Finite Group Databasebeta

← #11797971

Subgroups — #11797971

Order 1152.

353 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 80 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#10592892C211yesmaximal, characteristic
2C3×#5197C211yesmaximal, characteristic, complemented
3#10590768C211yesmaximal, characteristic, complemented
4C3×#4566C211yesmaximal, characteristic, complemented
5#10589138C211yesmaximal, characteristic, complemented
6#10589128C211yesmaximal, characteristic, complemented
7#10589105C211yesmaximal, characteristic, complemented
8#70929—31nomaximal
9#69249—31nomaximal
10C32×#135C2211yescharacteristic
11C2×#3745C2211yescharacteristic
12C3×#948—21no—
13#63894—21no—
14#63655C412yescomplemented
15#63653C412yescomplemented
16#63639C2211yescharacteristic
17C3×#837—21no—
18C3×#819C2211yescharacteristic
19C3×#817C2211yescharacteristic
20C3×#815C2211yescharacteristic
21C3×#812C2211yescharacteristic
22#63564—21no—
23C3×#444—31no—
24#5197D311yescharacteristic, complemented
25#5193—31no—
26#5193—31no—
27#5193—61no—
28#4869—31no—
29#4711—31no—
30C3×#360—31no—
31#4566D311yescharacteristic, complemented
32#4516—31no—
33#4512—31no—
34#4495—31no—
35#4494—31no—
36#4487—31no—
37#4466—31no—
38C32×#41—21no—
39C2×C4⋊(C3⋊D3)—21no—
40C2×#624—41no—
41C32×Q16C2×C411yescharacteristic, complemented
42C32×SD16—21no—
43C32×D8C2×C411yescharacteristic, complemented
44C2×C32×C8C2311yescharacteristic
45#3745—21no—
46#3745C2×C412yes—
47C3×C4⋊Dic3—21no—
48C3×C4×Dic3D411yescharacteristic
49C3×#208—21no—
50C3×#208—41no—
51C2×C3×C3⋊C8—41no—
52C2×C3×C3⋊C8D411yescharacteristic
53#3722—41no—
54C3⋊(C3⋊C16)C2×C412yescomplemented
55C3×C3⋊C16—21no—
56AGL(1,ℤ/16)—91noSylow
57C3×#135—21no—
58C3×#135C2×D311yescharacteristic
59C3×#135C2×D311yescharacteristic
60C3×#131—61no—
61#948—21no—
62C2×D24—31no—
63C2×D24—31no—
64C2×D24—61no—
65#893—61no—
66C4×D12—61no—
67C3×M32—31no—
68C3×#108—31no—
69C3×#106—31no—
70C3×#104—61no—
71C3×#97—31no—
72#837—21no—
73#828—32no—
74#828—32no—
75#828—62no—
76#826—32no—
77#826—32no—
78#826—62no—
79#821—61no—
80#819C2×D311yescharacteristic
81#817C2×D311yescharacteristic
82#815C2×D311yescharacteristic
83#812—31no—
84#812—31no—
85#812—61no—
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
94C4⋊(C3⋊D3)—22no—
95C4⋊(C3⋊D3)—41no—
96C2×C3×Dic3—21no—
97C2×C3×Dic3—41no—
98#624—82no—
99C32×C8C22×C411yescharacteristic
100C32×C8#4111yescharacteristic
101C3×C3⋊C8—21no—
102C3×C3⋊C8—41no—
103#509—91no—
104#444—31no—
105#442—91no—
106#365—91no—
107#361—91no—
108#360—31no—
109#347—91no—
110C3×#41—21no—
111C3×#41—21no—
112C3×#41—41no—
113C2×D12—61no—
114C2×D12—61no—
115C2×D12—121no—
116C2×C4×D3—121no—
117C2×C4×D3—121no—
118C3×Q16—21no—
119C3×Q16C4×D311yescharacteristic, complemented
120C3×Q16C4×D311yescharacteristic, complemented
121C3×SD16—21no—
122C3×SD16—21no—
123C3×SD16—41no—
124C3×D8—21no—
125C3×D8C4×D311yescharacteristic, complemented
126C3×D8C4×D311yescharacteristic, complemented
127C3×M16—61no—
128C3×M16—121no—
129C2×C3×C8—21no—
130C2×C3×C8—31no—
131C2×C3×C8—121no—
132C2×C3×C8C22×D311yescharacteristic
133C2×C3×C8C22×D311yescharacteristic
134C3×C4⋊C4—61no—
135C3×C42—31no—
136(C2×D3)⋊C4—121no—
137(C2×D3)⋊C4—121no—
138C4⋊Dic3—21no—
139C4×Dic3#6711yescharacteristic
140#208—21no—
141#208—41no—
142C2×C3⋊C8—41no—
143C2×C3⋊C8#6711yescharacteristic
144D24—32no—
145D24—32no—
146D24—62no—
147D24—61no—
148D24—61no—
149D24—121no—
150C3×C16—61no—
151C3⋊C16—21no—
152C3⋊C16—32no—
153C3⋊C16—32no—
154C3⋊C16—62no—
155C22×C32—41no—
156C22×C32AGL(1,ℤ/8)11yescharacteristic
157C2×C3⋊D3—42no—
158C2×C3⋊D3—81no—
159C2×C3⋊D3—81no—
160C32×C4—41no—
161C32×C4C4×D411yescharacteristic
162C32×C4AGL(1,ℤ/8)11yescharacteristic
163C3×Dic3—41no—
164C3×Dic3—81no—
165#135D3211yescharacteristic
166C2×D8—91no—
167#131—61no—
168C4×D4—181no—
169SD32—92no—
170D16—92no—
171M32—31no—
172M32—91no—
173#108—31no—
174#106—31no—
175#104—61no—
176#102—181no—
177#97—31no—
178C22×D3—121no—
179C22×D3—121no—
180C22×D3—241no—
181C3×Q8—21no—
182C3×Q8—21no—
183C3×Q8—41no—
184C3×D4—21no—
185C3×D4—21no—
186C3×D4—41no—
187C3×D4—41no—
188C3×D4—41no—
189C3×D4—81no—
190C2×C3×C4—21no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—61no—
194C2×C3×C4—81no—
195C2×C3×C4—121no—
196C2×C3×C4D3×D411yescharacteristic
197C2×C3×C4C2×#6711yescharacteristic
198C2×Dic3—21no—
199C2×Dic3—41no—
200D12—62no—
201D12—62no—
202D12—122no—
203D12—121no—
204D12—121no—
205D12—241no—
206C4×D3—242no—
207C4×D3—242no—
208C3×C8—21no—
209C3×C8—21no—
210C3×C8—61no—
211C3×C8—121no—
212C3×C8C2×C4×D311yescharacteristic
213C3×C8C2×C4×D311yescharacteristic
214C3×C8#23511yescharacteristic
215C3×C8#23711yescharacteristic
216C3⋊C8—21no—
217C3⋊C8—41no—
218C2×C32—21no—
219C2×C32—81no—
220C2×C32#44211yescharacteristic
221C3⋊D3—82no—
222#41—21no—
223C2×D4—181no—
224C22×C4—361no—
225Q16#62411yescharacteristic, complemented
226SD16—21no—
227D8—92no—
228D8—181no—
229D8#62411yescharacteristic, complemented
230M16—61no—
231M16—121no—
232C2×C8—31no—
233C2×C8—121no—
234C2×C8C2×D3211yescharacteristic
235C4⋊C4—61no—
236C22⋊C4—361no—
237C42—31no—
238C16—61no—
239C16—92no—
240C22×C3—21no—
241C22×C3—41no—
242C22×C3—41no—
243C22×C3—81no—
244C22×C3#91111yescharacteristic
245C22×C3#94911yescharacteristic
246C2×D3—122no—
247C2×D3—122no—
248C2×D3—242no—
249C2×D3—241no—
250C2×D3—241no—
251C2×D3—241no—
252C2×D3—241no—
253C2×D3—481no—
254C2×D3—481no—
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—41no—
258C3×C4—41no—
259C3×C4—81no—
260C3×C4—121no—
261C3×C4—241no—
262C3×C4#67⋊C411yescharacteristic
263C3×C4#91411yescharacteristic
264C3×C4C4×#6711yescharacteristic
265C3×C4#93211yescharacteristic
266Dic3—41no—
267Dic3—81no—
268C32AGL(1,ℤ/16)11yescharacteristic, complemented, Sylow
269C23—361no—
270Q8—21no—
271D4—21no—
272D4—41no—
273D4—182no—
274D4—361no—
275C2×C4—41no—
276C2×C4—61no—
277C2×C4—121no—
278C2×C4—722no—
279C2×C4D3×#6711yescharacteristic
280C8—61no—
281C8—121no—
282C8#380411yescharacteristic
283C8C2×#62411yescharacteristic
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—41no—
288C2×C3—81no—
289C2×C3—81no—
290C2×C3—161no—
291C2×C3#478511yescharacteristic
292C2×C3#501411yescharacteristic
293D3—242no—
294D3—242no—
295D3—482no—
296C22—41no—
297C22—362no—
298C22—721no—
299C22—721no—
300C22#6392511yescharacteristic
301C4—41no—
302C4—121no—
303C4—241no—
304C4#6393411yescharacteristic
305C4#6395211yescharacteristic
306C3—21no—
307C3#6924911yescharacteristic, complemented
308C3#7092911yescharacteristic, complemented
309C2—21no—
310C2—81no—
311C2—722no—
312C2#1059140411yescharacteristic
subgroupG/H
centerC2#10591404
commutatorC32×C8C22×C4
FrattiniC2×C8C2×D32
FittingC32×#135C22
socleC2×C32#442
prime radical#10592892C2
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
2. C3×#5197C2semidirect
3. #10590768C2semidirect
4. C3×#4566C2semidirect
5. #10589138C2semidirect
6. #10589128C2semidirect
7. #10589105C2semidirect
14. #63655C4semidirect
15. #63653C4semidirect
24. #5197D3semidirect
31. #4566D3semidirect
41. C32×Q16C2×C4semidirect
43. C32×D8C2×C4semidirect
54. C3⋊(C3⋊C16)C2×C4semidirect
119. C3×Q16C4×D3semidirect
120. C3×Q16C4×D3semidirect
125. C3×D8C4×D3semidirect
126. C3×D8C4×D3semidirect
225. Q16#624semidirect
229. D8#624semidirect
268. C32AGL(1,ℤ/16)semidirect
307. C3#69249semidirect
308. C3#70929semidirect

Maximal subgroups

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

subgroupindex
#105928922
C3×#51972
#105907682
C3×#45662
#105891382
#105891282
#105891052
#709293
#692493