Finite Group Databasebeta

← #22571562

Subgroups — #22571562

Order 8576.

360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 113 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#16863932C212yesmaximal, complemented
2#16863705C212yesmaximal, complemented
3#16863699C212yesmaximal, complemented
4#16863632C211yesmaximal, characteristic
5#16863627C212yesmaximal, complemented
6#16863620C212yesmaximal, complemented
7#16863614C212yesmaximal, complemented
8C2×Dic67⋊C8C211yesmaximal, characteristic, complemented
9#467×C67C211yesmaximal, characteristic, complemented
10C2×#11761615C2212yes—
11C2×D536C2211yescharacteristic
12C22×C67⋊C8C2211yescharacteristic
13#14524038—22no—
14#14524034C2212yes—
15#14524032C2212yes—
16#14524027—22no—
17#14524017C2212yes—
18Q8×Dic67C2212yes—
19#14524005C2212yes—
20(C22⋊D67)⋊C4—22no—
21#14524001—22no—
22C2×C4×Dic67C2211yescharacteristic
23#14523977C2211yescharacteristic
24#14523974C2212yes—
25Dic67⋊C8C2212yes—
26Dic67⋊C8C2212yes—
27#14523967C2212yes—
28#14523965C2212yes—
29#14523963C2212yes—
30C2×Q16×C67C2211yescharacteristic
31C22×C8×C67C2211yescharacteristic
32C22⋊Q8×C67C2212yes—
33#107×C67C2211yescharacteristic
34#103×C67C2212yes—
35C2×D268C2312yes—
36C2×C22⋊D67—22no—
37C22×Dic67D411yescharacteristic
38C22×Dic67D411yescharacteristic
39C2×C4×D67—22no—
40#11761615—42no—
41D536—41no—
42C2×C67⋊C8—41no—
43C2×C67⋊C8C2312yes—
44C4⋊Dic67—22no—
45C4⋊Dic67—22no—
46C4⋊Dic67C2312yes—
47Dic67⋊C4—22no—
48(C2×D67)⋊C4—22no—
49(C2×D67)⋊C4—22no—
50(C2×D67)⋊C4—22no—
51(C2×D67)⋊C4—22no—
52C4×Dic67—22no—
53C4×Dic67D412yes—
54C4×Dic67C2312yes—
55C2×Q8×C67C2312yes—
56C22×C4×C67C2311yescharacteristic
57Q16×C67—41no—
58C2×C8×C67—41no—
59C2×C8×C67C2311yescharacteristic
60C2×C8×C67C2311yescharacteristic
61C4⋊C4×C67—22no—
62C4⋊C4×C67C2312yes—
63C22⋊C4×C67—22no—
64C23×C67C2×D411yescharacteristic
65C22×D67—22no—
66Q8×C67—22no—
67Q8×C67—22no—
68C2×C4×C67—21no—
69C2×C4×C67—21no—
70C2×C4×C67—22no—
71C2×C4×C67—22no—
72C2×C4×C67C2×D411yescharacteristic
73C2×C4×C67C2411yescharacteristic
74C22⋊D67—44no—
75C2×Dic67—22no—
76C2×Dic67—22no—
77C2×Dic67—41no—
78C2×Dic67C2×D412yes—
79C2×Dic67C2×D412yes—
80D268—22no—
81D268—22no—
82C4×D67—44no—
83C8×C67—21no—
84C8×C67—21no—
85C67⋊C8—22no—
86C22×C67—21no—
87C22×C67—21no—
88C22×C67#13512yes—
89C22×C67C22×D411yescharacteristic
90C2×D67—42no—
91C2×D67—42no—
92C2×D67—42no—
93C4×C67—21no—
94C4×C67—42no—
95C4×C67—42no—
96C4×C67H(5,𝔽2)11yescharacteristic
97C4×C672−1+411yescharacteristic
98Dic67—22no—
99Dic67—22no—
100Dic67—42no—
101C2×C67—22no—
102C2×C67#53711yescharacteristic
103C2×C67C2×#13511yescharacteristic
104C2×C67#57511yescharacteristic
105D67—82no—
106#3250—671nomaximal, Sylow
107C67#325011yescharacteristic, complemented, Sylow
108#529—672no—
109#479—672no—
110#478—672no—
111#467D6711yescharacteristic, complemented
112#466—671no—
113#465—672no—
114#464—672no—
115#463—672no—
116C2×C4⋊C8—671no—
117C2×Q16C2×D6711yescharacteristic
118C2×SD16—672no—
119C2×D8—671no—
120C22×C8—671no—
121C22×C8C2×D6711yescharacteristic
122#126—1342no—
123#125—672no—
124#124—672no—
125#123—1342no—
126C22⋊Q8C2×D6712yes—
127#121—672no—
128C4×Q8—672no—
129C4×D4—672no—
130C4×D4—1342no—
131#117—1342no—
132C2×C42—671no—
133#107—671no—
134#107C2×D6711yescharacteristic
135#106—672no—
136C4⋊C8—672no—
137C4⋊C8—672no—
138#103—672no—
139#103C2×D6712yes—
140#102—672no—
141#102—672no—
142C2×Q8C22×D6712yes—
143C2×D4—672no—
144C2×D4—1342no—
145C22×C4—671no—
146C22×C4—671no—
147C22×C4—1342no—
148C22×C4C22×D6711yescharacteristic
149Q16—41no—
150SD16—2682no—
151D8—2681no—
152C2×C8—41no—
153C2×C8—2681no—
154C2×C8—672no—
155C2×C8C22×D6711yescharacteristic
156C2×C8C22×D6711yescharacteristic
157C4⋊C4—22no—
158C4⋊C4—672no—
159C4⋊C4—1342no—
160C4⋊C4—1342no—
161C4⋊C4—1342no—
162C4⋊C4C22×D6712yes—
163C22⋊C4—22no—
164C22⋊C4—1342no—
165C22⋊C4—1342no—
166C22⋊C4—1342no—
167C22⋊C4—1342no—
168C42—672no—
169C42—672no—
170C42—1342no—
171C23—1342no—
172C23D4×D6711yescharacteristic
173Q8—22no—
174Q8—22no—
175D4—2684no—
176D4—1342no—
177D4—1342no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—2684no—
183C2×C4—2681no—
184C2×C4—672no—
185C2×C4—672no—
186C2×C4—1342no—
187C2×C4—1342no—
188C2×C4D4×D6711yescharacteristic
189C2×C4C23×D6711yescharacteristic
190C8—21no—
191C8—21no—
192C8—1342no—
193C22—21no—
194C22—21no—
195C22—2682no—
196C22—2682no—
197C22—2682no—
198C22#1452406612yes—
199C22C2×D4×D6711yescharacteristic
200C4—21no—
201C4—42no—
202C4—42no—
203C4—2682no—
204C4—1342no—
205C4—1342no—
206C4#1452408811yescharacteristic
207C4#1452409011yescharacteristic
208C2—22no—
209C2—5362no—
210C2#1686397311yescharacteristic
211C2C2×#1452406611yescharacteristic
212C2#1686419311yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#467×C67C2
socleC22×C67C22×D4
hypercenter#467D67
hypocenterC67#3250
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #16863932C2semidirect
2. #16863705C2semidirect
3. #16863699C2semidirect
5. #16863627C2semidirect
6. #16863620C2semidirect
7. #16863614C2semidirect
8. C2×Dic67⋊C8C2semidirect
9. #467×C67C2semidirect
107. C67#3250semidirect
111. #467D67semidirect