Finite Group Databasebeta

← C22×#16862267 #30680688

Subgroups — C22×#16862267 #30680688

Order 16768.

1504 conjugacy classes of subgroups; only the 525 normal subgroups are shown.

normal subgroups only

This group has 1504 conjugacy classes of subgroups — too many to draw. Only its 525 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
1C22×#14523514C211yesmaximal, characteristic, complemented
2C22×#14523512C211yesmaximal, characteristic, complemented
3C2×#16862267C2124yesmaximal, direct product factor
4C22×#14523503C211yesmaximal, characteristic, complemented
5C22×#14523502C211yesmaximal, characteristic, complemented
6C22×#14523498C211yesmaximal, characteristic, complemented
7C22×C8×D131C211yesmaximal, characteristic, complemented
8C22×SD16×C131C211yesmaximal, characteristic, complemented
9C2×#14523514—26no—
10C2×#14523514C22112yescomplemented
11C2×#14523512—26no—
12C2×#14523512C22112yescomplemented
13C22×Dic262C2211yescharacteristic
14C22×D524—21no—
15C22×D524C2211yescharacteristic
16C22×C22⋊D131—21no—
17C23×Dic131—21no—
18C22×C4×D131—21no—
19C22×C4×D131C2211yescharacteristic
20#16862267C22164yesdirect product factor
21C2×#14523503C22112yescomplemented
22C2×#14523502C22112yescomplemented
23C2×#14523498C22112yescomplemented
24C22×C131⋊C8C2211yescharacteristic
25C2×C8×D131C22112yescomplemented
26C22×Q8×C131C2211yescharacteristic
27C22×D4×C131C2211yescharacteristic
28C2×SD16×C131C22112yescomplemented
29C22×C8×C131C2211yescharacteristic
30C23×D131—21no—
31C23×D131D411yescharacteristic
32#14523514—224no—
33#14523514C23116yescomplemented
34#14523512—224no—
35#14523512C23116yescomplemented
36C2×Dic262—23no—
37C2×Dic262C2316yes—
38C2×D524—212no—
39C2×D524—23no—
40C2×D524C2316yes—
41C2×C22⋊D131—212no—
42C22×Dic131—212no—
43C22×Dic131—21no—
44C22×Dic131D411yescharacteristic
45C2×C4×D131—212no—
46C2×C4×D131D416yes—
47C2×C4×D131C2316yes—
48#14523503C23116yescomplemented
49#14523502C23116yescomplemented
50#14523498C23116yescomplemented
51C2×C131⋊C8C2316yes—
52C8×D131C23116yescomplemented
53C24×C131—21no—
54C2×Q8×C131—23no—
55C2×Q8×C131C2316yes—
56C2×D4×C131—23no—
57C2×D4×C131C2316yes—
58C22×C4×C131—21no—
59C22×C4×C131C2311yescharacteristic
60SD16×C131C23116yescomplemented
61C2×C8×C131C2316yes—
62C22×D131—24no—
63C22×D131—26no—
64C22×D131—44no—
65C22×D131C2×D416yes—
66Dic262—26no—
67Dic262C2414yes—
68D524—216no—
69D524—26no—
70D524C2414yes—
71C22⋊D131—216no—
72C2×Dic131—26no—
73C2×Dic131—216no—
74C2×Dic131C2×D416yes—
75C4×D131—216no—
76C4×D131C2×D4112yes—
77C4×D131C2414yes—
78C131⋊C8C2414yes—
79C23×C131—26no—
80C23×C131—44no—
81C23×C131C2×D411yescharacteristic
82Q8×C131—26no—
83Q8×C131C2414yes—
84D4×C131—26no—
85D4×C131C2414yes—
86C2×C4×C131—26no—
87C2×C4×C131C2×D413yes—
88C2×C4×C131C2413yes—
89C8×C131C2414yes—
90C22×C131—24no—
91C22×C131—412no—
92C22×C131#13514yescomplemented
93C22×C131C22×D413yes—
94C2×D131—212no—
95C2×D131—24no—
96C2×D131—412no—
97C2×D131C22×D414yes—
98C4×C131—24no—
99C4×C131C22×D413yes—
100C4×C131C2511yescharacteristic
101Dic131—24no—
102Dic131C22×D414yes—
103C2×C131—44no—
104C2×C131C2×#13516yescomplemented
105C2×C131C23×D411yescharacteristic
106D131—24no—
107D131—44no—
108C131C22×#13511yescharacteristic, complemented, Sylow
109C22×#135—1311nomaximal, Sylow
110C22×#41—1311no—
111C22×#41—1311no—
112C2×#135—13124no—
113C22×Q16—1311no—
114C22×SD16—1311no—
115C22×SD16D13111yescharacteristic, complemented
116C22×D8—1311no—
117C23×C8—1311no—
118C2×#41—2626no—
119C2×#41—2626no—
120C2×#41—13112no—
121C2×#41—13112no—
122C22×Q8—1311no—
123C22×Q8C2×D13111yescharacteristic
124C22×D4—2621no—
125C22×D4—2621no—
126C22×D4—1311no—
127C22×D4C2×D13111yescharacteristic
128C23×C4—2621no—
129C23×C4—2621no—
130C23×C4—1311no—
131#135—13164no—
132C2×Q16—13112no—
133C2×SD16—13112no—
134C2×SD16C2×D131112yescomplemented
135C2×D8—13112no—
136C22×C8—1311no—
137C22×C8—13112no—
138C22×C8C2×D13111yescharacteristic
139C24—21no—
140C24—2621no—
141C24—1311no—
142#41—26224no—
143#41—26224no—
144#41—13116no—
145#41—13116no—
146C2×Q8—23no—
147C2×Q8—2623no—
148C2×Q8—1316no—
149C2×Q8C22×D13116yes—
150C2×D4—23no—
151C2×D4—26212no—
152C2×D4—26212no—
153C2×D4—2623no—
154C2×D4—1316no—
155C2×D4C22×D13116yes—
156C22×C4—21no—
157C22×C4—26212no—
158C22×C4—2621no—
159C22×C4—26212no—
160C22×C4—1311no—
161C22×C4—1316no—
162C22×C4—1316no—
163C22×C4C22×D13111yescharacteristic
164Q16—13116no—
165SD16—13116no—
166SD16C22×D131116yescomplemented
167D8—13116no—
168C2×C8—1316no—
169C2×C8—13116no—
170C2×C8C22×D13116yes—
171C23—26no—
172C23—44no—
173C23—2624no—
174C23—2626no—
175C23—5244no—
176C23—1316no—
177C23D4×D13111yescharacteristic
178Q8—26no—
179Q8—2626no—
180Q8—1314no—
181Q8C23×D13114yes—
182D4—26no—
183D4—26216no—
184D4—26216no—
185D4—2626no—
186D4—1314no—
187D4C23×D13114yes—
188C2×C4—26no—
189C2×C4—26216no—
190C2×C4—2626no—
191C2×C4—26216no—
192C2×C4—1316no—
193C2×C4—13112no—
194C2×C4—1314no—
195C2×C4D4×D13113yes—
196C2×C4C23×D13113yes—
197C8—1314no—
198C8C23×D13114yes—
199C22—24no—
200C22—412no—
201C22—26212no—
202C22—2624no—
203C22—52412no—
204C22—1314no—
205C22#1686226714yesdirect product factor
206C22C2×D4×D13113yes—
207C4—24no—
208C4—2624no—
209C4—1314no—
210C4C2×D4×D13113yes—
211C4C24×D13111yescharacteristic
212C2—44no—
213C2—2624no—
214C2—5244no—
215C2C2×#1686226716yesdirect product factor
216C2C22×D4×D13111yescharacteristic
subgroupG/H
centerC23D4×D131
commutator, cosocleC4×C131C25
FrattiniC4C24×D131
FittingC22×SD16×C131C2
socleC23×C131C2×D4
hypercenterC22×SD16D131
hypocenterC131C22×#135
LCSG ▷ C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#14523514C2semidirect
2. C22×#14523512C2semidirect
3. C2×#16862267C2direct (N × H)
4. C22×#14523503C2semidirect
5. C22×#14523502C2semidirect
6. C22×#14523498C2semidirect
7. C22×C8×D131C2semidirect
8. C22×SD16×C131C2semidirect
10. C2×#14523514C22semidirect
12. C2×#14523512C22semidirect
20. #16862267C22direct (N × H)
21. C2×#14523503C22semidirect
22. C2×#14523502C22semidirect
23. C2×#14523498C22semidirect
25. C2×C8×D131C22semidirect
28. C2×SD16×C131C22semidirect
33. #14523514C23semidirect
35. #14523512C23semidirect
48. #14523503C23semidirect
49. #14523502C23semidirect
50. #14523498C23semidirect
52. C8×D131C23semidirect
60. SD16×C131C23semidirect
92. C22×C131#135semidirect
104. C2×C131C2×#135semidirect
108. C131C22×#135semidirect
115. C22×SD16D131semidirect
134. C2×SD16C2×D131semidirect
166. SD16C22×D131semidirect
205. C22#16862267direct (N × H)
215. C2C2×#16862267direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145235142
C22×#145235122
C2×#168622672
C22×#145235032
C22×#145235022
C22×#145234982
C22×C8×D1312
C22×SD16×C1312
C22×#135131