Finite Group Databasebeta

← #11797967

Subgroups — #11797967

Order 1152.

374 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 76 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
1#10592568C211yesmaximal, characteristic, complemented
2#10590920C211yesmaximal, characteristic, complemented
3#10590634C211yesmaximal, characteristic
4C3×#4488C211yesmaximal, characteristic, complemented
5C3×#4487C211yesmaximal, characteristic, complemented
6#10589128C211yesmaximal, characteristic, complemented
7#10589106C211yesmaximal, characteristic, complemented
8#69310—31nomaximal
9#69249—31nomaximal
10C2×#3745C2211yescharacteristic
11#64102—21no—
12C3×#904C2211yescharacteristic
13#63815—21no—
14C32×#106C2211yescharacteristic
15#63642C2211yescharacteristic
16#63636—21no—
17C3×#819C2211yescharacteristic
18C3×#812C2211yescharacteristic
19C3×#812C2211yescharacteristic
20#63568—21no—
21#63549C412yescomplemented
22#63546C412yescomplemented
23#5197—31no—
24#5193—31no—
25#4908—31no—
26#4894—31no—
27#4869—31no—
28#4869—31no—
29#4869—61no—
30C3×#365—31no—
31C3×#360—31no—
32#4516—31no—
33#4512—31no—
34#4512—31no—
35#4495—31no—
36#4488D311yescharacteristic, complemented
37#4487D311yescharacteristic, complemented
38C2×C4⋊(C3⋊D3)—21no—
39C2×C4×C3⋊D3—41no—
40C3×#235—21no—
41C2×C32×C8C2311yescharacteristic
42C32×C4⋊C4—21no—
43#3752—41no—
44C4×C3⋊Dic3D411yescharacteristic
45C2×C3⋊(C3⋊C8)D411yescharacteristic
46#3745—21no—
47#3745C2×C412yes—
48C3×#208—21no—
49C3×Dic12C2×C411yescharacteristic, complemented
50C3×D24C2×C411yescharacteristic, complemented
51C3×#204—21no—
52#3711—41no—
53D3×C3⋊C8—41no—
54C3×C3⋊C16—21no—
55C3×C3⋊C16C2×C412yescomplemented
56AGL(1,ℤ/16)—91noSylow
57C3×#135—31no—
58#948—61no—
59#907—61no—
60#904C2×D311yescharacteristic
61C2×D24—31no—
62C2×D24—31no—
63C2×D24—61no—
64#893—61no—
65#893—61no—
66#893—62no—
67C3×M32—31no—
68C3×M32—31no—
69C3×#108—31no—
70C3×#106—21no—
71C3×#106C2×D311yescharacteristic
72C3×#106C2×D311yescharacteristic
73#837—61no—
74#828—32no—
75#826—32no—
76#825—61no—
77#819C2×D311yescharacteristic
78#815—31no—
79#815—31no—
80#815—61no—
81#812C2×D311yescharacteristic
82#812C2×D311yescharacteristic
83#809—61no—
84#809—61no—
85#809—62no—
86#800—32no—
87D48—32no—
88C22×C3⋊D3—41no—
89C2×C32×C4—41no—
90C2×C32×C4C2×D411yescharacteristic
91C2×C3⋊Dic3—21no—
92C4⋊(C3⋊D3)—22no—
93C4⋊(C3⋊D3)—41no—
94C4×C3⋊D3—82no—
95C3×#67—41no—
96C3×D12—21no—
97C3×C4×D3—41no—
98C3×Dic6—21no—
99C32×C8C22×C411yescharacteristic
100C32×C8#4111yescharacteristic
101C3⋊(C3⋊C8)—21no—
102C3×C3⋊C8—41no—
103#509—91no—
104#444—91no—
105#442—91no—
106#365—31no—
107#361—91no—
108#360—31no—
109#347—91no—
110C3×#41—61no—
111#235—21no—
112C2×D12—62no—
113C2×D12—61no—
114C2×D12—61no—
115C2×C4×D3—122no—
116C2×C4×D3—121no—
117C2×C4×D3—121no—
118C3×Q16—31no—
119C3×SD16—61no—
120C3×D8—31no—
121C3×M16—61no—
122C2×C3×C8—21no—
123C2×C3×C8C22×D311yescharacteristic
124C2×C3×C8C22×D311yescharacteristic
125C3×C4⋊C4—22no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128(C2×D3)⋊C4—122no—
129(C2×D3)⋊C4—121no—
130(C2×D3)⋊C4—121no—
131C4×Dic3—31no—
132C4×Dic3—31no—
133C4×Dic3—61no—
134#208—21no—
135#208—121no—
136C2×C3⋊C8—31no—
137C2×C3⋊C8—31no—
138C2×C3⋊C8—61no—
139C2×C3⋊C8—121no—
140Dic12C4×D311yescharacteristic, complemented
141D24—32no—
142D24—32no—
143D24—62no—
144D24—62no—
145D24—61no—
146D24—61no—
147D24C4×D311yescharacteristic, complemented
148#204—21no—
149ΓL(1,𝔽25)—121no—
150D3×C8—121no—
151C3×C16—32no—
152C3×C16—61no—
153C3⋊C16—21no—
154C3⋊C16C4×D312yescomplemented
155C22×C32AGL(1,ℤ/8)11yescharacteristic
156C2×C3⋊D3—42no—
157C2×C3⋊D3—81no—
158C2×C3⋊D3—81no—
159C2×C3×D3—41no—
160C32×C4—81no—
161C32×C4C4×D411yescharacteristic
162C32×C4AGL(1,ℤ/8)11yescharacteristic
163C3⋊Dic3—41no—
164C3×Dic3—41no—
165#135—31no—
166C2×D8—91no—
167#131—181no—
168C4×D4—181no—
169SD32—92no—
170D16—92no—
171M32—31no—
172M32—31no—
173#108—31no—
174#106D3211yescharacteristic
175#104—181no—
176#102—181no—
177#97—91no—
178C22×D3—122no—
179C22×D3—121no—
180C22×D3—121no—
181C3×Q8—61no—
182C3×D4—61no—
183C3×D4—121no—
184C2×C3×C4—21no—
185C2×C3×C4—42no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—121no—
189C2×C3×C4D3×D411yescharacteristic
190C2×C3×C4D3×D411yescharacteristic
191#67—41no—
192C2×Dic3—62no—
193C2×Dic3—61no—
194C2×Dic3—61no—
195D12—21no—
196D12—64no—
197D12—62no—
198D12—62no—
199D12—122no—
200D12—121no—
201D12—121no—
202C4×D3—41no—
203C4×D3—244no—
204C4×D3—242no—
205C4×D3—242no—
206Dic6—21no—
207C3×C8—21no—
208C3×C8—21no—
209C3×C8—121no—
210C3×C8C2×C4×D311yescharacteristic
211C3×C8C2×C4×D311yescharacteristic
212C3×C8#23711yescharacteristic
213C3×C8#23911yescharacteristic
214C3⋊C8—41no—
215C3⋊C8—61no—
216C3⋊C8—61no—
217C3⋊C8—121no—
218C2×C32—21no—
219C2×C32#44211yescharacteristic
220C3⋊D3—82no—
221C3×D3—81no—
222#41—61no—
223C2×D4—181no—
224C22×C4—361no—
225Q16—31no—
226SD16—61no—
227D8—31no—
228D8—92no—
229D8—181no—
230M16—61no—
231M16—361no—
232C2×C8—91no—
233C2×C8—361no—
234C2×C8C2×D3211yescharacteristic
235C4⋊C4—21no—
236C22⋊C4—361no—
237C42—91no—
238C16—32no—
239C16—61no—
240C22×C3—21no—
241C22×C3—121no—
242C22×C3#91111yescharacteristic
243C22×C3#91411yescharacteristic
244C2×D3—41no—
245C2×D3—124no—
246C2×D3—122no—
247C2×D3—122no—
248C2×D3—242no—
249C2×D3—241no—
250C2×D3—241no—
251C2×D3—242no—
252C2×D3—241no—
253C2×D3—241no—
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—82no—
257C3×C4—81no—
258C3×C4—81no—
259C3×C4—121no—
260C3×C4#67⋊C411yescharacteristic
261C3×C4#89311yescharacteristic
262C3×C4#91411yescharacteristic
263C3×C4#91411yescharacteristic
264Dic3—41no—
265Dic3—122no—
266Dic3—121no—
267Dic3—121no—
268C32AGL(1,ℤ/16)11yescharacteristic, complemented, Sylow
269C23—361no—
270Q8—61no—
271D4—61no—
272D4—121no—
273D4—182no—
274D4—361no—
275C2×C4—41no—
276C2×C4—121no—
277C2×C4—181no—
278C2×C4—722no—
279C2×C4#381111yescharacteristic
280C8—121no—
281C8—181no—
282C8#379611yescharacteristic
283C8C4×D3211yescharacteristic
284C2×C3—22no—
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—241no—
289C2×C3#478511yescharacteristic
290C2×C3#486911yescharacteristic
291D3—81no—
292D3—244no—
293D3—242no—
294D3—242no—
295C22—121no—
296C22—362no—
297C22—721no—
298C22—721no—
299C22#6393711yescharacteristic
300C4—81no—
301C4—121no—
302C4—361no—
303C4#6388111yescharacteristic
304C4#6394111yescharacteristic
305C3—21no—
306C3#6924911yescharacteristic, complemented
307C3#6931011yescharacteristic, complemented
308C2—21no—
309C2—241no—
310C2—722no—
311C2#1059046211yescharacteristic
subgroupG/H
centerC2#10590462
commutatorC32×C8C22×C4
FrattiniC2×C8C2×D32
FittingC32×#106C22
socleC2×C32#442
prime radical#10590634C2
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. #10592568C2semidirect
2. #10590920C2semidirect
4. C3×#4488C2semidirect
5. C3×#4487C2semidirect
6. #10589128C2semidirect
7. #10589106C2semidirect
21. #63549C4semidirect
22. #63546C4semidirect
36. #4488D3semidirect
37. #4487D3semidirect
49. C3×Dic12C2×C4semidirect
50. C3×D24C2×C4semidirect
55. C3×C3⋊C16C2×C4semidirect
140. Dic12C4×D3semidirect
147. D24C4×D3semidirect
154. C3⋊C16C4×D3semidirect
268. C32AGL(1,ℤ/16)semidirect
306. C3#69249semidirect
307. C3#69310semidirect

Maximal subgroups

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

subgroupindex
#105925682
#105909202
#105906342
C3×#44882
C3×#44872
#105891282
#105891062
#693103
#692493