Finite Group Databasebeta

← #22564479

Subgroups — #22564479

Order 8576.

672 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 153 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×(C2×D67)⋊C4C211yesmaximal, characteristic
2#16863331C212yesmaximal, complemented
3#16863322C212yesmaximal, complemented
4#16863287C214yesmaximal, complemented
5#16863268C212yesmaximal, complemented
6#16863259C211yesmaximal, characteristic, complemented
7#16863251C212yesmaximal
8C2×#117×C67C211yesmaximal, characteristic, complemented
9C24×D67C2211yescharacteristic
10C23×Dic67C2211yescharacteristic
11C2×Dic67⋊C4C2214yescomplemented
12C2×C22⋊Dic67C2212yes—
13C2×(C2×D67)⋊C4—24no—
14C2×(C2×D67)⋊C4C2214yes—
15C2×(C2×D67)⋊C4C2212yes—
16C2×(C2×D67)⋊C4C2212yes—
17C2×(C2×D67)⋊C4C2214yes—
18C67⋊#95C2212yescomplemented
19C67⋊#95C2212yescomplemented
20C67⋊#95C2214yescomplemented
21C23×C4×C67C2211yescharacteristic
22#117×C67—24no—
23C2×C4⋊C4×C67C2211yescharacteristic, complemented
24C2×C4⋊C4×C67C2211yescharacteristic, complemented
25C2×C22⋊C4×C67C2212yes—
26C2×C42×C67C2212yescomplemented
27C23×D67—22no—
28C23×D67—24no—
29C23×D67—24no—
30C23×D67—42no—
31C23×D67C2312yes—
32C22×Dic67—22no—
33C22×Dic67—42no—
34C22×Dic67C2314yes—
35C22×Dic67C2312yes—
36Dic67⋊C4—44no—
37C22⋊Dic67—24no—
38(C2×D67)⋊C4—28no—
39(C2×D67)⋊C4—24no—
40(C2×D67)⋊C4—24no—
41(C2×D67)⋊C4—44no—
42C24×C67C2311yescharacteristic
43C22×C4×C67—24no—
44C22×C4×C67D414yes—
45C22×C4×C67C2312yes—
46C22×C4×C67C2312yes—
47C22×C4×C67C2311yescharacteristic
48C22×C4×C67C2311yescharacteristic
49C4⋊C4×C67—22no—
50C4⋊C4×C67—22no—
51C22⋊C4×C67—24no—
52C42×C67—24no—
53C23×C67—22no—
54C23×C67—22no—
55C23×C67C2×D412yes—
56C23×C67#4114yes—
57C23×C67C2411yescharacteristic
58C22×D67—28no—
59C22×D67—22no—
60C22×D67—22no—
61C22×D67—24no—
62C22×D67—24no—
63C22×D67—416no—
64C22×D67—44no—
65C22×D67—44no—
66C22×D67#4114yes—
67C2×C4×C67—28no—
68C2×C4×C67—22no—
69C2×C4×C67—22no—
70C2×C4×C67—22no—
71C2×C4×C67—22no—
72C2×C4×C67—24no—
73C2×C4×C67—24no—
74C2×C4×C67C2×D412yes—
75C2×C4×C67C2×D412yes—
76C2×Dic67—22no—
77C2×Dic67—24no—
78C2×Dic67—24no—
79C2×Dic67—24no—
80C2×Dic67—24no—
81C2×Dic67—44no—
82C22×C67—22no—
83C22×C67—22no—
84C22×C67—28no—
85C22×C67#12314yes—
86C22×C67C22×D411yescharacteristic
87C22×C67C2×#4112yes—
88C22×C67C2×#4112yes—
89C22×C67H(5,𝔽2)12yes—
90C2×D67—24no—
91C2×D67—28no—
92C2×D67—416no—
93C2×D67—42no—
94C2×D67—42no—
95C2×D67—44no—
96C4×C67—22no—
97C4×C67—22no—
98C4×C67—42no—
99C4×C67—42no—
100Dic67—42no—
101Dic67—44no—
102C2×C67—24no—
103C2×C67C2×#12311yescharacteristic
104C2×C67#52511yescharacteristic
105C2×C67#53511yescharacteristic
106C2×C67#55114yes—
107D67—44no—
108#2482—671nomaximal, Sylow
109C67#248211yescharacteristic, complemented, Sylow
110C2×#117D6711yescharacteristic, complemented
111C22×C22⋊C4—671no—
112#399—672no—
113#397—672no—
114#388—674no—
115#385—672no—
116#381—671no—
117#379—672no—
118C25—671no—
119C23×C4—671no—
120C23×C4C2×D6711yescharacteristic
121#117—24no—
122C2×C4⋊C4—674no—
123C2×C4⋊C4C2×D6711yescharacteristic, complemented
124C2×C4⋊C4C2×D6711yescharacteristic, complemented
125C2×C22⋊C4—674no—
126C2×C22⋊C4—672no—
127C2×C22⋊C4—672no—
128C2×C22⋊C4—672no—
129C2×C22⋊C4—674no—
130C2×C22⋊C4—1344no—
131C2×C22⋊C4C2×D6712yes—
132C2×C42C2×D6712yescomplemented
133#95—672no—
134#95—672no—
135#95—674no—
136C24—2682no—
137C24—672no—
138C24—1344no—
139C24—1344no—
140C24—1342no—
141C24C22×D6711yescharacteristic
142C22×C4—24no—
143C22×C4—2682no—
144C22×C4—674no—
145C22×C4—672no—
146C22×C4—1342no—
147C22×C4C22⋊D6714yes—
148C22×C4C22×D6712yes—
149C22×C4C22×D6712yes—
150C22×C4C22×D6711yescharacteristic
151C22×C4C22×D6711yescharacteristic
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—2684no—
155C22⋊C4—24no—
156C22⋊C4—2684no—
157C22⋊C4—1348no—
158C22⋊C4—1344no—
159C22⋊C4—1344no—
160C22⋊C4—1344no—
161C42—24no—
162C23—22no—
163C23—22no—
164C23—2684no—
165C23—26816no—
166C23—2684no—
167C23—674no—
168C23—1348no—
169C23—1342no—
170C23—1342no—
171C23—1344no—
172C23—1344no—
173C23C2×C22⋊D6712yes—
174C23#1176162814yes—
175C23C23×D6711yescharacteristic
176C2×C4—28no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—24no—
183C2×C4—2684no—
184C2×C4—1342no—
185C2×C4—1344no—
186C2×C4—1344no—
187C2×C4—1344no—
188C2×C4—1344no—
189C2×C4C2×C22⋊D6712yes—
190C2×C4C2×C22⋊D6712yes—
191C22—28no—
192C22—22no—
193C22—22no—
194C22—26816no—
195C22—2684no—
196C22—2682no—
197C22—2682no—
198C22—1348no—
199C22—1344no—
200C22#1452402614yes—
201C22C22×C22⋊D6711yescharacteristic
202C22#41×D6712yes—
203C22C2×#1176162812yes—
204C22#1452408912yes—
205C4—22no—
206C4—22no—
207C4—42no—
208C4—42no—
209C4—2684no—
210C4—2682no—
211C2—24no—
212C2—2684no—
213C2C2×#1452402611yescharacteristic
214C2#1686390311yescharacteristic
215C2#1686396011yescharacteristic
216C2#1686407014yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
FittingC2×#117×C67C2
hypercenterC2×#117D67
hypocenterC67#2482
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #16863331C2semidirect
3. #16863322C2semidirect
4. #16863287C2semidirect
5. #16863268C2semidirect
6. #16863259C2semidirect
8. C2×#117×C67C2semidirect
11. C2×Dic67⋊C4C22semidirect
18. C67⋊#95C22semidirect
19. C67⋊#95C22semidirect
20. C67⋊#95C22semidirect
23. C2×C4⋊C4×C67C22semidirect
24. C2×C4⋊C4×C67C22semidirect
26. C2×C42×C67C22semidirect
109. C67#2482semidirect
110. C2×#117D67semidirect
123. C2×C4⋊C4C2×D67semidirect
124. C2×C4⋊C4C2×D67semidirect
132. C2×C42C2×D67semidirect