Finite Group Databasebeta

← #22568268

Subgroups — #22568268

Order 8576.

430 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14524037C211yesmaximal, characteristic
2#16863334C212yesmaximal, complemented
3#16863331C212yesmaximal
4#16863327C212yesmaximal, complemented
5#16863322C212yesmaximal
6#16863315C212yesmaximal, complemented
7#16863307C212yesmaximal
8#16863264C211yesmaximal, characteristic, complemented
9C2×C4⋊Q8×C67C211yesmaximal, characteristic, complemented
10#14524037—24no—
11C2×Dic67⋊C4C2211yescharacteristic
12C2×Dic67⋊C4C2211yescharacteristic
13C2×Dic67⋊C4C2211yescharacteristic
14C2×Dic67⋊C4C2214yes—
15C2×(C2×D67)⋊C4C2211yescharacteristic
16C2×(C2×D67)⋊C4C2211yescharacteristic
17C2×(C2×D67)⋊C4C2211yescharacteristic
18C2×(C2×D67)⋊C4C2214yes—
19C67⋊#95C2212yes—
20C67⋊#95C2212yes—
21C67⋊#95C2212yes—
22C67⋊#95C2214yes—
23C67⋊#95C2214yes—
24C22×Q8×C67C2212yes—
25C4⋊Q8×C67—24no—
26C2×C4⋊C4×C67C2212yes—
27C2×C4⋊C4×C67C2212yes—
28C2×C42×C67C2211yescharacteristic
29C23×D67C2311yescharacteristic
30C22×Dic67C2314yes—
31C22×Dic67C2311yescharacteristic
32C22×Dic67C2311yescharacteristic
33C22×Dic67C2311yescharacteristic
34Dic67⋊C4—22no—
35Dic67⋊C4—22no—
36Dic67⋊C4—22no—
37Dic67⋊C4—44no—
38(C2×D67)⋊C4—22no—
39(C2×D67)⋊C4—22no—
40(C2×D67)⋊C4—22no—
41(C2×D67)⋊C4—44no—
42C2×Q8×C67—24no—
43C2×Q8×C67—44no—
44C22×C4×C67C2314yes—
45C22×C4×C67C2311yescharacteristic
46C22×C4×C67C2311yescharacteristic
47C22×C4×C67C2311yescharacteristic
48C4⋊C4×C67—24no—
49C4⋊C4×C67—24no—
50C42×C67D414yes—
51C23×C67C2411yescharacteristic
52C22×D67—21no—
53C22×D67—22no—
54C22×D67—22no—
55C22×D67—22no—
56Q8×C67—48no—
57C2×C4×C67—22no—
58C2×C4×C67—22no—
59C2×C4×C67—22no—
60C2×C4×C67—24no—
61C2×C4×C67—24no—
62C2×C4×C67—24no—
63C2×C4×C67C2×D412yes—
64C2×C4×C67C2×D412yes—
65C2×C4×C67C2×D412yes—
66C2×Dic67—21no—
67C2×Dic67—21no—
68C2×Dic67—21no—
69C2×Dic67—24no—
70C2×Dic67—22no—
71C2×Dic67—22no—
72C2×Dic67—22no—
73C2×Dic67—24no—
74C2×Dic67—24no—
75C22×C67C22×D411yescharacteristic
76C22×C67H(5,𝔽2)12yes—
77C22×C672−1+412yes—
78C22×C672−1+412yes—
79C2×D67—41no—
80C2×D67—41no—
81C2×D67—41no—
82C2×D67—44no—
83C4×C67—22no—
84C4×C67—22no—
85C4×C67—22no—
86C4×C67—44no—
87Dic67—41no—
88Dic67—41no—
89Dic67—41no—
90Dic67—44no—
91C2×C67#53511yescharacteristic
92C2×C67#53611yescharacteristic
93C2×C67#53611yescharacteristic
94C2×C67#56314yes—
95D67—81no—
96#2821—671nomaximal, Sylow
97C67#282111yescharacteristic, complemented, Sylow
98C2×C4⋊Q8D6711yescharacteristic, complemented
99C2×#126—671no—
100#400—672no—
101#399—672no—
102#398—672no—
103#397—672no—
104#395—672no—
105#393—672no—
106#383—671no—
107C22×Q8C2×D6712yes—
108C4⋊Q8—24no—
109#126—1344no—
110C2×C4⋊C4—674no—
111C2×C4⋊C4—671no—
112C2×C4⋊C4—671no—
113C2×C4⋊C4—671no—
114C2×C4⋊C4C2×D6712yes—
115C2×C4⋊C4C2×D6712yes—
116C2×C22⋊C4—674no—
117C2×C22⋊C4—671no—
118C2×C22⋊C4—671no—
119C2×C22⋊C4—671no—
120C2×C42C2×D6711yescharacteristic
121#95—672no—
122#95—672no—
123#95—672no—
124#95—674no—
125#95—674no—
126C24—671no—
127C2×Q8—24no—
128C2×Q8—44no—
129C22×C4—674no—
130C22×C4—671no—
131C22×C4—671no—
132C22×C4—671no—
133C22×C4C22×D6714yes—
134C22×C4C22×D6711yescharacteristic
135C22×C4C22×D6711yescharacteristic
136C22×C4C22×D6711yescharacteristic
137C4⋊C4—24no—
138C4⋊C4—24no—
139C4⋊C4—2684no—
140C4⋊C4—1342no—
141C4⋊C4—1342no—
142C4⋊C4—1342no—
143C22⋊C4—2684no—
144C22⋊C4—1342no—
145C22⋊C4—1342no—
146C22⋊C4—1342no—
147C42C22⋊D6714yes—
148C23—1341no—
149C23—1342no—
150C23—1342no—
151C23—1342no—
152C23C23×D6711yescharacteristic
153Q8—48no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—24no—
160C2×C4—1341no—
161C2×C4—1341no—
162C2×C4—1341no—
163C2×C4—1344no—
164C2×C4—1342no—
165C2×C4—1344no—
166C2×C4—1344no—
167C2×C4—1342no—
168C2×C4—1342no—
169C2×C4C2×C22⋊D6712yes—
170C2×C4C2×C22⋊D6712yes—
171C2×C4C2×C22⋊D6712yes—
172C22—2684no—
173C22—2681no—
174C22—2681no—
175C22—2681no—
176C22C22×C22⋊D6711yescharacteristic
177C22#1452408812yes—
178C22#1452409112yes—
179C22#1452409112yes—
180C4—22no—
181C4—22no—
182C4—22no—
183C4—44no—
184C4—2684no—
185C4—2681no—
186C4—2681no—
187C4—2681no—
188C2—5361no—
189C2#1686396311yescharacteristic
190C2#1686397011yescharacteristic
191C2#1686397011yescharacteristic
192C2#1686414514yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
FittingC2×C4⋊Q8×C67C2
hypercenterC2×C4⋊Q8D67
hypocenterC67#2821
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #16863334C2semidirect
4. #16863327C2semidirect
6. #16863315C2semidirect
8. #16863264C2semidirect
9. C2×C4⋊Q8×C67C2semidirect
97. C67#2821semidirect
98. C2×C4⋊Q8D67semidirect