Finite Group Databasebeta

← #22566031

Subgroups — #22566031

Order 8576.

488 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 157 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×#14524000C211yesmaximal, characteristic
2#16863334C212yesmaximal, complemented
3#16863331C212yesmaximal
4#16863315C212yesmaximal, complemented
5#16863307C212yesmaximal
6#16863292C212yesmaximal, complemented
7#16863262C211yesmaximal, characteristic, complemented
8#16863253C212yesmaximal
9C2×C4⋊Q8×C67C211yesmaximal, characteristic, complemented
10C22×C4×D67C2211yescharacteristic
11#14524000—24no—
12C2×C4⋊Dic67C2212yes—
13C2×Dic67⋊C4C2211yescharacteristic
14C2×Dic67⋊C4C2211yescharacteristic
15C2×Dic67⋊C4C2214yes—
16C2×(C2×D67)⋊C4C2211yescharacteristic
17C2×(C2×D67)⋊C4C2211yescharacteristic
18C2×(C2×D67)⋊C4C2214yes—
19C2×C4×Dic67C2211yescharacteristic
20C67⋊#95C2214yes—
21C67⋊#95C2212yes—
22C67⋊#95C2212yes—
23C67⋊#95C2214yes—
24C22×Q8×C67C2212yes—
25C4⋊Q8×C67—24no—
26C2×C4⋊C4×C67C2212yes—
27C2×C4⋊C4×C67C2212yes—
28C2×C42×C67C2211yescharacteristic
29C23×D67C2311yescharacteristic
30C22×Dic67C2311yescharacteristic
31C22×Dic67C2314yes—
32C22×Dic67C2311yescharacteristic
33C22×Dic67C2311yescharacteristic
34C2×C4×D67—22no—
35C2×C4×D67—22no—
36C2×C4×D67Q814yescomplemented
37C4⋊Dic67—24no—
38Dic67⋊C4—22no—
39Dic67⋊C4—22no—
40Dic67⋊C4—44no—
41(C2×D67)⋊C4—22no—
42(C2×D67)⋊C4—22no—
43(C2×D67)⋊C4—44no—
44C4×Dic67—22no—
45C2×Q8×C67—24no—
46C2×Q8×C67—44no—
47C22×C4×C67C2314yes—
48C22×C4×C67C2311yescharacteristic
49C22×C4×C67C2311yescharacteristic
50C22×C4×C67C2311yescharacteristic
51C4⋊C4×C67—24no—
52C4⋊C4×C67—24no—
53C42×C67D414yes—
54C23×C67C2411yescharacteristic
55C22×D67—21no—
56C22×D67—21no—
57C22×D67—22no—
58C22×D67—22no—
59C22×D67C2×Q812yes—
60Q8×C67—48no—
61C2×C4×C67—22no—
62C2×C4×C67—22no—
63C2×C4×C67—24no—
64C2×C4×C67—24no—
65C2×C4×C67—24no—
66C2×C4×C67C2×D412yes—
67C2×C4×C67C2×D412yes—
68C2×C4×C67C2×D412yes—
69C2×C4×C67C2×Q812yes—
70C2×C4×C67#4112yes—
71C2×Dic67—21no—
72C2×Dic67—21no—
73C2×Dic67—21no—
74C2×Dic67—21no—
75C2×Dic67—24no—
76C2×Dic67—24no—
77C2×Dic67—22no—
78C2×Dic67—22no—
79C2×Dic67—24no—
80C2×Dic67C2×Q812yes—
81C4×D67—28no—
82C22×C67C22×D411yescharacteristic
83C22×C67C22×Q811yescharacteristic
84C22×C67C2×#4111yescharacteristic
85C22×C67H(5,𝔽2)12yes—
86C22×C672−1+412yes—
87C2×D67—22no—
88C2×D67—24no—
89C2×D67—41no—
90C2×D67—41no—
91C2×D67—42no—
92C4×C67—22no—
93C4×C67—22no—
94C4×C67—44no—
95C4×C67C22⋊Q814yes—
96Dic67—22no—
97Dic67—41no—
98Dic67—41no—
99Dic67—44no—
100C2×C67C2×C22⋊Q811yescharacteristic
101C2×C67#53511yescharacteristic
102C2×C67#53611yescharacteristic
103C2×C67#54212yes—
104C2×C67#54412yes—
105D67—42no—
106#2628—671nomaximal, Sylow
107C67#262811yescharacteristic, complemented, Sylow
108C2×C4⋊Q8D6711yescharacteristic, complemented
109C2×#117—671no—
110#400—672no—
111#399—672no—
112#395—672no—
113#393—672no—
114#389—672no—
115#382—671no—
116#380—672no—
117C22×Q8C2×D6712yes—
118C23×C4—671no—
119C4⋊Q8—24no—
120#117—1344no—
121C2×C4⋊C4—674no—
122C2×C4⋊C4—671no—
123C2×C4⋊C4—671no—
124C2×C4⋊C4—672no—
125C2×C4⋊C4C2×D6712yes—
126C2×C4⋊C4C2×D6712yes—
127C2×C22⋊C4—674no—
128C2×C22⋊C4—671no—
129C2×C22⋊C4—671no—
130C2×C42—671no—
131C2×C42C2×D6711yescharacteristic
132#95—674no—
133#95—672no—
134#95—672no—
135#95—674no—
136C24—671no—
137C2×Q8—24no—
138C2×Q8—44no—
139C22×C4—671no—
140C22×C4—674no—
141C22×C4—671no—
142C22×C4—674no—
143C22×C4—671no—
144C22×C4—1342no—
145C22×C4—1342no—
146C22×C4C22×D6714yes—
147C22×C4C22×D6711yescharacteristic
148C22×C4C22×D6711yescharacteristic
149C22×C4C22×D6711yescharacteristic
150C4⋊C4—24no—
151C4⋊C4—24no—
152C4⋊C4—2684no—
153C4⋊C4—1342no—
154C4⋊C4—1342no—
155C4⋊C4—1344no—
156C22⋊C4—2684no—
157C22⋊C4—1342no—
158C22⋊C4—1342no—
159C42—1342no—
160C42C22⋊D6714yes—
161C23—672no—
162C23—1341no—
163C23—1341no—
164C23—1342no—
165C23—1342no—
166C23C23×D6711yescharacteristic
167Q8—48no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—24no—
173C2×C4—672no—
174C2×C4—1341no—
175C2×C4—1341no—
176C2×C4—1344no—
177C2×C4—1344no—
178C2×C4—1342no—
179C2×C4—1344no—
180C2×C4—1342no—
181C2×C4—1341no—
182C2×C4—1341no—
183C2×C4—1348no—
184C2×C4C2×C22⋊D6712yes—
185C2×C4C2×C22⋊D6712yes—
186C2×C4C2×C22⋊D6712yes—
187C2×C4Q8×D6712yes—
188C2×C4#1176162912yes—
189C22—2682no—
190C22—2681no—
191C22—2681no—
192C22—1344no—
193C22—1342no—
194C22C22×C22⋊D6711yescharacteristic
195C22C2×Q8×D6711yescharacteristic
196C22C2×#1176162911yescharacteristic
197C22#1452408812yes—
198C22#1452409112yes—
199C4—22no—
200C4—22no—
201C4—44no—
202C4—2684no—
203C4—2681no—
204C4—2681no—
205C4—1342no—
206C4#1452401914yes—
207C2—2682no—
208C2C2×#1452401911yescharacteristic
209C2#1686396311yescharacteristic
210C2#1686397011yescharacteristic
211C2#1686402012yes—
212C2#1686402312yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
FittingC2×C4⋊Q8×C67C2
hypercenterC2×C4⋊Q8D67
hypocenterC67#2628
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. #16863315C2semidirect
6. #16863292C2semidirect
7. #16863262C2semidirect
9. C2×C4⋊Q8×C67C2semidirect
36. C2×C4×D67Q8semidirect
107. C67#2628semidirect
108. C2×C4⋊Q8D67semidirect