Finite Group Databasebeta

← #30675227

Subgroups — #30675227

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×#16862292C211yesmaximal, characteristic, complemented
2C2×#16862272C211yesmaximal, characteristic, complemented
3C22×#14523498C211yesmaximal, characteristic, complemented
4#22553455C214yesmaximal, complemented
5#22553445C214yesmaximal, complemented
6#22553316C211yesmaximal, characteristic, complemented
7#22553306C211yesmaximal, characteristic, complemented
8#22553279C211yesmaximal, characteristic, complemented
9#535×C131C211yesmaximal, characteristic, complemented
10#16862292—24no—
11#16862292—24no—
12C2×#14523514—21no—
13C2×#14523513C2211yescharacteristic
14C2×#14523513C2212yescomplemented
15C22×D524C2211yescharacteristic
16C22×D524C2212yescomplemented
17C2×D4×D131—22no—
18C2×D4×D131—21no—
19#16862272—24no—
20C2×#14523501C2212yescomplemented
21C2×#14523498—24no—
22C2×#14523498C2212yescomplemented
23C2×#14523498C2214yescomplemented
24C2×#14523497C2211yescharacteristic, complemented
25C22×C131⋊C8C2211yescharacteristic, complemented
26#16862169C2212yescomplemented
27#16862166C2212yescomplemented
28#16862166C2214yescomplemented
29C22⋊(C131⋊C8)C2214yescomplemented
30C2×#41×C131C2211yescharacteristic
31C22×D4×C131C2211yescharacteristic, complemented
32#127×C131—21no—
33#124×C131—21no—
34#121×C131C2214yescomplemented
35#120×C131—22no—
36#117×C131C2211yescharacteristic, complemented
37C23×D131—21no—
38C23×D131—22no—
39#14523514—28no—
40#14523513—24no—
41#14523513—24no—
42#14523513—22no—
43#14523513D414yescomplemented
44C2×Dic262C2311yescharacteristic
45C2×D524—24no—
46C2×D524—24no—
47C2×D524—21no—
48C2×D524—22no—
49C2×D524—22no—
50C2×D524D414yescomplemented
51C2×D524C2312yes—
52C2×D524C2311yescharacteristic
53C2×C22⋊D131—21no—
54C2×C22⋊D131—22no—
55D4×D131—216no—
56D4×D131—28no—
57C2×C4×D131—21no—
58C2×C4×D131—22no—
59#14523501—24no—
60#14523498—28no—
61#14523498—24no—
62#14523497—22no—
63C2×C131⋊C8—22no—
64C2×C131⋊C8C2312yes—
65C2×C131⋊C8C2312yes—
66C24×C131—21no—
67#41×C131—22no—
68#41×C131—22no—
69C2×Q8×C131D411yescharacteristic
70C2×D4×C131—24no—
71C2×D4×C131—22no—
72C2×D4×C131—22no—
73C2×D4×C131—21no—
74C2×D4×C131D412yes—
75C2×D4×C131D411yescharacteristic
76C2×D4×C131C2312yes—
77C22×C4×C131C2312yes—
78C22×C4×C131C2311yescharacteristic
79C4⋊C4×C131C2312yes—
80C22⋊C4×C131—21no—
81C22⋊C4×C131—24no—
82C42×C131—21no—
83C22×D131—21no—
84C22×D131—22no—
85C22×D131—28no—
86C22×D131—24no—
87C22×D131—82no—
88C22×D131—84no—
89Dic262—21no—
90Dic262C2×D412yes—
91D524—28no—
92D524—24no—
93D524—22no—
94D524—21no—
95D524C2×D412yes—
96D524C2×D414yes—
97C22⋊D131—28no—
98C22⋊D131—24no—
99C2×Dic131—21no—
100C4×D131—28no—
101C4×D131—24no—
102C131⋊C8—22no—
103C131⋊C8—22no—
104C23×C131—22no—
105C23×C131—42no—
106C23×C131—82no—
107C23×C131C2×D412yes—
108C23×C131C2×D411yescharacteristic
109Q8×C131—22no—
110D4×C131—24no—
111D4×C131—22no—
112D4×C131—22no—
113D4×C131—22no—
114D4×C131—42no—
115D4×C131—84no—
116C2×C4×C131—22no—
117C2×C4×C131—22no—
118C2×C4×C131—22no—
119C2×C4×C131C2×D412yes—
120C2×C4×C131C2×D412yes—
121C2×C4×C131C2×D412yes—
122C2×C4×C131C2×D411yescharacteristic
123C2×C4×C131C2411yescharacteristic
124C22×C131—22no—
125C22×C131—22no—
126C22×C131—24no—
127C22×C131—42no—
128C22×C131—44no—
129C22×C131—84no—
130C22×C131#12012yes—
131C22×C131C22×D411yescharacteristic
132C2×D131—24no—
133C2×D131—22no—
134C2×D131—44no—
135C2×D131—42no—
136C2×D131—84no—
137C2×D131—88no—
138C4×C131—22no—
139C4×C131—42no—
140C4×C131#12012yes—
141C4×C131C22×D411yescharacteristic
142C4×C131C22×D411yescharacteristic
143Dic131—22no—
144C2×C131—22no—
145C2×C131—42no—
146C2×C131—82no—
147C2×C131C2×#12011yescharacteristic
148C2×C131#57612yes—
149D131—44no—
150D131—42no—
151C131#300811yescharacteristic, complemented, Sylow
152#3008—1311nomaximal, Sylow
153C2×H(5,𝔽2)—1311no—
154C2×AGL(1,ℤ/8)—1311no—
155C22×D8—1311no—
156#535D13111yescharacteristic, complemented
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×#41C2×D13111yescharacteristic
168C22×D4—2622no—
169C22×D4—2621no—
170C22×D4—1312no—
171C22×D4—1311no—
172C22×D4C2×D13111yescharacteristic, complemented
173AGL(1,ℤ/8)—2624no—
174C2×SD16—1312no—
175C2×D8—2624no—
176C2×D8—1314no—
177C2×D8—1312no—
178C2×M16—1311no—
179C22×C8—1311no—
180#127—21no—
181#124—21no—
182#121C2×D13114yescomplemented
183#120—22no—
184#117C2×D13111yescharacteristic, complemented
185#103—1312no—
186#102—1312no—
187#102—1314no—
188C22⋊C8—1314no—
189C24—21no—
190C24—2622no—
191C24—2621no—
192#41—22no—
193#41—22no—
194#41—2624no—
195#41—2624no—
196#41—2628no—
197#41—2622no—
198#41—1314no—
199C2×Q8—1311no—
200C2×Q8C22⋊D13111yescharacteristic
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—2624no—
206C2×D4—2624no—
207C2×D4—26216no—
208C2×D4—2628no—
209C2×D4—2622no—
210C2×D4—2621no—
211C2×D4—2622no—
212C2×D4—2622no—
213C2×D4—2621no—
214C2×D4—1314no—
215C2×D4—1311no—
216C2×D4—1312no—
217C2×D4C22⋊D13112yes—
218C2×D4C22⋊D13111yescharacteristic
219C2×D4C22×D13112yes—
220C22×C4—2622no—
221C22×C4—2621no—
222C22×C4C22×D13112yes—
223C22×C4C22×D13111yescharacteristic
224SD16—2624no—
225D8—2628no—
226D8—2624no—
227M16—2622no—
228C2×C8—2622no—
229C2×C8—1312no—
230C2×C8—1312no—
231C4⋊C4C22×D13112yes—
232C22⋊C4—21no—
233C22⋊C4—24no—
234C42—21no—
235C23—22no—
236C23—42no—
237C23—2628no—
238C23—2622no—
239C23—2621no—
240C23—2624no—
241C23—82no—
242C23—10482no—
243C23—10484no—
244C23C2×C22⋊D13112yes—
245C23C2×C22⋊D13111yescharacteristic
246Q8—22no—
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—84no—
261D4—1312no—
262D4—1314no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—2628no—
267C2×C4—2621no—
268C2×C4—2624no—
269C2×C4D4×D13112yes—
270C2×C4D4×D13112yes—
271C2×C4C2×C22⋊D13112yes—
272C2×C4C2×C22⋊D13111yescharacteristic
273C2×C4C23×D13111yescharacteristic
274C8—2622no—
275C8—2622no—
276C22—22no—
277C22—24no—
278C22—22no—
279C22—44no—
280C22—42no—
281C22—2624no—
282C22—2622no—
283C22—84no—
284C22—5244no—
285C22—5242no—
286C22—10484no—
287C22—10488no—
288C22#1686221412yes—
289C22C22×C22⋊D13111yescharacteristic
290C4—22no—
291C4—42no—
292C4—2622no—
293C4#1686221412yes—
294C4C2×D4×D13111yescharacteristic
295C4C2×D4×D13111yescharacteristic
296C2—22no—
297C2—42no—
298C2—82no—
299C2—5244no—
300C2—5242no—
301C2C2×#1686221411yescharacteristic
302C2#2255411512yes—
subgroupG/H
centerC22C22×C22⋊D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#535×C131C2
socleC22×C131C22×D4
hypercenter#535D131
hypocenterC131#3008
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#16862292C2semidirect
2. C2×#16862272C2semidirect
3. C22×#14523498C2semidirect
4. #22553455C2semidirect
5. #22553445C2semidirect
6. #22553316C2semidirect
7. #22553306C2semidirect
8. #22553279C2semidirect
9. #535×C131C2semidirect
14. C2×#14523513C22semidirect
16. C22×D524C22semidirect
20. C2×#14523501C22semidirect
22. C2×#14523498C22semidirect
23. C2×#14523498C22semidirect
24. C2×#14523497C22semidirect
25. C22×C131⋊C8C22semidirect
26. #16862169C22semidirect
27. #16862166C22semidirect
28. #16862166C22semidirect
29. C22⋊(C131⋊C8)C22semidirect
31. C22×D4×C131C22semidirect
34. #121×C131C22semidirect
36. #117×C131C22semidirect
43. #14523513D4semidirect
50. C2×D524D4semidirect
151. C131#3008semidirect
156. #535D131semidirect
172. C22×D4C2×D131semidirect
182. #121C2×D131semidirect
184. #117C2×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622922
C2×#168622722
C22×#145234982
#225534552
#225534452
#225533162
#225533062
#225532792
#535×C1312
#3008131