Finite Group Databasebeta

← #30669311

Subgroups — #30669311

Order 16768.

1232 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 205 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×#16862217C212yesmaximal, complemented
2C2×#16862215C212yesmaximal, complemented
3C2×C4×C22⋊D131C211yesmaximal, characteristic, complemented
4#22553226C212yesmaximal, complemented
5#22553220C212yesmaximal, complemented
6#22553215C212yesmaximal, complemented
7#22553172C211yesmaximal, characteristic, complemented
8#22553168C212yesmaximal, complemented
9C23×D4×C131C211yesmaximal, characteristic, complemented
10C22×C22⋊D131C2211yescharacteristic
11C22×C22⋊D131C2214yescomplemented
12C22×C4×D131C2211yescharacteristic, complemented
13#16862217—28no—
14#16862215—28no—
15C4×C22⋊D131—24no—
16C2×C4⋊Dic131C2212yescomplemented
17C2×Dic131⋊C4C2211yescharacteristic, complemented
18C2×C22⋊Dic131C2211yescharacteristic
19C2×C22⋊Dic131C2212yes—
20C2×C22⋊Dic131C2214yescomplemented
21C2×C22⋊Dic131C2214yescomplemented
22C2×C22⋊Dic131C2214yescomplemented
23C2×(C2×D131)⋊C4C2211yescharacteristic, complemented
24C2×C4×Dic131C2211yescharacteristic, complemented
25C131⋊#95C2212yescomplemented
26C25×C131C2212yes—
27C22×D4×C131—28no—
28C22×D4×C131—24no—
29C22×D4×C131C2212yes—
30C22×D4×C131C2212yes—
31C23×C4×C131C2211yescharacteristic
32C23×D131C2311yescharacteristic
33C2×C22⋊D131—28no—
34C2×C22⋊D131—22no—
35C2×C22⋊D131—28no—
36C2×C22⋊D131—22no—
37C2×C22⋊D131—44no—
38C2×C22⋊D131D414yescomplemented
39C22×Dic131C2311yescharacteristic
40C22×Dic131C2314yes—
41C22×Dic131C2311yescharacteristic
42C22×Dic131C2311yescharacteristic
43C2×C4×D131—24no—
44C2×C4×D131—22no—
45C4⋊Dic131—24no—
46Dic131⋊C4—22no—
47C22⋊Dic131—28no—
48C22⋊Dic131—24no—
49C22⋊Dic131—28no—
50C22⋊Dic131—22no—
51C22⋊Dic131—44no—
52(C2×D131)⋊C4—22no—
53C4×Dic131—22no—
54C24×C131—28no—
55C24×C131—24no—
56C24×C131—24no—
57C24×C131—24no—
58C24×C131—44no—
59C24×C131C2314yes—
60C24×C131C2311yescharacteristic
61C2×D4×C131—232no—
62C2×D4×C131—28no—
63C2×D4×C131—24no—
64C2×D4×C131—24no—
65C2×D4×C131—24no—
66C2×D4×C131D418yescomplemented
67C22×C4×C131—24no—
68C22×C4×C131D414yescomplemented
69C22×C4×C131C2311yescharacteristic
70C22×C4×C131C2311yescharacteristic
71C22×D131—21no—
72C22×D131—22no—
73C22×D131—21no—
74C22×D131—22no—
75C22×D131C2×D412yescomplemented
76C22⋊D131—28no—
77C22⋊D131—416no—
78C2×Dic131—21no—
79C2×Dic131—21no—
80C2×Dic131—21no—
81C2×Dic131—22no—
82C2×Dic131—24no—
83C2×Dic131—21no—
84C2×Dic131—24no—
85C2×Dic131—22no—
86C2×Dic131—24no—
87C2×Dic131C2×D412yescomplemented
88C4×D131—44no—
89C23×C131—22no—
90C23×C131—22no—
91C23×C131—216no—
92C23×C131—24no—
93C23×C131—24no—
94C23×C131—24no—
95C23×C131—24no—
96C23×C131—24no—
97C23×C131—24no—
98C23×C131—416no—
99C23×C131—416no—
100C23×C131—48no—
101C23×C131—48no—
102C23×C131C2×D412yes—
103C23×C131C2×D418yes—
104C23×C131C2×D412yes—
105C23×C131#4112yes—
106C23×C131C2411yescharacteristic
107D4×C131—216no—
108D4×C131—216no—
109C2×C4×C131—22no—
110C2×C4×C131—28no—
111C2×C4×C131C2×D412yes—
112C2×C4×C131C2×D414yes—
113C2×C4×C131C2×D412yes—
114C22×C131—22no—
115C22×C131—22no—
116C22×C131—24no—
117C22×C131—24no—
118C22×C131—28no—
119C22×C131—28no—
120C22×C131—28no—
121C22×C131—432no—
122C22×C131—44no—
123C22×C131—44no—
124C22×C131—44no—
125C22×C131—44no—
126C22×C131#12114yescomplemented
127C22×C131C22×D411yescharacteristic
128C22×C131C22×D411yescharacteristic
129C22×C131C22×D412yes—
130C22×C131C2×#4111yescharacteristic
131C22×C131H(5,𝔽2)12yes—
132C2×D131—22no—
133C2×D131—24no—
134C2×D131—41no—
135C2×D131—41no—
136C2×D131—42no—
137C4×C131—22no—
138C4×C131#12014yescomplemented
139Dic131—22no—
140Dic131—41no—
141Dic131—41no—
142Dic131—44no—
143C2×C131—24no—
144C2×C131—48no—
145C2×C131C2×#12011yescharacteristic
146C2×C131C2×#12111yescharacteristic
147C2×C131#53511yescharacteristic
148C2×C131D4212yes—
149C2×C131#54612yes—
150D131—42no—
151C131#240811yescharacteristic, complemented, Sylow
152#2408—1311nomaximal, Sylow
153C23×D4D13111yescharacteristic, complemented
154C2×#121—1312no—
155C2×#120—1312no—
156C2×C4×D4—1311no—
157#394—1312no—
158#392—1312no—
159#390—1312no—
160#380—1311no—
161#379—1312no—
162C25C2×D13112yes—
163C22×D4—28no—
164C22×D4—24no—
165C22×D4—1314no—
166C22×D4—1311no—
167C22×D4C2×D13112yes—
168C22×D4C2×D13112yes—
169C23×C4—1311no—
170C23×C4C2×D13111yescharacteristic
171#121—2628no—
172#120—2628no—
173C4×D4—2624no—
174C2×C4⋊C4—1311no—
175C2×C4⋊C4—1312no—
176C2×C22⋊C4—1311no—
177C2×C22⋊C4—1312no—
178C2×C22⋊C4—1314no—
179C2×C22⋊C4—1314no—
180C2×C22⋊C4—1311no—
181C2×C22⋊C4—1314no—
182C2×C42—1311no—
183#95—1312no—
184C24—28no—
185C24—24no—
186C24—24no—
187C24—24no—
188C24—44no—
189C24—1311no—
190C24C22×D13114yes—
191C24C22×D13111yescharacteristic
192C2×D4—232no—
193C2×D4—28no—
194C2×D4—24no—
195C2×D4—24no—
196C2×D4—24no—
197C2×D4—2622no—
198C2×D4—2622no—
199C2×D4—2628no—
200C2×D4—2628no—
201C2×D4—5244no—
202C2×D4—1314no—
203C2×D4C22⋊D13118yescomplemented
204C22×C4—24no—
205C22×C4—2622no—
206C22×C4—2624no—
207C22×C4—1314no—
208C22×C4—1311no—
209C22×C4—1311no—
210C22×C4—1311no—
211C22×C4C22⋊D13114yescomplemented
212C22×C4C22×D13111yescharacteristic
213C22×C4C22×D13111yescharacteristic
214C4⋊C4—2622no—
215C4⋊C4—2624no—
216C22⋊C4—2624no—
217C22⋊C4—2622no—
218C22⋊C4—2628no—
219C22⋊C4—2628no—
220C22⋊C4—2622no—
221C22⋊C4—5244no—
222C42—2622no—
223C23—22no—
224C23—22no—
225C23—216no—
226C23—24no—
227C23—24no—
228C23—24no—
229C23—24no—
230C23—24no—
231C23—24no—
232C23—416no—
233C23—416no—
234C23—48no—
235C23—48no—
236C23—2621no—
237C23—2622no—
238C23—2621no—
239C23—2622no—
240C23—1312no—
241C23D4×D13112yes—
242C23C2×C22⋊D13118yes—
243C23C2×C22⋊D13112yes—
244C23#1452351212yes—
245C23C23×D13111yescharacteristic
246D4—216no—
247D4—216no—
248D4—2628no—
249D4—52416no—
250C2×C4—22no—
251C2×C4—28no—
252C2×C4—2621no—
253C2×C4—2621no—
254C2×C4—2624no—
255C2×C4—2621no—
256C2×C4—2622no—
257C2×C4—2621no—
258C2×C4—2624no—
259C2×C4—2622no—
260C2×C4—2624no—
261C2×C4—5244no—
262C2×C4—1312no—
263C2×C4C2×C22⋊D13112yes—
264C2×C4C2×C22⋊D13112yes—
265C2×C4C2×C22⋊D13114yes—
266C22—22no—
267C22—24no—
268C22—24no—
269C22—22no—
270C22—28no—
271C22—28no—
272C22—28no—
273C22—432no—
274C22—44no—
275C22—44no—
276C22—44no—
277C22—44no—
278C22—2624no—
279C22—2622no—
280C22—5241no—
281C22—5242no—
282C22—5241no—
283C22#1686221714yescomplemented
284C22C2×D4×D13111yescharacteristic
285C22C22×C22⋊D13111yescharacteristic
286C22C22×C22⋊D13112yes—
287C22C2×#1452351211yescharacteristic
288C22#1686229112yes—
289C4—22no—
290C4—2622no—
291C4—5241no—
292C4—5244no—
293C4—5241no—
294C4#1686221514yescomplemented
295C2—24no—
296C2—48no—
297C2—5242no—
298C2C2×#1686221511yescharacteristic
299C2C2×#1686221711yescharacteristic
300C2#2255387911yescharacteristic
301C2D4×C22⋊D13112yes—
302C2#2255395612yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
FittingC23×D4×C131C2
hypercenterC23×D4D131
hypocenterC131#2408
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16862217C2semidirect
2. C2×#16862215C2semidirect
3. C2×C4×C22⋊D131C2semidirect
4. #22553226C2semidirect
5. #22553220C2semidirect
6. #22553215C2semidirect
7. #22553172C2semidirect
8. #22553168C2semidirect
9. C23×D4×C131C2semidirect
11. C22×C22⋊D131C22semidirect
12. C22×C4×D131C22semidirect
16. C2×C4⋊Dic131C22semidirect
17. C2×Dic131⋊C4C22semidirect
20. C2×C22⋊Dic131C22semidirect
21. C2×C22⋊Dic131C22semidirect
22. C2×C22⋊Dic131C22semidirect
23. C2×(C2×D131)⋊C4C22semidirect
24. C2×C4×Dic131C22semidirect
25. C131⋊#95C22semidirect
38. C2×C22⋊D131D4semidirect
66. C2×D4×C131D4semidirect
68. C22×C4×C131D4semidirect
75. C22×D131C2×D4semidirect
87. C2×Dic131C2×D4semidirect
126. C22×C131#121semidirect
138. C4×C131#120semidirect
151. C131#2408semidirect
153. C23×D4D131semidirect
203. C2×D4C22⋊D131semidirect
211. C22×C4C22⋊D131semidirect
283. C22#16862217semidirect
294. C4#16862215semidirect

Maximal subgroups

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

subgroupindex
C2×#168622172
C2×#168622152
C2×C4×C22⋊D1312
#225532262
#225532202
#225532152
#225531722
#225531682
C23×D4×C1312
#2408131