Finite Group Databasebeta

← #22570951

Subgroups — #22570951

Order 8576.

358 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 119 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#16863950C212yesmaximal
2#16863793C211yesmaximal, characteristic, complemented
3#16863774C211yesmaximal, characteristic, complemented
4#16863745C212yesmaximal, complemented
5#16863655C211yesmaximal, characteristic
6#16863653C212yesmaximal
7#16863651C211yesmaximal, characteristic
8#16863504C211yesmaximal, characteristic, complemented
9#16863501C212yesmaximal, complemented
10#16863348C211yesmaximal, characteristic
11#441×C67C211yesmaximal, characteristic, complemented
12C2×Dic268C2211yescharacteristic
13C2×#11761618C2212yes—
14C2×#11761611C2211yescharacteristic
15C2×#11761611C2211yescharacteristic
16#14524042C2212yes—
17#14524041C2212yes—
18#14524033—22no—
19#14524023C2212yes—
20#14524021—22no—
21#14524019C2212yes—
22C4×Dic134—22no—
23C4×Dic134C2212yes—
24#14524000—22no—
25C42×D67C2211yescharacteristic
26#14523975C2211yescharacteristic
27#14523974C2212yes—
28#14523968C2212yes—
29#14523967C2212yes—
30#14523966C2212yes—
31#14523954C2211yescharacteristic
32C8×Dic67C2211yescharacteristic
33C8×Dic67C2211yescharacteristic
34C2×Q16×C67C2211yescharacteristic
35C4×Q8×C67C2212yes—
36#106×C67C2211yescharacteristic
37#103×C67C2212yes—
38#97×C67C2211yescharacteristic
39C2×Dic134—22no—
40C2×Dic134C2312yes—
41C2×C4×D67D412yes—
42C2×C4×D67C2311yescharacteristic
43Dic268—22no—
44#11761618—24no—
45#11761611—22no—
46#11761611—22no—
47C2×C67⋊C8C2311yescharacteristic
48C2×C67⋊C8C2311yescharacteristic
49C4⋊Dic67—22no—
50C4⋊Dic67C2312yes—
51Dic67⋊C4—22no—
52Dic67⋊C4—22no—
53Dic67⋊C4—22no—
54Dic67⋊C4—22no—
55Dic67⋊C4—22no—
56(C2×D67)⋊C4—22no—
57(C2×D67)⋊C4—22no—
58C4×Dic67—22no—
59C4×Dic67D412yes—
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×D411yescharacteristic
79C2×Dic67C2×D412yes—
80C4×D67—21no—
81C4×D67—21no—
82C4×D67—24no—
83Dic134—22no—
84Dic134—22no—
85Dic134—44no—
86C8×C67—21no—
87C8×C67#4112yes—
88C67⋊C8—21no—
89C67⋊C8#4112yes—
90C22×C67C22×D411yescharacteristic
91C2×D67—21no—
92C2×D67—22no—
93C4×C67—22no—
94C4×C67—42no—
95C4×C67—42no—
96C4×C67C2×#4111yescharacteristic
97C4×C67C2×#4111yescharacteristic
98Dic67—21no—
99Dic67—42no—
100Dic67—42no—
101Dic67#13714yes—
102C2×C67#53211yescharacteristic
103C2×C67C2×#13712yes—
104D67—41no—
105#3185—671nomaximal, Sylow
106C67#318511yescharacteristic, complemented, Sylow
107#533—672no—
108#498—671no—
109#494—671no—
110#487—672no—
111#470—671no—
112#470—671no—
113#470—672no—
114#441—672no—
115#441—671no—
116#441D6711yescharacteristic, complemented
117C4×M16—671no—
118C2×Q16—672no—
119C2×Q16—671no—
120C2×Q16C2×D6711yescharacteristic
121C2×M16—671no—
122C2×M16—671no—
123C4⋊Q8—672no—
124C4⋊Q8—672no—
125#125—1342no—
126C22⋊Q8—672no—
127C22⋊Q8—672no—
128C22⋊Q8—1342no—
129C4×Q8—672no—
130C4×Q8—1342no—
131C4×Q8C2×D6712yes—
132#117—1342no—
133C2×C42—671no—
134#106—671no—
135#106—672no—
136#106C2×D6711yescharacteristic
137#103—672no—
138#103—672no—
139#103—672no—
140#103C2×D6712yes—
141#97—671no—
142#97C2×D6711yescharacteristic
143C4×C8—671no—
144C4×C8—671no—
145C2×Q8—672no—
146C2×Q8—1342no—
147C2×Q8C22×D6712yes—
148C22×C4—672no—
149C22×C4—671no—
150Q16—22no—
151Q16—1344no—
152Q16—1342no—
153M16—1342no—
154M16—1342no—
155C2×C8—671no—
156C2×C8—671no—
157C2×C8C22×D6711yescharacteristic
158C2×C8C22×D6711yescharacteristic
159C4⋊C4—22no—
160C4⋊C4—672no—
161C4⋊C4—1342no—
162C4⋊C4—1342no—
163C4⋊C4—1342no—
164C4⋊C4—1342no—
165C4⋊C4—1342no—
166C4⋊C4—1342no—
167C4⋊C4C22×D6712yes—
168C22⋊C4—1342no—
169C22⋊C4—1342no—
170C42—22no—
171C42—672no—
172C42—671no—
173C42—1342no—
174C42C22×D6711yescharacteristic
175C23—671no—
176Q8—22no—
177Q8—22no—
178Q8—2684no—
179Q8—1342no—
180Q8—1342no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—672no—
184C2×C4—671no—
185C2×C4—1344no—
186C2×C4—1342no—
187C2×C4—1342no—
188C2×C4—1341no—
189C2×C4—1341no—
190C2×C4D4×D6712yes—
191C2×C4C23×D6711yescharacteristic
192C8—21no—
193C8—672no—
194C8—1341no—
195C8#1176162812yes—
196C22—1342no—
197C22—1341no—
198C22C2×D4×D6711yescharacteristic
199C4—22no—
200C4—42no—
201C4—42no—
202C4—2682no—
203C4—2682no—
204C4—674no—
205C4—1341no—
206C4#41×D6711yescharacteristic
207C4C2×#1176162811yescharacteristic
208C2—2681no—
209C2#1686394311yescharacteristic
210C2#137×D6712yes—
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#441×C67C2
socleC22×C67C22×D4
hypercenter#441D67
hypocenterC67#3185
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. #16863793C2semidirect
3. #16863774C2semidirect
4. #16863745C2semidirect
8. #16863504C2semidirect
9. #16863501C2semidirect
11. #441×C67C2semidirect
106. C67#3185semidirect
116. #441D67semidirect