Finite Group Databasebeta

← #11797970

Subgroups — #11797970

Order 1152.

360 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 86 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#10592896C211yesmaximal, characteristic, complemented
2C3×#5193C211yesmaximal, characteristic
3#10590781C211yesmaximal, characteristic, complemented
4C3×#4566C211yesmaximal, characteristic, complemented
5#10589136C211yesmaximal, characteristic, complemented
6#10589127C211yesmaximal, characteristic, complemented
7#10589106C211yesmaximal, characteristic, complemented
8#71515—31nomaximal
9#69249—31nomaximal
10C32×#135C2211yescharacteristic
11#64157—21no—
12C2×C3×D24C2211yescharacteristic
13#63898—21no—
14#63664—21no—
15#63646C2211yescharacteristic
16#63642C2211yescharacteristic
17C3×#828C412yescomplemented
18C3×#826C412yescomplemented
19C3×#817C2211yescharacteristic
20C3×#812C2211yescharacteristic
21C3×#812C2211yescharacteristic
22#63562—21no—
23C3×#509—31no—
24#5197—31no—
25#5197—31no—
26#5197—61no—
27#5193D311yescharacteristic
28#5152—31no—
29#4869—31no—
30C3×#360—31no—
31#4566D311yescharacteristic, complemented
32#4563—31no—
33#4516—31no—
34#4513—31no—
35#4512—31no—
36#4512—31no—
37#4487—31no—
38C32×#41—21no—
39C2×C3×D12—21no—
40C2×D3×Dic3—41no—
41C32×Q16C2×C411yescharacteristic, complemented
42C32×SD16—21no—
43C32×D8C2×C411yescharacteristic, complemented
44C2×C32×C8C2311yescharacteristic
45C4×C3⋊Dic3D411yescharacteristic
46#3748—21no—
47#3748—41no—
48C2×C3⋊(C3⋊C8)—41no—
49C2×C3⋊(C3⋊C8)D411yescharacteristic
50C3×C4⋊Dic3—21no—
51C3×D24—21no—
52C3×D24C2×C412yes—
53(C2×D3)⋊Dic3—41no—
54C3×C3⋊C16—21no—
55C3×C3⋊C16C2×C412yescomplemented
56AGL(1,ℤ/16)—91noSylow
57C3×#135—21no—
58C3×#135C2×D311yescharacteristic
59C3×#135C2×D311yescharacteristic
60C2×C3×D8—31no—
61#948—61no—
62#948—61no—
63#948—62no—
64D4×Dic3—61no—
65C2×D24C2×D311yescharacteristic
66#893—61no—
67C3×SD32—32no—
68C3×D16—32no—
69C3×M32—31no—
70C3×M32—31no—
71C3×#106—31no—
72#837—61no—
73#837—61no—
74#837—62no—
75#832—61no—
76#828Dic312yescomplemented
77#826Dic312yescomplemented
78#819—31no—
79#819—31no—
80#819—61no—
81#817C2×D311yescharacteristic
82#815—31no—
83#815—31no—
84#815—61no—
85#812C2×D311yescharacteristic
86#812C2×D311yescharacteristic
87#809—61no—
88C22×C3×D3—41no—
89C32×Q8—21no—
90C32×D4—21no—
91C32×D4—41no—
92C2×C32×C4—41no—
93C2×C32×C4C2×D411yescharacteristic
94C2×C3⋊Dic3—21no—
95C2×C3×Dic3—41no—
96C3×D12—22no—
97C3×D12—41no—
98D3×Dic3—82no—
99C32×C8C22×C411yescharacteristic
100C32×C8#4111yescharacteristic
101C3⋊(C3⋊C8)—21no—
102C3⋊(C3⋊C8)—41no—
103#509—31no—
104#444—91no—
105#442—91no—
106#365—91no—
107#361—91no—
108#360—31no—
109#347—91no—
110C3×#41—22no—
111C3×#41—21no—
112C3×#41—21no—
113C2×C3×D4—61no—
114C22×Dic3—121no—
115C2×D12—21no—
116C2×C4×D3—121no—
117C3×Q16—21no—
118C3×Q16C4×D311yescharacteristic, complemented
119C3×Q16C2×Dic311yescharacteristic, complemented
120C3×SD16—22no—
121C3×SD16—21no—
122C3×SD16—21no—
123C3×D8—21no—
124C3×D8—32no—
125C3×D8—61no—
126C3×D8C4×D311yescharacteristic, complemented
127C3×D8C2×Dic311yescharacteristic, complemented
128C2×C3×C8—21no—
129C2×C3×C8C22×D311yescharacteristic
130C2×C3×C8C22×D311yescharacteristic
131C3×C4⋊C4—61no—
132#217—121no—
133(C2×D3)⋊C4—121no—
134C4⋊Dic3—21no—
135C4×Dic3—31no—
136C4×Dic3—31no—
137C4×Dic3—61no—
138#208—62no—
139#208—61no—
140#208—61no—
141#208—122no—
142#208—121no—
143#208—121no—
144C2×C3⋊C8—31no—
145C2×C3⋊C8—31no—
146C2×C3⋊C8—61no—
147C2×C3⋊C8—122no—
148C2×C3⋊C8—121no—
149C2×C3⋊C8—121no—
150D24—21no—
151D24C2×Dic312yes—
152C3×C16—32no—
153C3×C16—61no—
154C3⋊C16—21no—
155C3⋊C16C2×Dic312yescomplemented
156C22×C32—41no—
157C22×C32AGL(1,ℤ/8)11yescharacteristic
158C2×C3×D3—42no—
159C2×C3×D3—81no—
160C2×C3×D3—81no—
161C32×C4—41no—
162C32×C4C4×D411yescharacteristic
163C32×C4AGL(1,ℤ/8)11yescharacteristic
164C3⋊Dic3—41no—
165C3×Dic3—81no—
166#135D3211yescharacteristic
167C2×D8—31no—
168#131—181no—
169C4×D4—181no—
170SD32—32no—
171D16—32no—
172M32—31no—
173M32—31no—
174#108—91no—
175#106—31no—
176#104—181no—
177#102—181no—
178#97—91no—
179C23×C3—121no—
180C22×D3—41no—
181C3×Q8—22no—
182C3×Q8—21no—
183C3×Q8—21no—
184C3×D4—22no—
185C3×D4—21no—
186C3×D4—21no—
187C3×D4—42no—
188C3×D4—41no—
189C3×D4—41no—
190C3×D4—62no—
191C3×D4—121no—
192C2×C3×C4—21no—
193C2×C3×C4—42no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—121no—
197C2×C3×C4D3×D411yescharacteristic
198C2×C3×C4D3×D411yescharacteristic
199C2×Dic3—41no—
200C2×Dic3—61no—
201C2×Dic3—61no—
202C2×Dic3—121no—
203C2×Dic3—242no—
204D12—22no—
205D12—41no—
206C4×D3—242no—
207C3×C8—21no—
208C3×C8—21no—
209C3×C8C2×C4×D311yescharacteristic
210C3×C8#23711yescharacteristic
211C3×C8#23711yescharacteristic
212C3×C8C22×Dic311yescharacteristic
213C3⋊C8—62no—
214C3⋊C8—61no—
215C3⋊C8—61no—
216C3⋊C8—122no—
217C3⋊C8—121no—
218C3⋊C8—121no—
219C2×C32—21no—
220C2×C32—81no—
221C2×C32#44211yescharacteristic
222C3×D3—82no—
223#41—21no—
224C2×D4—61no—
225C22×C4—361no—
226Q16D3×Dic311yescharacteristic, complemented
227SD16—21no—
228D8—32no—
229D8—61no—
230D8D3×Dic311yescharacteristic, complemented
231M16—181no—
232M16—361no—
233C2×C8—91no—
234C2×C8—361no—
235C2×C8C2×D3211yescharacteristic
236C4⋊C4—61no—
237C22⋊C4—361no—
238C42—91no—
239C16—32no—
240C16—61no—
241C22×C3—21no—
242C22×C3—42no—
243C22×C3—41no—
244C22×C3—41no—
245C22×C3—122no—
246C22×C3—241no—
247C22×C3—241no—
248C22×C3#91111yescharacteristic
249C22×C3#91111yescharacteristic
250C2×D3—42no—
251C2×D3—81no—
252C2×D3—81no—
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—42no—
256C3×C4—41no—
257C3×C4—41no—
258C3×C4—241no—
259C3×C4#67⋊C411yescharacteristic
260C3×C4#91111yescharacteristic
261C3×C4#91411yescharacteristic
262C3×C4D4×Dic311yescharacteristic
263Dic3—81no—
264Dic3—121no—
265Dic3—121no—
266Dic3—241no—
267C32AGL(1,ℤ/16)11yescharacteristic, complemented, Sylow
268C23—121no—
269Q8—21no—
270D4—21no—
271D4—41no—
272D4—62no—
273D4—121no—
274C2×C4—41no—
275C2×C4—121no—
276C2×C4—181no—
277C2×C4—722no—
278C2×C4#381111yescharacteristic
279C8—181no—
280C8—361no—
281C8C2×D3×Dic311yescharacteristic
282C8#380511yescharacteristic
283C2×C3—21no—
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—41no—
287C2×C3—82no—
288C2×C3—81no—
289C2×C3—81no—
290C2×C3—242no—
291C2×C3#478511yescharacteristic
292C2×C3#515211yescharacteristic
293D3—82no—
294C22—41no—
295C22—122no—
296C22—241no—
297C22—241no—
298C22#6392711yescharacteristic
299C4—41no—
300C4—241no—
301C4—361no—
302C4#6393711yescharacteristic
303C4#67⋊Dic311yescharacteristic
304C3—21no—
305C3#6924911yescharacteristic, complemented
306C3#7151511yescharacteristic, complemented
307C2—21no—
308C2—81no—
309C2—242no—
310C2#1059143611yescharacteristic
subgroupG/H
centerC2#10591436
commutatorC32×C8C22×C4
FrattiniC2×C8C2×D32
FittingC32×#135C22
socleC2×C32#442
prime radicalC3×#5193C2
cosocleC2×C32×C8C23
hypercenter#135D32
hypocenterC32AGL(1,ℤ/16)
LCSG ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #10592896C2semidirect
3. #10590781C2semidirect
4. C3×#4566C2semidirect
5. #10589136C2semidirect
6. #10589127C2semidirect
7. #10589106C2semidirect
17. C3×#828C4semidirect
18. C3×#826C4semidirect
31. #4566D3semidirect
41. C32×Q16C2×C4semidirect
43. C32×D8C2×C4semidirect
55. C3×C3⋊C16C2×C4semidirect
76. #828Dic3semidirect
77. #826Dic3semidirect
118. C3×Q16C4×D3semidirect
119. C3×Q16C2×Dic3semidirect
126. C3×D8C4×D3semidirect
127. C3×D8C2×Dic3semidirect
155. C3⋊C16C2×Dic3semidirect
226. Q16D3×Dic3semidirect
230. D8D3×Dic3semidirect
267. C32AGL(1,ℤ/16)semidirect
305. C3#69249semidirect
306. C3#71515semidirect

Maximal subgroups

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

subgroupindex
#105928962
C3×#51932
#105907812
C3×#45662
#105891362
#105891272
#105891062
#715153
#692493