Finite Group Databasebeta

← #30675569

Subgroups — #30675569

Order 16768.

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

normal subgroups only

This group has 500 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×#16862266C211yesmaximal, characteristic, complemented
2C22×#14523496C211yesmaximal, characteristic, complemented
3#22553824C211yesmaximal, characteristic, complemented
4#22553819C211yesmaximal, characteristic, complemented
5#22553569C212yesmaximal, complemented
6#22553564C212yesmaximal, complemented
7#22553561C212yesmaximal, complemented
8#22553557C212yesmaximal, complemented
9#22553348C211yesmaximal, characteristic, complemented
10#22553309C211yesmaximal, characteristic, complemented
11#416×C131C211yesmaximal, characteristic, complemented
12C2×#14523513C2211yescharacteristic
13C22×C4×D131C2211yescharacteristic
14#16862266—24no—
15C2×#14523503C2211yescharacteristic, complemented
16C2×#14523501C2211yescharacteristic, complemented
17C2×#14523500C2211yescharacteristic, complemented
18C2×#14523498C2211yescharacteristic, complemented
19C2×#14523496—24no—
20C2×#14523496C2212yes—
21C2×#14523496C2212yes—
22C22×C131⋊C8C2211yescharacteristic
23#16862230—21no—
24#16862227—21no—
25#16862226C2212yes—
26#16862222C2212yes—
27#16862221C2212yes—
28#16862217C2212yes—
29#16862216—21no—
30#16862214—21no—
31C4×D524—22no—
32C4×C22⋊D131—22no—
33#16862205C2211yescharacteristic
34#16862179C2212yescomplemented
35#16862177C2212yescomplemented
36#16862171C2212yescomplemented
37#16862168C2212yescomplemented
38C2×#41×C131C2211yescharacteristic
39C22×C8×C131C2211yescharacteristic
40#117×C131C2211yescharacteristic
41#103×C131C2212yescomplemented
42#102×C131C2212yescomplemented
43C23×D131D411yescharacteristic
44#14523513—22no—
45#14523513—22no—
46C2×Dic262C2311yescharacteristic
47C2×D524—22no—
48C2×D524C2311yescharacteristic
49C2×C22⋊D131—22no—
50C2×C22⋊D131—21no—
51C22×Dic131D411yescharacteristic
52C2×C4×D131—22no—
53C2×C4×D131—21no—
54C2×C4×D131—22no—
55C2×C4×D131D412yes—
56C2×C4×D131C2312yes—
57#14523503—22no—
58#14523501—22no—
59#14523500—22no—
60#14523498—22no—
61#14523496—24no—
62#14523496—24no—
63C2×C131⋊C8D414yescomplemented
64C2×C131⋊C8C2311yescharacteristic
65C2×C131⋊C8C2311yescharacteristic
66C4⋊Dic131—22no—
67C4⋊Dic131C2312yes—
68Dic131⋊C4—22no—
69C22⋊Dic131—22no—
70C22⋊Dic131—21no—
71(C2×D131)⋊C4—22no—
72(C2×D131)⋊C4—22no—
73(C2×D131)⋊C4—22no—
74C4×Dic131—21no—
75#41×C131—22no—
76#41×C131—22no—
77C2×Q8×C131C2311yescharacteristic
78C2×D4×C131—21no—
79C2×D4×C131C2311yescharacteristic
80C22×C4×C131—21no—
81C22×C4×C131C2311yescharacteristic
82C2×C8×C131—22no—
83C2×C8×C131C2312yes—
84C4⋊C4×C131C2312yes—
85C22⋊C4×C131—21no—
86C42×C131—21no—
87C22×D131—22no—
88C22×D131—21no—
89C22×D131—42no—
90C22×D131C2×D412yes—
91Dic262—21no—
92Dic262—21no—
93D524—21no—
94D524—21no—
95D524—44no—
96C22⋊D131—44no—
97C22⋊D131—42no—
98C2×Dic131—22no—
99C2×Dic131—22no—
100C2×Dic131—21no—
101C2×Dic131C2×D412yes—
102C4×D131—24no—
103C4×D131—22no—
104C4×D131—22no—
105C4×D131—42no—
106C131⋊C8C2×D412yes—
107C131⋊C8C2×D412yes—
108C23×C131—21no—
109C23×C131C2×D411yescharacteristic
110Q8×C131—21no—
111Q8×C131—21no—
112D4×C131—21no—
113D4×C131—21no—
114D4×C131—42no—
115C2×C4×C131—22no—
116C2×C4×C131—21no—
117C2×C4×C131—42no—
118C2×C4×C131C2×D412yes—
119C2×C4×C131C2×D411yescharacteristic
120C2×C4×C131#4112yes—
121C2×C4×C131C2411yescharacteristic
122C8×C131—22no—
123C22×C131—22no—
124C22×C131—41no—
125C22×C131—42no—
126C22×C131#12112yes—
127C22×C131C22×D411yescharacteristic
128C2×D131—22no—
129C2×D131—22no—
130C2×D131—22no—
131C2×D131—44no—
132C2×D131—41no—
133C2×D131—42no—
134C4×C131—41no—
135C4×C131—42no—
136C4×C131#12112yes—
137C4×C131C22×D411yescharacteristic
138C4×C131C2×#4111yescharacteristic
139Dic131—22no—
140Dic131—41no—
141Dic131—42no—
142C2×C131—22no—
143C2×C131—81no—
144C2×C131C2×#12111yescharacteristic
145C2×C131#57512yes—
146D131—42no—
147D131—81no—
148C131#305411yescharacteristic, complemented, Sylow
149#3054—1311nomaximal, Sylow
150C2×#135—1311no—
151C22×M16—1311no—
152#525—1311no—
153#525—1311no—
154#470—1312no—
155#469—1312no—
156#468—1312no—
157#468—1312no—
158#427—1311no—
159#416—1311no—
160#416D13111yescharacteristic, complemented
161C2×#41—1311no—
162C2×#41C2×D13111yescharacteristic
163C23×C4—1311no—
164#135—2624no—
165C2×Q16—1311no—
166C2×SD16—1311no—
167C2×SD16—1311no—
168C2×D8—1311no—
169C2×M16—2624no—
170C2×M16—1312no—
171C2×M16—1312no—
172C22×C8—1311no—
173C22×C8C2×D13111yescharacteristic
174#123—2621no—
175#123—2621no—
176C22⋊Q8—1312no—
177C22⋊Q8—1312no—
178#121—1312no—
179#121—1312no—
180#120—2621no—
181#120—2621no—
182C4×D4—2622no—
183C4×D4—2622no—
184#117—1311no—
185#117C2×D13111yescharacteristic
186#107—1312no—
187#106—1312no—
188#103—1312no—
189#103C2×D13112yescomplemented
190#102—1312no—
191#102C2×D13112yescomplemented
192C24—1311no—
193#41—22no—
194#41—22no—
195#41—2622no—
196#41—2622no—
197C2×Q8—1311no—
198C2×Q8C22×D13111yescharacteristic
199C2×D4—21no—
200C2×D4—2622no—
201C2×D4—2622no—
202C2×D4—2621no—
203C2×D4—1311no—
204C2×D4C22×D13111yescharacteristic
205C22×C4—21no—
206C22×C4—2622no—
207C22×C4—2622no—
208C22×C4—2621no—
209C22×C4—1311no—
210C22×C4—1312no—
211C22×C4—1312no—
212C22×C4C22×D13111yescharacteristic
213Q16—2622no—
214SD16—2622no—
215SD16—2622no—
216D8—2622no—
217M16—2624no—
218M16—2624no—
219C2×C8—22no—
220C2×C8—1314no—
221C2×C8—1311no—
222C2×C8—1311no—
223C2×C8C22×D13112yes—
224C4⋊C4—2622no—
225C4⋊C4—2622no—
226C4⋊C4—1312no—
227C4⋊C4C22×D13112yes—
228C22⋊C4—21no—
229C22⋊C4—2622no—
230C22⋊C4—2622no—
231C22⋊C4—2622no—
232C22⋊C4—2621no—
233C22⋊C4—2622no—
234C42—21no—
235C42—2621no—
236C23—21no—
237C23—2622no—
238C23—2621no—
239C23—5242no—
240C23—1312no—
241C23D4×D13111yescharacteristic
242Q8—21no—
243Q8—21no—
244Q8—2621no—
245Q8—2621no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—2621no—
250D4—2621no—
251D4—5244no—
252D4—5244no—
253D4—5242no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—2622no—
258C2×C4—2624no—
259C2×C4—2622no—
260C2×C4—2622no—
261C2×C4—2622no—
262C2×C4—2621no—
263C2×C4—5242no—
264C2×C4—1312no—
265C2×C4D4×D13112yes—
266C2×C4D4×D13111yescharacteristic
267C2×C4#1452351312yes—
268C2×C4C23×D13111yescharacteristic
269C8—22no—
270C8—1312no—
271C8—1312no—
272C22—22no—
273C22—41no—
274C22—42no—
275C22—2622no—
276C22—2622no—
277C22—2622no—
278C22—5244no—
279C22—5241no—
280C22—5242no—
281C22#1686221912yes—
282C22C2×D4×D13111yescharacteristic
283C4—41no—
284C4—42no—
285C4—2622no—
286C4—5241no—
287C4—5242no—
288C4#1686221912yes—
289C4C2×D4×D13111yescharacteristic
290C4C2×#1452351311yescharacteristic
291C2—22no—
292C2—81no—
293C2—5242no—
294C2—10481no—
295C2C2×#1686221911yescharacteristic
296C2#2255410912yes—
subgroupG/H
centerC2×C4D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#416×C131C2
socleC22×C131C22×D4
hypercenter#416D131
hypocenterC131#3054
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#16862266C2semidirect
2. C22×#14523496C2semidirect
3. #22553824C2semidirect
4. #22553819C2semidirect
5. #22553569C2semidirect
6. #22553564C2semidirect
7. #22553561C2semidirect
8. #22553557C2semidirect
9. #22553348C2semidirect
10. #22553309C2semidirect
11. #416×C131C2semidirect
15. C2×#14523503C22semidirect
16. C2×#14523501C22semidirect
17. C2×#14523500C22semidirect
18. C2×#14523498C22semidirect
34. #16862179C22semidirect
35. #16862177C22semidirect
36. #16862171C22semidirect
37. #16862168C22semidirect
41. #103×C131C22semidirect
42. #102×C131C22semidirect
63. C2×C131⋊C8D4semidirect
148. C131#3054semidirect
160. #416D131semidirect
189. #103C2×D131semidirect
191. #102C2×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622662
C22×#145234962
#225538242
#225538192
#225535692
#225535642
#225535612
#225535572
#225533482
#225533092
#416×C1312
#3054131