Finite Group Databasebeta

← #30675237

Subgroups — #30675237

Order 16768.

752 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16862292C211yesmaximal, characteristic
2C22×#14523500C211yesmaximal, characteristic, complemented
3#22553886C211yesmaximal, characteristic, complemented
4#22553473C214yesmaximal, complemented
5#22553461C214yesmaximal, complemented
6#22553318C211yesmaximal, characteristic, complemented
7#22553308C211yesmaximal, characteristic, complemented
8#22553280C211yesmaximal, characteristic, complemented
9C2×#137×C131C211yesmaximal, characteristic, complemented
10#16862292—24no—
11#16862292—24no—
12C2×#14523514C2212yes—
13C2×#14523513—21no—
14C2×#14523513C2211yescharacteristic
15C22×D524—21no—
16C22×D524C2211yescharacteristic
17C2×D4×D131—22no—
18C2×D4×D131C2212yes—
19C2×#14523500—24no—
20C2×#14523500C2214yescomplemented
21C22×C131⋊C8C2211yescharacteristic, complemented
22#16862244—21no—
23#16862235—21no—
24#16862229—22no—
25#16862222C2214yescomplemented
26#16862205C2211yescharacteristic, complemented
27#16862170C2212yescomplemented
28#16862168C2214yescomplemented
29#16862167C2212yescomplemented
30(C2×D131)⋊C8C2214yescomplemented
31C2×#41×C131C2211yescharacteristic
32C22×Q8×C131C2211yescharacteristic, complemented
33#137×C131—24no—
34C2×Q16×C131C2212yescomplemented
35C2×SD16×C131C2212yescomplemented
36C2×M16×C131C2211yescharacteristic, complemented
37C23×D131—22no—
38C23×D131—21no—
39#14523514—24no—
40#14523514—24no—
41#14523513—28no—
42#14523513—22no—
43#14523513—22no—
44C2×Dic262D411yescharacteristic
45C2×D524—28no—
46C2×D524—22no—
47C2×D524—21no—
48C2×D524—22no—
49C2×D524D411yescharacteristic
50C2×D524D414yes—
51C2×D524C2312yes—
52C2×C22⋊D131—22no—
53C2×C22⋊D131D412yes—
54D4×D131—216no—
55D4×D131—24no—
56D4×D131—24no—
57C2×C4×D131—22no—
58C2×C4×D131C2312yes—
59#14523500—28no—
60C2×C131⋊C8—22no—
61C2×C131⋊C8C2312yes—
62C4⋊Dic131C2312yes—
63Dic131⋊C4—24no—
64C22⋊Dic131—21no—
65(C2×D131)⋊C4—24no—
66C4×Dic131—21no—
67#41×C131—22no—
68#41×C131D414yes—
69C2×Q8×C131—21no—
70C2×Q8×C131—22no—
71C2×Q8×C131—22no—
72C2×Q8×C131C2312yes—
73C2×Q8×C131C2311yescharacteristic
74C2×D4×C131—21no—
75C2×D4×C131C2311yescharacteristic
76C22×C4×C131—21no—
77C22×C4×C131—21no—
78C22×C4×C131C2311yescharacteristic
79Q16×C131—24no—
80SD16×C131—24no—
81M16×C131—22no—
82C2×C8×C131C2312yes—
83C22×D131—22no—
84C22×D131—28no—
85C22×D131—24no—
86C22×D131—84no—
87C22×D131—82no—
88C22×D131C2×D412yes—
89Dic262—22no—
90D524—28no—
91D524—22no—
92D524—24no—
93D524—22no—
94D524C2×D414yes—
95C22⋊D131—28no—
96C22⋊D131—24no—
97C2×Dic131—22no—
98C2×Dic131C2×D412yes—
99C4×D131—28no—
100C4×D131—22no—
101C4×D131—22no—
102C131⋊C8—22no—
103C23×C131—21no—
104C23×C131C2×D411yescharacteristic
105Q8×C131—21no—
106Q8×C131—22no—
107Q8×C131—22no—
108Q8×C131—42no—
109Q8×C131C2×D412yes—
110D4×C131—24no—
111D4×C131—21no—
112D4×C131C2×D412yes—
113C2×C4×C131—21no—
114C2×C4×C131—24no—
115C2×C4×C131—22no—
116C2×C4×C131—42no—
117C2×C4×C131C2×D411yescharacteristic
118C2×C4×C131C2×D412yes—
119C2×C4×C131C2×D412yes—
120C2×C4×C131C2411yescharacteristic
121C8×C131—22no—
122C22×C131—22no—
123C22×C131—22no—
124C22×C131—42no—
125C22×C131#12012yes—
126C22×C131C22×D411yescharacteristic
127C2×D131—22no—
128C2×D131—24no—
129C2×D131—24no—
130C2×D131—44no—
131C2×D131—84no—
132C2×D131—88no—
133C4×C131—22no—
134C4×C131—42no—
135C4×C131#12012yes—
136C4×C131C22×D411yescharacteristic
137C4×C131C22×D411yescharacteristic
138Dic131—22no—
139Dic131—42no—
140C2×C131—22no—
141C2×C131—42no—
142C2×C131C2×#12011yescharacteristic
143C2×C131#57712yes—
144D131—42no—
145D131—44no—
146C131#300911yescharacteristic, complemented, Sylow
147#3009—1311nomaximal, Sylow
148C2×H(5,𝔽2)—1311no—
149C2×#137D13111yescharacteristic, complemented
150C22×SD16—1311no—
151#536—1311no—
152#452—1314no—
153#450—1314no—
154#418—1311no—
155#416—1311no—
156#408—1311no—
157H(5,𝔽2)—2624no—
158H(5,𝔽2)—2624no—
159C2×#41—2621no—
160C2×#41—1312no—
161C2×#41—1311no—
162C2×#41C2×D13111yescharacteristic
163C22×Q8C2×D13111yescharacteristic, complemented
164C22×D4—2621no—
165C22×D4—2622no—
166C22×D4—1312no—
167C22×D4—1311no—
168#137—24no—
169C2×Q16C2×D13112yescomplemented
170C2×SD16—2624no—
171C2×SD16—1314no—
172C2×SD16C2×D13112yescomplemented
173C2×M16C2×D13111yescharacteristic, complemented
174C22×C8—1311no—
175C4⋊Q8—2621no—
176#124—2621no—
177#123—2622no—
178C22⋊Q8—1314no—
179#117—1311no—
180#103—1312no—
181#102—1312no—
182#102—1314no—
183C22⋊C8—1314no—
184C24—2622no—
185C24—2621no—
186#41—22no—
187#41—2628no—
188#41—2624no—
189#41—2624no—
190#41—2622no—
191#41—2622no—
192#41C22⋊D13114yes—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—1311no—
197C2×Q8C22×D13112yes—
198C2×Q8C22×D13111yescharacteristic
199C2×D4—21no—
200C2×D4—2628no—
201C2×D4—2622no—
202C2×D4—26216no—
203C2×D4—2624no—
204C2×D4—2624no—
205C2×D4—2622no—
206C2×D4—2621no—
207C2×D4—2622no—
208C2×D4—1312no—
209C2×D4—1311no—
210C2×D4—1314no—
211C2×D4—1312no—
212C2×D4C22×D13111yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—2622no—
216C22×C4—1312no—
217C22×C4C22×D13111yescharacteristic
218Q16—24no—
219SD16—24no—
220SD16—2628no—
221M16—22no—
222C2×C8—2622no—
223C2×C8—1312no—
224C2×C8C22×D13112yes—
225C4⋊C4—2624no—
226C4⋊C4—1312no—
227C22⋊C4—2624no—
228C22⋊C4—2621no—
229C42—2621no—
230C23—21no—
231C23—2628no—
232C23—2622no—
233C23—2624no—
234C23—10482no—
235C23—10484no—
236C23—1312no—
237C23D4×D13111yescharacteristic
238Q8—21no—
239Q8—22no—
240Q8—22no—
241Q8—42no—
242Q8—2622no—
243Q8C2×C22⋊D13112yes—
244D4—24no—
245D4—21no—
246D4—2628no—
247D4—2628no—
248D4—2624no—
249D4—2622no—
250D4—2624no—
251D4—2622no—
252D4—1314no—
253D4C2×C22⋊D13112yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—2628no—
259C2×C4—2622no—
260C2×C4—2622no—
261C2×C4—2622no—
262C2×C4—1312no—
263C2×C4D4×D13111yescharacteristic
264C2×C4D4×D13112yes—
265C2×C4C2×C22⋊D13112yes—
266C2×C4C23×D13111yescharacteristic
267C8—22no—
268C8—2622no—
269C22—22no—
270C22—22no—
271C22—42no—
272C22—2624no—
273C22—2622no—
274C22—2624no—
275C22—5244no—
276C22—10484no—
277C22—10488no—
278C22#1686221412yes—
279C22C2×D4×D13111yescharacteristic
280C4—22no—
281C4—42no—
282C4—2622no—
283C4—5242no—
284C4#1686221412yes—
285C4C2×D4×D13111yescharacteristic
286C4C22×C22⋊D13111yescharacteristic
287C2—22no—
288C2—42no—
289C2—5242no—
290C2—5244no—
291C2C2×#1686221411yescharacteristic
292C2#2255412112yes—
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC2×#137×C131C2
socleC22×C131C22×D4
prime radicalC2×#16862292C2
hypercenterC2×#137D131
hypocenterC131#3009
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C22×#14523500C2semidirect
3. #22553886C2semidirect
4. #22553473C2semidirect
5. #22553461C2semidirect
6. #22553318C2semidirect
7. #22553308C2semidirect
8. #22553280C2semidirect
9. C2×#137×C131C2semidirect
20. C2×#14523500C22semidirect
21. C22×C131⋊C8C22semidirect
25. #16862222C22semidirect
26. #16862205C22semidirect
27. #16862170C22semidirect
28. #16862168C22semidirect
29. #16862167C22semidirect
30. (C2×D131)⋊C8C22semidirect
32. C22×Q8×C131C22semidirect
34. C2×Q16×C131C22semidirect
35. C2×SD16×C131C22semidirect
36. C2×M16×C131C22semidirect
146. C131#3009semidirect
149. C2×#137D131semidirect
163. C22×Q8C2×D131semidirect
169. C2×Q16C2×D131semidirect
172. C2×SD16C2×D131semidirect
173. C2×M16C2×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622922
C22×#145235002
#225538862
#225534732
#225534612
#225533182
#225533082
#225532802
C2×#137×C1312
#3009131