Finite Group Databasebeta

← #22570586

Subgroups — #22570586

Order 8576.

380 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 125 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#16863951C211yesmaximal, characteristic
2C4⋊Q8×D67C211yesmaximal, characteristic
3#16863763C214yesmaximal, complemented
4#16863748C211yesmaximal, characteristic, complemented
5#16863747C211yesmaximal, characteristic, complemented
6#16863746C211yesmaximal, characteristic, complemented
7#16863403C214yesmaximal
8C4×#11761611C211yesmaximal, characteristic
9#487×C67C211yesmaximal, characteristic, complemented
10C2×Q8×D67C2212yes—
11C2×#11761611C2212yes—
12#14524043C2211yescharacteristic
13#14524043C2211yescharacteristic
14#14524042—21no—
15#14524042—21no—
16#14524041C2212yes—
17#14524034C2212yes—
18#14524023—21no—
19#14524019—21no—
20Q8×Dic67—21no—
21#14524008—21no—
22#14524001C2212yes—
23C4⋊C4×D67—21no—
24C4⋊C4×D67—21no—
25C42×D67C2211yescharacteristic
26#14523968C2214yes—
27#14523967C2214yes—
28#14523966C2214yes—
29#14523953C2212yes—
30C4×C67⋊C8C2211yescharacteristic
31C4⋊Q8×C67C2211yescharacteristic
32C4⋊Q8×C67C2211yescharacteristic
33#103×C67C2214yes—
34C4×C8×C67C2211yescharacteristic
35C2×Dic134—21no—
36C2×Dic134—22no—
37C2×Dic134C2312yes—
38Q8×D67—28no—
39C2×C4×D67—22no—
40C2×C4×D67D411yescharacteristic
41C2×C4×D67D411yescharacteristic
42C2×C4×D67C2311yescharacteristic
43#11761611—24no—
44C2×C67⋊C8C2312yes—
45C4⋊Dic67—21no—
46C4⋊Dic67—21no—
47C4⋊Dic67—21no—
48C4⋊Dic67—22no—
49C4⋊Dic67C2312yes—
50Dic67⋊C4—22no—
51Dic67⋊C4—22no—
52Dic67⋊C4—22no—
53(C2×D67)⋊C4—22no—
54C4×Dic67—22no—
55C4×Dic67D411yescharacteristic
56C4×Dic67D411yescharacteristic
57C4×Dic67C2311yescharacteristic
58C2×Q8×C67—21no—
59C2×Q8×C67C2312yes—
60C2×C8×C67C2312yes—
61C4⋊C4×C67—21no—
62C4⋊C4×C67—21no—
63C4⋊C4×C67—21no—
64C4⋊C4×C67C2312yes—
65C42×C67C2311yescharacteristic
66C22×D67C2×D411yescharacteristic
67Q8×C67—24no—
68Q8×C67—42no—
69C2×C4×C67—22no—
70C2×C4×C67—22no—
71C2×C4×C67C2×D411yescharacteristic
72C2×C4×C67C2×D411yescharacteristic
73C2×C4×C67C2411yescharacteristic
74C2×Dic67—22no—
75C2×Dic67—22no—
76C2×Dic67C2×D411yescharacteristic
77C2×Dic67C2×D411yescharacteristic
78C2×Dic67C2×D411yescharacteristic
79C4×D67—22no—
80C4×D67—22no—
81C4×D67—44no—
82C4×D67#4114yes—
83Dic134—24no—
84Dic134—42no—
85Dic134—44no—
86C8×C67—22no—
87C67⋊C8—22no—
88C22×C67C22×D411yescharacteristic
89C2×D67—21no—
90C2×D67—21no—
91C2×D67#12412yes—
92C4×C67—42no—
93C4×C67—42no—
94C4×C67#13712yes—
95C4×C67#13712yes—
96C4×C67C2×#4112yes—
97Dic67—21no—
98Dic67—21no—
99Dic67—42no—
100Dic67—42no—
101Dic67#12412yes—
102C2×C67C2×#12411yescharacteristic
103C2×C67C2×#13711yescharacteristic
104C2×C67C2×#13711yescharacteristic
105D67—22no—
106#3139—671nomaximal, Sylow
107C67#313911yescharacteristic, complemented, Sylow
108#533—671no—
109C2×C4⋊Q8—671no—
110#491—674no—
111#487—671no—
112#487—671no—
113#487—671no—
114#487D6711yescharacteristic, complemented
115#419—674no—
116C4×M16—671no—
117C22×Q8—672no—
118C2×M16—672no—
119C4⋊Q8—672no—
120C4⋊Q8—671no—
121C4⋊Q8—671no—
122C4⋊Q8—1341no—
123C4⋊Q8—1341no—
124C4⋊Q8C2×D6711yescharacteristic
125C4⋊Q8C2×D6711yescharacteristic
126#125—672no—
127C22⋊Q8—1341no—
128C22⋊Q8—1341no—
129C4×Q8—1341no—
130C4×Q8—1341no—
131#117—672no—
132C2×C4⋊C4—1341no—
133C2×C4⋊C4—1341no—
134C2×C42—671no—
135#103—674no—
136#103—674no—
137#103—674no—
138#103C2×D6714yes—
139#97—672no—
140C4×C8—671no—
141C4×C8C2×D6711yescharacteristic
142C2×Q8—21no—
143C2×Q8—672no—
144C2×Q8—1342no—
145C2×Q8—1348no—
146C2×Q8—1341no—
147C2×Q8C22×D6712yes—
148C22×C4—671no—
149C22×C4—671no—
150C22×C4—671no—
151C22×C4—1342no—
152M16—1344no—
153C2×C8—672no—
154C2×C8C22×D6712yes—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—672no—
159C4⋊C4—1342no—
160C4⋊C4—1342no—
161C4⋊C4—1342no—
162C4⋊C4—1342no—
163C4⋊C4—1341no—
164C4⋊C4—1341no—
165C4⋊C4—1341no—
166C4⋊C4C22×D6712yes—
167C22⋊C4—1342no—
168C42—671no—
169C42—671no—
170C42—671no—
171C42—1342no—
172C42C22×D6711yescharacteristic
173C23—671no—
174Q8—24no—
175Q8—42no—
176Q8—2684no—
177Q8—2682no—
178Q8—1344no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—2684no—
182C2×C4—671no—
183C2×C4—674no—
184C2×C4—671no—
185C2×C4—671no—
186C2×C4—1342no—
187C2×C4—1342no—
188C2×C4—1342no—
189C2×C4—1342no—
190C2×C4D4×D6711yescharacteristic
191C2×C4D4×D6711yescharacteristic
192C2×C4C23×D6711yescharacteristic
193C8—22no—
194C8—1342no—
195C22—672no—
196C22—1341no—
197C22—1341no—
198C22C2×D4×D6711yescharacteristic
199C4—42no—
200C4—42no—
201C4—2682no—
202C4—2682no—
203C4—672no—
204C4—1341no—
205C4—1341no—
206C4#1452407212yes—
207C4#1452407512yes—
208C4#41×D6712yes—
209C2—1342no—
210C2#124×D6711yescharacteristic
211C2C2×#1452407211yescharacteristic
212C2C2×#1452407511yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#487×C67C2
socleC22×C67C22×D4
hypercenter#487D67
hypocenterC67#3139
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
3. #16863763C2semidirect
4. #16863748C2semidirect
5. #16863747C2semidirect
6. #16863746C2semidirect
9. #487×C67C2semidirect
107. C67#3139semidirect
114. #487D67semidirect