Finite Group Databasebeta

← #30676008

Subgroups — #30676008

Order 16768.

454 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 133 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#22553846C211yesmaximal, characteristic, complemented
2#22553647C211yesmaximal, characteristic, complemented
3#22553630C211yesmaximal, characteristic, complemented
4#22553612C212yesmaximal, complemented
5#22553527C212yesmaximal, complemented
6#22553463C211yesmaximal, characteristic, complemented
7#22553446C211yesmaximal, characteristic
8C4×#14523501C212yesmaximal, complemented
9C4×#14523498C212yesmaximal, complemented
10#22553365C211yesmaximal, characteristic, complemented
11C2×C4×D4×C131C211yesmaximal, characteristic, complemented
12C2×#14523501C2212yes—
13C2×#14523498C2212yes—
14#16862240—21no—
15#16862238C2211yescharacteristic, complemented
16#16862234C2211yescharacteristic, complemented
17#16862229—21no—
18#16862225C2211yescharacteristic
19#16862220C2211yescharacteristic
20C4×Dic262C2211yescharacteristic, complemented
21C4×D524C2211yescharacteristic, complemented
22C4×C22⋊D131—21no—
23#16862203—21no—
24#16862179C2212yescomplemented
25#16862177C2212yescomplemented
26C4⋊(C131⋊C8)C2212yescomplemented
27#16862169C2212yescomplemented
28#16862167C2212yes—
29#16862167C2212yes—
30#16862166C2212yescomplemented
31C22⋊(C131⋊C8)C2211yescharacteristic
32C22⋊(C131⋊C8)C2211yescharacteristic
33C4×C131⋊C8C2212yescomplemented
34C22×D4×C131C2211yescharacteristic
35C23×C4×C131—21no—
36C4×D4×C131—21no—
37C4×D4×C131—21no—
38C4×D4×C131C2212yes—
39C4×D4×C131C2212yes—
40C2×C4⋊C4×C131C2211yescharacteristic
41C2×C22⋊C4×C131—21no—
42C2×C42×C131C2211yescharacteristic
43C2×Dic262C2311yescharacteristic
44C2×D524C2311yescharacteristic
45C2×C22⋊D131—21no—
46C2×C4×D131—21no—
47#14523501—24no—
48#14523498—24no—
49C2×C131⋊C8C2312yes—
50C2×C131⋊C8C2312yes—
51C4⋊Dic131C2311yescharacteristic
52C4⋊Dic131C2311yescharacteristic
53Dic131⋊C4—21no—
54Dic131⋊C4—21no—
55Dic131⋊C4—21no—
56Dic131⋊C4—21no—
57C22⋊Dic131—21no—
58(C2×D131)⋊C4—21no—
59(C2×D131)⋊C4—21no—
60(C2×D131)⋊C4—21no—
61(C2×D131)⋊C4—21no—
62C4×Dic131—21no—
63C24×C131—21no—
64C2×D4×C131—24no—
65C2×D4×C131—21no—
66C2×D4×C131C2312yes—
67C22×C4×C131—21no—
68C22×C4×C131—22no—
69C22×C4×C131—22no—
70C22×C4×C131—44no—
71C22×C4×C131D411yescharacteristic
72C22×C4×C131D411yescharacteristic
73C22×C4×C131C2311yescharacteristic
74C4⋊C4×C131—21no—
75C4⋊C4×C131C2312yes—
76C22⋊C4×C131—22no—
77C22⋊C4×C131—22no—
78C42×C131D411yescharacteristic
79C42×C131D411yescharacteristic
80C42×C131C2311yescharacteristic
81C42×C131C2311yescharacteristic
82C22×D131—21no—
83Dic262—22no—
84D524—22no—
85C22⋊D131—42no—
86C2×Dic131—21no—
87C2×Dic131—21no—
88C2×Dic131—21no—
89C4×D131—42no—
90C131⋊C8—22no—
91C131⋊C8—22no—
92C23×C131—24no—
93C23×C131—22no—
94C23×C131—42no—
95C23×C131—42no—
96C23×C131C2×D411yescharacteristic
97D4×C131—24no—
98D4×C131—22no—
99D4×C131#4114yes—
100C2×C4×C131—22no—
101C2×C4×C131—22no—
102C2×C4×C131—22no—
103C2×C4×C131—22no—
104C2×C4×C131—44no—
105C2×C4×C131—42no—
106C2×C4×C131—44no—
107C2×C4×C131C2×D411yescharacteristic
108C2×C4×C131C2×D411yescharacteristic
109C2×C4×C131C2×D411yescharacteristic
110C2×C4×C131C2×D411yescharacteristic
111C2×C4×C131C2×D411yescharacteristic
112C2×C4×C131C2411yescharacteristic
113C22×C131—21no—
114C22×C131—21no—
115C22×C131—24no—
116C22×C131—48no—
117C22×C131—42no—
118C22×C131—42no—
119C22×C131#13512yes—
120C22×C131C22×D411yescharacteristic
121C2×D131—41no—
122C2×D131—41no—
123C2×D131—41no—
124C4×C131—21no—
125C4×C131—21no—
126C4×C131—21no—
127C4×C131—21no—
128C4×C131—42no—
129C4×C131AGL(1,ℤ/8)12yes—
130C4×C131C2×#4112yes—
131Dic131—41no—
132Dic131—41no—
133Dic131—41no—
134C2×C131—22no—
135C2×C131—44no—
136C2×C131#52511yescharacteristic
137C2×C131C2×#13511yescharacteristic
138C2×C131C2×AGL(1,ℤ/8)11yescharacteristic
139D131—81no—
140C131#310511yescharacteristic, complemented, Sylow
141#3105—1311nomaximal, Sylow
142#529—1311no—
143C2×C4×D4D13111yescharacteristic, complemented
144#484—1311no—
145#481—1311no—
146#478—1312no—
147#463—1312no—
148#450—1311no—
149#447—1311no—
150C4×SD16—1312no—
151C4×D8—1312no—
152#431—1311no—
153C22×D4C2×D13111yescharacteristic
154C23×C4—21no—
155C2×SD16—1312no—
156C2×D8—1312no—
157#126—2621no—
158#125—1311no—
159#124—1311no—
160#123—2621no—
161C22⋊Q8—1311no—
162#121—1311no—
163C4×Q8—1311no—
164C4×D4—21no—
165C4×D4—21no—
166C4×D4—2621no—
167C4×D4—1311no—
168C4×D4C2×D13112yes—
169C4×D4C2×D13112yes—
170#117—2621no—
171C2×C4⋊C4C2×D13111yescharacteristic
172C2×C22⋊C4—21no—
173C2×C42C2×D13111yescharacteristic
174#107—1312no—
175#106—1312no—
176C4⋊C8—1312no—
177#103—1312no—
178#102—1312no—
179#102—1312no—
180#102—1312no—
181C22⋊C8—1311no—
182C22⋊C8—1311no—
183C4×C8—1312no—
184C24—21no—
185C2×Q8—1311no—
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—2621no—
189C2×D4—1311no—
190C2×D4C22×D13112yes—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—44no—
195C22×C4—2621no—
196C22×C4C22⋊D13111yescharacteristic
197C22×C4C22⋊D13111yescharacteristic
198C22×C4C22×D13111yescharacteristic
199SD16—2624no—
200D8—2624no—
201C2×C8—1312no—
202C2×C8—1312no—
203C4⋊C4—21no—
204C4⋊C4—2621no—
205C4⋊C4—2621no—
206C4⋊C4—2621no—
207C4⋊C4—2621no—
208C4⋊C4—1311no—
209C4⋊C4—1311no—
210C4⋊C4C22×D13112yes—
211C22⋊C4—22no—
212C22⋊C4—22no—
213C22⋊C4—2621no—
214C22⋊C4—2621no—
215C22⋊C4—2621no—
216C22⋊C4—2621no—
217C22⋊C4—2621no—
218C42—2621no—
219C42C22⋊D13111yescharacteristic
220C42C22⋊D13111yescharacteristic
221C42C22×D13111yescharacteristic
222C42C22×D13111yescharacteristic
223C23—24no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—2621no—
228C23C2×C22⋊D13111yescharacteristic
229Q8—2622no—
230D4—24no—
231D4—22no—
232D4—2622no—
233D4—5242no—
234D4#1452351314yes—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—44no—
240C2×C4—42no—
241C2×C4—44no—
242C2×C4—2621no—
243C2×C4—2621no—
244C2×C4—2621no—
245C2×C4—5242no—
246C2×C4C2×C22⋊D13111yescharacteristic
247C2×C4C2×C22⋊D13111yescharacteristic
248C2×C4C2×C22⋊D13111yescharacteristic
249C2×C4C2×C22⋊D13111yescharacteristic
250C2×C4C2×C22⋊D13111yescharacteristic
251C2×C4C23×D13111yescharacteristic
252C8—2622no—
253C8—2622no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—5241no—
261C22—5241no—
262C22—5241no—
263C22#1686226612yes—
264C22C22×C22⋊D13111yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—42no—
270C4—5241no—
271C4—5241no—
272C4—5241no—
273C4#1686227212yes—
274C4C2×#1452351312yes—
275C2—22no—
276C2—44no—
277C2—10481no—
278C2#2255382311yescharacteristic
279C2C2×#1686226611yescharacteristic
280C2C2×#1686227211yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC2×C4×D4×C131C2
socleC22×C131C22×D4
prime radical#22553446C2
hypercenterC2×C4×D4D131
hypocenterC131#3105
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #22553846C2semidirect
2. #22553647C2semidirect
3. #22553630C2semidirect
4. #22553612C2semidirect
5. #22553527C2semidirect
6. #22553463C2semidirect
8. C4×#14523501C2semidirect
9. C4×#14523498C2semidirect
10. #22553365C2semidirect
11. C2×C4×D4×C131C2semidirect
15. #16862238C22semidirect
16. #16862234C22semidirect
20. C4×Dic262C22semidirect
21. C4×D524C22semidirect
24. #16862179C22semidirect
25. #16862177C22semidirect
26. C4⋊(C131⋊C8)C22semidirect
27. #16862169C22semidirect
30. #16862166C22semidirect
33. C4×C131⋊C8C22semidirect
140. C131#3105semidirect
143. C2×C4×D4D131semidirect

Maximal subgroups

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

subgroupindex
#225538462
#225536472
#225536302
#225536122
#225535272
#225534632
#225534462
C4×#145235012
C4×#145234982
#225533652
C2×C4×D4×C1312
#3105131