Finite Group Databasebeta

← #22570866

Subgroups — #22570866

Order 8576.

338 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 115 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#16864004C212yesmaximal
2#16863796C211yesmaximal, characteristic, complemented
3#16863774C211yesmaximal, characteristic, complemented
4#16863763C212yesmaximal, complemented
5#16863655C211yesmaximal, characteristic
6#16863651C211yesmaximal, characteristic
7#16863636C212yesmaximal
8#16863504C211yesmaximal, characteristic, complemented
9C4×#11761618C212yesmaximal, complemented
10#16863353C211yesmaximal, characteristic
11#441×C67C211yesmaximal, characteristic, complemented
12C2×Dic268C2211yescharacteristic
13C2×#11761618C2212yes—
14C2×#11761611C2211yescharacteristic
15C2×C8×D67C2211yescharacteristic
16#14524041C2212yes—
17#14524038—22no—
18#14524037—22no—
19#14524035—22no—
20#14524034C2212yes—
21#14524033—22no—
22#14524023C2212yes—
23#14524019C2212yes—
24C4×Dic134C2212yes—
25#14524000C2211yescharacteristic
26#14523976C2212yes—
27#14523975C2211yescharacteristic
28#14523968C2212yes—
29#14523967C2212yes—
30#14523966C2212yes—
31#14523953C2211yescharacteristic
32C4×C67⋊C8C2211yescharacteristic
33C8×Dic67C2211yescharacteristic
34C2×Q16×C67C2211yescharacteristic
35C4×Q8×C67C2212yes—
36#106×C67C2211yescharacteristic
37#103×C67C2212yes—
38#97×C67C2211yescharacteristic
39C2×Dic134C2312yes—
40C2×C4×D67C2311yescharacteristic
41Dic268—22no—
42#11761618—24no—
43#11761611—22no—
44C2×C67⋊C8C2311yescharacteristic
45C2×C67⋊C8C2311yescharacteristic
46C8×D67—22no—
47C4⋊Dic67—22no—
48C4⋊Dic67—22no—
49C4⋊Dic67C2312yes—
50Dic67⋊C4—22no—
51Dic67⋊C4—22no—
52Dic67⋊C4—22no—
53Dic67⋊C4—22no—
54Dic67⋊C4—22no—
55Dic67⋊C4D412yes—
56(C2×D67)⋊C4—22no—
57(C2×D67)⋊C4—22no—
58(C2×D67)⋊C4D412yes—
59C4×Dic67—22no—
60C4×Dic67C2311yescharacteristic
61C2×Q8×C67C2312yes—
62Q16×C67—22no—
63C2×C8×C67C2311yescharacteristic
64C2×C8×C67C2311yescharacteristic
65C4⋊C4×C67—22no—
66C4⋊C4×C67C2312yes—
67C42×C67—22no—
68C42×C67C2311yescharacteristic
69C22×D67C2×D411yescharacteristic
70Q8×C67—22no—
71Q8×C67—22no—
72C2×C4×C67—22no—
73C2×C4×C67—22no—
74C2×C4×C67C2×D412yes—
75C2×C4×C67C2411yescharacteristic
76C2×Dic67—22no—
77C2×Dic67—22no—
78C2×Dic67C2×D412yes—
79C2×Dic67C2×D411yescharacteristic
80C4×D67—21no—
81C4×D67—21no—
82Dic134—22no—
83Dic134—22no—
84C8×C67—21no—
85C8×C67#4112yes—
86C67⋊C8—21no—
87C67⋊C8#4112yes—
88C22×C67C22×D411yescharacteristic
89C2×D67—21no—
90C2×D67—22no—
91C4×C67—22no—
92C4×C67—42no—
93C4×C67—42no—
94C4×C67C2×#4111yescharacteristic
95C4×C67C2×#4111yescharacteristic
96Dic67—21no—
97Dic67—22no—
98Dic67—42no—
99Dic67—42no—
100C2×C67#53211yescharacteristic
101C2×C67#57812yes—
102D67—41no—
103#3177—671nomaximal, Sylow
104C67#317711yescharacteristic, complemented, Sylow
105#541—672no—
106#499—671no—
107#494—671no—
108#491—672no—
109#470—671no—
110#470—671no—
111#467—672no—
112#441—671no—
113#441D6711yescharacteristic, complemented
114C4×Q16—672no—
115#405—671no—
116C2×Q16—672no—
117C2×Q16—671no—
118C2×Q16C2×D6711yescharacteristic
119C2×M16—671no—
120C22×C8—671no—
121C4⋊Q8—672no—
122#126—1342no—
123#126—1342no—
124#125—672no—
125#125—1342no—
126#125—1342no—
127C22⋊Q8—672no—
128C22⋊Q8—672no—
129C4×Q8—672no—
130C4×Q8C2×D6712yes—
131#117—671no—
132#107—672no—
133#106—671no—
134#106C2×D6711yescharacteristic
135#103—672no—
136#103—672no—
137#103—672no—
138#103C2×D6712yes—
139#97—671no—
140#97C2×D6711yescharacteristic
141C4×C8—671no—
142C4×C8—671no—
143C2×Q8—672no—
144C2×Q8C22×D6712yes—
145C22×C4—671no—
146Q16—22no—
147Q16—1344no—
148Q16—1342no—
149M16—1342no—
150C2×C8—671no—
151C2×C8—671no—
152C2×C8—1342no—
153C2×C8C22×D6711yescharacteristic
154C2×C8C22×D6711yescharacteristic
155C4⋊C4—22no—
156C4⋊C4—672no—
157C4⋊C4—672no—
158C4⋊C4—1342no—
159C4⋊C4—1342no—
160C4⋊C4—1342no—
161C4⋊C4—1342no—
162C4⋊C4—1342no—
163C4⋊C4—1342no—
164C4⋊C4—1342no—
165C4⋊C4C22×D6712yes—
166C22⋊C4—672no—
167C22⋊C4—1342no—
168C22⋊C4—1342no—
169C42—22no—
170C42—671no—
171C42—1342no—
172C42C22×D6711yescharacteristic
173C23—671no—
174Q8—22no—
175Q8—22no—
176Q8—1342no—
177Q8—1342no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—672no—
181C2×C4—671no—
182C2×C4—1342no—
183C2×C4—1342no—
184C2×C4—1341no—
185C2×C4—1341no—
186C2×C4D4×D6712yes—
187C2×C4C23×D6711yescharacteristic
188C8—21no—
189C8—672no—
190C8—1341no—
191C8#1176162812yes—
192C22—1341no—
193C22—1342no—
194C22C2×D4×D6711yescharacteristic
195C4—22no—
196C4—42no—
197C4—42no—
198C4—2682no—
199C4—2682no—
200C4—1341no—
201C4—1342no—
202C4#41×D6711yescharacteristic
203C4C2×#1176162811yescharacteristic
204C2—2681no—
205C2#1686394311yescharacteristic
206C2#1686421212yes—
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#441×C67C2
socleC22×C67C22×D4
hypercenter#441D67
hypocenterC67#3177
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
2. #16863796C2semidirect
3. #16863774C2semidirect
4. #16863763C2semidirect
8. #16863504C2semidirect
9. C4×#11761618C2semidirect
11. #441×C67C2semidirect
104. C67#3177semidirect
113. #441D67semidirect