Finite Group Databasebeta

← #22570332

Subgroups — #22570332

Order 8576.

424 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 129 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#16863946C211yesmaximal, characteristic
2C2×C4×Dic134C211yesmaximal, characteristic, complemented
3#16863710C212yesmaximal, complemented
4#16863692C212yesmaximal, complemented
5#16863597C212yesmaximal, complemented
6#16863537C212yesmaximal, complemented
7#16863498C214yesmaximal, complemented
8#432×C67C211yesmaximal, characteristic, complemented
9C2×#11761628—21no—
10C22×Dic134C2211yescharacteristic
11C2×#11761617C2214yes—
12#14524043C2211yescharacteristic
13#14524040C2211yescharacteristic
14#14524031—21no—
15#14524017—21no—
16#14524017C2212yes—
17C4×Dic134—21no—
18C4×Dic134—21no—
19C4×Dic134C2212yes—
20C4×Dic134C2212yes—
21C4×D268—21no—
22C4×D268C2212yes—
23C2×C4⋊Dic67C2211yescharacteristic
24C2×Dic67⋊C4—21no—
25C2×C4×Dic67—21no—
26#14523977C2214yes—
27#14523968C2214yes—
28#14523965C2214yes—
29C2×C42×C67C2211yescharacteristic
30C4⋊C8×C67C2212yes—
31C22⋊C8×C67C2212yes—
32#97×C67C2212yes—
33#11761628—44no—
34C2×Dic134—21no—
35C2×Dic134—24no—
36C2×Dic134—21no—
37C2×Dic134C2312yes—
38C2×D268—21no—
39C2×D268—21no—
40C2×D268C2312yes—
41C2×C22⋊D67—22no—
42C22×Dic67—21no—
43C22×Dic67—21no—
44C2×C4×D67—22no—
45#11761617—28no—
46C4⋊Dic67—21no—
47C4⋊Dic67—21no—
48C4⋊Dic67C2312yes—
49C4⋊Dic67C2312yes—
50Dic67⋊C4—22no—
51Dic67⋊C4—22no—
52(C2×D67)⋊C4—22no—
53(C2×D67)⋊C4—22no—
54C4×Dic67—22no—
55C4×Dic67—22no—
56C22×C4×C67D411yescharacteristic
57C22×C4×C67D411yescharacteristic
58C22×C4×C67C2311yescharacteristic
59C2×C8×C67C2314yes—
60C42×C67D411yescharacteristic
61C42×C67D411yescharacteristic
62C42×C67C2311yescharacteristic
63C42×C67C2311yescharacteristic
64C23×C67C2×D411yescharacteristic
65C22×D67—22no—
66C2×C4×C67—22no—
67C2×C4×C67—22no—
68C2×C4×C67—22no—
69C2×C4×C67C2×D411yescharacteristic
70C2×C4×C67C2×D411yescharacteristic
71C2×C4×C67C2×D411yescharacteristic
72C2×C4×C67C2×D411yescharacteristic
73C2×C4×C67C2×D411yescharacteristic
74C2×C4×C67C2411yescharacteristic
75C22⋊D67—44no—
76C2×Dic67—24no—
77C2×Dic67—22no—
78C2×Dic67—22no—
79C2×Dic67—22no—
80C2×Dic67—42no—
81D268—24no—
82D268—42no—
83D268—42no—
84C4×D67—44no—
85Dic134—24no—
86Dic134—22no—
87Dic134—42no—
88Dic134#4114yescomplemented
89C8×C67—24no—
90C22×C67—21no—
91C22×C67—21no—
92C22×C67AGL(1,ℤ/8)12yes—
93C22×C67C22×D411yescharacteristic
94C2×D67—42no—
95C2×D67—42no—
96C2×D67—42no—
97C4×C67—21no—
98C4×C67—21no—
99C4×C67—21no—
100C4×C67—21no—
101C4×C67#13712yes—
102C4×C67C2×#4112yes—
103Dic67—24no—
104Dic67—42no—
105Dic67—42no—
106C2×C67—22no—
107C2×C67#52511yescharacteristic
108C2×C67C2×AGL(1,ℤ/8)11yescharacteristic
109C2×C67C2×#13711yescharacteristic
110D67—82no—
111#3112—671nomaximal, Sylow
112C67#311211yescharacteristic, complemented, Sylow
113#532—671no—
114C2×C4×Q8—671no—
115#480—672no—
116#477—672no—
117#460—672no—
118#448—672no—
119#440—674no—
120#432D6711yescharacteristic, complemented
121C2×#41—1341no—
122C22×Q8—671no—
123C2×SD16—674no—
124C4⋊Q8—671no—
125#127—671no—
126#124—1341no—
127#121—672no—
128#121—1341no—
129C4×Q8—672no—
130C4×Q8—672no—
131C4×Q8—1341no—
132C4×Q8—1341no—
133C4×D4—672no—
134C4×D4—1341no—
135C2×C4⋊C4—671no—
136C2×C4⋊C4—1341no—
137C2×C42—1341no—
138C2×C42C2×D6711yescharacteristic
139#107—674no—
140C4⋊C8C2×D6712yes—
141#103—674no—
142#102—674no—
143C22⋊C8C2×D6712yes—
144#97C2×D6712yes—
145#41—2684no—
146C2×Q8—672no—
147C2×Q8—1344no—
148C2×Q8—1341no—
149C2×Q8—1341no—
150C2×D4—672no—
151C2×D4—1342no—
152C2×D4—1341no—
153C2×D4—1341no—
154C22×C4—1341no—
155C22×C4—1342no—
156C22×C4—1341no—
157C22×C4D26811yescharacteristic
158C22×C4D26811yescharacteristic
159C22×C4C22×D6711yescharacteristic
160SD16—1348no—
161C2×C8C22×D6714yes—
162C4⋊C4—672no—
163C4⋊C4—672no—
164C4⋊C4—1341no—
165C4⋊C4—1342no—
166C4⋊C4—1342no—
167C4⋊C4—1341no—
168C22⋊C4—1342no—
169C22⋊C4—1342no—
170C42—1342no—
171C42—1342no—
172C42D26811yescharacteristic
173C42D26811yescharacteristic
174C42C22×D6711yescharacteristic
175C42C22×D6711yescharacteristic
176C23—1342no—
177C23C2×D26811yescharacteristic
178Q8—2682no—
179Q8—674no—
180Q8—1344no—
181Q8—1342no—
182D4—2682no—
183D4—2684no—
184D4—2682no—
185D4—1344no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—2684no—
190C2×C4—2682no—
191C2×C4—1342no—
192C2×C4—1342no—
193C2×C4—1344no—
194C2×C4—1342no—
195C2×C4C2×D26811yescharacteristic
196C2×C4C2×D26811yescharacteristic
197C2×C4C2×D26811yescharacteristic
198C2×C4C2×D26811yescharacteristic
199C2×C4C2×D26811yescharacteristic
200C2×C4C23×D6711yescharacteristic
201C8—24no—
202C22—21no—
203C22—21no—
204C22—2682no—
205C22—2682no—
206C22—2682no—
207C22#1452407112yes—
208C22C22×D26811yescharacteristic
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—2682no—
214C4—2682no—
215C4—1344no—
216C4#1452407612yes—
217C4#41×D6712yes—
218C2—22no—
219C2—5362no—
220C2#1686390811yescharacteristic
221C2C2×#1452407111yescharacteristic
222C2C2×#1452407611yescharacteristic
subgroupG/H
centerC22C22×D268
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#432×C67C2
socleC22×C67C22×D4
hypercenter#432D67
hypocenterC67#3112
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic134C2semidirect
3. #16863710C2semidirect
4. #16863692C2semidirect
5. #16863597C2semidirect
6. #16863537C2semidirect
7. #16863498C2semidirect
8. #432×C67C2semidirect
88. Dic134#41semidirect
112. C67#3112semidirect
120. #432D67semidirect