Finite Group Databasebeta

← #22566371

Subgroups — #22566371

Order 8576.

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

normal subgroups only

This group has 434 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×#14524035C211yesmaximal, characteristic, complemented
2#16863335C211yesmaximal, characteristic, complemented
3#16863332C211yesmaximal, characteristic
4#16863329C212yesmaximal, complemented
5#16863326C211yesmaximal, characteristic, complemented
6#16863321C211yesmaximal, characteristic
7#16863319C212yesmaximal
8#16863291C212yesmaximal, complemented
9#16863290C212yesmaximal
10C4×C4⋊Dic67C211yesmaximal, characteristic, complemented
11C2×#126×C67C211yesmaximal, characteristic
12#14524035—24no—
13C2×C4⋊Dic67C2212yes—
14C2×C4⋊Dic67C2211yescharacteristic
15C2×C4⋊Dic67C2211yescharacteristic
16C2×Dic67⋊C4C2212yes—
17C2×Dic67⋊C4C2212yes—
18C2×Dic67⋊C4C2212yes—
19C2×Dic67⋊C4C2212yes—
20C2×C22⋊Dic67C2212yes—
21C2×C22⋊Dic67C2212yes—
22C2×C4×Dic67C2212yes—
23C67⋊#95C2211yescharacteristic
24C67⋊#95C2211yescharacteristic
25C67⋊#95C2212yes—
26C67⋊#95C2212yes—
27C67⋊#95C2212yes—
28C67⋊#95C2212yes—
29#126×C67—24no—
30C2×C4⋊C4×C67C2211yescharacteristic
31C2×C4⋊C4×C67C2211yescharacteristic
32C2×C4⋊C4×C67C2211yescharacteristic
33C2×C22⋊C4×C67C2211yescharacteristic
34C2×C22⋊C4×C67C2211yescharacteristic
35C2×C22⋊C4×C67C2211yescharacteristic
36C2×C42×C67C2211yescharacteristic
37C22×Dic67C2312yes—
38C22×Dic67C2312yes—
39C22×Dic67C2312yes—
40C22×Dic67C2312yes—
41C4⋊Dic67—22no—
42C4⋊Dic67—22no—
43C4⋊Dic67—24no—
44Dic67⋊C4—24no—
45Dic67⋊C4—24no—
46Dic67⋊C4—24no—
47Dic67⋊C4—42no—
48C22⋊Dic67—24no—
49C22⋊Dic67—42no—
50C4×Dic67—42no—
51C24×C67C2311yescharacteristic
52C22×C4×C67C2311yescharacteristic
53C22×C4×C67C2311yescharacteristic
54C22×C4×C67C2311yescharacteristic
55C22×C4×C67C2311yescharacteristic
56C22×C4×C67C2311yescharacteristic
57C22×C4×C67C2311yescharacteristic
58C4⋊C4×C67—22no—
59C4⋊C4×C67—22no—
60C4⋊C4×C67—22no—
61C22⋊C4×C67—22no—
62C22⋊C4×C67—22no—
63C22⋊C4×C67—22no—
64C42×C67D414yes—
65C23×C67—21no—
66C23×C67—22no—
67C23×C67—21no—
68C23×C67—22no—
69C23×C67—21no—
70C23×C67C2411yescharacteristic
71C2×C4×C67—22no—
72C2×C4×C67—21no—
73C2×C4×C67—22no—
74C2×C4×C67—21no—
75C2×C4×C67—21no—
76C2×C4×C67—21no—
77C2×C4×C67—21no—
78C2×C4×C67—22no—
79C2×C4×C67—21no—
80C2×C4×C67—22no—
81C2×C4×C67—21no—
82C2×C4×C67C2×D412yes—
83C2×C4×C67C2×D412yes—
84C2×C4×C67C2×D412yes—
85C2×Dic67—22no—
86C2×Dic67—22no—
87C2×Dic67—22no—
88C2×Dic67—22no—
89C2×Dic67—22no—
90C2×Dic67—22no—
91C2×Dic67—22no—
92C2×Dic67—22no—
93C2×Dic67—22no—
94C2×Dic67—22no—
95C2×Dic67—22no—
96C2×Dic67#4114yes—
97C22×C67—41no—
98C22×C67—41no—
99C22×C67—41no—
100C22×C67—42no—
101C22×C67—42no—
102C22×C67C22×D411yescharacteristic
103C22×C67C2×#4112yes—
104C22×C67H(5,𝔽2)11yescharacteristic
105C22×C672−1+412yes—
106C22×C672−1+411yescharacteristic
107C4×C67—22no—
108C4×C67—22no—
109C4×C67—22no—
110C4×C67—41no—
111C4×C67—41no—
112C4×C67—41no—
113Dic67—24no—
114Dic67—42no—
115Dic67—42no—
116Dic67—42no—
117C2×C67—81no—
118C2×C67#53211yescharacteristic
119C2×C67#53611yescharacteristic
120C2×C67#53711yescharacteristic
121C2×C67#53912yes—
122C2×C67#54112yes—
123#2664—671nomaximal, Sylow
124C67#266411yescharacteristic, complemented, Sylow
125C2×#126D6711yescharacteristic
126C2×#125—671no—
127#400—671no—
128#399—671no—
129#398—671no—
130#398—672no—
131#397—671no—
132#396—672no—
133#389—672no—
134#388—672no—
135C4×C4⋊C4—671no—
136#126—24no—
137#125—1344no—
138C2×C4⋊C4—672no—
139C2×C4⋊C4—672no—
140C2×C4⋊C4—672no—
141C2×C4⋊C4—671no—
142C2×C4⋊C4—671no—
143C2×C4⋊C4—672no—
144C2×C4⋊C4—672no—
145C2×C4⋊C4C2×D6711yescharacteristic
146C2×C4⋊C4C2×D6711yescharacteristic
147C2×C4⋊C4C2×D6711yescharacteristic
148C2×C22⋊C4—672no—
149C2×C22⋊C4—672no—
150C2×C22⋊C4C2×D6711yescharacteristic
151C2×C22⋊C4C2×D6711yescharacteristic
152C2×C22⋊C4C2×D6711yescharacteristic
153C2×C42—672no—
154C2×C42C2×D6711yescharacteristic
155#95—671no—
156#95—671no—
157#95—672no—
158#95—672no—
159#95—672no—
160#95—672no—
161C24C22×D6711yescharacteristic
162C22×C4—672no—
163C22×C4—672no—
164C22×C4—672no—
165C22×C4—672no—
166C22×C4C22×D6711yescharacteristic
167C22×C4C22×D6711yescharacteristic
168C22×C4C22×D6711yescharacteristic
169C22×C4C22×D6711yescharacteristic
170C22×C4C22×D6711yescharacteristic
171C22×C4C22×D6711yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—2682no—
176C4⋊C4—1344no—
177C4⋊C4—1344no—
178C4⋊C4—1344no—
179C4⋊C4—1344no—
180C4⋊C4—1342no—
181C4⋊C4—1342no—
182C22⋊C4—22no—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—2682no—
186C22⋊C4—1344no—
187C42—2682no—
188C42C22⋊D6714yes—
189C23—21no—
190C23—22no—
191C23—21no—
192C23—21no—
193C23—22no—
194C23C23×D6711yescharacteristic
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—674no—
207C2×C4—1342no—
208C2×C4—1342no—
209C2×C4—1342no—
210C2×C4—1342no—
211C2×C4—1342no—
212C2×C4—1342no—
213C2×C4—1342no—
214C2×C4—1342no—
215C2×C4—1342no—
216C2×C4—1342no—
217C2×C4—1342no—
218C2×C4C2×C22⋊D6712yes—
219C2×C4C2×C22⋊D6712yes—
220C2×C4C2×C22⋊D6712yes—
221C22—42no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—42no—
226C22C22×C22⋊D6711yescharacteristic
227C22#41×D6712yes—
228C22#1452408911yescharacteristic
229C22#1452409011yescharacteristic
230C22#1452409012yes—
231C4—22no—
232C4—22no—
233C4—22no—
234C4—41no—
235C4—41no—
236C4—41no—
237C4—2682no—
238C4—2682no—
239C4—2682no—
240C4—1344no—
241C2—81no—
242C2#1686394211yescharacteristic
243C2#1686396711yescharacteristic
244C2#1686397411yescharacteristic
245C2#1686398712yes—
246C2#1686400312yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
FittingC2×#126×C67C2
hypercenterC2×#126D67
hypocenterC67#2664
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#14524035C2semidirect
2. #16863335C2semidirect
4. #16863329C2semidirect
5. #16863326C2semidirect
8. #16863291C2semidirect
10. C4×C4⋊Dic67C2semidirect
124. C67#2664semidirect