Finite Group Databasebeta

← #22570823

Subgroups — #22570823

Order 8576.

362 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 117 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#16864015C212yesmaximal, complemented
2#16863794C211yesmaximal, characteristic, complemented
3#16863755C212yesmaximal, complemented
4#16863651C211yesmaximal, characteristic, complemented
5#16863646C211yesmaximal, characteristic
6#16863622C212yesmaximal, complemented
7#16863508C211yesmaximal, characteristic, complemented
8#16863503C211yesmaximal, characteristic, complemented
9SD16×Dic67C212yesmaximal, complemented
10#16863350C211yesmaximal, characteristic, complemented
11#495×C67C211yesmaximal, characteristic, complemented
12C2×#11761611C2211yescharacteristic
13C2×C8×D67C2211yescharacteristic
14#14524043C2212yes—
15#14524036—22no—
16#14524029—22no—
17#14524027—22no—
18#14524020—22no—
19#14524019C2212yes—
20#14524014C2212yes—
21Q8×Dic67C2212yes—
22D4×Dic67C2212yes—
23#14524000C2211yescharacteristic
24#14523975C2211yescharacteristic
25#14523975C2211yescharacteristic
26#14523975C2212yes—
27#14523967C2212yes—
28#14523967C2212yes—
29#14523964C2212yes—
30#14523964C2212yes—
31#14523954C2211yescharacteristic
32#14523953C2211yescharacteristic
33C8×Dic67C2211yescharacteristic
34C2×Q16×C67C2211yescharacteristic
35C2×SD16×C67C2212yes—
36C2×D8×C67C2211yescharacteristic
37#124×C67C2212yes—
38C4×C8×C67C2211yescharacteristic
39C2×Dic134—22no—
40C2×C22⋊D67—22no—
41C22×Dic67—22no—
42C2×C4×D67C2311yescharacteristic
43#11761611—22no—
44C2×C67⋊C8C2311yescharacteristic
45C2×C67⋊C8C2311yescharacteristic
46C8×D67—22no—
47C4⋊Dic67—22no—
48C4⋊Dic67—22no—
49C4⋊Dic67C2312yes—
50C4⋊Dic67C2312yes—
51Dic67⋊C4—22no—
52Dic67⋊C4D412yes—
53C22⋊Dic67—22no—
54C22⋊Dic67—22no—
55(C2×D67)⋊C4—22no—
56(C2×D67)⋊C4D412yes—
57C4×Dic67—22no—
58C4×Dic67C2311yescharacteristic
59C2×Q8×C67C2312yes—
60C2×D4×C67C2312yes—
61Q16×C67—22no—
62SD16×C67—24no—
63D8×C67—22no—
64C2×C8×C67C2311yescharacteristic
65C2×C8×C67C2311yescharacteristic
66C22⋊C4×C67—22no—
67C22⋊C4×C67—22no—
68C42×C67C2311yescharacteristic
69C23×C67—22no—
70C22×D67C2×D411yescharacteristic
71Q8×C67—22no—
72Q8×C67—22no—
73D4×C67—22no—
74D4×C67—22no—
75C2×C4×C67—22no—
76C2×C4×C67C2×D412yes—
77C2×C4×C67C2411yescharacteristic
78C22⋊D67—82no—
79C2×Dic67—22no—
80C2×Dic67—22no—
81C2×Dic67—82no—
82C2×Dic67C2×D412yes—
83C2×Dic67C2×D411yescharacteristic
84C4×D67—21no—
85C4×D67—21no—
86Dic134—82no—
87C8×C67#4112yes—
88C8×C67#4112yes—
89C67⋊C8—21no—
90C67⋊C8—21no—
91C22×C67—42no—
92C22×C67—42no—
93C22×C67—42no—
94C22×C67C22×D411yescharacteristic
95C2×D67—21no—
96C2×D67—22no—
97C4×C67—22no—
98C4×C67—42no—
99C4×C67C2×#4111yescharacteristic
100C4×C67C2×#4111yescharacteristic
101Dic67—21no—
102Dic67—22no—
103Dic67—42no—
104Dic67—42no—
105C2×C67—82no—
106C2×C67#53211yescharacteristic
107C2×C67#57712yes—
108D67—41no—
109#3173—671nomaximal, Sylow
110C67#317311yescharacteristic, complemented, Sylow
111#542—672no—
112#498—671no—
113#495D6711yescharacteristic, complemented
114#489—672no—
115#470—671no—
116#469—671no—
117#465—672no—
118#442—671no—
119#441—671no—
120C4×SD16—672no—
121#405—671no—
122C2×Q16C2×D6711yescharacteristic
123C2×SD16C2×D6712yes—
124C2×D8C2×D6711yescharacteristic
125C2×M16—671no—
126C22×C8—671no—
127C4⋊Q8—672no—
128#126—1342no—
129#124—1342no—
130#124C2×D6712yes—
131#123—1342no—
132C22⋊Q8—672no—
133C22⋊Q8—1342no—
134#121—672no—
135C4×Q8—672no—
136C4×D4—672no—
137#117—671no—
138#106—672no—
139#106—671no—
140#106—671no—
141#103—672no—
142#103—672no—
143#102—672no—
144#102—672no—
145#97—671no—
146#97—671no—
147C4×C8—671no—
148C4×C8C2×D6711yescharacteristic
149C2×Q8—1342no—
150C2×Q8C22×D6712yes—
151C2×D4—1342no—
152C2×D4C22×D6712yes—
153C22×C4—671no—
154C22×C4—1342no—
155Q16—22no—
156SD16—24no—
157D8—22no—
158M16—1342no—
159C2×C8—671no—
160C2×C8—671no—
161C2×C8—1342no—
162C2×C8C22×D6711yescharacteristic
163C2×C8C22×D6711yescharacteristic
164C4⋊C4—672no—
165C4⋊C4—672no—
166C4⋊C4—672no—
167C4⋊C4—1342no—
168C4⋊C4—1342no—
169C4⋊C4—1342no—
170C22⋊C4—22no—
171C22⋊C4—22no—
172C22⋊C4—672no—
173C22⋊C4—1342no—
174C22⋊C4—1342no—
175C22⋊C4—1342no—
176C42—671no—
177C42—1342no—
178C42C22×D6711yescharacteristic
179C23—22no—
180C23—671no—
181Q8—22no—
182Q8—22no—
183Q8—5362no—
184D4—22no—
185D4—22no—
186D4—5362no—
187C2×C4—22no—
188C2×C4—5362no—
189C2×C4—672no—
190C2×C4—671no—
191C2×C4—1342no—
192C2×C4—1342no—
193C2×C4—1341no—
194C2×C4—1341no—
195C2×C4D4×D6712yes—
196C2×C4C23×D6711yescharacteristic
197C8—1341no—
198C8—1341no—
199C8#1176162712yes—
200C8#1176162712yes—
201C22—42no—
202C22—42no—
203C22—42no—
204C22—1341no—
205C22—1342no—
206C22C2×D4×D6711yescharacteristic
207C4—22no—
208C4—42no—
209C4—2682no—
210C4—2682no—
211C4—1341no—
212C4—1342no—
213C4C2×#1176162711yescharacteristic
214C4C2×#1176162711yescharacteristic
215C2—82no—
216C2—2681no—
217C2#1686394411yescharacteristic
218C2#1686420312yes—
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#495×C67C2
socleC22×C67C22×D4
hypercenter#495D67
hypocenterC67#3173
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #16864015C2semidirect
2. #16863794C2semidirect
3. #16863755C2semidirect
4. #16863651C2semidirect
6. #16863622C2semidirect
7. #16863508C2semidirect
8. #16863503C2semidirect
9. SD16×Dic67C2semidirect
10. #16863350C2semidirect
11. #495×C67C2semidirect
110. C67#3173semidirect
113. #495D67semidirect