Finite Group Databasebeta

← #30675360

Subgroups — #30675360

Order 16768.

800 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 145 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×#16862291C211yesmaximal, characteristic, complemented
2#22553874C212yesmaximal, complemented
3#22553871C211yesmaximal, characteristic, complemented
4#22553503C214yesmaximal, complemented
5#22553500C214yesmaximal, complemented
6C2×#16862160C212yesmaximal, complemented
7C2×#99×C131C211yesmaximal, characteristic, complemented
8#16862291—24no—
9#16862291—22no—
10#16862291C2214yes—
11C2×#14523513—21no—
12C2×#14523512—22no—
13C22×C22⋊D131—21no—
14C22×C22⋊D131C2212yes—
15C22×C22⋊D131C2211yescharacteristic, complemented
16C2×D4×D131—22no—
17#16862230—22no—
18#16862230C2214yescomplemented
19#16862229—21no—
20#16862227C2212yescomplemented
21#16862220—24no—
22#16862218—22no—
23#16862216—22no—
24#16862216C2214yescomplemented
25#16862215—21no—
26#16862214C2212yescomplemented
27C2×C22⋊Dic131C2211yescharacteristic, complemented
28#16862160C2218yescomplemented
29C22×D4×C131C2211yescharacteristic
30C2×C22⋊C4×C131C2212yescomplemented
31#99×C131C2214yescomplemented
32C23×D131—21no—
33C23×D131D412yescomplemented
34C23×D131C2311yescharacteristic, complemented
35#14523513—24no—
36#14523513—24no—
37#14523512—28no—
38#14523512—28no—
39C2×Dic262—21no—
40C2×D524—21no—
41C2×D524—24no—
42C2×C22⋊D131—22no—
43C2×C22⋊D131—24no—
44C2×C22⋊D131—22no—
45C2×C22⋊D131—22no—
46C2×C22⋊D131—24no—
47C2×C22⋊D131—22no—
48C2×C22⋊D131—22no—
49C2×C22⋊D131—21no—
50C2×C22⋊D131—22no—
51C2×C22⋊D131—21no—
52C2×C22⋊D131—22no—
53C2×C22⋊D131—22no—
54C2×C22⋊D131D414yescomplemented
55C2×C22⋊D131C2314yes—
56D4×D131—28no—
57D4×D131—28no—
58C22×Dic131—21no—
59C22×Dic131—21no—
60C22×Dic131D412yescomplemented
61C22×Dic131C2311yescharacteristic, complemented
62C2×C4×D131—22no—
63C4⋊Dic131—24no—
64Dic131⋊C4—22no—
65C22⋊Dic131—22no—
66C22⋊Dic131—21no—
67C22⋊Dic131C2312yescomplemented
68(C2×D131)⋊C4—24no—
69(C2×D131)⋊C4—24no—
70(C2×D131)⋊C4—22no—
71C24×C131D411yescharacteristic, complemented
72C24×C131C2311yescharacteristic
73C2×D4×C131—22no—
74C2×D4×C131—22no—
75C2×D4×C131D412yescomplemented
76C2×D4×C131C2312yes—
77C22×C4×C131—22no—
78C22×C4×C131D411yescharacteristic, complemented
79C22⋊C4×C131—22no—
80C22⋊C4×C131C2314yescomplemented
81C22×D131—22no—
82C22×D131—24no—
83C22×D131—22no—
84C22×D131—21no—
85C22×D131—22no—
86C22×D131—81no—
87C22×D131—82no—
88C22×D131—82no—
89C22×D131C2×D414yes—
90Dic262—24no—
91D524—24no—
92D524—84no—
93C22⋊D131—28no—
94C22⋊D131—24no—
95C22⋊D131—24no—
96C22⋊D131—24no—
97C22⋊D131—24no—
98C22⋊D131—41no—
99C22⋊D131—41no—
100C22⋊D131—42no—
101C22⋊D131—82no—
102C2×Dic131—24no—
103C2×Dic131—22no—
104C2×Dic131—22no—
105C2×Dic131—22no—
106C2×Dic131—22no—
107C2×Dic131—21no—
108C2×Dic131—81no—
109C2×Dic131C2×D414yes—
110C4×D131—28no—
111C23×C131—21no—
112C23×C131—21no—
113C23×C131—81no—
114C23×C131—81no—
115C23×C131C2×D412yes—
116C23×C131C2×D412yes—
117C23×C131C2×D412yes—
118C23×C131C2×D411yescharacteristic
119C23×C131C2×D411yescharacteristic
120C23×C131C2411yescharacteristic
121D4×C131—24no—
122D4×C131—24no—
123C2×C4×C131—21no—
124C2×C4×C131—21no—
125C2×C4×C131—24no—
126C2×C4×C131—82no—
127C2×C4×C131C2×D412yes—
128C22×C131—22no—
129C22×C131—42no—
130C22×C131—41no—
131C22×C131—41no—
132C22×C131—42no—
133C22×C131—41no—
134C22×C131—42no—
135C22×C131—42no—
136C22×C131—81no—
137C22×C131#12011yescharacteristic
138C22×C131#12011yescharacteristic
139C22×C131#12012yes—
140C22×C131C22×D412yes—
141C22×C131C22×D411yescharacteristic
142C2×D131—24no—
143C2×D131—44no—
144C2×D131—44no—
145C2×D131—42no—
146C2×D131—44no—
147C2×D131—41no—
148C2×D131—82no—
149C2×D131—84no—
150C4×C131—22no—
151C4×C131—44no—
152Dic131—24no—
153Dic131—42no—
154Dic131—41no—
155C2×C131—21no—
156C2×C131—22no—
157C2×C131—21no—
158C2×C131—22no—
159C2×C131—21no—
160C2×C131—42no—
161C2×C131C2×#12011yescharacteristic
162D131—44no—
163D131—81no—
164C131#302611yescharacteristic, complemented, Sylow
165#3026—1311nomaximal, Sylow
166C2×H(5,𝔽2)—1311no—
167#534—1311no—
168#534—1312no—
169#458—1314no—
170B(4,𝔽2)—1314no—
171C2×#99—1312no—
172C2×#99D13111yescharacteristic, complemented
173H(5,𝔽2)—2624no—
174H(5,𝔽2)—2622no—
175H(5,𝔽2)—1314no—
176C2×#41—2622no—
177C2×#41—2621no—
178C22×D4—2622no—
179C22×D4—2621no—
180C22×D4—1311no—
181C22×D4—1312no—
182C22×D4C2×D13111yescharacteristic
183#123—2622no—
184#123—2621no—
185#123—1312no—
186#123—1314no—
187#121—2624no—
188#121—2622no—
189#120—2622no—
190#120—2621no—
191#120—1312no—
192#120—1314no—
193C2×C22⋊C4—1311no—
194C2×C22⋊C4C2×D13112yescomplemented
195#99—1318no—
196#99C2×D13114yescomplemented
197C24—2621no—
198C24—1311no—
199C24—1312no—
200C24D52411yescharacteristic, complemented
201C24C22×D13111yescharacteristic
202#41—2624no—
203#41—2628no—
204#41—2628no—
205#41—2624no—
206C2×Q8—2621no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—2621no—
210C2×D4—2624no—
211C2×D4—2628no—
212C2×D4—2628no—
213C2×D4—2622no—
214C2×D4—2624no—
215C2×D4—2622no—
216C2×D4—2622no—
217C2×D4—2622no—
218C2×D4—2624no—
219C2×D4—2622no—
220C2×D4—2622no—
221C2×D4—2622no—
222C2×D4—2621no—
223C2×D4—2621no—
224C2×D4—2622no—
225C2×D4—1314no—
226C2×D4—1314no—
227C2×D4D52412yescomplemented
228C2×D4C22×D13112yes—
229C22×C4—22no—
230C22×C4—2622no—
231C22×C4—2621no—
232C22×C4—2621no—
233C22×C4—1311no—
234C22×C4—1312no—
235C22×C4D52411yescharacteristic, complemented
236C4⋊C4—2624no—
237C4⋊C4—2622no—
238C22⋊C4—22no—
239C22⋊C4—2624no—
240C22⋊C4—2624no—
241C22⋊C4—2621no—
242C22⋊C4—2622no—
243C22⋊C4—2622no—
244C22⋊C4—1312no—
245C22⋊C4C22×D13114yescomplemented
246C23—21no—
247C23—21no—
248C23—2624no—
249C23—2622no—
250C23—2622no—
251C23—2622no—
252C23—2621no—
253C23—81no—
254C23—81no—
255C23—10481no—
256C23—10482no—
257C23—10482no—
258C23—1314no—
259C23D4×D13112yes—
260C23D4×D13112yes—
261C23C2×D52412yes—
262C23C2×D52411yescharacteristic
263C23C2×D52411yescharacteristic
264C23C23×D13111yescharacteristic
265Q8—2624no—
266D4—24no—
267D4—24no—
268D4—2624no—
269D4—2628no—
270D4—2624no—
271D4—2624no—
272D4—2624no—
273D4—2624no—
274D4—5241no—
275D4—5241no—
276D4—5242no—
277D4—10484no—
278D4—10482no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—2628no—
283C2×C4—2624no—
284C2×C4—2622no—
285C2×C4—2622no—
286C2×C4—2622no—
287C2×C4—2622no—
288C2×C4—2621no—
289C2×C4—82no—
290C2×C4—10481no—
291C2×C4—1314no—
292C2×C4C2×D52412yes—
293C22—22no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—41no—
299C22—42no—
300C22—42no—
301C22—2624no—
302C22—81no—
303C22—5244no—
304C22—5244no—
305C22—5244no—
306C22—5242no—
307C22—5241no—
308C22—10482no—
309C22—10484no—
310C22#1686221611yescharacteristic
311C22#1686221611yescharacteristic
312C22#1686221612yes—
313C22C2×D4×D13112yes—
314C22C22×D52411yescharacteristic
315C4—22no—
316C4—44no—
317C4—2624no—
318C4—5242no—
319C4—5241no—
320C2—22no—
321C2—21no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—42no—
326C2—5244no—
327C2—10481no—
328C2C2×#1686221611yescharacteristic
subgroupG/H
centerC2C2×#16862216
commutator, cosocleC23×C131C24
FrattiniC23C23×D131
FittingC2×#99×C131C2
socleC2×C131C2×#120
hypercenterC2×#99D131
hypocenterC131#3026
LCSG ▷ C23×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
1. C2×#16862291C2semidirect
2. #22553874C2semidirect
3. #22553871C2semidirect
4. #22553503C2semidirect
5. #22553500C2semidirect
6. C2×#16862160C2semidirect
7. C2×#99×C131C2semidirect
15. C22×C22⋊D131C22semidirect
18. #16862230C22semidirect
20. #16862227C22semidirect
24. #16862216C22semidirect
26. #16862214C22semidirect
27. C2×C22⋊Dic131C22semidirect
28. #16862160C22semidirect
30. C2×C22⋊C4×C131C22semidirect
31. #99×C131C22semidirect
33. C23×D131D4semidirect
34. C23×D131C23semidirect
54. C2×C22⋊D131D4semidirect
60. C22×Dic131D4semidirect
61. C22×Dic131C23semidirect
67. C22⋊Dic131C23semidirect
71. C24×C131D4semidirect
75. C2×D4×C131D4semidirect
78. C22×C4×C131D4semidirect
80. C22⋊C4×C131C23semidirect
164. C131#3026semidirect
172. C2×#99D131semidirect
194. C2×C22⋊C4C2×D131semidirect
196. #99C2×D131semidirect
200. C24D524semidirect
227. C2×D4D524semidirect
235. C22×C4D524semidirect
245. C22⋊C4C22×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622912
#225538742
#225538712
#225535032
#225535002
C2×#168621602
C2×#99×C1312
#3026131