Finite Group Databasebeta

← #30675225

Subgroups — #30675225

Order 16768.

784 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16862291C211yesmaximal, characteristic, complemented
2C2×#16862272C211yesmaximal, characteristic, complemented
3#22553879C211yesmaximal, characteristic, complemented
4#22553454C214yesmaximal, complemented
5#22553444C214yesmaximal, complemented
6#22553317C211yesmaximal, characteristic, complemented
7#22553309C211yesmaximal, characteristic, complemented
8#22553278C211yesmaximal, characteristic, complemented
9C22×D8×C131C211yesmaximal, characteristic, complemented
10#16862291—24no—
11#16862291—24no—
12C2×#14523513C2211yescharacteristic
13C2×#14523513C2211yescharacteristic
14C2×#14523512—21no—
15C2×#14523512C2212yescomplemented
16C22×C22⋊D131—22no—
17C2×D4×D131—21no—
18C2×D4×D131C2212yescomplemented
19#16862272—24no—
20C2×#14523501C2212yescomplemented
21C2×#14523498C2212yescomplemented
22C2×#14523497C2211yescharacteristic, complemented
23#16862242—21no—
24#16862233—21no—
25#16862217C2214yescomplemented
26#16862215—22no—
27#16862205C2211yescharacteristic, complemented
28#16862171C2212yescomplemented
29#16862168C2212yescomplemented
30#16862167C2214yescomplemented
31(C2×D131)⋊C8C2214yescomplemented
32C22×D4×C131C2211yescharacteristic
33C22×D4×C131C2211yescharacteristic, complemented
34C2×D8×C131—24no—
35C2×D8×C131C2214yescomplemented
36C22×C8×C131C2211yescharacteristic, complemented
37C23×D131—22no—
38#14523513—22no—
39#14523513—22no—
40#14523513—22no—
41#14523513D414yescomplemented
42#14523512—28no—
43#14523512—24no—
44#14523512—24no—
45C2×Dic262D411yescharacteristic
46C2×Dic262C2311yescharacteristic
47C2×D524D411yescharacteristic
48C2×D524C2311yescharacteristic
49C2×C22⋊D131—216no—
50C2×C22⋊D131—22no—
51C2×C22⋊D131—24no—
52C2×C22⋊D131—22no—
53C2×C22⋊D131—21no—
54C2×C22⋊D131D412yes—
55D4×D131—28no—
56D4×D131—24no—
57D4×D131—24no—
58C22×Dic131—22no—
59C2×C4×D131—21no—
60C2×C4×D131C2312yes—
61#14523501—24no—
62#14523498—24no—
63#14523497—22no—
64C2×C131⋊C8C2312yes—
65C4⋊Dic131C2312yes—
66C22⋊Dic131—24no—
67C22⋊Dic131—21no—
68C4×Dic131—21no—
69C24×C131—21no—
70C24×C131—21no—
71C2×D4×C131—21no—
72C2×D4×C131—22no—
73C2×D4×C131—22no—
74C2×D4×C131—22no—
75C2×D4×C131—21no—
76C2×D4×C131D414yescomplemented
77C2×D4×C131C2312yes—
78C2×D4×C131C2312yes—
79C22×C4×C131C2311yescharacteristic
80D8×C131—28no—
81C2×C8×C131—22no—
82C2×C8×C131C2312yes—
83C22×D131—21no—
84C22×D131—28no—
85C22×D131—84no—
86C22×D131C2×D412yes—
87Dic262—22no—
88Dic262—21no—
89Dic262C2×D412yes—
90D524—22no—
91D524—21no—
92D524C2×D412yes—
93C22⋊D131—28no—
94C22⋊D131—28no—
95C22⋊D131—24no—
96C22⋊D131—24no—
97C22⋊D131—84no—
98C2×Dic131—28no—
99C2×Dic131—21no—
100C2×Dic131—22no—
101C2×Dic131C2×D412yes—
102C4×D131—24no—
103C4×D131—22no—
104C4×D131—22no—
105C131⋊C8—22no—
106C23×C131—22no—
107C23×C131—24no—
108C23×C131—22no—
109C23×C131—42no—
110C23×C131—82no—
111C23×C131—82no—
112C23×C131C2×D411yescharacteristic
113D4×C131—24no—
114D4×C131—22no—
115D4×C131—22no—
116D4×C131—22no—
117D4×C131—42no—
118D4×C131C2×D414yes—
119C2×C4×C131C2×D411yescharacteristic
120C2×C4×C131C2×D412yes—
121C2×C4×C131C2×D412yes—
122C2×C4×C131C2411yescharacteristic
123C8×C131—22no—
124C22×C131—22no—
125C22×C131—24no—
126C22×C131—44no—
127C22×C131—42no—
128C22×C131—44no—
129C22×C131—84no—
130C22×C131—84no—
131C22×C131#12012yes—
132C22×C131C22×D411yescharacteristic
133C2×D131—22no—
134C2×D131—22no—
135C2×D131—24no—
136C2×D131—42no—
137C2×D131—88no—
138C4×C131#12012yes—
139C4×C131C22×D411yescharacteristic
140C4×C131C22×D411yescharacteristic
141Dic131—22no—
142Dic131—22no—
143Dic131—42no—
144C2×C131—22no—
145C2×C131—44no—
146C2×C131—82no—
147C2×C131C2×#12011yescharacteristic
148C2×C131#57612yes—
149D131—42no—
150D131—42no—
151C131#300811yescharacteristic, complemented, Sylow
152#3008—1311nomaximal, Sylow
153C2×H(5,𝔽2)—1311no—
154C2×AGL(1,ℤ/8)—1311no—
155C22×D8D13111yescharacteristic, complemented
156#535—1311no—
157#449—1314no—
158#447—1314no—
159#418—1311no—
160#416—1311no—
161#408—1311no—
162H(5,𝔽2)—2624no—
163H(5,𝔽2)—2624no—
164C2×#41—2621no—
165C2×#41—1312no—
166C2×#41—1311no—
167C2×#41—1311no—
168C22×D4—2621no—
169C22×D4—2622no—
170C22×D4—1312no—
171C22×D4C2×D13111yescharacteristic
172C22×D4C2×D13111yescharacteristic, complemented
173AGL(1,ℤ/8)—2624no—
174C2×SD16—1312no—
175C2×D8—24no—
176C2×D8—1312no—
177C2×D8C2×D13114yescomplemented
178C2×M16—1311no—
179C22×C8C2×D13111yescharacteristic, complemented
180#127—2621no—
181#124—2621no—
182#121—1314no—
183#120—2622no—
184#117—1311no—
185#103—1312no—
186#102—1314no—
187#102—1312no—
188C22⋊C8—1314no—
189C24—21no—
190C24—21no—
191C24—2622no—
192#41—2628no—
193#41—2624no—
194#41—2624no—
195#41—2622no—
196#41—2622no—
197#41—2622no—
198#41—1314no—
199C2×Q8—1311no—
200C2×Q8—1311no—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—22no—
206C2×D4—2628no—
207C2×D4—2624no—
208C2×D4—26216no—
209C2×D4—2622no—
210C2×D4—2621no—
211C2×D4—2624no—
212C2×D4—2624no—
213C2×D4—2622no—
214C2×D4—1312no—
215C2×D4—1311no—
216C2×D4—1311no—
217C2×D4C22⋊D13114yescomplemented
218C2×D4C22×D13112yes—
219C2×D4C22×D13112yes—
220C22×C4—2622no—
221C22×C4—2621no—
222C22×C4—1312no—
223C22×C4C22×D13111yescharacteristic
224SD16—2624no—
225D8—28no—
226D8—2624no—
227M16—2622no—
228C2×C8—22no—
229C2×C8—1312no—
230C2×C8C22×D13112yes—
231C4⋊C4—1312no—
232C22⋊C4—2621no—
233C22⋊C4—2624no—
234C42—2621no—
235C23—22no—
236C23—24no—
237C23—22no—
238C23—42no—
239C23—2628no—
240C23—2621no—
241C23—82no—
242C23—82no—
243C23—10484no—
244C23—1312no—
245C23D4×D13111yescharacteristic
246Q8—2622no—
247Q8—2621no—
248Q8—1312no—
249D4—24no—
250D4—22no—
251D4—22no—
252D4—22no—
253D4—42no—
254D4—2628no—
255D4—2628no—
256D4—2624no—
257D4—2624no—
258D4—2622no—
259D4—2621no—
260D4—10484no—
261D4—1312no—
262D4C2×C22⋊D13114yes—
263C2×C4—2628no—
264C2×C4—2621no—
265C2×C4—2624no—
266C2×C4—2622no—
267C2×C4—2622no—
268C2×C4—2622no—
269C2×C4—1312no—
270C2×C4D4×D13111yescharacteristic
271C2×C4D4×D13112yes—
272C2×C4C2×C22⋊D13112yes—
273C2×C4C23×D13111yescharacteristic
274C8—22no—
275C8—2622no—
276C22—22no—
277C22—24no—
278C22—44no—
279C22—44no—
280C22—42no—
281C22—2624no—
282C22—2622no—
283C22—2622no—
284C22—84no—
285C22—84no—
286C22—5242no—
287C22—10488no—
288C22#1686221412yes—
289C22C2×D4×D13111yescharacteristic
290C4—2622no—
291C4—2622no—
292C4—5242no—
293C4#1686221412yes—
294C4C2×D4×D13111yescharacteristic
295C4C22×C22⋊D13111yescharacteristic
296C2—22no—
297C2—44no—
298C2—82no—
299C2—5242no—
300C2—5242no—
301C2C2×#1686221411yescharacteristic
302C2#2255411312yes—
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC22×D8×C131C2
socleC22×C131C22×D4
hypercenterC22×D8D131
hypocenterC131#3008
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#16862291C2semidirect
2. C2×#16862272C2semidirect
3. #22553879C2semidirect
4. #22553454C2semidirect
5. #22553444C2semidirect
6. #22553317C2semidirect
7. #22553309C2semidirect
8. #22553278C2semidirect
9. C22×D8×C131C2semidirect
15. C2×#14523512C22semidirect
18. C2×D4×D131C22semidirect
20. C2×#14523501C22semidirect
21. C2×#14523498C22semidirect
22. C2×#14523497C22semidirect
25. #16862217C22semidirect
27. #16862205C22semidirect
28. #16862171C22semidirect
29. #16862168C22semidirect
30. #16862167C22semidirect
31. (C2×D131)⋊C8C22semidirect
33. C22×D4×C131C22semidirect
35. C2×D8×C131C22semidirect
36. C22×C8×C131C22semidirect
41. #14523513D4semidirect
76. C2×D4×C131D4semidirect
151. C131#3008semidirect
155. C22×D8D131semidirect
172. C22×D4C2×D131semidirect
177. C2×D8C2×D131semidirect
179. C22×C8C2×D131semidirect
217. C2×D4C22⋊D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622912
C2×#168622722
#225538792
#225534542
#225534442
#225533172
#225533092
#225532782
C22×D8×C1312
#3008131