ℤ/8)11yescharacteristic
152C2×C3⋊D3—41no—
153C2×C3×D3—42no—
154C2×C3×D3—81no—
155C2×C3×D3—81no—
156C32×C4—81no—
157C32×C4C4×D411yescharacteristic
158C32×C4AGL(1,ℤ/8)11yescharacteristic
159C3⋊Dic3—41no—
160C3×Dic3—41no—
161#135—91no—
162C2×D8—31no—
163#131—181no—
164C4×D4—61no—
165SD32—92no—
166D16—92no—
167M32—31no—
168M32—91no—
169#108—31no—
170#106D3211yescharacteristic
171#104—181no—
172#102—61no—
173#97—31no—
174C23×C3—121no—
175C22×D3—41no—
176C3×D4—62no—
177C3×D4—121no—
178C2×C3×C4—21no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—61no—
182C2×C3×C4—81no—
183C2×C3×C4—242no—
184C2×C3×C4D3×D411yescharacteristic
185C2×C3×C4C2×#6711yescharacteristic
186#67—121no—
187#67—121no—
188#67—241no—
189C2×Dic3—21no—
190D12—22no—
191D12—41no—
192D12—61no—
193D12—61no—
194D12—121no—
195C4×D3—82no—
196C4×D3—121no—
197C4×D3—121no—
198C4×D3—241no—
199Dic6—61no—
200Dic6—61no—
201Dic6—121no—
202C3×C8—21no—
203C3×C8—21no—
204C3×C8—61no—
205C3×C8—121no—
206C3×C8C2×C4×D311yescharacteristic
207C3×C8C2×C4×D311yescharacteristic
208C3×C8#23511yescharacteristic
209C3×C8#23911yescharacteristic
210C3⋊C8—21no—
211C3⋊C8—41no—
212C2×C32—21no—
213C2×C32#44211yescharacteristic
214C3⋊D3—81no—
215C3×D3—82no—
216#41—181no—
217C2×D4—61no—
218C22×C4—121no—
219Q16—91no—
220SD16—181no—
221D8—32no—
222D8—61no—
223D8—91no—
224M16—61no—
225M16—361no—
226C2×C8—31no—
227C2×C8—361no—
228C2×C8C2×D3211yescharacteristic
229C4⋊C4—21no—
230C22⋊C4—121no—
231C42—31no—
232C16—32no—
233C16—181no—
234C22×C3—21no—
235C22×C3—122no—
236C22×C3—241no—
237C22×C3—241no—
238C22×C3#91411yescharacteristic
239C22×C3#93211yescharacteristic
240C2×D3—42no—
241C2×D3—81no—
242C2×D3—81no—
243C2×D3—121no—
244C2×D3—121no—
245C2×D3—241no—
246C3×C4—21no—
247C3×C4—21no—
248C3×C4—81no—
249C3×C4—81no—
250C3×C4—121no—
251C3×C4—161no—
252C3×C4#89311yescharacteristic
253C3×C4#91411yescharacteristic
254C3×C4C4×#6711yescharacteristic
255C3×C4#94911yescharacteristic
256Dic3—41no—
257Dic3—121no—
258Dic3—121no—
259Dic3—241no—
260C32AGL(1,ℤ/16)11yescharacteristic, complemented, Sylow
261C23—121no—
262Q8—181no—
263D4—62no—
264D4—121no—
265D4—181no—
266D4—361no—
267C2×C4—41no—
268C2×C4—61no—
269C2×C4—242no—
270C2×C4—361no—
271C2×C4D3×#6711yescharacteristic
272C8—61no—
273C8—121no—
274C8#379911yescharacteristic
275C8C4×D3211yescharacteristic
276C2×C3—21no—
277C2×C3—21no—
278C2×C3—21no—
279C2×C3—41no—
280C2×C3—242no—
281C2×C3#486911yescharacteristic
282C2×C3#501411yescharacteristic
283D3—82no—
284D3—241no—
285D3—241no—
286D3—481no—
287C22—122no—
288C22—241no—
289C22—241no—
290C22—361no—
291C22#6393411yescharacteristic
292C4—81no—
293C4—121no—
294C4—361no—
295C4#6390411yescharacteristic
296C4#6393911yescharacteristic
297C3—21no—
298C3#6931011yescharacteristic, complemented
299C3#7091811yescharacteristic, complemented
300C2—21no—
301C2—242no—
302C2—721no—
303C2#1059058911yescharacteristic
subgroupG/H
centerC2#10590589
commutatorC32×C8C22×C4
FrattiniC2×C8C2×D32
FittingC32×#106C22
socleC2×C32#442
prime radical#10590633C2
cosocleC2×C32×C8C23
hypercenter#106D32
hypocenterC32AGL(1,ℤ/16)
LCSG ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #106

Complements

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

NH ≅ G/Ntype
1. C3×#4869C2semidirect
2. #10590918C2semidirect
4. #10589354C2semidirect
5. C3×#4487C2semidirect
6. #10589126C2semidirect
7. #10589105C2semidirect
21. #63548C4semidirect
22. #63546C4semidirect
27. #4869D3semidirect
36. #4487D3semidirect
43. #3746C2×C4semidirect
44. #3745C2×C4semidirect
55. C3×C3⋊C16C2×C4semidirect
150. C3⋊C16C4×D3semidirect
260. C32AGL(1,ℤ/16)semidirect
298. C3#69310semidirect
299. C3#70918semidirect

Maximal subgroups

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

subgroupindex
C3×#48692
#105909182
#105906332
#105893542
C3×#44872
#105891262
#105891052
#709183
#693103