Finite Group Databasebeta

← #30675915

Subgroups — #30675915

Order 16768.

418 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#22553879C211yesmaximal, characteristic
2#22553793C211yesmaximal, characteristic, complemented
3#22553635C212yesmaximal, complemented
4#22553634C212yesmaximal, complemented
5#22553629C212yesmaximal, complemented
6#22553624C212yesmaximal, complemented
7#22553352C211yesmaximal, characteristic, complemented
8#22553350C211yesmaximal, characteristic, complemented
9#22553317C211yesmaximal, characteristic, complemented
10#22553278C211yesmaximal, characteristic, complemented
11C2×#102×C131C211yesmaximal, characteristic, complemented
12C2×#14523513C2211yescharacteristic
13C2×#14523497C2211yescharacteristic
14#16862242—21no—
15#16862233—21no—
16#16862217C2212yes—
17#16862217C2212yes—
18#16862215—22no—
19#16862211—21no—
20#16862208—21no—
21C4×C22⋊D131—22no—
22#16862205C2211yescharacteristic
23#16862205C2211yescharacteristic
24#16862204C2212yescomplemented
25C4⋊C4×D131C2212yescomplemented
26#16862180C2211yescharacteristic, complemented
27#16862180C2211yescharacteristic, complemented
28#16862179C2212yescomplemented
29#16862178C2211yescharacteristic, complemented
30#16862178C2211yescharacteristic, complemented
31#16862177C2212yescomplemented
32#16862167C2212yes—
33#16862167C2212yes—
34(C2×D131)⋊C8C2212yescomplemented
35(C2×D131)⋊C8C2212yescomplemented
36C22×D4×C131C2211yescharacteristic
37C22×C8×C131C2211yescharacteristic
38C2×C4⋊C4×C131C2211yescharacteristic
39#102×C131C2212yes—
40#102×C131C2212yes—
41#14523513—22no—
42#14523513—22no—
43C2×Dic262D411yescharacteristic
44C2×D524D411yescharacteristic
45C2×C22⋊D131—22no—
46C2×C22⋊D131—22no—
47C2×C22⋊D131D412yes—
48C2×C4×D131—22no—
49C2×C4×D131C2312yes—
50#14523497—22no—
51C2×C131⋊C8C2312yes—
52C4⋊Dic131C2311yescharacteristic
53C4⋊Dic131C2311yescharacteristic
54C4⋊Dic131C2311yescharacteristic
55C4⋊Dic131C2311yescharacteristic
56Dic131⋊C4—22no—
57C22⋊Dic131—22no—
58C22⋊Dic131—22no—
59C22⋊Dic131—21no—
60C22⋊Dic131—21no—
61(C2×D131)⋊C4—22no—
62C4×Dic131—22no—
63C4×Dic131—21no—
64C4×Dic131—21no—
65C24×C131—21no—
66C2×D4×C131—21no—
67C2×D4×C131—22no—
68C2×D4×C131—22no—
69C2×D4×C131C2312yes—
70C22×C4×C131—21no—
71C22×C4×C131C2311yescharacteristic
72C2×C8×C131—22no—
73C2×C8×C131C2311yescharacteristic
74C2×C8×C131C2311yescharacteristic
75C4⋊C4×C131—21no—
76C4⋊C4×C131C2312yes—
77C22×D131C2×D412yes—
78Dic262—22no—
79D524—22no—
80C22⋊D131—24no—
81C22⋊D131—82no—
82C22⋊D131—82no—
83C2×Dic131—21no—
84C2×Dic131—21no—
85C2×Dic131—21no—
86C2×Dic131—21no—
87C2×Dic131C2×D412yes—
88C4×D131—22no—
89C4×D131—22no—
90C4×D131—82no—
91C131⋊C8—22no—
92C23×C131—22no—
93C23×C131—42no—
94C23×C131—81no—
95C23×C131—81no—
96C23×C131C2×D411yescharacteristic
97D4×C131—22no—
98D4×C131—22no—
99D4×C131—42no—
100C2×C4×C131—22no—
101C2×C4×C131—81no—
102C2×C4×C131C2×D411yescharacteristic
103C2×C4×C131#4112yes—
104C2×C4×C131#4112yes—
105C2×C4×C131C2411yescharacteristic
106C8×C131—21no—
107C8×C131—21no—
108C22×C131—21no—
109C22×C131—21no—
110C22×C131—42no—
111C22×C131—42no—
112C22×C131—42no—
113C22×C131—84no—
114C22×C131#12312yes—
115C22×C131C22×D411yescharacteristic
116C2×D131—22no—
117C2×D131—22no—
118C2×D131—22no—
119C4×C131—42no—
120C4×C131#12312yes—
121C4×C131C2×#4111yescharacteristic
122C4×C131C2×#4111yescharacteristic
123Dic131—22no—
124Dic131—41no—
125Dic131—41no—
126Dic131—41no—
127Dic131—41no—
128C2×C131—22no—
129C2×C131—82no—
130C2×C131C2×#12311yescharacteristic
131C2×C131#57611yescharacteristic
132C2×C131#57711yescharacteristic
133D131—42no—
134C131#309611yescharacteristic, complemented, Sylow
135#3096—1311nomaximal, Sylow
136#535—1311no—
137#520—1311no—
138#482—1312no—
139#482—1312no—
140#481—1312no—
141#480—1312no—
142#428—1311no—
143#427—1311no—
144#418—1311no—
145C2×#102D13111yescharacteristic, complemented
146#408—1311no—
147C2×#41—1311no—
148C22×D4C2×D13111yescharacteristic
149C2×M16—1311no—
150C22×C8C2×D13111yescharacteristic
151#127—2621no—
152#124—2621no—
153#121—1312no—
154#121—1312no—
155#120—2622no—
156C4×Q8—2621no—
157C4×D4—2621no—
158C4×D4—2622no—
159#117—1311no—
160#117—1311no—
161#117—1312no—
162C2×C4⋊C4—1312no—
163C2×C4⋊C4C2×D13111yescharacteristic
164#107—1311no—
165#107—1311no—
166#107—1312no—
167#106—1311no—
168#106—1311no—
169#106—1312no—
170#102—1312no—
171#102—1312no—
172#102C2×D13112yes—
173#102C2×D13112yes—
174C22⋊C8—1312no—
175C22⋊C8—1312no—
176C24—21no—
177#41—2622no—
178#41—2622no—
179C2×Q8—1311no—
180C2×D4—21no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—2622no—
184C2×D4—2622no—
185C2×D4—1312no—
186C2×D4—1311no—
187C2×D4C22×D13112yes—
188C22×C4—21no—
189C22×C4—2622no—
190C22×C4—1312no—
191C22×C4C22×D13111yescharacteristic
192M16—2622no—
193C2×C8—22no—
194C2×C8—1312no—
195C2×C8C22×D13111yescharacteristic
196C2×C8C22×D13111yescharacteristic
197C4⋊C4—21no—
198C4⋊C4—2622no—
199C4⋊C4—1311no—
200C4⋊C4—1311no—
201C4⋊C4—1311no—
202C4⋊C4—1311no—
203C4⋊C4C22×D13112yes—
204C22⋊C4—2622no—
205C22⋊C4—2621no—
206C22⋊C4—2621no—
207C22⋊C4—2622no—
208C22⋊C4—2622no—
209C42—2622no—
210C42—2621no—
211C42—2621no—
212C23—22no—
213C23—42no—
214C23—81no—
215C23—81no—
216C23—1312no—
217C23D4×D13111yescharacteristic
218Q8—2622no—
219D4—22no—
220D4—22no—
221D4—42no—
222D4—2624no—
223D4—2622no—
224D4—10482no—
225D4—10482no—
226C2×C4—22no—
227C2×C4—2622no—
228C2×C4—2622no—
229C2×C4—2621no—
230C2×C4—2621no—
231C2×C4—2621no—
232C2×C4—2621no—
233C2×C4—81no—
234C2×C4—10482no—
235C2×C4—1312no—
236C2×C4D4×D13111yescharacteristic
237C2×C4#1452351212yes—
238C2×C4#1452351312yes—
239C2×C4C23×D13111yescharacteristic
240C8—21no—
241C8—21no—
242C8—2622no—
243C22—21no—
244C22—21no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—2622no—
249C22—2622no—
250C22—2622no—
251C22—84no—
252C22#1686222812yes—
253C22C2×D4×D13111yescharacteristic
254C4—42no—
255C4—2622no—
256C4—5241no—
257C4—5241no—
258C4—5241no—
259C4—5241no—
260C4#1686222812yes—
261C4C2×#1452351211yescharacteristic
262C4C2×#1452351311yescharacteristic
263C2—22no—
264C2—82no—
265C2—5242no—
266C2C2×#1686222811yescharacteristic
267C2#2255411311yescharacteristic
268C2#2255411811yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC2×#102×C131C2
socleC22×C131C22×D4
prime radical#22553879C2
hypercenterC2×#102D131
hypocenterC131#3096
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
2. #22553793C2semidirect
3. #22553635C2semidirect
4. #22553634C2semidirect
5. #22553629C2semidirect
6. #22553624C2semidirect
7. #22553352C2semidirect
8. #22553350C2semidirect
9. #22553317C2semidirect
10. #22553278C2semidirect
11. C2×#102×C131C2semidirect
24. #16862204C22semidirect
25. C4⋊C4×D131C22semidirect
26. #16862180C22semidirect
27. #16862180C22semidirect
28. #16862179C22semidirect
29. #16862178C22semidirect
30. #16862178C22semidirect
31. #16862177C22semidirect
34. (C2×D131)⋊C8C22semidirect
35. (C2×D131)⋊C8C22semidirect
134. C131#3096semidirect
145. C2×#102D131semidirect

Maximal subgroups

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

subgroupindex
#225538792
#225537932
#225536352
#225536342
#225536292
#225536242
#225533522
#225533502
#225533172
#225532782
C2×#102×C1312
#3096131