Finite Group Databasebeta

← #30675505

Subgroups — #30675505

Order 16768.

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

normal subgroups only

This group has 504 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×#16862272C211yesmaximal, characteristic, complemented
2C2×#16862268C211yesmaximal, characteristic, complemented
3#22553879C211yesmaximal, characteristic, complemented
4#22553793C211yesmaximal, characteristic, complemented
5#22553533C212yesmaximal, complemented
6#22553526C212yesmaximal, complemented
7#22553511C212yesmaximal, complemented
8#22553506C212yesmaximal, complemented
9#22553338C211yesmaximal, characteristic, complemented
10#22553311C211yesmaximal, characteristic, complemented
11C2×#102×C131C211yesmaximal, characteristic, complemented
12C2×#14523513C2211yescharacteristic
13C2×#14523513C2211yescharacteristic
14#16862272—24no—
15#16862268—24no—
16C2×Dic524C2211yescharacteristic, complemented
17C2×#14523502C2211yescharacteristic, complemented
18C2×#14523502C2211yescharacteristic, complemented
19C2×#14523501C2212yes—
20C2×D1048C2211yescharacteristic, complemented
21C2×#14523498C2212yes—
22C2×#14523497C2211yescharacteristic
23#16862242C2212yes—
24#16862233C2212yes—
25#16862217—21no—
26#16862217—21no—
27#16862215—22no—
28#16862211C2212yescomplemented
29#16862208C2212yescomplemented
30C4×C22⋊D131—22no—
31#16862205C2211yescharacteristic
32#16862204—21no—
33C4⋊C4×D131—21no—
34Dic131⋊C8C2212yescomplemented
35Dic131⋊C8C2212yescomplemented
36#16862169C2212yescomplemented
37#16862166C2212yescomplemented
38C22×D4×C131C2211yescharacteristic
39C22×C8×C131C2211yescharacteristic
40C2×C4⋊C4×C131C2211yescharacteristic
41#102×C131C2212yes—
42#102×C131C2212yes—
43#14523513—22no—
44#14523513—22no—
45#14523513—22no—
46#14523513D414yescomplemented
47C2×Dic262C2311yescharacteristic
48C2×Dic262C2311yescharacteristic
49C2×D524C2311yescharacteristic
50C2×D524C2311yescharacteristic
51C2×C22⋊D131—22no—
52C2×C22⋊D131—22no—
53C2×C22⋊D131—21no—
54C2×C22⋊D131—21no—
55C2×C4×D131—22no—
56C2×C4×D131—21no—
57C2×C4×D131—21no—
58Dic524—22no—
59#14523502—22no—
60#14523502—22no—
61#14523501—24no—
62D1048—22no—
63#14523498—24no—
64#14523497—22no—
65C2×C131⋊C8C2312yes—
66C4⋊Dic131D411yescharacteristic
67C4⋊Dic131D411yescharacteristic
68Dic131⋊C4—22no—
69C22⋊Dic131—22no—
70C22⋊Dic131—22no—
71C22⋊Dic131D412yes—
72(C2×D131)⋊C4—22no—
73C4×Dic131—22no—
74C4×Dic131C2312yes—
75C24×C131—21no—
76C2×D4×C131—22no—
77C2×D4×C131—21no—
78C2×D4×C131—22no—
79C2×D4×C131C2312yes—
80C22×C4×C131—21no—
81C22×C4×C131C2311yescharacteristic
82C2×C8×C131—22no—
83C2×C8×C131C2311yescharacteristic
84C2×C8×C131C2311yescharacteristic
85C4⋊C4×C131—21no—
86C4⋊C4×C131C2312yes—
87C22×D131—21no—
88C22×D131—21no—
89Dic262—21no—
90Dic262—21no—
91Dic262—21no—
92Dic262C2×D412yes—
93D524—21no—
94D524—21no—
95D524—21no—
96D524C2×D412yes—
97C22⋊D131—24no—
98C22⋊D131—42no—
99C22⋊D131—82no—
100C22⋊D131—82no—
101C2×Dic131—21no—
102C2×Dic131—21no—
103C2×Dic131C2×D412yes—
104C2×Dic131C2×D412yes—
105C4×D131—24no—
106C4×D131—42no—
107C4×D131—82no—
108C131⋊C8—22no—
109C23×C131—22no—
110C23×C131—42no—
111C23×C131—81no—
112C23×C131—81no—
113C23×C131C2×D411yescharacteristic
114D4×C131—22no—
115D4×C131—22no—
116D4×C131—42no—
117C2×C4×C131—22no—
118C2×C4×C131—81no—
119C2×C4×C131C2×D411yescharacteristic
120C2×C4×C131C2×D412yes—
121C2×C4×C131#4112yes—
122C2×C4×C131C2411yescharacteristic
123C8×C131—21no—
124C8×C131—21no—
125C22×C131—21no—
126C22×C131—21no—
127C22×C131—42no—
128C22×C131—42no—
129C22×C131—42no—
130C22×C131—84no—
131C22×C131#12112yes—
132C22×C131C22×D411yescharacteristic
133C2×D131—22no—
134C2×D131—41no—
135C2×D131—41no—
136C2×D131—41no—
137C2×D131—41no—
138C2×D131—41no—
139C4×C131—42no—
140C4×C131#12112yes—
141C4×C131C22×D411yescharacteristic
142C4×C131C2×#4111yescharacteristic
143Dic131—22no—
144Dic131—22no—
145Dic131—22no—
146Dic131—41no—
147C2×C131—22no—
148C2×C131—82no—
149C2×C131C2×#12111yescharacteristic
150C2×C131#57611yescharacteristic
151C2×C131#57711yescharacteristic
152D131—42no—
153D131—81no—
154C131#304511yescharacteristic, complemented, Sylow
155#3045—1311nomaximal, Sylow
156C2×AGL(1,ℤ/8)—1311no—
157C2×#135—1311no—
158#535—1311no—
159#520—1311no—
160#464—1312no—
161#463—1312no—
162#460—1312no—
163#459—1312no—
164#424—1311no—
165#417—1311no—
166C2×#102D13111yescharacteristic, complemented
167C2×#41—1311no—
168C2×#41—1311no—
169C22×D4C2×D13111yescharacteristic
170AGL(1,ℤ/8)—2624no—
171#135—2624no—
172C2×Q16—1311no—
173C2×SD16—1312no—
174C2×SD16—1311no—
175C2×SD16—1311no—
176C2×D8—1312no—
177C2×D8—1311no—
178C2×M16—1311no—
179C22×C8C2×D13111yescharacteristic
180#127—1312no—
181#124—1312no—
182#121—2621no—
183#121—2621no—
184#120—2622no—
185C4×Q8—1312no—
186C4×D4—2622no—
187C4×D4—1312no—
188#117—2621no—
189#117—1311no—
190C2×C4⋊C4—2621no—
191C2×C4⋊C4C2×D13111yescharacteristic
192C4⋊C8—1312no—
193C4⋊C8—1312no—
194#103—1312no—
195#102—1312no—
196#102C2×D13112yes—
197#102C2×D13112yes—
198C24—21no—
199#41—2622no—
200#41—2622no—
201#41—2622no—
202#41—1314no—
203C2×Q8—1311no—
204C2×Q8—1311no—
205C2×D4—22no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4—2621no—
209C2×D4—2621no—
210C2×D4—2622no—
211C2×D4—2622no—
212C2×D4—1311no—
213C2×D4—1311no—
214C2×D4C22×D13112yes—
215C22×C4—21no—
216C22×C4—2622no—
217C22×C4—2621no—
218C22×C4—2621no—
219C22×C4C22×D13111yescharacteristic
220Q16—2622no—
221SD16—2622no—
222SD16—2622no—
223SD16—2624no—
224D8—2622no—
225D8—2624no—
226M16—2622no—
227C2×C8—22no—
228C2×C8—1312no—
229C2×C8C22×D13111yescharacteristic
230C2×C8C22×D13111yescharacteristic
231C4⋊C4—21no—
232C4⋊C4—2622no—
233C4⋊C4—1311no—
234C4⋊C4—1311no—
235C4⋊C4C22×D13112yes—
236C22⋊C4—2622no—
237C22⋊C4—2622no—
238C22⋊C4—2622no—
239C22⋊C4—1312no—
240C42—2622no—
241C42—1312no—
242C23—22no—
243C23—42no—
244C23—2621no—
245C23—2621no—
246C23—81no—
247C23—81no—
248C23D4×D13111yescharacteristic
249Q8—2621no—
250Q8—2621no—
251Q8—2621no—
252Q8—1312no—
253D4—22no—
254D4—22no—
255D4—42no—
256D4—2624no—
257D4—2621no—
258D4—2621no—
259D4—2621no—
260D4—5242no—
261D4—10482no—
262D4—10482no—
263D4—1312no—
264C2×C4—22no—
265C2×C4—2621no—
266C2×C4—2624no—
267C2×C4—2621no—
268C2×C4—81no—
269C2×C4—5242no—
270C2×C4—10482no—
271C2×C4—1312no—
272C2×C4—1312no—
273C2×C4D4×D13111yescharacteristic
274C2×C4D4×D13112yes—
275C2×C4#1452351312yes—
276C2×C4C23×D13111yescharacteristic
277C8—21no—
278C8—21no—
279C8—2622no—
280C22—21no—
281C22—21no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—2622no—
286C22—84no—
287C22—5241no—
288C22—5241no—
289C22—5241no—
290C22—5241no—
291C22—5241no—
292C22#1686221912yes—
293C22C2×D4×D13111yescharacteristic
294C4—42no—
295C4—2622no—
296C4—2622no—
297C4—2622no—
298C4—5241no—
299C4#1686221912yes—
300C4C2×D4×D13111yescharacteristic
301C4C2×#1452351311yescharacteristic
302C2—22no—
303C2—82no—
304C2—5242no—
305C2—10481no—
306C2C2×#1686221911yescharacteristic
307C2#2255411311yescharacteristic
308C2#2255411811yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC2×#102×C131C2
socleC22×C131C22×D4
hypercenterC2×#102D131
hypocenterC131#3045
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#16862272C2semidirect
2. C2×#16862268C2semidirect
3. #22553879C2semidirect
4. #22553793C2semidirect
5. #22553533C2semidirect
6. #22553526C2semidirect
7. #22553511C2semidirect
8. #22553506C2semidirect
9. #22553338C2semidirect
10. #22553311C2semidirect
11. C2×#102×C131C2semidirect
16. C2×Dic524C22semidirect
17. C2×#14523502C22semidirect
18. C2×#14523502C22semidirect
20. C2×D1048C22semidirect
28. #16862211C22semidirect
29. #16862208C22semidirect
34. Dic131⋊C8C22semidirect
35. Dic131⋊C8C22semidirect
36. #16862169C22semidirect
37. #16862166C22semidirect
46. #14523513D4semidirect
154. C131#3045semidirect
166. C2×#102D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622722
C2×#168622682
#225538792
#225537932
#225535332
#225535262
#225535112
#225535062
#225533382
#225533112
C2×#102×C1312
#3045131