Finite Group Databasebeta

← #22566368

Subgroups — #22566368

Order 8576.

434 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 141 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
2C2×#14524035C211yesmaximal, characteristic, complemented
3#16863336C211yesmaximal, characteristic, complemented
4#16863333C211yesmaximal, characteristic
5#16863328C211yesmaximal, characteristic, complemented
6#16863326C212yesmaximal, complemented
7#16863323C211yesmaximal, characteristic
8#16863318C212yesmaximal
9#16863294C212yesmaximal, complemented
10#16863287C212yesmaximal
11C4×C4⋊C4×C67C211yesmaximal, characteristic, complemented
12#14524037—24no—
13#14524035—24no—
14C2×C4⋊Dic67C2212yes—
15C2×C4⋊Dic67C2211yescharacteristic
16C2×C4⋊Dic67C2211yescharacteristic
17C2×C4⋊Dic67C2212yes—
18C2×Dic67⋊C4C2212yes—
19C2×Dic67⋊C4C2211yescharacteristic
20C2×Dic67⋊C4C2212yes—
21C2×Dic67⋊C4C2211yescharacteristic
22C2×Dic67⋊C4C2211yescharacteristic
23C2×(C2×D67)⋊C4C2211yescharacteristic
24C2×(C2×D67)⋊C4C2212yes—
25C2×(C2×D67)⋊C4C2211yescharacteristic
26C2×(C2×D67)⋊C4C2211yescharacteristic
27C2×(C2×D67)⋊C4C2212yes—
28C67⋊#95C2212yes—
29C67⋊#95C2212yes—
30C67⋊#95C2212yes—
31C67⋊#95C2212yes—
32C2×C4⋊C4×C67C2212yes—
33C2×C42×C67C2212yes—
34C2×C42×C67C2211yescharacteristic
35#95×C67C2211yescharacteristic
36#95×C67C2211yescharacteristic
37C23×D67C2311yescharacteristic
38C22×Dic67C2311yescharacteristic
39C22×Dic67C2311yescharacteristic
40C22×Dic67C2311yescharacteristic
41C22×Dic67C2312yes—
42C22×Dic67C2312yes—
43C4⋊Dic67—22no—
44C4⋊Dic67—22no—
45C4⋊Dic67—24no—
46C4⋊Dic67—42no—
47Dic67⋊C4—24no—
48Dic67⋊C4—24no—
49Dic67⋊C4—22no—
50Dic67⋊C4—22no—
51Dic67⋊C4—22no—
52(C2×D67)⋊C4—22no—
53(C2×D67)⋊C4—22no—
54(C2×D67)⋊C4—22no—
55(C2×D67)⋊C4—24no—
56(C2×D67)⋊C4—42no—
57C22×C4×C67C2312yes—
58C22×C4×C67C2311yescharacteristic
59C22×C4×C67C2311yescharacteristic
60C22×C4×C67C2311yescharacteristic
61C22×C4×C67C2312yes—
62C4⋊C4×C67—24no—
63C42×C67—42no—
64C42×C67D414yes—
65C23×C67C2411yescharacteristic
66C22×D67—21no—
67C22×D67—22no—
68C22×D67—21no—
69C22×D67—22no—
70C22×D67—21no—
71C2×C4×C67—22no—
72C2×C4×C67—21no—
73C2×C4×C67—22no—
74C2×C4×C67—22no—
75C2×C4×C67—22no—
76C2×C4×C67—21no—
77C2×C4×C67—22no—
78C2×C4×C67—22no—
79C2×C4×C67—22no—
80C2×C4×C67C2×D412yes—
81C2×C4×C67C2×D412yes—
82C2×C4×C67C2×D412yes—
83C2×C4×C67#4114yes—
84C2×Dic67—21no—
85C2×Dic67—21no—
86C2×Dic67—21no—
87C2×Dic67—22no—
88C2×Dic67—22no—
89C2×Dic67—22no—
90C2×Dic67—22no—
91C2×Dic67—22no—
92C2×Dic67—22no—
93C2×Dic67—21no—
94C2×Dic67—22no—
95C2×Dic67—21no—
96C2×Dic67—22no—
97C22×C67C22×D411yescharacteristic
98C22×C67C2×#4112yes—
99C22×C67H(5,𝔽2)11yescharacteristic
100C22×C672−1+412yes—
101C22×C672−1+411yescharacteristic
102C2×D67—41no—
103C2×D67—41no—
104C2×D67—41no—
105C2×D67—42no—
106C2×D67—42no—
107C4×C67—22no—
108C4×C67—22no—
109C4×C67—22no—
110C4×C67—24no—
111C4×C67—42no—
112Dic67—41no—
113Dic67—42no—
114Dic67—41no—
115Dic67—41no—
116Dic67—42no—
117C2×C67#53211yescharacteristic
118C2×C67#53611yescharacteristic
119C2×C67#53711yescharacteristic
120C2×C67#53912yes—
121C2×C67#54112yes—
122D67—81no—
123#2664—671nomaximal, Sylow
124C67#266411yescharacteristic, complemented, Sylow
125C2×#126—671no—
126C2×#125—671no—
127#400—671no—
128#399—671no—
129#398—672no—
130#398—671no—
131#397—671no—
132#396—672no—
133#389—672no—
134#388—672no—
135C4×C4⋊C4D6711yescharacteristic, complemented
136#126—1344no—
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—671no—
144C2×C4⋊C4—671no—
145C2×C4⋊C4—671no—
146C2×C4⋊C4—672no—
147C2×C4⋊C4C2×D6712yes—
148C2×C22⋊C4—672no—
149C2×C22⋊C4—671no—
150C2×C22⋊C4—671no—
151C2×C22⋊C4—671no—
152C2×C22⋊C4—672no—
153C2×C42C2×D6712yes—
154C2×C42C2×D6711yescharacteristic
155#95—672no—
156#95—672no—
157#95—672no—
158#95—672no—
159#95C2×D6711yescharacteristic
160#95C2×D6711yescharacteristic
161C24—671no—
162C22×C4—671no—
163C22×C4—671no—
164C22×C4—671no—
165C22×C4—672no—
166C22×C4—672no—
167C22×C4C22×D6712yes—
168C22×C4C22×D6711yescharacteristic
169C22×C4C22×D6711yescharacteristic
170C22×C4C22×D6711yescharacteristic
171C22×C4C22×D6712yes—
172C4⋊C4—24no—
173C4⋊C4—2682no—
174C4⋊C4—1344no—
175C4⋊C4—1344no—
176C4⋊C4—1342no—
177C4⋊C4—1342no—
178C4⋊C4—1342no—
179C4⋊C4—1342no—
180C4⋊C4—1342no—
181C4⋊C4—1344no—
182C22⋊C4—2682no—
183C22⋊C4—1342no—
184C22⋊C4—1342no—
185C22⋊C4—1342no—
186C22⋊C4—1344no—
187C42—42no—
188C42D26814yes—
189C23—1341no—
190C23—1342no—
191C23—1341no—
192C23—1342no—
193C23—1341no—
194C23C23×D6711yescharacteristic
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—1341no—
205C2×C4—1341no—
206C2×C4—1341no—
207C2×C4—1342no—
208C2×C4—1342no—
209C2×C4—1342no—
210C2×C4—1342no—
211C2×C4—1342no—
212C2×C4—1342no—
213C2×C4—1341no—
214C2×C4—1342no—
215C2×C4—1341no—
216C2×C4—1342no—
217C2×C4C2×D26812yes—
218C2×C4C2×D26812yes—
219C2×C4C2×D26812yes—
220C2×C4#1176162814yes—
221C22—2682no—
222C22—2681no—
223C22—2681no—
224C22—2681no—
225C22—2682no—
226C22C22×D26811yescharacteristic
227C22C2×#1176162812yes—
228C22#1452408811yescharacteristic
229C22#1452409012yes—
230C22#1452409111yescharacteristic
231C4—22no—
232C4—22no—
233C4—22no—
234C4—24no—
235C4—42no—
236C4—2681no—
237C4—2682no—
238C4—2681no—
239C4—2681no—
240C4—2682no—
241C2—5361no—
242C2#1686394611yescharacteristic
243C2#1686397211yescharacteristic
244C2#1686397711yescharacteristic
245C2#1686398812yes—
246C2#1686400412yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
FittingC4×C4⋊C4×C67C2
hypercenterC4×C4⋊C4D67
hypocenterC67#2664
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#14524035C2semidirect
3. #16863336C2semidirect
5. #16863328C2semidirect
6. #16863326C2semidirect
9. #16863294C2semidirect
11. C4×C4⋊C4×C67C2semidirect
124. C67#2664semidirect
135. C4×C4⋊C4D67semidirect