Finite Group Databasebeta

← #30669284

Subgroups — #30669284

Order 16768.

1116 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 197 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×#16862227C212yesmaximal, complemented
2C22×C22⋊Dic131C211yesmaximal, characteristic, complemented
3#22553226C214yesmaximal, complemented
4#22553192C212yesmaximal, complemented
5#22553185C212yesmaximal, complemented
6#22553168C212yesmaximal, complemented
7#22553168C211yesmaximal, characteristic, complemented
8C23×D4×C131C211yesmaximal, characteristic
9C23×Dic131C2211yescharacteristic, complemented
10C23×Dic131C2211yescharacteristic, complemented
11#16862227—28no—
12C2×Dic131⋊C4C2214yescomplemented
13C2×C22⋊Dic131—24no—
14C2×C22⋊Dic131C2212yes—
15C2×C22⋊Dic131C2212yes—
16C2×C22⋊Dic131C2212yescomplemented
17C2×C22⋊Dic131C2214yescomplemented
18C2×C22⋊Dic131C2212yescomplemented
19C2×C22⋊Dic131C2214yescomplemented
20C2×C22⋊Dic131C2212yescomplemented
21C131⋊#95C2214yescomplemented
22C25×C131C2211yescharacteristic
23C25×C131C2211yescharacteristic
24C22×D4×C131—24no—
25C22×D4×C131—28no—
26C22×D4×C131C2212yes—
27C22×D4×C131C2212yes—
28C23×C4×C131C2211yescharacteristic
29C22×Dic131—24no—
30C22×Dic131—22no—
31C22×Dic131—22no—
32C22×Dic131—42no—
33C22×Dic131C2312yes—
34C22×Dic131C2312yes—
35C22×Dic131C2314yes—
36Dic131⋊C4—28no—
37C22⋊Dic131—28no—
38C22⋊Dic131—24no—
39C22⋊Dic131—24no—
40C22⋊Dic131—24no—
41C22⋊Dic131—24no—
42C22⋊Dic131—42no—
43C22⋊Dic131—44no—
44C24×C131—24no—
45C24×C131—24no—
46C24×C131—22no—
47C24×C131—24no—
48C24×C131—24no—
49C24×C131—42no—
50C24×C131—42no—
51C24×C131D414yes—
52C24×C131C2311yescharacteristic
53C24×C131C2312yes—
54C24×C131C2311yescharacteristic
55C24×C131C2311yescharacteristic
56C2×D4×C131—232no—
57C2×D4×C131—24no—
58C2×D4×C131—24no—
59C2×D4×C131—24no—
60C2×D4×C131—24no—
61C2×D4×C131—24no—
62C2×D4×C131D418yes—
63C22×C4×C131—22no—
64C22×C4×C131—24no—
65C22×C4×C131C2312yes—
66C2×Dic131—24no—
67C2×Dic131—22no—
68C2×Dic131—22no—
69C2×Dic131—24no—
70C2×Dic131—24no—
71C2×Dic131—24no—
72C2×Dic131—24no—
73C2×Dic131—44no—
74C2×Dic131—44no—
75C23×C131—22no—
76C23×C131—22no—
77C23×C131—22no—
78C23×C131—22no—
79C23×C131—28no—
80C23×C131—28no—
81C23×C131—28no—
82C23×C131—22no—
83C23×C131—22no—
84C23×C131—22no—
85C23×C131—22no—
86C23×C131—24no—
87C23×C131—22no—
88C23×C131—24no—
89C23×C131—44no—
90C23×C131—416no—
91C23×C131—416no—
92C23×C131—44no—
93C23×C131—44no—
94C23×C131C2×D412yes—
95C23×C131C2×D414yes—
96C23×C131C2×D412yes—
97C23×C131C2×D414yes—
98C23×C131C2×D412yes—
99C23×C131#4114yes—
100C23×C131C2411yescharacteristic
101D4×C131—216no—
102D4×C131—216no—
103C2×C4×C131—28no—
104C2×C4×C131—22no—
105C2×C4×C131—22no—
106C2×C4×C131C2×D414yes—
107C22×C131—22no—
108C22×C131—22no—
109C22×C131—22no—
110C22×C131—22no—
111C22×C131—28no—
112C22×C131—28no—
113C22×C131—22no—
114C22×C131—24no—
115C22×C131—22no—
116C22×C131—22no—
117C22×C131—28no—
118C22×C131—416no—
119C22×C131—416no—
120C22×C131—42no—
121C22×C131—42no—
122C22×C131—44no—
123C22×C131—44no—
124C22×C131#12014yes—
125C22×C131#12314yescomplemented
126C22×C131C22×D411yescharacteristic
127C22×C131C22×D412yes—
128C22×C131C2×#4112yes—
129C22×C131H(5,𝔽2)12yes—
130C4×C131—24no—
131Dic131—42no—
132Dic131—42no—
133Dic131—44no—
134C2×C131—24no—
135C2×C131—24no—
136C2×C131—42no—
137C2×C131—44no—
138C2×C131C2×#12011yescharacteristic
139C2×C131C2×#12311yescharacteristic
140C2×C131#53411yescharacteristic
141C2×C131#54614yes—
142C131#240511yescharacteristic, complemented, Sylow
143#2405—1311nomaximal, Sylow
144C23×D4D13111yescharacteristic
145C2×#123—1312no—
146C22×C22⋊C4—1311no—
147#394—1314no—
148#386—1312no—
149#385—1312no—
150#379—1312no—
151#379—1311no—
152C25C2×D13111yescharacteristic
153C25C2×D13111yescharacteristic
154C22×D4—24no—
155C22×D4—28no—
156C22×D4C2×D13112yes—
157C22×D4C2×D13112yes—
158C23×C4—1311no—
159C23×C4—1311no—
160C23×C4C2×D13111yescharacteristic
161#123—2628no—
162C2×C4⋊C4—1314no—
163C2×C22⋊C4—2624no—
164C2×C22⋊C4—1312no—
165C2×C22⋊C4—1312no—
166C2×C22⋊C4—1312no—
167C2×C22⋊C4—1314no—
168C2×C22⋊C4—1312no—
169C2×C22⋊C4—1314no—
170C2×C22⋊C4—1312no—
171#95—1314no—
172C24—24no—
173C24—24no—
174C24—22no—
175C24—24no—
176C24—24no—
177C24—42no—
178C24—42no—
179C24C22⋊D13114yes—
180C24C22×D13111yescharacteristic
181C24C22×D13112yes—
182C24C22×D13111yescharacteristic
183C24C22×D13111yescharacteristic
184C2×D4—232no—
185C2×D4—24no—
186C2×D4—24no—
187C2×D4—24no—
188C2×D4—24no—
189C2×D4—24no—
190C2×D4C22⋊D13118yes—
191C22×C4—22no—
192C22×C4—24no—
193C22×C4—2622no—
194C22×C4—2622no—
195C22×C4—2624no—
196C22×C4—5242no—
197C22×C4—1312no—
198C22×C4—1314no—
199C22×C4—1312no—
200C22×C4C22×D13112yes—
201C4⋊C4—2628no—
202C22⋊C4—2624no—
203C22⋊C4—2624no—
204C22⋊C4—2624no—
205C22⋊C4—2628no—
206C22⋊C4—2624no—
207C22⋊C4—5242no—
208C22⋊C4—5244no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—28no—
214C23—28no—
215C23—28no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—24no—
220C23—22no—
221C23—22no—
222C23—24no—
223C23—44no—
224C23—416no—
225C23—416no—
226C23—44no—
227C23—44no—
228C23C2×C22⋊D13112yes—
229C23C2×C22⋊D13114yes—
230C23C2×C22⋊D13114yes—
231C23C2×C22⋊D13112yes—
232C23C2×C22⋊D13112yes—
233C23#1452351214yes—
234C23C23×D13111yescharacteristic
235D4—216no—
236D4—216no—
237C2×C4—28no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—2624no—
241C2×C4—2622no—
242C2×C4—2622no—
243C2×C4—2624no—
244C2×C4—2624no—
245C2×C4—2624no—
246C2×C4—2624no—
247C2×C4—5244no—
248C2×C4—5244no—
249C2×C4C2×C22⋊D13114yes—
250C22—22no—
251C22—22no—
252C22—28no—
253C22—28no—
254C22—22no—
255C22—22no—
256C22—22no—
257C22—28no—
258C22—22no—
259C22—24no—
260C22—22no—
261C22—416no—
262C22—416no—
263C22—42no—
264C22—44no—
265C22—44no—
266C22—42no—
267C22#1686221514yes—
268C22#1686222714yescomplemented
269C22C22×C22⋊D13111yescharacteristic
270C22C22×C22⋊D13112yes—
271C22C2×#1452351212yes—
272C22#1686229112yes—
273C4—24no—
274C4—5242no—
275C4—5244no—
276C4—5242no—
277C2—24no—
278C2—24no—
279C2—42no—
280C2—44no—
281C2C2×#1686221511yescharacteristic
282C2C2×#1686222711yescharacteristic
283C2#2255387111yescharacteristic
284C2#2255395614yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting, prime radicalC23×D4×C131C2
hypercenterC23×D4D131
hypocenterC131#2405
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16862227C2semidirect
2. C22×C22⋊Dic131C2semidirect
3. #22553226C2semidirect
4. #22553192C2semidirect
5. #22553185C2semidirect
6. #22553168C2semidirect
7. #22553168C2semidirect
9. C23×Dic131C22semidirect
10. C23×Dic131C22semidirect
12. C2×Dic131⋊C4C22semidirect
16. C2×C22⋊Dic131C22semidirect
17. C2×C22⋊Dic131C22semidirect
18. C2×C22⋊Dic131C22semidirect
19. C2×C22⋊Dic131C22semidirect
20. C2×C22⋊Dic131C22semidirect
21. C131⋊#95C22semidirect
125. C22×C131#123semidirect
142. C131#2405semidirect
268. C22#16862227semidirect

Maximal subgroups

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

subgroupindex
C2×#168622272
C22×C22⋊Dic1312
#225532262
#225531922
#225531852
#225531682
C23×D4×C1312
#2405131