Finite Group Databasebeta

← #30676446

Subgroups — #30676446

Order 16768.

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

normal subgroups only

This group has 596 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×#16862274C211yesmaximal, characteristic, complemented
2C2×#16862266C211yesmaximal, characteristic, complemented
3C22×#14523498C211yesmaximal, characteristic, complemented
4#22553879C211yesmaximal, characteristic, complemented
5#22553876C211yesmaximal, characteristic, complemented
6#22553700C212yesmaximal, complemented
7#22553698C212yesmaximal, complemented
8#22553693C212yesmaximal, complemented
9#22553687C212yesmaximal, complemented
10#22553270C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C131C211yesmaximal, characteristic, complemented
12C2×#14523513C2211yescharacteristic
13C22×D524C2211yescharacteristic
14#16862274—24no—
15#16862266—24no—
16C2×#14523503C2211yescharacteristic, complemented
17C2×#14523502C2212yescomplemented
18C2×#14523501C2211yescharacteristic, complemented
19C2×#14523500C2211yescharacteristic, complemented
20C2×D1048C2212yescomplemented
21C2×#14523498—24no—
22C2×#14523498C2212yes—
23C2×#14523498C2212yes—
24C2×#14523498C2211yescharacteristic, complemented
25C22×C131⋊C8C2211yescharacteristic
26#16862242C2212yescomplemented
27#16862242C2212yescomplemented
28#16862235C2212yescomplemented
29#16862233C2212yescomplemented
30#16862220—22no—
31#16862217—22no—
32#16862215—22no—
33#16862214—22no—
34#16862205C2211yescharacteristic
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×D524C2312yes—
51C2×D524C2311yescharacteristic
52C2×C22⋊D131—24no—
53C2×C22⋊D131—24no—
54C2×C22⋊D131—21no—
55C2×C4×D131—21no—
56#14523503—22no—
57#14523502—24no—
58#14523501—22no—
59#14523500—22no—
60D1048—24no—
61#14523498—24no—
62#14523498—24no—
63#14523498—22no—
64C2×C131⋊C8D414yescomplemented
65C2×C131⋊C8C2311yescharacteristic
66C2×C131⋊C8C2311yescharacteristic
67C4⋊Dic131D412yes—
68C22⋊Dic131—24no—
69C22⋊Dic131D412yes—
70(C2×D131)⋊C4—24no—
71C4×Dic131C2312yes—
72C24×C131—21no—
73#41×C131—22no—
74#41×C131—22no—
75C2×Q8×C131C2311yescharacteristic
76C2×D4×C131—21no—
77C2×D4×C131—22no—
78C2×D4×C131—22no—
79C2×D4×C131—21no—
80C2×D4×C131C2312yes—
81C2×D4×C131C2311yescharacteristic
82C22×C4×C131—21no—
83C22×C4×C131C2311yescharacteristic
84SD16×C131—24no—
85D8×C131—24no—
86M16×C131D414yescomplemented
87C2×C8×C131C2312yes—
88C22×D131—22no—
89C22×D131—21no—
90C22×D131—42no—
91C22×D131—82no—
92Dic262—21no—
93Dic262—21no—
94D524—22no—
95D524—21no—
96D524—21no—
97D524—22no—
98D524—42no—
99C22⋊D131—42no—
100C22⋊D131—84no—
101C22⋊D131—84no—
102C2×Dic131—21no—
103C2×Dic131C2×D414yes—
104C4×D131—42no—
105C131⋊C8C2×D412yes—
106C131⋊C8C2×D412yes—
107C23×C131—21no—
108C23×C131—22no—
109C23×C131—42no—
110C23×C131—82no—
111C23×C131C2×D411yescharacteristic
112Q8×C131—21no—
113Q8×C131—21no—
114D4×C131—21no—
115D4×C131—22no—
116D4×C131—22no—
117D4×C131—21no—
118D4×C131—42no—
119D4×C131—42no—
120C2×C4×C131—21no—
121C2×C4×C131—42no—
122C2×C4×C131C2×D411yescharacteristic
123C2×C4×C131C2×D412yes—
124C2×C4×C131C2×D412yes—
125C2×C4×C131C2411yescharacteristic
126C8×C131C2×D414yes—
127C22×C131—22no—
128C22×C131—41no—
129C22×C131—42no—
130C22×C131—44no—
131C22×C131—42no—
132C22×C131—84no—
133C22×C131#12712yes—
134C22×C131C22×D411yescharacteristic
135C2×D131—41no—
136C2×D131—42no—
137C2×D131—42no—
138C2×D131—44no—
139C2×D131—84no—
140C4×C131—41no—
141C4×C131#12712yes—
142C4×C131C22×D411yescharacteristic
143C4×C131C22×D411yescharacteristic
144Dic131—24no—
145Dic131—41no—
146C2×C131—22no—
147C2×C131—81no—
148C2×C131—82no—
149C2×C131C2×#12711yescharacteristic
150C2×C131#57612yes—
151D131—81no—
152D131—82no—
153C131#315511yescharacteristic, complemented, Sylow
154#3155—1311nomaximal, Sylow
155C2×AGL(1,ℤ/8)—1311no—
156C2×AGL(1,ℤ/8)D13111yescharacteristic, complemented
157C2×#135—1311no—
158C22×D8—1311no—
159#535—1311no—
160#535—1311no—
161#496—1312no—
162#496—1312no—
163#495—1312no—
164#493—1312no—
165#405—1311no—
166C2×#41—1311no—
167C2×#41C2×D13111yescharacteristic
168C22×D4—1311no—
169C22×D4C2×D13111yescharacteristic
170AGL(1,ℤ/8)—24no—
171AGL(1,ℤ/8)—2624no—
172#135—2624no—
173C2×Q16—1311no—
174C2×SD16—1311no—
175C2×SD16—1311no—
176C2×SD16—1312no—
177C2×SD16C2×D13112yescomplemented
178C2×D8—2624no—
179C2×D8—1312no—
180C2×D8—1312no—
181C2×D8—1311no—
182C2×D8—1312no—
183C2×D8C2×D13112yescomplemented
184C2×M16C2×D13111yescharacteristic
185C22×C8—1311no—
186#127—1312no—
187#127—1312no—
188#124—1312no—
189#124—1312no—
190#121—2622no—
191#121—2622no—
192#120—2622no—
193#120—2622no—
194#117—1311no—
195#97—1312no—
196C4×C8—1312no—
197C24—21no—
198C24—2621no—
199#41—22no—
200#41—22no—
201#41—2622no—
202#41—2622no—
203C2×Q8—1311no—
204C2×Q8C22×D13111yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—2621no—
210C2×D4—2624no—
211C2×D4—2624no—
212C2×D4—2622no—
213C2×D4—2621no—
214C2×D4—2622no—
215C2×D4—1312no—
216C2×D4—1311no—
217C2×D4C22×D13112yes—
218C2×D4C22×D13111yescharacteristic
219C22×C4—21no—
220C22×C4—2621no—
221C22×C4C22×D13111yescharacteristic
222Q16—2622no—
223SD16—24no—
224SD16—2622no—
225SD16—2622no—
226SD16—2624no—
227D8—24no—
228D8—2624no—
229D8—2624no—
230D8—2622no—
231D8—2624no—
232M16C22⋊D13114yescomplemented
233C2×C8—1314no—
234C2×C8—1311no—
235C2×C8—1311no—
236C2×C8C22×D13112yes—
237C4⋊C4—1312no—
238C22⋊C4—2624no—
239C22⋊C4—2624no—
240C22⋊C4—1312no—
241C42—1312no—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—2622no—
246C23—2621no—
247C23—82no—
248C23—5242no—
249C23—10482no—
250C23D4×D13111yescharacteristic
251Q8—21no—
252Q8—21no—
253Q8—2621no—
254Q8—2621no—
255D4—22no—
256D4—21no—
257D4—21no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—2622no—
262D4—2621no—
263D4—2621no—
264D4—2622no—
265D4—5242no—
266D4—5242no—
267D4—10484no—
268D4—10484no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—2621no—
272C2×C4—5242no—
273C2×C4—1314no—
274C2×C4D4×D13111yescharacteristic
275C2×C4D4×D13112yes—
276C2×C4C2×C22⋊D13112yes—
277C2×C4C23×D13111yescharacteristic
278C8—1312no—
279C8—1312no—
280C8C2×C22⋊D13114yes—
281C22—22no—
282C22—42no—
283C22—44no—
284C22—41no—
285C22—42no—
286C22—84no—
287C22—5242no—
288C22—5244no—
289C22—5241no—
290C22—5242no—
291C22—10484no—
292C22#1686224212yes—
293C22C2×D4×D13111yescharacteristic
294C4—41no—
295C4—2624no—
296C4—5241no—
297C4#1686224212yes—
298C4C2×D4×D13111yescharacteristic
299C4C22×C22⋊D13111yescharacteristic
300C2—22no—
301C2—82no—
302C2—81no—
303C2—10482no—
304C2—10481no—
305C2C2×#1686224211yescharacteristic
306C2#2255411612yes—
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#3155
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×#16862266C2semidirect
3. C22×#14523498C2semidirect
4. #22553879C2semidirect
5. #22553876C2semidirect
6. #22553700C2semidirect
7. #22553698C2semidirect
8. #22553693C2semidirect
9. #22553687C2semidirect
10. #22553270C2semidirect
11. C2×AGL(1,ℤ/8)×C131C2semidirect
16. C2×#14523503C22semidirect
17. C2×#14523502C22semidirect
18. C2×#14523501C22semidirect
19. C2×#14523500C22semidirect
20. C2×D1048C22semidirect
24. C2×#14523498C22semidirect
26. #16862242C22semidirect
27. #16862242C22semidirect
28. #16862235C22semidirect
29. #16862233C22semidirect
35. #16862156C22semidirect
36. C8×Dic131C22semidirect
40. C2×SD16×C131C22semidirect
41. C2×D8×C131C22semidirect
64. C2×C131⋊C8D4semidirect
86. M16×C131D4semidirect
153. C131#3155semidirect
156. C2×AGL(1,ℤ/8)D131semidirect
177. C2×SD16C2×D131semidirect
183. C2×D8C2×D131semidirect
232. M16C22⋊D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622742
C2×#168622662
C22×#145234982
#225538792
#225538762
#225537002
#225536982
#225536932
#225536872
#225532702
C2×AGL(1,ℤ/8)×C1312
#3155131