Finite Group Databasebeta

← #30675310

Subgroups — #30675310

Order 16768.

724 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 161 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
1#22554110C212yesmaximal, complemented
2#22553858C211yesmaximal, characteristic, complemented
3#22553495C212yesmaximal, complemented
4#22553490C212yesmaximal, complemented
5#22553485C212yesmaximal, complemented
6#22553480C212yesmaximal, complemented
7C2×#16862173C212yesmaximal, complemented
8#22553296C211yesmaximal, characteristic, complemented
9#585×C131C211yesmaximal, characteristic, complemented
10C2×#14523513C2211yescharacteristic, complemented
11#16862277—22no—
12#16862276C2214yescomplemented
13#16862272—22no—
14#16862271C2214yescomplemented
15#16862266—24no—
16C2×#14523497C2212yescomplemented
17#16862244C2211yescharacteristic, complemented
18#16862242C2211yescharacteristic, complemented
19#16862235C2211yescharacteristic, complemented
20#16862233C2211yescharacteristic, complemented
21#16862218—22no—
22C4×C22⋊D131—22no—
23C2×C4×Dic131C2211yescharacteristic, complemented
24#16862173C2214yescomplemented
25#16862173C2214yescomplemented
26#16862165C2212yescomplemented
27#16862163C2212yescomplemented
282−1+4×C131—22no—
292−1+4×C131C2212yes—
30H(5,𝔽2)×C131—22no—
31H(5,𝔽2)×C131—22no—
32H(5,𝔽2)×C131C2212yes—
33C2×#41×C131—24no—
34C2×#41×C131—22no—
35C2×#41×C131C2212yes—
36C2×#41×C131C2211yescharacteristic
37#14523513—22no—
38#14523513—22no—
39C2×Dic262C2311yescharacteristic
40C2×D524C2311yescharacteristic
41C2×C22⋊D131—21no—
42C2×C22⋊D131—22no—
43C22×Dic131—21no—
44C2×C4×D131—21no—
45#14523503—24no—
46#14523501—24no—
47#14523500—24no—
48#14523498—24no—
49#14523497—22no—
50#14523497C2314yes—
51C2×C131⋊C8—22no—
52Dic131⋊C4—22no—
53C22⋊Dic131—22no—
54(C2×D131)⋊C4—22no—
55C4×Dic131—21no—
56C4×Dic131C2311yescharacteristic
57C4×Dic131C2311yescharacteristic
58#41×C131—24no—
59#41×C131—28no—
60#41×C131—24no—
61#41×C131—24no—
62#41×C131—24no—
63#41×C131—22no—
64#41×C131—22no—
65#41×C131—24no—
66#41×C131—22no—
67#41×C131—22no—
68#41×C131D414yescomplemented
69#41×C131C2314yes—
70C2×Q8×C131—22no—
71C2×Q8×C131—24no—
72C2×Q8×C131D412yescomplemented
73C2×Q8×C131C2311yescharacteristic
74C2×D4×C131—28no—
75C2×D4×C131—22no—
76C2×D4×C131—24no—
77C2×D4×C131—22no—
78C2×D4×C131—24no—
79C2×D4×C131D412yes—
80C2×D4×C131D412yescomplemented
81C2×D4×C131C2311yescharacteristic
82C22×C4×C131—24no—
83C22×C4×C131—22no—
84C22×C4×C131D412yes—
85C22×C4×C131C2311yescharacteristic
86C22×D131—21no—
87Dic262—22no—
88D524—22no—
89C22⋊D131—44no—
90C22⋊D131—42no—
91C2×Dic131—21no—
92C2×Dic131—21no—
93C2×Dic131—21no—
94C2×Dic131—42no—
95C4×D131—41no—
96C4×D131—41no—
97C131⋊C8—24no—
98C23×C131—24no—
99C23×C131—22no—
100C23×C131C2×D412yes—
101C23×C131C2×D411yescharacteristic
102Q8×C131—24no—
103Q8×C131—22no—
104Q8×C131—22no—
105Q8×C131C2×D414yes—
106D4×C131—28no—
107D4×C131—24no—
108D4×C131—24no—
109D4×C131—24no—
110D4×C131—24no—
111D4×C131—22no—
112D4×C131—22no—
113D4×C131C2×D414yes—
114C2×C4×C131—24no—
115C2×C4×C131—28no—
116C2×C4×C131—22no—
117C2×C4×C131—24no—
118C2×C4×C131—22no—
119C2×C4×C131—22no—
120C2×C4×C131—24no—
121C2×C4×C131C2×D412yes—
122C2×C4×C131C2×D411yescharacteristic
123C2×C4×C131C2×D412yes—
124C2×C4×C131C2×D412yes—
125C2×C4×C131C2411yescharacteristic
126C22×C131—22no—
127C22×C131—24no—
128C22×C131—41no—
129C22×C131—42no—
130C22×C131—82no—
131C22×C131—84no—
132C22×C131#12012yes—
133C22×C131C22×D411yescharacteristic
134C2×D131—42no—
135C2×D131—41no—
136C4×C131—24no—
137C4×C131—22no—
138C4×C131#12011yescharacteristic
139C4×C131#12011yescharacteristic
140C4×C131C22×D412yes—
141Dic131—22no—
142Dic131—41no—
143Dic131—41no—
144C2×C131—21no—
145C2×C131—22no—
146C2×C131—42no—
147C2×C131—44no—
148C2×C131C2×#12011yescharacteristic
149D131—81no—
150C131#301811yescharacteristic, complemented, Sylow
151#3018—1311nomaximal, Sylow
152#585D13111yescharacteristic, complemented
153#575—1312no—
154#532—1311no—
155#456—1312no—
156#455—1312no—
157#454—1312no—
158#453—1312no—
159C2×#104—1312no—
160#413—1311no—
1612−1+4—22no—
1622−1+4C2×D13112yes—
163H(5,𝔽2)—22no—
164H(5,𝔽2)—22no—
165H(5,𝔽2)C2×D13112yes—
166C2×#41—24no—
167C2×#41—22no—
168C2×#41—1311no—
169C2×#41C2×D13112yes—
170C2×#41C2×D13111yescharacteristic
171#137—2622no—
172#137—1314no—
173AGL(1,ℤ/8)—2622no—
174AGL(1,ℤ/8)—1314no—
175#135—2624no—
176C2×M16—1312no—
177C4⋊Q8—1311no—
178#127—1311no—
179#124—1311no—
180#124—1311no—
181#121—2622no—
182C4×D4—2622no—
183C2×C42—1311no—
184#104—1314no—
185#104—1314no—
186#101—1312no—
187#100—1312no—
188#41—24no—
189#41—28no—
190#41—24no—
191#41—24no—
192#41—24no—
193#41—22no—
194#41—22no—
195#41—24no—
196#41—22no—
197#41—22no—
198#41—2622no—
199#41—2622no—
200#41C22⋊D13114yescomplemented
201#41C22×D13114yes—
202C2×Q8—22no—
203C2×Q8—24no—
204C2×Q8—1311no—
205C2×Q8C22⋊D13112yescomplemented
206C2×Q8C22×D13111yescharacteristic
207C2×D4—28no—
208C2×D4—22no—
209C2×D4—24no—
210C2×D4—22no—
211C2×D4—24no—
212C2×D4—2622no—
213C2×D4—2621no—
214C2×D4—1311no—
215C2×D4C22⋊D13112yes—
216C2×D4C22⋊D13112yescomplemented
217C2×D4C22×D13111yescharacteristic
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—2621no—
221C22×C4—2621no—
222C22×C4C22⋊D13112yes—
223C22×C4C22×D13111yescharacteristic
224Q16—2624no—
225SD16—2624no—
226SD16—2624no—
227D8—2624no—
228M16—2622no—
229M16—1314no—
230C2×C8—2622no—
231C4⋊C4—2622no—
232C22⋊C4—2622no—
233C22⋊C4—2622no—
234C42—2621no—
235C42—1311no—
236C42—1311no—
237C23—24no—
238C23—22no—
239C23—2621no—
240C23C2×C22⋊D13112yes—
241C23C2×C22⋊D13111yescharacteristic
242Q8—24no—
243Q8—22no—
244Q8—22no—
245Q8—2622no—
246Q8C2×C22⋊D13114yes—
247D4—28no—
248D4—24no—
249D4—24no—
250D4—24no—
251D4—24no—
252D4—22no—
253D4—22no—
254D4—2622no—
255D4—5244no—
256D4—5242no—
257D4C2×C22⋊D13114yes—
258C2×C4—24no—
259C2×C4—28no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—2621no—
266C2×C4—2621no—
267C2×C4—2621no—
268C2×C4—5242no—
269C2×C4—5241no—
270C2×C4—5241no—
271C2×C4C2×C22⋊D13112yes—
272C2×C4C2×C22⋊D13111yescharacteristic
273C2×C4C2×C22⋊D13112yes—
274C2×C4C2×C22⋊D13112yes—
275C2×C4C23×D13111yescharacteristic
276C8—2624no—
277C22—22no—
278C22—24no—
279C22—41no—
280C22—42no—
281C22—82no—
282C22—84no—
283C22—5242no—
284C22—5241no—
285C22#1686221512yes—
286C22C22×C22⋊D13111yescharacteristic
287C4—24no—
288C4—22no—
289C4—2622no—
290C4—5241no—
291C4—5241no—
292C4#1686221511yescharacteristic
293C4#1686221511yescharacteristic
294C4C22×C22⋊D13112yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—44no—
299C2—10481no—
300C2C2×#1686221511yescharacteristic
subgroupG/H
centerC4#16862215
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#585×C131C2
socleC2×C131C2×#120
hypercenter#585D131
hypocenterC131#3018
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C4 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #22554110C2semidirect
2. #22553858C2semidirect
3. #22553495C2semidirect
4. #22553490C2semidirect
5. #22553485C2semidirect
6. #22553480C2semidirect
7. C2×#16862173C2semidirect
8. #22553296C2semidirect
9. #585×C131C2semidirect
10. C2×#14523513C22semidirect
12. #16862276C22semidirect
14. #16862271C22semidirect
16. C2×#14523497C22semidirect
17. #16862244C22semidirect
18. #16862242C22semidirect
19. #16862235C22semidirect
20. #16862233C22semidirect
23. C2×C4×Dic131C22semidirect
24. #16862173C22semidirect
25. #16862173C22semidirect
26. #16862165C22semidirect
27. #16862163C22semidirect
68. #41×C131D4semidirect
72. C2×Q8×C131D4semidirect
80. C2×D4×C131D4semidirect
150. C131#3018semidirect
152. #585D131semidirect
200. #41C22⋊D131semidirect
205. C2×Q8C22⋊D131semidirect
216. C2×D4C22⋊D131semidirect

Maximal subgroups

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

subgroupindex
#225541102
#225538582
#225534952
#225534902
#225534852
#225534802
C2×#168621732
#225532962
#585×C1312
#3018131