Finite Group Databasebeta

← #30675240

Subgroups — #30675240

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
2C2×#16862279C211yesmaximal, characteristic, complemented
3C22×#14523502C211yesmaximal, characteristic, complemented
4#22553883C211yesmaximal, characteristic, complemented
5#22553476C214yesmaximal, complemented
6#22553465C214yesmaximal, complemented
7#22553318C211yesmaximal, characteristic, complemented
8#22553309C211yesmaximal, characteristic, complemented
9#408×C131C211yesmaximal, characteristic, complemented
10#16862292—24no—
11#16862292—24no—
12C2×#14523514—21no—
13C2×#14523513C2212yes—
14C2×#14523513C2211yescharacteristic
15C22×Dic262C2211yescharacteristic, complemented
16C22×D524C2212yes—
17C22×D524C2211yescharacteristic
18C2×D4×D131—22no—
19C2×D4×D131—21no—
20#16862279—24no—
21C2×Dic524C2212yescomplemented
22C2×#14523502—24no—
23C2×#14523502C2212yescomplemented
24C2×#14523502C2214yescomplemented
25#16862244—21no—
26#16862233—21no—
27#16862227—22no—
28#16862225C2214yescomplemented
29#16862205C2211yescharacteristic, complemented
30#16862171C2212yescomplemented
31#16862168C2212yescomplemented
32#16862168C2214yescomplemented
33C2×#41×C131C2211yescharacteristic
34C2×M16×C131C2211yescharacteristic, complemented
35C22×C8×C131C2211yescharacteristic, complemented
36C22⋊C8×C131C2214yescomplemented
37C23×D131—22no—
38C23×D131—21no—
39#14523514—28no—
40#14523513—24no—
41#14523513—24no—
42#14523513—22no—
43#14523513D414yes—
44C2×Dic262—21no—
45C2×Dic262—22no—
46C2×Dic262—22no—
47C2×Dic262C2312yes—
48C2×Dic262C2311yescharacteristic
49C2×D524—24no—
50C2×D524—24no—
51C2×D524—22no—
52C2×D524—21no—
53C2×D524—22no—
54C2×D524D414yes—
55C2×D524C2311yescharacteristic
56C2×D524C2312yes—
57C2×C22⋊D131—22no—
58C2×C22⋊D131—21no—
59D4×D131—216no—
60D4×D131—28no—
61C22×Dic131—21no—
62C2×C4×D131—22no—
63C2×C4×D131—21no—
64Dic524—24no—
65#14523502—28no—
66#14523502—24no—
67C4⋊Dic131C2312yes—
68Dic131⋊C4—24no—
69C22⋊Dic131—24no—
70C22⋊Dic131—21no—
71C4×Dic131—21no—
72#41×C131—22no—
73#41×C131—22no—
74C2×Q8×C131D411yescharacteristic
75C2×D4×C131D412yes—
76C2×D4×C131D411yescharacteristic
77C22×C4×C131C2312yes—
78C22×C4×C131C2311yescharacteristic
79M16×C131—22no—
80C2×C8×C131—22no—
81C2×C8×C131C2312yes—
82C2×C8×C131C2312yes—
83C22×D131—22no—
84C22×D131—21no—
85C22×D131—28no—
86C22×D131—24no—
87C22×D131—82no—
88C22×D131—84no—
89Dic262—21no—
90Dic262—22no—
91Dic262—22no—
92Dic262—42no—
93Dic262C2×D412yes—
94D524—28no—
95D524—24no—
96D524—22no—
97D524—21no—
98D524C2×D412yes—
99D524C2×D414yes—
100C22⋊D131—28no—
101C22⋊D131—24no—
102C2×Dic131—21no—
103C2×Dic131—22no—
104C2×Dic131—22no—
105C2×Dic131—42no—
106C4×D131—28no—
107C4×D131—24no—
108C23×C131C2×D412yes—
109C23×C131C2×D411yescharacteristic
110Q8×C131—22no—
111D4×C131—24no—
112D4×C131—22no—
113C2×C4×C131—22no—
114C2×C4×C131—22no—
115C2×C4×C131C2×D412yes—
116C2×C4×C131C2×D412yes—
117C2×C4×C131C2×D412yes—
118C2×C4×C131C2×D411yescharacteristic
119C2×C4×C131C2411yescharacteristic
120C8×C131—22no—
121C8×C131—22no—
122C22×C131—22no—
123C22×C131—22no—
124C22×C131—24no—
125C22×C131#12012yes—
126C22×C131C22×D411yescharacteristic
127C2×D131—24no—
128C2×D131—22no—
129C2×D131—44no—
130C2×D131—42no—
131C2×D131—84no—
132C2×D131—88no—
133C4×C131—22no—
134C4×C131#12012yes—
135C4×C131C22×D411yescharacteristic
136C4×C131C22×D411yescharacteristic
137Dic131—22no—
138Dic131—42no—
139Dic131—42no—
140C2×C131—22no—
141C2×C131—42no—
142C2×C131C2×#12011yescharacteristic
143C2×C131#57712yes—
144D131—44no—
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#418—1311no—
155#416—1311no—
156#408D13111yescharacteristic, complemented
157H(5,𝔽2)—2624no—
158H(5,𝔽2)—2624no—
159C2×#41—2621no—
160C2×#41—1312no—
161C2×#41—1311no—
162C2×#41C2×D13111yescharacteristic
163C22×Q8—1311no—
164C22×D4—2622no—
165C22×D4—2621no—
166C22×D4—1312no—
167C22×D4—1311no—
168#137—2624no—
169C2×Q16—1312no—
170C2×SD16—2624no—
171C2×SD16—1314no—
172C2×SD16—1312no—
173C2×M16C2×D13111yescharacteristic, complemented
174C22×C8C2×D13111yescharacteristic, complemented
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⋊C8C2×D13114yescomplemented
184C24—2622no—
185C24—2621no—
186#41—22no—
187#41—22no—
188#41—2624no—
189#41—2624no—
190#41—2628no—
191#41—2622no—
192#41—1314no—
193C2×Q8—2621no—
194C2×Q8—2622no—
195C2×Q8—2622no—
196C2×Q8—1312no—
197C2×Q8—1311no—
198C2×Q8D52411yescharacteristic
199C2×D4—2624no—
200C2×D4—2624no—
201C2×D4—26216no—
202C2×D4—2628no—
203C2×D4—2622no—
204C2×D4—2621no—
205C2×D4—2622no—
206C2×D4—2622no—
207C2×D4—2621no—
208C2×D4—1314no—
209C2×D4—1311no—
210C2×D4—1312no—
211C2×D4D52412yes—
212C2×D4D52411yescharacteristic
213C22×C4—2622no—
214C22×C4—2621no—
215C22×C4—2621no—
216C22×C4C22×D13112yes—
217C22×C4C22×D13111yescharacteristic
218Q16—2624no—
219SD16—2628no—
220SD16—2624no—
221M16—22no—
222C2×C8—22no—
223C2×C8C22×D13112yes—
224C2×C8C22×D13112yes—
225C4⋊C4—2624no—
226C4⋊C4—1312no—
227C22⋊C4—2621no—
228C22⋊C4—2624no—
229C42—2621no—
230C23—2628no—
231C23—2622no—
232C23—2621no—
233C23—2624no—
234C23—10482no—
235C23—10484no—
236C23C2×D52412yes—
237C23C2×D52411yescharacteristic
238Q8—22no—
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—
253D4—1314no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—2628no—
257C2×C4—2621no—
258C2×C4—2624no—
259C2×C4—2622no—
260C2×C4—2622no—
261C2×C4—5242no—
262C2×C4D4×D13112yes—
263C2×C4D4×D13112yes—
264C2×C4C2×D52412yes—
265C2×C4C2×D52411yescharacteristic
266C2×C4C23×D13111yescharacteristic
267C8—22no—
268C8—22no—
269C22—22no—
270C22—24no—
271C22—22no—
272C22—2624no—
273C22—2622no—
274C22—5244no—
275C22—5242no—
276C22—10484no—
277C22—10488no—
278C22#1686221612yes—
279C22C22×D52411yescharacteristic
280C4—22no—
281C4—2622no—
282C4—5242no—
283C4—5242no—
284C4#1686221612yes—
285C4C2×D4×D13111yescharacteristic
286C4C2×D4×D13111yescharacteristic
287C2—22no—
288C2—42no—
289C2—5244no—
290C2—5242no—
291C2C2×#1686221611yescharacteristic
292C2#2255412412yes—
subgroupG/H
centerC22C22×D524
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#408×C131C2
socleC22×C131C22×D4
prime radicalC2×#16862292C2
hypercenter#408D131
hypocenterC131#3009
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. C2×#16862279C2semidirect
3. C22×#14523502C2semidirect
4. #22553883C2semidirect
5. #22553476C2semidirect
6. #22553465C2semidirect
7. #22553318C2semidirect
8. #22553309C2semidirect
9. #408×C131C2semidirect
15. C22×Dic262C22semidirect
21. C2×Dic524C22semidirect
23. C2×#14523502C22semidirect
24. C2×#14523502C22semidirect
28. #16862225C22semidirect
29. #16862205C22semidirect
30. #16862171C22semidirect
31. #16862168C22semidirect
32. #16862168C22semidirect
34. C2×M16×C131C22semidirect
35. C22×C8×C131C22semidirect
36. C22⋊C8×C131C22semidirect
146. C131#3009semidirect
156. #408D131semidirect
173. C2×M16C2×D131semidirect
174. C22×C8C2×D131semidirect
183. C22⋊C8C2×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622922
C2×#168622792
C22×#145235022
#225538832
#225534762
#225534652
#225533182
#225533092
#408×C1312
#3009131