Finite Group Databasebeta

← #30675540

Subgroups — #30675540

Order 16768.

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

normal subgroups only

This group has 520 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
1C2×#16862265C211yesmaximal, characteristic, complemented
2C22×C8×D131C211yesmaximal, characteristic, complemented
3#22553822C212yesmaximal, complemented
4#22553554C211yesmaximal, characteristic, complemented
5#22553553C211yesmaximal, characteristic, complemented
6#22553545C211yesmaximal, characteristic, complemented
7#22553540C212yesmaximal, complemented
8#22553539C212yesmaximal, complemented
9#22553349C211yesmaximal, characteristic, complemented
10#22553305C212yesmaximal, complemented
11#466×C131C211yesmaximal, characteristic, complemented
12C2×#14523512C2212yes—
13C22×C4×D131C2211yescharacteristic
14#16862265—24no—
15C2×Dic524C2211yescharacteristic, complemented
16C2×#14523501C2212yescomplemented
17C22×C131⋊C8C2211yescharacteristic
18C2×C8×D131—24no—
19C2×C8×D131C2211yescharacteristic
20C2×C8×D131C2211yescharacteristic
21C2×C8×D131C2211yescharacteristic
22C2×C8×D131C2211yescharacteristic
23#16862228—22no—
24#16862226C2212yes—
25#16862225C2212yes—
26#16862217C2212yes—
27#16862214—22no—
28D4×Dic131—22no—
29(C22⋊D131)⋊C4—22no—
30#16862204C2212yes—
31#16862180C2211yescharacteristic, complemented
32#16862177C2212yescomplemented
33#16862171C2212yescomplemented
34#16862169C2212yescomplemented
35#16862167C2212yescomplemented
36C2×D8×C131C2211yescharacteristic, complemented
37C22×C8×C131C2211yescharacteristic
38#121×C131C2212yes—
39#107×C131C2211yescharacteristic, complemented
40#102×C131C2212yescomplemented
41C23×D131D411yescharacteristic
42#14523512—24no—
43#14523512—24no—
44C2×Dic262C2312yes—
45C2×C22⋊D131—22no—
46C2×C22⋊D131—22no—
47C22×Dic131—22no—
48C22×Dic131D411yescharacteristic
49C2×C4×D131—22no—
50C2×C4×D131—22no—
51C2×C4×D131D411yescharacteristic
52C2×C4×D131D411yescharacteristic
53C2×C4×D131C2311yescharacteristic
54C2×C4×D131C2311yescharacteristic
55Dic524—22no—
56#14523501—24no—
57C2×C131⋊C8—22no—
58C2×C131⋊C8C2311yescharacteristic
59C2×C131⋊C8C2311yescharacteristic
60C8×D131—22no—
61C8×D131—22no—
62C8×D131—22no—
63C8×D131D414yescomplemented
64C4⋊Dic131—22no—
65C4⋊Dic131C2312yes—
66Dic131⋊C4—22no—
67C22⋊Dic131—22no—
68C22⋊Dic131—22no—
69(C2×D131)⋊C4—22no—
70(C2×D131)⋊C4—22no—
71C4×Dic131—22no—
72C2×D4×C131—22no—
73C2×D4×C131C2312yes—
74C22×C4×C131C2311yescharacteristic
75D8×C131—22no—
76C2×C8×C131—22no—
77C2×C8×C131C2311yescharacteristic
78C2×C8×C131C2311yescharacteristic
79C4⋊C4×C131C2312yes—
80C22⋊C4×C131—22no—
81C22×D131—22no—
82C22×D131—22no—
83C22×D131—22no—
84C22×D131C2×D411yescharacteristic
85C22×D131C2×D411yescharacteristic
86Dic262—22no—
87Dic262—22no—
88C22⋊D131—44no—
89C22⋊D131—44no—
90C2×Dic131—22no—
91C2×Dic131—22no—
92C2×Dic131—22no—
93C2×Dic131—44no—
94C2×Dic131C2×D411yescharacteristic
95C2×Dic131C2×D411yescharacteristic
96C4×D131—22no—
97C4×D131—21no—
98C4×D131—21no—
99C4×D131—22no—
100C4×D131C2×D412yes—
101C4×D131#4112yes—
102C131⋊C8—21no—
103C131⋊C8C2×D412yes—
104C23×C131—22no—
105C23×C131C2×D411yescharacteristic
106D4×C131—22no—
107D4×C131—22no—
108D4×C131—44no—
109C2×C4×C131—21no—
110C2×C4×C131—21no—
111C2×C4×C131—22no—
112C2×C4×C131C2×D411yescharacteristic
113C2×C4×C131C2411yescharacteristic
114C8×C131—21no—
115C8×C131C2×D412yes—
116C22×C131—21no—
117C22×C131—21no—
118C22×C131—42no—
119C22×C131—42no—
120C22×C131—42no—
121C22×C131#13512yescomplemented
122C22×C131C22×D411yescharacteristic
123C2×D131—28no—
124C2×D131—21no—
125C2×D131—21no—
126C2×D131—21no—
127C2×D131—21no—
128C2×D131#12112yes—
129C2×D131#13512yescomplemented
130C4×C131—21no—
131C4×C131—42no—
132C4×C131C22×D411yescharacteristic
133C4×C131C2×#4111yescharacteristic
134Dic131—21no—
135Dic131—42no—
136Dic131—42no—
137Dic131#12112yes—
138C2×C131—22no—
139C2×C131—82no—
140C2×C131C2×#12111yescharacteristic
141C2×C131C2×#13511yescharacteristic
142C2×C131C2×#13511yescharacteristic
143D131—22no—
144D131—22no—
145C131#305011yescharacteristic, complemented, Sylow
146#3050—1311nomaximal, Sylow
147C2×#135—1311no—
148C23×C8—1311no—
149#525—1312no—
150#467—1311no—
151#467—1311no—
152#466—1311no—
153#466D13111yescharacteristic, complemented
154#465—1312no—
155#465—1312no—
156#427—1311no—
157#416—1312no—
158C2×#41—1312no—
159C23×C4—1311no—
160#135—2624no—
161C2×Q16—1311no—
162C2×SD16—1312no—
163C2×D8C2×D13111yescharacteristic, complemented
164C22×C8—2624no—
165C22×C8—1311no—
166C22×C8—1311no—
167C22×C8—1311no—
168C22×C8—1311no—
169C22×C8—1311no—
170C22×C8C2×D13111yescharacteristic
171#123—2622no—
172C22⋊Q8—1312no—
173C22⋊Q8—1312no—
174#121—1312no—
175#121C2×D13112yes—
176#120—2622no—
177C4×D4—2622no—
178C4×D4—2622no—
179#117—1312no—
180#107—1311no—
181#107C2×D13111yescharacteristic, complemented
182#106—1312no—
183#103—1312no—
184#103—1312no—
185#102—1312no—
186#102C2×D13112yescomplemented
187C24—1311no—
188#41—2624no—
189#41—2624no—
190C2×Q8—1312no—
191C2×D4—22no—
192C2×D4—2622no—
193C2×D4—2622no—
194C2×D4C22×D13112yes—
195C22×C4—2622no—
196C22×C4—2622no—
197C22×C4—2622no—
198C22×C4—1311no—
199C22×C4—1311no—
200C22×C4—1311no—
201C22×C4—1311no—
202C22×C4—1311no—
203C22×C4C22×D13111yescharacteristic
204Q16—2622no—
205SD16—2624no—
206D8—22no—
207C2×C8—22no—
208C2×C8—2622no—
209C2×C8—2622no—
210C2×C8—2622no—
211C2×C8—2622no—
212C2×C8—1311no—
213C2×C8—1311no—
214C2×C8—1314no—
215C2×C8C22×D13111yescharacteristic
216C2×C8C22×D13111yescharacteristic
217C4⋊C4—2622no—
218C4⋊C4—2622no—
219C4⋊C4—1312no—
220C4⋊C4C22×D13112yes—
221C22⋊C4—22no—
222C22⋊C4—2622no—
223C22⋊C4—2622no—
224C22⋊C4—2622no—
225C22⋊C4—2622no—
226C42—2622no—
227C23—22no—
228C23—2622no—
229C23—2622no—
230C23—2622no—
231C23—1311no—
232C23—1311no—
233C23D4×D13111yescharacteristic
234Q8—2622no—
235Q8—2622no—
236D4—22no—
237D4—22no—
238D4—44no—
239D4—5244no—
240D4—5244no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—2622no—
245C2×C4—2622no—
246C2×C4—2622no—
247C2×C4—2621no—
248C2×C4—2621no—
249C2×C4—2622no—
250C2×C4—2622no—
251C2×C4—5244no—
252C2×C4—1311no—
253C2×C4—1311no—
254C2×C4—1312no—
255C2×C4—1312no—
256C2×C4D4×D13111yescharacteristic
257C2×C4C23×D13111yescharacteristic
258C8—21no—
259C8—2621no—
260C8—1312no—
261C8D4×D13112yes—
262C22—21no—
263C22—21no—
264C22—42no—
265C22—42no—
266C22—42no—
267C22—2628no—
268C22—2621no—
269C22—2621no—
270C22—2621no—
271C22—2621no—
272C22—1312no—
273C22—1312no—
274C22#1686226512yescomplemented
275C22C2×D4×D13111yescharacteristic
276C4—21no—
277C4—42no—
278C4—2621no—
279C4—5242no—
280C4—5242no—
281C4—1312no—
282C4C2×D4×D13111yescharacteristic
283C4#41×D13111yescharacteristic
284C2—22no—
285C2—2622no—
286C2—2622no—
287C2—82no—
288C2#121×D13111yescharacteristic
289C2#135×D13111yescharacteristic
290C2C2×#1686226511yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#466×C131C2
socleC22×C131C22×D4
hypercenter#466D131
hypocenterC131#3050
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#16862265C2semidirect
2. C22×C8×D131C2semidirect
3. #22553822C2semidirect
4. #22553554C2semidirect
5. #22553553C2semidirect
6. #22553545C2semidirect
7. #22553540C2semidirect
8. #22553539C2semidirect
9. #22553349C2semidirect
10. #22553305C2semidirect
11. #466×C131C2semidirect
15. C2×Dic524C22semidirect
16. C2×#14523501C22semidirect
31. #16862180C22semidirect
32. #16862177C22semidirect
33. #16862171C22semidirect
34. #16862169C22semidirect
35. #16862167C22semidirect
36. C2×D8×C131C22semidirect
39. #107×C131C22semidirect
40. #102×C131C22semidirect
63. C8×D131D4semidirect
121. C22×C131#135semidirect
129. C2×D131#135semidirect
145. C131#3050semidirect
153. #466D131semidirect
163. C2×D8C2×D131semidirect
181. #107C2×D131semidirect
186. #102C2×D131semidirect
274. C22#16862265semidirect

Maximal subgroups

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

subgroupindex
C2×#168622652
C22×C8×D1312
#225538222
#225535542
#225535532
#225535452
#225535402
#225535392
#225533492
#225533052
#466×C1312
#3050131