Finite Group Databasebeta

← #30675243

Subgroups — #30675243

Order 16768.

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

normal subgroups only

This group has 732 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×#16862294C211yesmaximal, characteristic, complemented
2C2×#16862272C211yesmaximal, characteristic, complemented
3#22553879C211yesmaximal, characteristic, complemented
4#22553456C214yesmaximal, complemented
5#22553449C214yesmaximal, complemented
6#22553321C211yesmaximal, characteristic, complemented
7#22553309C211yesmaximal, characteristic, complemented
8#22553278C211yesmaximal, characteristic, complemented
9C22×SD16×C131C211yesmaximal, characteristic, complemented
10#16862294—24no—
11#16862294—24no—
12C2×#14523514—21no—
13C2×#14523514C2212yescomplemented
14C2×#14523513—22no—
15C2×#14523513C2211yescharacteristic
16C2×#14523513C2211yescharacteristic
17C2×Q8×D131—21no—
18C2×Q8×D131C2212yescomplemented
19#16862272—24no—
20C2×#14523501C2212yes—
21C2×#14523498C2212yes—
22C2×#14523497C2211yescharacteristic
23#16862242—21no—
24#16862233—21no—
25#16862217C2214yes—
26#16862215—22no—
27#16862205C2211yescharacteristic
28#16862171C2212yescomplemented
29#16862170C2214yescomplemented
30#16862168C2212yescomplemented
31(C2×D131)⋊C8C2214yescomplemented
32C22×Q8×C131C2211yescharacteristic
33C22×D4×C131C2211yescharacteristic
34C2×SD16×C131—24no—
35C2×SD16×C131C2214yes—
36C22×C8×C131C2211yescharacteristic
37#14523514—28no—
38#14523514—24no—
39#14523514—24no—
40#14523513—216no—
41#14523513—22no—
42#14523513—22no—
43#14523513—22no—
44#14523513D414yescomplemented
45C2×Dic262—22no—
46C2×Dic262D411yescharacteristic
47C2×Dic262C2311yescharacteristic
48Q8×D131—28no—
49Q8×D131—24no—
50Q8×D131—24no—
51C2×D524—22no—
52C2×D524D411yescharacteristic
53C2×D524C2311yescharacteristic
54C2×C22⋊D131—24no—
55C2×C22⋊D131—21no—
56C2×C22⋊D131D412yes—
57C2×C4×D131—22no—
58C2×C4×D131—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—
70C2×Q8×C131—21no—
71C2×Q8×C131—22no—
72C2×Q8×C131D414yescomplemented
73C2×Q8×C131C2312yes—
74C2×D4×C131—22no—
75C2×D4×C131—22no—
76C2×D4×C131—21no—
77C2×D4×C131C2312yes—
78C22×C4×C131—21no—
79C22×C4×C131C2311yescharacteristic
80SD16×C131—28no—
81C2×C8×C131—22no—
82C2×C8×C131C2312yes—
83C22×D131—21no—
84C22×D131C2×D412yes—
85Dic262—28no—
86Dic262—22no—
87Dic262—21no—
88Dic262C2×D412yes—
89D524—28no—
90D524—22no—
91D524—21no—
92D524C2×D412yes—
93C22⋊D131—24no—
94C22⋊D131—24no—
95C22⋊D131—84no—
96C2×Dic131—21no—
97C2×Dic131—22no—
98C2×Dic131C2×D412yes—
99C4×D131—28no—
100C4×D131—28no—
101C4×D131—24no—
102C4×D131—22no—
103C4×D131—22no—
104C131⋊C8—22no—
105C23×C131—22no—
106C23×C131—42no—
107C23×C131—82no—
108C23×C131C2×D411yescharacteristic
109Q8×C131—24no—
110Q8×C131—22no—
111Q8×C131C2×D414yes—
112D4×C131—22no—
113D4×C131—22no—
114D4×C131—42no—
115C2×C4×C131—24no—
116C2×C4×C131—22no—
117C2×C4×C131C2×D411yescharacteristic
118C2×C4×C131C2×D412yes—
119C2×C4×C131C2×D412yes—
120C2×C4×C131C2411yescharacteristic
121C8×C131—22no—
122C22×C131—22no—
123C22×C131—42no—
124C22×C131—44no—
125C22×C131—84no—
126C22×C131#12012yes—
127C22×C131C22×D411yescharacteristic
128C2×D131—22no—
129C2×D131—22no—
130C2×D131—24no—
131C2×D131—42no—
132C4×C131—24no—
133C4×C131#12012yes—
134C4×C131C22×D411yescharacteristic
135C4×C131C22×D411yescharacteristic
136Dic131—22no—
137Dic131—22no—
138Dic131—42no—
139C2×C131—22no—
140C2×C131—82no—
141C2×C131C2×#12011yescharacteristic
142C2×C131#57712yes—
143D131—42no—
144D131—42no—
145C131#301011yescharacteristic, complemented, Sylow
146#3010—1311nomaximal, Sylow
147C2×2−1+4—1311no—
148C2×AGL(1,ℤ/8)—1311no—
149C22×SD16D13111yescharacteristic, complemented
150#535—1311no—
151#449—1314no—
152#448—1314no—
153#419—1311no—
154#416—1311no—
155#408—1311no—
1562−1+4—2624no—
1572−1+4—2624no—
158C2×#41—2622no—
159C2×#41—2621no—
160C2×#41—1312no—
161C2×#41—1311no—
162C2×#41—1311no—
163C22×Q8—2621no—
164C22×Q8—1312no—
165C22×Q8C2×D13111yescharacteristic
166C22×D4C2×D13111yescharacteristic
167AGL(1,ℤ/8)—2624no—
168C2×SD16—24no—
169C2×SD16—1312no—
170C2×SD16C2×D13114yes—
171C2×D8—1312no—
172C2×M16—1311no—
173C22×C8C2×D13111yescharacteristic
174#127—2621no—
175#124—2621no—
176#121—1314no—
177#120—2622no—
178#117—1311no—
179#103—1314no—
180#103—1312no—
181#102—1312no—
182C22⋊C8—1314no—
183C24—21no—
184#41—26216no—
185#41—2628no—
186#41—2624no—
187#41—2624no—
188#41—2622no—
189#41—2622no—
190#41—2622no—
191#41—1314no—
192C2×Q8—21no—
193C2×Q8—22no—
194C2×Q8—2622no—
195C2×Q8—2628no—
196C2×Q8—2624no—
197C2×Q8—2624no—
198C2×Q8—1311no—
199C2×Q8—1311no—
200C2×Q8C22⋊D13114yescomplemented
201C2×Q8C22×D13112yes—
202C2×D4—22no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—2622no—
206C2×D4—2621no—
207C2×D4—2624no—
208C2×D4—1312no—
209C2×D4—1311no—
210C2×D4—1311no—
211C2×D4C22×D13112yes—
212C22×C4—21no—
213C22×C4—2622no—
214C22×C4—2622no—
215C22×C4—2621no—
216C22×C4—1312no—
217C22×C4C22×D13111yescharacteristic
218SD16—28no—
219SD16—2624no—
220D8—2624no—
221M16—2622no—
222C2×C8—22no—
223C2×C8—1312no—
224C2×C8C22×D13112yes—
225C4⋊C4—1312no—
226C22⋊C4—2621no—
227C22⋊C4—2624no—
228C42—2621no—
229C23—22no—
230C23—42no—
231C23—2621no—
232C23—82no—
233C23—1312no—
234C23D4×D13111yescharacteristic
235Q8—24no—
236Q8—22no—
237Q8—2628no—
238Q8—2622no—
239Q8—2621no—
240Q8—1312no—
241Q8C2×C22⋊D13114yes—
242D4—22no—
243D4—22no—
244D4—42no—
245D4—2628no—
246D4—2624no—
247D4—2624no—
248D4—2622no—
249D4—2621no—
250D4—10484no—
251D4—1312no—
252C2×C4—24no—
253C2×C4—22no—
254C2×C4—2628no—
255C2×C4—2628no—
256C2×C4—2621no—
257C2×C4—2624no—
258C2×C4—2622no—
259C2×C4—2622no—
260C2×C4—2622no—
261C2×C4—1312no—
262C2×C4D4×D13111yescharacteristic
263C2×C4D4×D13112yes—
264C2×C4C2×C22⋊D13112yes—
265C2×C4C23×D13111yescharacteristic
266C8—22no—
267C8—2622no—
268C22—22no—
269C22—44no—
270C22—42no—
271C22—2624no—
272C22—2622no—
273C22—2622no—
274C22—84no—
275C22—5242no—
276C22#1686221412yes—
277C22C2×D4×D13111yescharacteristic
278C4—24no—
279C4—2622no—
280C4—2622no—
281C4—5242no—
282C4#1686221412yes—
283C4C2×D4×D13111yescharacteristic
284C4C22×C22⋊D13111yescharacteristic
285C2—22no—
286C2—82no—
287C2—5242no—
288C2—5242no—
289C2C2×#1686221411yescharacteristic
290C2#2255411812yes—
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC22×SD16×C131C2
socleC22×C131C22×D4
hypercenterC22×SD16D131
hypocenterC131#3010
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#16862294C2semidirect
2. C2×#16862272C2semidirect
3. #22553879C2semidirect
4. #22553456C2semidirect
5. #22553449C2semidirect
6. #22553321C2semidirect
7. #22553309C2semidirect
8. #22553278C2semidirect
9. C22×SD16×C131C2semidirect
13. C2×#14523514C22semidirect
18. C2×Q8×D131C22semidirect
28. #16862171C22semidirect
29. #16862170C22semidirect
30. #16862168C22semidirect
31. (C2×D131)⋊C8C22semidirect
44. #14523513D4semidirect
72. C2×Q8×C131D4semidirect
145. C131#3010semidirect
149. C22×SD16D131semidirect
200. C2×Q8C22⋊D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622942
C2×#168622722
#225538792
#225534562
#225534492
#225533212
#225533092
#225532782
C22×SD16×C1312
#3010131