Finite Group Databasebeta

← #30675239

Subgroups — #30675239

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×#16862291C211yesmaximal, characteristic
2C2×#16862279C211yesmaximal, characteristic, complemented
3C22×#14523501C211yesmaximal, characteristic, complemented
4#22553888C211yesmaximal, characteristic, complemented
5#22553475C214yesmaximal, complemented
6#22553464C214yesmaximal, complemented
7#22553306C211yesmaximal, characteristic, complemented
8#22553280C211yesmaximal, characteristic, complemented
9#418×C131C211yesmaximal, characteristic, complemented
10#16862291—24no—
11#16862291—24no—
12C2×#14523513C2211yescharacteristic
13C2×#14523513C2211yescharacteristic
14C2×#14523512—21no—
15C2×#14523512C2212yes—
16C22×Dic262C2211yescharacteristic, complemented
17C22×C22⋊D131—22no—
18C2×D4×D131—21no—
19C2×D4×D131C2212yes—
20#16862279—24no—
21C2×Dic524C2212yescomplemented
22C2×#14523502C2212yescomplemented
23C2×#14523501—24no—
24C2×#14523501C2214yescomplemented
25C22×C131⋊C8C2211yescharacteristic, complemented
26#16862246—21no—
27#16862234—21no—
28#16862230—22no—
29#16862226C2214yescomplemented
30#16862169C2212yescomplemented
31#16862166C2212yescomplemented
32(C2×D131)⋊C8C2214yescomplemented
33C22×D4×C131C2211yescharacteristic
34C2×M16×C131C2211yescharacteristic, complemented
35#117×C131C2211yescharacteristic, complemented
36#102×C131C2214yescomplemented
37C23×D131—22no—
38#14523513—22no—
39#14523513—22no—
40#14523513—22no—
41#14523513D414yes—
42#14523512—28no—
43#14523512—24no—
44#14523512—24no—
45C2×Dic262—21no—
46C2×Dic262—22no—
47C2×Dic262—22no—
48C2×Dic262D411yescharacteristic
49C2×Dic262C2312yes—
50C2×Dic262C2311yescharacteristic
51C2×D524D411yescharacteristic
52C2×D524C2311yescharacteristic
53C2×C22⋊D131—216no—
54C2×C22⋊D131—22no—
55C2×C22⋊D131—22no—
56C2×C22⋊D131—21no—
57C2×C22⋊D131D412yes—
58D4×D131—28no—
59D4×D131—24no—
60D4×D131—24no—
61C22×Dic131—22no—
62C22×Dic131—21no—
63C2×C4×D131—21no—
64C2×C4×D131C2312yes—
65Dic524—24no—
66#14523502—24no—
67#14523501—28no—
68C2×C131⋊C8—22no—
69C2×C131⋊C8C2312yes—
70C4⋊Dic131—24no—
71(C2×D131)⋊C4—24no—
72C24×C131—21no—
73C2×D4×C131—21no—
74C2×D4×C131—22no—
75C2×D4×C131D414yes—
76C2×D4×C131C2312yes—
77C22×C4×C131C2311yescharacteristic
78M16×C131—22no—
79C2×C8×C131C2312yes—
80C4⋊C4×C131C2312yes—
81C22⋊C4×C131—21no—
82C42×C131—21no—
83C22×D131—21no—
84C22×D131—28no—
85C22×D131—84no—
86C22×D131C2×D412yes—
87Dic262—22no—
88Dic262—21no—
89Dic262—22no—
90Dic262—22no—
91Dic262—42no—
92Dic262C2×D412yes—
93D524—22no—
94D524—21no—
95D524C2×D412yes—
96C22⋊D131—28no—
97C22⋊D131—28no—
98C22⋊D131—24no—
99C22⋊D131—24no—
100C2×Dic131—28no—
101C2×Dic131—21no—
102C2×Dic131—22no—
103C2×Dic131—42no—
104C2×Dic131C2×D412yes—
105C4×D131—24no—
106C4×D131—22no—
107C4×D131—22no—
108C131⋊C8—22no—
109C23×C131—22no—
110C23×C131—24no—
111C23×C131—82no—
112C23×C131C2×D411yescharacteristic
113D4×C131—24no—
114D4×C131—22no—
115D4×C131C2×D414yes—
116C2×C4×C131—22no—
117C2×C4×C131C2×D411yescharacteristic
118C2×C4×C131C2×D412yes—
119C2×C4×C131C2×D412yes—
120C2×C4×C131C2411yescharacteristic
121C8×C131—22no—
122C22×C131—22no—
123C22×C131—24no—
124C22×C131—44no—
125C22×C131—84no—
126C22×C131#12012yes—
127C22×C131C22×D411yescharacteristic
128C2×D131—22no—
129C2×D131—22no—
130C2×D131—24no—
131C2×D131—42no—
132C2×D131—88no—
133C4×C131—42no—
134C4×C131#12012yes—
135C4×C131C22×D411yescharacteristic
136C4×C131C22×D411yescharacteristic
137Dic131—22no—
138Dic131—22no—
139Dic131—42no—
140C2×C131—22no—
141C2×C131—44no—
142C2×C131C2×#12011yescharacteristic
143C2×C131#57712yes—
144D131—42no—
145D131—42no—
146C131#300911yescharacteristic, complemented, Sylow
147#3009—1311nomaximal, Sylow
148C2×H(5,𝔽2)—1311no—
149C2×#137—1311no—
150C22×SD16—1311no—
151#536—1311no—
152#452—1314no—
153#450—1314no—
154#418D13111yescharacteristic, complemented
155#416—1311no—
156#408—1311no—
157H(5,𝔽2)—2624no—
158H(5,𝔽2)—2624no—
159C2×#41—2621no—
160C2×#41—1312no—
161C2×#41—1311no—
162C2×#41—1311no—
163C22×Q8—1311no—
164C22×D4—2621no—
165C22×D4—2622no—
166C22×D4—1312no—
167C22×D4C2×D13111yescharacteristic
168#137—2624no—
169C2×Q16—1312no—
170C2×SD16—2624no—
171C2×SD16—1314no—
172C2×SD16—1312no—
173C2×M16C2×D13111yescharacteristic, complemented
174C22×C8—1311no—
175C4⋊Q8—2621no—
176#124—2621no—
177#123—2622no—
178C22⋊Q8—1314no—
179#117C2×D13111yescharacteristic, complemented
180#103—1312no—
181#102—1312no—
182#102C2×D13114yescomplemented
183C22⋊C8—1314no—
184C24—21no—
185C24—2622no—
186#41—2628no—
187#41—2624no—
188#41—2624no—
189#41—2622no—
190#41—2622no—
191#41—2622no—
192#41—1314no—
193C2×Q8—2621no—
194C2×Q8—2622no—
195C2×Q8—2622no—
196C2×Q8—1312no—
197C2×Q8—1311no—
198C2×Q8—1311no—
199C2×D4—21no—
200C2×D4—22no—
201C2×D4—2628no—
202C2×D4—2624no—
203C2×D4—26216no—
204C2×D4—2622no—
205C2×D4—2621no—
206C2×D4—2624no—
207C2×D4—2622no—
208C2×D4—1312no—
209C2×D4—1311no—
210C2×D4—1311no—
211C2×D4D52414yes—
212C2×D4C22×D13112yes—
213C22×C4—2622no—
214C22×C4—2621no—
215C22×C4—2621no—
216C22×C4—1312no—
217C22×C4C22×D13111yescharacteristic
218Q16—2624no—
219SD16—2624no—
220SD16—2628no—
221M16—22no—
222C2×C8—2622no—
223C2×C8—1312no—
224C2×C8C22×D13112yes—
225C4⋊C4—2624no—
226C4⋊C4C22×D13112yes—
227C22⋊C4—21no—
228C22⋊C4—2624no—
229C42—21no—
230C23—22no—
231C23—24no—
232C23—2628no—
233C23—2621no—
234C23—82no—
235C23—10484no—
236C23—1312no—
237C23D4×D13111yescharacteristic
238Q8—2622no—
239Q8—2621no—
240Q8—2622no—
241Q8—2622no—
242Q8—5242no—
243Q8—1312no—
244D4—24no—
245D4—22no—
246D4—2628no—
247D4—2628no—
248D4—2624no—
249D4—2624no—
250D4—2622no—
251D4—2621no—
252D4—1312no—
253D4C2×D52414yes—
254C2×C4—22no—
255C2×C4—2628no—
256C2×C4—2621no—
257C2×C4—2624no—
258C2×C4—2622no—
259C2×C4—2622no—
260C2×C4—2622no—
261C2×C4—5242no—
262C2×C4—1312no—
263C2×C4D4×D13111yescharacteristic
264C2×C4D4×D13112yes—
265C2×C4C2×D52412yes—
266C2×C4C23×D13111yescharacteristic
267C8—22no—
268C8—2622no—
269C22—22no—
270C22—24no—
271C22—44no—
272C22—2624no—
273C22—2622no—
274C22—2622no—
275C22—84no—
276C22—5242no—
277C22—10488no—
278C22#1686221612yes—
279C22C2×D4×D13111yescharacteristic
280C4—42no—
281C4—2622no—
282C4—2622no—
283C4—5242no—
284C4#1686221612yes—
285C4C2×D4×D13111yescharacteristic
286C4C22×D52411yescharacteristic
287C2—22no—
288C2—44no—
289C2—5242no—
290C2—5242no—
291C2C2×#1686221611yescharacteristic
292C2#2255412212yes—
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#418×C131C2
socleC22×C131C22×D4
prime radicalC2×#16862291C2
hypercenter#418D131
hypocenterC131#3009
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. C2×#16862279C2semidirect
3. C22×#14523501C2semidirect
4. #22553888C2semidirect
5. #22553475C2semidirect
6. #22553464C2semidirect
7. #22553306C2semidirect
8. #22553280C2semidirect
9. #418×C131C2semidirect
16. C22×Dic262C22semidirect
21. C2×Dic524C22semidirect
22. C2×#14523502C22semidirect
24. C2×#14523501C22semidirect
25. C22×C131⋊C8C22semidirect
29. #16862226C22semidirect
30. #16862169C22semidirect
31. #16862166C22semidirect
32. (C2×D131)⋊C8C22semidirect
34. C2×M16×C131C22semidirect
35. #117×C131C22semidirect
36. #102×C131C22semidirect
146. C131#3009semidirect
154. #418D131semidirect
173. C2×M16C2×D131semidirect
179. #117C2×D131semidirect
182. #102C2×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622912
C2×#168622792
C22×#145235012
#225538882
#225534752
#225534642
#225533062
#225532802
#418×C1312
#3009131