Finite Group Databasebeta

← #30675440

Subgroups — #30675440

Order 16768.

584 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 137 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×#16862274C211yesmaximal, characteristic, complemented
2C2×#16862272C211yesmaximal, characteristic, complemented
3C2×#16862242C211yesmaximal, characteristic, complemented
4#22553780C211yesmaximal, characteristic, complemented
5#22553511C214yesmaximal, complemented
6#22553506C214yesmaximal, complemented
7#22553335C211yesmaximal, characteristic, complemented
8#22553316C211yesmaximal, characteristic, complemented
9#418×C131C211yesmaximal, characteristic, complemented
10C2×#14523513C2211yescharacteristic
11C22×D524C2211yescharacteristic
12C22×C22⋊D131—22no—
13#16862274—24no—
14#16862272—24no—
15C2×#14523502C2212yescomplemented
16C2×#14523501C2212yescomplemented
17C2×D1048C2212yescomplemented
18C2×#14523498C2212yescomplemented
19C2×#14523497C2211yescharacteristic
20#16862242—21no—
21#16862242—21no—
22#16862242C2214yes—
23#16862211C2212yescomplemented
24#16862208C2212yescomplemented
25(C22⋊D131)⋊C4—22no—
26#16862203—21no—
27C2×C4×Dic131C2211yescharacteristic
28C42×D131—21no—
29Dic131⋊C8C2214yescomplemented
30#16862166C2214yescomplemented
31C22×D4×C131C2211yescharacteristic
32C2×M16×C131C2211yescharacteristic
33#117×C131C2211yescharacteristic
34#102×C131C2214yescomplemented
35C23×D131—21no—
36#14523513—22no—
37#14523513D414yescomplemented
38C2×Dic262C2311yescharacteristic
39C2×D524—22no—
40C2×D524—21no—
41C2×D524—22no—
42C2×D524C2312yes—
43C2×D524C2311yescharacteristic
44C2×C22⋊D131—24no—
45C2×C22⋊D131—24no—
46C2×C22⋊D131—21no—
47C2×C22⋊D131—48no—
48C22×Dic131D412yes—
49C2×C4×D131—22no—
50C2×C4×D131—21no—
51#14523502—24no—
52#14523501—24no—
53D1048—24no—
54#14523498—24no—
55#14523497—22no—
56C2×C131⋊C8C2312yes—
57Dic131⋊C4—22no—
58(C2×D131)⋊C4—22no—
59C4×Dic131—22no—
60C4×Dic131D411yescharacteristic
61C4×Dic131D411yescharacteristic
62C4×Dic131C2312yes—
63C24×C131—21no—
64C2×D4×C131—22no—
65C2×D4×C131—21no—
66C2×D4×C131—22no—
67C2×D4×C131C2312yes—
68C22×C4×C131C2311yescharacteristic
69M16×C131—22no—
70C2×C8×C131C2312yes—
71C4⋊C4×C131C2312yes—
72C22⋊C4×C131—21no—
73C42×C131—21no—
74C22×D131—21no—
75C22×D131—22no—
76C22×D131—44no—
77C22×D131—42no—
78Dic262—21no—
79Dic262C2×D412yescomplemented
80D524—22no—
81D524—21no—
82D524—22no—
83D524—42no—
84D524C2×D412yescomplemented
85C22⋊D131—24no—
86C22⋊D131—48no—
87C22⋊D131—48no—
88C2×Dic131—21no—
89C2×Dic131—24no—
90C2×Dic131C2×D412yes—
91C2×Dic131C2×D412yes—
92C4×D131—24no—
93C4×D131—44no—
94C131⋊C8—22no—
95C23×C131—22no—
96C23×C131—44no—
97C23×C131—42no—
98C23×C131C2×D411yescharacteristic
99D4×C131—22no—
100D4×C131—22no—
101D4×C131—42no—
102C2×C4×C131—22no—
103C2×C4×C131C2×D411yescharacteristic
104C2×C4×C131C2×D412yes—
105C2×C4×C131#4112yes—
106C2×C4×C131C2411yescharacteristic
107C8×C131—22no—
108C22×C131—22no—
109C22×C131—42no—
110C22×C131—44no—
111C22×C131—84no—
112C22×C131#12112yes—
113C22×C131C22×D411yescharacteristic
114C2×D131—22no—
115C2×D131—42no—
116C2×D131—42no—
117C2×D131—44no—
118C2×D131—84no—
119C4×C131—42no—
120C4×C131#12112yes—
121C4×C131C22×D411yescharacteristic
122C4×C131C2×#4111yescharacteristic
123Dic131—22no—
124Dic131—22no—
125Dic131AGL(1,ℤ/8)14yescomplemented
126C2×C131—22no—
127C2×C131—82no—
128C2×C131C2×#12111yescharacteristic
129C2×C131C2×AGL(1,ℤ/8)12yes—
130D131—42no—
131D131—82no—
132C131#303811yescharacteristic, complemented, Sylow
133#3038—1311nomaximal, Sylow
134C2×AGL(1,ℤ/8)—1311no—
135C2×AGL(1,ℤ/8)—1311no—
136C2×#127—1311no—
137C4×#41—1311no—
138#460—1314no—
139#459—1314no—
140#423—1311no—
141#418—1311no—
142#418D13111yescharacteristic, complemented
143C2×#41—1311no—
144C22×D4—2622no—
145C22×D4—1311no—
146C22×D4C2×D13111yescharacteristic
147AGL(1,ℤ/8)—2624no—
148AGL(1,ℤ/8)—2624no—
149C2×SD16—1312no—
150C2×SD16—1312no—
151C2×D8—1312no—
152C2×D8—1312no—
153C2×M16—1311no—
154C2×M16C2×D13111yescharacteristic
155#127—2621no—
156#127—2621no—
157#127—1314no—
158C4×Q8—1312no—
159C4×D4—2622no—
160C4×D4—1312no—
161#117—2621no—
162#117C2×D13111yescharacteristic
163C2×C42—2621no—
164C2×C42—1311no—
165C4⋊C8—1314no—
166#102—1314no—
167#102C2×D13114yescomplemented
168C24—21no—
169C24—2621no—
170#41—2622no—
171#41—1314no—
172C2×Q8—1311no—
173C2×D4—22no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—2621no—
177C2×D4—2624no—
178C2×D4—2624no—
179C2×D4—2622no—
180C2×D4—2621no—
181C2×D4—2622no—
182C2×D4—5248no—
183C2×D4—1312no—
184C2×D4—1311no—
185C2×D4C22×D13112yes—
186C22×C4—2622no—
187C22×C4—2621no—
188C22×C4—1312no—
189C22×C4C22×D13111yescharacteristic
190SD16—2624no—
191SD16—2624no—
192D8—2624no—
193D8—2624no—
194M16—22no—
195M16—2622no—
196C2×C8—1312no—
197C2×C8C22×D13112yes—
198C4⋊C4—2622no—
199C4⋊C4C22×D13112yes—
200C22⋊C4—21no—
201C22⋊C4—2622no—
202C42—21no—
203C42—2622no—
204C42—1311no—
205C42—1311no—
206C42—1312no—
207C23—22no—
208C23—44no—
209C23—42no—
210C23—2621no—
211C23—2622no—
212C23—5244no—
213C23—5242no—
214C23D4×D13111yescharacteristic
215Q8—2621no—
216Q8—1312no—
217D4—22no—
218D4—22no—
219D4—42no—
220D4—2624no—
221D4—2622no—
222D4—2621no—
223D4—2622no—
224D4—5248no—
225D4—5248no—
226D4—5242no—
227D4—1312no—
228C2×C4—22no—
229C2×C4—2621no—
230C2×C4—2624no—
231C2×C4—2624no—
232C2×C4—5244no—
233C2×C4—1312no—
234C2×C4—1312no—
235C2×C4D4×D13111yescharacteristic
236C2×C4D4×D13112yes—
237C2×C4#1452351312yes—
238C2×C4C23×D13111yescharacteristic
239C8—22no—
240C8—2622no—
241C22—22no—
242C22—42no—
243C22—44no—
244C22—2622no—
245C22—84no—
246C22—5242no—
247C22—5242no—
248C22—5244no—
249C22—10484no—
250C22#1686221912yes—
251C22C2×D4×D13111yescharacteristic
252C4—42no—
253C4—2622no—
254C4—2622no—
255C4—1314no—
256C4#1686221912yes—
257C4C2×D4×D13111yescharacteristic
258C4C2×#1452351311yescharacteristic
259C2—22no—
260C2—82no—
261C2—5242no—
262C2—10482no—
263C2C2×#1686221911yescharacteristic
264C2AGL(1,ℤ/8)×D13112yes—
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#418×C131C2
socleC22×C131C22×D4
hypercenter#418D131
hypocenterC131#3038
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
1. C2×#16862274C2semidirect
2. C2×#16862272C2semidirect
3. C2×#16862242C2semidirect
4. #22553780C2semidirect
5. #22553511C2semidirect
6. #22553506C2semidirect
7. #22553335C2semidirect
8. #22553316C2semidirect
9. #418×C131C2semidirect
15. C2×#14523502C22semidirect
16. C2×#14523501C22semidirect
17. C2×D1048C22semidirect
18. C2×#14523498C22semidirect
23. #16862211C22semidirect
24. #16862208C22semidirect
29. Dic131⋊C8C22semidirect
30. #16862166C22semidirect
34. #102×C131C22semidirect
37. #14523513D4semidirect
79. Dic262C2×D4semidirect
84. D524C2×D4semidirect
125. Dic131AGL(1,ℤ/8)semidirect
132. C131#3038semidirect
142. #418D131semidirect
167. #102C2×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622742
C2×#168622722
C2×#168622422
#225537802
#225535112
#225535062
#225533352
#225533162
#418×C1312
#3038131