Finite Group Databasebeta

← #30676440

Subgroups — #30676440

Order 16768.

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

normal subgroups only

This group has 564 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×#14523500C211yesmaximal, characteristic, complemented
4#22553886C211yesmaximal, characteristic, complemented
5#22553876C211yesmaximal, characteristic, complemented
6#22553705C212yesmaximal, complemented
7#22553698C212yesmaximal, complemented
8#22553697C212yesmaximal, complemented
9#22553683C212yesmaximal, complemented
10#22553270C211yesmaximal, characteristic, complemented
11C2×#137×C131C211yesmaximal, characteristic, complemented
12C2×#14523513C2211yescharacteristic
13C22×D524C2211yescharacteristic
14#16862274—24no—
15#16862266—24no—
16C2×#14523503C2211yescharacteristic, complemented
17C2×#14523502C2212yes—
18C2×#14523501C2211yescharacteristic, complemented
19C2×#14523500—24no—
20C2×#14523500C2212yes—
21C2×#14523500C2212yes—
22C2×#14523500C2211yescharacteristic, complemented
23C2×D1048C2212yes—
24C2×#14523498C2211yescharacteristic, complemented
25C22×C131⋊C8C2211yescharacteristic
26#16862244C2212yescomplemented
27#16862242C2212yes—
28#16862235C2212yes—
29#16862235C2212yescomplemented
30#16862229—22no—
31#16862222—22no—
32#16862220—22no—
33#16862214—22no—
34#16862205C2211yescharacteristic
35#16862156C2212yescomplemented
36C8×Dic131C2212yescomplemented
37C2×#41×C131C2211yescharacteristic
38C22×Q8×C131C2211yescharacteristic
39#137×C131—24no—
40C2×Q16×C131C2212yescomplemented
41C2×SD16×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—21no—
54C2×C4×D131—21no—
55#14523503—22no—
56#14523502—24no—
57#14523501—22no—
58#14523500—24no—
59#14523500—24no—
60#14523500—22no—
61D1048—24no—
62#14523498—22no—
63C2×C131⋊C8D414yescomplemented
64C2×C131⋊C8C2311yescharacteristic
65C2×C131⋊C8C2311yescharacteristic
66C4⋊Dic131D412yes—
67Dic131⋊C4—24no—
68C22⋊Dic131D412yes—
69(C2×D131)⋊C4—24no—
70(C2×D131)⋊C4—24no—
71C4×Dic131C2312yes—
72#41×C131—22no—
73#41×C131—22no—
74C2×Q8×C131—21no—
75C2×Q8×C131—22no—
76C2×Q8×C131—22no—
77C2×Q8×C131C2311yescharacteristic
78C2×Q8×C131C2312yes—
79C2×D4×C131—21no—
80C2×D4×C131C2311yescharacteristic
81C22×C4×C131—21no—
82C22×C4×C131—21no—
83C22×C4×C131C2311yescharacteristic
84Q16×C131—24no—
85SD16×C131—24no—
86M16×C131D414yescomplemented
87C2×C8×C131C2312yes—
88C22×D131—21no—
89C22×D131—22no—
90C22×D131—42no—
91C22×D131—82no—
92Dic262—21no—
93Dic262—21no—
94D524—21no—
95D524—21no—
96D524—22no—
97D524—22no—
98D524—42no—
99C22⋊D131—42no—
100C22⋊D131—84no—
101C2×Dic131—21no—
102C2×Dic131C2×D414yes—
103C4×D131—42no—
104C131⋊C8C2×D412yes—
105C131⋊C8C2×D412yes—
106C23×C131—21no—
107C23×C131C2×D411yescharacteristic
108Q8×C131—22no—
109Q8×C131—21no—
110Q8×C131—22no—
111Q8×C131—21no—
112Q8×C131—42no—
113D4×C131—21no—
114D4×C131—21no—
115D4×C131—42no—
116C2×C4×C131—21no—
117C2×C4×C131—22no—
118C2×C4×C131—42no—
119C2×C4×C131—42no—
120C2×C4×C131C2×D411yescharacteristic
121C2×C4×C131C2×D412yes—
122C2×C4×C131C2×D412yes—
123C2×C4×C131C2411yescharacteristic
124C8×C131C2×D414yes—
125C22×C131—22no—
126C22×C131—41no—
127C22×C131—42no—
128C22×C131#12712yes—
129C22×C131C22×D411yescharacteristic
130C2×D131—41no—
131C2×D131—42no—
132C2×D131—44no—
133C2×D131—42no—
134C2×D131—84no—
135C4×C131—41no—
136C4×C131—42no—
137C4×C131#12712yes—
138C4×C131C22×D411yescharacteristic
139C4×C131C22×D411yescharacteristic
140Dic131—24no—
141Dic131—41no—
142C2×C131—22no—
143C2×C131—81no—
144C2×C131C2×#12711yescharacteristic
145C2×C131#57712yes—
146D131—81no—
147D131—82no—
148C131#315411yescharacteristic, complemented, Sylow
149#3154—1311nomaximal, Sylow
150C2×#137D13111yescharacteristic, complemented
151C2×AGL(1,ℤ/8)—1311no—
152C2×#135—1311no—
153C22×SD16—1311no—
154#536—1311no—
155#535—1311no—
156#497—1312no—
157#496—1312no—
158#495—1312no—
159#492—1312no—
160#405—1311no—
161C2×#41—1311no—
162C2×#41C2×D13111yescharacteristic
163C22×Q8C2×D13111yescharacteristic
164C22×D4—1311no—
165#137—24no—
166AGL(1,ℤ/8)—2624no—
167#135—2624no—
168C2×Q16—1311no—
169C2×Q16C2×D13112yescomplemented
170C2×SD16—2624no—
171C2×SD16—1312no—
172C2×SD16—1312no—
173C2×SD16—1311no—
174C2×SD16—1311no—
175C2×SD16—1312no—
176C2×SD16C2×D13112yescomplemented
177C2×D8—1311no—
178C2×D8—1312no—
179C2×M16C2×D13111yescharacteristic
180C22×C8—1311no—
181C4⋊Q8—1312no—
182#127—1312no—
183#124—1312no—
184#124—1312no—
185#123—2622no—
186C22⋊Q8—2622no—
187#121—2622no—
188#120—2622no—
189#117—1311no—
190#97—1312no—
191C4×C8—1312no—
192C24—2621no—
193#41—22no—
194#41—22no—
195#41—2622no—
196#41—2622no—
197C2×Q8—21no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—1311no—
201C2×Q8C22×D13111yescharacteristic
202C2×Q8C22×D13112yes—
203C2×D4—21no—
204C2×D4—2621no—
205C2×D4—2624no—
206C2×D4—2622no—
207C2×D4—2621no—
208C2×D4—2622no—
209C2×D4—1311no—
210C2×D4—1312no—
211C2×D4C22×D13111yescharacteristic
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—2621no—
215C22×C4C22×D13111yescharacteristic
216Q16—24no—
217Q16—2622no—
218SD16—24no—
219SD16—2622no—
220SD16—2624no—
221SD16—2624no—
222SD16—2622no—
223SD16—2624no—
224D8—2622no—
225D8—2624no—
226M16C22⋊D13114yescomplemented
227C2×C8—1314no—
228C2×C8—1311no—
229C2×C8—1311no—
230C2×C8C22×D13112yes—
231C4⋊C4—2624no—
232C4⋊C4—1312no—
233C22⋊C4—2624no—
234C22⋊C4—2624no—
235C22⋊C4—1312no—
236C42—1312no—
237C23—21no—
238C23—2621no—
239C23—2622no—
240C23—5242no—
241C23—10482no—
242C23D4×D13111yescharacteristic
243Q8—22no—
244Q8—21no—
245Q8—22no—
246Q8—21no—
247Q8—42no—
248Q8—2621no—
249Q8—2621no—
250D4—21no—
251D4—21no—
252D4—42no—
253D4—2621no—
254D4—2622no—
255D4—2621no—
256D4—2622no—
257D4—5242no—
258D4—5242no—
259D4—10484no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—2621no—
265C2×C4—5242no—
266C2×C4—1314no—
267C2×C4D4×D13111yescharacteristic
268C2×C4D4×D13112yes—
269C2×C4C2×C22⋊D13112yes—
270C2×C4C23×D13111yescharacteristic
271C8—1312no—
272C8—1312no—
273C8C2×C22⋊D13114yes—
274C22—22no—
275C22—42no—
276C22—41no—
277C22—5244no—
278C22—5242no—
279C22—5241no—
280C22—5242no—
281C22—10484no—
282C22#1686224212yes—
283C22C2×D4×D13111yescharacteristic
284C4—41no—
285C4—42no—
286C4—2624no—
287C4—5241no—
288C4#1686224212yes—
289C4C2×D4×D13111yescharacteristic
290C4C22×C22⋊D13111yescharacteristic
291C2—22no—
292C2—81no—
293C2—10482no—
294C2—10481no—
295C2C2×#1686224211yescharacteristic
296C2#2255412112yes—
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC2×#137×C131C2
socleC22×C131C22×D4
hypercenterC2×#137D131
hypocenterC131#3154
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#16862274C2semidirect
2. C2×#16862266C2semidirect
3. C22×#14523500C2semidirect
4. #22553886C2semidirect
5. #22553876C2semidirect
6. #22553705C2semidirect
7. #22553698C2semidirect
8. #22553697C2semidirect
9. #22553683C2semidirect
10. #22553270C2semidirect
11. C2×#137×C131C2semidirect
16. C2×#14523503C22semidirect
18. C2×#14523501C22semidirect
22. C2×#14523500C22semidirect
24. C2×#14523498C22semidirect
26. #16862244C22semidirect
29. #16862235C22semidirect
35. #16862156C22semidirect
36. C8×Dic131C22semidirect
40. C2×Q16×C131C22semidirect
41. C2×SD16×C131C22semidirect
63. C2×C131⋊C8D4semidirect
86. M16×C131D4semidirect
148. C131#3154semidirect
150. C2×#137D131semidirect
169. C2×Q16C2×D131semidirect
176. C2×SD16C2×D131semidirect
226. 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×#145235002
#225538862
#225538762
#225537052
#225536982
#225536972
#225536832
#225532702
C2×#137×C1312
#3154131