Finite Group Databasebeta

← #30676441

Subgroups — #30676441

Order 16768.

596 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 137 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×#16862270C212yesmaximal, complemented
2C2×#16862265C211yesmaximal, characteristic, complemented
3C2×D8×D131C211yesmaximal, characteristic, complemented
4#22553875C212yesmaximal, complemented
5#22553700C212yesmaximal, complemented
6#22553699C212yesmaximal, complemented
7#22553696C211yesmaximal, characteristic, complemented
8#22553695C211yesmaximal, characteristic, complemented
9#22553687C211yesmaximal, characteristic, complemented
10#22553266C211yesmaximal, characteristic, complemented
11#493×C131C211yesmaximal, characteristic, complemented
12C2×#14523512C2212yes—
13C2×D4×D131C2212yes—
14#16862270—28no—
15#16862265—24no—
16C2×Dic524C2211yescharacteristic, complemented
17C2×#14523502C2212yescomplemented
18C2×#14523501C2212yescomplemented
19C2×#14523501C2212yescomplemented
20C2×D1048C2211yescharacteristic
21C2×#14523498C2212yes—
22C2×#14523498C2212yescomplemented
23D8×D131—24no—
24C2×#14523496C2211yescharacteristic
25C2×C8×D131C2211yescharacteristic
26#16862242C2212yescomplemented
27#16862234C2212yescomplemented
28#16862233C2212yescomplemented
29#16862218—24no—
30#16862214—24no—
31#16862203C2211yescharacteristic
32#16862157C2211yescharacteristic, complemented
33#16862156C2211yescharacteristic, complemented
34C8×Dic131C2211yescharacteristic, complemented
35C2×D8×C131C2211yescharacteristic
36C2×D8×C131C2211yescharacteristic, complemented
37C2×D8×C131C2212yescomplemented
38#127×C131C2212yescomplemented
39C4×C8×C131C2211yescharacteristic, complemented
40C23×D131—22no—
41#14523512—24no—
42#14523512—24no—
43C2×Dic262C2312yes—
44C2×D524C2312yes—
45C2×C22⋊D131—22no—
46C2×C22⋊D131—22no—
47C2×C22⋊D131—24no—
48D4×D131—24no—
49D4×D131—24no—
50C22×Dic131—22no—
51C2×C4×D131C2311yescharacteristic
52Dic524—22no—
53#14523502—24no—
54#14523501—24no—
55#14523501—24no—
56D1048—22no—
57#14523498—24no—
58#14523498—24no—
59#14523496D414yescomplemented
60C2×C131⋊C8C2311yescharacteristic
61C2×C131⋊C8C2311yescharacteristic
62C8×D131D414yescomplemented
63Dic131⋊C4D412yes—
64C22⋊Dic131—24no—
65(C2×D131)⋊C4—24no—
66(C2×D131)⋊C4D412yes—
67C4×Dic131C2311yescharacteristic
68C2×D4×C131—24no—
69C2×D4×C131C2312yes—
70C2×D4×C131C2312yes—
71D8×C131—22no—
72D8×C131—22no—
73D8×C131—24no—
74C2×C8×C131C2311yescharacteristic
75C2×C8×C131C2311yescharacteristic
76C42×C131C2311yescharacteristic
77C22×D131—22no—
78C22×D131—44no—
79C22×D131—84no—
80C22×D131C2×D411yescharacteristic
81Dic262—22no—
82Dic262—22no—
83D524—22no—
84D524—22no—
85C22⋊D131—44no—
86C22⋊D131—44no—
87C22⋊D131—84no—
88C2×Dic131—22no—
89C2×Dic131—44no—
90C2×Dic131C2×D412yes—
91C2×Dic131C2×D411yescharacteristic
92C4×D131C2×D412yes—
93C4×D131C2×D412yes—
94C131⋊C8C2×D412yes—
95C131⋊C8C2×D412yes—
96C23×C131—22no—
97C23×C131—22no—
98D4×C131—22no—
99D4×C131—22no—
100D4×C131—22no—
101D4×C131—22no—
102D4×C131—84no—
103C2×C4×C131C2×D412yes—
104C2×C4×C131C2411yescharacteristic
105C8×C131C2×D412yes—
106C8×C131C2×D412yes—
107C22×C131—42no—
108C22×C131—42no—
109C22×C131—44no—
110C22×C131—44no—
111C22×C131C22×D411yescharacteristic
112C2×D131—22no—
113C2×D131—42no—
114C2×D131—44no—
115C2×D131—88no—
116C2×D131#12712yes—
117C4×C131—22no—
118C4×C131C22×D411yescharacteristic
119C4×C131C22×D411yescharacteristic
120Dic131—22no—
121Dic131—42no—
122Dic131#12712yes—
123C2×C131—82no—
124C2×C131—82no—
125C2×C131C2×#12711yescharacteristic
126C2×C131#57612yes—
127D131—22no—
128D131—82no—
129C131#315511yescharacteristic, complemented, Sylow
130#3155—1311nomaximal, Sylow
131C2×AGL(1,ℤ/8)—1312no—
132C2×#135—1311no—
133C22×D8—1311no—
134#535—1312no—
135#496—1312no—
136#496—1312no—
137#495—1311no—
138#495—1311no—
139#493—1311no—
140#493D13111yescharacteristic, complemented
141#405—1311no—
142C2×#41—1312no—
143C22×D4—1312no—
144AGL(1,ℤ/8)—2628no—
145#135—2624no—
146C2×Q16—1311no—
147C2×SD16—1312no—
148C2×SD16—1312no—
149C2×SD16—1312no—
150C2×D8—2624no—
151C2×D8—1312no—
152C2×D8—1312no—
153C2×D8—1311no—
154C2×D8C2×D13111yescharacteristic
155C2×D8C2×D13111yescharacteristic, complemented
156C2×D8C2×D13112yescomplemented
157C2×M16—1311no—
158C22×C8—1311no—
159#127—1312no—
160#127C2×D13112yescomplemented
161#124—1312no—
162#124—1312no—
163#121—2624no—
164#120—2624no—
165#117—1311no—
166#97—1311no—
167#97—1311no—
168C4×C8—1311no—
169C4×C8C2×D13111yescharacteristic, complemented
170C24—2622no—
171#41—2624no—
172#41—2624no—
173C2×Q8—1312no—
174C2×D4—24no—
175C2×D4—2622no—
176C2×D4—2624no—
177C2×D4—2624no—
178C2×D4—2622no—
179C2×D4—2624no—
180C2×D4—1312no—
181C2×D4C22×D13112yes—
182C2×D4C22×D13112yes—
183C22×C4—2622no—
184C22×C4—1311no—
185Q16—2622no—
186SD16—2624no—
187SD16—2624no—
188SD16—2624no—
189D8—22no—
190D8—22no—
191D8—24no—
192D8—2624no—
193D8—2624no—
194D8—2622no—
195M16—1314no—
196C2×C8—1311no—
197C2×C8—1311no—
198C2×C8—1314no—
199C2×C8C22×D13111yescharacteristic
200C2×C8C22×D13111yescharacteristic
201C4⋊C4—1312no—
202C22⋊C4—2624no—
203C22⋊C4—2624no—
204C22⋊C4—1312no—
205C42—1311no—
206C42C22×D13111yescharacteristic
207C23—22no—
208C23—22no—
209C23—2622no—
210C23—5244no—
211C23—10484no—
212C23—1311no—
213Q8—2622no—
214Q8—2622no—
215D4—22no—
216D4—22no—
217D4—22no—
218D4—22no—
219D4—2622no—
220D4—2622no—
221D4—84no—
222D4—5244no—
223D4—5244no—
224D4—10484no—
225C2×C4—2622no—
226C2×C4—5244no—
227C2×C4—1312no—
228C2×C4—1311no—
229C2×C4—1312no—
230C2×C4—1312no—
231C2×C4D4×D13112yes—
232C2×C4C23×D13111yescharacteristic
233C8—1312no—
234C8—1312no—
235C8D4×D13112yes—
236C8D4×D13112yes—
237C22—44no—
238C22—44no—
239C22—42no—
240C22—42no—
241C22—2622no—
242C22—5244no—
243C22—5242no—
244C22—10488no—
245C22—1312no—
246C22C2×D4×D13111yescharacteristic
247C4—22no—
248C4—2622no—
249C4—5242no—
250C4—1312no—
251C4C2×D4×D13111yescharacteristic
252C4C2×D4×D13111yescharacteristic
253C2—2622no—
254C2—82no—
255C2—82no—
256C2—10482no—
257C2#127×D13111yescharacteristic
258C2#2255411312yes—
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#493×C131C2
socleC22×C131C22×D4
hypercenter#493D131
hypocenterC131#3155
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#16862270C2semidirect
2. C2×#16862265C2semidirect
3. C2×D8×D131C2semidirect
4. #22553875C2semidirect
5. #22553700C2semidirect
6. #22553699C2semidirect
7. #22553696C2semidirect
8. #22553695C2semidirect
9. #22553687C2semidirect
10. #22553266C2semidirect
11. #493×C131C2semidirect
16. C2×Dic524C22semidirect
17. C2×#14523502C22semidirect
18. C2×#14523501C22semidirect
19. C2×#14523501C22semidirect
22. C2×#14523498C22semidirect
26. #16862242C22semidirect
27. #16862234C22semidirect
28. #16862233C22semidirect
32. #16862157C22semidirect
33. #16862156C22semidirect
34. C8×Dic131C22semidirect
36. C2×D8×C131C22semidirect
37. C2×D8×C131C22semidirect
38. #127×C131C22semidirect
39. C4×C8×C131C22semidirect
59. #14523496D4semidirect
62. C8×D131D4semidirect
129. C131#3155semidirect
140. #493D131semidirect
155. C2×D8C2×D131semidirect
156. C2×D8C2×D131semidirect
160. #127C2×D131semidirect
169. C4×C8C2×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622702
C2×#168622652
C2×D8×D1312
#225538752
#225537002
#225536992
#225536962
#225536952
#225536872
#225532662
#493×C1312
#3155131