Finite Group Databasebeta

← #22567339

Subgroups — #22567339

Order 8576.

416 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 129 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#16863337C212yesmaximal, complemented
2#16863316C212yesmaximal, complemented
3#16863308C211yesmaximal, characteristic
4#16863291C212yesmaximal, complemented
5#16863290C212yesmaximal
6#16863277C212yesmaximal, complemented
7#16863268C211yesmaximal, characteristic
8#16863254C212yesmaximal
9#399×C67C211yesmaximal, characteristic
10C22×Dic134C2211yescharacteristic
11C23×Dic67C2211yescharacteristic
12C2×Dic67⋊C4C2212yes—
13C2×Dic67⋊C4C2212yes—
14C2×Dic67⋊C4C2212yes—
15C2×Dic67⋊C4C2212yes—
16C2×C22⋊Dic67C2212yes—
17C2×C22⋊Dic67C2211yescharacteristic
18C2×C4×Dic67C2212yes—
19C67⋊#95C2212yes—
20C67⋊#95C2212yes—
21C67⋊#95C2212yes—
22C67⋊#95C2211yescharacteristic
23C67⋊#95C2212yes—
24C67⋊#95C2212yes—
25C67⋊#95C2212yes—
26C2×C4⋊C4×C67C2212yes—
27C2×C4⋊C4×C67C2211yescharacteristic
28C2×C22⋊C4×C67C2212yes—
29C2×C22⋊C4×C67C2211yescharacteristic
30#95×C67C2211yescharacteristic
31C2×Dic134—42no—
32C2×Dic134—41no—
33C22×Dic67—24no—
34C22×Dic67—41no—
35C22×Dic67C2312yes—
36C22×Dic67C2312yes—
37C22×Dic67C2312yes—
38C22×Dic67C2312yes—
39Dic67⋊C4—24no—
40Dic67⋊C4—24no—
41Dic67⋊C4—42no—
42Dic67⋊C4—42no—
43C22⋊Dic67—42no—
44C22⋊Dic67Q814yescomplemented
45C4×Dic67—42no—
46C24×C67C2311yescharacteristic
47C22×C4×C67C2311yescharacteristic
48C22×C4×C67C2312yes—
49C22×C4×C67C2312yes—
50C22×C4×C67C2311yescharacteristic
51C4⋊C4×C67—42no—
52C4⋊C4×C67—41no—
53C22⋊C4×C67—42no—
54C22⋊C4×C67—41no—
55C23×C67—22no—
56C23×C67—22no—
57C23×C67—21no—
58C23×C67—21no—
59C23×C67C2×Q812yes—
60C23×C67C2411yescharacteristic
61C2×C4×C67—22no—
62C2×C4×C67—22no—
63C2×C4×C67—22no—
64C2×C4×C67—21no—
65C2×C4×C67—22no—
66C2×C4×C67—21no—
67C2×C4×C67—22no—
68C2×C4×C67—22no—
69C2×C4×C67—22no—
70C2×C4×C67—22no—
71C2×Dic67—22no—
72C2×Dic67—22no—
73C2×Dic67—22no—
74C2×Dic67—22no—
75C2×Dic67—22no—
76C2×Dic67—22no—
77C2×Dic67—22no—
78C2×Dic67—22no—
79C2×Dic67—22no—
80C2×Dic67—22no—
81C2×Dic67—44no—
82C2×Dic67C2×Q814yes—
83C2×Dic67#4114yes—
84Dic134—44no—
85C22×C67—22no—
86C22×C67—24no—
87C22×C67—42no—
88C22×C67—41no—
89C22×C67—41no—
90C22×C67C22×Q811yescharacteristic
91C22×C67C2×#4112yes—
92C22×C67H(5,𝔽2)12yes—
93C22×C67H(5,𝔽2)11yescharacteristic
94C22×C672−1+411yescharacteristic
95C4×C67—42no—
96C4×C67—41no—
97C4×C67—41no—
98C4×C67—42no—
99Dic67—24no—
100Dic67—24no—
101Dic67—42no—
102Dic67—42no—
103C2×C67—42no—
104C2×C67#54212yes—
105C2×C67#54411yescharacteristic
106C2×C67#55111yescharacteristic
107C2×C67#55412yes—
108C2×C67#56011yescharacteristic
109#2742—671nomaximal, Sylow
110C67#274211yescharacteristic, complemented, Sylow
111#400—672no—
112#399D6711yescharacteristic
113#395—672no—
114#393—671no—
115#389—672no—
116#388—672no—
117#387—672no—
118#385—671no—
119#380—672no—
120C22×Q8—671no—
121C23×C4—671no—
122C2×C4⋊C4—672no—
123C2×C4⋊C4—672no—
124C2×C4⋊C4—672no—
125C2×C4⋊C4—672no—
126C2×C4⋊C4C2×D6712yes—
127C2×C4⋊C4C2×D6711yescharacteristic
128C2×C22⋊C4—672no—
129C2×C22⋊C4—671no—
130C2×C22⋊C4C2×D6712yes—
131C2×C22⋊C4C2×D6711yescharacteristic
132C2×C42—672no—
133#95—672no—
134#95—672no—
135#95—672no—
136#95—671no—
137#95—672no—
138#95—672no—
139#95—672no—
140#95C2×D6711yescharacteristic
141C24C22×D6711yescharacteristic
142C2×Q8—2681no—
143C2×Q8—2682no—
144C22×C4—2681no—
145C22×C4—672no—
146C22×C4—672no—
147C22×C4—672no—
148C22×C4—672no—
149C22×C4—1344no—
150C22×C4C22×D6712yes—
151C22×C4C22×D6711yescharacteristic
152C22×C4C22×D6711yescharacteristic
153C22×C4C22×D6712yes—
154C4⋊C4—42no—
155C4⋊C4—41no—
156C4⋊C4—2682no—
157C4⋊C4—2682no—
158C4⋊C4—1344no—
159C4⋊C4—1344no—
160C22⋊C4—42no—
161C22⋊C4—41no—
162C22⋊C4—2682no—
163C22⋊C4—674no—
164C42—2682no—
165C23—22no—
166C23—22no—
167C23—21no—
168C23—21no—
169C23Q8×D6712yes—
170C23C23×D6711yescharacteristic
171Q8—2684no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—2684no—
183C2×C4—674no—
184C2×C4—674no—
185C2×C4—1342no—
186C2×C4—1342no—
187C2×C4—1342no—
188C2×C4—1342no—
189C2×C4—1342no—
190C2×C4—1342no—
191C2×C4—1342no—
192C2×C4—1342no—
193C2×C4—1342no—
194C2×C4—1342no—
195C22—22no—
196C22—24no—
197C22—42no—
198C22—41no—
199C22—41no—
200C22C2×Q8×D6711yescharacteristic
201C22#41×D6712yes—
202C22#1452408812yes—
203C22#1452408811yescharacteristic
204C22#1452409011yescharacteristic
205C4—42no—
206C4—41no—
207C4—41no—
208C4—42no—
209C4—2682no—
210C4—2682no—
211C4—1344no—
212C4—1344no—
213C2—42no—
214C2#1686401412yes—
215C2#1686402511yescharacteristic
216C2#1686407411yescharacteristic
217C2#1686410012yes—
218C2#1686413311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#399×C67C2
hypercenter#399D67
hypocenterC67#2742
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. #16863337C2semidirect
2. #16863316C2semidirect
4. #16863291C2semidirect
6. #16863277C2semidirect
44. C22⋊Dic67Q8semidirect
110. C67#2742semidirect