Finite Group Databasebeta

← #30676409

Subgroups — #30676409

Order 16768.

656 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 153 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×#16862274C211yesmaximal, characteristic, complemented
2C2×#16862272C211yesmaximal, characteristic, complemented
3C2×#16862271C211yesmaximal, characteristic, complemented
4#22553858C211yesmaximal, characteristic, complemented
5C2×#16862242C211yesmaximal, characteristic, complemented
6#22553700C212yesmaximal, complemented
7#22553698C212yesmaximal, complemented
8#22553687C212yesmaximal, complemented
9#22553683C212yesmaximal, complemented
10M16×Dic131C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C131C211yesmaximal, characteristic, complemented
12C2×#14523513C2211yescharacteristic
13C22×D524C2211yescharacteristic
14C22×C22⋊D131—22no—
15#16862274—24no—
16#16862272—24no—
17#16862271—24no—
18C2×#14523502C2212yescomplemented
19C2×#14523501C2212yescomplemented
20C2×#14523500C2212yescomplemented
21C2×D1048C2212yescomplemented
22C2×#14523498C2212yescomplemented
23C2×#14523498C2212yescomplemented
24C2×#14523497C2211yescharacteristic
25#16862244C2211yescharacteristic, complemented
26#16862242—22no—
27#16862242C2212yes—
28#16862242C2212yes—
29#16862242C2211yescharacteristic, complemented
30#16862235C2211yescharacteristic, complemented
31#16862233C2211yescharacteristic, complemented
32#16862218—22no—
33C4×C22⋊D131—22no—
34C2×C4×Dic131C2211yescharacteristic
35#16862156C2212yescomplemented
36C8×Dic131C2212yescomplemented
37C2×#41×C131C2211yescharacteristic
38C22×D4×C131C2211yescharacteristic
39AGL(1,ℤ/8)×C131—24no—
40C2×SD16×C131C2212yescomplemented
41C2×D8×C131C2212yescomplemented
42C2×M16×C131C2211yescharacteristic
43C23×D131—21no—
44#14523513—22no—
45#14523513—22no—
46C2×Dic262C2311yescharacteristic
47C2×D524—22no—
48C2×D524—22no—
49C2×D524—21no—
50C2×D524C2311yescharacteristic
51C2×D524C2312yes—
52C2×C22⋊D131—22no—
53C2×C22⋊D131—24no—
54C2×C22⋊D131—24no—
55C2×C22⋊D131—21no—
56C2×C22⋊D131—48no—
57C22×Dic131D412yes—
58C2×C4×D131—21no—
59#14523502—24no—
60#14523501—24no—
61#14523500—24no—
62D1048—24no—
63#14523498—24no—
64#14523498—24no—
65#14523497D414yescomplemented
66C2×C131⋊C8C2312yes—
67Dic131⋊C4—22no—
68C22⋊Dic131—22no—
69(C2×D131)⋊C4—22no—
70C4×Dic131D412yes—
71C4×Dic131C2311yescharacteristic
72C4×Dic131C2311yescharacteristic
73C24×C131—21no—
74#41×C131—22no—
75#41×C131—22no—
76C2×Q8×C131C2311yescharacteristic
77C2×D4×C131—21no—
78C2×D4×C131—22no—
79C2×D4×C131—22no—
80C2×D4×C131—21no—
81C2×D4×C131C2311yescharacteristic
82C2×D4×C131C2312yes—
83C22×C4×C131—21no—
84C22×C4×C131C2311yescharacteristic
85SD16×C131—24no—
86D8×C131—24no—
87M16×C131D414yescomplemented
88C2×C8×C131C2312yes—
89C22×D131—21no—
90C22×D131—22no—
91C22×D131—44no—
92C22×D131—42no—
93Dic262—21no—
94Dic262—21no—
95D524—21no—
96D524—22no—
97D524—21no—
98D524—22no—
99D524—42no—
100C22⋊D131—48no—
101C22⋊D131—42no—
102C22⋊D131—44no—
103C22⋊D131—48no—
104C2×Dic131—24no—
105C2×Dic131—21no—
106C2×Dic131C2×D412yes—
107C2×Dic131C2×D412yes—
108C4×D131—42no—
109C131⋊C8C2×D414yescomplemented
110C23×C131—21no—
111C23×C131—22no—
112C23×C131—44no—
113C23×C131—42no—
114C23×C131C2×D411yescharacteristic
115Q8×C131—21no—
116Q8×C131—21no—
117D4×C131—21no—
118D4×C131—22no—
119D4×C131—22no—
120D4×C131—21no—
121D4×C131—42no—
122D4×C131—42no—
123C2×C4×C131—21no—
124C2×C4×C131—42no—
125C2×C4×C131C2×D411yescharacteristic
126C2×C4×C131C2×D412yes—
127C2×C4×C131C2×D412yes—
128C2×C4×C131C2411yescharacteristic
129C8×C131C2×D414yescomplemented
130C22×C131—22no—
131C22×C131—41no—
132C22×C131—42no—
133C22×C131—44no—
134C22×C131—42no—
135C22×C131—84no—
136C22×C131#12712yes—
137C22×C131C22×D411yescharacteristic
138C2×D131—41no—
139C2×D131—42no—
140C2×D131—44no—
141C2×D131—42no—
142C2×D131—84no—
143C4×C131—41no—
144C4×C131#12712yes—
145C4×C131C22×D411yescharacteristic
146C4×C131C22×D411yescharacteristic
147Dic131—22no—
148Dic131—41no—
149Dic131AGL(1,ℤ/8)14yescomplemented
150C2×C131—22no—
151C2×C131—81no—
152C2×C131—82no—
153C2×C131C2×#12711yescharacteristic
154C2×C131C2×AGL(1,ℤ/8)12yes—
155D131—81no—
156D131—82no—
157C131#315111yescharacteristic, complemented, Sylow
158#3151—1311nomaximal, Sylow
159C2×AGL(1,ℤ/8)—1311no—
160C2×AGL(1,ℤ/8)—1311no—
161C2×AGL(1,ℤ/8)—1311no—
162C2×AGL(1,ℤ/8)D13111yescharacteristic, complemented
163#532—1311no—
164C2×#127—1311no—
165#496—1312no—
166#496—1312no—
167#493—1312no—
168#492—1312no—
169C4×M16—1311no—
170C2×#41—1311no—
171C2×#41C2×D13111yescharacteristic
172C22×D4—2622no—
173C22×D4—1311no—
174C22×D4C2×D13111yescharacteristic
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—2624no—
177AGL(1,ℤ/8)—2624no—
178AGL(1,ℤ/8)—2624no—
179C2×SD16—1312no—
180C2×SD16—1312no—
181C2×SD16—1312no—
182C2×SD16C2×D13112yescomplemented
183C2×D8—1312no—
184C2×D8—1312no—
185C2×D8—1312no—
186C2×D8C2×D13112yescomplemented
187C2×M16—1311no—
188C2×M16C2×D13111yescharacteristic
189C4⋊Q8—1311no—
190#127—2622no—
191#127—1312no—
192#127—1312no—
193#127—1311no—
194#124—1311no—
195#124—1311no—
196#121—2622no—
197C4×D4—2622no—
198C2×C42—1311no—
199#97—1312no—
200C4×C8—1312no—
201C24—21no—
202C24—2621no—
203#41—22no—
204#41—22no—
205#41—2622no—
206#41—2622no—
207C2×Q8—1311no—
208C2×Q8C22×D13111yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—2621no—
214C2×D4—2622no—
215C2×D4—2624no—
216C2×D4—2624no—
217C2×D4—2622no—
218C2×D4—2621no—
219C2×D4—2622no—
220C2×D4—5248no—
221C2×D4—1311no—
222C2×D4—1312no—
223C2×D4C22×D13111yescharacteristic
224C2×D4C22×D13112yes—
225C22×C4—21no—
226C22×C4—2621no—
227C22×C4—1312no—
228C22×C4C22×D13111yescharacteristic
229SD16—24no—
230SD16—2624no—
231SD16—2624no—
232SD16—2624no—
233D8—24no—
234D8—2624no—
235D8—2624no—
236D8—2624no—
237M16—1314no—
238M16C22⋊D13114yescomplemented
239C2×C8—1312no—
240C2×C8C22×D13112yes—
241C4⋊C4—2622no—
242C22⋊C4—2622no—
243C22⋊C4—2622no—
244C42—1312no—
245C42—1311no—
246C42—1311no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—2621no—
252C23—2622no—
253C23—5244no—
254C23—5242no—
255C23D4×D13111yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—2621no—
259Q8—2621no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—2621no—
267D4—2622no—
268D4—2621no—
269D4—2622no—
270D4—5248no—
271D4—5242no—
272D4—5248no—
273D4—5244no—
274D4—5242no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—2624no—
278C2×C4—2621no—
279C2×C4—5242no—
280C2×C4—1312no—
281C2×C4—1312no—
282C2×C4D4×D13111yescharacteristic
283C2×C4D4×D13112yes—
284C2×C4C2×C22⋊D13112yes—
285C2×C4C23×D13111yescharacteristic
286C8—1314no—
287C8C2×C22⋊D13114yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—5244no—
295C22—5242no—
296C22—5241no—
297C22—5242no—
298C22—10484no—
299C22#1686224212yes—
300C22C2×D4×D13111yescharacteristic
301C4—41no—
302C4—2622no—
303C4—5241no—
304C4—1314no—
305C4#1686224212yes—
306C4C2×D4×D13111yescharacteristic
307C4C22×C22⋊D13111yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—10482no—
312C2—10481no—
313C2C2×#1686224211yescharacteristic
314C2AGL(1,ℤ/8)×D13112yes—
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC2×AGL(1,ℤ/8)×C131C2
socleC22×C131C22×D4
hypercenterC2×AGL(1,ℤ/8)D131
hypocenterC131#3151
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16862274C2semidirect
2. C2×#16862272C2semidirect
3. C2×#16862271C2semidirect
4. #22553858C2semidirect
5. C2×#16862242C2semidirect
6. #22553700C2semidirect
7. #22553698C2semidirect
8. #22553687C2semidirect
9. #22553683C2semidirect
10. M16×Dic131C2semidirect
11. C2×AGL(1,ℤ/8)×C131C2semidirect
18. C2×#14523502C22semidirect
19. C2×#14523501C22semidirect
20. C2×#14523500C22semidirect
21. C2×D1048C22semidirect
22. C2×#14523498C22semidirect
23. C2×#14523498C22semidirect
25. #16862244C22semidirect
29. #16862242C22semidirect
30. #16862235C22semidirect
31. #16862233C22semidirect
35. #16862156C22semidirect
36. C8×Dic131C22semidirect
40. C2×SD16×C131C22semidirect
41. C2×D8×C131C22semidirect
65. #14523497D4semidirect
87. M16×C131D4semidirect
109. C131⋊C8C2×D4semidirect
129. C8×C131C2×D4semidirect
149. Dic131AGL(1,ℤ/8)semidirect
157. C131#3151semidirect
162. C2×AGL(1,ℤ/8)D131semidirect
182. C2×SD16C2×D131semidirect
186. C2×D8C2×D131semidirect
238. M16C22⋊D131semidirect
287. C8C2×C22⋊D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622742
C2×#168622722
C2×#168622712
#225538582
C2×#168622422
#225537002
#225536982
#225536872
#225536832
M16×Dic1312
C2×AGL(1,ℤ/8)×C1312
#3151131