Finite Group Databasebeta

← #22567912

Subgroups — #22567912

Order 8576.

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

normal subgroups only

This group has 472 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
1#16863335C211yesmaximal, characteristic, complemented
2#16863318C212yesmaximal, complemented
3#16863307C212yesmaximal, complemented
4#16863297C211yesmaximal, characteristic
5#16863283C212yesmaximal, complemented
6#16863274C212yesmaximal
7#16863270C212yesmaximal, complemented
8#16863253C212yesmaximal, complemented
9#395×C67C211yesmaximal, characteristic, complemented
10C22×D268C2211yescharacteristic
11C22×C4×D67C2212yes—
12C2×C4⋊Dic67C2212yes—
13C2×C4⋊Dic67C2211yescharacteristic
14C2×Dic67⋊C4C2212yes—
15C2×(C2×D67)⋊C4C2212yes—
16C2×(C2×D67)⋊C4C2212yes—
17C2×(C2×D67)⋊C4C2212yes—
18C2×(C2×D67)⋊C4C2212yes—
19C2×(C2×D67)⋊C4C2212yes—
20C2×C4×Dic67C2211yescharacteristic
21C67⋊#95C2211yescharacteristic
22C67⋊#95C2212yes—
23C67⋊#95C2212yes—
24C67⋊#95C2212yes—
25C67⋊#95C2212yes—
26C22×Q8×C67C2211yescharacteristic
27C2×C4⋊C4×C67C2212yes—
28C2×C4⋊C4×C67C2211yescharacteristic
29#95×C67C2211yescharacteristic
30#95×C67C2212yes—
31C23×D67C2312yes—
32C2×D268—41no—
33C2×D268—41no—
34C2×D268Q814yescomplemented
35C22×Dic67C2312yes—
36C22×Dic67C2312yes—
37C22×Dic67C2312yes—
38C2×C4×D67—24no—
39C2×C4×D67—42no—
40C2×C4×D67—42no—
41C4⋊Dic67—42no—
42C4⋊Dic67—41no—
43Dic67⋊C4—24no—
44(C2×D67)⋊C4—24no—
45(C2×D67)⋊C4—42no—
46(C2×D67)⋊C4—42no—
47(C2×D67)⋊C4—42no—
48(C2×D67)⋊C4—42no—
49C4×Dic67—41no—
50C2×Q8×C67—42no—
51C2×Q8×C67—41no—
52C22×C4×C67C2311yescharacteristic
53C22×C4×C67C2312yes—
54C22×C4×C67C2311yescharacteristic
55C22×C4×C67C2311yescharacteristic
56C22×C4×C67C2312yes—
57C4⋊C4×C67—42no—
58C4⋊C4×C67—41no—
59C23×C67C2411yescharacteristic
60C22×D67—22no—
61C22×D67—22no—
62C22×D67—22no—
63C22×D67—22no—
64C22×D67—22no—
65C22×D67—22no—
66C22×D67C2×Q814yes—
67Q8×C67—44no—
68C2×C4×C67—22no—
69C2×C4×C67—21no—
70C2×C4×C67—21no—
71C2×C4×C67—22no—
72C2×C4×C67—22no—
73C2×C4×C67—21no—
74C2×C4×C67—22no—
75C2×C4×C67—21no—
76C2×C4×C67—22no—
77C2×C4×C67—22no—
78C2×C4×C67—22no—
79C2×C4×C67C2×Q812yes—
80C2×C4×C67#4114yes—
81C2×Dic67—22no—
82C2×Dic67—22no—
83C2×Dic67—22no—
84C2×Dic67—22no—
85C2×Dic67—22no—
86C2×Dic67—22no—
87C2×Dic67—22no—
88C2×Dic67—22no—
89C2×Dic67—22no—
90D268—44no—
91C4×D67—48no—
92C22×C67C22×Q811yescharacteristic
93C22×C67C2×#4112yes—
94C22×C67H(5,𝔽2)12yes—
95C22×C67H(5,𝔽2)11yescharacteristic
96C22×C67H(5,𝔽2)11yescharacteristic
97C2×D67—24no—
98C2×D67—24no—
99C2×D67—24no—
100C2×D67—42no—
101C2×D67—42no—
102C2×D67—42no—
103C2×D67—42no—
104C4×C67—24no—
105C4×C67—22no—
106C4×C67—41no—
107C4×C67—42no—
108C4×C67—41no—
109Dic67—42no—
110Dic67—42no—
111Dic67—42no—
112C2×C67#53812yes—
113C2×C67#54311yescharacteristic
114C2×C67#55412yes—
115C2×C67#55511yescharacteristic
116C2×C67#56011yescharacteristic
117D67—44no—
118#2784—671nomaximal, Sylow
119C67#278411yescharacteristic, complemented, Sylow
120#400—671no—
121#396—672no—
122#395D6711yescharacteristic, complemented
123#393—672no—
124#390—671no—
125#387—672no—
126#386—672no—
127#385—672no—
128#380—672no—
129C22×Q8C2×D6711yescharacteristic
130C22×D4—671no—
131C23×C4—672no—
132C2×C4⋊C4—672no—
133C2×C4⋊C4—671no—
134C2×C4⋊C4—672no—
135C2×C4⋊C4C2×D6712yes—
136C2×C4⋊C4C2×D6711yescharacteristic
137C2×C22⋊C4—672no—
138C2×C22⋊C4—672no—
139C2×C22⋊C4—672no—
140C2×C22⋊C4—672no—
141C2×C22⋊C4—672no—
142C2×C42—671no—
143#95—671no—
144#95—672no—
145#95—672no—
146#95—672no—
147#95—672no—
148#95C2×D6711yescharacteristic
149#95C2×D6712yes—
150C24—672no—
151C2×Q8—41no—
152C2×Q8—42no—
153C2×D4—2681no—
154C2×D4—2681no—
155C2×D4—674no—
156C22×C4—2682no—
157C22×C4—2682no—
158C22×C4—672no—
159C22×C4—672no—
160C22×C4—672no—
161C22×C4—1344no—
162C22×C4C22×D6712yes—
163C22×C4C22×D6711yescharacteristic
164C22×C4C22×D6711yescharacteristic
165C22×C4C22×D6711yescharacteristic
166C22×C4C22×D6712yes—
167C4⋊C4—42no—
168C4⋊C4—41no—
169C4⋊C4—2682no—
170C4⋊C4—2681no—
171C4⋊C4—1344no—
172C22⋊C4—2682no—
173C22⋊C4—2682no—
174C22⋊C4—2682no—
175C22⋊C4—2682no—
176C22⋊C4—1344no—
177C42—2681no—
178C23—674no—
179C23—1342no—
180C23—1342no—
181C23—1342no—
182C23—1342no—
183C23—1342no—
184C23—1342no—
185C23C23×D6711yescharacteristic
186Q8—44no—
187D4—2684no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—2688no—
200C2×C4—1342no—
201C2×C4—1342no—
202C2×C4—1342no—
203C2×C4—1342no—
204C2×C4—1342no—
205C2×C4—1342no—
206C2×C4—1342no—
207C2×C4—1342no—
208C2×C4—1342no—
209C2×C4Q8×D6712yes—
210C2×C4#1176162914yes—
211C22—2682no—
212C22—2682no—
213C22—2682no—
214C22—2682no—
215C22—1344no—
216C22—1344no—
217C22—1344no—
218C22C2×Q8×D6711yescharacteristic
219C22C2×#1176162912yes—
220C22#1452408811yescharacteristic
221C22#1452408812yes—
222C22#1452408911yescharacteristic
223C4—22no—
224C4—24no—
225C4—42no—
226C4—41no—
227C4—41no—
228C4—2682no—
229C4—2682no—
230C4—2682no—
231C2—2684no—
232C2#1686398612yes—
233C2#1686402111yescharacteristic
234C2#1686410712yes—
235C2#1686411211yescharacteristic
236C2#1686413511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#395×C67C2
hypercenter#395D67
hypocenterC67#2784
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. #16863335C2semidirect
2. #16863318C2semidirect
3. #16863307C2semidirect
5. #16863283C2semidirect
7. #16863270C2semidirect
8. #16863253C2semidirect
9. #395×C67C2semidirect
34. C2×D268Q8semidirect
119. C67#2784semidirect
122. #395D67semidirect